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Adaptive insertion of cohesive elements for simulation of delamination in laminated composite materials Shor, Ofir 2016

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Adaptive Insertion of CohesiveElements for Simulation ofDelamination in Laminated CompositeMaterialsbyOfir ShorB.Sc., Ben-Gurion University of the Negev, 2001M.E, Technion, Israel Institute of Technology, 2006A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Civil Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)September 2016© Ofir Shor 2016AbstractComposite materials are increasingly being used in advanced structural ap-plications. Debonding of adjacent laminate layers, also known as delami-nation, is considered to be one of the most dominant damage mechanismsaffecting the behavior of composite laminates. Various numerical methodsfor simulating delamination in composite materials do exist, but they aregenerally limited to small-scale structures due to their complexity and highnumerical cost.In this thesis, a novel technique aimed to allow efficient simulation of de-lamination in large-scale laminated composite structures is presented. Dur-ing the transient analysis, continuum elements within regions where de-lamination has the potential to initiate are adaptively split through theirthickness into two shell elements sandwiching a cohesive element. By elimi-nating the a priori requirement to implant cohesive elements at all possiblespatial locations, the computational efforts are reduced, thus lending themethod suitable for treatment of practical size structures. The methodol-ogy, called the local cohesive zone method (LCZ), is verified here throughits application to Mode-I, Mode-II and Mixed-Mode loading conditions, andis validated using a dynamic tube-crushing loading case and plate impactevents. Good agreement between the numerical results and the availableexperimental data is obtained. The results obtained using the LCZ methodare compared favourably with the numerical results obtained using the con-ventional cohesive zone method (CZM).The numerical performance of the method and its efficiency is investi-gated. The efficiency of the method was found to be superior compared tothat of the conventional CZM, and was found to increase with increasingmodel size. The LCZ method is shown to have a lower effect on reducingthe structural stiffness of the structure, compared to the conventional CZM.The results obtained from the application of the LCZ method to thevarious cases tested are encouraging, and prove that the local and adaptiveinsertion of cohesive zones into a finite element mesh can effectively capturethe delamination crack propagation in laminated composite structures. It isexpected that further improvements in speed and accuracy will be attainediiAbstractonce the algorithm is embedded within commercial finite element solvers asa built-in feature.iiiPrefaceThis thesis entitled "Adaptive Insertion of Cohesive Elements for Simulationof Delamination in Laminated Composite Materials" presents the researchperformed by Ofir Shor. The research was supervised by Dr. Reza Vaziri atthe University of British Columbia.A version of the contents of Chapter 2, Chapter 3, and Chapter 4, waspublished in Shor and Vaziri [118], "Adaptive insertion of cohesive elementsfor simulation of delamination in laminated composite materials", Engineer-ing Fracture Mechanics. These sections, include the original verificationwork of the LCZ algorithm developed during the research.Figure 5.6.a, Figure 5.7.b, Figure 5.8.b, and Figure 5.9.b are courtesy ofDr. Stephen Hallett from the University of Bristol. The results from tensileexperiments performed by Dr. Hallett are shown in these figures, in orderto validate the failure patterns in a double-notched test coupon, which wasalso predicted by the algorithm developed by the thesis’ author.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Computer Programs . . . . . . . . . . . . . . . . . . . xviList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . xxDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Numerical Simulation of Delamination in Composites . 42.1. The Cohesive-Zone Method ...................................................... 52.2. Estimating the Cohesive Zone Length ..................................... 93 A Local Cohesive Zone Method for Simulation of Inter-laminar Damage in Laminated Composite Materials . 163.1. Introduction .............................................................................. 163.2. Main Principles of the LCZ Method ........................................ 173.3. LCZ Algorithm Overview......................................................... 183.3.1. Problem Initialization....................................................... 183.3.2. Element-Splitting Criteria ............................................... 213.3.3. Radial Neighbour Search ................................................. 223.3.4. Threshold Neighbour Search............................................ 23vTable of Contents3.3.5. Through-Thickness Element Splitting and Local Inser-tion of Cohesive Elements .......................................................... 263.3.6. Propagation of the Local Cohesive Zones........................ 293.4. Multi-Delamination Capability ................................................ 314 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.1. Mode-I delamination ................................................................. 384.1.1. Obtaining the Cohesive Properties ................................... 404.1.2. Applying the LCZ Method to the DCB VerificationProblem...................................................................................... 484.1.3. Mesh-Size Sensitivity ........................................................ 504.1.4. R-Size Sensitivity.............................................................. 514.1.5. Sensitivity to the Element Splitting Criterion ................. 524.1.6. Energy Balance................................................................. 544.2. Mode-II Delamination .............................................................. 584.3. Mixed-Mode Delamination ....................................................... 624.4. Summary and Conclusions ....................................................... 635 Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.1. Tensile Loading of a Notched Coupon...................................... 675.1.1. Material and Test Specification ........................................ 675.1.2. Finite Element Model....................................................... 695.1.3. Intralaminar Damage Modelling ...................................... 725.1.4. Interlaminar Damage Modelling....................................... 745.2. Dynamic Tube Crush Simulation ............................................. 825.2.1. Introduction...................................................................... 825.2.2. Material and Test Specimens........................................... 825.2.3. Finite Element Model ...................................................... 865.2.4. Intralaminar Damage Modelling ...................................... 885.2.5. Interlaminar Damage Modelling ...................................... 895.2.6. Results and Discussion..................................................... 935.3. Dynamic Plate Impact Simulations.......................................... 975.3.1. Material and Test Specifications ...................................... 985.3.2. Finite Element Model ...................................................... 1005.3.3. Intralaminar Damage Modelling...................................... 1005.3.4. Interlaminar Damage Modelling ...................................... 1035.3.5. Results and Discussion..................................................... 1045.4. Summary and Conclusions ....................................................... 122viTable of Contents6 Numerical Performance of the LCZ method . . . . . . . 1326.1. Introduction .............................................................................. 1326.2. Solution of Larger Models ........................................................ 1326.3. Effect of CZM on the Structural Stiffness................................ 1366.3.1. Simply Supported Beam Under Bending Load ................ 1396.3.2. Static Plate Loading ........................................................ 1406.3.3. Dynamic Plate Impact..................................................... 1447 Summary, Conclusions and Future Work . . . . . . . . . 1467.1. Summary ................................................................................... 1467.2. Conclusions ............................................................................... 1487.3. Future Work ............................................................................. 149Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151AppendicesA Flowchart of the LCZ Algorithm . . . . . . . . . . . . . . 166B Execution of the LCZ Algorithm . . . . . . . . . . . . . . 174C General Description of Composite Tube Crushing Pro-cess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186D LS-DYNA Material Cards . . . . . . . . . . . . . . . . . . 193E Brief Description of the CODAM2 Material Model . . 202F Calibrating the CODAM2 Material Model for the Tube-Crushing Simulation . . . . . . . . . . . . . . . . . . . . . . 205viiList of Tables2.1 Estimated cohesive zone length (lpz) and equivalent value forM in Equation 2.17 . . . . . . . . . . . . . . . . . . . . . . 124.1 Material properties used in the numerical verification prob-lems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.1 Elastic Hexcel E-glass/913 material properties . . . . . . . 685.2 LS-DYNA’s *MAT 54 Material model damage parametersused in the [90/0]s double-notched coupon simulation . . 755.3 Cohesive properties used in the [90/0]s double-notched couponsimulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.4 Manufacturer’s Constituent properties for tube braiding ma-terial. Source: [85] . . . . . . . . . . . . . . . . . . . . . . 845.5 Model input parameters for the [0◦/±45◦] braided compositetube. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.6 Model input parameters for the T800/ 3900-2 CFRP sub-laminate [45◦/90◦/−45◦/0◦] (Williams et al. [129] and Forghaniand Vaziri [36]) . . . . . . . . . . . . . . . . . . . . . . . . 1035.7 Predicted delamination patterns for a 4.29 m/s, 6.33 kg im-pactor, plate-impact event. . . . . . . . . . . . . . . . . . 1115.8 Predicted delamination pattern for a 18.97 m/s, 0.314 kg pro-jectile, plate-impact event. . . . . . . . . . . . . . . . . . . 1175.9 Predicted delamination pattern for a 14.59 m/s, 0.314 kg pro-jectile, plate-impact event. . . . . . . . . . . . . . . . . . . 1316.1 LS-DYNA run-time in seconds, using solid cohesive elements 1346.2 LS-DYNA run-time in seconds obtained using the conven-tional CZM . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.3 Isotropic material and interface properties for the simply-supported beam benchmark problem. . . . . . . . . . . . . 140viiiList of Figures2.1 A schematic diagram demonstrating a typical cohesive zonein a continuum material. . . . . . . . . . . . . . . . . . . . 62.2 A typical bilinear traction-separation law used in a cohesiveinterface model. . . . . . . . . . . . . . . . . . . . . . . . . 72.3 A schematic side view of a crack under mode-I loading con-dition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.4 Uniformly mode I loaded infinite geometry . . . . . . . . . 92.5 Typical σzz profiles obtained using Equation 2.2 with φ = 0,as a function of the distance from the crack tip, r. . . . . 153.1 A schematic comparison for the application of the conven-tional CZM vs. the LCZ method, for a single cohesive inter-face. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Typical cohesive-band created when solving a DCB problemusing the LCZ algorithm. . . . . . . . . . . . . . . . . . . 203.3 A schematic flow diagram demonstrating the relation betweenthe LCZ algorithm and the finite element solver (LS-DYNA). 203.4 A simple numerical model used to demonstrate the LCZ al-gorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.5 Isometric view of the detailed cantilever beam model used fordemonstrating the LCZ algorithm . . . . . . . . . . . . . . 223.6 Isometric view of a double cantilever beam (DCB) subjectedsubjected to an evenly distributed splitting displacement ∆. 243.7 A threshold value is applied to the element splitting crite-rion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.8 A side view demonstrating a schematic element splitting pro-cess. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.9 Addition of overlapping offset shell elements. . . . . . . . 283.10 Schematic side view of a DCB subjected to a splitting loadmodelled using the LCZ method. . . . . . . . . . . . . . . 30ixList of Figures3.11 Schematic progression of the cohesive zones using the LCZalgorithm, for a cantilever beam-splitting example. . . . . 333.12 Typical cohesive-band created when solving a DCB problemusing the LCZ algorithm. . . . . . . . . . . . . . . . . . . 343.13 Simulating a structure having multiple delamination inter-faces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.14 Using the LCZ method, multiple delamination cracks can betreated simultaneously. . . . . . . . . . . . . . . . . . . . . 353.15 Schematic performance of the LCZ search algorithm for amultiple delamination problem. . . . . . . . . . . . . . . . 364.1 Normalized traction-separation law used in the cohesive ma-terial model. . . . . . . . . . . . . . . . . . . . . . . . . . . 384.2 Schematic description of the DCB test case, using the con-ventional CZM. . . . . . . . . . . . . . . . . . . . . . . . . 394.3 Schematic description of the DCB test case, using the LCZmethod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.4 Force vs.crack opening displacement plot, obtained for a beammodeled using σmax values of 57. . . . . . . . . . . . . . . 424.5 Isometric view of the DCB finite element model. . . . . . 434.6 Typical cohesive behavior obtained for the DCB mode-I load-ing case, using σmax = 57 MPa. . . . . . . . . . . . . . . . 454.7 Typical cohesive behavior obtained for the DCB mode-I load-ing case, using σmax = 8 MPa. . . . . . . . . . . . . . . . 464.8 Typical cohesive behavior obtained for the DCB mode-I load-ing case, using σmax = 1 MPa. . . . . . . . . . . . . . . . 474.9 DCB test case 2. . . . . . . . . . . . . . . . . . . . . . . . 494.10 DCB loading case (Mode-I delamination) - Reaction force atend of beam vs. crack opening displacement 2∆. . . . . . 504.11 DCB loading case (Mode-I delamination) - Reaction force atend of beam vs. crack opening displacement ∆, for elementsizes of 1, 2 and 4mm. . . . . . . . . . . . . . . . . . . . . 514.12 DCB loading case - Reaction force at end of beam vs. crackopening displacement, for different values of R. . . . . . . 524.13 DCB loading case - Reaction force at end of beam vs. crackopening displacement, for an element size of 2mm and differ-ent values of R. . . . . . . . . . . . . . . . . . . . . . . . . 534.14 Force vs. crack opening displacement results for various ele-ment sizes used to simulate the DCB test case. . . . . . . 54xList of Figures4.15 Typical cohesive band obtained for the DCB mode-I loadingcase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.16 DCB loading case - Reaction force at end of beam vs. crackopening displacement, for different threshold values. R=12mm 564.17 Cohesive-band migration obtained for the DCB loading case. 574.18 DCB loading case - sum of internal energies vs. crack openingdisplacement. . . . . . . . . . . . . . . . . . . . . . . . . . 574.19 End Notch Flexure (ENF) test case. . . . . . . . . . . . . 584.20 Typical cohesive behavior obtained for the ENF mode-II load-ing case, using a 1mm mesh size and σmax = 57 MPa. . . 604.21 ENF loading case (Mode-II delamination) - Reaction force atloading point vs. z-displacement.0– . . . . . . . . . . . . . 614.22 Mixed Mode Bending (MMB) test case. . . . . . . . . . . 634.23 Mixed Mode Bending (MMB) test case finite element modelduring the loading process. . . . . . . . . . . . . . . . . . 644.24 MMB loading case - Reaction force at end of beam vs. verticaldeflection at the beam end. . . . . . . . . . . . . . . . . . 655.1 A double-notched test coupon geometry used in the tensileexperiments performed by Hallett and Wisnom [48]. . . . 685.2 Finite element model used for solving the double-notch spec-imen under tensile loading condition. . . . . . . . . . . . . 715.3 Typical finite-element model used to calibrate the discretecohesive element material model. . . . . . . . . . . . . . . 735.4 Typical stress vs. displacement obtained using the finite ele-ment model in Figure 5.3. . . . . . . . . . . . . . . . . . . 745.5 Far stress vs. displacement obtained from double-notchedtest coupon using the LCZ method. . . . . . . . . . . . . . 775.6 Experimental damage, and damage obtained using the LCZalgorithm, applied to a double-notched test coupon. . . . 785.7 Delamination damage in a double-notched test coupon. . 795.8 90◦ ply matrix damage in a double-notched test coupon. . 805.9 0◦ ply matrix damage in a double-notched test coupon. . 815.10 Drop tower assembly together with composite tube. . . . 835.11 Square-profile Composite tube with a braided architectureused in the tube-crushing experiment. . . . . . . . . . . . 845.12 Force vs. displacement results obtained from the tube-crushexperiments. . . . . . . . . . . . . . . . . . . . . . . . . . 855.13 An isometric view of the LS-DYNA finite-element model usedfor the tube-crush analysis. . . . . . . . . . . . . . . . . . 87xiList of Figures5.14 An isometric view of the LS-DYNA finite-element model usedfor the tube-crush analysis. . . . . . . . . . . . . . . . . . 885.15 Stress - strain curves used as an input for the CODAM2 ma-terial model, calibrated for a 2.5 mm element size. . . . . 905.16 Force vs. displacement results obtained from the numericalsimulation of the tube crush tests using the combined CO-DAM2 and the LCZ algorithm. . . . . . . . . . . . . . . . 935.17 Typical topology of the crushed tube geometry. . . . . . . 945.18 Propagation of the cohesive-band when the LCZ method isapplied to the tube-crushing simulation. . . . . . . . . . . 965.19 Model topology and CODAM2 intralmainar damage valuesin the axial direction for the tube-crush problem. . . . . . 975.20 Model topology and CODAM2 intralmainar damage valuesin the hoop direction for the tube crush problem. . . . . . 985.21 A side view of the plate-impact experiment configuration. 995.22 A top view of the plate-impact test configuration. . . . . . 995.23 An isometric view of the plate-impact finite-element model. 1015.24 Typical strain-softening behavior of LS-DYNA’s *MAT PLAS-TICITY WITH DAMAGE material model. . . . . . . . . 1025.25 Predicted and experimental impact force vs. time for a 6.33kg impactor, impacting the plate at 4.29 m/s. . . . . . . . 1065.26 Predicted and experimental impact force vs. plate displace-ment, for a 6.33 kg impactor, impacting the plate at 4.29m/s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.27 Delamination propagation predicted by the conventional CZ,for a 4.29 m/s, 6.33 kg projectile using a 5 interface model. 1095.28 Delamination propagation predicted by the LCZ method, fora 4.29 m/s, 6.33 kg projectile using a 5 interface model. . 1105.29 Projected delamination area, for an 6.33 kg impactor impact-ing the plate at 4.29 m/s. . . . . . . . . . . . . . . . . . . 1125.30 Predicted and experimental impactor’s kinetic energy vs. time,for a 4.29m/s impact velocity and impactor mass of 6.33 kg. 1145.31 Predicted and experimental impact force vs. time for a 0.314kg impactor, impacting the plate at 18.97 m/s. . . . . . . 1155.32 Predicted and experimental impact force vs. plate displace-ment, for a 0.314 kg impactor, impacting the plate at 18.97m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.33 Damage in a 3 interface model, impact velocity of 18.97 m/s,at time of 0.3 ms. . . . . . . . . . . . . . . . . . . . . . . 118xiiList of Figures5.34 Damage in a 3 interface model, impact velocity of 18.97 m/s,at time of 1.8 ms. . . . . . . . . . . . . . . . . . . . . . . . 1195.35 Projected delamination area, for an 0.314 kg impactor im-pacting the plate at 18.97 m/s . . . . . . . . . . . . . . . 1205.36 Predicted and experimental impactor’s kinetic energy vs. time,for a 18.97 m/s impact velocity and impactor mass of 0.314kg. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.37 Predicted and experimental impact force vs. time for a 0.314kg impactor, impacting the plate at 14.59 m/s. . . . . . . 1235.38 Predicted and experimental impact force vs. plate displace-ment, for a 0.314 kg impactor, impacting the plate at 14.59m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.39 Predicted and experimental impactor’s kinetic energy vs. time,for a 14.59 m/s impact velocity and impactor mass of 0.314kg. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255.40 Projected delamination area, for an 0.314 kg impactor im-pacting the plate at 14.59 m/s. . . . . . . . . . . . . . . . 1265.41 Stress vs. strain for plate impact model, 2mm and 1mmmesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.42 Stress vs. COD obtained from Tie-Break contact, Mode-Icrack opening . . . . . . . . . . . . . . . . . . . . . . . . 1295.43 Stress vs. COD obtained from Tie-Break contact, Mode-IIcrack opening . . . . . . . . . . . . . . . . . . . . . . . . 1306.1 Isometric view of three finite element models used to test theLCZ method’s efficiency over the conventional CZM. . . . 1336.2 Ratio between the LS-DYNA run-time using the conventionalapplication of CZM (using solid cohesive elements a-prioriseeded along all of the cohesive interfaces). . . . . . . . . 1356.3 Ratio between the LS-DYNA run-time using the conventionalapplication of CZM using a cohesive contact algorithm alongall of the cohesive interface . . . . . . . . . . . . . . . . . 1366.4 A simple finite-element topology demonstrating the increaseof numerical complexity when cohesive elements are intro-duced into the model. . . . . . . . . . . . . . . . . . . . . 1386.5 Schematic 3Dmodel demonstrating the behavior of a cohesiveinterface. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1386.6 Schematic view of a simply supported beam under a bendingload. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139xiiiList of Figures6.7 Isometric view of simply supported beam, modeled using threedifferent configurations. . . . . . . . . . . . . . . . . . . . 1416.8 Load at maximum displacement for a simply-supported beamunder central bending load. . . . . . . . . . . . . . . . . . 1426.9 Load at maximum displacement for the plate bending ex-ample, as a function of through-thickness discretization andnumerical solution method. . . . . . . . . . . . . . . . . . 1436.10 Predicted and experimental impact force vs. plate displace-ment, for a 6.33 kg impactor, impacting the plate at 4.29m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145A.1 Schematic LCZ algorithm flowchart, image 1 out of 7 . . . 167A.2 Schematic LCZ algorithm flowchart, image 2 out of 7 . . . 168A.3 Schematic LCZ algorithm flowchart, image 3 out of 7 . . . 169A.4 Schematic LCZ algorithm flowchart, image 4 out of 7 . . . 170A.5 Schematic LCZ algorithm flowchart, image 5 out of 7 . . . 171A.6 Schematic LCZ algorithm flowchart, image 6 out of 7 . . . 172A.7 Schematic LCZ algorithm flowchart, image 7 out of 7 . . . 173B.1 Schematic directory architecture required for the correct ex-ecution of the LCZ algorithm. . . . . . . . . . . . . . . . . 175B.2 Schematic directory architecture following execution of theLCZ algorithm. . . . . . . . . . . . . . . . . . . . . . . . . 185C.1 Schematic crushing morphologies obtained during a compos-ite tube-crushing event. . . . . . . . . . . . . . . . . . . . 187C.2 A schematic load vs. displacement profile obtained during astable composite tube crushing process. . . . . . . . . . . 189C.3 Failure morphologies of composite-tube walls undergoing pro-gressive, axial crushing. . . . . . . . . . . . . . . . . . . . 190C.4 Typical failure morphology obtained in a composite tube dur-ing a dynamic crushing process. . . . . . . . . . . . . . . . 191E.1 Stiffness reduction coefficients R , as a function of the damageparameter ω, for the CODAM2 material model. . . . . . . 204F.1 A simple single-element model used for the CODAM2 cali-bration process. . . . . . . . . . . . . . . . . . . . . . . . . 206F.2 Stress vs. strain plot obtained from a single shell-elementsimulation, under axial tensile loading, using the CODAM1and CODAM2 material models. . . . . . . . . . . . . . . . 207xivList of FiguresF.3 Stress vs. strain plot obtained from s a single shell-elementsimulation, under axial compressive loading, using the CO-DAM1 and CODAM2 material models. . . . . . . . . . . . 208F.4 Stress vs. strain plot obtained from a single shell-element un-der tensile loading in the transverse material direction, usingthe CODAM1 and CODAM2 material models. . . . . . . 209F.5 Stress vs. strain plot obtained from a single shell-element un-der compressive loading in the transverse material direction,using the CODAM1 and CODAM2 material models. . . . 210xvList of Computer ProgramsB.1 Typical content of a parameters.txt file, used to control theexecution of the LCZ algorithm. . . . . . . . . . . . . . . 182B.2 Content of the 1_dynascr text file used to batch-process theLCZ execution. . . . . . . . . . . . . . . . . . . . . . . . . 184D.1 LS-DYNA MAT_54 card used in the double-notched [90/0]stest coupon (Chapter 5.1), for the 0◦ ply . . . . . . . . . . 194D.2 Material definitions used in the double-notch [90/0]s for the90◦ ply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195D.3 Material definitions (CODAM1) used in the tube crush anal-ysis (Defined using an LS-DYNA user-defined material card 196D.4 Strain-softening parameters used in the tube crush analysis(This is the first part of the ssparam.dat file, which is calledby the CODAM1 material model . . . . . . . . . . . . . . 197D.5 Strain-softening parameters used in the tube crush analysis(This is the first part of the ssparam.dat file, which is calledby the CODAM1 material model . . . . . . . . . . . . . . 198D.6 Strain-softening parameters used in the tube crush analysis(This is the first part of the ssparam.dat file, which is calledby the CODAM1 material model . . . . . . . . . . . . . . 199D.7 CODAM2 card used in tube-crush analysis . . . . . . . . 200D.8 Cohesive material card used in tube-crush analysis . . . . 201xviList of SymbolsA element-set used by the LCZ algorithm during the element splittingprocessa length of initial crackATSLC area under the normalized traction-separation law, used as an inputto the cohesive material model.C0 structural stiffness of the undamaged materialCd structural stiffness of the damaged materiald damage parameterE11 Young’s modulus, axial directionE22 Young’s modulus, transverse directionE33 Young’s modulus, normal directionExx longitudinal elastic modulusEyy transverse elastic modulusEzz out-of-plane elastic modulusG12 shear modulus, 12 directionG23 shear modulus, 23 directionG31 shear modulus, 31 directionGIc critical strain-energy release rate, mode IGIIc critical strain-energy release rate, mode IIh thickness of beami first element to reach the critical value of the element-splitting criteriaKI stress intensity factor for mode-I crack openingKII stress intensity factor for mode-II crack openingl length of beamxviiList of SymbolsL length of beamP reaction force monitored during the transient analysisQ element satisfying the radial search criterion,r radial distance from a crack tipR geometrical radius used by the radial-search algorithmR1 geometrical radius used by the radial-search algorithmR2 geometrical radius used by the radial-search algorithmS parameter used for the element splitting criterionSc parameter used for the critical value of the element-splitting criteriont thickness of beamt¯ normalized traction-stressXMU failure parameter used in the cohesive material lawα user-defined interaction term for cohesive damage growthβ mode-mixity of the cohesive loadδI crack-opening in the normal directionδII crack-opening in the shear direction∆ applied displacementεf intralaminar damage saturation strain[εiy]tinitiation strain for damage under tension in the transverse direction[εiy]cinitiation strain for damage under compression in the transverse di-rection[εsy]tsaturation strain for damage under tension in the transverse direction[εsy]csaturation strain for damage under compression in the transversedirection[εix]tinitiation strain for damage under tension in the axial direction[εix]cinitiation strain for damage under compression in the axial direction[εsx]tsaturation strain for damage under tension in the axial directionxviiiList of Symbols[εsx]csaturation strain for damage under compression in the axial directionφ angle in a radial coordinate system, relative to the crack interfaceλ normalized crack openingν12 major Poisson’s ratio, (in-plane)ν13 major Poisson’s ratio, (out-of-plane)ν23 Poisson’s ratio, (transverse plane)σ13 out-of-plane shear stress in the 13 local material’s coordinate systemdirectionσ23 out-of-plane shear stress in the 23 local material’s coordinate systemdirectionσ33 out-of-plane normal stress in the local material’s coordinate systemdirectionσmax maximum normal stress in the cohesive interface, before softeningbeginsσzz stress in the global zz directionσczz critical value of the stress in the global zz direction, at which elementsplitting takes place, in the numerical DCB verification case.τxz shear stress in the global zx planeτmax maximum shear stress in the cohesive interface, before softening be-ginsxixAcknowledgementsI would like to express my special appreciation and thanks to my supervi-sor Professor Reza Vaziri, for his continuous support, guidance and criti-cism throughout my study, serving as a lighthouse directing me toward theachievement of the challenging research goals.I would like to thank members of the of the UBC Composites Group -Mina Shahbazi, Alireza Forghani and Navid Zobeiry for their helpful dis-cussions, useful tips and friendly company during these years.I acknowledge Professor Stephen R. Hallett for allowing me to use thefigure from his work which is shown in Figure 5.7, Figure 5.8 and 5.9 of thisthesis.I gratefully acknowledge the Natural Sciences and Engineering ResearchCouncil (NSERC) of Canada, for their financial support of this research.I would like to thank the technical support team of the Livermore Soft-ware Technology Corporation, and in particular, Mr. Jim Day, for their as-sistance in responding to numerous technical questions related to LS-DYNA.I would like to express my gratitude to my dear friends in Vancouver,who have been my second family and provided great support - Roee andHadas Diamant, Chen and Itai Bavli, Nir and Michal Simon, Ofira and IdoRoll, Dorit and Eran Treister.This work would not be possible without the support of my family mem-bers - my dear parents and sister, for their unconditional love, encourage-ment and support. And above all - to my love Ronit who walked right nextto me during this challenging journey, supporting me through the good anddifficult times as well.xxDedicationTo my parents, Etta and Menachem, who always gave me the best educationthey could.xxiChapter 1IntroductionThe use of composite materials in advanced engineering applications is grow-ing rapidly due to their excellent specific strength, durability, fatigue andcorrosion resistance. While originally used in limited number of militaryand aerospace applications due to their high manufacturing cost and thelack of scientific knowledge related to their processing and mechanical be-havior, the need to develop lighter and yet stronger structures, togetherwith the accumulated knowledge related to their processing and mechanicalbehavior, allowed composites to become widely used in numerous industrialapplications requiring peak performance and superior reliability.Composite materials can now be found in a wide range of commercialproducts, ranging from sports equipment such as bikes and skis, to loadbearing structures in the automotive, aerospace and ship-building industries.The use of composites as the main structural material in the fuselage ofthe Boeing 787 commercial jet airliner, demonstrates the huge leap in thedevelopment of composites during the last decades.Despite their benefits, composites pose great engineering challenges, bothfrom the manufacturing standpoint as well as their mechanical behavior.As their physical architecture consists of thin layers (plies) of relativelyhigh-strength anisotropic material stacked to the required thickness usingrelatively weak bonding materials (matrix), their structural performanceoften derives not only from the behavior of a single ply, but also from theinteraction and mechanical behavior of the ply-bonding interface. Failure ofthe interface between the composite plies, often known as delamination, canlead to premature strength reduction and reduced load carrying capacityof the structure, hence the need to develop reliable methods to predict itsinitiation and growth within these materials.Despite the vast amount of research, a unified damage theory that cancapture all of the complex damage mechanisms in composites and describetheir behavior up to final failure is still beyond reach. Yet, various damagetheories are constantly being developed, and the rapid growth in computa-tional power allows more sophisticated theories and computational methodsto be applied, increasing the predictive capability of their behavior.1Chapter 1. IntroductionThe UBC Composites Group, part of the Composite Research Network,a consortium founded and supported by the government of Canada, servesas a fertile growing ground for composites research. The Group’s visionhas always been development of physically based and numerically robustnumerical tools, that would give engineers practical tools to develop theircomposite products with greater reliability, higher performance, and lowercost.Recognizing the importance of modelling delamination in a discrete man-ner, yet aiming toward the development of a numerically efficient modellingmethodology, the author’s contribution of the present work is in the develop-ment of a new method which allows simulating the initiation and growth ofdelamination damage in an adaptive manner. Using this method, the struc-ture can be modelled with only one layer of elements through the thickness.During the transient analysis, the continuum elements are locally and adap-tively split through their thickness, and cohesive elements are introducedin regions where delamination has the potential to initiate and grow. De-lamination can thus propagate in the structure as the simulation progresses.Reducing the number of cohesive elements present in a model contributes tothe reduction of the computational cost, as well as alleviating the unwantedartificial stiffness reduction caused when cohesive-elements are embedded inlarge regions of the model. The method can be combined with the other nu-merical models developed within the UBC Composites Group for intralam-inar (in-plane) damage behavior, and thus serves as a continuation of thework.The details of the method are presented in Chapter 3, followed by verifi-cation of the method for Mode-I, Mode-II, and Mixed-Mode loading condi-tions (Chapter 4). In Chapter 5, the method is applied to various engineer-ing applications including dynamic loading of composite structures. Theresults obtained using the current methodology are compared to experimen-tal results, as well as to results obtained using other numerical techniques.The numerical advantages of the method are described in Chapter 6. Chap-ter 7 presents the conclusions drawn from the research and suggest topicsfor future work. Appendix A presents a flowchart of the LCZ algorithmdeveloped during the research. Detailed instructions for setting and exe-cuting the LCZ algorithm is outlined in Appendix B. A general descriptionof composite tube crushing process is provided in Appendix B. AppendixD provides the important keywords used during the simulations describedwithin this thesis. A brief description of the CODAM2 material model,which is used in Chapter 5 for simulating a dynamic tube crushing event,is given in Appendix E. Description of the process used in order to cali-2Chapter 1. Introductionbrate the CODAM2 material parameters for the tube crushing simulationsis presented in Appendix F.3Chapter 2Numerical Simulation ofDelamination in CompositesComposite materials are increasingly being used in advanced structural ap-plications. Failure of these materials involves evolution of various damagemechanisms, such as fibre breakage and matrix cracks (Zobeiry, Vaziri, andPoursartip [137], Green et al. [44]), where the debonding of adjacent lami-nate layers, also known as delamination, is considered to be one of the mostdominant damage mechanisms affecting the behavior of composite lami-nates. Delamination will usually lead to a reduction in structural stiffnessand load carrying capability, and can also lead to instability and prematurestructural failure under compressive loading (Bolotin [11]). This raises thenecessity to predict its initiation and propagation.Early methods aimed at simulating delamination in composites werebased on stress-based criteria, where the inter-laminar and out-of-planestresses (σ13, σ23, σ33, with x1 and x2 as in-plane coordinates and x3 beingthe out-of-plane coordinate) were used to predict the initiation and growthof delamination damage in the material ([61]). These models were proven tobe effective in capturing the initiation of delamination, but could not cap-ture the scale-effects as in a fracture-based model (Davies and Zhang [21]).Since the delamination damage mode is discrete in nature, it is widely ac-cepted in the scientific community that fracture mechanics principles shouldbe implemented in order to accurately predict delamination initiation andgrowth.The Virtual Crack Closure Technique (VCCT), originally developed byRybicki et al. [113],[112], is based on fracture-mechanics principles. Usingthis method, the strain energy release rate G is calculated numerically, andis compared to some critical value Gc in order to determine whether or notthe delamination crack propagates in a given timestep. VCCT was provento be capable of predicting the evolution of delamination damage under var-ious loading conditions (Rybicki et al. [113], Raju et al. [108], Zheng and Sun[136]). Complex delamination patterns were also predicted by the VCCTmethod, where the strain-energy release rate was used to predict delami-42.1. The Cohesive-Zone Methodnation induced damage during a low velocity impact event ([72], [73]). Anoverview of the VCCT method and its numerical implementation into finiteelement codes can be found in [69], [70]. An evaluation of the capabilitiesof two commercial finite element solvers - ABAQUS and MARC, to predictdelamination growth and the strain-energy release rate in ENF and DCBloading cases, appears in Ori [1]. A major drawback of the VCCT methodis that it requires the presence of an initial crack in the finite element meshprior to the analysis, which makes the method useful for cases where theexact location of the delamination crack is explicitly known. For cases in-volving large structures where delamination crack location is unknown, themethod becomes less favourable. In addition, since VCCT is based on linearelastic fracture mechanics (LEFM), it is limited to cases where the size ofthe fracture process zone is negligibly small compared to the other structuraldimensions. This assumption is not valid for many quasi-brittle materials[10]. In such cases, the fracture process zone, as well as the embedded co-hesive tractions have to be modelled explicitly. Cohesive zone models havebeen developed over the past decades to address the above issues.2.1 The Cohesive-Zone MethodThe cohesive zone method (CZM) is based on a concept originally proposedby Dugdale [27] and Barenblatt [8], which stated that unlike perfectly elasticmaterial behavior which predicts infinite stress values at the crack tip, thereis a region of material ahead of the crack tip within which the materialbehavior is not linear elastic, thus yielding a state of stress with finite values.This region of damaged material is often referred to as the Cohesive DamageZone (CDZ), or the Cohesive Zone (CZ), (Figure 2.1.a). This idea was usedby Dugdale [27] to determine plastic zone sizes in steel panels containingslits. In fiber reinforced plastic composite materials (FRPs) this "damagezone" consists of matrix cracking, fibre breakage, interface separation andfibre pullout. In a cohesive crack approach, a relationship between tractionand displacement is defined along a potential crack surface, ahead of anactual crack or notch tip.When implementing the cohesive-zone method in a finite element solver,the cohesive zone is embedded into the model using a cohesive interface(Figure 2.1.b), where the need to calculate the non-physical singular stressfield at the crack tip is eliminated by using a force-displacement relationbetween the nodes in the finite element mesh (traction-separation law) [28].This law is the basis for computing the delamination crack initiation, prop-52.1. The Cohesive-Zone MethodCohesive InterfaceCrack TipCohesiveDamage Zonea) b)Figure 2.1: A schematic diagram demonstrating a typical cohesive zone ina continuum material. a) A volume of damaged material which extendsbeyond the crack tip, is referred to as the "Cohesive Damage Zone". b)In a finite element model, the cohesive zone is modelled using a cohesiveinterface realized using a cohesive law, via discrete (Hallett and Wisnom[46]) or solid cohesive elements (Hu et al. [56],[122],), a cohesive contactformulation (Borg et al. [12], Borg et al. [13]), or cohesive properties directlyembedded into the element formulations [60]. As the crack opens, tractionforces are applied across the interface, defined using a traction vs. separationlaw.agation, and opening. A typical bilinear traction-separation law is shown inFigure 2.2. The stress at the interface, σ, is plotted against the interfaceopening, λ. The interface is linear elastic up to a maximum stress, σmax,at a displacement λ0. As the interface is further loaded, the interface stresslinearly decreases until a critical displacement λ0 is reached, at which theinterface stress reaches 0 and the crack interface is traction-free.CZMs incorporates fracture mechanics principles, as the area under thetraction-separation curve is equal to the critical strain-energy release rate ofthe material Gc (Figure 2.2). By defining multiple traction-separation laws,the cohesive behavior of the material can be described under mode-I andmode-II loading directions. In this case, a separate traction-separation lawis defined for each mode of loading, together with a mode-mixity relation,between the two modes (Mi et al. [91] Alfano and Crisfield [5] ,[52]).The relation between the state of stress in a crack under mode-I loadingcondition and the crack opening displacement is shown in Figure 2.3. Inthis figure, a schematic side view of a crack under mode-I loading conditionis presented, together with a traction-separation law, representing the state62.1. The Cohesive-Zone MethodλσGcσmaxλ0 λcrFigure 2.2: A typical bilinear traction-separation law used in a cohesiveinterface model. The stress at the interface, σ, is plotted against the interfaceopening, λ.of traction stress along the crack interface. The loaded geometry is dividedinto three distinct regions or zones - A region of homogeneous material, thecohesive zone, and a region consisting of the crack’s free surfaces. The ho-mogeneous material consists of material at an unloaded state, or material ata loaded state having a traction level lower than σmax. The geometrical lo-cation where the traction level reaches σmax and the cohesive zone begins, istermed the mathematical crack front. At this location, the traction stressesdecrease with an increased crack opening. The cohesive zone spans alongthe positive x direction, from the mathematical crack front, to the pointwhere the traction stresses reach a value of 0, often termed the physicalcrack front. The physical crack front is the location where a traction freeregion begins, consisting of the crack’s free surfaces.For mode-II loading condition, the schematic stress distribution is similarto the one shown in Figure 2.3, with the maximum traction stress equal toτmax, and the critical fracture energy equal to GIIc.72.1. The Cohesive-Zone MethodxzHomogeneous MaterialCohesive Zonelcz Free Surfaceλσσmaxλ0 λcrλcrλ0MathematicalCrack FrontPhysicalCrack FrontGIcFigure 2.3: A schematic side view of a crack under mode-I loading condition.A traction-separation law is plotted below the crack, representing the stateof traction stress within the material. Three distinct regions span across thecrack interface - A region of homogeneous material, the cohesive zone, andthe crack interface where free surfaces have been created. The cohesive zonespans from the point of maximum stress (σmax) to the location where thefree-surfaces of the crack begin and the stress level has reached a value of0. The geometrical location where the traction level reaches σmax and thecohesive zone begins, is termed the mathematical crack front. The physicalcrack front is the location where a traction free region begins (λcr, σ = 0),consisting of the crack’s free surfaces.82.2. Estimating the Cohesive Zone Length2.2 Estimating the Cohesive Zone LengthDiferent models have been proposed in the literature to estimate the lengthof the cohesive zone. Irwin [59] presented a simplified model for the deter-mination of the plastic zone ahead of a crack tip, for an elastic, perfectlyplastic material. Based on his approach, crack growth can be predictedusing linear-elastic fracture-mechanics principles. Figure 2.4 presents an in-finite geometry, having an initial crack of length 2a uniformly loaded withan applied stress σ.Interfacexzrφ2aσσFigure 2.4: Uniformly mode I loaded infinite geometry with an linear crackof an initial length 2a in the interface.The stress components around the crack tip are given by [128]:σxx =KI√2pir[cos(12φ)(1− sin(12φ)sin(32φ))](2.1)92.2. Estimating the Cohesive Zone Lengthσzz =KI√2pir[cos(12φ)(1 + sin(12φ)sin(32φ))](2.2)σxz =KI√2pir[cos(12φ)sin(12φ)cos(32φ)](2.3)σyy = ν (σxx + σzz) (2.4)where KI is the stress intensity factor for mode-I crack propagation, ris the radial distance from the crack tip, and φ is the angle relative to thecrack interface (x axis).It can be inferred from Equation 2.2, that in the vicinity of the cracktip (r = 0), σzz reaches a value of ∞ along the crack propagation direction(φ = 0), (Figure 2.5(a). Irwin [59] stated that since yielding or further crackpropagation will occur prior for the stress levels reaching infinite values, thestress values are limited by a maximum limiting value (σmax, Figure 2.5(b)).Thus, under this assumption, using Equation 2.2 to evaluate σzz at φ = 0yields:σzz = σmax =KI√2pila(2.5)where la (Figure 2.5(a)) is the length of the geometrical region along thecrack propagation direction in which the relation σzz = σmax is satisfied.Equation 2.5 can be re-written to evaluate la:la =K2I2piσ2max(2.6)Using la as suggested by Equation 2.6 will not satisfy the global equi-librium. Irwin suggested a correction - defining a new plastic zone size lpz,which will, for a given value of σmax, satisfy the following equation:σmaxlpz =KI√2pi∫ la0r−12dr = 2KI√2pi√la (2.7)The geometrical interpretation of Equation 2.7 is schematically presentedin Figure 2.5(c) and Figure 2.5(d), where lpz is obtained by equating thehatched area in Figure 2.5(d) to the hatched area in Figure 2.5(c).Combining Equation 2.7 and Equation 2.6 yields:lpz =K2Ipiσ2max= 2la (2.8)102.2. Estimating the Cohesive Zone LengthIrwin derived a relation between the stress intensity factor K and theenergy release rate G as:K2 = GE′ (2.9)where in plane stress E′ = E, and in plane strain E′ = E/(1− ν2).Assuming a mode-I loading condition, lpz can be obtained by rewritingEquation 2.8 as:lpz =E′GIcpiσ2max(2.10)In case of mode-II loading condition, the stress components around thecrack tip are given by [128]:σxx =KII√2pir[−sin(12φ)(2 + cos(12φ)cos(32φ))](2.11)σzz =KII√2pir[sin(12φ)cos(12φ)cos(32φ)](2.12)σxz =KII√2pir[cos(12φ)(1− sin(12φ)sin(32φ))](2.13)σyy = ν (σxx + σzz) (2.14)Where KII is the stress intensity factor for mode-II crack propagation.Evaluating Equation 2.13 using φ = 0 and r = la, yields:σxz =KII√2pila(2.15)Similar to the assumptions made in Irwin’s corrections for mode-I crackpropagation, Equation 2.7 can be used to obtain the plastic zone size undermode-II loading conditions:lpz =E′GIIcpiτ2max(2.16)where GIIc is the critical strain energy release rate under mode-II load-ing, and τmax is the maximum shear stress within the plastic zone.Equation 2.8 can be rewritten using a more generalized format:112.2. Estimating the Cohesive Zone Lengthlpz = ME′GIcτ2max(2.17)where M is a scaling factor, in which according to Irwin is equal to1/pi. Other models predict different values for M , and are listed in Table2.1. Dugdale [27] estimated the size of the plastic zone ahead of a crack tipby treating the plastic region as a narrow strip that is loaded by the yieldtraction. Hui [57] estimated the length of the cohesive zone for soft elasticmaterials, while Rice [109] and Falk et al. [30] estimated the length of thecohesive zone as a function of the crack growth velocity. Barenblatt [9] usedassumptions similar to Dugdale [27] for ideally brittle materials.Table 2.1: Estimated cohesive zone length (lpz) and equivalent value for Min Equation 2.17Source lpz MHui [57] 23piEGIcσ2max0.21Irwin [59] 1piEGIcσ2max0.31Dugdale [27], Barenblatt [9] pi8EGIcσ2max0.4Rice [109], Falk et al. [30] 9pi32EGIcσ2max0.88Hillerborg et al. [54] E GIcσ2max1In order to obtain a reliable numerical solution when using the CZMto predict crack propagation, a minimum number of finite-elements shouldlie within the cohesive zone along the crack propagation direction [126].Moes and Belytschko [92] suggested using more than 10 elements along thecohesive zone, while Falk et al. [30] used 2-5 elements in their work.Turon et al. [126] and Alfano and Crisfield [5] showed that by reducingthe maximum interfacial strength (σmax in case of mode-I crack propagation,or τmax in case of mode-II crack propagation), the cohesive zone length isincreased, resulting in more elements spanning across the cohesive zone.This allows coarser meshes to be used in the analysis, still yielding reliableresults. Alfano and Crisfield [5] showed that since artificially scaling themaximum traction stresses will also alter the stress distribution within the122.2. Estimating the Cohesive Zone Lengthstructure, there is a limitation to the allowable artificial cohesive zone lengththat will still be acceptable before the solution will no longer reflect thecorrect mechanical behavior of the structure. This effect is demonstrated inSection 4.1.1, where scaling of the maximum cohesive strength is performedin order to investigate its effect on the solution of a mode-I verificationproblem.Describing the complex material behavior using traction-separation laws,allows cohesive zone models to deal with the nonlinear zone ahead of thecrack tip without necessitating the use of a refined mesh around the crack-tipregion, and without introducing delamination cracks into the model prior tothe analysis. CZM was successfully used by Camacho and Ortiz [14] to pre-dict crack propagation during a steel/rock impact. The method was provento be capable of predicting delamination growth in composite materials un-der static (Liu et al. [77],Yang and Cox [134]), as well as dynamic loadingand impact conditions (Feng and Aymerich [33], Sokolinsky et al. [120],Olsson et al. [97], Gonzalez et al. [42]). Camanho [15] developed a zero-thickness volumetric cohesive element able to capture delamination onsetand growth under mixed-mode loading condition. Forghani and Vaziri [36]and Menna et al. [90] used a cohesive contact interface to simulate delami-nation propagation in plates subjected to transverse impacts. Abisset et al.[3] used solid cohesive elements constrained using a contact algorithm tiedto the continuum elements in the model in order to simulate plate indenta-tion experiments. Cohesive elements were used to model both interlaminaras well as intralaminar damage within the material. By using a tied con-tact algorithm to constrain the cohesive elements nodes to the continuumelements, the need to have adjoining nodes between these different topolo-gies is removed, thus allowing high flexibility from the modelling standpoint.Abisset et al. [3] were able to correctly depict the damage evolution withinthe specimens, for different material lay-ups.Some models aim to describe the rate-dependency of the cohesive inter-face. Musto and Alfano [93] and Makhecha et al. [80] implemented cohesiveelements formulations with strain-rate dependency, for mode-I DCB load-ing scenario. Anvari et al. [6] used rate-sensitive and triaxiality-dependentcohesive elements, to simulate crack growth under quasi-static and dynamicloading conditions.Since the use of CZM in its standard form requires introduction of largenumber of cohesive elements in all possible locations where delamination islikely to grow, using this method to predict delamination crack growth inlarge structures is not practical from the numerical standpoint. Further-more, using cohesive elements in wide regions of the model will artificially132.2. Estimating the Cohesive Zone Lengthrender the structure more compliant (Kaliske et al. [62]). Increasing thestiffness of the cohesive interface in order to address this can in many caseslead to numerical noise and instability. A method to adaptively reduce thecohesive stiffness in the vicinity of the crack tip, in order to stabilize andreduce the noise in the numerical analysis, was devised by Elmarakbi et al.[29] and Hu et al. [56]. Although this method was capable of reducingthe numerical noise, it still required embedding cohesive elements in largeregions of the model, where delamination is expected to take place.One solution to the above problems is to use an adaptive approach,where cohesive elements could be locally and adaptively inserted into thefinite element model as the delamination crack propagates. Park et al. [105]used an adaptive technique, where a two-dimensional triangular mesh wasadaptively refined and coarsened around the crack tip, and cohesive elementswere inserted into the finite element mesh, in order to simulate 2D in-planecrack propagation in isotropic materials. Kawashita, Bedos, and Hallett [64]used an approach to simulate in-plane matrix crack propagation throughadaptive element splitting and insertion of in-plane cohesive zones into themodel.A novel adaptive method, which allows adaptively seeding cohesive el-ements into the structure and is intended to simulate delamination in anadaptive manner, will be presented in Chapter 3, followed by verificationof the method for Mode-I, Mode-II, and Mixed-Mode loading conditions(Chapter 4). In Chapter 5, the method is applied to various engineeringapplications. The results obtained using the current methodology are com-pared against available experimental results, as well as to results obtainedusing other numerical techniques. The numerical advantages of the methodare described in Chapter 6.142.2. Estimating the Cohesive Zone Lengthrσzz(a)rσmaxσzzla(b)rσmaxσzzla(c)rσmaxσzzlalpz(d)Figure 2.5: Typical σzz profiles obtained using Equation 2.2 with φ = 0, as afunction of the distance from the crack tip, r. a.) Stress profile as suggestedby [128] (Equation 2.2), predict infinite σzz values at r = 0. a.) Accordingto Irwin [59], due to the material’s yielding and crack propagation, the stresslevels are limited by some limiting value, σmax. c.) Irwin introduced thegeometrical distance la, in which the relation σzz = σmax is satisfied. Sincethe maximum stress levels are limited to σmax, equilibrium is not conserved.c.) Irwin suggested a correction to la, which is now increased and termed theplastic zone size (lpz). lpz is determined such that the hatched area underthe corrected stress profile (d), will be equal to the hatched area in (c), thusmaintaining the global equilibrium (Equation 2.7).15Chapter 3A Local Cohesive ZoneMethod for Simulation ofInterlaminar Damage inLaminated CompositeMaterials3.1 IntroductionThe increasing use of composites in advanced engineering applications, to-gether with the improvement of the available computational power and fi-nite element codes, raises the need to develop efficient and reliable numericaltools to predict their behavior under various loading conditions. Numericallypredicting failure and damage in these materials, requires correct numericalrepresentation of the various damage mechanisms within which contribute totheir behavior. Chapter 1 emphasized the need to simulate delamination, oneof the most dominant damage mechanisms in composites. A brief review ofthe dominant numerical methods aimed at simulating delamination was pre-sented in Chapter 2. It was shown that although various numerical methodsto simulate this failure mode do exist, their application to larger numericalmodels still presents a considerable computational challenge. Amongst thevarious numerical methods aimed at simulating delamination in compositematerials, the cohesive-zone method is becoming the method of choice byresearchers and engineers, due to its reliability, its applicability to commer-cial finite element solvers, and its relative numerical simplicity [4]. Despiteits benefits, the high computational cost of the method when an increasednumber of cohesive interfaces is used, renders the method unfeasible forsolving large engineering models. In addition, the method can effect thecompliance of the structure, thus introducing unwanted numerical error tothe analysis. Some adaptive cohesive approaches were discussed in Chap-163.2. Main Principles of the LCZ Methodter 2, in which adaptive introduction of cohesive zones is used in order toalleviate the drawbacks of the method when applied to larger models.The contribution of the present work is in the development of a newmethod which allows simulating the initiation and growth of delaminationdamage in an adaptive manner. Using this method, the structure can bemodelled with only one layer of elements through the thickness. During thetransient analysis, the continuum elements are locally and adaptively splitthrough their thickness, and cohesive elements are introduced in regionswhere delamination has the potential to initiate and grow. Delaminationcan thus propagate in the structure as the simulation progresses. Reduc-ing the number of cohesive elements present in a model contributes to thereduction of the computational cost, as well as alleviating the unwanted ar-tificial stiffness reduction caused when cohesive-elements are embedded inlarge regions of the model.3.2 Main Principles of the LCZ MethodThe adaptive element splitting technique presented here is based on the no-tion that only a minimal number of cohesive elements need to be be presentin a finite element model in order to correctly describe the delaminationcrack propagation, and cohesive elements should only be introduced whereand when needed during the analysis. Using this method, the structuremade of the composite material can be modelled using only one layer ofstructural elements. Delamination cracks and crack-growth paths do notneed to be defined in the model prior to the analysis, as they are createdadaptively during the course of the transient simulation. . This is demon-strated in Figure 3.1.a and Figure 3.1.b: Figure 3.1.a shows the conventionalimplementation of the CZM. A cohesive interface is defined apriori of thefinite element analysis, along the potential delamination crack interface. Asdelamination propagates, nodes located along the interface are released andnew free surfaces are created (Figure 3.1.b). Simulating delamination usingthe Local Cohesive Zone method, requires no cohesive interface to be definedapriori of the finite element analysis (Figure 3.1.c). As the finite elementanalysis progresses, a user-defined element splitting criterion is evaluatedwithin the continuum elements. When this criterion is satisfied, the contin-uum finite elements are split through their thickness, and cohesive elementsare locally seeded along a narrow band ("cohesive band") that is wide enoughto capture the delamination crack propagation (Figure 3.1.d). The cohesiveband can propagate through the structure as the transient analysis pro-173.3. LCZ Algorithm Overviewgresses. This is demonstrated in Figure 3.2, where a DCB problem is solvedusing the LCZ algorithm. The cohesive-band migrates as the delaminationcrack propagates along the structure, shown here in three subsequent stages- starting at stage a), propagating to stage (Figure 3.2.a, Figure 3.2.b andFigure 3.2.c).The method is implemented through the development of a computercode governing the transient finite element analysis, written in Python. Thestructural problem is solved using the commercial explicit finite elementcode LS-DYNA [18], while the LCZ algorithm monitors the solution andperforms the various operations as the transient analysis progresses, such aselement splitting and introduction of new cohesive elements into the model.The following section will give a brief description of the main operationsperformed by the LCZ algorithm, and the main variables controlling itsbehavior.A schematic flow-diagram demonstrating the relation between the LCZalgorithm and the finite element solver is shown in Figure 3.3.The following sections describe the main execution stages. Two relatedmodels were chosen in order to demonstrate the method. Both models de-scribe a similar structural problem - loading of a cantilever beam under a"splitting" type load. The first model is presented in a simple two dimen-sional view (Figure 3.4). Here, a cantilever beam of length l and thicknessh, is subjected to a splitting displacement ∆ at its other end. A three-dimensional model, with a larger number of elements, is shown in Figure3.5, in order to demonstrate other features in the model, not visible in thetwo dimensional view. A schematic flow chart demonstrating the algorithmbehavior is presented in Appendix A.3.3 LCZ Algorithm Overview3.3.1 Problem InitializationDuring the initialization phase, the LCZ algorithm receives the LS-DYNAinput file, which includes the material definitions, as well as the boundaryand initial conditions. Once initiated, the algorithm executes independentlyand controls the transient analysis until the final termination time is reached.As opposed to the implementation of the conventional CZM, the currentmethodology does not require explicit modelling of the cohesive interfaceand cohesive elements prior to the analysis. In most examples described inthis thesis, only a single layer of continuum elements was used through thethickness of the composite part, without a priori introduction of cohesive183.3. LCZ Algorithm Overviewy xCohesive interface(a)y xDelamination front(b)y x(c)y xCohesive band frontDelamination frontCohesive band(d)Figure 3.1: a). When simulating delamination using the conventional im-plementation of the cohesive zone method, the cohesive interface is definedprior to the finite element analysis, along the potential delamination crackinterface. b). As delamination propagates, interface nodes are released andnew free surfaces are created. c). When simulating delamination usingthe Local Cohesive Zone method, no cohesive interface is defined a priori.d). When an element splitting criterion is satisfied, the continuum finiteelements are split through their thickness and cohesive elements are locallyseeded along a narrow band ("cohesive band").193.3. LCZ Algorithm Overview(a) (b) (c)Figure 3.2: Typical cohesive-band created when solving a DCB problemusing the LCZ algorithm. The cohesive-band migrates as the delaminationcrack propagates along the structure, shown here in three subsequent stages- starting at stage a), propagating to stage b) and finally c).LCZ codeFinite Element meshUser control variablesMaterial parameters, boundaryand loading conditionsLS-DYNAAnalysisFigure 3.3: A schematic flow diagram demonstrating the relation betweenthe LCZ algorithm and the finite element solver (LS-DYNA).elements in the model. In this case, the continuum elements chosen for thesolution were LS-DYNA’s built-in Thick-Shell elements [18], which allowthe user to specify a different material angle for each through-thicknessintegration point. These elements assume a parabolic distribution of theout-of plane shear stresses (τzx) through the thickness of the element. Whilethis assumption might not be valid for every composite laminate system, forthe cases considered here, it did not seem to result in significant errors.203.3. LCZ Algorithm OverviewxzhL∆∆iFigure 3.4: A simple numerical model used to demonstrate the LCZ algo-rithm - a cantilever beam with a length L and thickness h under splittingloading condition. One end of the beam is subjected to two opposing applieddisplacements, ∆.3.3.2 Element-Splitting CriteriaOnce the transient analysis has been initialized, the continuum elementscontinue to be monitored in order to determine whether or not cohesiveelements should be introduced into the finite element mesh. An elementsplitting criterion, based on a critical value Sc of a quantity S, is defined inorder to evaluate the specific location where the cohesive elements shouldbe introduced. Whenever S reaches the value of Sc in a specific element,a through-thickness element splitting operation is performed, and cohesiveelements are locally introduced at the splitting interface. It is important tonote that the value of Sc should not be viewed as a delamination threshold,as the actual delamination and delamination crack propagation will be gov-erned by the cohesive element traction-separation law. Instead, Sc servesas a flag to determine when and where the potential for the delaminationgrowth should be seeded into the model, by introducing the cohesive ele-ments into the finite element mesh. For simplicity, let us now consider thiscriterion to be a normal out-of-plane stress value in the global z direction(3.1), where z denotes the direction normal to the surface of the compositepart, and σczz denotes the critical σzz value. Splitting takes place wheneverS is equal to or exceeds Sc (Equation 3.2).Sc = σczz (3.1)S ≥ Sc (3.2)213.3. LCZ Algorithm Overviewxyz∆Figure 3.5: Isometric view of the detailed cantilever beam model used fordemonstrating the LCZ algorithm - supported cantilever beam under split-ting loading condition. One end of the beam is subjected to two oppositedisplacement constraints ∆.As the applied displacement, ∆, is increased, at some instance in time, Sreaches a user-defined critical value of Sc within the thick-shell element i inFigure 3.6. Thick shell element i is now flagged as a parent-element flaggedfor splitting, and the algorithm now proceeds to the next phase, which isdescribed in the following section.3.3.3 Radial Neighbour SearchIn order for CZMs to correctly capture the mechanical crack propagation, anumber of cohesive elements should be included across the active cohesivezone (Mi et al. [91], Harper et al. [52]). In order to increase the numberof thick-shell elements which will be split by the code, a radial geometri-cal search is now performed for all neighbouring thick-shell elements withcentroids that lie within a user-defined distance R1 from the centroid of theelement i (Figure 3.6.a). All of the thick-shell elements satisfying this ge-ometrical search, including element i, are added into an initially empty setof elements, A, which will include all thick-shell elements flagged for split-ting, and are shaded in Figure 3.6.a using a green color. Since the valueof R1 used in this example is small, thick-shell elements j and k are notdepicted by the radial search, and are therefore not added to the elementset A. This is an undesirable outcome - assuming that the load ∆ is evenly223.3. LCZ Algorithm Overviewdistributed along the edge of the beam, the value of S in these elementsapproaches Sc, and it is mechanically feasible that these elements will besplit as well. Increasing the value of R from R1 to R2 (Figure 3.6.b) willnow result in a larger region of thick-shell elements to be included withinelement set A (Shaded in red and green in Figure 3.6.b). Element set A willnow include thick-shell element i, as well as thick-shell elements j and k.However, the boundaries of the geometrical region bounding element set A(and is marked by line a¯b in Figure 3.6.b), do not cross the beam along itswidth perpendicular to the beam’s main axis. It is expected that given theevenly distributed load ∆, the element set A will be bounded by a straightline (similar to line a¯c in Figure 3.6.b). A second search operation which isdescribed in the following section, is now performed in order to improve theresults.3.3.4 Threshold Neighbour SearchIr order to improve the mechanical feasibility of the results obtained by theradial search, a threshold filter is now added to the search (Figure 3.7):Once S reaches a value of Sc within a thick-shell element i, all thick-shellelements which satisfy the following equation are flagged for splitting andare included in element set A:S ≥ threshold× Sc (3.3)where threshold is a user-defined scalar, having a value which is higherthan 0 and lower than 1.Adding the threshold filter, and choosing an appropriate value for thethreshold parameter, will result in all thick-shell elements located along theedge of the beam where ∆ is applied, to be included in element set A (Figure3.7.a). The geometrical search is now performed separately, for each elementincluded in element set A. The thick-shell element in set A for which thesearch is performed is termed the parent element. Each thick-shell elementwho has its centroid lie within a distance R of the parent element, will beadded to element set A. this will result in a mechanically feasible region ofelements which will flagged for splitting, shown in Figure 3.7.b. This regionconsists of thick-shell element i, which was the first thick-shell element toreach a value of S = Sc, a row of thick-shell elements shaded in green, whichsatisfied the threshold search (Equation B.4), and a region of thick-shellelements shaded in red, which satisfied the radial search.233.3. LCZ Algorithm OverviewMeaning'of'R'35'xyzFigure 4: Isometric view of the detailed cantilever beam model used for demonstrating the LCZ algorithm - supported cantileverbeam under splitting loading condition. One end of the beam is subjected to two opposite displacement constraints .found to have a negligible e↵ect on the final results obtained in the analysis. This will be demonstrated inthe Application and Verification section of this paper (Section 3).In both Figure 6.a and Figure 6.b, the left side of the beam consists of continuum elements that werenot split, as they were not included in the element set created by the radial search. The right side of thebeam, consists of two layers of o↵set-shell elements, connected to each other via cohesive elements. Duringthe splitting process, all loads, stresses and strains are mapped from the continuum elements to the newlycreated elements, and the transient analysis continues until more elements satisfy the splitting criterion (2).The created cohesive interface can be realized in LS DYNA using Solid Continuum Cohesive Elements(*ELEMENT SOLID together with ELFORM = 20), Discrete Cohesive elements (*ELEMENT BEAM) ora cohesive contact interface (i.e TIEBREAK type contact). As stated before, it is important to note thatthe splitting process performed during this stage is not the actual delamination in the material. It is merelya means to locally ”seed” the delamination potential in the structure. The delamination growth will begoverned by the cohesive laws related to the cohesive-element/cohesive-contact.2.6. Propagation of the local cohesive zonesOnce the splitting process is completed and the dynamic analysis continues, cohesive zones can furtherpropagate into the structure. A schematic example of the cohesive zone propagation is shown in Figure 7.A side view of the cantilever beam at the beginning of the analysis can be seen in Figure 7.a. The beam issubjected to a displacement splitting constraint  applied to end of the beam. Figure 7.b shows the beamright after the first splitting cycle. To the left of the beam are 3 thick-shell elements (labeled D) which werenot split during the splitting step. Next to them lies a single thick-shell element (labeled E) overlappedby two o↵set-shell elements. The rest of the beam consists of two layers of o↵set-shell elements, and solidcohesive elements that connect these two layers.Figure 7.c shows the states of the cohesive zones in each cohesive element, during the initial stages of theloading process right after splitting has occurred. Each cohesive element is marked with a number. Cohesiveelements 8 to 5 do not undergo any normal peeling tension, and therefore remain at a 0 cohesive state andexert no cohesive force on the structure. Cohesive element 4 just begins the loading phase, and cohesiveelements 3 and 2 have already been strained by the loads applied to them. Cohesive element 1, located at theend of the beam, has the largest crack-opening displacement, therefore the force transmited by its cohesivestate is about to diminish. When the force transmited by this element reaches a value of 0, the element willbe deleted.7R1#!!!!!!ijk(a)Meaning'of'R'xyzFigure 4: Isometric view of the detailed cantilev r beam mod l used for demonstrating he LCZ algor thm - supported cantileverbeam under splitting loading condition. One end of the beam is subje ted to two opposite displacement constraints .found to have a negligible e↵ect on the final results obtained in the analysis. This will be demonstrated inthe Application and Verification section of this paper (Section 3).In both Figure 6.a and Figure 6.b, the left side of the beam consists of continuum elements that werenot split, as they were not included in the element set created by the radial search. The right side of thebeam, consists of two layers of o↵set-shell elements, onnected to each other via cohesive elements. Duringthe splitting pr cess, all loa s, stresses and strains are mappe from th cont uum elements to the ewlycreated elements, and th transien analysis c ntinues until more eleme ts satisfy the splitting criterion (2).The created cohesive interfac can be realized in LS DYNA using Solid Continuum Cohesive Elem nts(*ELEMENT SOLID together with ELFORM = 20), Discrete Cohesive elements (*ELEMENT BEAM) ora cohesive contact interface (i.e TIEBREAK type contact). As stated before, it is important to note thatthe splitting process performed during this stage is not the actual delamination in the material. It is merelya means to locally ”seed” the delamination potential in the structure. The delamination growth will begoverned by the cohesive laws related to the cohesive-element/cohesive-contact.2.6. Propagation of the local cohesive zonesOnce the splitting process is completed and the dynamic analysis continues, cohesive zones can furtherpropagate into the structure. A schematic example of the cohesive zone propagation is shown in Figure 7.A side view of the cantilever beam at the beginning of the analysis can be seen in Figure 7.a. The beam issubjected to a displacement splitting constraint  applied to end of the beam. Figure 7.b shows the beamright after the first splitting cycle. To the left of the beam are 3 thick-shell elements (labeled D) which werenot split during the splitting step. Next to them lies a single thick-shell element (labeled E) overlappedby two o↵set-shell elements. The rest of the beam consists of two layers of o↵set-shell elements, and solidcohesive elements that connect these two layers.Figure 7.c shows the states of the cohesive zones in each cohesive element, during the initial stages of theloading process right after splitting has occurred. Each cohesive element is marked with a number. Cohesiveelements 8 to 5 do not undergo any normal peeling tension, and therefore remain at a 0 cohesive state andexert no cohesive force on the structure. Cohesive element 4 just begins the loading phase, and cohesiveelements 3 and 2 have already been strained by the loads applied to them. Cohesive element 1, located at theend of the beam, has the largest crack-opening displacement, therefore the force transmited by its cohesivestate is about to diminish. When the force transmited by this element reaches a value of 0, the element willbe deleted.7R2#!!!!!!R1#abci(b)Figure 3.6: a). Isometric view of a double cantilever beam (DCB) subjectedsubjected to an evenly dis ributed splitt displac m nt ∆. Delaminationpropagation through the beam is odelled using the LCZ method. A radialsearch is performed to find all neighbouring elements which lie within aradius R1 of element i satisfayi g t e el m nt splitting criterion. b). Theboundaries of the geometrical region fou d using this search process do notcross the beam along its width, which would be more feasible given theevenly distributed splitting displacement.243.3. LCZ Algorithm OverviewNeighbor)Spli-ng)criterion)threshold!39!xyzFigure 4: Isometric view of the detailed cantilever beam model used for demonstrating the LCZ algorithm - supported cantileverbeam under splitting loading condition. One end of the beam is subjected to two opposite displacement constraints .found to have a negligible e↵ect on the final results obtained in the analysis. This will be demonstrated inthe Application and Verification section of this paper (Section 3).In both Figure 6.a and Figure 6.b, the left side of the beam consists of continuum elements that werenot split, as they were not included in the element set created by the radial search. The right side of thebeam, consists of two layers of o↵set-shell elements, connected to each other via cohesive elements. Duringthe splitting process, all loads, stresses and strains are mapped from the continuum elements to the newlycreated elements, and the transient analysis continues until more elements satisfy the splitting criterion (2).The created cohesive interface can be realized in LS DYNA using Solid Continuum Cohesive Elements(*ELEMENT SOLID together with ELFORM = 20), Discrete Cohesive elements (*ELEMENT BEAM) ora cohesive contact interface (i.e TIEBREAK type contact). As stated before, it is important to note thatthe splitting process performed during this stage is not the actual delamination in the material. It is merelya means to locally ”seed” the delamination potential in the structure. The delamination growth will begoverned by the cohesive laws related to the cohesive-element/cohesive-contact.2.6. Propagation of the local cohesive zonesOnce the splitting process is completed and the dynamic analysis continues, cohesive zones can furtherpropagate into the structure. A schematic example of the cohesive zone propagation is shown in Figure 7.A side view of the cantilever beam at the beginning of the analysis can be seen in Figure 7.a. The beam issubjected to a displacement splitting constraint  applied to end of the beam. Figure 7.b shows the beamright after the first splitting cycle. To the left of the beam are 3 thick-shell elements (labeled D) which werenot split during the splitting step. Next to them lies a single thick-shell element (labeled E) overlappedby two o↵set-shell elements. The rest of the beam consists of two layers of o↵set-shell elements, and solidcohesive elements that connect these two layers.Figure 7.c shows the states of the cohesive zones in each cohesive element, during the initial stages of theloading process right after splitting has occurred. Each cohesive element is marked with a number. Cohesiveelements 8 to 5 do not undergo any normal peeling tension, and therefore remain at a 0 cohesive state andexert no cohesive force on the structure. Cohesive element 4 just begins the loading phase, and cohesiveelements 3 and 2 have already been strained by the loads applied to them. Cohesive element 1, located at theend of the beam, has the largest crack-opening displacement, therefore the force transmited by its cohesivestate is about to diminish. When the force transmited by this element reaches a value of 0, the element willbe deleted.7Combining!both!filters!!!!!!!i S ≥ threshold× Sc(a)Neighbor)Spli-ng)criterion)threshold!39!xyzFigure 4: Isometric view of the detailed cantilever beam model used for demonstrating the LCZ algorithm - supported cantileverbeam under splitting loading condition. One end of the beam is subjected to two opposite displacement constraints .found to have a negligible e↵ect on the final results obtained in the analysis. This will be demonstrated inthe Application and Verification section of this paper (Section 3).In both Figure 6. and Figure 6.b, the left side of the beam consists of conti um elements that werenot split, as they were not included in the lement set crea ed by th radi l searc . The right side of thebeam, consists of two layers of o↵set-shell elem nts, connected to each other via cohesive elements. Duringthe splitting process, all loads, stresses and strains are mapped from the continuum elements to the newlycreated elements, and the transient analysis continues until more elements satisfy the splitting criterion (2).The created cohesive interface can be realized in LS DYNA using Solid Continuum Cohesive Elements(*ELEMENT SOLID together with ELFORM = 20), Discrete Cohesive elements (*ELEMENT BEAM) ora cohesive contact interface (i.e TIEBREAK type contact). As stated before, it is important to note thatthe splitting process performed during this stage is not the actual delamination in the material. It is merelya means to locally ”seed” the delamination potential in the structure. The delamination growth will begoverned by the cohesive laws related to the cohesive-element/cohesive-contact.2.6. Propagation of the local cohesive zonesOnce the splitting process is completed and the dynamic analysis continues, cohesive zones can furtherpropagate into the structure. A schematic example of the cohesive zone propagation is shown in Figure 7.A side view of the cantilever beam at the beginning of the analysis can be seen in Figure 7.a. The beam issubjected to a displacement splitting constraint  applied to end of the beam. Figure 7.b shows the beamright after the first splitting cycle. To the left of the beam are 3 thick-shell elements (labeled D) which werenot split during the splitting step. Next to them lies a single thick-shell element (labeled E) overlappedby two o↵set-shell elements. The rest of the beam consists of two layers of o↵set-shell elements, and solidcohesive elements that connect these two layers.Figure 7.c shows the states of the cohesive zones in each cohesive element, during the initial stages of theloading process right after splitting has occurred. Each cohesive element is marked with a number. Cohesiveelements 8 to 5 do not undergo any normal peeling tension, and therefore remain at a 0 cohesive state andexert no cohesive force on the structure. Cohesive element 4 just begins the loading phase, and cohesiveelements 3 and 2 have already been strained by the loads applied to them. Cohesive element 1, located at theend of the beam, has the largest crack-opening displacement, therefore the force transmited by its cohesivestate is about to diminish. When the force transmited by this element reaches a value of 0, the element willbe deleted.7R1)Combining!both!filters!!!!!!!i(b)Figure 3.7: a). A threshold value is applied to the element splitting crite-rion. b). Applying the threshold results in a physically feasible region ofthick-s ell element to be flagged for splitting.253.3. LCZ Algorithm Overview3.3.5 Through-Thickness Element Splitting and LocalInsertion of Cohesive ElementsOnce element-set A is obtained, through-thickness element splitting andlocal insertion of cohesive elements are performed. A side view of a schematicelement splitting process is shown in Figure 3.8. Here, LS-DYNA’s thick-shell element is being split into two offset shell elements (Figure 3.8.a).Offset-shell elements have their nodes lie on the virtual element-surface,and not on the mid-plane of the shell. The element formulation takes thisoffset into account when computing the stresses and strains in the elements.The offset shell elements are sharing the same nodes as the parent thick-shellelement, i.e, no nodes are deleted or added during the process. During thesplitting operation, all history variables belonging to the integration pointsof the parent thick-shell element (i.e stresses, damage variables, etc.), aremapped to the integration points of the new offset-shell elements createdduring the splitting process (Figure 3.8.b). Once splitting is performed, asolid cohesive element connecting the two offset shell elements is created(Figure 3.8.c). The solid cohesive element behaves as a system of springswhich transmits traction forces across the cohesive interface, based on thetraction-separation law defined within the cohesive material model, and therelative displacement of the nodes belonging to the offset shell elements(Figure 3.8.d). A contact algorithm, schematically represented here usingorange marks, is defined between the new offset shell elements, in order toaccount for contact between the offset shell elements in case delaminationwill develop (Figure 3.8.e).Connecting meshes having different element topologies, i.e offset shellelements and thick-shell elements, poses some numerical difficulties. Thisis demonstrated in Figure 3.9. LS-DYNA’s thick-shell elements have trans-lational DOF and no rotational DOF, whereas LS-DYNA’s offset shell el-ements have translational, as well as rotational DOF (Figure 3.9.a). Dueto the lack of rotational DOF in the thick-shell elements, moments can notbe transmitted at locations where offset shell elements are connected to athick-shell element. This is demonstrated at Figure 3.9.b, where load P willcause rotation of offset shell element i, as thick-shell element j can not resistits rotation, and node w acts as a pivot point. By adding offset shell ele-ments k and l which overlap thick-shell element j (Figure 3.9.c), momentscould be transferred between the two different mesh topologies, resisting therotational movement of offset shell element i due to the load P . Adding theoffset shell elements on top of the thick-shell elements results in an unrealis-tic numerical representation of the material. However, since the overlapped263.3. LCZ Algorithm Overviewa)b)c) d)e)Figure 3.8: A side view of a schematic element splitting process. a.) LS-DYNA’s thick-shell element is being split into two offset shell elements. Thetwo offset shell elements created during the splitting process, are definedusing the nodes of the parent thick-shell element. b.) During the splittingoperation, all history variables belonging to the integration points of theparent thick-shell element, are mapped to the integration points of the newoffset-shell elements created during the splitting process. c.) A solid cohe-sive element connecting the two offset shell elements is created, and is definedsuch that it shares all of its nodes with the offset shell elements. d.) Thesolid cohesive element behaves as a system of springs which transmits trac-tion forces across the cohesive interface, based on the traction-separation lawdefined within the cohesive material model, and the relative displacement ofthe nodes belonging to the offset shell elements. e.). A contact algorithm,schematically represented here using orange marks, is defined between thenew offset shell elements, in order to account for contact between the offsetshell elements in case delamination will develop.273.3. LCZ Algorithm Overviewregion is created over a very narrow band in the model, only at the sharedboundaries connecting the two mesh topologies, it is expected that the errorintroduced due to this unrealistic numerical representation will be localizedand minimal. For the cases being investigated within this work, it was foundto have a negligible effect on the final results obtained from the analysis.PiwjPijkla)x, y, z x, y, z, rx, ry, rzb)c)Figure 3.9: A schematic side view demonstrating some numerical limitationsof connecting LS-DYNA’s thick-shell elements with LS-DYNA’s offset shellelements. a.) LS-DYNA’s thick-shell elements have translational DOF andno rotational DOF, whereas LS-DYNA’s offset shell elements have transla-tional, as well as rotational DOF. b.) Due to the lack of rotational DOF inthe thick-shell elements, moments can not be transmitted at locations whereoffset shell elements are connected to a thick-shell element. This is demon-strated here, where load P will cause the rotation of offset shell element i,as thick-shell element j can not resist its rotation. c.) Adding offset shellelements k and l which overlap thick-shell element j, will allow momentsto be transferred between the two different mesh topologies, resisting therotational movement of offset shell element i due to the load P .Figure 3.10.a shows a schematic beam topology, modeled using a singlethrough-thickness layer of thick-shell elements. The beam is constrained at283.3. LCZ Algorithm Overviewone of it ends, and is loaded via a splitting load ∆ along its free end. Fig-ure 3.10.b shows the beam immediately after a splitting operation has beenperformed. During the element splitting process, the continuum elementsflagged for splitting (Thick-shell elements in set A) are split into two layersof offset-shell elements, and solid cohesive elements to which LS-DYNA’s*MAT COHESIVE GENERAL is assigned, are implanted along the newlycreated interface. A mapping operation is performed in order to transmitall history variables from the thick-shell elements to the offset shell elementsreplacing the thick-shell elements during the element splitting process. Over-lapping offset-shell elements are created on top of the thick-shell elements,but only for the thick-shell elements that share a boundary with the splitregion (Figure 3.10.b). The overlapping offset-shell elements, generated bythe LCZ algorithm, share the same nodes as the parent thick-shell element,thus occupying the same volume.The splitting process performed during this stage is not the actual de-lamination in the material. It is merely a means to locally "seed" the delami-nation potential in the structure. The delamination growth will be governedby the cohesive laws related to the cohesive-elements seeded into the struc-ture.Once splitting and insertion of cohesive elements is performed, the tran-sient analysis is resumed using the new element topology, and the continuumthick-shell elements are again monitored for further splitting.3.3.6 Propagation of the Local Cohesive ZonesOnce the splitting process is completed and the dynamic analysis contin-ues, cohesive zones can further propagate into the structure. A schematicexample of the cohesive zone propagation is shown in Figure 3.11. A sideview of the cantilever beam at the beginning of the analysis can be seen inFigure 3.11.a. The beam is subjected to a displacement splitting constraint∆ applied to end of the beam. Figure 3.11.b shows the beam right afterthe first splitting cycle. To the left of the beam are 3 thick-shell elements(labeled D) which were not split during the splitting step. Next to them liesa single thick-shell element (labeled E) overlapped by two offset-shell ele-ments. The rest of the beam consists of two layers of offset-shell elements,and solid cohesive elements that connect these two layers.Figure 3.11.c shows the states of the cohesive zones in each cohesiveelement, during the initial stages of the loading process right after splittinghas occurred. Each cohesive element is marked with a number. Cohesiveelements 8 to 5 do not undergo any normal peeling tension, and therefore293.3. LCZ Algorithm Overviewa)xz∆∆thick-shell elements (3 DOF per-node )b)xzOverlapping offset-shell and thick-shell elementsthick-shell elements(3 DOF per node)offset-shell elements(6 DOF per node)Cohesive elementsElement Splitting InterfaceFigure 3.10: Schematic side view of a DCB subjected to a splitting loadmodelled using the LCZ method. a.) The beam is modelled using a singlethrough-thickness layer of LS-DYNA’s thick shell elements, with 3 DOF pernode. A splitting displacement ∆ is applied to the end of the beam. b.)Once splitting takes place, thick-shell elements are replaced by offset-shellelements, connected using solid cohesive elements. In order to correctlytransmit the bending-moment from the offset shell elements to the thick-shell elements, the resulting topology includes a single thick-shell elementoverlapped by two offset-shell elements. A contact algorithm is defined be-tween the newly created surfaces to account for contact between the cracksurfaces.remain at a 0 cohesive state and exert no cohesive force on the structure.Cohesive element 4 just begins the loading phase, and cohesive elements 3and 2 have already been strained by the loads applied to them. Cohesiveelement 1, located at the end of the beam, has the largest crack-openingdisplacement, therefore the force transmited by its cohesive state is aboutto diminish. When the force transmited by this element reaches a value of0, the element will be deleted.The first cohesive element deletion is shown in Figure 3.11.d. The cohe-303.4. Multi-Delamination Capabilitysive elements 1 and 2 have gone through their complete softening path inthe traction-separation law, and have therefore been deleted. The cohesiveelements 3 and 4 have partially softened while cohesive elements 5 and 6 arestill in their elastic loading state, and continue to absorb energy as the crackpropagates. At this stage, the cohesive elements 7 and 8 are not loaded. Atsome point in time, as the applied loading increases, cohesive elements 7 and8 will be loaded, and the critical element splitting criterion (Equation 3.2)will be satisfied in thick-shell element E. At this moment, the next splittingphase will take place, as can be seen in Figure 3.11.e, and more cohesiveelements are seeded along the length of the beam (elements 10 to 12). Ascohesive elements 3 and 4 are deleted, the crack opening increases further.The cohesive elements 5 to 8 each undergo a different state of loading alongthe traction-separation curve.Typical results from a three-dimensional beam splitting simulation areshown in Figure 3.12. Here, the resulting cohesive band is identified withdarker shade. The figure shows the manner in which the local cohesive bandmigrates and propagates into the structure as the crack opens. Cohesiveelements are locally and adaptively introduced in the model, and only atspecific locations where delamination is about to take place, thus reducingthe computational costs.3.4 Multi-Delamination CapabilityIn order to simulate multiple through-thickness delamination cracks usingthe conventional CZM, cohesive interfaces should be defined along all poten-tial delamination crack paths prior to the analysis (Figure 3.13.a). Recentdevelopments of the LCZ algorithm allow multiple through-thickness delam-ination cracks to propagate through the laminate. Using the LCZ method,no a priori cohesive elements should be present in the model (Figure 3.13.b).Multiple delamination cracks can simultaneously and independently propa-gate through the thickness of the structure (Figure 3.13.c).In order to allow multiple delamination cracks to be simulated using theLCZ method, the number of through-thickness thick-shell elements shouldbe equal to the number of potential delamination interfaces in the struc-ture. Independent cohesive properties can be defined for each interface andfor each mode of loading. In this case, the behavior of the LCZ algorithmis similar to the case where a single delamination crack is present in themodel, except for the geometrical search performed on all thick-shell ele-ments satisfying the threshold search. This is demonstrated in Figure 3.15,313.4. Multi-Delamination Capabilitywhere a schematic of the LCZ search algorithm performance is presentedfor a case of multi-delamination analysis. Figure 3.15.a shows a side viewof a beam modelled using 3 through-thickness layers of thick-shell elements.The beam is subjected to a displacement ∆, applied at various locationsalong the beam. As ∆ is increased, thick-shell element i is the first ele-ment in which S reaches Sc. This element is added to an initially emptyset of thick-shell elements A. A threshold search is now performed, for allthick-shell elements which satisfy Equation B.4 (Figure 3.15.b). Supposethick-shell element j satisfies Equation B.4, and is thus added to elementset A. A geometrical radial search is now performed (Figure 3.15.c), for allthick-shell elements which are included in element set A (thick-shell elementi and thick-shell element j). For each element in set A, a radial search isperformed using a user-specified geometrical radius R. The thick-shell ele-ment in set A for which the search is performed is termed the parent element.Each thick-element found in the search, whos its centroid fall within a radialdistance R of the centroid of the parent element, is added to element set A,only if it shares the same thick-shell element ply with the parent element.In this example, thick shell elements i1, i2, i3 and i4 all satisfy the radialsearch with respect to the parent thick-shell element i, as well as share thesame thick-shell layer as element i, and are therefore added to element setA. The centroid of thick-shell element m is within a distance R from thecentroid of thick-shell element i, but since it does not belong to the originalthick-shell element layer of element i, it is not added to element set A. Sim-ilarly, thick-shell elements j1 and j2 satisfy the radial search with respect toparent thick-shell element j, as well as share the same thick-shell layer aselement j, and are therefore added to the element set A. Similar to above,since element n does not belong to the original thick-shell element layer ofelement j, it is not added to the element set A even though its centroid fallswithin the radial distance R of element j’s centroid.Once the element set A is populated, the algorithmic details follow a sim-ilar process to that involving a single delamination interface, i.e, mappingand element splitting are performed, together with the insertion of cohesiveelements. The transient analysis is resumed using the new element topol-ogy, and the continuum thick-shell elements are monitored again for furthersplitting.323.4. Multi-Delamination Capabilityxza)thick shell elements ∆∆ixzb)D D D EOverlapping offset shell and thick shell elementsthick shell elements offset shell elementsSolid Cohesive elementsSolid Cohesive elementsElement Splitting Interfacexzc)∆∆D D D Exzd)∆∆D D D Eλλλλλλλλσσσσσσσσ12345678DeletedCohesiveelementsλλλλλλσσσσσσ12345678xze)∆∆Deleted Cohesiveelementsλλλλλλλλσσσσσσσσ123456789101112Figure 3.11: Schematic progression of the cohesive zones using the LCZ al-gorithm, for a cantilever beam-splitting example. a). Before splitting, themodel consists of a single layer of thick-shell elements. A splitting displace-ment constraint, ∆, is applied at the end of the beam. b). First splittingstep. c). Solid cohesive elements are being loaded as the crack is opened.Schematic representation of the cohesive state for each cohesive element canbe seen above the beam. d). Cohesive elements are deleted as the crackpropagates along the beam. e). Second splitting step is performed and thecrack propagates further into the beam.333.4. Multi-Delamination Capability(a) (b) (c)Figure 3.12: Typical cohesive-band created when solving a DCB problemusing the LCZ algorithm. The cohesive-band migrates as the delaminationcrack propagates along the structure, shown here in three subsequent stages-starting at stage a), propagating to stage b) and finally c).y xCohesive Interfaces(a)y x(b)y xCohesive bands(c)Figure 3.13: Simulating a structure having multiple delamination interfaces:a).Using conventional CZM, cohesive interfaces should be defined along allpotential delamination crack paths prior of the analysis. b). Using theLCZ method, no cohesive elements should be present in the model prior ofthe analysis. c). Using the LCZ method, Multiple delamination cracks cansimultaneously and independently propagate through the thickness of thestructure.343.4. Multi-Delamination Capabilityy x1st potential interface2nd potential interface.......nth potential interfaceG1stIc , G1stIIc , σ1stmax , τ1stmaxG2ndIc , G2ndIIc , σ2ndmax , τ2ndmaxGnthIc , GnthIIc , σnthmax , τnthmax......................Figure 3.14: Using the LCZ method, multiple delamination cracks can betreated simultaneously, by defining independent fracture properties for eachpotential delamination interface.353.4. Multi-Delamination Capabilitya)xz∆∆∆ib)xz∆∆∆jic)xz∆∆∆nj1j2i3 i1 i2i4mRjRiFigure 3.15: Schematic performance of the LCZ search algorithm. a). Side viewof a beam modelled using 3 through-thickness layers of thick-shell elements. Thebeam is subjected to a displacement ∆, applied at various locations along the beam.As ∆ is increased, thick-shell element i is the first element in which S reaches Sc.b). A threshold search is now performed, for all thick-shell elements which satisfyEquation B.4. Thick-shell element j satisfies the threshold criterion (EquationB.4), and is thus added to element set A. c). A geometrical radial search is nowperformed, for all thick-shell elements which are included in element set A. Thickshell elements i1, i2, i3 and i4 all satisfy the radial search with respect to the parentthick-shell element i, as well as share the same thick-shell layer as element i, and aretherefore added to element set A. Similarly, thick-shell elements j1 and j2 satisfythe radial search with respect to parent thick-shell element j, as well as share thesame thick-shell layer as element j, and are therefore added to the element set A.Thick-shell elements m and n are not added to the element set A, since they donot share the same thick-shell layer as the parent elements i or j.36Chapter 4VerificationThe correct numerical implementation of the LCZ algorithm was verifiedusing its application to simple numerical problems, involving pure delami-nation crack propagation, i.e, not involving any intralaminar damage. Thischapter focuses on the solution of these simple problems, where in Chapter5, the algorithm is validated against engineering applications involving morecomplicated loading scenarios, as well as intralaminar damage.The following sections describe the solution of pure Mode-I, Mode-II, andMixed-Mode loading conditions, used as a benchmark verification problems.In all configurations tested, LS-DYNA’s built-in material model (*MAT CO-HESIVE GENERAL) was chosen as the material law for the solid cohesiveelements. This material model allows modelling cohesive materials usingarbitrary traction-separation laws. The critical strain-energy release ratesfor Mode-I (GIc ) and II (GIIc), are specified for each loading case, togetherwith the maximum normal (σmax) and shear (τmax) stresses at the cohesiveelements. A normalized traction-separation curve is used (Figure 4.1), inwhich the normalized traction in the cohesive interface is plotted againstthe normalized crack opening.The following relation was chosen as the maximum displacement at fail-ure of the cohesive elements [18]:δf = 1 + β2ATSLC(σmaxGIc)XMU+(τmax β2GIIc)XMU− 1XMU (4.1)where β is the "mode mixity", and is defined as δII/δI , δI and δII are thecrack-opening in the normal and shear directions, respectively, ATSLC is thearea under the normalized traction-separation curve, and XMU is a failureparameter, which has a default value of 1.0 in LS-DYNA. This default valuewas used for both the Mode-I and Mode-II loading cases. However, for themixed-mode loading scenario, XMU was set equal to 1.5 in order to obtainmeaningful results.374.1. Mode-I delamination0 0.2 0.4 0.6 0.8 1 1.200.20.40.60.811.2ATSLCt¯λFigure 4.1: Normalized traction-separation law used in the cohesive materialmodel. The normalized crack opening, λ, is defined as: λ = δδf, where δ isthe crack opening, and δf is the crack opening to failure. The normalizedtraction, t¯, is defined as t/tmax, where t is the traction, and tmax is themaximum traction stress, taken as σmax or τmax from Table 4.1, dependingon the opening mode. ATSLC is the area under the normalized traction-separation curve.4.1 Mode-I delaminationA Double Cantilever-Beam example (DCB), described in detail in [5], wassimulated using the LCZ algorithm (Figure 4.2). The example consists ofa beam of length L = 100mm, thickness h = 3mm, and width of 20mm.A crack of an initial length a = 30mm is present in the beam. A splittingdisplacement ∆, in the global z direction, is applied at the end of the beam.The material and cohesive interface parameters used in the analysis arelisted in Table 4.1.The beam was modelled using two layers of thick-shell elements, withthe expected crack growth path pre-defined prior to the analysis (Figure4.3). The crack path was defined using a layer of solid cohesive elements,located along the interface layer of length L−a along the beam. The initialcrack, of length a, was defined in the model by using two layers of thick-shellelements with no cohesive elements in between.In order to allow reasonable solution times, the typical element size usedin the finite element model along the crack propagation direction was 2mm. Alfano and Crisfield [5] used 0.25 mm element size, together witha maximum cohesive traction stress value, σmax, equal to 57 MPa. The384.1. Mode-I delaminationxz haL∆∆Figure 4.2: DCB test case, consists of a beam with length L = 100mm,thickness h = 3mm, and width of 20mm. A crack of an initial lengtha = 30mm is present in the beam. A splitting displacement ∆, in the globalz direction, is applied at the end of the beam.following section describes the the investigation that was performed in orderto obtain the value of σmax that will allow obtaining reliable results using a2 mm mesh.xz haL∆∆1st layer of thick-shell elements2nd layer of thick-shell elementsSolid cohesive elements (Crack potential growth path)Initial crackFigure 4.3: Schematic side view of the DCB finite element model. The beamis modelled using two layers of thick shell elements, with cohesive elementspre-defined along the potential delamination crack path.394.1. Mode-I delaminationTable 4.1: Material properties used in the numerical verification problemsDCB a ENF b MMB bProperty (Mode-I) (Mode-II) (Mixed-Mode) UnitDensity 1.34× 10−3 1.34× 10−3 1.34× 10−3 g/mm3ElasticLongitudinal elastic modulus (Exx) 135.3 135.3 135.3 GPaTransverse elastic modulus (Eyy) 9 135.3 9 GPaOut-of-plane elastic modulus(Ezz) 9 135.3 9 GPaMajor Poisson’s ratio, in plane (νxy) 0.24 0.25 0.24 (-)Major Poisson’s ratio, out-of-plane (νxz) 0.24 0.25 0.24 (-)Poisson’s ratio, transverse plane (νyz) 0.46 0.25 0.46 (-)In-plane shear modulus (Gxy) 5.2 − 5.2 GPaTransverse shear modulus (Gxz) 3.08 − 3.08 GPaOut-of-plane shear modulus (Gzx) 5.2 − 5.2 GPaInterlaminar damageInterlaminar normal strength (σmax) 8 c - 57 MPaInterlaminar shear strength (τmax) - 57 57 MPaMode I critical energy release rate, (GIc) 0.28 − 4 kJ/m2Mode II critical energy release rate, (GIIc) − 4 4 kJ/m2a Source: [5].b Source: [91].c Source: Value obtained from preliminary simulations, described in Section 4.1.1.4.1.1 Obtaining the Cohesive PropertiesIn Section 2.2, a brief description of the CZM was given. It was stated thatin order for the CZM to correctly describe delamination crack propagationwithin an interface, the cohesive zone should span across several cohesiveelements along the crack propagation direction. Using the out-of-plane elas-tic modulus, together with a maximum traction stress of σmax = 57 MPa(which is the value used by Alfano and Crisfield [5] to solve a similar DCBproblem), the obtained value of the cohesive zone size is 0.16 < lpz < 0.77,depending on the approximation method which appear on Table 2.1. As-suming that the element size along the crack propagation direction shouldallow several elements to span across the cohesive zone, a mesh size muchfiner than 2 mm should be used.It was shown by Alfano and Crisfield [5] that the maximum tractionstress, σmax, can be reduced in order to increase the cohesive zone length,404.1. Mode-I delaminationlpz, thus allowing coarser mesh to be used in the analysis. Alfano andCrisfield [5] used a value of σmax = 37 MPa together with a 1 mm mesh forthe solution of the DCB problem under investigation, which still yielded areasonable solution.In order to investigate the effect of σmax on the numerical solution ob-tained using a 2 mm mesh, the problem was solved using different valuesof σmax, while keeping GIc = 0.28 kJ/m2 constant. All models were solvedusing LS-DYNA’s explicit solver together with the default time step size.The average run-times for each model was approximately 7 minutes, whenusing a cluster of 12 Intel Xeon 2.40GHz processors having 6 cores each forthe solution.Figure 4.4 shows the force vs. crack opening displacement plot, obtainedfrom σmax values of 57 MPa, 8 MPa and 1 MPa, compared to the numericalsolution obtained by Alfano and Crisfield [5] using a 0.25 mm fine meshand the VCCT method. It can be seen that σmax value of 57 MPa resultsin over-prediction of the force profile, both at the elastic loading stage, aswell as the post peak, crack-propagation stage. Numerical noise is present aswell, which is due to the crack front moving from one cohesive element to theother. Using a value of σmax = 8 MPa, results in better agreement with theVCCT solution. Although the response is slightly more compliant duringthe elastic loading stage, better prediction is obtained for the maximumpeak force, and the results follow the Alfano and Crisfield’s solution duringthe crack propagation stage, with slight numerical noise which is presentas the crack propagates. When the cohesive interface is modeled using avalue of σmax = 1 MPa, the compliance of the structure during the elasticloading phase is further increased, and under-estimation of the peak forceis noticeable as well. The response, however, exhibits less noise from thehigher values of σmax tested.In order to shed more light on the results shown in Figure4.4, the normal(out-of-plane) traction stress in a row of neighbouring cohesive elementslocated along the crack propagation direction was monitored as the interfacewas loaded. Figure 4.5 shows the numbering method used in order to analyzethe results. The solid cohesive elements are numbered in an increasing order,such that the 1st element is the element adjacent to the initial crack. Theresults from this study appear in Figure 4.6, Figure 4.7 and Figure 4.8.Figure 4.6.a shows a typical state of the normalized traction stress inthe cohesive elements, using σmax = 57 MPa. It can be seen that thecohesive zone spans across a single element along the crack propagationdirection. Figure 4.6.b shows the normal traction stress in the first 7 cohesiveelements, vs. the crack opening displacement. The results are displayed for414.1. Mode-I delaminationFigure 4.4: Force vs.crack opening displacement plot, obtained for a beammodeled using 2 mm mesh, and σmax values of 57, 8 and 1 MPa.solid cohesive elements located along the crack propagation path which isshown in Figure 4.4. Since the element size is too coarse with respect to thecohesive zone size, a maximum of two cohesive elements are in a loaded stateat each time instant. Failure of these two elements results in a sudden loaddrop, which is the cause for the numerical noise shown in Figure 4.4 for theσmax = 57 MPa loading case. Similar phenomena is also reported by Alfanoand Crisfield [5], when this DCB problem was solved using σmax = 57 MPaand 1 mm mesh.Figure 4.7.a shows a typical state of the normalized traction stress in thecohesive elements, using σmax = 8 MPa. It can be seen that the cohesivezone spans across approximately two elements along the crack propagationdirection. Figure 4.7.b shows the normal traction stress in the first 10 cohe-sive elements, located along the crack propagation path. Since the cohesivezone size is sufficiently large, the normal traction load is shared betweenseveral cohesive elements at each time instant. During the dynamic crackpropagation, as σmax is reached at one cohesive element, followed by thecohesive softening phase, neighbouring cohesive elements which are still inthe state of elastic loading, carry the load. This "load shifting" results in a424.1. Mode-I delaminationCrackPropagationDirection1st2nd3rdInitial CrackFigure 4.5: Isometric view of the DCB finite element model. The solidcohesive elements are shown in green color. A row of neighbouring cohesiveelements located along the crack propagation path was used in order toinvestigate the cohesive behavior as the crack propagates. The results fromthis study appear in Figure 4.6, Figure 4.7 and Figure 4.8 for different σmaxvalues. The solid cohesive elements are numbered in an increasing ordersuch that the 1st element is the element adjacent to the initial crack.smoother force displacement curve and better prediction of the structuralbehavior, compared to the case where σmax was set equal to 57 MPa.Figure 4.8.a shows a typical state of the normalized traction stress in thecohesive elements, using σmax = 1 MPa. It can be seen that the cohesivezone spans across approximately six elements along the crack propagationpath. Figure 4.8.b shows the normal traction stress in the first 8 cohe-sive elements, located along the crack propagation direction. The extendedlength of the cohesive zone allows approximately six cohesive elements tobe loaded simultaneously, a fact which yields a smooth load-displacementcurve. However, the low value of the maximum traction stress results inincreased compliance of the structure. It can be noticed from Figure 4.8.athat the effect of the cohesive zone extends approximately 32 mm ahead ofthe crack tip, which renders the value of σmax = 1 MPa to be unrealistic.Similar behavior was reported by Alfano and Crisfield [5], when applying a434.1. Mode-I delaminationvalue of σmax = 1.7 MPa to the problem. This important finding suggeststhat there is a limit to the amount of scaling that could be applied to σmax,since, below a certain value, the stress distribution within cohesive interfacediverges from the realistic stress distribution to an extent which makes theCZ method innaplicable to the problem.Following the investigation process described above, a value of σmaxwhich is equal to 8 MPa was chosen when applying the LCZ method to theproblem. This value allows obtaining a reasonable solution using a relativelycoarse, 2 mm mesh, thus shortening the LS-DYNA and LCZ algorithm run-times.Ö£444.1. Mode-I delamination(a)(b)Figure 4.6: Typical cohesive behavior obtained for the DCB mode-I loadingcase, using a 2mm mesh size and σmax = 57 MPa. a). Normalized out-of-plane traction stress within the cohesive interface. Only the cohesiveelements are presented in this figure. b). Normal traction stress within a rowof neighbouring cohesive elements located along the crack propagation path.The cohesive elements are located at the center of the cohesive interfacewidth, and span along the beam’s axial direction, with the first elementbeing the first element loaded as the initial crack opens.454.1. Mode-I delamination(a)(b)Figure 4.7: Typical cohesive behavior obtained for the DCB mode-I loadingcase, using a 2mm mesh size and σmax = 8 MPa. a). Normalized out-of-plane traction stress within the cohesive interface. Only the cohesiveelements are presented in this figure. b). Normal traction stress within a rowof neighbouring cohesive elements located along the crack propagation path.The cohesive elements are located at the center of the cohesive interfacewidth, and span along the beam’s axial direction, with the first elementbeing the first element loaded as the initial crack opens.464.1. Mode-I delamination(a)(b)Figure 4.8: Typical cohesive behavior obtained for the DCB mode-I loadingcase, using a 2mm mesh size and σmax = 1 MPa. a). Normalized out-of-plane traction stress within the cohesive interface. Only the cohesiveelements are presented in this figure. b). Normal traction stress within a rowof neighbouring cohesive elements located along the crack propagation path.The cohesive elements are located at the center of the cohesive interfacewidth, and span along the beam’s axial direction, with the first elementbeing the first element loaded as the initial crack opens.474.1. Mode-I delamination4.1.2 Applying the LCZ Method to the DCB VerificationProblemIn order to apply the LCZ method for the solution of the DCB problem, themodel was simulated using the following two configurations:• Case 1: (Figure 4.3) A configuration similar to the model solved inSection 4.1.1, with the maximum traction stress, σmax set equal to8 MPa. The crack path was defined using a layer of solid cohesiveelements, located along the interface layer of length L − a along thebeam. The LCZ algorithm was not used in this analysis. Thus, thiscase is identical to the conventional CZM, where the crack path isdefined prior to the analysis.• Case 2: (Figure 4.9) The beam was modelled using only one layer ofthick-shell elements through its thickness, along the un-cracked sectionof the beam. Only the initial crack of length a was defined in themodel. The cracked region was modelled using two layers of regularshell elements, each describing one surface of the cracked section. Nocohesive elements are present in the model prior to the analysis, andthe crack-growth path is not pre-defined. The LCZ algorithm wasimplemented to predict the delamination crack growth, and embed thelocal cohesive zones where and when needed. The splitting criterionfor this case was defined as:S = σzz ≥ Sc (4.2)where Sc was set equal to 0.8 MPa ( i.e. 10% of σmax - the maximumstress of the cohesive interface) in order to allow early introductionof cohesive elements into the model before the out-of-plane stressesdeveloped in the thick-shell elements become appreciable. The valueof R was set equal to 12mm.For both Cases 1 and 2, the in-plane dimensions of the thick-shell el-ements were 2mm×2mm. The analysis was carried out using LS-DYNA’sexplicit solver with a time step size of 8× 10−5 ms.The force vs. displacement results from the analysis of Cases 1 and2, together with the VCCT model’s prediction [5], are shown in Figure4.10. Both the LCZ method and the conventional CZM results, are in goodagreement with the results obtained in [5]. The curve corresponding to the484.1. Mode-I delaminationxz hA crack of aninitial length a isdefined using twolayers of regularshell elementsL∆∆One layer ofthick-shell elements alongthe un-cracked length of the beamNo cohesive elements are pre-defined in the modelInitial crackFigure 4.9: DCB test case 2. The beam is modelled using one layer ofthick-shell elements, along the un-cracked section of the beam. No cohesiveelements are pre-defined in the model. The LCZ algorithm was implementedto predict the crack growth.LCZ method exhibits jaggedness due to the fact that the LCZ algorithm,in its present form, is not an integral part of LS-DYNA, thus resulting innumerical noise whenever element splitting takes place during the analysis.The evolution of the crack, and the migration of the cohesive-band for Case2 where the LCZ algorithm is used, can be seen in Figure 3.12. The solidcohesive elements embedded in the model are presented using a darker shade.494.1. Mode-I delamination0 1 2 3 4 5 6 7Crack Opening Displacement (∆) [mm]010203040506070Force [N]Case 2 (LCZ method)Case 1 (Conventional Cohesive   Zone method)VCCT (Alfano et al., 2001)Figure 4.10: DCB loading case (Mode-I delamination) - Reaction force atend of beam vs. crack opening displacement 2∆. Results obtained usingthe conventional CZM (Case 1) vs. the LCZ algorithm prediction (Case2), as well as the Virtual Crack Closure Technique predictions (Alfano andCrisfield [5]).4.1.3 Mesh-Size SensitivityIn order to investigate the effect of the mesh size on the results, the simula-tion performed in Case 2, was repeated using three different element sizes -1, 2, and 4mm. The value of R was kept constant in these simulations, andwas chosen to be 24mm, in order to be sufficient for the coarser element size.Figure 4.11 shows the force-displacement results for the various mesh sizes,compared to the VCCT results obtained in [5]. It can be seen that the LCZmethod correctly captures the crack propagation in the structure, even fora relatively coarse mesh size of 4mm. However, the onset of delamination issomewhat under-predicted.504.1. Mode-I delamination0 1 2 3 4 5 6 7 8Crack Opening Displacement (∆) [mm]010203040506070Force [N]LCZ method, 1mm mesh, R=24mmLCZ method, 2mm mesh, R=24mmLCZ method, 4mm mesh, R=24mmVCCT (Alfano et al., 2001)Figure 4.11: DCB loading case (Mode-I delamination) - Reaction force atend of beam vs. crack opening displacement ∆, for element sizes of 1, 2and 4mm. In all cases R = 24mm. The results are compared to the VirtualCrack Closure Technique predictions (Alfano and Crisfield [5]).4.1.4 R-Size SensitivityUsing the LCZ method, the number of the cohesive elements in the model isdirectly linked to R. Since a lower number of cohesive elements in the finiteelement mesh will reduce the computational cost of the problem, an effortshould be made to use smaller values of R, such that reliable results canstill be obtained. Figures 4.12 and 4.13 show the results of solving the DCBproblem, using a 1mm and 2mm mesh element size, together with varying Rvalues, respectively. The results are compared to those obtained using theVCCT method ([5]). It can be seen that when R is equal to the element size,the results do not agree with the VCCT method. This is expected, sincefor the CZM to correctly capture the crack propagation in the material, thecohesive zone should span across a number of elements. Increasing R allowsmore cohesive elements to participate in the cohesive zone, thus obtaininga more reliable solution.Figure 4.14 shows that fairly accurate results can be obtained when R514.1. Mode-I delamination0 1 2 3 4 5 6 7Crack Opening Displacement (∆) [mm]1001020304050607080Force [N]LCZ method, 1mm mesh, R=1mmLCZ method, 1mm mesh, R=3mmLCZ method, 1mm mesh, R=6mmLCZ method, 1mm mesh, R=12mmVCCT (Alfano et al., 2001)Figure 4.12: DCB loading case (Mode-I delamination) - Reaction force atend of beam vs. crack opening displacement ∆, for different values of R,and an element size of 1mm. The results are compared to the Virtual CrackClosure Technique predictions (Alfano and Crisfield [5]).is set to a value which is 6 times the element size. In this figure, variouselement size models were tested, while keeping the ratio between R to theelement size constant and equal to 6.The geometrical effect of R on the obtained cohesive-band for 1mmmesh,using R = 1mm and R = 6mm is depicted in Figure 4.20(a) and Figure4.20(b). It can be seen clearly that reducing the value of R will narrow thecohesive-band, and thereby reduce the number of cohesive elements presentin the model.4.1.5 Sensitivity to the Element Splitting CriterionIn order to investigate the effect of the splitting criterion on the results, the2mm mesh sized DCB model was solved using varying Sc values. S wasidentical to the previous cases (σzz). The following equation was used tovary Sc:524.1. Mode-I delamination0 1 2 3 4 5 6 7Crack Opening Displacement (∆) [mm]1001020304050607080Force [N]LCZ method, 2mm mesh, R=2mmLCZ method, 2mm mesh, R=6mmLCZ method, 2mm mesh, R=12mmLCZ method, 2mm mesh, R=24mmVCCT (Alfano et al., 2001)Figure 4.13: DCB loading case (Mode-I delamination) - Reaction force atend of beam vs. crack opening displacement 2∆, for an element size of 2mmand different values of R. The results are compared to the Virtual CrackClosure Technique predictions (Alfano and Crisfield [5]).Sc = threshold× σmax (4.3)where threshold is a scale factor on the maximum stress value in thetraction-separation law of the cohesive zone, σmax. A range of values wasused for this scaling factor. The results are shown in Figure 4.16. It can beseen that for higher values of this scaling factor, the obtained results exhibitsome noticable load-drops, mainly around a crack opening of 1.8mm-3mm,compared to the VCCT solution. The reason for this can be seen in Figure4.17, which shows the cohesive-band migration for a threshold value of 0.5,and R = 12mm. Figure 6.1(c) shows the state of the cohesive band ata certain point in time, where the crack has already some initial openingdisplacement. As the crack is being opened, the value of S in the structuralelements adjacent to the cohesive-band front increases. Further splitting isnot performed as S does not reach Sc. As the crack continues to open (FigureE.1(b)), further cohesive elements are deleted from the cohesive-band, after534.1. Mode-I delamination0 1 2 3 4 5 6 7Crack Opening Displacement (∆) [mm]010203040506070Force [N]LCZ method, 1mm mesh, R=6mmLCZ method, 2mm mesh, R=12mmLCZ method, 4mm mesh, R=24mmVCCT (Alfano et al., 2001)Figure 4.14: Force vs. crack opening displacement results for various elementsizes used to simulate the DCB test case. R is set to be 6 times the elementsize. The results are compared to the Virtual Crack Closure Techniquepredictions (Alfano and Crisfield [5]).having gone through their complete softening path in the traction-separationlaw. Only when one row of cohesive elements remains in the model, the valueof S in the structural elements reaches Sc, and the next splitting operationis performed (Figure E.1(a)). Having only one row of cohesive elementsin the model cannot capture the crack propagation correctly, since CZMrequires a number of cohesive elements to be included across the cohesivezone. It is therefore important, when using the LCZ method, to chooseR and Sc values such that sufficient number of cohesive elements will beincluded in the cohesive-band as the delamination crack propagates throughthe material.4.1.6 Energy BalanceIn order for the LCZ method to correctly capture the crack propagation,energy should be conserved during the element splitting process. Figure 4.18shows the total internal (strain) energy in the structure for the DCB loading544.1. Mode-I delaminationZYX(a) R=1mmZYX(b) R=6mmFigure 4.15: Typical cohesive band obtained for the DCB mode-I loadingcase, using a 1mm mesh size and varying R values. a). R = 1mm, b).R = 6mm.cases (Cases 1 and 2). It can be seen that the strain energy in the modelwhile using the LCZ method is slightly lower. This is due to the fact thatthe algorithm is currently not an internal part of the finite element solver(LS-DYNA), and the various operations performed by the code lead to somenumerical errors. Nevertheless, the overall energy balance is encouraging,as the error in the internal energy prediction seems to be acceptable.554.1. Mode-I delamination0 1 2 3 4 5Crack Opening Displacement (∆) [mm]20100102030405060Force [N]LCZ method, 2mm mesh, threshhold=0.1LCZ method, 2mm mesh, threshhold=0.25LCZ method, 2mm mesh, threshhold=0.5LCZ method, 2mm mesh, threshhold=0.75VCCT (Alfano et al., 2001)Figure 4.16: DCB loading case (Mode-I delamination) - Reaction force atend of beam vs. crack opening displacement ∆, for an element size of 2mmand different different threshold values. R=12mm. The results are com-pared to the Virtual Crack Closure Technique predictions (Alfano and Cr-isfield [5]).564.1. Mode-I delaminationX YZ(a)ZYX(b) (c)Figure 4.17: Cohesive-band migration obtained for the DCB loading case,using a 2mm mesh size and threshold = 0.5. a). The crack is being openedand the value of S in the structural elements adjacent to the cohesive-bandfront increases. Further splitting is not performed as S does not reach Sc.b). As the crack continues to open, further cohesive elements are deletedfrom the cohesive-band. c). Only when one layer of cohesive elementsremains in the model, the value of S in the structural elements reaches Sc,and the next splitting operation is performed.Figure 4.18: DCB loading case - sum of internal energies vs. crack openingdisplacement, for Case 1 (conventional CZM) and Case 2 (LCZ method).574.2. Mode-II Delamination4.2 Mode-II DelaminationTo further verify the capabilities of the proposed method, a Mode-II loadingcase (Mi et al. [91]) was analyzed (Figure 4.19).The example describes an End-Notch Flexure (ENF) test, and consistsof a beam of total length 2L = 100mm, thickness h = 3mm, and widthof 1mm. A crack of an initial length a = 30mm is present in the beam. Aspecified displacement ∆, in the global negative z direction, is applied at thecenter of the beam. The material and cohesive interface parameters used inthis ENF analysis can be found in Table 4.1.xz∆haL LFigure 4.19: End Notch Flexure (ENF) test case, consists of a beam of totallength 2L = 100mm, thickness h = 3mm, and width of 1 mm. A crack ofan initial length a = 30mm is present in the beam. A displacement ∆, inthe global negative z direction, is applied at the center of the beam.584.2. Mode-II DelaminationThe beam was modelled using one thick-shell element through the thick-ness of the beam along the un-cracked section, and one element across itswidth. The element size along the axis of the beam was 1 mm. Figure 4.20shows results from a preliminary analysis, performed in order to verify thatthe 1 mm element size is sufficient in terms of the cohesive zone length. Fig-ure 4.20.a shows a fringe plot of the normalized shear traction stress withinthe cohesive elements. It can be seen from the figure that the cohesive zonespans across approximately 12 solid cohesive elements. Figure 4.20.b showsthe shear traction stress vs. time within a row of neighbouring cohesiveelements located along the crack propagation path. The first element is lo-cated 18 mm from the initial crack tip. It can be seen that the load is sharedbetween several cohesive elements along the crack propagation path, whichis required in order to correctly describe the crack propagation, and reducethe numerical noise of the numerical solution.Similar to the Mode-I benchmark problem, the problem was solved usingtwo modelling approaches: (i) the conventional modelling approach, i.e withcohesive elements existing along the delamination crack prior to the analysis("Case 1"),and (ii) the LCZ method, where no cohesive elements were presentin the model prior to the analysis ("Case 2").Element splitting occurs in Case 2 when the following criterion is satis-fied:S = τzx ≥ Sc (4.4)Sc = 0.1× τmax (4.5)where the value of τmax was set to 57.0 MPa. The value of R wasset equal to 6mm. LS-DYNA’s explicit solver was used for obtaining thesolution, using a time step size of 1.7× 10−5 ms.Figure 4.21 shows the obtained reaction force at the loading point, vs.the z-displacement (∆), for the LCZ algorithm prediction (Case 2), as wellas the conventional CZM (Case 1). Numerical results obtained by Mi et al.[91], Liu et al. [77], and the analytical model results (Mi et al. [91]) areshown as well. Reasonable agreement between the LCZ algorithm and otherresults is obtained. The visible noise present in the results obtained usingthe LCZ algorithm is due to the fact that the algorithm is not a built-infeature of the finite element solver, and the various numerical operationsperformed by the algorithm on the finite element mesh lead to some numer-ical noise. Nevertheless, the algorithm is still able to represent the essenceof the mechanical behavior correctly.594.2. Mode-II Delamination   1.080e+01 _  -1.070e+01 _  -6.400e+00 _  -2.100e+00 _   2.200e+00 _   6.500e+00 _Fringe Levels   1.510e+01 _   1.940e+01 _   2.370e+01 _   2.800e+01 _  -1.500e+01 _1.0 0.9 0.8 0.7 0.6 0.3 0.5 0.4 0.2 0.1 0.0 Normalized Traction Stress Crack Propagation Direction (a)(b)Figure 4.20: Typical cohesive behavior obtained for the ENF mode-II load-ing case, using a 1mm mesh size and σmax = 57 MPa. a). Normalized sheartraction stress within the cohesive interface. b). Shear traction stress withina row of neighbouring cohesive elements located along the crack propagationpath. The first element is located 18 mm from the initial crack tip.604.2. Mode-II Delamination0 2 4 6 8 10Displacement ∆ [mm]020406080100Force [N]Case 2 (LCZ method)Case 1 (Conventional CZM)Numerical solution, (Xia et al, 2011)Numerical solution, (Mi et al, 1998)Analytical solution (Mi et al, 1998)Figure 4.21: ENF loading case (Mode-II delamination) - Reaction force atloading point vs. z-displacement. The LCZ algorithm prediction is pre-sented vs. numerical results obtained by Mi et al. [91], Liu et al. [77], andthe analytical model (Mi et al. [91]).614.3. Mixed-Mode Delamination4.3 Mixed-Mode DelaminationA mixed-mode-bending (MMB) (Mi et al. [91]) was modelled using the LCZalgorithm (Figure 4.22). The loading case consists of a beam of total length2L = 100mm, thickness h = 3mm, and width of 1mm. A crack of an initiallength a = 30mm is present in the beam. A loading lever is placed abovethe beam and is attached to the beam’s tip via a pivot connection. A rollingmechanism allows sliding of the loading lever at its point of interaction withthe center of the beam.A displacement constraint in the global negative z direction, is appliedat the the loading lever at a point which is at a distance c = 42mm from thebeam’s center. The deflection ∆ and the reaction force P at the end of thebeam are monitored during the run. This loading setup results in a mixedmode ratio GI/GII = 0.909. The loading lever is modelled as a rigid part.The material and cohesive interface parameters used in this analysis can befound in Table 4.1. Element splitting occurs when the following criterion issatisfied:SSc=√(σzσmax)2+(τzxτmax)2≥ threshold (4.6)The value of threshold was set equal to 0.1 to allow seeding the cohesiveelements into the model early enough before significant out-of-plane normaland shear stresses develop within the thick-shell elements. The value of Rwas set equal to 15mm.The beam was modelled using one thick-shell element through the thick-ness of the beam along the un-cracked section, and one element across itswidth. Offset-shell elements were used to model the initially cracked sectionof the beam. The element size along the axis of the beam was 0.25mm,fine enough to allow several cohesive elements to span across the cohesivezone. It was found that using a coarser mesh for this loading case, causedan unrealistic sliding of the roller along the beam, due to the coarse meshsize that did not allow the roller to roll smoothly along the beam’s surface.The analysis was carried out using LS-DYNA’s explicit solver with a timestep size of 4.6× 10−6 ms.Similar to the Mode-I and Mode-II benchmark problems, two differentmodel configurations were investigated: Case 1, where cohesive elementsexisted along the delamination crack prior to the analysis, and Case 2, where624.4. Summary and Conclusionsthe LCZ method was applied, and no cohesive elements were present in themodel prior to the analysis.xzPrescribedDisplacementchaL L∆, PFigure 4.22: Mixed Mode Bending (MMB) test case. The loading caseconsists of a beam of total length 2L = 100mm, thickness h = 3mm, andwidth of 1mm. A crack of an initial length a = 30mm is present in thebeam. A displacement constraint ∆, in the global negative z direction, isapplied at the loading lever at a point which is at a distance c = 42mm fromthe beam’s center.Figure 4.23 shows the beam’s deformed geometry during the loadingprocess, while Figure 4.24 shows the obtained reaction force at the beam’send (P ), vs. the beam’s end displacement in the global z direction (∆),using the LCZ algorithm (Case 2) as well as the conventional CZM (Case1). The numerical and analytical solutions obtained by Mi et al. [91] are alsosuperposed on the graph for comparison. While the LCZ method leads to amore jagged response during the pre-peak regime of loading (attributed tothe fact that it is not a built-in feature of LS-DYNA), the predicted curveis in reasonable agreement with those obtained using the conventional CZM(Case 1) and the various solutions by Mi et al. [91].4.4 Summary and ConclusionsIn this chapter, the LCZ method was applied to pure delamination crackpropagation under mode I, mode II, and mixed mode loading scenarios. Theresults were compared to analytical results, as well as to results obtainedusing other numerical methods.634.4. Summary and ConclusionsYZMixed mode simulation                                                   Time =       14.59P , ∆PrescribedDisplacementxzFigure 4.23: Mixed Mode Bending (MMB) test case finite element modelduring the loading process. The load at the end of the beam (P ) vs. thebeam’s end displacement (∆) is shown in Figure 4.24.For the mode I loading scenario, good agreement was obtained betweenthe force vs. displacement profile obtained using the LCZ method and theVCCT method. Mesh sensitivity analysis shoed little influence on the re-sults, for the range of element size tested. For all mesh size tested, themaximum force predicted was slightly lower (5%-8%) than the maximumforce predicted by the VCCT method. A slight decrease in the maximumload was noticed for the 4mm element size, which is an expected outcomefor the CZM, which shows a decrees in the maximum load with increasedelement size.For the numerical case under investigation, a value of R which was setequal to at least 6 times the element size, was required in order to obtainreliable results, and the threshold value should be small enough with respectto the maximum load of the traction-separation law. These findings areexpected, otherwise, the resulting cohesive band does not span across aminimum number of cohesive elements required in order to obtain reliableresults from the CZM.Reasonable results were obtained for the mode-II and mixed-mode prob-lems solved using the LCZ method. Owing to the fact that the algorithm iscurrently not an internal part of the finite element solver (LS-DYNA) used inthis study, the various numerical operations performed by the algorithm onthe finite element mesh lead to some numerical noise. The limited elementformulations that are currently available in LS-DYNA pose some challenges644.4. Summary and Conclusions0 2 4 6 8 10 12 14 16 18 Deflection at the beam end  (∆) [mm]05101520253035Force at the beam end (P) [N]Case 2 (LCZ method)Case 1 (Conventional Cohesive Zone method)Numerical results (Mi et al, 1998)Analytical solution, linear part (Mi et al, 1998)Analytical delamination (a>L), linear interaction (Mi et al, 1998)Analytical delamination (a<L), linear interaction  (Mi et al, 1998)Figure 4.24: MMB loading case (Mixed-Mode delamination) - Reaction forceat end of beam, P , vs. vertical deflection at the beam end, ∆. The LCZalgorithm prediction is shown in comparison to the results obtained usingthe conventional CZM as well as the analytical results obtained by Mi et al.[91].in achieving full compatibility between the offset-shell elements (in the splitregion) and the thick-shell elements (in the unsplit region) of the mesh. Inthe current LCZ method this difficulty is overcome by introducing narrowregions of overlapping shell elements in the transition region. Nevertheless,the algorithm is able to capture the delamination crack propagation cor-rectly. It is expected that further improvements in speed and accuracy ofthe computations will be attained once the algorithm is embedded withinthe finite element solver, and a layered thick-shell formulation with rota-tional nodal degrees of freedom is implemented in LS-DYNA. This would654.4. Summary and Conclusionsallow a smoother connectivity between the split and neighbouring unsplitregions of the mesh.66Chapter 5ValidationFollowing the verification process, which was described in Chapter 4, thischapter describes the validation process performed to the LCZ method. Vali-dation of a numerical code is often referred to as the procedure taken in orderto establish its legitimacy, i.e its ability to correctly represent the physicsof the problem it intends to solve. Here, the LCZ method was validatedby its application to loading cases combining delamination crack propaga-tion together with intralaminar damage growth. Section 5.1 will describeits application to static loading of a double-notched coupon, Section 5.2will describe its application to a dynamic tube crushing loading case, andSection 5.3 will describe its application to a dynamic plate-impact event.5.1 Tensile Loading of a Notched CouponIn order to test the LCZ method capability to predict delamination combinedwith in-plane damage growth, a simple double-notched tensile experimentwas chosen as a benchmark problem. A brief description of the experimentis brought bellow, followed by the description of the numerical model. Theresults obtained by applying the LCZ method for the simulation of theexperiment are presented thereafter.5.1.1 Material and Test SpecificationA [90/0]s Hexcel E-glass/913 double-notched tensile coupon, was experi-mentally loaded to complete failure by Hallett and Wisnom [48]. The test-specimens were prepared using 0.125mm thick pre-preg tapes cured in anautoclave, resulting in a nominal specimen thickness of 0.5 mm. The spec-imen, shown in Figure 5.1, had a length of 200 mm, where a region havinglength L2 = 50 mm was clamped at each end of the specimen to an Instronuniversal tensile machine, resulting in an effective tensile length of 100 mm(L1). Two symmetric notches, cut from the specimen at its center, resultedin a narrow region a, 10 mm in width.The elastic material sublaminate parameters, are brought in Table 5.1.675.1. Tensile Loading of a Notched CouponTable 5.1: Elastic Hexcel E-glass/913 material propertiesParameter Unit Value SourceDensity (ρ) (g/mm3) 1.97E-3 [20]Longitudinal Modulus (E11) GPa 43.9 [46]Transverse Modulus (E22) GPa 15.4 [46]Transverse Modulus (E33) GPa 15.4 [46]Minor Poisson’s ratio (ν21) (-) 0.11 [46]Minor Poisson’s ratio (ν31) (-) 0.11 [46]Transverse Poisson’s ratio (ν32) (-) 0.3 [46]Shear Modulus (G12) GPa 4.34 [46]Shear Modulus (G23) GPa 4.34 [46]Shear Modulus (G31) GPa 4.34 [46]The experimental results are described in detail in [48]. In all specimenstested, damage initiated at the notch tip. Transverse cracks in the 0◦ plygrew simultaneously with transverse cracks along the 90◦ ply. Triangularshaped delamination grew along the 90/0 interface as the plies continued tosplit. Complete failure of the specimen occurred at the ultimate load dueto the final failure of the 0◦ fibers. Fibers closer to the notch tip failed first,and subsequent failure of fibers quickly followed with a sudden drop in theload-carying capability of the specimen.L1 = 100 mmW = 20 mm a = 10 mmL2 = 50 mmL2 = 50 mm60◦Figure 5.1: A double-notched [90/0]s Hexcel E-glass/913 test coupon ge-ometry used in the tensile experiments performed by Hallett and Wisnom[48]. This experiment was used as a benchmark problem to test the capabil-ity of the LCZ algorithm to simulate delamination combined with in-planedamage growth.685.1. Tensile Loading of a Notched Coupon5.1.2 Finite Element ModelThe finite element model of the coupon is shown in Figure 5.2. Threeplanes of symmetry exist in the model - yz, xz and xy planes. Thus, itis sufficient to model 1/8 of the coupon in order to correctly capture itsbehavior during the tensile experiment. The darker shaded region in Figure5.2.a, which is bounded by the zy and xz planes, resembles 1/4 of the testcoupon. The actual finite-element model was further simplified, taking intoaccount the symmetry of the xy plane, allowing the specimen to be modelledusing a single through-thickness layer of LS-DYNA’s thick-shell elements,each element representing two material plies - a single 0◦ ply, as well asa single 90◦ ply. This was achieved by taking advantage of LS-DYNA’s*PART COMPOSITE keyword, which allows multiple material models andmaterial directions to be assigned to a single element. Each of the thick-shell elements contained 8 through-thickness integration points, to whichseparate material angles were assigned, defining the appropriate materialorientations through the thickness of the element (Figure 5.2.c and Figure5.2.d). In their work, Hallett and Wisnom [46] showed the importance ofintroducing discrete elements within the plies, to capture the correct stressdistribution during the failure process, and to allow discrete failure modeswhich were also found in the experiments. A similar approach was usedhere, where two sets of LS-DYNA’s discrete beam elements were used tomodel the in-plane matrix splitting cracks. One set of beams was used tomodel the matrix splitting cracks along the 0◦ fiber direction within the 0◦ply (Figure 5.2.(c.2)), and a second set of beam elements was used to modelthe matrix splitting cracks along the 90◦ fibers direction within the 90◦ ply(Figure 5.2.(c.3)). It is important to note that the nodes of the thick-shellelements bounding the splitting cracks were artificially displaced in orderto create this figure and visualize the beam elements, as the discrete beamelements have a zero initial length at t = 0, and could not be visualizedotherwise.A closer isometric view at the crack tip, showing the connectivity ofthe thick-shell elements to the discrete beams, is shown in Figure 5.2.d.Similar to Figure 5.2.c, the nodes of the thick-shell elements bounding thesplitting cracks were artificially displaced in order to visualize the discretebeam elements in this figure.The elements edge length in the in-plane direction is approximately 0.2mm long. Appropriate boundary conditions were applied to all nodes whichlie on one of the symmetry planes, and displacement constraint was pre-scribed to the end of the coupon, where the load was applied by the Instron695.1. Tensile Loading of a Notched Couponmachine clamps. The model was solved using the explicit numerical solverof LS-DYNA, and care was taken to ensure that the load was applied at arate which allows a numerical solution within a reasonable amount of time,yet will ensure that the inertia effects on the results were negligible. Thefinite element model, at t = 0, contained 12, 685 thick shell elements and245 discrete beam elements. When applying the LCZ algorithm to solve theproblem, it is expected that the thick-shell elements will adaptively split ascohesive elements will be seeded within the coupon, and delamination couldthus propagate in the coupon as the splitting cracks grow in the in-planedimension.705.1. Tensile Loading of a Notched CouponZ[90/0]s coupon under tensile loadingYZ[90/0]s coupon under tensile loadingY[90/0]s coupon under tensile loading[90/0]s coupon under tensile loading90° 0° x z YXZ x y a) b) c)d)12345Figure 5.2: Finite element model used for solving the double-notch [90/0]sspecimen under tensile loading condition, simulated using the LCZ algo-rithm. a). Front view of a the unclamped section of the test coupon. b).Only 1/8 of the test coupon was modeled, taking advantage of the couponand the ply orientation symmetry. c). A closer isometric view of the cracktip. The specimen was modelled using a single through-thickness layer of LS-DYNA’s thick-shell elements. Discrete beam elements were used to modelthe in-plane splitting cracks - where one set of beams was used to model thesplits in the 0◦ ply (2), and a second set of beams was used to model thesplitting cracks in the 90◦ ply (3). The nodes of the thick-shell elementsbounding the splitting cracks were artificially displaced in order to createthis figure, and visualize the beam elements, which have zero initial lengthin the actual model at t = 0. d). A closer isometric view of the crack tip,showing the connectivity of the thick-shell element to the discrete beams.Here, too, the nodes of the elements bounding the splitting cracks wereartificially displaced in order to create this figure, and visualize the beamelements. Each of the thick-shell elements contained 8 through-thicknessintegration points, to which separate material angles were assigned. Theelements edges which lie within the xy plane are approximately 0.2 mm inlength.715.1. Tensile Loading of a Notched Coupon5.1.3 Intralaminar Damage ModellingTwo of LS-DYNA’s built-in material models were simultaneously used dur-ing the analysis - *MAT ENHANCED COMPOSITE DAMAGE (*MAT54) to model the homogenized damage in the continuum elements, and*MAT GENERAL NONLINEAR 6DOF DISCRETE BEAM (*MAT 119)to model the discrete damage within the in-plane direction. When appliedto thick-shells, *MAT ENHANCED COMPOSITE DAMAGE material be-haves as an elastic, perfectly plastic orthotropic material. Maximum stressfor each direction of loading can be defined, as well as the failure strain foreach mode of loading.Hallett and Wisnom [46] reported that during the experiments, a trans-verse crack density of approximately 30 cracks/cm was measured in the 90◦ply. In order to compensate for the influence of these cracks on the overallstiffness reduction of this ply , the Young’s modulus of the 90◦ ply was re-duced by 50% in the transverse direction only. This was backed by analysismade by Kashtalyan and Soutis [63]. Similar reduction factor was used herefor the Young’s modulus of the 90◦ ply in the transverse direction.Table 5.1 and Table 5.2 summarize the elastic and damage propertiesused during the analysis, respectively, for the 0◦ and 90◦ material directions.The LS-DYNA material cards used in the analysis for the 0◦ and 90◦ plies,appear in Appendix D, Program D.1 and Program D.2, for the 0◦ and 90◦material directions, respectively.The strain-to-failure values for each mode of loading, were chosen suchthat based on the element size used in the analysis, realistic fracture energyvalues would be obtained. Thus, failure strain values of 0.14 was chosenfor the transverse (matrix) loading direction in tension, yielding a fractureenergy of 1 kJ/m2, 0.5 for transverse shear (yielding a fracture energy of7.7 kJ/m2) , and 0.1 for the axial direction, in tension (yielding a fractureenergy of 19.83 kJ/m2).In order to capture the discrete nature of the in-plane splitting, discretebeam elements were used (*ELEMENT BEAM THICKNESS), to which*MAT GENERAL NONLINEAR 6DOF DISCRETE BEAM was assigned.This material model allows arbitrary force vs. displacement curves to be de-fined for the axial as well as the transverse loading directions of the beam,thus, traction-separation curves can be defined, resulting in a "cohesive-like"behavior of the elements. Hallett and Wisnom [46] used a user-defined co-hesive material model, to which the strain energy release rate values of thematerial for mode-I and mode-II loading directions (GIc and GIIc) were in-put directly. Since *MAT GENERAL NONLINEAR 6DOF DISCRETE725.1. Tensile Loading of a Notched CouponBEAM does not support the use of these values directly, traction-separationcurves were to be used instead. Load-curves that yield the appropriate strainenergy release rate for the material were applied, and a simple finite ele-ment models was used in order to verify this transition, where the behaviorof the discrete beams to which *MAT GENERAL NONLINEAR 6DOFDISCRETE BEAM was compared to the behavior of standard solid cohe-sive elements modelled using *MAT COHESIVE GENERAL. Figure 5.3shows the finite element model used for calibrating the discrete beam mate-rial model, and Figure 5.5 shows the stress vs. strain plots obtained in thecohesive interface during axial loading of the interface, using both modellingapproaches. It can be noted that both modelling approaches yield similarresults, as well as similar strain-energy release rate values.111123Figure 5.3: Typical finite-element model used to calibrate the discrete co-hesive element material model. 1. Thick shell elements to which cohesiveelements were connected and were displaced using a prescribed displace-ment constraint. 2. Solid cohesive element to which *MAT COHESIVGENERAL was assigned. 3. Discrete beam elements in which *MATGENERAL NONLINEAR 6DOF DISCRETE BEAM was used.735.1. Tensile Loading of a Notched Coupon0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08Displacement  [mm]05101520253035Stress [Mpa]GIIc=0.9 N/mmSolid Cohesive ElementDiscrete beam ElementFigure 5.4: Typical stress vs. displacement obtained using the finite elementmodel in Figure 5.3. In this case, the solid cohesive element and the discretebeam elements were loaded in shear. The obtained strain energy release ratefor both configurations (GIIc) is 0.9 N/mm or kJ/m2).Table 5.3summarizes the cohesive properties assigned to the discretebeam elements for mode-I and mode-II loading directions. The critical strainenergy release rate values are taken from Hallett and Wisnom [46], wherethe maximum normal and shear stress values were adjusted to allow damageto accumulate within the discrete elements. Choosing higher values for thenormal and shear stresses did not allow the discrete elements to deform ina manner that correctly distributed the stress within the coupon.5.1.4 Interlaminar Damage ModellingFor interlaminar damage modelling, the LCZ method was applied. Thisallowed the finite element model to include no cohesive interface for thethrough-thickness direction prior to the analysis. The following mixed-modeelement splitting criterion was used in order to trigger through-thicknesselement splitting and adaptive insertion of solid cohesive elements into themodel:S =√(σzσmax)2+(τzxτmax)2≥ Sc (5.1)745.1. Tensile Loading of a Notched Couponwhere σmax was defined as 50 Mpa, and τmax was defined as 25 Mpa.A critical value Sc = 0.6 was chosen. It is important to note that thevalues of the parameters related to the element splitting criterion has littlephysical meaning, as they merely serve as a flag to seed the cohesive elementswhen the potential for delamination exist.LS-DYNA’s *MAT COHESIVE GENERAL material model was assignedto the solid-cohesive elements (8-noded solid elements (*ELEMENT SOLID,ELFORM=19) inserted during the analysis, in order to capture the interfacefailure.The interlaminar cohesive properties used in the analysis are listed inTable Table 5.3. It can be noted that although the fracture energy valuesused for the mode-I and mode-II intralaminar cohesive model are identicalto the values used in the interlaminar cohesive model, the maximum stressvalues for both normal and shear direction are different. It was found thatusing values similar to the values used for the interlaminar cohesive interfacesresulted in unreliable results, even when the conventional application of theCZM was used to model the intralaminar damage.The radial distance R used by the LCZ algorithm was set to 1mm, whichis approximately 5 times the in-plane element size used in the analysis.Table 5.2: LS-DYNA’s *MAT 54 Material model damage parameters usedin the [90/0]s double-notched coupon simulationParameter Unit Value *MAT 54variableMaximum stress in the axial direction, under compression a (σf1c) MPa 620 xtMaximum stress in the axial direction, under tension a (σf1t) MPa 1140 xtMaximum stress in the transverse direction, under compression a (σf2c) MPa 128 ycMaximum stress in the transverse direction, under tension a (σf2t) MPa 39 ytMaximum in-plane shear stress (τ f) a MPa 80 scMaximum strain for matrix straining in tension or compression (εf2) (-) 0.14 dfailmMaximum in-plane shear strain (εfshear) (-) 0.5 dfailsMaximum strain for fiber tension (εf1t (-) 0.1 dfailta Source: [20].b Source: [91]755.1. Tensile Loading of a Notched CouponTable 5.3: Cohesive properties used in the [90/0]s double-notched couponsimulationIntralaminar damage (In-plane discrete cohesive elements)Parameter σmax τmax GIc GIIcUnit MPa MPa N/mm N/mm33 33 0.25 0.9Interlaminar damage (Solid cohesive elements created by the LCZ algorithm)Parameter σmax τmax GIc GIIcUnit MPa MPa N/mm N/mm50 25 0.25 0.9Results and discussionFigure 5.5 shows the resulting force vs. displacement curve obtained duringthe run, as well as typical cohesive bands created as the loading of thecoupon increases. The noticeable noise in the loading curve obtained duringthe simulation is due to the fact that the LCZ algorithm is not integrated intoLS-DYNA, and the frequent external interaction of the LCZ algorithm withthe finite element solver introduces some numerical noise into the results.Nevertheless, the simulation is able to predict the maximum load beforefailure and the displacement at failure with good accuracy.As the specimen is loaded and the load is further increased, the zxcomponent of the shear stress (out-of-plane shear stress) within the thick-shell elements located at the notch-tip is increased to levels which causethe element-splitting criteria to be satisfied. Adaptive insertion of cohesiveelements is then automatically performed, as can be seen in Figure 5.5, for acrack-opening displacement of approximately 0.1mm. As the loading furtherincreases, beam elements across the splitting interfaces deform. Cohesiveelements are seeded along the edges of the splitting cracks, as well as acrossthe free edges of the coupon.Figure 5.6 shows the experimentally and numerically obtained damagepattern, at 25.8% of the maximum load. A still image from the experiment[48] is shown in Figure 5.6.a. Small matrix cracks begin to develop at thenotch tip, in both 0◦ and 90◦ plies. Figure 5.6.b and Figure 5.6.c, showsthe LCZ algorithm damage prediction for the matrix cracks at the 90◦ and0◦ ply, respectively, where red color indicates fully damaged material in thetransverse direction, and blue color indicates undamaged material.765.1. Tensile Loading of a Notched CouponAs the load increases, the matrix damage, which originally originated atthe notch tip, spreads across the coupon, in both 0◦ and 90◦ plies. Figure 5.7shows the experimentally and numerically obtained delamination damage,at the vicinity of the notch at the maximum load before failure. Figure5.7.a shows the delamination pattern predicted by the LCZ algorithm, at the0◦/90◦ interface. Figure 5.7.b shows the experimental results from [48]. Thepredicted delamination pattern follows a somewhat narrow triangular shape,which emerges from the notch tip and spreads along the 0◦ fiber direction.Further increasing the load causes the stress at the 0◦ fibers to increasebeyond their load bearing capacity, and the fibers fail almost instantaneouslyacross the coupon, which leads to a total failure of the specimen.Figure 5.5: Far stress vs. displacement obtained from double-notched[90/0]s E-glass/913 test coupon using the LCZ method, together with theexperimental results [48]. Typical cohesive bands created adaptively duringthe analysis by the LCZ algorithm are shown in brown colour. Final failureof the coupon occurs due to failure of the 0◦ plies, accompanied by triangulardelamination patterns (shown in green color).Figure 5.8.a shows the numerically predicted 90◦ ply matrix damage at775.1. Tensile Loading of a Notched Couponthatseenbeforethefiberfailure.Thefailuremodewassimilaracrossallspecimenstestedatallsizes.Figure4showstheresultsobtainedfromthethermaldeplyoftheabovespecimen.Theareaofthedelaminationcanbeseenasthelightercoloredregionandthefiberhasbeenhighlightedwithalineforclarity.ThetopsurfaceofeachplyhasbeenshownFigure3.Progressivefailureofa20mmwide[90/0]sspecimen(%maximumloadshown).Figure4.Deplied[90/0]sspecimenshowingdelaminationandfiberfailure.ProgressiveDamageandtheEffectofLayupinNotchedTensileTests123a) b) c)Figure 5.6: Experimental damage, and damage obtained using the LCZ al-gorithm, applied to a double-notched [90/0]s E-glass/913 test coupon. Im-ages shown are for 25.8% of the maximum load. a). Experimental damageobtained using stills from digital video footage [46] b). Transverse matrixdamage in the 90◦ ply. Red colour indicates a fully damaged material, whileblue colour indicates an undamaged material. (c) Transverse matrix damageat the 0◦ ply.the vicinity of the notch at the maximum load before failure. Damagedis predicted to be located along a 10 mm wide narrow section of the testspecimen. The experimental results, shown in Figure 5.8.b, agree well withthis prediction - the image presents large number of matrix crack locatedalong an area having the width of the narrow section of the coupon (10 mm).Some of these cracks are highlighted in Figure 5.8.b using red rectangles.Figure 5.9.a shows the numerically predicted matrix damage within the0◦ ply. The simulations predict a narrow band approximately 14 mm inwide, of fully developed matrix damage at the 0◦ ply. Figure 5.9.a showsthe experimental obtained damage. Although less noticeable than the cracksin the 90◦ ply, cracks in the 0◦ ply are still visible. Several cracks arehighlighted using a red rectangle.785.1. Tensile Loading of a Notched Coupon(a) (b)Figure 5.7: Delamination damage in a double-notched [90/0]s E-glass/913test coupon, highlighted using red rectangles. a). Delamination damagepredicted using the LCZ algorithm. b). Experimental damage obtainedusing stills from digital video footage [46].795.1. Tensile Loading of a Notched Coupon(a) (b)Figure 5.8: 90◦ ply matrix damage in a double-notched [90/0]s E-glass/913test coupon. a). Damage predicted using the LCZ algorithm. Fully dam-aged material is represented using a red color, while undamaged material iscolored in blue. b). Experimental damage obtained using stills from digitalvideo footage [46]. Several matrix cracks in the 90◦ ply are highlighted usingred rectangles.805.1. Tensile Loading of a Notched Coupon(a) (b)Figure 5.9: 0◦ ply matrix damage in a double-notched [90/0]s E-glass/913test coupon. a). Damage predicted using the LCZ algorithm. Fully dam-aged material is represented using a red color, while undamaged material iscolored in blue. b). Experimental damage obtained using stills from digitalvideo footage [46]. Several matrix cracks in the 0◦ ply are highlighted usingred rectangles.815.2. Dynamic Tube Crush Simulation5.2 Dynamic Tube Crush Simulation5.2.1 IntroductionTo demonstrate the capability of the LCZ method to model progressivedamage in composite structures undergoing impact, a test case involvingdynamic axial crushing of composite tubes will be investigated. This prob-lem has been tackled previously using the first generation of the compositedamage models, CODAM, developed at the University of British Columbia,and implemented as a user-defined material model in LS-DYNA, in tandemwith the built-in cohesive based tie-break contact interface for modelling de-lamination ([84]). The work performed here, is a continuation of this work- it involved using the second generation of the continuum damage modeldeveloped at the UBC Composite group (CODAM2), and applying the LCZmethod to test the method’s capability to predict delamination under dy-namic loading conditions. The cohesive properties were kept constant withrespect to the strain-rate of the problem.Although in some cases the cohesive interface properties might exhibitrate-dependencies, they were treated here as constant with respect to theloading rate of the interface. Introducing strain-rate dependencies to thecohesive model and the LCZ method might be the topic of future research.5.2.2 Material and Test SpecimensThe experiments performed by McGregor et al. [85], included dynamiccrushing of braided tubes having a rectangular cross-section. Althoughseveral tube dimensions were tested, the work performed here focuses ona two-ply tube configuration, with an initial length of 360 mm having asquare cross section, with an outside dimensions of 55 mm and wall thick-nesses of 2.3 mm (Figure 5.11.a). The tubes were braided using Fortafil 55680K carbon as the axial tows, and Grafil 700 12K carbon as the biaxial tows,using Ashland Hetron 922 resin, with each ply having a [0◦/ ± 45◦] braidarchitecture. The 0◦ denotes an angle which is parallel to the tube’s mainaxis. The manufacturer’s properties for the resin and braiding materials arelisted in Table 5.4.Dynamic testing of the tubes was conducted in a 10 kJ drop tower usinga drop-mass of 535 kg and a maximum drop height of 2.0 m (Figure 5.10).Prior to performing the experiment, each tube was glued to a steel mountingplate using a standard hot melt adhesive, and the mounting plate was boltedto the bottom of the drop mass, with the tube pointing downward. Thedrop-mass was then allowed to fall freely, and impact a dynamic load-cell825.2. Dynamic Tube Crush Simulationmounted at the bottom of the drop-tower assembly. Two experiments withimpact velocities of 2.5 m/s and 2.9 m/s are considered here for simulationpurposes For the current study. Figure 5.10.a shows the drop tower assemblywith the drop-mass located at the upper position, just before being releasedtoward impact with the load cell. The composite tube, connected to thedrop mass via the connecting plate, is visible at the center of the image.Figure 5.10.b shows an image taken by a fast-speed video camera, with aresolution of 512×512 pixels and 2200 frames per second, just at the momentwhen the tube-assembly impacted the load-cell located at the bottom of thedrop-tower, and before any noticeable deformations are visible. In order toinitialize a stable and progressive crushing process, the leading edge of thetubes (located at the tubes-end impacting the load cell) was chamfered at a45◦ angle. In addition, a metallic plug was inserted into the bottom of thetube prior to the experiment, to serve as a fracture initiator and to allow thedebris/fronds formed during the crushing process to flow smoothly and notaccumulate between the tube and the impact plane. The tube, mountingplate, and plug-initiator are shown in Figure 5.11.Load CellTube specimentMounting plateLoad CellTube specimen connectedto the drop-massvia mounting plateDrop-Massa) b)Figure 5.10: Drop tower assembly together with composite tube a). Thedrop-mass is located at the top of the drop tower, to which the composite-tube is connected via the mounting plate. When released, the mass isdropped and the composite tube impacts the load cell. b). The compositetube is shown in an image taken using a high-speed video camera, just atthe moment of impact with the load cell, before any noticeable deformationis obtained.835.2. Dynamic Tube Crush Simulation(a)(b)(c)Figure 5.11: a). Square-profile Composite tube with a [0◦/ ± 45◦] braidarchitecture used in the tube-crushing experiment. b). Mounting plateused to connect the tube to the drop-mass c). Plug initiator.Table 5.4: Manufacturer’s Constituent properties for tube braiding material.Source: [85]Property Fortafil Grafil Hetron#556 80K #34− 700 12k 992Number Of Filaments 80, 000 12, 000 -Strength (MPa) 3, 790 4, 820 86.2Modulus (GPa) 231 234 3.17Density (g/cm3) 1.8 1.8 1.14Tow cross-sectional area (mm2) 2.34 0.444 -Elongation At Break (%) 1.64 2 6.7Filament Diameter (µm) 6 7 -Force vs. displacement results from the tube-crush experiments arebrought in Figure 5.12, for impact velocities of 2.5 m/s and 2.9 m/s. It845.2. Dynamic Tube Crush Simulationis noticeable that the plug-initiator and the tube leading edge which was in-tentionally chamfered at 45◦, successfully eliminated the initial load-peak, astheir presence initialized the fracture at an early stage and allowed progress-ing and stable crushing to develop without the presence of a maximum-loadlevel which is considerably higher than the average load during the stablecrushing process.The work performed during the crushing process was calculated usingEquation C.1. Values ofWf = 2629.05 J andWf = 3541.57 J were computedfor the 2.5 m/s and 2.9 m/s impact velocities, respectively. Using a tubecross-sectional area of 475.87 mm2, material density ρ = 1.3× 10−3 g/mm3in Equation C.2, resulted in values of SEA which are equal to 23.11 J/gand 23.29 J/g for the 2.5 m/s and 2.9 m/s impact velocities, respectively.0 50 100 150 200 250 300Displacement [mm]01020304050Force  [KN]Wf, 2.5 m/s=2629.05 J  SAE2.5 m/s=23.11 J/gWf, 2.9 m/s=3541.57 J  SAE2.9 m/s=23.29 J/gExperiment, 2.5 m/s (Mc Gregor at al, 2010)Experiment, 2.9 m/s (Mc Gregor at al, 2010)Figure 5.12: Force vs. displacement results obtained from the tube-crushexperiments, for impact velocities of 2.5 m/s and 2.9 m/s (McGregor et al.[85]). The work performed during the crushing process, as well as the specificEnergy Absorption values were calculated using Equation C.1 and EquationC.2, and are displayed as well.855.2. Dynamic Tube Crush Simulation5.2.3 Finite Element ModelIn order to apply the LCZ method to the problem, and provide an alter-nate and comparative numerical solution to the LCZ method’s results, twosimulation approaches are used. In the first approach (Figure 5.13), whichis referred to as the conventional cohesive zone method (conventional CZM)within this thesis, the 2-ply braided composite tube is represented by twolayers of regular shell elements with an element size of 2.5mm, each with 4through-thickness integration points and tied together using a conventionalcohesive type tie-break contact interface available in LS-DYNA. The mate-rial behavior of each shell element is governed by CODAM2 (MAT219) inLS-DYNA, which is the second-generation of the sub-laminate based con-tinuum damage mechanics models developed at the University of BritishColumbia.In the second modelling approach where the adaptive LCZ algorithm isapplied to the problem, the tube is modelled using a single layer of thick-shell elements (ELFORM=5 in LS-DYNA) through the thickness of the tube(Figure 5.14), with 8 integration points through the element thickness. Here,too, the element size used was 2.5mm. No cohesive elements are introducedin the model prior to the analysis, as they are added adaptively to thestructure during the transient simulation.In both modelling approaches, the chamfer at the end of the tube ismodelled using a row of shell elements with a thickness equal to half thethickness of the tube wall, connected to the end of the tube. These shellelements are the first to come in contact with the plug initiator. Theseelements are required in order to initiate a stable crushing process.In both modelling approaches, the finite element model consists of 3main components: a drop weight, a tube, and a plug to initiate the crushingprocess. Only a quarter of each component is modelled as shown in Figure5.13 and in Figure 5.14, and the required symmetry boundary conditionsare enforced accordingly. The drop weight is modelled as a rigid body usingsolid elements with a mass of a quarter of the total mass (i.e. 134 kg). Theplug initiator is also modelled as a rigid body using solid elements and isfixed in space.865.2. Dynamic Tube Crush SimulationYXZ2-ply without plug 0/+45/-45 TUBE CRUSH - shellYXZ112234Figure 5.13: An isometric view of the LS-DYNA finite-element model usedfor the tube-crush analysis in Phase I and Phase II, and concisted of thefollowing parts: Composite tube (1), plug (2), dropped mass (3). The 45◦chamfer at the leading-edge of the tube was modelled using shell elementswith varying cross-section thickness (4)An initial velocity of 2.7 m/s (corresponding to the average of the twovelocities, 2.5 m/s and 2.9 m/s, used in the tests) is assigned to the dropweight causing the tube to impact the plug.875.2. Dynamic Tube Crush SimulationYXZYXZ2-ply with plug 0/+45/-45 TUBE CRUSH - 2X TshellsYXZ2-ply with plug 0/+45/-45 TUBE CRUSH - 2X TshellsXZY112234Figure 5.14: An isometric view of the LS-DYNA finite-element model usedfor the tube-crush analysis, consisting of the following parts: Composite tube(1), plug (2), dropped mass (3), and a row of shell elements, resembling the45◦ chamfer which is present in the experimental tube, which is required inorder to initiate a stable crushing process (4).5.2.4 Intralaminar Damage ModellingCODAM2 (Forghani [35], Forghani et al. [37]), served as the intralaminardamage model for the braided tube . Within CODAM2, the intra-laminardamage consisting of fibre breakage and matrix cracking is modelled usinga sub-laminate based approach, which acknowledges the existence of or-thotropic layers within the sub-laminate and casts the damage formulationin terms of the strain components in those directions. Another feature ofthis material model, compared to its predecessor, is its non-local averagingcapability. This is to alleviate the problem of mesh size and orientationdependency which is commonly encountered in continuum damage modelsthat lead to a strain softening response. CODAM2 has been implemented885.2. Dynamic Tube Crush Simulationas a built-in material model (MAT219) in LS-DYNA, thus making it pos-sible to take advantage of the efficiencies that come with all of the built-infeatures of the code, a requirement for the LCZ method to be applied to theproblem.The CODAM2 parameters required to identify the in-plane orthotropicresponse of the braided material were estimated using constituent propertiesand information obtained from standard and specialized coupon tests. Thedetails of material characterization can be found in McGregor et al. [87].Based on the failure mechanisms of the tube, tension along the local y-axis(transverse or hoop direction) and compression along the local x-axis (axialor longitudinal direction) were identified as the primary loading directions.Therefore, the input material damage properties were calibrated assumingthe braided material system to be an equivalent (lumped) single layer oforthotropic composite.The input model parameters are calibrated based on the characterizedproperties for a representative volume element (RVE) for each damage mode,together with the physical and elastic material properties, are listed in Table5.5. The strain values for the damage saturation associated with all damagemodes are scaled according to the size of the element, such that for eachmode of loading, the fracture energy is element size independent, in keep-ing with Bazant’s crack band scaling method (Bazant and Planas [10]). Adetailed description of the calibration process is brought in Appendix F.Figure 5.15.a and Figure 5.15.b show the resulting stress-strain profilethat result from applying the calibrated material model to a 2.5mm elementunder axial and hoop loading directions, respectively.5.2.5 Interlaminar Damage ModellingIn the standard modelling approach, where delamination damage is simu-lated using a conventional tie-break interface, the interface damage initiateswhen a quadratic traction-based criterion is satisfied. When this criterion ismet, the interface tractions are gradually decreased to zero at a user-definedcritical crack opening displacement. The traction-separation law used forthis model is considered to be identical for both the normal (Mode I) andshear (Mode II) crack openings (see Table 5.5 for the input variables used).These values are the same as those used by McGregor et al. [85] to simulatethe tube crushing experiments where a tie-break contact was also used tomodel delamination.In the second approach where the LCZ algorithm is applied for delami-nation modelling, the cohesive elements which are adaptively added to the895.2. Dynamic Tube Crush Simulation1.0 0.5 0.0Strain60040020002004006008001000Stress [MPa][ε ix ]t[ε sx ]t[ε ix ]c[ε sx ]c(a)0.4 0.3 0.2 0.1 0.0 0.1 0.2 0.3 0.4Strain2001000100Stress [MPa][ε iy ]t[ε sy ]t[ε iy ]c[ε sy ]c(b)Figure 5.15: Stress - strain curves used as an input for the CODAM2 mate-rial model, calibrated for a 2.5 mm element size. a. Axial loading directionwhere [εix]t and [εsx]t denote the damage initiation and saturation strains intension, [εix]c and [εsx]c denote the damage initiation and saturation strainsin compression. b. Transverse loading direction where [εiy]t and [εsy]t de-note the damage initiation and saturation strains in tension, [εiy]c and [εsy]cdenote the damage initiation and saturations strains in compression.905.2. Dynamic Tube Crush SimulationTable 5.5: Model input parameters for the [0◦/ ± 45◦] braided compositetube.Property Value UnitDensitya 1.3× 10−3 g/mm3Ply thicknessa 1.15 mmElasticLongitudinal elastic modulusa (Exx) 60 GPaTransverse elastic modulusa (Eyy) 12.5 GPaOut-of-plane elastic modulusc (Ezz) 8 GPaMajor Poisson’s ratiosc, (νyx = νzx = νzy) 0.3 (-)In-plane shear modulusa (Gxy) 9 GPaTransverse shear modulusc (Gxz = Gyz) 9 GPaIntralaminar damageInitiation strain for damage under tension in the transverse direction a[εiy]t9× 10−3 (-)Saturation strain for damage under tension in the transverse directionb[εsy]t3.47× 10−1 (-)Initiation strain for damage under tension in the axial directiona[εix]t1.5× 10−2 (-)Saturation strain for damage under tension in the axial directionb[εsx]t2.24× 10−1 (-)Initiation strain for damage under compression in the transverse directiona[εiy]c2× 10−2 (-)Saturation strain for damage under compression in the transverse directionb[εsy]c2.85× 10−1 (-)Initiation strain for damage under compression in the axial directiona[εix]c5× 10−3 (-)Saturation strain for damage under compression in the axial directionb[εsx]c1.29 (-)Interlaminar damageInterlaminar normal strengtha (σmax) 50 MPaInterlaminar shear strengtha (τmax) 50 MPaMode I critical energy release ratea, (GIc) 1.75 kJ/m2Mode II critical energy release ratea, (GIIc) 1.75 kJ/m2a Source: [85].b Calibrated values, as described in Section 5.2.4.c Assumed value in this study.structure, are 8-noded solid elements (*ELEMENT SOLID, ELFORM=19).LS-DYNA’s *MAT COHESIVE GENERAL material model is assigned tothese elements in order to capture the interface failure. The cohesive pa-rameters used in this model are identical to those listed in Table 5.5 forthe tie-break interface. In order to assess the potential for splitting thestructural thick-shell elements and seeding the solid cohesive elements, thefollowing interactive stress-based element-splitting criterion was used in theanalysis:S =√(σnσmax)2+(σsτmax)2≥ Sc (5.2)915.2. Dynamic Tube Crush Simulationwhere σn and σs are the through-thickness normal and shear stress com-ponents with σmax and τmax being the respective maximum values of thesequantities. The critical value for splitting the thick-shell elements, Sc, isassumed to be 0.5, which is sufficiently large to allow the cohesive elementsto be introduced in relatively small regions of the model. The maximumvalues of the out-of-plane normal and shear stresses, σmax and τmax, aretaken to be 50 MPa, and the radius of the splitting region, R is assumed tobe 8 mm, slightly more than three times the element size used in the ana-lyis, which is recommended in order to capture the correct behavior of thecrack propagation using CZM. A coefficient of friction of 0.22 was definedfor the contact between the newly created surfaces, and a value of 0.32 wasused between the tube and the plug. This value is slightly higher than thevalue, 0.22, used by McGregor et al. [85], as preliminary simulations usingsolid cohesive elements showed that a slightly higher value of the frictioncoefficient was necessary in order to obtain accurate results.925.2. Dynamic Tube Crush Simulation5.2.6 Results and DiscussionThe force vs. displacement results obtained from the numerical simula-tion (CODAM2 / LCZ) is presented in Figure 5.16, together with resultsobtained using classical shells and a tie-break contact algorithm. The cor-responding experimental results reported in McGregor et al. [86] are alsoshown for comparison. Good agreement between the numerical and exper-imental results is obtained. The final obtained displacements at which thedropped mass was brought to a halt is 224.16 mm, which is within the rangeof the maximum displacement measured for the lower and higher impact ve-locities used in the tests (183.87 mm and 245.83 mm, for impact velocitiesof 2.5 m/s and 2.9 m/s, respectively).0 50 100 150 200 250Displacement [mm]0510152025303540Force  [KN]Simulation, CODAM2 + LCZ method, 2.7 m/sSimulation, CODAM2 + Tie-Break contact, 2.7 m/sExperiment, 2.5 m/s (Mc Gregor at al, 2010)Experiment, 2.9 m/s (Mc Gregor at al, 2010)Figure 5.16: Force vs. displacement results obtained from the numericalsimulation of the tube crush tests using the combined CODAM2 and theLCZ algorithm. Also shown for comparison are the results obtained usingthe conventional delamination modelling approach that employs the tie-break contact interface. These results for impact velocity of 2.7 m/s areshown together with the experimental results for impact velocities of 2.5m/s and 2.9 m/s.The predicted initial peak force, which is noticeable before stable crush-ing begins, exceeds the corresponding force measured in the experiments,935.2. Dynamic Tube Crush Simulationprobably due to the inaccurate discretization of the chamfer which plays animportant role in initializing a stable and progressive crushing. The specificenergy absorption (SEA) calculated from the numerical simulation (23.21J/g) is in very good agreement with the experimentally determined valuesof 23.11 J/g and 23.29 J/g corresponding to impact velocities of 2.5 m/sand 2.9 m/s, respectively.Both the conventional modeling approach, as well as the LCZ method,yield a similar topology of the crushed tube geometry, which is shown forthe bottom of the tube (crushed zone) in Figure 5.17. As the tube is forcedagainst the plug, its plies fail due to a combination of several damage mech-anisms, mainly tension along the y-axis and compression along the x-axis,and delamination which results in complete separation of the plies. Frondsare created as the progressive crushing process continues and the tube isfurther pushed against the plug.YXZ2-PLY WITHOUT PLUG 0/+45/-45 TUBE CRUSHTime =        40.8a abccFigure 5.17: Typical topology of the crushed tube geometry, demonstratingthe dominant damage mechanisms. Transverse (hoop) damage develops atthe tube’s corners (a), Delamination results in complete separation of theplies (b). Fronds are created as the progressive crushing process continuesand the tube is further pushed against the plug (c).Figure 5.18 shows a close-up view at the bottom of the tube, demonstrat-ing the propagation of the cohesive-band when the LCZ method is applied.The initial finite element model contains no cohesive elements, and the tubeis modelled using a single through-thickness layer of thick-shell elements945.2. Dynamic Tube Crush Simulation(Figure 5.18.a). As the element-splitting criterion is satisfied within the firstthick-shell element, a splitting operation is performed and solid-cohesive el-ements are introduced into the model, shown in a darker colour in Figure5.18.b. As the tube is pushed against the plug, elements located at thetube’s corners fail, and fronds consisting of shell elements begin to develop(Figure 5.18.c). Progressive and stable crushing process develops, at whichelongated fronds are created as the tube is pushed against the plug andthe cohesive band further propagates into the tube. New cohesive elementsare created along the leading edge of the cohesive band, and as cohesiveelements are deleted due to delamination, shell elements are separated andnew, free surfaces, are created (Figure 5.18.c).955.2. Dynamic Tube Crush Simulation(a) (b)(c) (d)Figure 5.18: Propagation of the cohesive-band when the LCZ method isapplied to the tube-crushing simulation. a). Initial finite element model.b). Solid-cohesive elements are introduced into the model as the element-splitting criterion is satisfied, shown here in a darker colour. c). Frondsconsisting of shell elements develop. d). Progressive and stable crushingprocess results in separation of shell elements as new surfaces are created.The model topology and CODAM2 damage values for the axial and hoopdirections, reported at t = 40 ms after initial impact, are shown in Figure5.20 and Figure 5.19, respectively, for both the conventional CZM, as wellas results obtained from applying the LCZ method to the problem. Thetopology obtained from both methods are similar but not identical - the965.3. Dynamic Plate Impact Simulationsfronds created using the LCZ method are somewhat more symmetric com-pared to the fronds obtained using the conventional CZM. A slight differencecan be seen in the intralaminar (CODAM2) damage values reported, wherethe LCZ method yields some regions of lower axial damage value comparedto the conventional CZM.This example demonstrates the ability of the LCZ method to model theprogression of delamination in a dynamic event, without prior introductionof cohesive elements or cohesive contact at all possible ply interfaces in thefinite element mesh. Although some minor differences are observed whenthe method is compared to the standard application of the cohesive-zonemethod to the problem, the LCZ method is able to correctly predict theforce vs. displacement profile, the SEA of the material, as well as the totaldisplacement of the dropped-mass.YXZFringe Levels min=0, at elem# 21max IP. valueContours of History Variable#5Time =        40.82-ply with plug 0/+45/-45 TUBE CRUSH - shell                               0.000e+00 _   1.000e-01 _   2.000e-01 _ max=1, at elem# 2915   4.000e-01 _   5.000e-01 _   6.000e-01 _   7.000e-01 _   8.000e-01 _   9.000e-01 _   1.000e+00 _   3.000e-01 _   6.000e-01 _   1.000e-01 _Fringe Levels   2.000e-01 _   4.000e-01 _   0.000e+00 _   3.000e-01 _   7.000e-01 _   8.000e-01 _   9.000e-01 _   5.000e-01 _   1.000e+00 _(a) (b) YXZFringe Levels min=0, at elem# 21max IP. valueContours of History Variable#5Time =       40.82-ply with plug 0/+45/-45 TUBE CRUSH - shell                               0.000e+00 _   1.000e-01 _   2.000e-01 _ max=1, at elem# 2915   4.000e-01 _   5.000e-01 _   6.000e-01 _   7.000e-01 _   8.000e-01 _   9.000e-01 _   1.000e+00 _   3.000e-01 _   6.000e-01 _   1.000e-01 _Fringe Levels   2.000e-01 _   4.000e-01 _   0.000e+00 _   3.000e-01 _   7.000e-01 _   8.000e-01 _   9.000e-01 _   5.000e-01 _   1.000e+00 _(a) (b) 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 YXZFringe Levels min=0, at elem# 21max IP. valueContours of History Variable#5Time =        40.82-ply with plug 0/+45/-45 TUBE CRUSH - shell                            0 +00 _12 max=1, at elem# 2915456789 -01 _   1.000e+00 _3   6.000e-01 _   1.000e-01 _Fringe Levels   2.000e-01 _   4.000e-01 _   0.000e+00 _   3.000e-01 _   7.000e-01 _   8.000e-01 _   9.000e-01 _   5.000e-01 _   1.000e+00 _(a) (b) Figure 5.19: Model topology and CODAM2 intralmainar damage valuesin the axial direction (compression), at t = 40ms. Damage value of 0.0represents an undamaged material, and a damage value of 1.0 represents afully damaged material. a). Results obtained using the conventional CZM.b). Results obtained using the LCZ method.5.3 Dynamic Plate Impact SimulationsIn order to validate the predictive capability of the LCZ method to simulatea dynamic loading case involving multiple through-thickness delaminationcrack propagation, non-penetrating impact response of T800/3900-2 CFRPlaminates with quasi-isotropic stacking sequence of [45/90/−45/0]3s was in-vestigated. The impact experiments, performed by Delfosse et al. [24], cov-ered a wide range of impact energies obtained using high-mass drop-weight975.3. Dynamic Plate Impact SimulationsYXZFringe Levels min=0, at elem# 21max IP. valueContours of History Variable#5Time =        40.82-ply with plug 0/+45/-45 TUBE CRUSH - shell                               0.000e+00 _   1.000e-01 _   2.000e-01 _ max=1, at elem# 2915   4.000e-01 _   5.000e-01 _   6.000e-01 _   7.000e-01 _   8.000e-01 _   9.000e-01 _   1.000e+00 _   3.000e-01 _(a) (b)    6.000e-01 _   1.000e-01 _Fringe Levels   2.000e-01 _   4.000e-01 _   0.000e+00 _   3.000e-01 _   7.000e-01 _   8.000e-01 _   9.000e-01 _   5.000e-01 _   1.000e+00 _YXZFringe Levels min=0, at e em# 21max IP. valueContours of History Variable#5Time =        40.82-ply with plug 0/+45/-45 TUBE CRUSH - shell                               0.000e+00 _   1.000e-01 _   2.000e-01 _ max=1, at elem# 29 5   4.000e-01 _   5.000e-01 _   6.000e-01 _   7.000e-01 _   8.000e-01 _   9.000e-01 _   1.000e+00 _   3.000e-01 _(a) (b)    6.000e-01 _   1.000e-01 _Fringe Levels   2.000e-01 _   4.000e-01 _   0.000e+00 _   3.000e-01 _   7.000e-01 _   8.000e-01 _   9.000e-01 _   5.000e-01 _   1.000e+00 _1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 YXZFringe Levels min=0, at elem# 21max IP. valueContours of History Variable#5Time =        40.82-ply with plug 0/+45/-45 TUBE CRUSH - shell                               0.000e+00 _   1.000e-01 _   2.000e-01 _ max=1, at elem# 2915   4.000e-01 _   5.000e-01 _   6.000e-01 _   7.000e-01 _   8.000e-01 _   9.000e-01 _   1.000e+00 _   3.000e-01 _   6.000e-01 _   1.000e-01 _Fringe Levels   2.000e-01 _   4.000e-01 _   0.000e+00 _   3.000e-01 _   7.000e-01 _   8.000e-01 _   9.000e-01 _   5.000e-01 _   1.000e+00 _(a) (b) Figure 5.20: Model topology and CODAM2 intralmainar damage values inthe hoop direction (tension), at t= 40 ms . Damage value of 0.0 repre-sents an undamaged material, and a damage value of 1.0 represents a fullydamaged material. a). Results obtained using the conventional CZM. b).Results obtained using the LCZ method.and low-mass gas-gun impact tests. Similar to the case of the dynamic tube-crush validation problem, the cohesive interface properties were treated hereas constant with respect to the loading rate of the interface. Introducingstrain-rate dependencies to the cohesive model and the LCZ method mightbe the topic of future research.5.3.1 Material and Test SpecificationsA rectangular composite plate, 152.4 mm ×101.6 mm ×4.65 mm in size(Figure 5.21 and Figure 5.22), served as a target for the experiments. Theplate was clamped to an aluminum backing plate having a 76.2 × 127.0mm2 rectangular opening, using four rubber fasteners (Figure 5.22), andimpacted using a 25.4 mm diameter hemispherical shaped hardened steelprojectile.The experiments performed by Delfosse et al. [24] consisted of low-velocity impact tests, where a 6.33 kg impactor was dropped from a droptower from various heights, impacting the plate at velocities ranging from1.76 m/s to 4.29 m/s (Resulting in impactor’s kinetic energies of 9J to 56J,for the lowest and highest impact velocities, respectively), and high-velocitytests, in which a 0.314 kg projectile was launched using a gas gun, and im-pacted the plate at velocities ranging from 7.74 m/s to 23.19 m/s (Resultingin projectile’s kinetic energies of 9J to 58J, for the lowest and highest impactvelocities, respectively). The contact force between the plate and the pro-985.3. Dynamic Plate Impact Simulations101.6 mm × 152.4 mm76.2 mm × 127.0 mmTarget PlateProjectileBacking PlatePlate FastnerV0Figure 5.21: A side view of the plate-impact experiment configuration. Acomposite target plate is attached to an aluminum backing plate using fourrubber fasteners (of which only two are shown in the figure). The projectilehas an initial velocity V0 as it strikes the plate at normal incidence.Backing Plate Opening EdgeTarget Plate EdgeProjectilePlate FastnerPlate FastnerPlate FastnerPlate FastnerFigure 5.22: A top view of the plate-impact test configuration. The targetplate is attached to the aluminum backing plate using four rubber fasteners.The opening in the aluminum backing plate is marked using a dashed line.995.3. Dynamic Plate Impact Simulationsjectile was measured using a load-cell, and impact velocities were measuredusing three pairs of optical gates placed right before the point of impact. Theresulting projected internal delamination area was mapped using pulse-echoultrasonics. Since the equipment was capable of mapping the first level ofdelamination encountered through the thickness of the panel, a destructiveinspection process was carried out to determine the total delamination areaDelfosse et al. [24].5.3.2 Finite Element ModelFigure 5.23 shows an isometric view of the finite-element model. Due tosymmetry, only a quarter of the test configuration is modelled, and therequired boundary conditions are applied to the finite-element model ac-cordingly. Similar to an assumption made by Forghani and Vaziri [36] andWilliams et al. [129], only the portion of the plate which is positioned abovethe aluminum-backing opening is modelled (i.e the size of the actual platemodelled is 76.2 mm×127 mm), and simply supported boundary conditionare applied around the free edges of the plate. The plate is modelled usingLS-DYNA’s thick-shell elements, with an in-plane square element size of 1mm. Several models were tested, with varying number of through-thicknessthick-shell elements (1, 3, 4 and 5), in order to assess the sensitivity of theresults to the number of potential through-thickness delamination cracks.The projectile was modelled using perfectly-rigid solid elements, with anappropriate material density that would yield masses of 6.33 and 0.314 kg,based on the test conditions. An initial projectile velocity was defined in thenegative z-direction, with a value suiting the case under investigation. Themodel was analyzed using the default time step size calculated by the explicitsolver, resulting in a time step size ranging from 1×10−5 to 6×10−5 ms, de-pending on the number of through-thickness elements in the model (wherea smaller time step size relates to a higher number of through-thicknessthick-shell elements).5.3.3 Intralaminar Damage ModellingThe intralaminar modelling approach used in the analysis, is based on theprevious work by Forghani and Vaziri [36]. LS-DYNA’s built-in isotropicmaterial model, *MAT PLASTICITYWITH DAMAGE (*MAT_81), whichcombines both damage and plasticity, is used in order to simulate damageevolution within each sublaminate ([45◦/90◦/ − 45◦/0◦]). Since the lam-inate (and its sublaminates) investigated in this study are quasi-isotropic1005.3. Dynamic Plate Impact SimulationsSphericalProjectileΦ = 25.4 mmTargetPlateV0Figure 5.23: An isometric view of the plate-impact finite-element model.Due to the problem’s symmetry, a quarter of the test geometry is modelled.Similar to an assumption made by Forghani and Vaziri [36] and Williamset al. [129], only the portion of the plate which is positioned above thealuminum-backed opening is modelled (i.e the size of the actual plate mod-elled is 76.2mm ×127 mm), and simply supported boundary conditions areapplied to the free edges of the plateand therefore exhibit an isotropic behavior in-plane, this material model isconsidered to be a suitable choice.A typical behavior of *MAT_81 under cyclic loading is shown in Figure5.24. Upon loading, stress increases linearly along the loading path o¯a, untilthe value of the stress reaches its maximum value, σu. As the loading furtherincreases, the stress drops linearly, until it reaches point b, where unloadingbegins. The unloading path b¯c does not follow a secant path to the origin,and it crosses the state of zero stress at point e. It is characterized by areduced slope compared to the loading path o¯a, driven by some damagethat has already developed in the material. During a second loading cycle,a decrease in the ultimate strength of the material is noticeable, where thevalue of the maximum stress now reaches point f .1015.3. Dynamic Plate Impact SimulationsWithin *MAT_81 , the maximum stress σu, and the stiffness of the dam-aged material, Cd, are scaled based on the growth of a damage parameter,d, which is a user-defined function of the equivalent plastic strain. The valueof the maximum stress σ decreases based on the following equation:σ = σeff (1− d) (5.3)where σeff is the value of the effective stress.The structural stiffness of the damaged material, Cd, decreases with agrowth of d, according to the relation:Cd = C0(1− d) (5.4)where C0 is the undamaged-material stiffness. For more informationregarding the behavior of *MAT_81, the reader is referred to LS-DYNAusers manual (2013) and [26].εσσuεfoabecfFigure 5.24: Typical strain-softening behavior of LS-DYNA’s *MAT PLAS-TICITY WITH DAMAGE material model (*MAT_81) during a fullload/unload cycle. This material model is used to simulate the in-planedamage behavior of the sublaminate of the T800/3900-2 CFRP laminatewith layup of [45/90/− 45/0]3s1025.3. Dynamic Plate Impact SimulationsThe material parameters used in the analysis are based on the valuesused by Forghani and Vaziri [36] for simulating the plate impact event.The lamina elastic properties of the T800/3900-2 CFRP and the effectivesublaminate properties are listed in Table 5.6.Table 5.6: Model input parameters for the T800/ 3900-2 CFRP sublami-nate [45◦/90◦/−45◦/0◦] (Williams et al. [129] and Forghani and Vaziri [36])Property Value UnitDensity 1.543× 10−3 g/mm3Sublaminate thickness 0.775 mmElasticEffective elastic modulusa (Exx = Eyy = Ezz) 48.37 GPaEffective shear modulusa (Gxy = Gyz = Gzx) 18.36 GPaEffective Poisson ratiosa (νxy = νyz = νxz) 0.32 (-)Intralaminar damageIntralaminar peak stress (σu) 800 MPaIntralaminar damage saturation strain (εf ) 0.148 (-)Interlaminar damageMode I critical energy release rate (GIc) 0.8 kJ/m2Mode II critical energy release rate (GIIc) 2.0 kJ/m2Interlaminar normal strength (σmax) 80 MPaInterlaminar shear strength (τmax) 150 MPaa Out-of-plane elastic properties are assumed to be the same as the in-plane propertiesbecause of the isotropic limitations of *MAT_81 material model in LS-DYNA .5.3.4 Interlaminar Damage ModellingInterlaminar damage is modelled using LS-DYNA’s cohesive solid elements,to which *MAT_81 is assigned. Similar to the tube crush benchmark prob-lem, a mixed mode criterion is used, in order to account for the interactiveeffect of damage in mode-I and mode-II loading. The interlaminar damageproperties used in the analysis are listed in Table 5.6.Similar to the tube crush benchmark problem, two modelling approacheswere used: The conventional application of the CZM, where cohesive el-ements were present along all potential delamination interfaces prior tothe analysis, and the LCZ method, where cohesive elements were adap-tively seeded along the delamination propagation interface using the LCZ1035.3. Dynamic Plate Impact Simulationsalgorithm. An interactive stress-based element-splitting criterion was used(Equation 5.2) in order to assess the potential for splitting the structuralthick-shell elements and seeding the solid cohesive elements . The criticalvalue for splitting the thick-shell elements, Sc, is assumed to be 0.4. Choos-ing a small value for this parameter will cause large number of thick-shellelements to satisfy the element splitting criterion, thus increasing the sizeof the geometrical region being split in each splitting step, making the LCZmethod less favourable compared to the conventional CZM. Increasing thevalue of Sc, however, will result in smaller regions of thick-shell elementsto be split, up to a point where excessive numerical noise is introduced.The maximum values of the out-of-plane normal and shear stresses, σmaxand τmax, are taken to be 80 MPa and 150 MPa, respectively, and the ra-dius of the splitting region, R is assumed to be 8 mm, to allow sufficientnumber of cohesive elements to be included within the cohesive band. Acoefficient of friction of 0.4 was defined for the contact between the newlycreated surfaces, which is an acceptable value for rough composites. Themulti-delamination capability of the LCZ method is applied to the problem,i.e, each thick-shell element can adaptively split through its thickness, andsince the plate was modelled using multiple number of through-thicknesselements, multiple delamination cracks can propagate independently withinthe material.5.3.5 Results and DiscussionIn order to examine the capability of the LCZ method to simulate the plate-impact event, impact force and kinetic energy time histories were computedand compared to the results obtained from the application of the conven-tional CZM to the problem, as well as the results obtained from the ex-perimental data. Similarly, the predicted damage patterns were examinedand compared to available experimental measurements. The results are pre-sented for different number of through-thickness cohesive interfaces, whenusing the conventional CZM, the FE models included the a-priori placed co-hesive interfaces, while when using the LCZ method, the cohesive elementswere automatically generated and seeded during the computational run.4.29 m/s Impact EventFor an impactor mass of 0.314 kg impacting the plate at 14.59 m/s, Fig-ure 5.37 and Figure 5.38 show the impact-force vs. time and impact-forcevs. displacement profiles, respectively, obtained from the simulations using1045.3. Dynamic Plate Impact Simulationsdifferent number of through-thickness cohesive interfaces.When a single cohesive interface is used, unstable growth of delamina-tion occurs, resulting in a sudden load drop which is visible in Figure 5.25.aat a time of approximately 1.1 ms from impact, and in Figure 5.26.a ata displacement of approximately 4.5 mm. According to the conventionalCZ prediction, this sudden growth of delamination leads to a delamina-tion crack that spans across most of the plate (Figure 5.7), where the LCZmethod predicts complete failure of the cohesive interface. Neither of thesebehaviors was observed in the experiment, as can bee seen by examiningthe experimental delamination pattern which is shown at the bottom ofTable 5.7. It is believed that this unstable growth can be driven by an in-sufficient through-thickness mesh refinement, which does not allow correctdistribution of the energy through the interlaminar and intralaminar damagemechanisms. However, the divergence of the delamination area predicted bythe LCZ solution from the experimental findings is smaller compared to theresults predicted by the conventional CZ method.As the number of cohesive interfaces is increased, better prediction isobtained. Both of the LCZ’s method and the conventional CZ’s methodpredictions yield stable results for the 3, 4, and 5 cohesive-interface modelsinvestigated. For the first 2 ms after impact, as the impactor is bendingand pushing the plate downwards, the 3, 4 and 5 interface models showgood agreement between the experiment and the numerical predictions, forboth conventional CZ as well as the LCZ method’s predictions. As themaximum load is reached, and the impactor begins to rebound from theplate, both conventional and LCZ method over predict the force comparedto the experimental results, for the 3 and 5 interface models. For the 4interface model, the LCZ method’s prediction is higher in the rebound phasecompared to the numerical prediction of the conventional CZ, up to a timeof approximately 4 ms from impact, and is then lower compared to theconventional CZ prediction until complete separation of the impactor fromthe plate.Examining the delamination patterns predicted by the simulation for dif-ferent number of interfaces (Table 5.7), it can be seen that better predictionof the delamination pattern is obtained as the number of cohesive interfaceis increased, for both conventional CZ and the LCZ method. The experi-mental delamination pattern is somewhat oval, and increasing the numberof cohesive interfaces results in a better depiction of this oval contour bythe simulations. The contour predicted by the LCZ method using the 4interface model is an exception for this behavior, where a somewhat lessoval pattern is obtained compared to the pattern predicted by the 3 inter-1055.3. Dynamic Plate Impact Simulations(a) (b)(c) (d)Figure 5.25: Predicted and experimental impact force vs. time for a 6.33kg impactor, impacting the plate at 4.29 m/s. The numerical results wereobtained using the LCZ method and the conventional CZM, for differentnumber of through-thickness cohesive interfaces: a). 1 cohesive interfaceb).3 cohesive interfaces c). 4 cohesive interfaces d). 5 cohesive interfaces.face model. Further increasing the number of cohesive interfaces from 4 to5 will improve the results, and yield a contour that better resembles theexperimental findings.The predicted propagations of delamination at different time states, areshown in Figure 5.27.a and b, for the conventional CZM and LCZ methods,respectively, where both methods are solved using a model that employs5 cohesive interfaces. The figures show a local cross-sectional view of theplate in the vicinity of the impactor, where the coloured regions mark the1065.3. Dynamic Plate Impact Simulations(a) (b)(c) (d)Figure 5.26: Predicted and experimental impact force vs. plate displace-ment, for a 6.33 kg impactor, impacting the plate at 4.29 m/s. The nu-merical results were obtained using the LCZ method and the conventionalCZM, for different number of through-thickness cohesive interfaces: a). 1cohesive interface b). 3 cohesive interfaces c). 4 cohesive interfaces d). 5cohesive interfaces.delamination crack along the interfaces. The cross-section is taken alongthe length of the plate. Delamination damage first appears at around t =0.6ms from impact. Delamination damage predicted by the LCZ method,initiates at the central interface, whereas delamination predicted by theconventional CZM initiates at interface numbers 2 to 4 (where "1" denotesthe interface closest to the impactor and "5" denotes the interface closestto the distal surface of the plate). As the load increases, delamination1075.3. Dynamic Plate Impact Simulationsgrows further and at 1 ms the delamination crack spans across interfaces2,3 and 4, according to both the conventional CZM as well as the LCZpredictions. Delamination does not propagate beyond t = 2.8ms, which isthe point in time when rebound of the impactor begins. At this state of fullydeveloped delamination damage, only a slight difference is observed betweenthe conventional CZ and the LCZ prediction. While the conventional CZpredicts delamination damage at interfaces 2,3,4 and 5, the LCZ methodpredicts delamination damage at all interfaces, including a relatively smallarea of delamination right under the point of contact with the sphericalsurface of the impactor, located at the first interface.Figure 5.30.a and Figure 5.30.b show the impactor’s kinetic energy, pre-dicted by the conventional CZ method and the LCZ method, respectively.Both methods under-predict the kinetic energy when a single-cohesive-interfaceis used. This is probably a result of the unstable delamination crack propa-gation and decrease in the structural stiffness due to the size of the resultingcrack. As the number of cohesive interfaces is increased, both conventionalCZ and LCZ methods show an improvement in the kinetic energy predic-tion, where better convergence is achieved using the conventional CZ. Inboth methods, 5 interface models yield results which agree very well withthe experimental findings.1085.3. Dynamic Plate Impact Simulationst=0.4 ms t=0.6 ms t=1.0 mst=1.6 mst=2.8 msFigure 5.27: Delamination propagation predicted by the conventional CZ,for a 4.29 m/s, 6.33 kg projectile using a 5 interface model.1095.3. Dynamic Plate Impact Simulationst=0.4 ms t=0.6 ms t=1.0 mst=1.6 mst=2.8 mst=0.4 ms t=0.6 ms t=1.0 mst=1.6 mst=2.8 mst=0. s t=0. t 1.0t=1.6t=2.8Figure 5.28: Delamination propagation predicted by the LCZ method, fora 4.29 m/s, 6.33 kg projectile using a 5 interface model.18.97 m/s Impact EventFigure 5.31 and Figure 5.32 show the impact-force vs. time and impact-force vs. displacement profiles obtained from the simulations, respectively,for an impactor mass of 0.314 kg impacting the plate at 18.97 m/s. Table5.8 shows the delamination patterns predicted by the conventional and LCZmethod, as well as the image obtained from the experimental ultrasonic scan.Similar to the 4.29 m/s impact event, a single-cohesive interface results ina delamination crack growth which reaches the edges of the plate (Table5.8). Increasing the number of cohesive interfaces to 3, 4 and 5 interfaces,will result in stable crack growth which better resembles the delaminationdamage found in the experiment. The oscillations which are noticeablein the impact force for all cases tested, are caused mainly by the naturaloscillations of the plate which are driven by the dynamic impact event.Figure 5.33 and Figure 5.34 show a schematic representation of the in-1105.3. Dynamic Plate Impact SimulationsTable 5.7: Predicted delamination patterns for a 4.29 m/s, 6.33 kg impactor,plate-impact event, obtained using the conventional CZ method, comparedto the experimental and LCZ’s method prediction. Fringe colors representdifferent through-thickness interfaces.Numberofthrough-thicknesscohesiveinter-facesConventional CZ method LCZ method1345Experimental result [129]r Chapter 4: Numerical Case Studies Figure 4.22 Comparison of predicted projected matrix/delamination damage and experimental C-scan images for high mass impact events on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. Results presented are for (a) 9.5 J (v = 1.76 m/s, m = 6141 g), (d) 46.2 J (v = 3.82 m/s, m = 6330 g), (c) 58.2 J (v = 4.29 m/s, m = 6330 g) impacts. 182 r Chapter 4: Numerical Case Studies Figure 4.22 Comparison of predicted projected matrix/delamination damage and experimental C-scan images for high mass impact events on a [45/90/-45/0]3S T800H/3900-2 CFRP plate. Results presented are for (a) 9.5 J (v = 1.76 m/s, m = 6141 g), (d) 46.2 J (v = 3.82 m/s, m = 6330 g), (c) 58.2 J (v = 4.29 m/s, m = 6330 g) impacts. 182 1115.3. Dynamic Plate Impact Simulations1 2 3 4 500.511.5·104Number of through-thickness cohesive interfacesProjectedDelaminationArea(mm2)LCZ methodConventional CZ methodExperiment (Delfosse et al, 1995)Figure 5.29: Projected delamination area, for an 6.33 kg impactor impactingthe plate at 4.29 m/s, as a function of the number of through thicknesscohesive interfaces, predicted by the Conventional CZ and LCZ methods, aswell as the experimental results.plane and delamination damage patterns for a time of 0.3 milliseconds and1.3 milliseconds, respectively, obtained using a 3 cohesive interface modeland the LCZ method. Initially, the plate is modelled using 3 through-thickness thick-shell elements. Assuming that delamination occurs in allof the potential interfaces, each thick shell element splits into two offsetshell elements, thus resulting in plate consisting of of 6 offset shell elementsthrough its thickness. The fringe plots located to the left side of Figure5.33 and Figure 5.34 show the values of LS-DYNA’s *MAT_81 damageparameter for each of the offset shell layers created during the splittingprocess, where a value of 1 resembles a fully damaged material, and a valueof 0 resembles an undamaged material. The right side of Figure 5.33 andFigure 5.34 shows a schematic through-thickness place locator of each offsetshell layers, together with the delamination damage of each of the threecohesive interfaces in the model. The largest delamination crack is predictedto take place within the second (middle) interface, while the in-plane damage1125.3. Dynamic Plate Impact Simulationsspans across a smaller region compared to the delamination damage of theneighbouring interfaces.Figure 5.35 shows the predicted projected delamination area as a func-tion of the number of through-thickness cohesive interfaces. Similar to the4.29 m/s impact event, both of the conventional CZ, as well as the LCZmethod over-predict the delamination area for the number of interfacestested, and the LCZ method converges faster toward the experimental solu-tion. Here, too, it is believed that the number of cohesive interfaces, which islower in the numerical models compared to the actual experimental laminatearchitecture results in delamination cracks which cover a larger area.1135.3. Dynamic Plate Impact Simulations0 1 2 3 4 5 6Time [msec]0102030405060Kinetic Energy  [J]1 interface3 interfaces4 interfaces5 interfacesExperiment (Delfosse et al (1995)(a)0 1 2 3 4 5 6Time [msec]0102030405060Kinetic Energy  [J]1 interface3 interfaces4 interfaces5 interfacesExperiment (Delfosse et al (1995)(b)Figure 5.30: Predicted and experimental impactor’s kinetic energy vs. time,for a 4.29 m/s impact velocity and impactor mass of 6.33 kg. a). The pre-dicted results are shown for different number of through-thickness cohesiveinterfaces using the conventional CZM .a). Results obtained using the LCZmethod.1145.3. Dynamic Plate Impact Simulations(a) (b)(c) (d)Figure 5.31: Predicted and experimental impact force vs. time for a 0.314kg impactor, impacting the plate at 18.97 m/s. The numerical results wereobtained using the LCZ method and the conventional CZM, for differentnumber of through-thickness cohesive interfaces: a). 1 cohesive interface,b).3 cohesive interfaces, c). 4 cohesive interfaces, d). 5 cohesive interfaces.1155.3. Dynamic Plate Impact Simulations(a) (b)(c) (d)Figure 5.32: Predicted and experimental impact force vs. plate displace-ment, for a 0.314 kg impactor, impacting the plate at 18.97 m/s. The nu-merical results were obtained using the LCZ method and the conventionalCZM, for different number of through-thickness cohesive interfaces: a). 1cohesive interface,b). 3 cohesive interfaces, c). 4 cohesive interfaces, d). 5cohesive interfaces.1165.3. Dynamic Plate Impact SimulationsTable 5.8: Predicted delamination pattern for a 18.97 m/s, 0.314 kg pro-jectile, plate-impact event, obtained using the conventional CZM, comparedto the experimental and LCZ’s method prediction. Fringe colors representdifferent through-thickness interfaces.Numberofthrough-thicknesscohesiveinter-facesConventional CZ method LCZ method1345Experimental result [129]1175.3. Dynamic Plate Impact Simulations   6.000e-01 _   1.000e-01 _   2.000e-01 _   3.000e-01 _   4.000e-01 _   5.000e-01 _Fringe Levels   7.000e-01 _   8.000e-01 _   9.000e-01 _   1.000e+00 _   0.000e+00 _YXZFringe Levels min=0, at elem# 21max IP. valueContours of History Variable#5Time =        40.82-ply with plug 0/+45/-45 TUBE CRUSH - shell                               0.000e+00 _   1.000e-01 _   2.000e-01 _ max=1, at elem# 2915   4.000e-01 _   5.000e-01 _   6.000e-01 _   7.000e-01 _   8.000e-01 _   9.000e-01 _   1.000e+00 _   3.000e-01 _(a) (b)    6.000e-01 _   1.000e-01 _Fringe Levels   2.000e-01 _   4.000e-01 _   0.000e+00 _   3.000e-01 _   7.000e-01 _   8.000e-01 _   9.000e-01 _   5.000e-01 _   1.000e+00 _1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 YXZFringe Levels min=0, at elem# 21max IP. valueContours of History Variable#5Time =        40.82-ply with plug 0/+45/-45 TUBE CRUSH - shell                               0.000e+00 _   1.000e-01 _   2.000e-01 _ max=1, at elem# 2915   4.000e-01 _   5.000e-01 _   6.000e-01 _   7.000e-01 _   8.000e-01 _   9.000e-01 _   1.000e+00 _   3.000e-01 _   6.000e-01 _   1.000e-01 _Fringe Levels   2.000e-01 _   4.000e-01 _   0.000e+00 _   3.000e-01 _   7.000e-01 _   8.000e-01 _   9.000e-01 _   5.000e-01 _   1.000e+00 _(a) (b) 1234561st interface2nd interface3rd interfaceFigure 5.33: Damage in a 3 interface model, impact velocity of 18.97 m/s,at time of 0.3 ms.Figure 5.36.a and Figure 5.36.b show the impactor’s kinetic energy, pre-dicted by the conventional CZ method and the LCZ method, respectively.Except for the case where the LCZ method was used to solve a 5 interfacemodel, all models tested resulted in under-prediction of the impactor’s ki-netic energy. When the LCZ method was used, increasing the number ofcohesive interfaces in the finite element model lead to better convergenceof the results. This trend was less pronounced when the conventional CZmethod was applied to the problem, where the improvement of the resultswas either negligible or even worsened by an increased number of interfaces(A 3 interface model yielded inferior results compared to a single interfacemodel).1185.3. Dynamic Plate Impact Simulations   6.000e-01 _   1.000e-01 _   2.000e-01 _   3.000e-01 _   4.000e-01 _   5.000e-01 _Fringe Levels   7.000e-01 _   8.000e-01 _   9.000e-01 _   1.000e+00 _   0.000e+00 _YXZFringe Levels min=0, at elem# 21max IP. valueContours of History Variable#5Time =        40.82-ply with plug 0/+45/-45 TUBE CRUSH - shell                               0.000e+00 _   1.000e-01 _   2.000e-01 _ max=1, at elem# 2915   4.000e-01 _   5.000e-01 _   6.000e-01 _   7.000e-01 _   8.000e-01 _   9.000e-01 _   1.000e+00 _   3.000e-01 _(a) (b)    6.000e-01 _   1.000e-01 _Fringe Levels   2.000e-01 _   4.000e-01 _   0.000e+00 _   3.000e-01 _   7.000e-01 _   8.000e-01 _   9.000e-01 _   5.000e-01 _   1.000e+00 _1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 YXZFringe Levels min=0, at elem# 21max IP. valueContours of History Variable#5Time =        40.82-ply with plug 0/+45/-45 TUBE CRUSH - shell                               0.000e+00 _   1.000e- 1 _   2.000e- 1 _ max=1, at elem# 2915   4.000e- 1 _   5.000e- 1 _   6.000e- 1 _   7.000e- 1 _   8.000e- 1 _   9.000e- 1 _   1.000e+00 _   3.000e- 1 _   6.000e-01 _   1.000e-01 _Fringe Levels   2.000e-01 _   4.000e-01 _   0.000e+00 _   3.000e-01 _   7.000e-01 _   8.000e-01 _   9.000e-01 _   5.000e-01 _   1.000e+00 _(a) (b) 1234561st interface2nd interface3rd interfaceFigure 5.34: Damage in a 3 interface model, impact velocity of 18.97 m/s,at time of 1.8 ms.1195.3. Dynamic Plate Impact Simulations1 2 3 4 500.511.5·104Number of through-thickness cohesive interfacesProjectedDelaminationArea(mm2)LCZ methodConventional CZ methodExperiment (Delfosse et al, 1995)Figure 5.35: Projected delamination area, for an 0.314 kg impactor impact-ing the plate at 18.97 m/s, as a function of the number of through thicknesscohesive interfaces, predicted by the Conventional CZ and LCZ methods, aswell as the experimental results.1205.3. Dynamic Plate Impact Simulations0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8Time [msec]0102030405060Kinetic Energy  [J]1 interface3 interfaces4 interfaces5 interfacesExperiment (Delfosse et al (1995)(a)0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8Time [msec]0102030405060Kinetic Energy  [J]1 interface3 interfaces4 interfaces5 interfacesExperiment (Delfosse et al (1995)(b)Figure 5.36: Predicted and experimental impactor’s kinetic energy vs. time,for a 18.97 m/s impact velocity and impactor mass of 0.314 kg. a). Thepredicted results are shown for different number of through-thickness cohe-sive interfaces, using the conventional CZM. b). Results obtained using theLCZ method.1215.4. Summary and Conclusions14.59 m/s Impact EventFigure 5.37 and Figure 5.38 show the impact-force vs. time and impact-force vs. displacement profiles obtained from the simulations, respectively,for an impactor mass of 0.314 kg impacting the plate at 14.59 m/s. Table5.9 shows the delamination patterns predicted by the conventional and LCZmethod, as well as the image obtained from the experimental ultrasonic scan,and Figure 5.39 show the impactor’s kinetic energy as a function of time.The force vs. time profiles show good agreement with the experimentaldata, for both the conventional CZ as well as the LCZ method’s solution.The force vs. displacement plots show lower agreement, particularly forthe single cohesive interface models, where overall the numerical predictionsunder-predict the impact force. As with all other plate impact scenariostested, using a single cohesive interface to describe the plate, results in anunstable delamination crack growth, yielding a crack that reaches the plate’sboundaries (Table 5.9). However, the divergence of the LCZ solution’s pre-diction from the experimental findings is smaller compared to the resultspredicted by the conventional CZ method. Figure 5.40 shows the predictedprojected delamination area depending on the number of through-thicknesscohesive interfaces. Similar to all other cases tested, it can be seen that theconventional CZ, as well as the LCZ method, over-predict the delaminationarea for the number of interfaces used and that both converge to the exper-imental value as the number of interfaces increases, with the LCZ methodrequiring fewer number of interfaces for convergence.It is believed that the number of cohesive interfaces throughout the thick-ness which is lower in the numerical models than the actual number of in-terfaces results in delamination cracks which cover a larger area.Figure 5.39 shows the impactor’s kinetic energy as a function of time.The final predicted kinetic energy of the impactor (Figure 5.39), convergesto the experimental result with an increased number of cohesive interfaces,for both of the LCZ and conventional CZ method’s predictions.5.4 Summary and ConclusionsThe LCZ method, which was verified for the solution of pure delaminationcrack propagation in Chapter 4, was applied here for the solution of a tensileloading of a double-notched coupon, dynamic tube crushing experiments, aswell as dynamic plate impact events.For the tensile loading of the double-notch coupon, the method wascombined with LS-DYNA’s *MAT_54, to capture the in-plane behavior1225.4. Summary and Conclusions(a) (b)(c)0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6Time [msec]05101520Force [KN]LCZ methodConventional CZ methodExperiment (Delfosse et al, 1995)(d)Figure 5.37: Predicted and experimental impact force vs. time for a 0.314kg impactor, impacting the plate at 14.59 m/s. The numerical results wereobtained using the LCZ method and the conventional CZM, for differentnumber of through-thickness cohesive interfaces: a). 1 cohesive interface,b). 3 cohesive interfaces, c). 4 cohesive interfaces, d). 5 cohesive interfaces.and overall strength reduction of the coupon, such that both in-plane aswell as out of plane damage could be simulated simultaneously.For the tube crush loading scenario, a continuum damage model de-veloped at the University of British Columbia (CODAM2), was applied tocapture the in-plane damage within the tube’s wall, and the LCZ methodwas applied to the model in order to capture interlaminar damage. Impact-force profiles were compared to the experimental data, as well as to resultsobtained using the conventional CZM. Good agreement was obtained for1235.4. Summary and Conclusions(a) (b)(c)0 1 2 3 4 5 6Displacement [mm]05101520Force [KN]LCZ methodConventional CZ methodExperiment (Delfosse et al, 1995)(d)Figure 5.38: Predicted and experimental impact force vs. plate displace-ment, for a 0.314 kg impactor, impacting the plate at 14.59 m/s. The nu-merical results were obtained using the LCZ method and the conventionalCZM, for different number of through-thickness cohesive interfaces: a). 1cohesive interface,b). 3 cohesive interfaces, c). 4 cohesive interfaces, d). 5cohesive interfaces.both numerical methods compared to the experiments. Slight differenceswere observed between the prediction of damage using the two numericalapproaches. The LCZ algorithm was able to adaptively split the structuralelements through their thickness during the dynamic tube-crushing process,and seed the cohesive elements along the required locations.A new capability of the LCZ method, which allows adaptive introduc-tion of multiple through-thickness delamination cracks into the structure,1245.4. Summary and Conclusions0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8Time [msec]50510152025303540Kinetic Energy  [J]1 interface3 interfaces4 interfaces5 interfacesExperiment (Delfosse et al (1995)(a)0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8Time [msec]50510152025303540Kinetic Energy  [J]1 interface3 interfaces4 interfaces5 interfacesExperiment (Delfosse et al (1995)(b)Figure 5.39: Predicted and experimental impactor’s kinetic energy vs. time,for a 14.59 m/s impact velocity and impactor mass of 0.314 kg. a). Thepredicted results are shown for different number of through-thickness cohe-sive interfaces, using the conventional CZM. b). Results obtained using theLCZ method.1255.4. Summary and Conclusions1 2 3 4 500.20.40.60.81·104Number of through-thickness cohesive interfacesProjectedDelaminationArea(mm2)LCZ methodConventional CZ methodExperiment(Delfosse et al, 1995)Figure 5.40: Projected delamination area, for an 0.314 kg impactor impact-ing the plate at 14.59 m/s, as a function of the number of through-thicknesscohesive interfaces, predicted by the Conventional CZ and LCZ methods, aswell as the experimental results.was applied to a plate-impact event used as a benchmark problem. The testconfiguration was modelled using various number of through-thickness co-hesive interfaces, and the problem was solved using both the LCZ method,as well as the conventional CZ method. Impact-force vs. displacementsprofiles, as well as impact force vs. time histories, were compared to theexperimental data, together with the impactor’s kinetic energy, predicteddelamination patterns, and predicted delamination area. The results werepresented for three impact velocities and different values of the impactormass.For the range of velocities tested, the impact force profiles obtained fromthe numerical simulations were with reasonable agreement of the experimen-tal data, for 3, 4, and 5 through-thickness cohesive interfaces. Good resultswas obtained from the LCZ method, as well as the conventional CZ method.When a single cohesive interface was used, the delamination crack propa-gated in an unstable manner, for both conventional CZ as well as the LCZ1265.4. Summary and Conclusionssolution. This is believed to be caused by the insufficient through-thicknessdiscretization of the finite element model, which requires a larger area ofdelamination crack to be formed in order to absorb enough energy by thisdamage mechanism. The through-thickness discretization is believed to havean effect on the projected delamination area, which although resembled theexperimental pattern for all number of through-thickness cohesive interfacestested, covered a larger area in the simulations compared to the experimen-tal results. Increasing the number of cohesive interfaces in order to evaluatethis assumption was not performed in this study, as it will result in a finiteelement model which exceeds the number of interfaces found in the exper-imental setup, given that each sublaminate is treated as an isotropic andhomogenized material.Kinetic energy profiles of the impactor were with reasonable agreementof the experimental data, for all cases tested, for both LCZ simulation as wellas the numerical solution obtained using the conventional CZM. Increasingthe number of cohesive interfaces improved the results for both numericalmethods.The work performed here proved the ability of the LCZ method to becombined with other in-plane damage theories, where intra-laminar damageis treated in a smeared manner, and the element stiffness is reduced grad-ually as a function of damage evolution within the finite element volume.Such methodology combines the numerical advantages of smeared modellingtechniques with the need to model delamination in a discrete manner.The numerical advantages and performance of the method will be dis-cussed in the next chapter.1275.4. Summary and Conclusions0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18Strain1000100200300400500600700800Stress [Mpa]Gf 1mm=41.46 kJ/m2Gf 2mm=41.48 kJ/m21mm2mmFigure 5.41: Stress vs. strain for plate impact model, 2mm and 1mm mesh1285.4. Summary and Conclusions0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035COD [mm]0102030405060708090Stress [MPa]Gf=0.94 kJ/m2Figure 5.42: Stress vs. COD obtained from Tie-Break contact, Mode-I crackopening1295.4. Summary and Conclusions0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035COD [mm]020406080100120140160Stress [MPa]Gf=1.67 kJ/m2Figure 5.43: Stress vs. COD obtained from Tie-Break contact, Mode-IIcrack opening1305.4. Summary and ConclusionsTable 5.9: Predicted delamination pattern for a 14.59 m/s, 0.314 kg projec-tile, plate-impact event, obtained using the conventional CZM, compared tothe experimental and LCZ’s method prediction.Numberofthrough-thicknesscohesiveinter-facesConventional CZ method LCZ method1345Experimental result [129]131Chapter 6Numerical Performance ofthe LCZ method6.1 IntroductionIn line with the motivation for the development of the LCZ method, andtaking into account the fact the the LCZ algorithm is not yet an internalpart of a commercial finite element code, some preliminary simulations wereperformed in order to evaluate the numerical performance of the algorithmcompared to the conventional CZM. The following sections will describethe performance of the LCZ method and its efficiency when solving largermodels, and its overall effect on the structural stiffness.6.2 Solution of Larger ModelsSolving larger models using the conventional CZM, often requires largenumber of numerically-expensive cohesive interfaces to be present in themodel, which can increase the computational load to an extent which ren-ders this method inapplicable to large engineering applications. Using theLCZ method, however, requires no cohesive elements to be present in themodel a priori of the finite element analysis, as they are seeded only at spe-cific locations where delamination has the potential to initiate and grow. Inorder to compare the numerical efficiency of the LCZ method with respect tothe conventional CZM, a series of numerical models with increased numberof elements, were solved. These models were based on the tube-crush model- for each model tested, the tube portion of the model was duplicated, whereeach model had a different number of tube duplications. Models containing2, 6 and 10 tubes are shown in Figure 6.1.a, Figure 6.1.b, and Figure 6.1.c,respectively. It is important to note that only the tube portion of the modelwas duplicated, i.e, only a single plug and dropped mass was used in eachof the models, thus in each of the models tested, only a single tube wasundergoing active crushing.1326.2. Solution of Larger ModelsThe models were then solved using six CPUs for each model. Giventhe fact that the LCZ algorithm is currently in a development state and isnot an integrated part of LS-DYNA, only the actual LS-DYNA run-timeobtained from solving the models using the LCZ algorithm was comparedto the LS-DYNA run-time using the conventional CZM.The resulting LS-DYNA run-time, in seconds, for each of the modelssolved, is listed in Table 6.1. The ratio between the LS-DYNA run-timeusing the conventional application of CZM (using solid cohesive elementsa-priori seeded along all of the cohesive interfaces), tconv, and the LS-DYNArun-time using the LCZ method tLCZ , is plotted vs. the number of tubesin the model in Figure 6.2. Even for a single tube model, the LS-DYNArun-time using the conventional cohesive-zone method is 1.79 times longerthan the run-time using the LCZ method, where for a model containing 10tubes, the ratio between the run-times of the two methods is 5.4 in favourof the LCZ method.XYZ(a)XYZ(b)Y XZ(c)Figure 6.1: Isometric view of three finite element models used to test the LCZmethod’s efficiency over the conventional CZM, using increasing number oftubes: a). Two tubes. b). Six tubes. (c) Ten tubes.To compare the efficiency of the LCZ method (which is based on solidcohesive elements as the numerical representation of the cohesive interface)against the conventional CZM using a cohesive contact interface, the sim-ulations for the conventional CZM were repeated using a cohesive contactalgorithm (LS-DYNA’s TIEBREAK contact) replacing all of the solid cohe-1336.2. Solution of Larger ModelsTable 6.1: LS-DYNA run-time in seconds, using solid cohesive elements,obtained using the conventional CZM, as well as using the LCZ method.The ratio between the LS-DYNA run-time using both methods is presentedas well.Number of tubes1 2 4 6 8 10Conventionalcohesive-zonemethod (tconv)52,371 85,188 193,500 214,225 339,976 483,520LCZ method(tLCZ)29,300 51,307 79,342 81,023 79,241 89,502tconv/tLCZ 1.79 2.07 2.7 3.12 4.3 5.4sive elements in the model. The LS-DYNA resulting run-time, in seconds,for each of the models solved, is listed in Table 6.1. The ratio between theLS-DYNA run-time using the conventional application of CZM (using a co-hesive contact algorithm a-priori defined along all of the cohesive interfacesin the model), tconv, and the LS-DYNA run-time using the LCZ method,is plotted vs. the number of tubes in the model in Figure 6.3. For a singletube model, the LS-DYNA run-time using the conventional cohesive-zonemethod is 1.73 times longer than the run-time using the LCZ method. Asthe number of tubes is further increased, a meandering trend can e noticed.It is believed that the internal treatment of the contact algorithm within thefinite element solver is the main cause for the slight decrease in the efficiencyfor some of the cases tested. For all cases tested, however, the run-time usingthe LCZ method was considerably shorter compared to the run-time usingthe conventional CZM.1346.2. Solution of Larger ModelsTable 6.2: LS-DYNA run-time in seconds obtained using the conventionalCZM, together with a TIEBREAK contact algorithm, and the LS-DYNArun-time using the LCZ method with solid cohesive elements. The ratiobetween the LS-DYNA run-time using both methods is presented as well.Number of tubes1 2 4 6 8 10Conventionalcohesive-zonemethod (tconv)50,762 85,188 161,026 231,630 308,147 376,564LCZ method(tLCZ)29,300 51,307 79,342 81,023 79,241 89,502tconv/tLCZ 1.73 1.66 3.02 2.86 3.90 4.210 2 4 6 8 10 1223456Number of Tubest conv/tLCZFigure 6.2: Ratio between the LS-DYNA run-time using the conventionalapplication of CZM (using solid cohesive elements a-priori seeded along allof the cohesive interfaces), tconv, and the LS-DYNA run-time using the LCZmethod, tLCZ , vs. the number of tubes in the models.1356.3. Effect of CZM on the Structural Stiffness0 2 4 6 8 10 122345Number of Tubest conv/tLCZFigure 6.3: Ratio between the LS-DYNA run-time using the conventionalapplication of CZM using a cohesive contact algorithm along all of the co-hesive interface, tconv, and the LS-DYNA run-time using the LCZ method,tLCZ , vs. the number of tubes in the models.6.3 Effect of CZM on the Structural StiffnessSince the use of CZM in its standard form requires introduction of largenumber of cohesive elements in all possible locations where delaminationis likely to grow, using this method to predict delamination crack growthin large structures is not practical from the numerical standpoint. In theirwork, Kaliske et al. [62] investigated the effect of the CZM on the numericalcomplexity of a finite element model. It was shown that for a two dimen-sional uniform finite-element mesh consisting of 4-node elements (Figure6.4.a), introducing cohesive interfaces in all possible crack-growth paths canlead to a 4-times increase in the number of DOF within the model (Figure6.4.b). Furthermore, using cohesive elements in wide regions of the modelwill artificially render the structure more compliant. This is demonstrated1366.3. Effect of CZM on the Structural Stiffnessin Figure 6.5, where a schematic 3D model consisting of two continuumelements with an initial thickness t, which are connected to each other us-ing a cohesive interface of an initial zero thickness is loaded normal to thecohesive interface. Once a load F is applied, (Figure 6.5.b), the continuumelements deform to a thickness t+ δt, and the cohesive interface opens to adisplacement ∆. In this loaded state, the traction continuity requires that:σ = E3ε = K∆ (6.1)where σ is the stress resulting from the applied force F , E3 is the con-tinuum element’s Young’s modulus in the normal (out-of-plane) direction, εis the strain of the continuum element, and K is the stiffness of the cohesiveinterface.The effective strain, εeff , of the material is:εeff =δtt+ ∆t= ε+ ∆t(6.2)Combining Equation 6.1 and Equation 6.2 yields:εeff = E3(11 + E3Kt)(6.3)Thus, in order for the cohesive interface stiffness to have a lower effecton the effective stiffness of the laminate, the condition E3 << Kt needs tobe satisfied. Since E3 and t are given material and geometrical properties,and the sublaminate thickness t is generally small (on the order of tenths ofmillimetres), this implies that in order to reduce the unwanted complianceintroduced to the model by the cohesive interface, the cohesive stiffness needsto be much higher than the stiffness of the sublaminate plies. However, largevalues of the interface stiffness may cause numerical noise and loss of stability[115].1376.3. Effect of CZM on the Structural Stiffnessrepresent failure processes. The mesh dependence of continuum approaches arises from the representation of the crack as asmeared discontinuity of the displacement field. In the same way, the fracture energy Gc or energy dissipation due to crackface opening is distributed over a particular volume. The effective dissipated energy depends therefore on the volume of thebulk elements which have been assigned to the crack. This disadvantage can only be overcome by the application of regu-larisation techniques taking into account the characteristic element volume. In contrast, the cohesive element method al-lows not only to model the crack in a discrete manner but also to account for the energy release rate during crackpropagation realistically.A common procedure for the simulation of crack propagation with the help of cohesive elements is based on a priori con-sidered cohesive surfaces. Since the crack path has to be incorporated into the initial finite element mesh, the path of thediscontinuity has to be known in advance. Consequently, the application range of this strategy is restricted to structures withan identified interface, e.g. composite materials or glued structures, as well as structures subjected to boundary conditions orloads which lead to predefined crack paths. In case of arbitrary and complex unknown fracture patterns, a priori consideredcohesive surfaces have to be provided between all internal bulk element boundaries as exemplarily shown in Xu and Nee-dleman [6] and Tijssens et al. [7]. This leads to an increase of the number of unknowns depending both on the spatial dimen-sion of the problem and the element type. Fig. 1 shows the number of degrees of freedom before and after the incorporationof cohesive surfaces, n1 and n2, respectively. The resulting effect for an increasing number of bulk elements ne is depicted inFig. 2a for the special case of a regular mesh on a rectangular domain. In a two-dimensional simulation, the degrees of free-dom after the cohesive element insertion will be four times higher, in case of a three-dimensional analyses even eight timeshigher.Furthermore, the conventional strategy requires the use of an initially traction free (also referred to as initially elastic)traction separation law. In case of an initially elastic polynomial traction separation law (Fig. 3a), the initial stiffness yieldsfor example (cf. Geißler [8])K0 ¼ 274T0d0: ð2ÞThis initially traction free state of the a priori considered cohesive elements leads to a significant reduction of the structure’seffective stiffness Eeff. As shown in Fig. 4, a one-dimensional analysis yields(a) (b)Fig. 1. Degrees of freedom (a) before and (b) after incorporation of cohesive surfaces.(a) (b)Fig. 2. Conventional cohesive finite element method: (a) ratio of degrees of freedom for elastic and cohesive computation and (b) reduction of the effectivestiffness for increasing contribution of the cohesive phaseG. Geißler et al. / Engineering Fracture Mechanics 77 (2010) 3541–3557 3543(a)represent failure processes. The mesh depend nce of continuum approa hes arises from the representation of the crack as asmeared discontinuity of the displacement field. In the same way, the fracture energy Gc or energy dissipation due to crackface opening is distributed over a particular volume. The effective dissip ted energy depends therefore on the volume of thebulk elements which have been as igned to he crack. This disadvantage can only be overcom b the application f egu-larisation tech iques taking into account the characteristic element volume. I contrast, the coh sive element method al-lows not only to model the crack in a discrete manner but also to account for the energy release rate during crackpropagation realistically.A common procedure for the simulation of crack propagation with the help of cohesive elements is based on a priori con-sidered cohesive surfaces. Since the crack path has to be incorporated into the initial finite element mesh, the path of thediscontinuity has to be known in advance. Consequently, the application range of this strategy is restricted to structures withan identified interface, e.g. composite materials or glued structures, as well as structures subjected to boundary conditions orloads which lead to predefined crack paths. In case of arbitrary and complex unknown fracture patterns, a priori consideredcohesive surfaces have to be provided between all internal bulk element boundaries as exemplarily shown in Xu and Nee-dleman [6] and Tijssens et al. [7]. This leads to an increase of the number of unknowns depending both on the spatial dimen-sion of the problem and the element type. Fig. 1 shows the number of degrees of freedom before and after the incorporationof cohesive surfaces, n1 and n2, respectively. The resulting effect for an increasing number of bulk elements ne is depicted inFig. 2a for the special case of a regular mesh on a rectangular domain. In a two-dimensional simulation, the degrees of free-dom after the cohesive element insertion will be four times higher, in case of a three-dimensional analyses even eight timeshigher.Furthermore, the conventional strategy requires the use of an initially traction free (also referred to as initially elastic)traction separati n law. In case of an initiall elast c polynomial traction separa law Fig. 3a), the initial stiffness yieldsfor example (cf. Geißler [8])K0 ¼ 274T0d0: ð2ÞThis initially traction free state of the a priori considered cohesive elements leads to a significant reduction of the structure’seffective stiffness Eeff. As shown in Fig. 4, a one-dimensional analysis yields(a) (b)Fig. 1. Degre s of freedom (a) before and (b) after incorporation of cohesive surfaces.(a) (b)Fig. 2. Conventional cohesive finite element method: (a) ratio of degrees of freedom for elastic and cohesive computation and (b) reduction of the effectivestiffness for increasing contribution of the cohesive phaseG. Geißler et al. / Engineering Fracture Mechanics 77 (2010) 3541–3557 3543(b)Figure 6.4: A simple finite-element topology demonstrating the increase of numer-ical complexity when cohesive elements are introduced into the model. a.) A simple2D model consisting of 4-node elements. b.) Cohesive elements are introduced inbetween the continuum elements, shown here in darker shade.The eÆective strain of the composite is:"eÆ =±tt+¢t= "+¢t(11)tt+ tdDFFFig. 3. Influence of the cohesive surface on the deformation.Since the traction continuity condition requires that æ = EeÆ"eÆ, the equivalentYoung’s modulus EeÆ can be written as a function of the Young’s modulus ofthe material, the mesh size, and the interface stiÆness. Using equations (10)and (11), the eÆective Young’s modulus can be written as:EeÆ = E3√11 + E3Kt!(12)The eÆective elastic properties of the composite will not be aÆected by thecohesive surface whenever the inequality E3 ø Kt is being accomplished, i.e:K =ÆE3t(13)where Æ is a parameter much larger than 1 (Æ¿ 1). However, large values ofthe interface stiÆness may cause numerical problems, such as spurious oscilla-tions of the tractions [17]. Thus, the interface stiÆness should be large enoughto provide a reasonable stiÆness but small enough to avoid numerical problemssuch as spurious oscillations of the tractions in an element.The ratio between the value of the Young modulus obtained with equation(12) and the Young modulus of the material, as a function of the parameter9FFtt+ δt∆ta)b)Figure 6.5: Schematic 3D model demonstrating the behavior of a cohesive inter-face. a.) Two continuum elements with an initial thickness t, connected to eachother using a cohesive interface of an initial zero thickness, are shown in this fig-ure in an unloaded state. b.) Once a load F is applied normal to the cohesiveinterface, the continuum elements deform to a thickness t + δt, and the cohesiveinterface opens to a displacement ∆.1386.3. Effect of CZM on the Structural Stiffness6.3.1 Simply Supported Beam Under Bending LoadThe effect of the CZM on the structural stiffness of a simply supported beamwas investigated. The model geometry is shown in Figure 6.6. A simplysupported beam of length 2L = 99 mm, thickness h = 3 mm, and width of1 mm, is subjected to a 1 mm displacement ∆ in the negative z direction,which is applied at the center of the beam. Three beam configurationswere numerically tested: the first, which is shown is Figure 6.7.a, consistsof a beam having no cohesive interfaces within the finite element model.The second configuration, shown in 6.7.b, consists of 5 cohesive interfacesthrough the thickness of the beam, with solid cohesive elements located alongall potential delamination crack paths within the interfaces. The cohesiveinterfaces were modelled using LS-DYNA’s *MAT COHESIVE GENERALis applied, and are shaded in brown color. The third configuration, shown inFigure 6.7.c, consists of two 15 mm long cohesive interfaces, locally placedalong the edges of the beam. Each configuration was solved using 2, 6 and18 through-thickness thick-shell elements. The beam was modeled usingan isotropic elastic material model (*MAT ELASTIC), using the propertieslisted in Table 6.3. A typical element size of 1 mm was used along the axialdirection of the beam. The through-thickness element size was determinedbased on the configuration and element discretization described above.xz∆hL LFigure 6.6: Simply supported beam of total length 2L = 99 mm, thicknessh = 3 mm, and width of 1 mm. A displacement ∆, in the global negative zdirection, is applied at the center of the beam.The predicted load at 1 mm displacement as a function of the cohesivetopology and number of through-thickness discretization, is shown in Fig-ure 6.8. The dashed line represents the analytical solution. It can be seenthat the numerical models containing two through-thickness elements (thusa single cohesive interface), yielded results which are close to the analyti-1396.3. Effect of CZM on the Structural Stiffnesscal solution, for all cohesive configurations tested. Increasing the numberof through-thickness elements from 2 to 6 (thus increasing the number ofthrough-thickness cohesive interfaces from 1 to 5), resulted in a decrease ofthe structural stiffness, for the case where the conventional CZM was used.Further reduction of stiffness occurred as the number of through-thicknesselements increased from 6 to 18 (increasing the number of through-thicknesscohesive interfaces from 5 to 17). Localizing the cohesive interface along theedge of the beam did not decrease the peak force, for any of the configura-tions tested.Table 6.3: Isotropic material and interface properties for the simply-supported beam benchmark problem.Property Value UnitDensity (ρ) 1.543× 10−3 g/mm3Elastic PropertiesElastic modulus (Exx = Eyy = Ezz) 150 GPaPoisson ratios (νxy = νyz = νxz) 0.32 (-)Interlaminar propertiesMode I critical energy release rate (GIc) 0.8 kJ/m2Mode II critical energy release rate (GIIc) 2.0 kJ/m2Interlaminar normal strength (σmax) 80 MPaInterlaminar shear strength (τmax) 150 MPa6.3.2 Static Plate LoadingIn order to investigate the effect of the LCZ method on the structural stiff-ness of a plate under a bending load, a finite element model, similar to theplate-impact model described and solved in Section 5.3, was quasi stati-cally loaded by prescribing a 1 mm displacement to the spherical impactor.The value of all material and cohesive parameters, remained identical to thevalues used in the plate-impact simulations described in Section 5.3. Themodel was solved using the LCZ method, as well as the conventional CZM.1406.3. Effect of CZM on the Structural Stiffness(a)(b)(c)Figure 6.7: Isometric view of simply supported beam, modeled using threedifferent configurations. Cohesive interfaces are shaded in brown color: a).No cohesive interface is present b). 5 through-thickness cohesive interfaces,distributed along all potential delamination crack-paths c). 4 cohesive in-terfaces d). Local cohesive interface model, concisting of two 15 mm longcohesive interfaces located at both ends of the beam.Several models were tested, with increasing number of through-thickness co-hesive interfaces. For comparative purposes, additional models containingno cohesive interfaces were solved as well. These models contained increas-ing number of through-thickness thick-shell elements, without any cohesiveinterfaces present.The load at 1mm displacement for each of the models, as a function of thenumber of through-thickness cohesive interfaces, as well as a function of thenumber of through-thickness thick-shell elements, is shown in Figure 6.9. It1416.3. Effect of CZM on the Structural StiffnessNocohesiveinterfaceLocalcohesiveinterfaceConventionalCZM10121416Loadatmaximumdisplacement[kN]2 through-thicknesselements6 through-thicknesselements18 through-thicknesselementsFigure 6.8: Load at maximum displacement for a simply-supported beamunder central bending load, as a function of the cohesive topology and num-ber of through-thickness elements. Dashed line represents the analyticalsolution.can be seen that the stiffness reduction using the LCZ method is negligible,where an noticeable stiffness reduction is exhibited by the conventional CZmethod’s results for the range of models tested.1426.3. Effect of CZM on the Structural Stiffness3 4 5 6 7 8 94006008001,0001,2001,4001,600Number of Through-Thickness Cohesive Interfaces orNumber of Through-Thickness Thick-Shell ElementsLoadatMaximumDisplacement[mm] LCZ methodConventional CZNo Cohesive InterfaceFigure 6.9: Load at maximum displacement for the plate bending exam-ple, as a function of through-thickness discretization and numerical solutionmethod.1436.3. Effect of CZM on the Structural Stiffness6.3.3 Dynamic Plate ImpactIn order to compare the effect of the LCZ method to cohesive interfacesdefined using a cohesive contact algorithm, the dynamic plate-impact simu-lations described in Section 5.3, for the case of a 6.33 kg impactor impact-ing the plate at 4.29 m/s, were repeated using LS-DYNA’s TIEBREAKcontact algorithm (*CONTACT AUTOMATIC ONE WAY SURFACE TOSURFACE TIEBREAK), where the contact interface was defined along allpotential delamination crack paths (similar to the conventional CZM).The cohesive parameters defined in the cohesive contact algorithm weresimilar to the values listed in Table 5.6. Several models were tested, withincreasing number of through-thickness cohesive interfaces. All other modelparameters, as well as boundary conditions, remained similar to the param-eters described in Section 5.3.Figure 6.10.a shows the impact force vs. displacement when the tiebreakcohesive interface is used to model the cohesive interface, and Figure 6.10.bshows the results obtained using the LCZ method. When a tiebreak contactis globally applied to all potential delamination crack propagation cracks,the structural stiffness of the plate decreases with increasing number ofthrough-thickness cohesive interfaces. This effect is not noticeable when theLCZ method is used, and the cohesive interfaces are locally seeded withinthe model.It can be concluded, that lower stiffness reduction was obtained when ap-plying the LCZ method for the solution of the cases investigated, comparedto the stiffness reduction caused by the application of the conventional CZMto the same problem. The lower effect of the LCZ method on the structuralstiffness is in-line with the motivation for the development of the method,and it expected that this advantage will be more pronounced when the al-gorithm will be an integrated part of a finite-element solver, allowing themethod to be applied to more complex structures.1446.3. Effect of CZM on the Structural Stiffness0 2 4 6 8 10Displacement [mm]05101520Force [KN]1 cohesive interface3 cohesive interfaces5 cohesive interfacesExperiment (Delfosse et al, 1995)(a)0 2 4 6 8 10Displacement [mm]05101520Force  [KN]1 Cohesive interface3 Cohesive interfaces5 Cohesive interfacesExperiment (Delfosse et al, 1995)(b)Figure 6.10: Predicted and experimental impact force vs. plate displacement, fora 6.33 kg impactor, impacting the plate at 4.29 m/s. a.) Results obtained usingthe LCZ method b.). Results obtained using the conventional CZ method. Inboth figures, results are shown for different number of through-thickness cohesiveinterfaces.145Chapter 7Summary, Conclusions andFuture Work7.1 SummaryComputational modelling of delamination in laminated composite struc-tures is challenging, due to the interaction of this damage mechanism withthe other complex damage mechanisms in the material. Several numericalmethods intended to simulate delamination in composites do exist, but theyare still limited to the solution of relatively small-sized structures. Due tothe relatively high numerical cost of the available methods, applying thesemethods to the solution of larger models is often not practical. Amongstthe various numerical methods aimed at simulating delamination in compos-ites, CZM is gaining increased popularity amongst scientists and engineersalike, due to its reliability and relatively simple numerical implementationin existing commercial finite element codes. Pioneering researchers aroundthe world have suggested various approaches to allow using CZM in largermodels, ranging from automatic scaling of the cohesive stiffness, in order toreduce the effect of the method on the stiffness of the structure, to adaptiveapproaches where cohesive interfaces are locally introduced into the finiteelement mesh, in order to reduce the computational cost. A novel approach,aimed at simulating delamination in composites using an adaptive manner,is presented here.Chapter 2 presents a brief overview of the available numerical techniques,aimed at simulating delamination in composites, together with their benefitsand limitations.A novel technique aimed at simulating delamination in laminated com-posites in an adaptive manner, is presented in Chapter 3. The methodallows modelling the structure without a priori knowledge or definition ofthe delamination location in the analysis, i.e. delaminations initiate andevolve as the simulation progresses. Using this method, no cohesive ele-ments nor initial cracks need to be introduced in the finite element mesh1467.1. Summaryprior to the analysis. The continuum elements are split through their thick-ness and potential paths for delamination growth are seeded into the modeladaptively.In Chapter 4, the method is verified against the solution of pure delami-nation crack propagation, under mode-I, mode-II, and mixed-mode loadingconditions. The method was shown capable of predicting the delaminationcrack in these cases with good to reasonable agreement with the analyti-cal data, as well as with respect to results obtained using other numericalmethods.In Chapter 5, the method is validated against engineering applications,involving impact and dynamic loading scenarios. First, the method is ap-plied for the static loading of a [90/0]s glass/epoxy double-notched tensilecoupon (Chapter 5.1). The LCZ algorithm was able to predict the maximumload bearing capacity of the coupon, together with the deflection at completerupture. Reasonable agreement was obtained when comparing the damagepredicted in the coupon using the LCZ algorithm with the experimentalresults. In Section 5.2, the method is applied for a loading case involvingdynamic tube crushing. Good agreement was obtained between the impactforce profile predicted by the LCZ method, compared to an impact force pro-file obtained experimentally. When solving a dynamic plate-impact problem(Section 5.3), the LCZ method was successful in predicting the force vs. im-pactor’s displacement profiles, as well as predicting the impactor’s energyloss. The overall delamination patterns predicted using the LCZ methodcovered a larger area compared to the experimental observation, althoughthe overall shape of the predicted delamination area agreed well with theexperimental findings. Similar behavior was observed when the conventionalCZM was applied to the problem, suggesting that the homogenization of thematerial and the lower number of cohesive interfaces in the finite elementmodel had a negative effect on the predictive capability of this quantity.In Chapter 6, the LCZ method was applied to larger numerical models.The LS-DYNA run-time when using the LCZ method, was shorter comparedto the run-time when the conventional CZ method is applied to the problem.The efficiency of the LCZ method over conventional CZ method improveswith increasing model size, which is an encouraging finding (A factor of 5.4was measured in favour of the LCZ method, for the model size tested). Inorder to investigate the effect of the LCZ method on the structural stiffnessof the structure, quasi-static bending loads were applied to a simple plategeometry, when the LCZ was applied to the problem. It was found that theresults obtained using the LCZ method, were closer to the results obtainedusing a model containing no cohesive interface, compared to the results1477.2. Conclusionsobtained using the conventional CZM.7.2 ConclusionsA new and robust computational method suited for efficient simulation ofprogression of delamination in laminated composite structures has been pre-sented here. The following highlights the salient features and benefits of thisnewly developed local cohesive zone (LCZ) methodology:• The method allows modelling the structure without a priori knowl-edge or definition of the delamination location in the analysis, i.e.delaminations initiate and evolve as the simulation progresses. Us-ing this method, no cohesive elements nor initial cracks need to beintroduced in the finite element mesh prior to the analysis. The con-tinuum elements are split through their thickness and potential pathsfor delamination growth are seeded into the model adaptively.• The method has only a minor effect on the overall structural stiff-ness before the onset of delamination, as the cohesive zone is locallyembedded in the structure only where and when needed.• The method uses a narrow band of cohesive elements that is suffi-cient to capture the mechanical behavior of the fracture process zonerequired for predicting delamination crack propagation.• The method has the potential to be combined with other in-planedamage theories (e.g. continuum damage models developed at theUBC Composites Group ([89], [85], [37]), where intra-laminar dam-age is treated in a smeared manner, and the element stiffness is re-duced gradually as a function of damage evolution within the finite el-ement volume. Such methodology combines the numerical advantagesof smeared modelling techniques with the need to model delaminationin a discrete manner. The interaction of delamination damage withother damage mechanisms in composite materials is the subject of ourongoing research and further development of the algorithm.In this paper, the LCZ method has been verified for Mode-I, Mode-II, andMixed-Mode loading conditions. The obtained force-displacement results, aswell as the overall energy balance, are shown to be in good agreement withthe results predicted using other numerical and analytical methods avail-able in the literature. Owing to the fact that the algorithm is currently1487.3. Future Worknot an internal part of the finite element solver (LS-DYNA) used in thisstudy, the various numerical operations performed by the algorithm on thefinite element mesh lead to some numerical noise. The limited element for-mulations that are currently available in LS-DYNA pose some challenges inachieving full compatibility between the offset-shell elements (in the splitregion) and the thick-shell elements (in the unsplit region) of the mesh. Inthe current LCZ method this difficulty is overcome by introducing narrowregions of overlapping shell elements in the transition region. Nevertheless,the algorithm is able to capture the delamination crack propagation cor-rectly. It is expected that further improvements in speed and accuracy ofthe computations will be attained once the algorithm is embedded withinthe finite element solver, and a layered thick-shell formulation with rota-tional nodal degrees of freedom is implemented in LS-DYNA. This wouldallow a smoother connectivity between the split and neighbouring unsplitregions of the mesh.The initial results obtained from the application of the LCZ methodto the various loading cases are encouraging, and prove that the local andadaptive insertion of cohesive zones into a finite element mesh can effectivelycapture the delamination crack propagation in laminated composite struc-tures. Ongoing research is being carried out to verify the implementationand application of the method to more complex loading cases, involving acombination of in-plane damage together with delamination crack propaga-tion, as well as simultaneous, multiple through-thickness delamination crackgrowth. Numerical issues such as the scalability of the current methodol-ogy and its computational efficiency relative to the conventional CZM is thesubject of ongoing investigation.7.3 Future WorkIn order to allow the method to be applied to larger industrial applicationsand to improve its predictive capability, the following procedures should betaken:• Compatibility with higher DOF multilayered elements: Thecurrent version of LS-DYNA does not support multilayered thick-shellelements with rotational degrees of freedom. This requires an artificial,non-physical solution to be enforced on the finite element model in theform of overlapping elements, in order to perform the element splittingoperation. This overlapping, described in Section 3.3.5, is required in1497.3. Future Workorder to transfer moment between the thick-shell elements and theoffset shell elements created during the element splitting process.• Integrating the LCZ algorithm within LS-DYNA: In its cur-rent form, the LCZ algorithm is an external Python code, written inPython, not embedded into LS-DYNA. Its execution is based on read-ing large volume of numerical results from the LS-DYNA simulations,a process which slows its execution and will be eliminated while thealgorithm will be embedded into LS-DYNA.• Introducing strain-rate dependencies to the cohesive model and theLCZ method in order to take account of the possible variation of thecohesive properties as a function of the interface loading rate.150Bibliography[1] Benchmark assessment of automated delamination propagation capa-bilities in finite element codes for static loading. Finite Elements inAnalysis and Design, 54:28 – 36, 2012.[2] LS-DYNA Aerospace Working Group Modeling Guidelines Document.Technical report, 08 2014.[3] E. 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PhD thesis, The Universityof British Columbia, 2010.165Appendix AFlowchart of the LCZAlgorithmThis Appendix presents a schematic flow-chart description of the LCZ al-gorithm, which is shown in Figure A.1, Figure A.2, Figure A.3 and FigureA.4.166AppendixA.FlowchartoftheLCZAlgorithm•  Read	  LCZ	  algorithm-­‐related	  parameters	  and	  control	  variables.	  •  Read	  finite-­‐element	  input	  file,	  search	  for	  composite	  part	  defini=ons	  and	  material	  orienta=ons.	  •  Read	  user-­‐defined	  element	  spli>ng	  criterion	  Ini=alize/	  con=nue	  	  the	  finite-­‐element	  transient	  analysis.	  Normal	  termina=on	  occurs	  if	  termina=on	  =me	  is	  reached	  or	  element-­‐spli>ng	  criterion	  is	  met	  (next	  step)	  •  Create	  an	  empty	  set	  of	  elements	  A	  which	  will	  include	  all	  of	  the	  element	  numbers	  flagged	  for	  spli>ng.	  •  Search	  for	  the	  first	  element	  in	  the	  mesh	  that	  sa=sfies	  the	  user-­‐defined	  element	  spli>ng	  criterion.	  •  Add	  this	  element	  number	  to	  set	  A	  (Figure	  1).	  •  Search	  for	  more	  elements	  that	  were	  close	  to	  sa=sfying	  the	  user-­‐defined	  element	  spli>ng	  criterion	  iden=fied	  by	  a	  user-­‐defined	  threshold	  (Figure	  2)	  •  Add	  these	  element	  numbers	  to	  set	  A	  Schema=c	  flowchart	  of	  the	  LCZ	  algorithm	  Figure	  1:	  First	  element	  to	  sa=sfy	  the	  element	  spli>ng	  criterion	  Figure	  2:	  Iden=fy	  elements	  close	  to	  sa=sfying	  the	  element	  spli>ng	  criterion	  by	  a	  user-­‐defined	  threshold	  	  Step	  A	  Proceed	  to	  step	  B	  Figure A.1: Schematic LCZ algorithm flowchart, image 1 out of 7167AppendixA.FlowchartoftheLCZAlgorithm•  Geometrical	  searches	  –	  for	  each	  element,	  i	  ,	  in	  set	  A,	  find	  all	  elements	  that	  belong	  to	  the	  same	  source	  ply	  as	  element	  i,	  and	  have	  their	  centroids	  lie	  within	  a	  user-­‐defined	  radius	  R	  from	  the	  centroid	  of	  element	  i.	  •  Add	  these	  element	  numbers	  to	  set	  A	  (Figures	  3a	  ,	  3b	  and	  3c)	  	  Figure	  3a:	  Perform	  ply-­‐based	  geometrical	  search	  	  Figure	  3b:	  IdenEfy	  elements	  	  that	  saEsfy	  the	  geometrical	  search	  Proceed	  to	  step	  C	  R	  Ply I Ply II Ply III Element	  I	  which	  belongs	  to	  set	  A	  An	  element	  whose	  centroid	  	  lies	  within	  a	  distance	  R	  from	  the	  centroid	  of	  element	  i	  and	  shares	  the	  same	  ply	  as	  element	  i.	  Elements	  which	  have	  their	  centroid	  	  lie	  within	  a	  distance	  R	  from	  the	  centroid	  of	  element	  i	  ,	  	  but	  DO	  NOT	  	  share	  the	  same	  ply	  as	  element	  i.	  Figure	  3c	  –	  A	  schemaEc	  cross	  secEon	  in	  a	  composite	  laminate	  consisEng	  of	  3	  plies	  -­‐	  only	  elements	  which	  saEsfy	  the	  radial	  search	  and	  share	  the	  same	  ply	  as	  element	  i	  are	  included	  in	  set	  A.	  Step	  B	  Figure A.2: Schematic LCZ algorithm flowchart, image 2 out of 7168AppendixA.FlowchartoftheLCZAlgorithm•  Spli%ng	  opera-on	  –	  for	  each	  element	  i	  in	  set	  A,	  create	  two	  elements	  through	  the	  thickness	  of	  the	  parent	  element	  i,	  	  using	  the	  nodes	  originally	  used	  for	  the	  defini-on	  of	  element	  i.	  •  The	  thickness	  of	  each	  newly	  created	  element	  is	  obtained	  based	  on	  the	  through-­‐thickness	  spli%ng	  loca-on.	  	  Do	  the	  elements	  in	  set	  A	  have	  rota-onal	  DOF	  as	  well	  as	  transla-onal	  DOF?	  Cohesive	  element/interface	  crea-on	  •  For	  each	  element	  i	  in	  set	  A,	  insert	  cohesive	  elements/interfaces	  connec-ng	  each	  pair	  of	  elements	  that	  were	  created	  during	  the	  spli%ng	  opera-on,	  	  unless	  the	  element	  lies	  on	  the	  edge	  of	  the	  split	  region,	  and	  is	  neighboring	  with	  other	  element(s)	  which	  are	  not	  to	  be	  split	  (are	  not	  in	  set	  A)	  (Figure	  4a	  and	  Figure	  4b)	  Figure	  4a:	  Cohesive	  elements/interfaces	  are	  NOT	  created	  on	  the	  marked	  elements	  belonging	  to	  set	  A,	  as	  they	  neighbor	  composite	  elements	  not	  belonging	  to	  set	  A	  Figure	  4b:	  Cohesive	  elements/interfaces	  are	  created	  on	  the	  marked	  elements	  which	  belong	  to	  element	  set	  A,	  as	  they	  do	  not	  neighbor	  with	  composite	  elements	  not	  belonging	  to	  set	  A	  Step	  C	  yes	  no	  Proceed	  to	  step	  D	  Proceed	  to	  step	  E	  yes	  Figure A.3: Schematic LCZ algorithm flowchart, image 3 out of 7169AppendixA.FlowchartoftheLCZAlgorithmCohesive	  element/interface	  crea2on	  •  For	  each	  element	  i	  in	  set	  A,	  insert	  cohesive	  elements/interfaces	  connec2ng	  each	  pair	  of	  elements	  that	  were	  created	  during	  the	  spli;ng	  opera2on.	  	  Cohesive	  element/interface	  crea2on	  -­‐	  con2nued	  	  •  The	  proper2es	  for	  each	  cohesive	  element/interface	  being	  created	  is	  derived	  from	  a	  list	  of	  user	  defined	  interface	  proper2es,	  which	  are	  defined	  at	  the	  beginning	  of	  the	  run.	  Mul2ple	  proper2es	  could	  be	  specified	  for	  mul2ple	  delamina2on	  crack	  propaga2on,	  such	  that	  the	  appropriate	  cohesive	  proper2es	  are	  assigned	  to	  specific	  delamina2on	  cracks.	  Proceed	  to	  step	  F	  step	  D	  step	  E	  Figure A.4: Schematic LCZ algorithm flowchart, image 4 out of 7170AppendixA.FlowchartoftheLCZAlgorithmDo	  the	  elements	  in	  set	  A	  have	  rota/onal	  DOF	  as	  well	  as	  transla/onal	  DOF?	  yes	  no	  Mapping	  •  For	  each	  parent	  element	  i	  in	  set	  A,	  map	  the	  stresses/strains	  and	  state	  variables	  onto	  the	  newly	  created	  elements.	  	  •  Mapping	  is	  performed	  by	  keeping	  consistency	  with	  the	  integra/on	  point	  orienta/on,	  such	  that	  the	  informa/on	  from	  each	  integra/on	  point	  belonging	  to	  the	  parent	  element	  is	  mapped	  to	  the	  appropriate	  integra/on	  point	  in	  the	  newly	  created	  element.	  	  Step	  F	  Proceed	  to	  step	  J	  Remove	  all	  elements	  in	  set	  A	  from	  the	  finite	  element	  mesh	  Proceed	  to	  step	  G	  Figure	  5:	  A	  schema/c	  cross	  sec/onal	  view	  demonstra/ng	  the	  mapping	  process.	  Mapping	  is	  performed	  from	  each	  integra/on	  point	  in	  the	  parent	  element	  to	  the	  appropriate	  integra/on	  point	  in	  the	  newly	  created	  elements.	  The	  virtual	  spliKng	  loca/on	  is	  marked	  with	  a	  dashed	  line.	  Figure A.5: Schematic LCZ algorithm flowchart, image 5 out of 7171AppendixA.FlowchartoftheLCZAlgorithmFor	  each	  element	  i	  in	  set	  A,	  check	  if	  it	  lies	  on	  the	  edge	  of	  the	  region	  defined	  by	  set	  A,	  AND	  is	  also	  neighboring	  with	  elements	  not	  belonging	  to	  set	  A	  (Figure	  6a	  and	  Figure	  6b)	  •  Con<nue	  performing	  this	  check	  for	  all	  elements	  in	  set	  A.	  	  yes	  no	  Figure	  6a:	  Elements	  belonging	  to	  set	  A	  which	  lie	  on	  the	  edge	  of	  the	  region	  defined	  by	  set	  A,	  BUT	  have	  all	  their	  neighboring	  elements	  belonging	  to	  set	  A.	  Figure	  6b:	  Elements	  belonging	  to	  set	  A	  which	  lie	  on	  the	  edge	  of	  the	  region	  defined	  by	  set	  A,	  AND	  have	  at	  least	  one	  neighbor	  that	  does	  not	  belong	  to	  set	  A.	  These	  elements	  labeled	  with	  green	  circles	  are	  added	  to	  set	  B.	  •  Add	  these	  elements	  to	  element	  set	  B	  •  Con<nue	  performing	  this	  check	  for	  all	  elements	  in	  set	  A	  Step	  G	  Proceed	  to	  step	  H	  Figure A.6: Schematic LCZ algorithm flowchart, image 6 out of 7172AppendixA.FlowchartoftheLCZAlgorithm•  Define	  contact	  between	  the	  surfaces	  of	  the	  newly	  created	  elements.	  •  Contact	  parameters	  (fric:on,	  etc.)	  are	  taken	  from	  the	  user-­‐defined	  parameters	  which	  were	  read	  during	  the	  ini:aliza:on	  of	  the	  run.	  Remove	  all	  elements	  in	  set	  A	  which	  DO	  NOT	  belong	  to	  set	  B	  from	  the	  finite	  element	  mesh	  (Figure	  7)	  Figure	  7:	  Remove	  all	  elements	  in	  set	  A	  which	  DO	  NOT	  belong	  to	  set	  B	  (marked	  here	  with	  blue	  circles)	  from	  the	  FE	  mesh	  Return	  to	  step	  A	  Step	  H	  Proceed	  to	  step	  J	  Step	  J	  Figure A.7: Schematic LCZ algorithm flowchart, image 7 out of 7173Appendix BExecution of the LCZAlgorithmThis appendix describes the procedure required in order to correctly set-upand perform a mechanical analysis using the LCZ algorithm. Currently, inits BETA version, the algorithm can be executed using a Linux operatingsystem having a Python installation.In order to execute the code, the following steps should be performed:1. Setting the working directories: A working directory is to be cre-ated, in which the execution will take place. Within this directory,two subdirectories should be created: build directory, containing allof the binary files of the algorithm. The user must not alter the con-tent of this directory as it contains all of the binary files requiredfor the correct execution of the code. A second directory, titled dist,serves as the directory which the actual algorithm execution will beperformed. The dist directory should include the following files: theLCZ algorithm executable, the parameters.txt text file described bel-low, containing all of the required user-defined parameters controllingthe execution of the LCZ algorithm, and the LS-DYNA keyword file/sof the mechanical problem to be solved during the analysis.2. The parameters.txt file: This file which is placed right beside theLCZ executable, allows the user to control various aspects of the run.A typical file is brought in Program B.1. The file format should notchange from the format brought here. The following is a descriptionof the different fields within the file.• runfile : Name of LS-DYNA’s keyword file describing the me-chanical problem under investigation, which will be solved usingthe LCZ algorithm, in this example, FILENAME.• instant_split : A flag to perform an automatic split and insertionof cohesive elements through all of the interfaces defined in the174Appendix B. Execution of the LCZ Algorithmdist!Parameters.txt!LCZ_executable.py!Ls-Dyna_keyword.key!build!Running	  directory	  	  Figure B.1: Schematic directory architecture required for the correct execu-tion of the LCZ algorithm.model, without performing an initial LS-DYNA run. Optionsfor this parameter are either "true" or "false", in this example,it is set to false. Setting this parameter to "true", will causethe LCZ algorithm to perform a single LS-DYNA run, in whichall thick-shell elements will be split and converted into offset-shellelements with solid-cohesive elements embedded in between theseoffset-shells. Such a model is equivalent to solving the problemusing the conventional CZ method. Thus, this parameter allowsa useful mean of comparing the results obtained using the LCZmethod to the conventional CZ method’s results. Setting thisparameter to "false" will perform a standard LCZ simulation inwhich solid cohesive elements will be adaptively seeded into thestructure.• interface_springback_NSHV : Number of history variables to bemapped during the mapping process, for each integration point,during the element splitting operation. In this example, this num-ber is set equal to 5. This value is material-model dependent, aseach of LS-DYNA’s material model has a different number of his-175Appendix B. Execution of the LCZ Algorithmtory variables. More information can be found in LS-DYNA’skeyword manual [18].• radial_split_distance : Distance in length units used by theradials-search algorithm during the element splitting process. Inthis example, this number is set equal to 8.• exe : Path to LS-DYNA’s executable which will be used for thesolution of the mechanical analysis. In this example, this path isdefined as /home/username/./lsdyna_exe.• termination_factor : Currently, the LCZ algorithm is not em-bedded into LS-DYNA, and is based on reading LS-DYNA’s ELOUTASCII files for obtaining the thick-shell elements outputs. Read-ing the ASCII files are performed only once the LS-DYNA sim-ulation has terminated. In order to shorten the run-time, thesimulation is performed in multiple time-segments, where eachtime-segment is a fraction of the final run-time specified in LS-DYNA’s *CONTROL TERMINATION card. The number oftime-segments is determined by the value of the terminationfactor parameter. If, for example, the termination time spec-ified in LS-DYNA’s *CONTROL TERMINATION card is 15,and the value of the termination factor is 10, the LCZ algorithmwill execute the first LS-DYNA simulation from time 0 to a timeof 1.5 (15/10). The code will then search for the first thick-shellelement to satisfy the element splitting criteria during this run. Ifno element satisfied this criteria, the simulation will be resumedfrom a time of 1.5 to a time of 3, and so on, until the final ter-mination time of 15 is reached.• parts_to_skip : Part numbers which are not composite partsand are not taking part during the splitting operation, such asrigid parts, parts made of solid elements, etc. These parts willbe ignored by the LCZ algorithm during its execution. In thisexamples, these are part numbers 2, 3 and 4.• direc : Type of element-splitting criterion. The following optionsare available:5 - τzx values only, either negative or positive.S =√(τzxτmax)2(B.1)176Appendix B. Execution of the LCZ Algorithm6 - mixed-mode, taking into account either positive or negativevalues of τzx and σz.S =√(σzσmax)2+(τzxτmax)2(B.2)7 - mixed-mode, taking into account either positive or negativevalues of τzx, and positive values of σz. Negative values of σz(compressive normal stress) are neglected.S =√(σzσmax)2+(τzxτmax)2, if σz ≥ 0√(τzxτmax)2, otherwise(B.3)• sig_zz_max : Maximum normal stress values (σmax) used in theelement splitting criterion, in this example, 80.• tau_zx_max : Maximum shear stress values (τmax) used in theelement splitting criterion, in this example, 80.• threshhold_initial : Critical value for the element splitting cri-teria, Sc. Once an LS-DYNA analysis is performed, the LCZ al-gorithm searches for the first thick-shell element to have a valueof S which is equal or greater than Sc. In this example, thisthreshold is set equal to 0.4.• neighbour_trheshhold_scaling_factor : A neighbour thresholdvalue for the element splitting criteria. Once the first thick-shellelement reaches Sc, the code will search for more thick shell ele-ments which satisfy:S ≥ threshold× Sc (B.4)where threshold is the neighbour_trheshhold_scaling_factordefined above, having a value which is higher than 0 and lowerthan 1, in this example, 0.95.• initial_shock_time : An option to allow the code to "ignore"critical element splitting criterion values which are satisfied ear-lier than the time specified using this parameter. In many load-ing scenarios using an explicit time integration scheme, the ini-tial loading of the structure causes an unrealistic stress wave totravel through the structure, which will cause the element split-ting criterion to be satisfied, leading to premature splitting of177Appendix B. Execution of the LCZ Algorithmthick-shell elements. Such an example could be during an initial-ization phase where the structure is subjected to a gravitationalload. This parameter supplies a mean to overcome this limita-tion by ignoring element-splitting criterion values at early stagesof the analysis. In this example, this parameter is set to 0.04,thus element splitting values satisfied from time 0 to 0.04 will beignored.• shock_time : Similar to the above option, this parameter sup-plies a mean to ignore critical element-splitting criterion valueswhich are satisfied right after a restart or an element-splittingstep. In this example, assuming that an element splitting stepoccurs at a time of 3, the algorithm will search for element sat-isfying the element splitting criteria at a time of 3+0.04. Thisoption allows to mitigate the numerical noise introduced into theanalysis following a restart.• global_damping_value : A global damping value which will beused in the analysis, using LS-DYNA’s *DAMPING_GLOBALcard. The *DAMPING_GLOBAL card will be automaticallycreated by the code and does not need to be added to the LS-DYNA keyword used in the analysis. In this example, the globaldamping factor is set equal to 0.• contact_type : Contact type to be created between the offset-shell elements generated using the element splitting process. Avail-able options are: ’single : A single AUTOMATIC_SINGLE_SURFACE contactalgorithm between all shell parts generated during the splittingprocess will be created.single_sep : A separate AUTOMATIC_SINGLE_SURFACEcontact algorithm between each pair of offset-shell element partsgenerated during the splitting process will be created.s2s : A separate AUTOMATIC_SURFACE_TO_SURFACE con-tact between each pair of offset-shell element parts generated dur-ing the splitting process will be created.• extra_parts_to_single_contact : This parameter allows speci-fying more parts that will be added to the single surface contact,in case a single surface contact is used between the new shells.This is useful, for example, in case non-composite parts may in-teract with the offset-shells generated during the execution of the178Appendix B. Execution of the LCZ AlgorithmLCZ algorithm, such as the rigid sphere in a plate impact loadingscenario. In the above example, the sphere part number is 200.Any number of parts can be listed as needed in a list format, suchas [’121’,’123’,”300’,...] if no parts are to be specified, an emptyset should be used [”].• fs : Static coefficient of friction between the offset-shell elementsgenerated by the LCZ algorithm generated during the run, in thisexample, 0.2.• fd : Dynamic coefficient of friction between the offset-shell ele-ments generated by the LCZ algorithm during the run, in thisexample, 0.2.• sfs : Scale factor on the default LS-DYNA slave penalty stiffness,in this example, 1. More information can be found under the*CONTACT keyword description in LS-DYNA’s user’s manual[18].• sfm : Scale factor on the default LS-DYNA master penalty stiff-ness, in this example, 1. More information can be found underthe *CONTACT keyword description in LS-DYNA’s user’s man-ual [18].• dc : Exponential decay coefficient used in the contact betweenthe offset-shell elements generated by the LCZ algorithm duringthe run, in this example, 3.0. More information can be foundunder the *CONTACT keyword description in LS-DYNA’s user’smanual [18].• vc : Coefficient of viscous friction used in the contact betweenthe offset-shell elements generated by the LCZ algorithm duringthe run, in this example, 0.0. More information can be foundunder the *CONTACT keyword description in LS-DYNA’s user’smanual [18].• vdc : Viscous damping coefficient in percent of critical, used inthe contact between the offset-shell elements generated by theLCZ algorithm during the run, in this example, 0.0. More infor-mation can be found under the *CONTACT keyword descriptionin LS-DYNA’s user’s manual [18].• soft : Soft constraint option used in the contact between theoffset-shell elements generated by the LCZ algorithm during therun, in this example, 2. More information can be found under the179Appendix B. Execution of the LCZ Algorithm*CONTACT keyword description in LS-DYNA’s user’s manual[18].• add_Ö£control_shell_card : This parameter can be set to ei-ther true or false. Setting this parameter to "true", will cause theLCZ algorithm to add an LS-DYNA’s *CONTROL_SHELL cardto the main keyword file, with the CNTCO parameter within thiscard set to 1, thus making LS-DYNA consider the shell offset andthickness in the contact algorithms. It was found, however, thatin some cases, this can cause bugs during the restarts performedby the code, and was thus left as an option for the user to eitherset it on or off.• mapping_flag : This parameter allows performing the LCZ anal-ysis without mapping, by setting its value to false. A value oftrue for this variable will cause the LCZ algorithm to performthe mapping process during the element splitting operation.• ply_list : A list containing all thick-shell parts numbers to besplit during the execution of the LCZ algorithm. The parts shouldbe listed in their through thickness order. In this example, part1 is followed by part 5, then part 6 and finally part 7.• cohesive_mat_deff : A list of cohesive material definitions forthe cohesive interfaces generated by the LCZ algorithm. By usinga list structure for this parameter, different cohesive propertiescan be defined for each cohesive interface in the model. The num-ber of cohesive materials defined should be equal to the number ofparts in the ply_list parameter defined above. This list consistsof several sub-lists, where each sub-list is a definition of a singlecohesive material. In this example, the first material definition isgiven using the list [’MID=21’,’G1c=0.8’,’G2c=2’,’T_normal=80’,’S_shear=150’,’intfall=4’], the second cohesive material is de-fined using the list [’MID=22’,’G1c=0.8’,’G2c=2’,’T_normal=80’,’S_shear=150’,’intfall=4’], and so on. Each sub list containsthe cohesive material number (MID), the critical energy releaserate under mode-I loading condition (G1c), the critical energyrelease rate under mode-II loading condition (G2c), the maxi-mum normal stress in the cohesive interface (T_normal), andthe maximum shear stress in the cohesive interface (T_shear).The order of the material input should be equal to the order ofthick-shell parts defined in the ply_list parameter above, suchthat the first cohesive material defined in the list will relate to180Appendix B. Execution of the LCZ Algorithmthe interface of the first thick-shell part defined in the ply_listparameter, and so on. The cohesive material definitions shouldbe enclosed in square brackets, and separated by commas, asin the following format: cohesive_mat_deff=[[first_material],[second_material],[third_material], .... ].• split_location_list : A list containing the relative through-thicknesssplitting location, for each thick-shell part defined in the ply_listparameter. In this example, each thick-shell part will be split inits mid-plane location, as the values are set to 0.5. Valid valuesshould be grater than 0 and lower than 1.• ply_thickness_list: A list containing the thickness of each thick-shell part defined in the ply_list parameter. In this example,each thick-shell part has a thickness of 1.1625.3. General guidelines for setting the LCZ analysis:(a) It is advised to run the LS-DYNA keyword file for several timesteps prior to the execution of the LCZ algorithm, in order toverify that the keyword is set up correctly and does not containany errors.(b) The composite thick-shell parts should be defined using LS-DYNA’s*PART_COMPOSITE card, and the number of through-thicknessintegration points in the *PART_COMPOSITE keyword de-scribing the composite thick-shell parts should be sufficient toallow correct splitting using the LCZ algorithm, and should beequal in all composite components taking part in the analysis.(c) All nodes and thick-shell elements defined within the initial LS-DYNA keyword file have to be numbered such that the lowest IDof the first node and thick-shell element will be 1. The numberingof the nodes and thick-shell elements should be continuous in anincreasing order, and should not contain any number-jumps.(d) Composite part should e defined prior to the non-composite andrigid parts in the LS-DYNA keyword file.(e) No thick-shell history output should be requested in the LS-DYNA keyword file, as this will cause errors during the executionof the LCZ algorithm. The required history database files will beautomatically created by the code during the run.(f) In order to allow a correct mapping process, an LS-DYNA181Appendix B. Execution of the LCZ AlgorithmComputer Program B.1 Typical content of a parameters.txt file, usedto control the execution of the LCZ algorithm.runfile=1_plate_impact_4_layers.kinstant_split=falseinterface_springback_NSHV=5radial_split_distance=8exe=/home/lsdyna/executables/./lsdyna_smp_971termination_factor=24parts_to_skip=2,3,4direc=7sig_zz_max=80sig_zx_max=150threshhold_initial=0.4neighbour_trheshhold_scaling_factor=0.95initial_shock_time=0.04shock_time=0.125global_damping_value=0contact_type=single_sepextra_parts_to_single_contact=[’’]fs=0.2fd=0.2sfs=1sfm=1dc=3.0vc=3.0vdc=80soft=2add_control_shell_card=truemapping_flag=trueply_list=[[’1’],[’5’],[’6’],[’7’]]cohesive_mat_deff=[[’MID=21’,’G1c=0.8’,’G2c=2’,’T_normal=80’,’S_shear=150’,’intfall=4’],[’MID=22’,’G1c=0.8’,’G2c=2’,’T_normal=80’,’S_shear=150’,’intfall=4’],[’MID=23’,’G1c=0.8’,’G2c=2’,’T_normal=80’,’S_shear=150’,’intfall=4’],[’MID=24’,’G1c=0.8’,’G2c=2’,’T_normal=80’,’S_shear=150’,’intfall=4’]]split_location_list=[’0.5’,’0.5’,’0.5’,’0.5’]ply_thickness_list=[’1.1625’,’1.1625’,’1.1625’,’1.1625’]182Appendix B. Execution of the LCZ Algorithm*DATABASE_EXTENT_BINARY card should be specified withinthe original LS-DYNA keyword, with the INTOUT parameter setto ALL. This will cause LS-DYNA to write both stress and straindata into the the eloutdet file, which is necessary for the mappingprocess to be performed correctly.Ö£(g) An LS-DYNA *SECTION_TSHELL card is to be defined in theoriginal keyword, even if only *PART_COMPOSITE_TSHELLcards are used and no *SECTION_TSHELL card is required forthe analysis. The card does not need to be referred by any of theother parts in the model.4. Performing the numerical simulation using the LCZ algorithmOnce the parameters.txt file was modified and placed in the dist run-ning directory next to the LS-DYNA keyword and LCZ executable,the following command should be executed in order to initialize theLCZ analysis:» python LCZ_executable.pyOnce initialized, the LCZ algorithm will read the parameters text file,and a log file (logfile.txt) will be created, in which important opera-tions performed by the code will be documented during the executionof the LCZ code. A subdirectory titled Ö£0 will be created, in whichthe first LS-DYNA simulation will be performed. The subsequent LS-DYNA simulations will be performed in adjacent directories numberedincreasingly, as can be seen in Figure B.2.In addition to executing the LCZ algorithm locally on a Linux shell,the job can be batch processed across a high performance computingcluster, by submitting the job to the queuing system. A typical pro-cedure to batch process the LCZ analysis would be to create a scriptwhich will be submitted to the queuing system. In this exmaple, thescript name would be 1_dynascr, and its content is brought in Pro-gram B.2.In order to send the job to be executed across the cluset, one shouldchange to the working directory using the command:» cd PATH_TO_WORKING_DIRECTORYand then send the job to the queuing system using the following com-mand:183Appendix B. Execution of the LCZ AlgorithmComputer Program B.2 Content of the 1_dynascr text file used to batch-process the LCZ execution.#!/bin/sh#PBS -N my_run_name#PBS -l select=1:ncpus=12cd PATH_TO_WORKING_DIRECTORY/dist./LCZ_executable.py» qsub 1_dynascr5. Reading the results following the execution of the LCZ algorithmEach subdirectory created by the LCZ algorithm (shown in FigureB.2), contains the LS-DYNA result files from the simulation performedwithin this directory. These results include the binary three dimen-sional result files (d3plot), as well as results in ASCII format, suchas LS-DYNA’s rcforc and nodout files, which contain time historiesof the contact forces and nodal information requested during the run.It is important to note that these files will be created only if this wasspecifically requested during the LS-DYNA simulation by the user,using the appropriate commands in the original LS-DYNA keywordfile.Generating continuous time history plots, such as the ones presentedin these thesis, from the results obtained following the execution of theLCZ algorithm, is a tedious task, as it requires manually "stitching"all of the results from all running directories under the main runningdirectory. This task can be much simplified using automated scripts,which are able to find the required results in each working directoryand stitch all of the results together, generating meaningful and con-tinuous plots. Such scripts are problem dependent and are thus notpresented here, but were used to generate the various plots presentedin this theses.184Appendix B. Execution of the LCZ Algorithmdist!logfile.txt!Running	  directory	  	  1!2!3!…	  build!Figure B.2: Schematic directory architecture following execution of the LCZalgorithm.185Appendix CGeneral Description ofComposite Tube CrushingProcessThe composite tubes analyzed in Chapter 5.2 intended to be used as energyabsorbing structures, more specifically, to help reduce the deceleration loadstransmitted to the passengers during a car accident, and thus lower the riskor severity of an injury.Initial design of these energy-absorbing members consisted of metallictubes of either a circular or a rectangular cross sections, which were under-going plastic deformation and progressive plastic folding during an impactevent [127]. In recent years, however, environmental regulations are forcingcar manufacturers to design cars with improved fuel efficiency, constructedof lighter materials. Composites pose a great potential as a constructive ma-terial due to their excellent specific strength properties. Thus, an effort ismade to design energy absorbing components using light-weight compositematerials.While there is a considerable amount of published data on the response ofmetallic tubes to dynamic, axial crushing, and the response can be predictedwith reasonable accuracy, predicting the crushing response of compositetubes is far more difficult, for a number of reasons. Most composites aremade from brittle fibres embedded in a polymer matrix, which may be brittleor ductile depending on the choice of polymer. This means that extensiveplastic deformation cannot occur, and collapse by progressive plastic foldingof the type observed in metal and plastic tubes is impossible. In addition,the properties of composite materials are strongly dependent on the fibrearrangement, the fibre volume fraction and the properties of the fibre-matrixinterface. Thus, simulating the behavior of these structures requires theability to correctly describe the damage growth mechanisms in the compositematerial, which is driven by the microstructure of the material.The global response of composite-tubes undergoing dynamic crushing186Appendix C. General Description of Composite Tube Crushing Processcan be categorized as belonging to one of three categories: Stable progres-sive folding (Fig C.1.a), Stable progressive crushing (Fig C.1.b), or an un-stable folding / crushing (Fig C.1.c). The stable progressive folding mode,resembles the progressive folding mode observed in ductile metal and plastictubes, and occurs mainly in crushing of thin walled tubes made of relativelyductile composite materials, and is less common in tubes made of brittlefibers or having a thicker wall thickness ([125] , [31]). Stable, progressivecrushing mode, involves the formation of a zone of micro-fracture at one endof the tube which then propagates along the tube at the same speed as thecrushing front.In most engineering applications, stable behavior is desired, as this leadsto smoother deceleration profiles, as well as relatively confined crushed ge-ometry. The stability of the crushing process is governed by the tube’sgeometrical dimensions, as well as ply layup, the loading symmetry, and thetype of boundary conditions which are present at the end of the tube. In or-der to ensure a stable crushing process, it is usually desired for the crushingto be initialized at a specific location rather than at a random location alongthe tube. This location is most often chosen to be at the tube end, and inthis case fracture initialization can be achieved by chamfering the tube end,thus weakening and reducing the cross-section, or by using external surfacesthat initiates fracture and allow the tube material to flow in a stable manner[58],[32].(a) (b) (c)Figure C.1: Schematic crushing morphologies obtained during a compositetube-crushing event, with an initial tube axis of symmetry shown in a dashedline. a). Progressive folding b). Progressive crushing. (c) Unstable foldingand crushing.A schematic load vs. displacement of a stable crushing process is shownin Figure C.2. Here, the crushing load P is shown as a function of the187Appendix C. General Description of Composite Tube Crushing Processtube-end displacement, S. The load profile can be characterized by threedistinct regions: I - the initial loading state, II - progressive stable crushingstate, and III - post progressive crushing state. As the tube is loaded, theloading force increases until crushing of the tube is initialized, which resultsin a sharp load drop. The maximum load, just before the initialization ofthe crushing process, is denoted by Pmax. This load is usually limited bythe maximum compressive strength of the composite reinforcing material.Following the load-drop, is a plateau of an oscillating, but stable, forceprofile, with an average load value P¯ . The region of stable crushing isdenoted by II. SI denotes the displacement at which the stable crushingprocess begins. As the displacement increases, more energy is absorbedby the tube, until one of two conditions are met - if the load is removedbefore reaching the tube’s possible maximum crushing displacement, theload will eventually drop to zero, with a final displacement SB. On theother hand, if loading continues to be applied, at some displacement SBthe tube will reach a solid state which will result in a sudden increase ofthe loading force. If the tube is to be used as an energy absorber, it isusually preferable to design the tube such that this region (denoted by III)will be outside the performance envelope of the tube, as reaching this statewill lead to large deceleration values. SB is therefore considered to be themaximum displacement of the tube, that is still within the stable crushingphase, wether the stable crushing phase was terminated due to the reductionof the load, or reaching a solid-state of the tube.188Appendix C. General Description of Composite Tube Crushing ProcessSPI II IIIP¯PmaxSbSiFigure C.2: A schematic load vs. displacement profile obtained during astable composite tube crushing process. The crushing load P is shown as afunction of the tube-end displacement, S. Three distinct regions are visible:I - the initial loading state, II - progressive stable crushing phase, and III- post progressive crushing phase. Pmax is the maximum load, P¯ denotesthe average load value during the stable crushing state, and Si denotes thedisplacement at which the stable behavior begins. At displacement Sb, thetube reaches a solid state which will result in a sudden increase of the loadingforce.The global response of the tube is influenced by failure mechanisms whichevolve at tube’s tip within the ply level, and have a direct effect on the globalfailure topology. Hull [58] and Farley [32] studied the failure morphologiesof composite-tube walls undergoing progressive, axial crushing. They cate-gorized the ply-level failure modes as to be belonging to one of the followingthree groups: Transverse Shear (Figure C.3.a), which involves transversematrix-crack growth and extensive failure of the plies with fibers in the trans-verse directions, lamina bending, or splaying (Figure C.3.b) or ply buckling(Figure C.3.c). In most crushing scenarios, however, multiple failure modesare present, as can be seen in Figure C.4. Here, a schematic cross-section of189Appendix C. General Description of Composite Tube Crushing Processa tube with a [0◦/90◦/0¯◦]s ply layup undergoes progressive crushing. Trans-verse shearing cracks develop in the 90◦ plies, where delamination developsbetween the 0◦ and 90◦ plies as bending and splaying develop at the 0◦ plies.In some cases, a debris wedge, a wedge-shaped volume of confined, crushedmaterial is formed between the tube and the crushing surface. Simulatingthe behavior of composite tubes undergoing crushing requires the ability tocorrectly describe the various damage growth mechanisms and their inter-action.(a) (b) (c)Figure C.3: Failure morphologies of composite-tube walls undergoing pro-gressive, axial crushing. a). Transverse Shear , b). Lamina bending, orsplaying, c). Ply buckling.190Appendix C. General Description of Composite Tube Crushing Process0◦ 90◦ 0◦ 90◦ 0◦1234Figure C.4: Typical failure morphology obtained in a composite tube duringa dynamic crushing process, shown here for a [0◦/90◦/0¯◦]s ply layup. Across section through the wall of the tube demonstrates the following failuremodes: 1). Transverse Shear in the 90◦ plies. 2). Lamina bending in the0◦ plies. 3). Splitting within the 0◦ ply and delamination between the 0◦and the 90◦ plies. 4). Debris Wedge, consisting of crushed fibres and resin,which is formed at the between the tube and the impacted plane.Work and Specific Energy during a tube-crushing processThe work performed during the crushing process (neglecting the post pro-gressive crushing phase) is [85]:Wf =∫ Sb0PdS (C.1)Where P is the crushing load, S is the crushing displacement, and Sb isthe displacement at which the tube reaches a solid state.The specific energy absorption, SEA, is defined as the energy absorbedor work done in forming a unit mass of crushed material [85]:191Appendix C. General Description of Composite Tube Crushing ProcessSEA = energy absorbedmass of damaged material =WfASbρ(C.2)where A is the cross-sectional area of the tube, and ρ is the tube’s density.192Appendix DLS-DYNA Material CardsThis appendix brings the LS-DYNA material cards used during the valida-tion process described in Chapter 5.193AppendixD.LS-DYNAMaterialCardsComputer Program D.1 LS-DYNA MAT_54 card used in the double-notched [90/0]s test coupon (Chapter5.1), for the 0◦ ply*MAT_ENHANCED_COMPOSITE_DAMAGE$# mid ro ea eb (ec) prba (prca) (prcb)6 1.9700E-3 43900.000 15400.000 15400.000 0.105239 0.105239 0.300000$# gab gbc gca (kf) aopt4340.0000 4340.0000 4340.0000 0.000 -3$# xp yp zp a1 a2 a3 mangle0.000 0.000 0.000 0.000 0.000 0.000 0.000$# v1 v2 v3 d1 d2 d3 dfailm dfails0.000 0.000 0.000 0.000 0.000 0.000 0.14 0.50$# tfail alph soft fbrt ycfac dfailt dfailc efs0.000 0.000 1.000000 0.000 2.000000 0.100 -1E24 0.50$# xc xt yc yt sc crit beta620.0000 1140.0000 128.00000 39.00000 80.000000 54.000000 0.000$# pel epsf epsr tsmd soft20.000 1E24 2E24 0.000 1.000000$# slimt1 slimc1 slimt2 slimc2 slims ncyred softg0.000 0.000 0.000 0.000 0.000 0.000 1.000000194AppendixD.LS-DYNAMaterialCardsComputer Program D.2 Material definitions used in the double-notch [90/0]s for the 90◦ ply*MAT_ENHANCED_COMPOSITE_DAMAGE$# mid ro ea eb (ec) prba (prca) (prcb)69 1.9700E-3 43900.000 7700.0000 7700.0000 5.2620E-2 5.2620E-2 0.300000$# gab gbc gca (kf) aopt4340.0000 4340.0000 4340.0000 0.000 -33$# xp yp zp a1 a2 a3 mangle0.000 0.000 0.000 0.000 0.000 0.000 0.000$# v1 v2 v3 d1 d2 d3 dfailm dfails0.000 0.000 0.000 0.000 0.000 0.000 0.14 0.50$# tfail alph soft fbrt ycfac dfailt dfailc efs0.000 0.000 1.000000 0.000 2.000000 0.100 -1E24 0.50$# xc xt yc yt sc crit beta620.00000 1140.0000 128.00000 39.000000 80.000 54.000000 0.0000$# pel epsf epsr tsmd soft20.000 1E24 2E24 0.000 1.000000$# slimt1 slimc1 slimt2 slimc2 slims ncyred softg0.000 0.000 0.000 0.000 0.000 0.000 1.000000195AppendixD.LS-DYNAMaterialCardsComputer Program D.3 Material definitions (CODAM1) used in the tube crush analysis (Defined using anLS-DYNA user-defined material card*MAT_USER_DEFINED_MATERIAL_MODELS$ MID RO MT LMC NHV IORTHO IBULK IG2 1.3E-3 45 32 32 1 29 30$ IVECT IFAIL1 1$ AOPT MAXC XP YP ZP A1 A2 A33.0$ V1 V2 V3 D1 D2 D3 BETA0.0 0.0 1.0 90$ EX VXY GXY NOZCF SRFLAG E1TMAX E2TMAX E3TMAX60000.0 0.0 9000.0 1$ EY VYZ GYZ EWSF E1CMAX E2CMAX E3CMAX12500.0 0.0 9000.0$ EZ VXZ GZX EWSN E4CMAX E5CMAX E6CMAX8000.0 0.0 9000.0$ ERODE EPSMAX DAMT BSFLAG BMOD GMOD DAMDT DEBUG2.0 2.0 1 20000.0 30000.0196Appendix D. LS-DYNA Material CardsComputer Program D.4 Strain-softening parameters used in the tubecrush analysis (This is the first part of the ssparam.dat file, which is calledby the CODAM1 material modelc Strain Softening Parameterscc X Parameterscc F-w-RE t30.000 0.000 1.0000.015 0.000 1.0000.030 1.000 0.000c F-w-RE c x normal40.000 0.000 1.0000.005 0.000 1.0000.008 0.500 0.0010.750 1.000 0.000c Y Parameters Normalcc F-w-RE t30.000 0.000 1.0000.009 0.000 1.0000.020 1.000 0.000c F-w-RE c30.000 0.000 1.0000.020 0.000 1.0000.040 1.000 0.000197Appendix D. LS-DYNA Material CardsComputer Program D.5 Strain-softening parameters used in the tubecrush analysis (This is the first part of the ssparam.dat file, which is calledby the CODAM1 material modelc Z Parameterscc F-w-RE t30.000 0.000 1.0000.009 0.000 1.0000.020 1.000 0.000c F-w-RE c30.000 0.000 1.0000.500 0.000 1.0001.000 1.000 0.000c XY Parameterscc w-RE30.000 1.0000.500 0.5001.000 0.000c YZ Parameterscc w-RE30.000 1.0000.500 0.5001.000 0.000c ZX Parameterscc w-RE30.000 1.0000.500 0.5001.000 0.000198Appendix D. LS-DYNA Material CardsComputer Program D.6 Strain-softening parameters used in the tubecrush analysis (This is the first part of the ssparam.dat file, which is calledby the CODAM1 material modelcc Plateauc sigx fxci fxcs-250. 0.005 0.008c sigy fyci fycs0.0c sigz fzci fzcs0.0cc damage potential function constantsc kxt kxc lxt lxc mxt mxc1.000E+00 1.000E+00 1.000E+10 1.000E+10 1.000E+10 1.000E+10c kyt kyc lyt lyc myt myc1.000E+10 1.000E+10 1.000E+00 1.000E+00 1.000E+10 1.000E+10c kzt kzc lzt lzc mzt mzc1.000E+10 1.000E+10 1.000E+10 1.000E+10 1.000E+00 1.000E+00c fxs fxt fxu1.000E+10 1.000E+10 1.000E+10c fys fyt fyu1.000E+10 1.000E+10 1.000E+10c fzs fzt fzu1.000E+10 1.000E+10 1.000E+10cc bazant scaling factorsc KXT KXC KYT KYC KZT KZC10.50 1.00 25.00 10.00 25.00 1.00199AppendixD.LS-DYNAMaterialCardsComputer Program D.7 CODAM2 card used in tube-crush analysis*MAT_CODAM2_TITLEMAT219$# mid ro ea eb - prba - prcb2 1.3000E-3 60000.000 12500.000 8000.0000 0.062499 0.062499 0.062499$# gab - - nlayer r1 r2 nfreq9000.0000 9000.0000 9000.0000 1 0.000 0.000 0$# xp yp zp a1 a2 a3 aopt0.000 0.000 0.000 0.000 0.000 0.000 3.000000$# v1 v2 v3 d1 d2 d3 beta macf0.000 0.000 1.000000 0.000 0.000 0.000 90.000000 1$# angle1 angle2 angle3 angle4 angle5 angle6 angle7 angle80.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000$# imatt ifibt iloct idelt smatt sfibt sloct sdelt9.0000E-3 1.5000E-2 1.000000 1.000000 0.347000 0.224000 0.000 0.000$# imatc ifibc ilocc idelc smatc sfibc slocc sdelc2.0000E-2 5.0000E-3 1.000000 1.000000 0.285000 1.290000 0.000 0.000$# erode erpar1 erpar2 resids2 0.000 1.000000 0.000200AppendixD.LS-DYNAMaterialCardsComputer Program D.8 Cohesive material card used in tube-crush analysis*MAT_COHESIVE_GENERAL$# mid ro roflg intfall tes tslc gic giic21,1.34e-6,0,2,0.000,8713,1.75,1.75$# xmu t s stfsf1.000000,50,50,0.000201Appendix EBrief Description of theCODAM2 Material ModelCODAM2, which is the second generation of the composite damage materialmodels developed at the UBC Composites Group, and is described in detailin Forghani [35]. CODAM2 is a sub-laminate based model, that is designedto simulate the behavior of laminated composites at the macro (structural)scale.Using the CODAM2 material formulation, the in-plane stiffness matrixof the damaged laminate, Ad, is written as the summation of the effectivecontributions of the layers in the laminate as shown in Equation E.1 [35]:Ad =n∑k=1TTkQdkTktk (E.1)where Tk is the transformation matrix for the strain vector of the kthlayer, tk is the thickness of the kth layer, n is the number of layers in thesub-laminate, and Qdk is the in-plane stiffness matrix of the kth layer in theprincipal orthotropic plane, which is written as:Qdk =RfE11−RfRmν12ν21RfRmν12E21−RfRmν12ν21 0RfE21−RfRmν12ν21 0SYM RmG12 (E.2)where E1 is the Young’s modulus in the fiber direction, E2 is the Young’smodulus in the matrix direction, Rf and Rm are two reduction coefficients,that represent the reduction of stiffness in the longitudinal (fibre) and trans-verse (matrix) directions, respectively. The reduction coefficients are equalto 1 in the undamaged condition, and gradually decrease to 0 for a saturateddamage condition. Thus, for a fully damaged material (i.e Rf = Rm = 0,202Appendix E. Brief Description of the CODAM2 Material Modelfor all n material layers), all members of the in-plane stiffness matrix are setto 0.The stiffness reduction parameters (Rf and Rm) are defined as linearfunctions of damage parameter ω, as shown in Figure E.1.a, which by itselfis assumed to grow as a hyperbolic function of the equivalent averaged (non-local) strains, ε¯ eq (Figure E.1.b), and Equation E.3:ωα =(|ε¯eqα |)− εiα(εsα − εiα)εsα|ε¯eqα | ; for(|ε¯eqα | − εiα)> 0 (E.3)where superscripts i and s denote the damage initiation and saturationvalues, respectively. The initiation and saturation parameters are definedin material cards #6 and #7 of the LS-DYNA *MAT_CODAM2 materialinput. Damage is considered to be monotonically increasing as a functionof time.The averaged, non-local strains (ε¯eqα ) are evaluated using the followinggeometrical averaging equation [35]:ε¯ eqα =∫Ωxεeqα (x)wα(X − x)dΩ (E.4)where α denotes damage for fiber (α = f) or matrix (α = m), withinthe kth layer of the sub-laminate. X represents the position vector of theoriginal point of interest, and x denotes the position vector of all other points(Gauss points) in a spherical averaging zone denoted by Ω. The radius ofthe spherical averaging zone Ω is defined as rd, and is linked directly tothe predicted size of the damage-zone in the material. rd is defined using auser-defined variable R1 in the LS-DYNA material card. wa is a bell-shapedweight function, which is evaluated by:wa =[1−(drd)2]2(E.5)where d is the distance from the integration point of interest to anotherintegration point within the averaging zone.The initiation and saturation strains, which appear in Equation E.3 andgovern the behavior of ω, are defined by the user for both the longitudinal(fibre) and transverse (matrix) directions, and are used to control the strainsat which damage initiates and saturates at the material.For further details regarding the CODAM2 material model formulation,the reader is referred to [35].203Appendix E. Brief Description of the CODAM2 Material Model11Rω(a)εsεi1ωε¯ eq(b)Figure E.1: a). Stiffness reduction coefficients R , as a function of the dam-age parameter ω. R is equal to 1 in an undamaged material, and 0 in afully damaged material. b). Value of damage parameter ω as a functionof the averaged (non-local) equivalent strain, ε¯ eq. εi and εs are the av-eraged (non-local) damage initiation and saturations strains, respectively,defined by the user for both the longitudinal (fibre) and transverse (matrix)directions, using the LS-DYNA CODAM2 material card.204Appendix FCalibrating the CODAM2Material Model for theTube-Crushing SimulationThe CODAM2 parameters required to identify the in-plane orthotropic re-sponse of the braided material were calibrated based on previous numer-ical work performed by McGregor et al. [84], where the first generationof the Composite Damage Models (CODAM1) developed at the Universityof British Columbia, was successfully applied to the tube-crushing problemstudied here, and thus served as a datum baseline for the calibration processdescribed bellow.A single element model, which is schematically shown in Figure F.1,served for the purpose of the calibration process. Here, a single LS-DYNAshell element (*ELEMENT_SHELL, ELFORM=16) was loaded in the pos-itive y direction. All x translational degrees of freedom were constrained,and the degrees of freedom of the unloaded nodes were constrained in theglobal y direction as well. The model was first simulated using the originalCODAM1 material card used by McGregor et al. [84]. In this case, thematerial axial direction was defined to be along the global y direction, thusresembling a pure tensile test along the fiber direction, with 0 strain in thematrix (transverse) direction. The strain energy per unit volume of materialprior to complete fracture, γ, was then calculated using:γ =∫ εf0σdε (F.1)where εf is the strain to complete fracture, and σ is the stress in theelement in the loading (global y) direction.The fracture energy of the material, Gf , for the loading direction tested,was then obtained using:Gf = γ × he (F.2)205Appendix F. Calibrating the CODAM2 Material Model for the Tube-Crushing Simulation∆ ∆xyFigure F.1: A simple single-element model used for the CODAM2 cali-bration process. LS-DYNA shell element is used (*ELEMENT_SHELL,ELFORM=16). All x translational degrees of freedom were constrained,and the degrees of freedom of the unloaded nodes were constrained in theglobal y direction as well. Prescribed displacement is applied for the topnodes (∆) in the global y direction.where he is the element size used in the analysis (2.5 mm).Once Gf was obtained, the simulation was repeated using CODAM2as the in-plane damage model. The value of the initiation and saturationstrains, which are required as a user input for the CODAM2 material model,were obtained as follows: The initiation strain was set equal to the value ofthe initiation strain used in the CODAM1 material model. The value of thedamage saturation strain in CODAM2 was then adjusted, until the fractureenergy obtained from the analysis using Equation F.1 and Equation F.2,was close to the value of the fracture energy obtained when the simulationwas performed using CODAM1. The resulting stress vs. strain plot obtainedfrom the calibration process, is brought in Figure F.2. Once loading initiates,there is a linear dependency between the stress and strain for both models,up to the point of damage initiation. Once damage initiates, while CODAM1material model exhibits a parabolic reduction of stress until complete failure206Appendix F. Calibrating the CODAM2 Material Model for the Tube-Crushing Simulationof the material, CODAM2 exhibits a linear softening behavior. The resultingfracture energy obtained following the calibration process, is 253.12 kJ/m2and 253.47 kJ/m2 for CODAM1 and CODAM2, respectively.The same calibration process was now repeated for the following loadingscenarios: compression in the fiber direction (Figure F.3), tension in thetransverse direction (Figure F.4), and compression in the transverse direc-tion (Figure F.5). The prescribed displacement was applied in the positivey direction in case of tensile loading, and along the negative y direction incase of compression, and the material direction was adjusted in accordanceof the material direction tested.Following the calibration process, the calibrated material card was thenused in tandem with the LCZ method to simulate the tube-crushing exper-iment. The CODAM2 material parameters obtained from the calibrationprocess, are listed in Table 5.5.0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45Strain20002004006008001000Stress [Mpa]Gf, CODAM 1=253.12 kJ/m2Gf, CODAM 2=253.47 kJ/m2CODAM1CODAM2Figure F.2: Stress vs. strain plot obtained from a single shell-element sim-ulation, under axial tensile loading, using the CODAM1 and CODAM2 ma-terial models. The fibres are aligned in the global y direction. The fractureenergies obtained are 253.12 kJ/m2 and 253.47 kJ/m2, for CODAM1 andCODAM2, respectively.207Appendix F. Calibrating the CODAM2 Material Model for the Tube-Crushing Simulation1.5 1.0 0.5 0.0Strain4003002001000100200Stress [Mpa]Gf, CODAM 1=482.01 kJ/m2Gf, CODAM 2=482.17 kJ/m2CODAM1CODAM2Figure F.3: Stress vs. strain plot obtained from s a single shell-element sim-ulation, under axial compressive loading, using the CODAM1 and CODAM2material models. The fibres are aligned in the global y direction. The frac-ture energies obtained are 482.01 kJ/m2 and 482.17 kJ/m2, for CODAM1and CODAM2, respectively.208Appendix F. Calibrating the CODAM2 Material Model for the Tube-Crushing Simulation0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45Strain020406080100120Stress [Mpa] Gf, CODAM 1=48.89 kJ/m2Gf, CODAM 2=48.95 kJ/m2CODAM1CODAM2Figure F.4: Stress vs. strain plot obtained from a single shell-element undertensile loading in the transverse material direction, using the CODAM1 andCODAM2 material models. The main axis of the material is parallel tothe global x direction. The fracture energies obtained are 48.89 kJ/m2 and48.95 kJ/m2, for CODAM1 and CODAM2, respectively.209Appendix F. Calibrating the CODAM2 Material Model for the Tube-Crushing Simulation0.5 0.4 0.3 0.2 0.1 0.0Strain30025020015010050050Stress [Mpa]Gf, CODAM 1=89.59 kJ/m2Gf, CODAM 2=89.61 kJ/m2CODAM1CODAM2Figure F.5: Stress vs. strain plot obtained from a single shell-element undercompressive loading in the transverse material direction, using the CODAM1and CODAM2 material models. The main axis of the material is parallel tothe global x direction. The fracture energies obtained are 89.59 kJ/m2 and89.61 kJ/m2, for CODAM1 and CODAM2, respectively.210

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