MODELLING THE IMPACT BEHAVIOUR OFFIBRE REINFORCED COMPOSITE MATERIALSbyMICHAEL 0. PIERSONB.A.Sc., The University ofWaterloo, 1991A THESIS SUBMITTED IN PARTIAL FULFILMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIES(Department ofMetals and Materials Engineering)We accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIASeptember 1994© Michael 0. Pierson, 1994In presenting this thesis in partial fulfillment of therequirements for an advanced degree at the University of BritishColumbia, I agree that the Library shall make it freely availablefor reference and study. I further agree that permission forextensive copying of this thesis for scholarly purposes may begranted by the head of my department or by his or herrepresentatives. It is understood that copying or publication ofthis thesis for financial gain shall not be allowed without mywritten permission.(Signature)Department of Metals and Materials EngineeringThe University of British ColumbiaVancouver, CanadaDate Sentember 8. 1994AbstractThree analytical models describing the behaviour of composite targets under impact, and suitable forengineering applications are developed herein. Each model assumes that the impacting projectiles are rigid,and that the targets are fibre reinforced laminated plates.One model is concerned with the non-penetrating impact of hemispherical projectiles. A modal seriessolution which includes the effects of shear deformation and rotary inertia is developed to describe thetarget deformations in this model.Another model predicts the penetration behaviour of blunt projectiles. Results of static penetration tests areused by this model to characterize the damage caused by impacting blunt projectiles.The third model describes the penetration due to impacting conical shaped projectiles. The progression ofdamage as it is described by Zhu et al [1992] is used as a basis for the characterization of damage in thismodel.Each model is compared with experimental results obtained from low velocity instrumented impact tests,and high velocity ballistic tests.11Table of ContentsAbstract iiTable of Contents iiiList of Tables vList of Figures viAcknowledgements ixCHAPTER ONEIntroduction 11.1 Previous Work 11.2 Present Work 51.3 Figures 7CHAPTER TWONon-Penetrating Impact 82.1 Background 82.2 Theory of Non-Penetrating Impact 92.2.1 Indentation 92.2.2 Target Response 112.2.1 Convergence of Target Response 162.3 Impact Model 202.3.1 Stepwise Form of Target Deflection 212.3.2 Stepwise Form of Indentation 222.3.3 Impact Model Expressions 222.3.4 Impact Model Convergence 242.4 Results of Non-Penetrating Model and Discussion 242.5 Tables and Figures 27111CHAPTER THREEImpact of Blunt Projectiles 373.1 Background 373.2 Theory of Blunt Impact 393.2.1 Target Deflection 393.2.2 Penetration Force 403.3 Blunt Impact Model 423.3.1 Inertia Force 433.3.2 Friction Force 443.4 Results of Blunt Model and Discussion 443.5 Tables and Figures 47CHAPTER FOURImpact of Conical Projectiles 574.1 Background 584.2 Theory of Conical Impact 584.2.1 Target Deflection 594.2.2 Penetration Force 594.2.3 Effective Strength 604.3 Conical Impact Model 644.3.1 Friction Force 644.4 Results of Conical Model and Discussion 654.5 Tables and Figures 68CHAPTER FIVEConclusions and Recommendations 80CHAPTER SIXReferences 84APPENDIX AStatics and Dynamics of Laminated Plates 87APPENDIX BUBC Impact Software 111ivList of TablesTable 2.1: Material used in experiments: Material A. 27Table 2.2: Static indentation best fit data, as used in Figure 2.3. 27Table 2.3: Data used for target deflection convergence study, from Qian and Swanson [1990]. 28Table 2.4: Calculated natural frequencies for target described in Table 2.3. 28Table 2.5 : Impact conditions used for convergence study of impact model, from Qian andSwanson [19901. 28Table 2.6: Convergence of peak contact force for Qian and Swanson [1990] and present impactmodels, as a function of time step (0.1 us used as reference). 29Table 2.7: Natural frequencies calculated for Material A. 29Table 3.1: Material used in experiments: Material B. 47Table 3.2: Impact conditions used for Awerbuch and Bodner model simulation. 47Table 3.3: Characteristic force and displacement data derived from Figure 3.4, and used by model. 47Table 3.4: Model parameters derived from Figure 3.4; used in impact model to define staticindentation force. 48Table 3.5: Experimental and model results for various blunt impact conditions, Material B. 48Table 4.1: Experimental and model results for various conical impact conditions (025.4 mm opening,Material B). 68vList of FiguresFigure 1.1: Digital image of instrumented projectile, shown with blunt tip. 7Figure 1.2: Schematic drawing of instrumented projectile. 7Figure 2.1: Non-penetrating impact nomenclature. 30Figure 2.2: Static indentation experiment compared with Hertzian indentation law (hemisphericalindenter, 025.4 mm). 30Figure 2.3: Static indentation experiment compared with modified Hertzian indentation law, and withbest fit power law (hemispherical indenter, 025.4 mm). 31Figure 2.4: Target plate nomenclature. 31Figure 2.5: Convergence of target deflection, step load applied at center of target, 160 mode solutionused as reference. 32Figure 2.6: Convergence of target deflection, step load applied over small patch, 160 mode solutionused as reference. 32Figure 2.7: Convergence of target deflection, step load applied over entire target area, 160 modesolution used as reference. 33Figure 2.8: Target deflection history of impact event (see Table 2.5 ) and resulting free vibration;present model compared with published results. 33Figure 2.9: Contact force history of impact event (see Table 2.5 ); present model compared with Qianand Swanson [1990] results. 34Figure 2.10: Contact force history for 6150 g, 1.76 mIs impact event; present model compared withexperimental results (no target damage). 35Figure 2.11: Contact force history for 6150 g, 2.68 mIs impact event; present model compared withexperimental results (target damaged). 35Figure 2.12: Contact force history for 314 g, 7.70 mIs impact event; present model compared withexperimental results (no target damage). 36Figure 2.13: Contact force history for 314 g, 11.85 mIs impact event; present model compared withexperimental results (target damaged). 36viFigure 3.1: Digital image of a Material B specimen following the static penetration test with bluntindenter. 49Figure 3.2: Nomenclature for three stage Awerbuch and Bodner model: (a) plug initiation, (b) plugformation, (c) plug ejection. 49Figure 3.3: Contact force and velocity histories as predicted by Awerbuch and Bodner model for 400mIs, 4.2 g impact event. 50Figure 3.4: Force-displacement relations for static penetration, including results as measured fromstatic flexure experiment, results adjusted to remove target deflection, and characteristicrelation used by model. 50Figure 3.5: Force displacement results, 320 g, 20.5 mIs blunt impact: experiment and present model. 51Figure 3.6: Force displacement results, 320 g, 26.9 mIs blunt impact: experiment and present model. 51Figure 3.7: Force displacement results, 308 g, 28.1 mIs blunt impact: experiment and present model. 52Figure 3.8: Force displacement results, 308 g, 32.0 m/s blunt impact: experiment and present model. 52Figure 3.9: Contact force history for high velocity blunt impacts, as predicted by impact model. 53Figure 3.10: Comparison of predicted vs. measured absorbed energies in various impact events. 53Figure 3.11: Energy absorbed versus impact energy for various impact conditions, as predicted bymodel. 54Figure 3.12: Image of section through specimen following a static penetration test; blunt indenter. 55Figure 3.13: Image of section through specimen following a ballistic impact event; blunt projectile,265 rn/s. 56Figure 4.1: Material B specimen following static penetration test with conical indenter. 69Figure 4.2: Conical penetration nomenclature and typical contact area projections. 70Figure 4.3: Schematic showing damage grid used for conical model. 71Figure 4.4: Average contact stress as a function of conical indenter displacement, for various statictests. 72Figure 4.5: Nomenclature used for calculation of local strain surrounding conical projectile. 72Figure 4.6: Nomenclature used for calculation of bulge radius. 73viiFigure 4.7: Nomenclature used for calculation of bulging strain through the target thickness. 74Figure 4.8: Contact force history of 320g, 20.5 mIs, conical impact event; experiment and presentmodel. 75Figure 4.9: Contact force history of 320g, 30.2 mIs, conical impact event; experiment and presentmodel. 75Figure 4.10: Contact force history of 308g, 29.1 mIs, conical impact event; experiment and presentmodel. 76Figure 4.11: Contact force history of 308 g, 31.5 mIs, conical impact event; experiment and presentmodel. 76Figure 4.12: Model prediction of contact force history of 316 mIs, 4.2g conical impact event. 77Figure 4.13: Comparison of measured and predicted absorbed energies for various conical impacts. 77Figure 4.14: Average stress surrounding penetrator, due to conical penetration/impact. 78Figure 4.15: Energy absorbed due to impact of conical projectile: experiment and model. 78Figure 4.16: Section through specimen following non-perforating impact event; conical projectile,V0=156m1s. 79Figure 4.17: Section through specimen following ballistic event; conical projectile, V0 = 316 mIs,Vexit = 228 mIs. 79viiiAcknowledgementsI would like to take this opportunity to thank my supervisor Dr. Reza Vaziri for his patience and support,and for making this investigation possible. As well, the support and guidance of Dr. Anoush Poursartip, andDr. Daniel Delfosse have been invaluable.I wish to acknowledge the financial support of the Canadian Department of National Defence, through acontract with the Directorate Research Development Land (DRDL), and the Cy and Emerald KeyesFellowship in Metals and Materials Engineering.I would also like to thank the many members of the Composites Group, past and present, who have mademy graduate work thoroughly enjoyable.Finally, a loving thank-you to Mrs. Frances M. Prychitka.ixCHAPTER ONEIntroductionAdvances in the design and manufacture of fibre reinforced plastics have enabled these materials to be usedin place of traditional homogeneous materials. The behaviour of composite materials under extreme loadconditions is of great interest as their use in ballistic protection and aerospace applications has increased.Composite materials are also being implemented as critical structural components in which reliability anddamage detection is important.Design with regard to impact performance is obviously necessary for protective systems and aircraftcomponents, however, low velocity impact behaviour is also important for critical components that mayexperience ‘tool drop’ impacts. An understanding of the damage caused by these incidental impacts, whichis often difficult to detect, is required when predicting post impact strength and reliability.Three types of damage can be expected in fibre reinforced laminated materials: delaminations, matrixcracking, and fibre breakage. Delaminations occur between the layers of the laminate and are the result ofshear stresses between these layers. Matrix cracks are often associated with delaminations, as they developthrough layers, allowing the delaminations to grow throughout the laminate. Fibre breakage can result fromhigh tensile strains in the plane of the laminate, or high shear loads applied in the transverse directions.The study of composite materials, including damage states, has garnered much interest recently.1.1 Previous WorkFor information concerning the design and manufacture of composite materials, the reader is directed to anynumber of introductory texts. Jones [1975] provides insight into the mechanics of fibre reinforced1composites, including micro-mechanical determination of lamina stiffness, and strength theories of lamina.Selection of fibre and matrix with respect to performance and compatibility is examined by Hull [19811.Impact DynamicsThe mechanics of impacting bodies and isotropic targets was examined early in this century byTimoshenko [19131 and others. The deflection of beams impacted by non-penetrating rigid spheres wasanalyzed by Timoshenko, using an integral form of Newton’s law of motion and the Hertz contact relation.This simple analysis contrasts with recent work concerning penetrating projectiles with velocities in excessof 1 kmls, and non-homogeneous composite targets. The method used by Timoshenko for the analysis oftarget response is used extensively in this thesis, for both low and high velocity impact events.The contact problem has been examined by many authors, most notably Willis [19671, who presented aFourier transform solution to the indentation problem of a rigid flat punch and an anisotropic half-space.Dahan and Zarka [1977] adapted the Hertzian type indentation relations for an isotropic materials totransversely isotropic media.Herrman [1955] presented a general solution to the problem of the forced vibration of Timoshenko beams.A series solution and corresponding property of orthoganality is developed, analogous to the sheardeformable plate solution presented in Chapter 2.The application of finite element analysis to impact problems was investigated by Trowbridge [1991]. Anon-linear finite element routine is used to model the low velocity impact of a spherical ball on a cylindricalrod. A specialized element is developed which accounts for the non-linear contact relationship and thelinear target response. Analysis results are shown to be reasonably close to experimentally measured contactforce histories.den Reijer [15] presented an analytical model in the form of a computer program called ALARM. Theimpact response of isotropic armour with a ceramic facing is modelled and predictions are compared toexperimental results. Mechanisms such as plastic hinging, membrane action, and shear failure of the2backing plate are considered. Model predictions are highly dependent on the behaviour of the ceramicfacing and are thus not applicable to uncoated composite targets.Impact ofMetallic TargetsExtensive analytical investigations into the behaviour of metals under impact have been published.Woodward [1978] examined the penetration behaviour of conical shaped projectiles as they impact metallictargets. Ductile hole formation associated with thicker targets is compared to the dishing type failure oftenseen in thin target plates. The work done by the penetrator in each type of failure is used to predict theimpact behaviour of the target.Woodward and Crouch [1989] developed a simple analytical model for the impact of flat shaped projectileson layered metallic targets. An energy balance is applied to the projectile and target which includes globaldeformations of the target, although a contact force history is not generated. The model is presented in theform of a computer program called LAMP.Awerbuch and Bodner [1974] presented an analytical model for the penetration of metallic targets by bluntprojectiles. The process is divided into three distinct phases: initial plug formation, shear growth of theplug, and exit of the plug and projectile. This model is adapted to composite targets in Chapter 3.A five stage model for the perforation of isotropic metallic targets was developed by Ravid andBodner [19831 and extended to include the perforation of layered metallic targets by Ravid et al [19871.Plastic flow of the target material forms a bulge on the impact face of the target in the preliminary stage.The second stage considers the formation of a bulge on the distal face of the target. This distal bulge growsin depth but not in width during the third stage. Plug formation begins in stage four, and stage five considersthe exit of plug and projectile. In all stages the projectile is assumed to be rigid, target deflections areignored, and frictional forces are assumed to be negligible. This five stage model is complex and notsuitable for an engineering type of approach. As well the model does not predict a contact force history,making it difficult to compare to experiments.3Impact of Composite MaterialsAbrate [1991] presented a survey article of existing experimental studies and analytical models dealing withthe impact of composite materials. Articles describing experimental apparatus and procedure for gas gun,drop weight, flyer plate, and other types of testing methods, are listed. Simple spring-mass models, energybalance models, and wave propagation models are described. Studies concerning local indentation laws forhemispherical and flat shaped indenters are reviewed. Cantwell and Morton [1991] presented a similarreview.Sun and Chattopadhyay [1975] presented a complete formulation for the elastic impact of an isotropicsphere and a laminated specially orthotropic plate. The Hertz contact law is used in conjunction with asimplified series type solution which includes shear deformation, but neglects rotary inertia. An incrementalapproach in time is used to solve the non-linear equations of motion. As well, a simplification is used toperform the time dependent integration, required by the series solution. Results are presented for deflection,bending stress, and shear stress over the duration of impact. These results are used for comparison inChapter 3.Zhu et al [1992] developed a penetration model for conical shaped projectiles impacting Kevlar/polyesterlaminated plates and compared the model predictions to experiment. Experimental projectile velocities weretypically about 300 mIs, to a maximum value of 800 mIs. A finite difference analysis is used in the model topredict the global target response. A local penetration model, which includes a prediction of damage, isused. This work is used extensively to develop the penetration model described in Chapter 4.Langlie and Cheng [1990] developed a comprehensive model for high-velocity penetrations, andincorporated it into a non-linear transient finite element code. A punching shear type of failure is assumedfor flat shaped projectiles. Contact force is used to predict the onset of damage, and a shear wavepropagation model is used to model the damaged target behaviour.4Experimental StudyExperimental results and observations from tests performed at UBC provide a basis for the impact modelsdeveloped in this thesis. Three types of tests have been performed: static tests, instrumented low velocityimpact tests, and high velocity ballistic tests. A complete and more detailed description of the experimentalset-up is contained in Delfosse et al [July 1993].The instrumented impact tests provide a valuable information about impact events, and are essential for thedevelopment and evaluation of impact models. The apparatus consists of a gas gun, and an instrumentedprojectile which records a contact force history of the impact event. The heavy projectile can be fired atvelocities up to 50 mIs, resulting in high impact energies.A picture and schematic of the instrumented projectile are shown in Figure 1.1 and Figure 1.2. A small,light weight piezoelectric load cell is used to measure loads between an aluminum mass and the projectileshaft. Hardened steel (30R) cylindrical impactor heads of diameter 7.6 mm with flat and conical (37° tipangle) nose shapes are threaded onto the tip of the projectile. The total mass of the projectile, depending onthe configuration, is either 320 g, or 308 g.1.2 Present WorkThis thesis investigates the behaviour of laminated composite targets as they are impacted by a variety ofprojectiles. To simplify the modelling of dynamic target behaviour, carbon fibre reinforced epoxy materials,which are rate insensitive, are studied. Both non-penetrating and penetrating events are modelled.Hemispherical shaped projectiles are examined in the context of non-penetrating impact events, whilepenetrating events are modelled with respect to blunt and conical shaped projectiles. Both ballistic and lowvelocity ‘tool drop’ impacts, including the impact of fragments and objects with unconventional shapes, willbe applicable to one of the three models presented.Each model is analytical in nature, and has been implemented into a user friendly computer code, designedto operate in the Windows environment. This type of engineering software will complement other types of5impact studies, including experimental and numerical techniques. A description of the software, UBCImpact, is included in Appendix B.The foundation of each model is the separation of local and global effects. Local effects generally includeindentation or penetration, and are unaffected by the target behaviour away from the point of impact. Globaleffects include the vibrations and elastic deflections of the target plate. Global and local effects are a part ofany impact event, however, the local effects tend to dominate at higher impact velocities.The modelling of global target behaviour is required to predict low energy impact events, and also leads toa better understanding of local effects. The non-penetrating impact model, described in Chapter 2, combinesa modal solution for global target deflections, with a well known relation describing the indentation ofhemispherical indenters. This model allows validation of the global analysis through comparison withpreviously published analytical and numerical results, and with experimentally measured results.The global model developed in Chapter 2 is used in the blunt model presented in Chapter 3, and in theconical model presented in Chapter 4. The blunt and conical models use static penetration behaviour as abasis for dynamic impact predictions. This approach is reasonable due to the strain rate independence of thecomposite material considered here.A composite material, for the purposes of this thesis, is defined as a material in which evenly distributed,continuous fibres are aligned within a homogeneous matrix material. Thin slices, or laminae, of such amaterial are stacked with specific fibre orientations to form a laminate. A detailed examination of the elasticproperties and behaviour of such laminates is presented in Appendix A.61.3 FiguresFigure 1.1: Digital image of instrumented projectile, shown with blunt tip.load cellaluminum shaft _\\\\\aluminum holderinterchangeable impactor torlon caseheadFigure 1.2: Schematic drawing of instrumented projectile.cable to digitaloscilloscope7CHAPThR TWONon-Penetrating ImpactA model describing the non-penetrating impact of rigid, hemispherical projectiles on fibre reinforcedlaminates has been developed. Analytical techniques are applied to local indentation and global targetdeflection, which together define the impact problem.In contrast with existing models, an analytical form of the non-linear Hertzian contact relation is included inthe solution. A modal series solution for the target deformation, which includes all first order shear andinertia effects, is developed. The accuracy and convergence of the plate solution and impact model arestudied and compared with published results.The non-penetrating impact model is applicable to low energy impact events, where inelastic effects due todamage are insignificant. This model is important because it serves to validate the modal laminate solutionused to calculate target deflections, which can also be used to describe the global behaviour in penetratingmodels.2.1 BackgroundThe study of non-penetrating impacts has produced many solutions based on the small increment methodfirst presented by Timoshenko [1913]. This numerical method applies rigid body dynamics to the projectile,and equates projectile displacement to the sum of local deformation and plate bending.The analysis of hemispherical projectiles allows the well known Hertzian type relation to be used in thesolution for local indentation. Sun and Chattopadhyay [1975] applied a recursive relation to deal with thenon-linearity introduced by the Hertzian relation. Christoforou and Swanson [19911 used an approximate,8linearized form of the Hertzian relation. Both of these numerical methods simplify the analysis, butintroduce an unnecessary error into the solution.The dynamic response of orthotropic, fibre reinforced targets can be solved with a modal series solution.Dobyns [19801 and others have used the plate equations developed by Whitney and Pagano [19701 to modelthe target deflection. These equations include first order shear deformation and inertia terms, but the rotaryinertia effects are often ignored. It has been shown that shear deformation effects are significant, but rotaryinertia terms are not, when considering low velocity impact events. The plate equations apply to rectangulartargets with simply supported boundary conditions.A new model has been developed, which includes a non-linear analytical solution to the Hertzian contactproblem. A plate solution which includes shear deformation and rotary inertia effects, has been developedfor this model, so that the solution may be applied to high velocity impact events.2.2 Theory of Non-Penetrating ImpactAs in previous approaches, the basis of the new model is the recognition of two distinct responses toimpacting projectiles. A hemispherical projectile causes global deformation of the target mid-plane, andlocal indentation of the target surface. Total projectile displacement is equal to the sum of globaldeformation and local indentation.Thus the compatibility condition is:(2.1)where is the projectile displacement, a is the indentation, and w is the target deflection (see Figure 2.1).2.2.1 IndentationLocal indentation of the target is generally assumed to follow a Hertzian type relation. Hertz suggested thatcontact between a sphere and an elastic medium could be described in terms of the total applied force F andthe resulting indentation a.9The Hertzian relation is as follows:F=kHcL312 (2.2)The Hertz contact stiffness, kH, for a transversely isotropic material, is’:kH= / 2 (2.3)31—VpE Ewhere and v, are the modulus and Poisson’s ratio of the projectile, E is the transverse modulus of thecomposite, v, and are the transverse Poisson’s ratio of the top layer of the composite, and D is the tipdiameter of the hemispherical projectile.The force-displacement results for a static indentation test on Material A, are plotted in Figure 2.2. Theproperties of Material A are listed in Table 2.1. Superimposed on Figure 2.2, is the theoretical indentationcurve calculated using Eq. (2.2) and Eq. (2.3). The stiffness calculated according to Eq. (2.3) is 14.1 N/rn213.The obvious disagreement shown in Figure 2.2 is not surprising due to the nature of the elastic medium, inthis case, a laminated plate. In the static indentation test the plate is placed on a steel backing plate, makingit much more stiff than the half space assumed by Hertz. In an impact event, there is a stress free conditionon the distal side of the target, making it even less stiff than the Hertz assumption of a half-space. For smallamounts of penetration it is assumed that the error due to the stress free condition, is small.Figure 2.3 shows the same experimental result, with two ‘best fit’ curves superimposed. Curve ‘a)’ uses anexponent of 3/2 as in Eq. (2.2), with a stiffness adjusted to match the experiment. Curve ‘b)’ is fitted withrespect to both exponent and stiffness. The best fit parameters are summarized in Table 2.2.See Willis [1967].10The most accurate contact law is curve ‘b)’ in Figure 2.3, which uses an exponent of 1.7, instead of theHertzian exponent of 1.5. However, the mathematical advantage of using the Hertzian exponent will beapparent when the impact problem is formulated, therefore an exponent of 3/2 is used..Thus the contact law is:F=kOL312 (2.4)where ke is the non-linear contact stiffness obtained as in Figure 2.3.2.2.2 Target ResponseThe target laminate is modelled as a specially orthotropic, rectangular laminate with simply supportedboundary conditions. A dynamic solution for this type of target is available, and is presented in detail inAppendix A.The laminate equations as developed by Whitney and Pagano, including the effects of transverse sheardeformation and rotary inertia are solved.The assumed displacement field takes the following form:u= u°(x,y,t)+z’qi(x,y,t) (2.5a)v=v°(x,y,t)+ziji(x,y,t) (2.5b)w = w(x,y,t) (2.5c)where u°, v° and w are the laminate displacements in the x, y, and z directions at the mid-plane, ‘qlx and ly’are the cross-sectional rotations in the x and y directions, respectively.11The differential equations of motion are as follows:÷(D1266)’+D66a(2.6a)____ ____aw (‘ aD11 ÷(D12+66) a2 _kANJy+_J=Iliiy (2.6b)kA55(_+J+kAL+J+pZ(X,y,t)= phi (2.6c)where the overdot indicates differentiation with respect to time.A solution to the homogeneous form of Eq. (2.6) is obtained by assuming rotations and lateral displacementas follows:= e’° Umn cos(mlc x/a) sin(rnt y/b) (2.7a)= e’° Vmn sin(mm x/a) cos(nit y/b) (2.7b)w = ebo)t Wmn sin(mlt x/a) sin(nlt y/b) (2.7c)where Umn, Vmn, and Wmn are undetermined constant coefficients, and a, b are the length and width asshown in Figure 2.4.12Substituting Eq. (2.7) into the governing differential equations, results in a set of three linear algebraicequations:‘11’rnn ‘12 ‘13 Umn 0‘12 21Wrnn L23 Vmn = 0 (2.8)‘13 L23 L3—ph1 Wmn 0whereL =D11(mit/a)2+D66(nir/b)2+kA55 (2.9a)‘12 = (D12 +D66)(mm/a)(n it/b) (2.9b)‘13 = kA55(mit/a) (2.9c)‘P22 = D66 (mit/a)2+D22(nic/b) +kA (2.9d)L23 = kA(nic/b) (2.9e)L33 =kA55(mit/a)2+kA(nit/b)2 (2.90Each solution set m,n, results in three eigenvalues and their associated eigenvectors Umnj, Vmnj, Wmnj,where the subscript j = 1,2,3. Only two components of each eigenvector are independent, thus theeigenvectors are normalized with respect to Wmnj.The particular solutions are assumed to be separable into functions of position and time, as follows:3= Umnj •cos(mitx/a)sin(n7uy/b).Tj(t) (2.lOa)m=1 n=1 j=13= Vmnj .sin(mitx/a)cos(nity/b).Tmnj(t) (2.lOb)m=1 n=1 j=I3w= Wmnj.sin(mitX/a)sin(nity/b).Tmnj(t) (2.lOc)m=1 n=1 j=I13Substituting Eq. (2.10) into the equations of motion, and applying the homogeneous solution yields:cos(m1u/a)sin(n7ty/b)[coj Tmnj + Tmnj] = 0 (2.lla)m=1 n=1 j=lsin(mirx/a) cos(nmy/b)[w T,i + Tmnjj = 0 (2.1 ib)m=1 n=1 j=1sin(mtx/a) sin(nlry/b)[w Tmnj + Tmnj] =- (2.1 lc)m=1 n=1 j=l PIn order to solve for the time dependent variable Tmnj, a single equation without summation, i.e. anorthogonal set, is required. The orthogonality condition for the solution sets is obtained by applyingClebsch’ s Theorem.Clebsch’s Theorem, in a general form, is 2:IIIP[UrU8+VrVc+WrWcldXdydZ=0 rs (2.12)where the displacements u, v, w, have solutions of the form:{u,v,w} = {Ur,Vr,Wr}•f(t) (2.13)Using the assumed solutions of Eq. (2.10), which are in the above form, and inserting into Clebsch’sequation, the orthogonality condition becomes:jf0bf1jg,jdydx=0 e,fgm,n,j (2.14)whereF’efg,mnj = IUefg cos(e/a)sin(ftry/b)U cos(rrntc/a)sin(nity/b)+1 Vejg sin(eltx/a) cos(ftcy/b) . V sin(mltx/a) cos(nity/b) (2.15)÷ph Wefg sin(eltx/a) sin (flcy/b) . W,,. sin(mtx/a) sin (nicy/b)2 See Love [1926].14Multiplying Eq. (2.7a) by:IUefg cos(eicx/a)sin(fxty/b); (2.16a)Eq. (2.7b) by:I Vefg sin(elrx/a) cos(ficy/b); (2.1 6b)and Eq. (2.7c) by:PM’’efg sin(eitx/a)sin(ftty/b) (2.16c)then summing the three results, one equation in terms of Tmnj is obtained:‘1efg,mnj Tmnj + efg mnj Tmj] — PhWfg sin(eirx/a)sin(flty/b). = 0 (2.17)m=1 n=1 j=1Integrating over the laminate area meets the orthogonality condition, leaving only one non-zero term whene,J g = m, n,j, and allowing the summation to be dropped:ab ab.2—-W,nnjf0sin(m7rx/a)sin(nity/b)pz dydx0)mnjTmnj+Tmnj = (2.18)whereMmnj fcJ1mnj,mnjdtYdtx(2.19)= IU ÷IV, ÷phW,Note that Wmnj = 1 because of the eigenvector normalization, and the integrated term in Eq. (2.18) is simplythe Fourier transform of the load, PzAs well, the load is assumed to be stationary, i.e.:= F(t) q(x, y) (2.20)15The Fourier transform of the load is then:Jalbi(ltx/)i(rnt/b) dydx = F(t).q (2.21)Values of for different types of loading can be found in Appendix A.Solving the differential equation of Eq. (2.18), using Eq.’s (2.20) and (2.21), yields:Tmnj = cos(omnj t)+- sin(wmnj t)+4a;J.F(t)o) sin(t—t)dt (2.22)mnj mnj mnjwhere the nought superscript indicates a value @ t = 0.The complete solution for lateral displacement of the target laminate is then:w = sin(mitx/a)sin(rnty/b).m=1 n=1(2.23)±{T71 cos(wmnjt) + Sifl(O)it) + ab q,, f F(t) sin °mnj (t — t) dt}j=1 mnj -0mnj mnj2.2.1 Convergence of Target ResponseA closed form convergence criterion for the Fourier solution above is not available. Convergence of thesolution is demonstrated numerically for a typical laminate with three loading conditions: point load, loaddistributed over a small patch, and uniformly distributed load. Properties of the target are shown inTable 2.3.For each type of loading, responses at the center of the target were calculated using 5, 10, 20, 40, 80, and160 non-zero modes in both the x and y dimensions. Equation 2.23 was evaluated for a range oft values, onan Indigo workstation. In each case the load began at t = 0 and continued to act for the duration. Theintegration with respect to time in Eq. (2.23) is exact for this type of step loading.16Solution error is calculated using the solution for 160 modes as a reference, i.e.:%e(t)=w(t)100% ?=5, 10, 20, 40, 80 (2.24)w160(t)where(2—1) (2—1)w= sin(mt/2)sin(rnt/2).m=1 n=1(2.25)±{T11 cos(omnjt) + sin(U)mpjt) + f F(’c) sin mnj (t — t) dr}j1 0mnj t0mnj mnjFigures 2.5, 2.6, and 2.7 plot this error function vs. the logarithm of a normalized time variable:= log(L” (2.26)Vt)where(2.27)The expression of Eq. (2.27) is chosen as a normalizing parameter because it represents the time taken forone transverse shock wave to travel through the thickness of the laminate. Laminate theory assumes that allshock waves travelling through the thickness of the laminate have dissipated and are insignificant. Resultscalculated at t 1 are likely invalid because of these shock wave effects.A value of t = 0 indicates t = t, which, for the target described by Table 2.3 , is 1.2 ts.For each set in the solution, i.e. each m,n pair, there is a set of three natural frequencies and three modalamplitudes, as shown by Eq. (2.8). One frequency is associated with the lateral motion of the target, and twofrequencies are attributed to the cross sectional rotations. In general the frequency associated with lateralmotion is an order of magnitude smaller than the others.17An approximate value of this dominant frequency can be calculated with the following3:=(‘ +2L1L3-L22I13 -ziz3)/(phQ) (2.28)where(2.29)The (m,n)th periods are calculated using the frequencies described by Eq. (2.28), i.e.:Tmn=_iL (2.30)0mnNatural frequencies for the material described in Table 2.3 are listed in Table 2.4.In the target solution, all even numbered modes are zero because of the sinusoidal terms, thus the modenumber, x’ corresponds to the mode indices m,n as follows:m,n=2X—1 (2.31)The accuracy of a modal solution is often discussed in terms of the last significant natural period ofvibration as it compares to the time step. The last significant natural period is shown on Figures 2.5, 2.6,and 2.7 with a vertical line.Point LoadingFigure 2.5 shows the convergence of the laminate solution for a step load applied as a single point force atthe centre of the plate. It is apparent that the number of modes required to reach reasonable accuracy, ishighly dependent on the time of interest.Convergence for this type of loading is poor due to the nature of the solution. Early in the event, the pointforce causes shear and flexural waves to travel outward toward the boundaries. As these waves traveloutward, they act over a larger area, and the wave fronts increases in circumference, resulting in reduced3FromDobyns [1981].18intensities. However, at a time close to zero, before these waves have time to travel, the point load issupported by an infinitely small region of the target, resulting in infinitely large plate deformations. Thus attimes close to zero the solution for target deflection is indeterminate.Convergence is obtained for normalized times in excess of 3.0, for a reasonably small number of modes.For any given number of modes, the solution converges at a time much greater than the last significantnatural period.This type of loading is not recommended for time steps of the same order of magnitude as the normalizingparameter, t.Patch LoadingFigure 2.6 shows the convergence of the laminate solution for a patch load applied at the center of thelaminate. The patch area is equal to ten percent of the laminate area.As each curve in Figure 2.6 approaches the ordinate axis, an amount of ‘noise’ appears. This fluctuation isdue to the dynamic nature of the problem. The solutions using a small number of modes are slightly out ofphase with the reference solution, causing deceptively high percentage error even after the solution hasconverged.Applying the load over a patch eliminates the indeterminate problem seen in the point load solution. Asexpected the patch solution converges much more rapidly. Each curve shown in Figure 2.6 converges priorto the last significant natural period.Typically, reasonable accuracy (0.1%) is obtained at the last significant natural period. The small patchsolution is the most appropriate solution for impact events, as reasonable accuracy can be achieved whilerealistically modelling the force applied to the target.Distributed LoadingFor completeness, behaviour of the uniformly distributed solution is presented in Figure 2.7. A unit load hasbeen applied over the entire surface of the laminate.19Convergence of the distributed load is quite similar to the patch load solution, with reasonable accuracyobtained at the last significant natural period. The uniformly distributed load is obviously impractical formodelling impact events, although other dynamic loads such as blasting or fluid pressure may be modelledas such.2.3 Impact ModelCombining the indentation law and the target deformation solution, a complete impact model can beformulated. The non-linear nature of the indentation law requires a model that is stepwise in the timedomain, as in Timoshenko’s incremental method.The impact event is divided into equal segments or time steps, and the contact force is assumed to beconstant through each time step. The non-linear equations describing the impactor/target system are solvedin terms of the contact force.The displacement of the rigid impactor, at any given time, in terms of the applied force, is described by:z,=V0.t__-$F(t)(t—t)dt (2.32)where V is the impactor velocity, and M is the impactor mass.During any time increment, in which the contact force is constant, Eq. (2.32) can be rewritten:= A(t—zt)÷vQ—At).At— F(t)&2 (2.33)where & is the time increment.In order to solve the displacement constraint equation, Eq. (2.1), using Eq. (2.33), both the target deflectionand indentation must be expressed in terms of the contact force.2.3.1 Stepwise Form of Target DeflectionThe modal solution for target deflection requires an integration over the entire history of loading. In orderto simplify this integration, the solution is evaluated with a moving time scale.20Evaluating the target deflection is simplified by applying a transformation in the time domain, as follows:I = & (2.34)The initial conditions then become:=Tmnj(I=O)=Tmnj(t=t_JXt) (2.35a)1n0inj mj(0)’mnj(tt_1-t) (2.35b)The solution for target deflection is rewritten in a stepwise form by evaluating at I =w = sin(mitx/a)sin(nicy/b)m=1 n=1± cos(omnj&) + --‘—sin(o,,&) + ab qfl f FQc) sin0mnj (& — T) dt} (2.36)j1 0mnj -°mnj ‘“mnjThis stepwise form requires the initial conditions to be used at each interval, and not just at t = 0. The initialconditions are re-evaluated at the end of each interval. This is equivalent to using initial conditions at t = 0,and performing the integration over the entire load history.Equation (2.36) evaluated to reflect the assumption of constant contact force is:w = sin(mtx/a)sin(rnty/b)m=1 n=1±{T1 cos(wmnj&) + sin(omnj&) + F(t) ab q,, [i — cos(comni&)]} (2.37)j=1 mnj mnj mnj21Noting that the contact force is independent of the modal summation, a simpler form of Eqn. (2.37) can bewritten:w=w°(t)+CF(t) (2.38)wherew° (t) = sin(mc/2) sin(nlt/2) [mni cos(omnj&) + sin(wmni&)] (2.39)m=1 n=1 j1 0mnjis the displacement due to the loading F(t = 0 —* t — , andC = sin(mt/2) sin(nit/2)abq,{i — cos(o mnjtt)] (2.40)m=1 n=1 j=1 (0 mnj Mmnjis the dynamic compliance of the target.The load is assumed, for optimum convergence, to act over a small patch equal in size to the diameter of thehemispherical projectile. The load term is evaluated according to Appendix A.2.3.2 Stepwise Form of IndentationThe local indentation is assumed to be independent of the load history, therefore Eq. (2.4) can be useddirectly. Thus the local response, in terms of the contact force, is:2/3o(t)= [F(t)](2.41)2.3.3 Impact Model ExpressionsThe integer fraction exponent in Eq. (2.41), and the linear nature of Eq. (2.37) allow the impact model to beformulated analytically.22Substituting the displacement expressions of Eq.’s. (2.33), (2.38), and (2.41) into the constraint condition,Eq. (2.1), one equation in terms of a single unknown variable, F =F(t), is obtained:0F3+12÷F÷=0 (2.42)where= _c+J (2.43a)=— l/k + 3(z — w° + v0At)C&2J(2.43b)2=— w° + V0&)[c+(2.43c)=— W + V°At (2.43d)The above cubic equation can be solved as follows4:F= 2fZcos(0/3)_?± (2.44)whereQ=(2.45a)cosO— 27?3 — 22(2 45b)54E2.3.4 Impact Model ConvergenceEach of the displacement solutions (indentation and target deflection) are exact for a given contact force.However, because the contact force history is not known a priori, some error is incurred. Convergence ofthe full impact model is shown in Table 2.6.From Speigel [19681.23Peak force is chosen as the defining characteristic, and the error listed in Table 2.6 is defined as follows:9boE(t)=1’At •100% (2.46)F01where is the peak force calculated with time step &, and & = 0.1 jis is used as the reference solution.Qian and Swanson [19901 chose to use a 100 mode solution to calculate target deflection. However thepresent model uses a number of modes appropriate for the time step. The convergence suggested inSection 3.1.1 is applied, thus the last significant natural period is less than the chosen time step. Note thatthe natural periods listed in Table 2.4 are applicable here.The different solution method applied to the present model results in different peak contact force values,and an improved convergence rate. Qian and Swanson used a patch of changing size corresponding to theamount of indentation. As well, they used the approximate, non-linear contact law as presented byChristoforou and Swanson [19911.The present model attains a reasonable level of accuracy (0.1%) with a time step of 1 JIs, which correspondsto approximately 200 calculations. This is performed in a matter of seconds on a 486-50 MHz personalcomputer.2.4 Results of Non-Penetrating Model and DiscussionThe impact model is compared to published results and to experimental measurements with goodagreement. A comparison with published results is shown in Figures 2.8 and 2.9. Qian and Swansondeveloped a Rayleigh-Ritz solution in addition to the analytical routine described above, and Sun andChen [19851 used a finite element code to simulate an impact event. The impact conditions are listed inTable 2.5.The present model agrees quite well considering the approximations involved. Each of the previous modelsused a changing patch size when calculating the target deflection; the present model does not. Qian andSwanson used a 100 mode solution, and a time step of 0.1 ts. In accordance with the convergence results,24the present model used a 20 mode solution and a time step of 1 j.ts resulting in slightly more than 200 datapoints.Contact force histories collected from instrumented impact tests are compared with model predictions inFigures 2.10 through 2.13. Each impact event involved a 025.4 mm hemispherical shaped projectile, and a76.2 mm x 127.0 mm target of Material A. The impact tests were carried out with the gas gun andinstrumented projectile.For each simulation the time step was chosen in order to provide sufficient detail in the resulting contactforce history. In each case at least two hundred points were generated by the model, with the low massimpact events requiring slightly more points to model the sharp peaks. The number of modes used tocalculate target deflection was determined according to Table 2.7, ensuring that the last significant naturalperiod is not larger than the time step. The shock wave parameter for Material A is r = 2.07 J.Is,from Eq. (2.27).Figure 2.10 compares results of a 1.76 mIs, 6.14 kg impact. The simulation used a time step of 20 j.is, and a10 mode solution (Tm,n = 11 .ts) for target deflection. The analysis agrees closely with the experimentalresults in most aspects of the contact force history. The rate of loading and unloading as well as the peakforce are accurately predicted. Oscillations observed early in the impact event are also indicated, to someextent, in the prediction. Post impact examination of the target indicated that the target incurred nopermanent damage.Figure 2.11 compares results of a 2.68 mIs, 6.14 kg impact. Again, the simulation used a time step of 20 ts,and a 10 mode solution (x = 10), for target deflection. The analysis agrees closely with the experimentalresult in the early loading stage of impact. At a contact force of 8 500 N the predicted response is muchstiffer than measured by the instrumented projectile. Subsequent examination of the target indicated thatsome damage, in the form of delaminations had occurred due to the impact. The impact model assumes aperfectly undamaged target, and thus will over-estimate stiffness if softening of the target occurs due todamage.25Figure 2.12 compares results of a 7.70 mIs, 0.314 kg impact. The simulation was performed with a time stepof 5 us, and a 25 mode solution (Tm,n = 4.2 us) for target deflection. This impact event generated a muchmore articulated contact force history, however the analysis was again quite accurate. Oscillations in thecontact force history are the result of higher frequency modes in the target plate being excited by the highervelocity projectile. The target did not incur permanent damage due to the impact.Figure 2.13 compares results of a 11.85 mIs, 0.314 kg impact. The simulation used a time step of 5 JIs, anda 25 mode solution for target deflection. As in Figure 2.11, the analysis agrees closely with the experimentalresult early in the impact event. At times greater than 0.4 ms, the predicted response does not agree withexperiment. This target experienced damage due to the impact event.The model is not able to predict the dynamic local response of the target in the early stages of the low massimpact events. The measured responses shown in Figures 2.12 and 2.13 show a peak in the contact force ata time less than 0.1 ms, that is not reflected in the model predictions. Early in the impact event, the responseis dominated by local indentation, as the target has not had sufficient time to deflect. The amount ofindentation increases rapidly in this phase, likely giving rise to large inertial forces. The Hertzian relationused by the model to describe local indentation, does not include any dynamic effects.Damage in this type of impact may be initiated when the contact force exceeds a critical value. Themeasured results for impacts where damage has occurred, deviate noticeably form the model predictionsafter the contact force exceeds 10 000 N. Figures 2.11 and 2.13 provide examples of this. In the impactevents not leading to damage, the contact force does not exceed 10 000 N.The model is able to accurately predict target response due to non-penetrating events, if damage caused bythe impact is minimal. Results suggest that a contact force in excess of a critical value will cause damage.The model is able to predict contact forces up until the onset of damage. Agreement with experimentalresults and with published results validates the solution for target deflection, as well as the non-penetratingimpact model. The target solution can now be used to describe the global behaviour in models that simulatepenetrating impacts.262.5 Tables and FiguresTable 2.1: Material used in experiments: Material A.System: T800H/3900-2 CFRPLay-up: [45/90/-45/0]3sE11: 129 GPa E22: 7.5 GPaG12: 3.5 GPa G23: 2.6 GPaV12: 0.33 p: 1540 kg/rn3h: 4.65 mmTable 2.2: Static indentation best fit data, as used in Figure 2.3.Curve Exponent Stiffness(N/mn)a 1.5 6.Ox 108b 1.7 2.5x109Table 2.3: Data used for target deflection convergence study, from Qian and Swanson [19901.Material: T300/934 carbon - epoxyLay-up: [0/90/0/90/0]Size: 200mm x 200 mm x 2.69 mmE11: 120 GPa E22: 7.9 GPaG12: 5.5 GPa G23: 5.5 GPaV12: 0.30 p: 1580 kg/rn3‘U: 1.20 jis27Table 2.4: Calculated natural frequencies for target described in Table 2.3.Mode # Tm,n Normalized Tm,nx (rad/sec) (jis) t1 302 3310.77 3.4405 7324 136.54 2.05510 26923 37.14 1.49020 84802 11.79 0.99140 214679 4.66 0.58880 466777 2.14 0.251160 955815 1.05 -0.061Table 2.5 : Impact conditions used for convergence study of impact model, from Qian and Swanson [1990].Target As in Table 2.3Impactor Diameter: 12.7 mm p: 7960 kg/m3Mass: 8.537 g V0: 3.0 mIs28Table 2.6: Convergence of peak contact force for Qian and Swanson [1990] and present impact models, as afunction of time step (0.1 jis used as reference).Qian and Swanson [1990] Present ModelAt # Modes’ Fm Error # Modes’ Fm Error(ps) (N) (%) (N) (%)10 100 179.3 37.42 5 302.2 1.105 100 224.8 21.54 10 300.7 0.602 100 255.4 10.86 15 299.5 0.221 100 269.7 5.86 20 299.3 0.130.5 100 279.4 2.48 30 299.1 0.060.2 100 285.3 0.42 50 298.9 0.010.1 100 286.5 0.00 100 298.9 0.001 Indicates number of non-zero modes used in each directionTable 2.7: Natural frequencies calculated for Material A.Mode # Tm,nX (rad/s) (j.ts)1 18088 347.365 253048 24.8310 572781 10.9720 1189696 5.2840 4107555 1.5329ØDw+15000z100005000Displacement (mm)Figure 2.2: Static indentation experiment compared with Hertzian indentation law (hemisphericalindenter, 025.4 mm).hFigure 2.1: Non-penetrating impact nomenclature.20000II025.4 mm steel indenterMaterial ‘A’• Static indentation experiment•—Hertzian indentation•••••••••00.00 0.25 0.50 0.75 1.00302000015000z10000‘SC500000.00 1.00Figure 2.3: Static indentation experiment compared with modified Hertzian indentation law, and with bestfit power law (hemispherical indenter, 025.4 mm).h/2h12025.4 mm steel indenterMaterial ‘A’Static indentation experiment—a) Hertzian best-fitb) Non-Hertzian best-fit0.25 0.50 0.75Displacement (mm)z, w------a1Figure 2.4: Target plate nomenclature.31200C0.5 1.0 1.5 2.0 2.5 3.0Normalized Time t*1510500.0Figure 2.5: Convergence of target deflection, step load applied at center of target, 160 mode solution usedas reference.543210-0.5 0.0 0.5 1.0 1.5 2.0Normalized Time t*Figure 2.6: Convergence of target deflection, step load applied over small patch, 160 mode solution used asreference.326Normalized Time tFigure 2.7: Convergence of target deflection, step load applied over entire target area, 160 mode solutionused as reference.20-0.5 0.0 0.5 1.0 1.5 2.033EE0C300‘ 200000100loading freephase vibration0.50.40.30.20.10.0025.4mm hemispherical projectile200mm x 200mm x 2.69mm target ‘Qian and Swanson: Rayleigh-RitzSun and Chen: FEMPresent model0.0 0.5 1.0 1.5Time (ms)Figure 2.8: Target deflection history of impact event (see Table 2.5 ) and resulting free vibration; presentmodel compared with published results.I‘‘I‘I‘I025.4nun hemispherical projectile200mm x 200mm x 2.69mm target- - -- Sun and Chen: FEM\, \ — — — Qian and Swanson: analytical\\\ Qian and Swanson: Rayleigh-RitzPresent model/100.00 0.05 0.10 0.15 0.20Time (ms)Figure 2.9: Contact force history of impact event (see Table 2.5 ); present model compared with Qian andSwanson [1990] results.34Figure 2.10: Contact force history for 6150 g, 1.76 mIs impact event; present model compared withexperimental results (no target damage).1250010000Time (ms)Figure 2.11: Contact force history for 6150 g, 2.68 mIs impact event; present model compared withexperimental results (target damaged).76.2 mm x 127.0 mm target025.4 mm hemispherical projectileM=6.lSkg Vo=1.76m/sza0c-)12500100007500500025000• Experiment—Model prediction0 1 2 3 4Time (ms)z0a0c-)75005000250000 1 2 3 43512500zC.?ci0z0C.?‘S00.25 0.50 0.75Time (ms)1000075005000250000.00 1.00Figure 2.12: Contact force history for 314 g, 7.70 mIs impact event; present model compared withexperimental results (no target damage).125001000075005000250000.00 1.00Figure 2.13: Contact force history for 314 g, 11.85 mIs impact event; present model compared withexperimental results (target damaged).0.25 0.50 0.75Time (ms)36CHAPTER THREEImpact of Blunt ProjectilesThe impact behaviour of blunt projectiles is modelled in this chapter. A blunt projectile is one thatpenetrates a target by forming a plug of target material, and then ejecting the plug through the distal side ofthe target. Hemispherical projectiles, as examined in Chapter 2, impacting a target with sufficient energycan behave in a blunt manner, as can projectiles with large angle conical tips. The model presented will dealspecifically with cylindrically shaped shaped projectiles.Awerbuch and Bodner[ 1974] presented an analytical model describing a plugging type of penetration. Thismodel, intended for metallic targets, is based on failure mechanisms that do not correspond directly withthose observed in composite targets. A new model is presented that includes some aspects of the Awerbuchand Bodner model such as inertia force and added mass due to plug formation.The static penetration behaviour of blunt indenters will be examined and used to model the damage causedby impact events. The target deflection analysis presented in Chapter 2 will be applied to the new bluntmodel. Predictions made by the new model will be compared to experimental results, with varying degreesof success.3.1 BackgroundBlunt projectiles, as they penetrate laminates in a static or impact event, can cause a plugging type offailure. Figure 3.1 provides a good example of such an event, in this case a static penetration test usingMaterial B (see Table 3.1). The plug is visible and still intact at the front of the indenter. The plug length isroughly equal to the laminate thickness, and the diameter of the plug matches that of the indenter. A bluntimpact model for this type of material should agree with these observations.37The Awerbuch and Bodner model developed for homogeneous metallic target is simple and appropriateconsidering Figure 3.1. A three stage penetration process composed of plug initiation, plug fonnation andplug ejection, is assumed (Figure 3.2). Developed for targets impacted by rigid, high velocity projectiles,the model uses a simple approach to describe the plugging mechanism, and is formulated in terms of thecontact force between projectile and target. The Awerbuch and Bodner model does not consider globaltarget deformation.In stage one (Figure 3.2a), target material in front of the projectile becomes completely sheared away fromthe surrounding material, and is accelerated to the projectile velocity. Movement of the projectile and plugis resisted with a stress equal to the ultimate compressive stress, c5 , of the target material. Through allstages, the diameter of the cylindrical plug is equal to the diameter of the projectile.Stage two (Figure 3.2b), begins when the sides of the plug are no longer sheared away from the surroundingmaterial. The plug depth, x, as stage two begins is:x=h—b (3.1)where h is the target thickness, and b is defined in Figure 3.2. The value of b must be determinedexperimentally.The plug formed in stage two remains continuous with the target, and plastic shear flow at the plugboundary resists plug movement. This shear flow occurs at a stress, t, equal to the yield stress of the targetmaterial in shear. The normal compressive stress continues to act as in stage one.The plug grows to a maximum size h, after which the normal compressive stress no longer resists projectileand plug movement. In this third stage the area over which shear flow acts remains constant, as inFigure 3.2c. When the strain in the material connecting the plug and the target exceeds the ultimate shearstrain of the material, the plug separates from the target completely. The penetration process is complete atthis point.Throughout stages one and two, of the Awerbuch and Bodner model, an inertial force is assumed to act overthe contact area of the projectile. The inertia arises because as the plug grows, work is done by the38projectile in order to accelerate target material to the velocity of the projectile. This force is a function ofthe projectile velocity and can be significant at high impact velocities.A typical force-time history for a high velocity event, as predicted by the Awerbuch and Bodner model isshown in Figure 3.3. The model conditions are for Material B, impacted by a 4.2 g, 07.82 mm cylindricalprojectile with an initial velocity of 200 mIs. An ultimate stress of 300 MPa, and a shear strength of 200MPa were used as input to the model (see Table 3.2). Predictions of low velocity events yield resultssimilar to Figure 4.3, and do not correspond to measured results from impact tests. Because predictions ofboth low and high velocity impact events is desired, a new model has been developed. The concept of alocal inertia force, and the increasing projectile mass due to plugging, are included in the new model.3.2 Theory of Blunt ImpactA force-displacement curve from a static flexure experiment, using Material B and a blunt indenter(Figure 3.4), provides the basis for a penetration model. Similar to Awerbuch and Bodner, the proposedmodel assumes plug formation, and an inertia force. The new model also includes global targetdeformations, and an initial elastic loading phase.As with the non-penetrating model, a displacement constraint defines the impact problem. The constraintequation is:(3.2)3.2.1 Target DeflectionCalculation of plate deformation, w, is performed as with the non-penetrating model.In stepwise form, the solution for target deflection is:w=w°Q)+C.FQ) (3.3)where w°, and C are as defined in Chapter 2.393.2.2 Penetration ForceIn contrast to the non-penetrating model, an analytical relation describing indentation, or in this casepenetration, is not available. Instead, force-displacement results from static penetration tests are used tocharacterize the local behaviour. Although damage will progress at different rates, it is assumed that thedamage mechanisms leading to penetration in static events also occur in impact events. It is important tonote that the carbon fibre\epoxy system being examined here (Material B), has been shown to be strain rateinsensitive.A typical force-displacement curve, from a static flexure test using a 76.2 mm x 127.0 mm target, is shownin Figure 3.4. The figure shows the experimental results as measured, as well as results adjusted to removethe target deflection component, w.The deflection measured in the static flexure test is adjusted as follows:(3.4)kçwhere W is the adjusted displacement, ö is the measured displacement, F is the contact force, and k, is thestatic flexural stiffness of the target.The static flexural stiffness is calculated according to (see Appendix A):kç =1(3.5)_Cmn sin(mt/2)sin(nit/2)m nBy approximating the shape of the static flexure results with a simple relation consisting of linear segments,the characteristic damage can easily be incorporated into the blunt model. Four points, A through D, definethe static indentation force, as shown in Figure 3.4.40Preliminary examination of specimens following static and impact testing resulted in specific assumptionsregarding the damage states as they occur in penetration. The initial loading stage, up to point A, isconsidered the elastic loading phase, similar to the indentation found in the non-penetrating model.Point A is defined by a displacement ö, and a stress a1:i =-- (3.6)where FA is the contact force at point A, and Apis the frontal area of the projectile.At point A a critical force, FA, is reached, and failure of target material begins. Fibre breakage and matrixcracking cause the material directly under the penetrator to form a plug, and move independently of thesurrounding material. The plug formation continues with a driving force FA, from point A to point B. Thecontact force increases in this region because of a shear force acting on the plug surface. This shear force t,is due to mechanical interlock between the newly formed plug surface and the surrounding target material.As the plug size increases, so does the contact force. Friction between the penetrator and target materialalso begins to act in this phase of penetration.Displacement 2’ and shear stress ‘r define point B:FB—FA(3.7)Awhere FB is the contact force at point B, and A is the maximum plug surface area embedded in the target,i.e.:A=itD.h (3.8)At point B the damage has progressed to the extent that plug size equals target thickness. The sharp drop incontact force, from point B to point C, is due to the removal of FA, which no longer acts once the plug iscompletely formed. Only the shear stress due to mechanical interlock, and friction resist the penetrator andplug. As the plug is pushed out the distal side of the target, the surface area in contact with surroundingtarget material is reduced. A corresponding reduction in contact force is seen from point C to point D.41Point D occurs at a displacement equal to the target thickness, i.e., the plug has been ejected, and at a forceequal to the maximum friction load. Friction begins to act at point A, and increases to a maximum atpoint D, where the maximum amount of penetrator surface is in contact with target material.Point D defines the friction stress, aj, as follows:(3.9)An effort has been made in choosing points A through D to ensure that the area under the equivalentcharacteristic curve, i.e., the energy, is equal to the area under the adjusted static flexure curve. The pointschosen for the indentation test shown in Figure 3.4, are listed in Table 3.3. The energy absorbed by thestatic penetration process, up to point D, bending excluded, is 58 3, which corresponds to the area under theassumed model curve. The resulting model parameters are summarized in Table 3.4.Subsequent examination of specimens has suggested that the damage mechanisms described above may notbe completely valid. Delaminations in the target material local to the penetrator may be a significant causeof the softening or change in slope occurring at point A. Regardless of what damage is occurring in thetarget, the indentation force modelled by points A through D is a valid representation.The parameters listed in Table 3.4 apply only to targets of Material B impacted by 7.84 mm cylindricalprojectiles. Changes in lay-up or thickness, or changes in the size of the projectile require new parametersfor the model. Using stresses instead of contact force to define the points A through D, may reduce themodels dependency on geometry, but static testing is required for different impact conditions.3.3 Blunt Impact ModelIn the new model, the projectile is treated as a rigid body with a changing mass, similar to the Awerbuchand Bodner model. The plug essentially becomes a part of the projectile, allowing rigid body dynamics tobe used to define the motion of the projectile.42The equation of motion of the projectile is:m(t)iX = F (3.10)where m is the mass of the combined projectile and plug, and F is the applied force. This model ignores theacceleration component due to the rate of change of mass.The total applied force, F, is:F=P1(cL)+I+Ff (3.11)where Fq is the force due to indentation, as defined by the static indentation curve, F, is the local inertiaforce, and Ff is the frictional force.3.3.1 Inertia ForceThe inertia force F, is a measure of the force required to accelerate material surrounding the growing plug’,and is calculated as follows:i=Ltpr2i2 (3.12)where p is the density of the target material, r is the radius of the projectile, and x is the position of the plugface.The above expression can be simplified by assuming that x changes at a rate equal to the projectile velocity,thus the inertia force becomes:I =‘-itprc2 (3.13)See Awerbuch and Bodner [1974].433.3.2 Friction ForceFriction acts on the surface of the projectile as it passes through the target, and is calculated as follows:Ff—GfAf (3.14)where Af is the surface area of the portion of projectile which is embedded in the target, and a1 is the frictionstress.For each projectile and target, experimental observation provides an estimate of friction stress.3.4 Results of Blunt Model and DiscussionLow velocity instrumented impact tests were performed using cylindrical 07.82 mm tips. Two targetopening sizes: rectangular 76.2 mm x 127.0 mm, and 025.4 mm round, with specimens all cut fromMaterial B, were used. Force-deflection results from these tests are shown in Figures 3.5 through 3.8.Ballistic tests using 4.2 g, 07.52 mm projectiles, and 025.4 mm round openings (Material B) were alsoperformed. The new impact model was used to predict each of the low velocity and ballistic impact events.In each case the model parameters used, are those listed in Table 3.4. The number of modes and the timestep used in the model were chosen according to the criterion described in Chapter 2. The results aresummarized in Table 3.5.Comparison of measured low velocity results with model predictions indicate that the damage mechanismshave been accurately characterized by the model. The contact force histories in Figures 3.5 through 3.8,indicate that the peak values of contact force and the amount of penetration have been predicted by themodel. As well, the exit velocities in each low velocity impact were accurately predicted, see Figure 3.10.The impact events involving the larger 76.2 mm x 127.0 mm targets indicate that target size has an effect onthe local behaviour, and/or that delaminations are affecting the global response. Figures 3.5 and 3.6 show aninitial peak in contact, at approximately 1.5 mm of projectile displacement, a peak that is not predicted bythe model. Results from impact tests involving the smaller, 025.4 mm targets, show only one peak in thecontact force. Impact events using the larger rectangular targets result in an increased amount of target44deflection, compared to impact events using the 025.4 mm targets. Larger deflections result in larger in-plane strains, which may cause splitting in layers at the distal side of the target. This type of splittingdamage will reduce the local stiffness of the target. Delaminations resulting from high shear stresses willreduce the global stiffness of the target. Both sizes of targets experience high shear stresses that can causedelaminations, however the target size can affect how the stiffness changes with such damage.Each of the impact events resulting in perforation show considerable oscillation in the contact force, atdisplacements in excess of five millimetres. At this stage in the penetration process it is reasonable toassume that most of the plug has been pushed through the target, and that friction between projectile andtarget dominates the behaviour. Vibrations in the target are thought to cause both positive and negativefriction forces. Projectile velocities remain positive in these events, however the target plate may indeed bemoving faster than the projectile. Impulsive types of loading, such as the high contact forces measured earlyin the impact event, will cause a great deal of vibration in the target. The type of friction assumed by themodel does not consider the velocity of the target relative to the projectile, and thus predicts only positivefriction forces. The amount of energy absorbed by friction can be significant (note peaks of up to 6000 N),and is thus important for the model.Although only residual velocities are measured in the ballistic tests, it is apparent that the damagemechanisms are very different compared to the low velocity impact events. The model, which was able topredict the impact response at lower velocities, is obviously misrepresenting the damage at ballisticvelocities. Model predictions of absorbed energy are much higher, in some cases 40 1 higher, than theabsorbed energies measured in the ballistic tests, see Figure 3.10.Figure 3.11 shows the effects of changing target size on the impact response as predicted by the model.Increasing the impact velocity of the high mass projectiles when a large target is used, leads to an increasein the predicted absorbed energy. In contrast, the predictions involving high mass projectiles and smalltargets, show almost no change in absorbed energy. This trend can be explained with reference to globaltarget response. The impact events involving large targets absorb more energy at higher velocities because45of the energy stored as elastic strain energy due to bending. The small target absorbs energy mainly via theindentation force and friction force, both of which are not functions of velocity.The ballistic predictions shown in Figure 3.11 follow a trend similar to the large target, low velocity impactevents, while absorbing much less energy. The ballistic predictions are much lower than the low velocitypredictions because the ballistic events involve very little target deflection. The ballistic event occurs soquickly that the target does not have time to deflect, thus no energy is stored in bending. The increase inabsorbed energy, as ballistic velocity increases, is due to the inertia force described in Eq. (3.13). Thisforce, negligible at lower velocities, becomes significant in the ballistic predictions.Sections through specimens following a static penetration test and a ballistic impact event are shown inFigures 3.12 and 3.13. Extensive delamination has resulted from the static test indicating that the localdamage may indeed be affecting the global behavior. Extensive delamination can also be seen in theballistic specimen, although the plug formation is much more complete indicating a difference in energyabsorption. The ballistic results of this model, although poor, have revealed a lack of understandingconcerning damage as it occurs in high velocity events. The low velocity model predictions are reasonablyaccurate, but have also shown that delaminations and/or splitting in the distal target layers may play a rolein the penetration of blunt projectiles.3.5 Tables and FiguresTable 3.1: Material used in experiments: Material B.System: 17/8551-7 CFRPLay-up: [-45/90145/02]4sE11: 142 GPa E22: 7.9 GPaG12: 4.1 GPa G23: 3.0 GPaV12: 0.34 p: 1540 kg/m3h: 6.15 mm46Table 3.2: Impact conditions used for Awerbuch and Bodner model simulation.Material BTargetG: 300 MPa b: 0.5 mm t: 200 MPaProjectile Dia.: 7.82 mm Mass: 4.2 g V0: 400 mIsTable 3.3: Characteristic force and displacement data derived from Figure 3.4, and used by model.Projectile ContactDisplacement Contact Force StressPoint # (mm) (N) (MPa)0 0 0 0A 0.64 12820 267B 2.23 26110 544C 2.23 13290 277D 6.15 2370 49Table 3.4: Model parameters derived from Figure 3.4; used in impact model to define static indentationforce.47Table 3.5: Experimental and model results for various blunt impact conditions, Material B.Impact Conditions Analysis ExperimentM V0 E0 Target & # modes Vr Ea V Ea(g) (mis) (I) size1 (.ts) x (mis) (J) (mis) (I)1 320 20.5 67 a 20.0 5 0.0 67 0.0 672 320 20.9 70 a 10 10 0.02 703 320 26.9 116 a 5.0 15 15.2 79 15.6 774 308 20.1 62 b 2.0 30 0.02 625 308 28.1 122 b 2.0 30 19.2 65 18.0 726 308 32.0 158 b 2.0 30 24.6 65 23.4 737 4.2 148 46 b 2.0 30 02 468 4.2 189 75 b 1.0 45 112 49 0 759 4.2 265 147 b 0.5 80 211 54 159 9410 4.2 299 188 b 0.5 80 251 56 204 1001 Symbol ‘a’ corresponds to the 76.2mm x 127.0mm target, and ‘b’ corresponds to the25.4 mm target.2 Indicates ballistic limit.48(b)Figure 3.2: Nomenclature for three stage Awerbuch and Bodner model: (a) plug initiation, (b) plugformation, (c) plug ejection.Figure 3.1: Digital image of a Material B specimen following the static penetration test with blunt indenter.a,,(a) (c)49- 300 10- 200400300200zQ00400003000020000100000- 40007.82 mm blunt projectileM=4.2g Vo=400m/s—Contact forceProjectile velocity0.000 0.005 0.010 0.015 0.0201000Time (ms)Figure 3.3: Contact force and velocity histories as predicted by Awerbuch and Bodner model for 400 m/s,4.2 g impact event.\0z0-4-0c-)600500300002500020000150001000050000I\IB 76.2 mm x 127.0 mm target07.82 mm blunt indenter0 Static flexure results0Adjusted static results— Characteristic curve\0I I210004 6 8 10Displacement (mm)Figure 3.4: Force-displacement relations for static penetration, including results as measured from staticflexure experiment, results adjusted to remove target deflection, and characteristic relation used by model.502500076.2 mm x 127.0 mm target7.82 mm blunt projectileM = 320 g Vo = 20.5 mIs• Experiment (did not perforate)—Model (did not perforate)10 15 2Projectile Displacement (mm)Figure 3.5: Force displacement results, 320 g, 20.5 mIs blunt impact: experiment and present model.25000200001500010000 e50000-5000Projectile Displacement (nun)Figure 3.6: Force displacement results, 320 g, 26.9 mIs blunt impact: experiment and present model.)4z0020000150001000050000-50005 )z00c-)76.2 mm x 127.0 mm target07.82 nun blunt projectile) M=320g Vo=26.9m/s•.. I• • / • Experiment (Vexit = 15.6 m/s)• •J•—Model (Vexit = 15.2 mIs)••5125000200001500010000•5000-5000Projectile Displacement (mm)Figure 3.7: Force displacement results, 308 g, 28.1 mIs blunt impact: experiment and present model.2500020000150001000050000-5000Projectile Displacement (mm)Figure 3.8: Force displacement results, 308 g, 32.0 mIs blunt impact: experiment and present model.025.4 mm target07.82 mm blunt projectileM=308g Vo28.lnt!s• Experiment (Vexit = 18.0 m/s)—Model (Vexit = 19.2 m/s)0r523000025000‘ 2000015000c 10000500000 5 10 15Projectile Displacement (mm)Figure 3.9: Contact force history for high velocity blunt impacts, as predicted by impact model.10090a060504040 50 60 70 80 90 100Energy Absorbed in Experiment (J)Figure 3.10: Comparison of predicted vs. measured absorbed energies in various impact events.Vo=189m1s ——— Vo=265m1s Vo=299m/s025.4 mm targetProjectile (to scale)2F.QLength = 12.7 mmM = 4.2 g2032Og projectile, 76.2mm x 127.0mm target308g projectile, 25.4mm dia. target4.2g projectile, 25.4mmtarge-model over-predictsunder-predicts53225200• 320g projectile, 76.2mm x 127.0mm targetA 308g projectile, 25.4mm dia. target175 • 4.2g projectile, 25.4mm dia. target150125- non-perforating —perforating100 -75-- ————.A- —50-- ——--.250— iiiilii,iIiiti I I I I I I I0 25 50 75 100 125 150 175 200 225Impact Energy (J)Figure 3.11: Energy absorbed versus impact energy for various impact conditions, as predicted by model.54Figure 3.12: Image of section through specimen following a static penetration test; blunt indenter.55Figure 3.13: Image of section through specimen following a ballistic impact event; blunt projectile,V0=265 rn/s.56CHAPTER FOURImpact of Conical ProjectilesThe impact behaviour of conical tip projectiles is modelled in this chapter. The penetration of sharplypointed projectiles causes a unique type of damage in the target material, characterized by movement ofmaterial laterally and along the axis of penetration. Fragments, or other irregularly shaped projectiles cancause similar types of damage, although only regular shaped cones are examined in this thesis.A model describing the impact of conical projectiles on Kevlar\polyester targets, presented byZhu et al [19901, is examined. This model assumes plastic type material behaviour with degradation ofmaterial strength due to damage. Global target deflections are calculated using a finite difference solution.The Zhu model requires a mainframe computer to perform simulations, thus a new model is developedwhich runs efficiently on a personal computer.The new model assumes the same type of plastic material behaviour, and includes the same damagemechanisms as the Zhu model. As well, the bulge mechanism suggested by Zhu will be extended, anddamage due to target deflection will be added in the new model. The ‘bluntness’ inherent in all real conicaltipped projectiles is also considered. The global model, presented in Chapter 2, and included in the bluntmodel, will also be used in the new conical model.Experimental results from low velocity and ballistic impact tests will be compared to predictions made bythe new model.574.1 BackgroundThe Zhu model was developed for unidirectional Kevlar/polyester type laminates with specially orthotropiclay-ups. Three stages of penetration are assumed: initial undamaged penetration, damaged penetration, andperforation. In each stage a perfectly plastic type behaviour is assumed locally in the target material.Undamaged penetration is governed by a simple force balance. Damaged penetration is initiated by fibrefailure at the distal side of the target caused by bulging. Subsequent damage is caused by deformation of thetarget material surrounding the penetrating projectile. The increasing damage reduces the strength of thetarget, until only friction resists projectile movement.Zhu calculated target deflection using a finite difference scheme. This method requires a mainframecomputer because of the large amounts of memory needed, and thus cannot be used on a personal computer.Results from experimental studies conducted by Zhu et al were not suitable for comparisons with the Zhumodel or the new model presented in this chapter, because of the inherent difficulties in the measurement ofhigh velocity impact events. Projectile displacements were measured by Zhu using an optical technique, andthen differentiated twice to produce contact force histories. Sufficient experimental data is available frominstrumented impact tests (Delfosse et al) for comparisons with the new model.4.2 Theory of Conical ImpactThe result of a static penetration test involving a conical indenter is shown in Figure 4.1. Material has beenpushed away from the indenter, and considerable splitting has occurred at the back face of the specimen.The penetration mechanisms used in the Zhu model have been adapted to the new model because theyattempt to describe the type of penetration seen in Figure 4.1.In the new model damage occurs in the target material throughout the penetration process, instead of only inthe latter stages of penetration, as assumed by Zhu. Three types of strain contribute to damage: strain due totarget deflection, strain local to the projectile, and strain due to bulging of the target. The analytical solution58for target deflection described in Chapter 2 is used. As well, the model takes into account the “bluntness” ofreal conical projectiles.As with the non-penetrating and blunt impact models, the local and global displacements define the impactproblem. Therefore, the constraint equation is:(4.1)where t, is the projectile displacement, c’ is the penetration, and w is the central deflection of the target.4.2.1 Target DeflectionCalculation of plate deformation, w, is performed as with the non-penetrating model.In stepwise form, the solution for target deflection is:w=w°Q)+C.FQ) (4.2)where w°, and C are as defined in Chapter 2.4.2.2 Penetration ForceThe penetrating projectile causes plastic flow in the surrounding target material. The target is assumed tobehave in a perfectly plastic manner, resisting penetration with a stress equal to the effective strength of thematerial.Resolving the stresses applied to the projectile in the direction of travel (normal to the target surface), thefollowing force balance applies:p3eAp (4.1)where F is the force due to penetration, Ge is the effective strength of the target material, and A is theprojected area of the embedded penetrator, as in Figure 4.2.The projected area determines the contact force to a great extent, and is a function of target and projectilegeometries, as well as the amount of penetration.594.2.3 Effective StrengthThe target material resists movement of the projectile with an effective strength, which is a function of thedamage in the target. In an undamaged target, the effective strength is equal to the ultimate strength of thematerial.The model that assumes damage occurs only in the region directly in front of the projectile, and is the resultof tensile strains. The damage mechanisms assumed by the model are much like fibre breakage. However,no attention is paid to the orientation of fibres in the laminate, thus damage is attributed, in a general way,to target material only. Three distinct deformations cause damage in the target: displacement of materialdirectly surrounding the projectile, bulge formation on the distal side of the target, and target deflection.Damage caused by these deformations are considered equal when calculating degradations in the effectivestrength.Each of the three strains are calculated at evenly spaced points in the path of the projectile. Figure 4.3shows how a damage grid, of n2 points, is plotted on the target material in the path of the projectile. Eachpoint on the grid represents the surrounding area of material (shaded region of Figure 4.3). The sum of allthree strains is compared with an ultimate or breaking strain of the target material. Points experiencingstrain in excess of the fibre breaking strain are considered to be damaged, and exhibit no strength. Theability of the target to resist the projectile is due to undamaged material only.The effective strength of the target material is:Ge =(ldm)Gu (4.2)where is the ultimate strength of the material, and the damage factor dm is calculated according to:dm= (4.3)where Nb is the number of points that are considered damaged, and N is the total number of points.The ultimate strength of the target material can be estimated from a variety of static tests. Figure 4.4 plotsthe average stress seen by the indenter in a number of static penetration tests. The early stages of60penetration, i.e., 0-4 mm, are examined so that the maximum, or undamaged, material strength can beestimated. A penetrator tip with one millimetre of bluntness, i.e., one millimetre truncated from the end ofthe cone, is assumed for each test.In each case, the measured contact force is used to determine the effective stress as follows:(4.4)The estimated effective strength of Material B is 1350 MPa. This value is highly dependent on the amountof bluntness assumed. Static and dynamic compression tests, have shown the ultimate strength forMaterial B is approximately 850 MPa. This provides a lower bound for this model because the specimens inthe compression tests are small and unconfined, whereas the material resisting conical penetration isconfined by rest of the target.The total strain is calculated as follows:E=Eg+El+Eb (4.5)where Lg is the strain due to global target deformations, e is the local penetrator strain, and Lb is the straindue to bulging.Global StrainThe in-plane strain due to global deformations in the target plate is calculated according to:(4.6)where v is the in-plane displacement of the target plate in the y direction.The in-plane displacement, v, is defined in Chapter 2.Local Penetrator SfrainStrains are induced in the target material directly surrounding the penetrator, as material is displacedlaterally and in the direction of penetrator movement.61A line in the target material is examined as it is displaced by the indenter, Figure 4.5. The line is initiallystraight, passing through points A and B, at a distance z from the mid-plane of the target, and r0 from theaxis of the penetrator.The initial length of this line is:10 =2I(zotanI3)2_, (47)where z0 is the distance of the line from the tip of the conical penetrator, and 13 is the half-angle of theprojectile cone.The line is assumed to remain fixed at the points A and B, the intermediate length being strained as thepenetrator moves into the target.The length of the line when the penetrator is at a depth c from the top surface of the target, is:1 = 2—---sin sin 13cos’“(4.8)cos13 [ z0tani3 JThe strain in the target material is then easily found using:(4.9)Bulging SfrainThrough the process of indentation, the conical shape displaces target material toward the back face of thetarget. Experimental observations by Zhu et al and others have shown that a bulge forms on the distal sideof the target.Assuming incompressibility, the volume displaced by the cone, Va., must be equal to the volume containedin the bulge, V. The shape of the bulge is assumed to be spherical with a varying radius.The volume displaced by the cone, as shown in Figure 4.6, is:vjtw.2 =x3tan213 (4.10)62The volume of the spherical cap is:y.=zi2(3R_z1) (4.11)where z1 is the height of the bulge, and R3 is the radius of the sphere, as shown in Figure 4.6.The target material deforms spherically from the tip of the projectile to the bulge on the distal side of thetarget. The radius of deformation R’ varies linearly from zero at the penetrator tip, to R at the distal side ofthe target, see Figure 4.7. Thus each fibre lying between the projectile tip and the distal side of the targetwill experience a unique strain due to bulging, depending on its position.An initially straight line from points A to B, has a length:10 =2y =2Jij2_x0 (4.12)where r1 is defined in Figure 4.6, and y,, x0 are defined in Figure 4.7.Once again the end points of line AB remain fixed.The length of the line as it displaced to the surface of a sphere with radius R is:1=2•Rc (4.13)where yis defined in Figure 4.7.The strain due to bulging is:E1l0’h1 (4.14)10 Y0634.3 Conical Impact ModelThe projectile is assumed to behave as a rigid body, similar to the non-penetrating and blunt models.The equation of motion of the projectile is:mA=F (4.15)The applied force, for this model, is as follows:F=F(cL)÷Ff (4.16)where F1., is the force due to penetration as described in Section 4.2.2, andF1is the friction force.The concept of an inertial force, as used in the blunt model, can be applied to the conical impactor as well.Awerbuch and Bodner suggested an inertial force for conical projectiles as follows:j=Ø.-pAV2 (4.17)where A is the projected area of the embedded projectile, as in Eq. (4.1), and is a shape factor defined as:(4.18)where 1 is the half-angle of the conical projectile.The new model is intended for conical projectiles with relatively small half angles, leading to negligibleinertia forces. The conical projectiles used for comparisons in this chapter have sharp tips with half-anglesof 17.5°, resulting in shape factors of 0.09. The inertia force is ignored in the new model.4.3.1 Friction ForceFriction acts on the surface of the projectile as it passes through the target. The friction stress is a parameterof the projectile /target system and is assumed to acts only on the shaft of the projectile.64The total friction force is a function of the surface area imbedded in the target:Ffaf.Af (4.19)where A is the surface area of the portion of projectile shaft which is imbedded in the target, and a is thefriction stress.4.4 Results of Conical Model and DiscussionLow velocity instrumented impact tests using conical tips with a shaft diameter of 7.62 mm and total coneangle of 35°, were performed. Two target sizes: rectangular 76.2 mm x 127.0 mm specimens, and025.4 mm round specimens, cut from Material B, were used. Ballistic tests using 4.2 g, 07.52 mm, 37°projectiles, and 025.4 mm round openings (Material B), were also performed. The new conical model isused to predict each of the impact events. In each case the model uses an ultimate strength of 1350 MPa, asobtained from Figure 4.4. All other material parameters can be found in Table 3.1. The number of modesand the time step used in the model were chosen according to the criterion described in Chapter 2.Experimental results and predictions are summarized in Table 4.1. Low velocity model predictions arecompared to the contact force as measured by the instrumented projectile, in Figures 4.8 through 4.11.Contact force histories are not available from the ballistic tests, and are thus only model predictions areshown in Figure 4.12. Figure 4.13 summarizes the model predictions in terms of the energy absorbed. Incontrast to the blunt model predictions, the conical model accurately predicts the ballistic events, but not thelow velocity impacts.Examining the force-displacement results of the low velocity impact events, it is apparent that the modeldoes not accurately predict damage. All peak forces are considerably over predicted by the model. Targetmaterial effective strength Ge, as predicted by the model, is compared with the average contact stressmeasured in static experiments (Figure 4.14). Predicted target strengths cannot be compared with resultsfrom impact tests because only the projectile displacement As,, and not the penetration a, is measured by the65instrumented projectile. The predicted target strength, initially agrees with the measured values (about1350 MPa) because the parameter Ge is obtained from the static flexure test (Figures 4.4 and 4.14).The disagreement in Figure 4.14, between one and seven millimetres of penetration, indicates that thedamage due to bulging as assumed by the model is not valid. A majority of the strength reduction predictedby the model, at these levels of penetration, is due to bulging strain. Experimental observations from staticpenetration tests have also suggested that bulging on the distal side of the target does not form until three tofour millimetres of penetration has occurred. Much of the material displaced by the penetrator forms a bulgeon the impact side of the target, and this type of bulge is not considered by the model.At penetration levels in excess of seven millimetres, the single damage factor used to degrade the targetstrength, may not be sufficient. According to Figure 4.14, at, the model prediction of effective strength isonly slightly higher than the measured value, however examination of the contact force histories reveal howimportant this error is. The target seems to have a load carrying capacity of approximately 7 000 N to8 000 N. Although the material in contact with the penetrator likely still has sufficient strength to resist thepenetrator, the supporting target structure may be failing due to other mechanisms.As in the blunt impacts, different damage mechanisms appear to result from ballistic type impacts,compared to low velocity events. This is evident from the increase in measured absorbed energy in theballistic tests (Table 4.1). The model accurately predicts these higher absorbed energies, however it isdifficult to gauge the validity of the model without a method of determining damage in the impactedspecimens.Figure 4.15 shows the effects of changing target size on the impact response as predicted by the model. Aswith the blunt model, increasing the impact velocity of the high mass projectiles when a large target is used,leads to an increase in the predicted absorbed energy. The predictions involving high mass projectiles andsmall targets, show almost no change in absorbed energy. As explained in Chapter 3 this trend is due toglobal target response.66In contrast to the blunt model, the ballistic predictions in Figure 4.15 show a slight decrease in absorbedenergy as the impact velocity is increased. Note that the inertia force causing an increasing trend for theblunt model is not used in the conical model. The ballistic predictions are lower than the low velocitypredictions because the ballistic events involve very little target deflection.Examination of specimens following impact events as shown in Figures 4.16 and 4.17, reveal weaknesses inthe model assumptions. The damage resulting from the non-perforating impact, Figure 4.16, is verylocalized and fibre breakage is evident, however the bulge formation assumed by the model is not evident.The specimen shown in Figure 4.17, having undergone a perforating impact, has been damaged in theregion beyond the path of the projectile. Assumptions made by the new conical model concerning damageoccurring in the path of the projectile, and as a result of bulge formation are not accurate, indicating that theballistic impact predictions made by the model are not reliable.674.5 Tables and FiguresTable 4.1: Experimental and model results for various conical impact conditions (025.4 mm opening,Material B).Impact Conditions Analysis ExperimentM V0 E0 Target lit # modes Vr Ea Vr Ea(g) (mis) (J) size1 (jis) x (mis) (J) (mis) (I)1 320 20.3 66 a 5 20 0 66 0 662 320 26.8 115 a 5 20 02 1153 320 30.2 146 a 5 20 10.5 128 19.5 854 308 26.8 111 b 5 20 02 1115 308 29.1 130 b 5 20 11.4 110 15.3 946 308 31.5 153 b 5 20 15.2 110 19.9 927 4.2 156 51 b 0.5 80 0 51 0 518 4.2 235 116 b 0.5 80 02 1169 4.2 316 210 b 0.5 80 217 111 228 1011 Symbol ‘a’ corresponds to the 76.2 mm x 127.0 mm target, and ‘b’ corresponds to the25.4 mm target.2 Indicates ballistic limit.68Figure 4.1: Material B specimen following static penetration test’ h conical indenter.6913a(a) (b)A11,(c) (d)Figure 4.2: Conical penetration nomenclature and typical contact area projections.70c)projectileDamage grid: n x n pointsn-i @ D/2(n-i)Figure 4.3: Schematic showing damage grid used for conical model.71Static indentation0 Static flexure - 76.2mm x 127.0mm targetStatic flexure - 25.4mm dia. target0 1 2 3 4Figure 4.5: Nomenclature used for calculation of local strain surrounding conical projectile.160013501200A800AA4000Displacement (nun)Figure 4.4: Average contact stress as a function of conical indenter displacement, for various static tests.zacx72Dtarget plate•h/2h/2xf3 conical indenterzFigure 4.6: Nomenclature used for calculation of bulge radius.73xinitial positiondeflected positionzFigure 4.7: Nomenclature used for calculation of bulging strain through the target thickness.7410 15 20Projectile Displacement (nun)Figure 4.8: Contact force history of 320g, 20.5 mIs, conical impact event; experiment and present model.125001000010 15 20Projectile Displacement (mm)Figure 4.9: Contact force history of 320g, 30.2 mIs, conical impact event; experiment and present model./z01250010000750050002500076.2 mm x 127.0 mm target07.82 mm conical projectileM = 320 g Vo =20.3 mIs• Experiment (did not perforate)—Model (did not perforate)/0 5 25 30z0Cc-)76.2 mm x 127..0 mm target07.82 mm conical projectileM=320g Vo=30.2mIs• Experiment (Vexit = 19.5 mIs)Model (Vexit = 10.5 m/s)7500500025000//0 5 25 30751250010000z7500500025000-0 5 10 15 20 25 30Projectile Displacement (mm)Figure 4.10: Contact force history of 308g, 29.1 mIs, conical impact event; experiment and present model.1200010000‘ 80006000c.) 4000200000 5 10 15 20 25 30Projectile Displacement (nun)Figure 4.11: Contact force history of 308 g, 31.5 mIs, conical impact event; experiment and present model.025.4 nun target07.82 mm conical projectileM=308g Vo=29.lmIs• Experiment (Vexit = 15.3 m/s)—Model (Vexit = 11.4 m/s)7612500z001000075005000250000 5 10 15 20 25Projectile Displacement (mm)Figure 4.12: Model prediction of contact force history of 316 mIs, 4.2g conical impact event.3014012010080604040 60 80 100 120 140Energy Absorbed in Experiment (J)Figure 4.13: Comparison of measured and predicted absorbed energies for various conical impacts.7715001000•00 5 10 15 20 25Penetration (mm)Figure 4.14: Average stress surrounding penetrator, due to conical penetration/impact.225• 320g projectile, 76.2mm x 127.0mm target200A 308g projectile, 25.4mm dia. target175 • 4.2g projectile, 25.4mm dia. target150125 .Er_ -*--—-——100 non-perforating N75 irforating50250 I I I I I.0 25 50 75 100 125 150 175 200 225Impact Energy (J)Figure 4.15: Energy absorbed due to impact of conical projectile: experiment and model.Static flexure experiment: 76.2mm x127.0mm targetStatic flexure experiment: 25.4mm dia.targetModel: 30.2m/s - 320g projectile,76.2mm x 127.0mm targetA__sa\‘_‘- - - - Model: 29.lmIs - 308g projectile,25.4nuu dia. target— —— Model: 316m1s - 4.2g projectile,25.4mm dia. target78Figure 4.16: Section through specimen following non-perforating impact event; conical projectile,V0= 156 mIs.Figure 4.17: Section through specimen following ballistic event; conical projectile, V0 = 316 mIs,Vexit = 228 mIs.79CHAPTER FIVEConclusions and RecommendationsThree analytical models describing the impact response of composite targets, have been developed. Eachmodel is simple and efficient, appropriate for use in an engineering environment. To this end all threemodels have been implemented in a user-friendly computer code.Each model assumes that the impact event can be divided into two distinct regimes: local behaviour,characterized by indentation or penetration depending on the model, and global target deflection. The localand global solutions are coupled by the models only through the contact force. The target material is alsoassumed, by the models, to behave in a strain rate independent manner. This assumption allows datagathered from static penetration and low velocity impact tests to be applied directly to high velocity ballisticmodels. The materials studied in this thesis are carbon fibre epoxy systems which have been shown to bestrain rate independent.One model simulates low energy, non-penetrating impacts, while two other models attempt to predictpenetrating impacts. The two penetrating models predict the impact of different shaped projectiles, onedealing with blunt or cylindrical shapes, and the other with sharp conical projectiles. These two shapes arestudied in order to provide bounds for the behaviour of projectiles with unknown or irregular shapes.The first model, which simulates non-penetrating impacts involving hemispherical projectiles, is useful forlow energy impacts. This model is more efficient then previous non-penetrating models, and is the first touse a closed form solution that includes the non-linear Hertzian type indentation law. The model is able toaccurately predict the force-time histories of impact events up to the onset of substantial damage in thetarget. The non-penetrating model is also important in the validation of a modal series solution describing80global target behaviour. The global solution developed includes shear deformation and rotary inertia effects,and was found to agree with published data and experimental results.Convergence of the global solution, and non-penetrating impact model, is much improved in comparisonwith previously published solutions. Using a patch type of loading, the new model converges if anappropriate number of modes are used in the modal plate solution. A simple criterion, in terms of thenatural periods of vibration of the target, is used to ensure convergence.The validity of the non-penetrating model was found to be limited to impacts were damage in the target isminimal. It was found that impact events where a critical contact force is exceeded resulted in damagedspecimens. A validated criteria based on the contact force may enable the model to predict the onset ofdamage. Prediction of the post-damage behaviour may be possible if a plate solution that is capable ofreducing the material properties during a simulation, is developed.A model describing the impact behaviour of penetrating blunt projectiles has been developed, andcompared to experimental results. Aspects of the Awerbuch and Bodner model for metallic targets havebeen adapted to composite targets, and applied to the new model. A characteristic force-displacement curve,obtained from a static penetration test, is used to describe the local penetration response. The globalsolution, verified with the non-penetrating model, is used in this model as well.Model predictions of low velocity impact events agree with experimentally measured results. Absorbedenergies, as well as contact force histories are accurately predicted by the new blunt model. Thecharacteristic force-displacement curve, estimated from static tests, is a reasonable representation of damagein low velocity penetration events.At higher velocities, the blunt model does not predict impact behaviour as it is measured in experiments.The model consistently underpredicts the amount of energy absorbed in high velocity events. It is apparentthat different damage mechanisms play a role in high velocity penetration events, making the characteristicforce-displacement curve invalid. An examination of targets after they have undergone impact tests, may81provide a method of adapting the force-displacement curve so that it may be used to represent the damage inhigh velocity impact events.The effects of friction are also not accurately predicted by the blunt model. An advanced method ofanalyzing the friction between projectile and target which accounts for the negative friction forces observedin the experiments, is required. These type of friction models have likely been previously investigated and aliterature search of this topic may be helpful.The final model describes the impact behaviour of conical projectiles. A model developed by Zhu et al forthe impact of Kevlar/polyester systems, is used as a basis for this new model. A strain based criterion isused to predict the progression of damage as the projectile travels through the target. The initial, undamagedstrength of the target is estimated from static penetration tests. The global target solution developed for thenon-penetrating model is also applied to the conical model. Predictions made by the blunt model for highvelocity impact events are reasonably accurate, compared to experiment.Low velocity impact predictions made by the model do not agree with experiment, indicating that thedamage mechanisms assumed by the model are not valid. In particular, the model predicts a great deal ofdamage due to bulging in the early stages of penetration. Experimental observations have suggested that thetype of bulging assumed by the model does not occur until the later stages of penetration. Accounting for adifferent type of bulge, one that forms on the impact face of the target instead of on the distal side, mayaddress this problem.The concept of a degraded effective material strength, as used by the blunt model, may not be sufficient inthe later stages of penetration. Evidence suggests that the target structure is capable of sustaining amaximum load, regardless of the residual material strength. The model requires another failure mechanismto predict this type of behaviour.Further work and improvements to these three models should focus on the understanding of damage.Examination of the damage states in impacted specimens is critical, with the sectioning and visualinspection of specimens being a simple method. As well, the models must be compared to experimentsinvolving a range of target and projectile geometries. This is particularly important for the blunt model,82which at this point requires static testing for each projectile and target combination. The effect ofgeometrical changes in both the projectile and target should be dealt with by each model.83CHAPTER SIXReferencesAbrate, S., “Impact on Laminated Composite Materials,” Appl. Mech. Rev. Vol. 44, no. 4, pp. 155-190,April 1991.Awerbuch, J. and Bodner, S.R., “Analysis of the Mechanics of Perforation of Projectiles in Metallic Plates,”mt. J. Solids and Structures, Vol. 10, 1974, pp. 67 1-684.Cantwell, W.J., and Morton, I., “The Impact Resistance of Composite Materials - A Review,” Composites,September, 1991, pp. 347-362.Christoforou, A.P. and Swanson, S.R., “Analysis of Impact Response in Composite Plates,” mt. J. Solidsand Structures, Vol. 27, No. 2, 1991, pp. 161-170.Dahan, M. and Zarka, J., “Elastic Contact Between a Sphere and a Semi-Infinite Transversely IsotropicBody,” mt. J. Solids and Structures, Vol. 13, 1977, pp. 229-238.Delfosse, D., Pageau, G., Bennet, R., and Poursartip, A., “Instrumented Impact Testing at High Velocities,”Journal of Composites Technology and Research, Vol. 15, No. 1, Spring 1993, pp. 38-45.Delfosse, D., Vaziri, R., Pierson, M.O., and Poursartip, A., “Analysis of The Non-Penetrating ImpactBehaviour of CFRP Laminates,” Proc. 9th mt. Conf Composite Materials, Madrid, Spain, July 1993.den Reijer, P.C., “Impact on Ceramic Faced Armour,” Technische Universiteit Delft, 1991.Dobyns, A.L., “Analysis of Simply-Supported Orthotropic Plates Subject to Static and Dynamic Loads,”AIAA Journal, Vol. 19, May 1981, pp. 642-650.Goldsmith, Werner, Impact, Edward Arnold Publishers Inc., London, 1960.Herrman, G., “Forced Motion of Timoshenko Beams,” Journal ofApplied Mechanics, March, 1955, pp. 53-56.Hull, Derek, An Introduction to Composite Materials, Cambridge University Press, Cambridge, 1981.Jones, Robert M., Mechanics of Composite Materials, 1975.84Langlie, S. and Cheng, W., “A Simplified Analytical Model for ImpactlPenetration Process in Thick Fibre-Reinforced Composites,” Composite Materials: Design and Analysis, Proc. Second mt. Conference onComputer Aided Design in Composite Material Technology, April, 1990, pp. 429-448Love, A.E.H., A Treatise on the Mathematical Theory ofElasticity, Dover, New York, 1926, pp. 180-18 1.Qian, Yibo, and Swanson, Stephen R., “A Comparison of Solution Techniques for Impact Response ofComposite Plates,” Composite Structures, Vol. 14, 1990, pp. 177-192.Ravid, M., and Bodner, S.R., “Dynamic Perforation of Viscoplastic Plates by Rigid Projectiles,” mt. J.Engng. Sci., Vol. 21, 1983, pp. 577-591.Ravid, M., Bodner, S.R., and Holcman, I., “A Two-Dimensional Engineering Model for Perforation ofLayered Targets,” Proc. Tenth mt. Symposium on Ballistics, San Diego, 1987.Spiegel, M.R., Mathematical Handbook ofFormulas and Tables, McGraw-Hill, New York, 1968.Sun, C.T. and Chattopadhyay, S., “Dynamic response of Anisotropic Laminated Plates Under Initial Stressto Impact of a Mass,” Journal ofApplied Mechanics, Sept. 1975, pp. 693-698.Sun, C.T., and Chen, J.K., “On the Impact of Initially Stressed Composite Laminates,” Journal ofComposite Materials, Vol. 19, 1985,pp.490-504.Timoshenko, S.P., “Zur Frage nach der Wirkung eines Stosses auf einen Balken,” Zeitschrft fürMathematik und Physik, Vol. 62, No. 2, 1913, p.198.Trowbridge, D.A., Grady, J.E., and Aiello, R.A., “Low Velocity Impact Analysis With NASTRAN,”Computers and Structures, Vol. 40, No.4, 1991, pp. 977-984.Whitney, J.M., and Pagano, N.J., “Shear Deformation in Heterogeneous Anisotropic Plates,” J. ofAppliedMechanics, Dec. 1970, pp. 103 1-1036.Willis, J.R., “Boussinesq Problems for an Anisotropic Half-Space,” J. Mech. Phys. Solids, Vol. 15, 1967,pp. 331-339.Woodward, R.L. and Crouch, I.G., “A Computational Model of the Perforation of Multi-Layer MetallicLaminates,” Materials Research Laboratory Research Report, MRL-RR-9-89, Oct. 1989.Woodward, R.L., “The Penetration of Metal Targets By Conical Projectiles,” mt. J. Mech. Sci., Vol. 20,1978, pp. 349-359.Yu, Y. “Forced Flexural Vibrations of Sandwich Plates in Plane Strain,” Journal of Applied Mechanics,Sept. 1960, pp. 535-540.85Zhu, G, Goldsmith, W, and Dharan, C.K.H., “Penetration of Laminated Kevlar by Projectiles - I.Experimental Investigation,” mt. J. Solids Structures, Vol. 29, No. 4, 1992, pp. 399-420.Zhu, G, Goldsmith, W, and Dharan, C.K.H., “Penetration of Laminated Keviar by Projectiles - II.Analytical Model,” mt. J. Solids Structures, Vol. 29, No. 4, 1992, pp. 42 1-436.86APPENDIX AStatics and Dynamics of Laminated PlatesIn order to analyze the behaviour of laminated plates we first examine the individual layers, or laminae ofthe plate.A.1 Stress-Strain Behaviour of a LaminaA fibre reinforced lamina is a special case of a generally anisotropic material.Anisotropic, elastic materials behave according to the following three dimensional stress-strain relationship:Gil C11 C12 C13 C14 C15 C16 £a22 C22 C23 C24 C25 C26 £22— C33 C34 C35 C36 £33A 11G23 — C44 C45 C46 2E3(a31 sym C55 C56 2E31C66 2e12in which there are 21 independent elastic constants C. The subscripts i, j, refer to the principle directionsof FigureA.1.87In the case of a unidirectional lamina, i.e. fibres aligned in just one direction, the material has threeorthogonal planes of symmetry. The interdependence of normal stresses and shear strains in Eq. (A. 1.1) iseliminated due to symmetry, and the number of independent constants is reduced to nine:a11 C11 C12 C13 0 0 0 ElIG22 C22 C23 0 0 0 E22C33 0 0 0 E33(A.l.2)C44 0 0 2e3a31 sym C55 0 2E31C66 21For most applications, a lamina can be considered transversely isotropic. In a transversely isotropic materialthere exists one plane in which properties are equal in all directions. In a reinforced lamina, the 2-3 plane isisotropic.The stress strain relation for a transversely isotropic material is as follows:a1 C11 C12 C12 0 0 0 EHa22 C2 C23 0 0 0 s22C22 0 0 0 E(A.l.3)G23 (C22—C3)/ 0 0 2e3a31 sym C66 0 2c31a12 C66 2e12Note that only five independent constants remain in Eq. (A. 1.3).88The stiffness matrix components, C,, can be expressed in terms of standard engineering constantsE1, v, G,:= 1—v23(A.1.3a)C12=”12” 3”13 (A.1.3b)= 1—vv(A.1.3c)E13AC23 23+21\?13 (A.1.3d)E12zC66 =G12 (A.1.3e)whereA1v12v23l3v (A13f)E1In general only in-plane properties of a lamina in plane stress conditions are of interest. This allows us tofurther simplify the stiffness matrix of Eq. (A. 1.3), and define the reduced stiffness, or [Q] matrix:an Qii Q12 0 E11= Q12 Q22 0 e22 (A.1.4)a12 0 0 Q66 2812where the components in terms of engineering constants, are as follows:= E1(A.1.4a)= vE1(A.1.4b)1v 12’ 21= E2(A. 1 .4c)1-v2v21Q66 =G12 (A.1.4d)89Now we consider the behaviour of a lamina in which the direction of reinforcement does not coincide with anatural co-ordinate system {x,y,z}, as shown in Figure A.2.Generating the stress-strain relation involves a co-ordinate transformation, with the result:axx iI Q12 Q16 E0yy = Q12 Q22 Q26 E (A.1.5)0xy Q16 Q26 Q66 2Eywhere the transformed reduced stiffnesses are as follows:Q = Q11 cos’O +2(Q12+2Q66)sinecos2O+Q22sin4O (A.1.5a)= (Qii +Q22—4Q66)sinOcos28+Q12(sin4o +cos4O) (A.1.5b)= Q11 sin8 + 2(Q12 +2Q66)sinOcos2O+Q22 cos4O (A.1.5c)= (Qn —Q12—2Q66)sinecos3O+(Q12—Q22+2Q66)sin3Ocos9 (A.L5d)Q26 =(Q11—Q12—2Q6)sin3OcosO ÷(Q12—Q22 +2Q)sinO cos3O (A.l.5e)Q66 =(Q11 +Q22—2Q1—2Q66)sinOcos2O+Q(sin4O +cos4O) (A.l.5f)A.2 Load-Displacement Behaviour of a LaminateMost applications of fibre reinforced materials require strength and stiffness in more than one direction.Laminates consisting of a series of differently oriented laminae provide a more versatile component. Theanalysis of these laminates can be performed with varying degrees of complexity, however, a few basicassumptions are required.First, the thickness of the laminate must be small compared to the in-plane dimensions, and thedisplacements, in turn, are small compared to the thickness. The laminae are assumed to be perfectlybonded, i.e. the strain field is continuous through layer interfaces. As well, the laminae are considered to bein a plane stress state, and the transverse strain is negligible.90A.2.1 Kirchoff TheoryThe study of isotropic plates by Kirchoff led to a simple type of plate analysis, essentially a two-dimensional study of the plate mid-plane. The extension of this theory to laminates is direct.The in-plane displacement field is of the form:U(X,y,Z,t)=U0_Z— (A.2.1)V(X,y,Z,t)=v0_Z_ (A.2.2)w(x, y, z, t) = w (A.2.3)where u and v are the in-plane displacements in the x and y directions respectively (the naught superscriptindicates a displacement of the mid-plane), and w is the transverse displacement.The definition of strain, in terms of the displacement, is as follows:Ers (A.2.4)For Kirchoff plate theory, only the in-plane strains are needed, thus the strain field can be written:,2ic} (A.2.5)where the middle surface curvatures are:12w 2w a2wl(A.2.6)Given the strains in each lamina we can find the resultant forces and moments in the laminate.91The resultants are calculated in a manner similar to an isotropic analysis, i.e., integrated through thethickness:h/2{N,N,N}= f{,a,a}dz (A.2.7)—h12h/2{M,, M} = z dz (A.2.8)—h/2The stresses can be expressed in terms of strains using the transformed reduced stiffnesses of Eq. (A. 1.5).For a stacked laminate the stresses are not continuous, thus the integration of Eq.’s (A.2.7) and (A.2.8) mustbe performed stepwise through each lamina, e.g. for the normal forces:{N,N,N}= f{ dz (A.2.9)where n is the total number of laminae.Noting that the mid-plane strains are not functions of z, Eq.’s (A.1.5) and (A.2.5) are substituted into theexpression for normal forces, with the result:N Qi 1 Qi2 QI 6 XX Zk K XX ZkN = Q12 Q22 Q26 jdz+ ic,, fzdz (A.2.1O)k=li6 Q26 66 k 2EZk_l 2KZk_I92Performing the integration and arranging in matrix form, the normal forces and moments are as follows:N A11 A12 A16 B11 B12 B16 eA22 A26 B12 B22 B26 £= A66 B16 B26 B66 2e(A 2 11)M D11 D12 D16 ic,M sym D22 D26 icM D66 2iç3,whereA =t( (Zk —Zk_1) (A.2.12)B = t(’)k (z — (A.2.13)D =t(1 (z—z_1) (A.2.14)Components of the [A] matrix are referred to as extensional stiffnesses, as they describe the mid planeextension of a laminate under normal tractions. Bending stiffnesses are contained in the [Dl matrix, and the[B] matrix components are referred to as coupling stiffnesses.For particular lay-ups, in which the lamina properties and orientations are symmetric about the mid-plane,coupling stiffnesses reduce to zero. Even more important, is the case when the coupling stiffnesses and thebend-twist stiffnesses D16,D26, reduce to zero. These laminates are termed specially orthotropic. Normallylaminates do not meet the criterion for D16 = D26 = 0, however, for symmetric lay-ups with n > 5, the bend-twist stifnesses are often negligible.It is these specially orthotropic laminates that will be analyzed.Equations ofMotionGoverning equations of motion are derived in a manner similar to the isotropic case, by examining adifferential element, Figure A.3. For this analysis we will ignore the inertia terms arising from the in-plane93displacements u, v, rotary inertia terms arising from changes in the mid-plane slope and other higherorder terms.Equilibrium is first satisfied for the forces in the z direction, with the following result:aQ aQ(A.2.15)ax aywhere p is the density of the laminate material, h is the thickness of the laminate, and the overdot indicatesdifferentiation with respect to time.Similarly for equilibrium in the x and y directions:aN aN(A.2.16)ax ayaN aN(A.2.17)ay axFor the purposes of this development, we will assume that no in-plane tractions are applied, thusEq.’s (A.2.16) and (A.2.17) are identically satisfied by N = N = = 0.Satisfying equilibrium of moments yields:aM aM(A.2.18)ax ayaM aMQ=—-+------ (A.2.19)ay axDifferentiating Eq.’ s (A.2. 18) and (A.2. 19) and substituting the results into Eq. (A.2. 15) yields a singleexpression for motion of the laminate:a2M a2M a2M2 +2+ 2 +p(x,y,t)=ph’ (A.2.20)ax axay ay94We can now use the load-displacement result of Eq.(A.2.1 1), along with the definitions of plate curvatures,Eq. (A.2.6), to re-write the equation of motion in terms of the displacement of the middle surface w:D11 .-4+2(D1+2D66)a2:2 +D22-= p(x,y,t)— ph (A.2.21)Compare with the isotropic formulation, in which the isotropic bending stiffness D replaces the termsabove:D[4+ 2 :;2 +—] = p(x, y, t) — phi’ (A.2.22)It now remains to solve the equation of motion for a laminate.Static LoadingIn general the loading function p(x,y,t) is, as indicated, a function of time and space. Static loading is aspecial case, when the loading function does not vary in time.A simpler form of Eq. (A.2.21) is then considered:D11 f+2(D12+2D66)22+D22 = p(x, y) (A.2.23)Boundary conditions are necessary to complete the problem, and identical to the isotropic case, one memberof each of the following two pairs must be described along each boundary:w;-+Q,1 (A.2.24)an aswhere the subscripts n and s indicate normal and tangential to the boundary, respectively.95Common boundary conditions include:(1) Simply Supportedw = M =0 (A.2.25a)(2) Clampedw = =0 (A.2.25b)Jn(3) FreeM =-+Q =0 (A.2.25c)A rectangular plate simply supported on all sides, Figure A.5, will have the following boundary conditions:at x = 0 and x = aw = M =0 (A.2.26)at y = 0 and y = bw = M =0 (A.2.27)The static solution for a simply supported laminate is a double Fourier series:=5in(m/a)5in(y/I)4(A.2.28),n=1 (J + 2(D12 +2D66)( ) +D() ]This solution requires that the load can be expressed in a Fourier series as follows:Pmn = s:sp(x, y) sin(micx/a) sin(rnty/b) dx dy (A.2.29)Loads of common interest include uniform loading, i.e. p(x,y) = p0, patch loads, and point loading (seeFigure A.6).96The evaluated results of Eq. (A.2.29) for these types of loading follows:A solution of the form:D114+2(D+2D66)aa:+D22-4=p(x,y).F(t)—phw = w0 e’°(A.2.30a)(A.2.30b)(A.2.30c)(A.2.3 1)(A.2.32)D11 —+2(D1 +2D66):’O+D22-°—phw2w0= 0 (A.2.33)Loading Condition Pmnuniform ‘6Po; m,n =1,3,5,...It mnpatch 16p . m . nItrl . mtc . rntd° sin—sin---———sin——-—sin—--——lc2mn a b a bpoint 4P mitT. . nItrl— sin—sinab a bDynamic LoadingIf the loading function remains fixed in space, but changes in magnitude, a dynamic analysis must beperformed. The loading function is rewritten as the product of a function of space p(x,y), and a function oftime F(t).The equation of motion is then:Again, the procedure is identical to the isotropic case. The homogeneous form of Eq. (A.2.3 1) isconsidered, and a solution is sought that satisfies both the boundary and initial conditions. A rectangularsimply supported specially orthotropic laminate will be considered.is assumed, where 0 is a natural frequency of vibration.Substitution of the assumed form of solution into Eq. (A.2.3 1) yields a governing equation independent oftime:97With regard to the boundary conditions of Eq. (A.2.26) and Eq. (A.2.27), the solution is of the form:w0 = Amn sin(mtx/a)sin(nity/b) (A.2.34)m=1 n=1which becomes the orthogonality function for the plate.Substituting this solution into the governing equation of Eq. (A.2.33), an expression for the naturalfrequency is obtained:Wmn = + 2(D1 +2D66)( + (A.2.35)The homogeneous solution must be complemented by a particular solution involving the loading functionp(x,y)•F(t). The normalizing properties of the orthogonality function are used to find the particular solution.The particular solution will be of the form:w= (A.2.36)m=1 n=1where0 mn = sin(mtx/a) sin(niry/b) (A.2.37)and Tmn(t) is an arbitrary function of time.Noting that:fo”J’0rnn0rs dxdy=O; m,nr,s (A.2.38)we can normalize the equation of motion with respect to the homogeneous solution.This requires multiplying each term in Eq. (A.2.33) by the orthogonality function and integrating throughthe thickness of the plate:[fpIuo nn0 mn0 rs dx dy •Tmn t + s:s:pho mn0 rs dx dy Tmn (t)]m=ln=1 (A.2.39)= s:s:px. Y)ø rs dx dy F(t)98The orthogonality property allows us to drop the summation in Eq. (A.2.39), and we then only need to solvea series of ordinary differential equations.The normalized equation is:(onnTmn(t)+Tmn(t)=_i.__s:j:p(X, y) •Ømfl dx dy (A.2.40)where the homogeneous solution has been applied, and the results of the integration:s:s: dxdy=.!has also been used.Note that in this case the integral on the right hand side of Eq. (A.2.40) matches the Fourier integral used toevaluate Pmfl in the static solution.The solution for Tmn(t) is the well known form for one degree of freedom dynamic systems:w = Tmn(O).sin(Omnt+Tmn(O).cosO),,j,t+ Pmn IF(t)SiflO:Imn(t—T)dt (A.2.42)m=ln=1 P 0mnFor the case when the system is initially at rest, and the applied load is constant with respect to time, thefollowing solution applies:1’m(1—coswmnt) (A.2.43)m=1 n=1 phw mnSubstituting the expression of Eq. (A.2.35), into the solution above, yields:Wpmn(1050mnt)2 4(A.2.44)m=1 n=14[Di !J + 2(D12 +2D66)(f) (-) +D(-) ]99Comparing this result to the static solution of Eq. (A.2.28), we can write:w = w .(i— CoSf:Omnt) (A.2.45)where w. is the equivalent static deflection.A.2.2 Whitney and Pagano TheoryThe Kirchoff theory reduced the three dimensional elasticity problem of a loaded laminate, to a twodimensional analysis of the laminate mid-plane. Whitney and Pagano [19701 investigated the extension ofMindlin’s isotropic plate bending theory to laminated plates.The assumed strain field takes the form:u(x, y, z, t) = u0(x, y, t) + zj’ (x, y, t) (A.2.46a)v(x, y, z,t) = v0(x, y,t)+ ZN’ (x, y,t) (A.2.46b)w = w(x, y, t) (A.2.46c)where ‘qJ and are the cross sectional rotations in the x and y directions respectively (see Figure A.7).The strains are defined as in the Kirchoff type theory, Eq. (A.2.5), and the transverse shear strains aretherefore:au2E1 =——=N’--— (A.2.47)az a atav a=—+—=qi,,+— (A.2.48)az a.100The in-plane forces, and moments are described by the constitutive equation used for the Kirchoffderivation, where the curvatures are redefined as:= J’- (A.2.49a)aNJlC =—s- (A.2.49b)a))K3, (A.2.49c)2ay ax)In addition the transverse shear force is related to the shear strains by the following:IQ ‘1 rA A45112 1= k K YZ (A.2.50)Qj LA45 A55ji2ejwhere k is a parameter first used in the Mindlin isotropic, shear deformation theory.Satisfying equilibrium in the transverse direction yields the following:ph (A.2.51)ax ayEquilibrium of moments in the x and y directions yields:aM aM(A.2.52a)ax ayM M—-+-—--—Q =I’ji, (A.2.52b)ay axwhere the rotary inertia is defined as I=Again, we assume that no in-plane tractions are applied thus the in-plane equilibrium is identically satisfied.101Substituting the constitutive relations into Eq.’s (A.2.51) through (A.2.52b) yields three equations ofmotion:aW +(D12 +D66 —kA55(14J+J=Iiix (A.2.53a)____ ____( aD11 2’ ÷(D12+D66) 2 —kA4J +-—J = (A.2.53b)aN’ aw (aw awkA55(__L++++ p(x, y, t) = phi (A.2.53c)Static LoadingFor a constant applied load, we can ignore the inertial terms of the equations of motion. Solving theequations requires boundary conditions, which we will assume to be hinged on all sides. In this case, theshear rotations normal to the boundary are zero, and the slope is unrestrained.For the rectangular plate of Figure A.5 the prescribed boundary conditions are:at x = 0 and x = aw=M=W=O (A.2.54)at y = 0 and y = bw=M=W=O (A.2.55)The following solutions satisfy the simply supported boundary conditions:11J=U,,cos(micx/a)sin(n’Jt y/b)N’ = EVmn sin(mic x/a)cos(nic y/b) (A.2.56b)w = Wmn sin(mit x/a) sin(nit y/b) (A.2.56c)102Substituting the above solutions into the equations of motion, Eq. (A.2.53a-c), the following set of linearequations results:111 112 L13 Umn 0‘12 ‘p22 1.23 Vmn = 0 (A.2.57)‘13 1.23 ‘13 Wmn Pmnwhere the load has again been expressed as a Fourier series.The components of the [L] matrix are as follows:‘11 = D11 (mit/a)2+D(nit/b)2+kA55 (A.2.57a)‘12 = (D12+D66)(mit/a)(nit/b) (A.2.57b)‘13 = kA55(mit/a) (A.2.57c)L22 =D66 (mit/a)2+D22 (n it/b)2 +kA44 (A.2.57d)1.23 = kA(n it/b) (A.2.57e)‘13 =kA55(mic/a)2+kA(nit/b)2 (A.2.57f)Solving for the constants U,,, V,, W,,m, yields the following:Umn= (‘132’13)’mn (A.2.58a)Vmn= (‘12’13 1i11.23)Pmn (A.2.58b)w = (‘1l2)Pmn (A.2.58c)where= Lu (1. ‘13 — L.3) L2(L123—L2313)+L3(L23—L.2213) (A.2.58d)Recall the solution for lateral deflection:w =Wmn sin(mit x/a)sin(nit y/b) (A.2.59)mn103Dynamic LoadingWe begin by solving the homogeneous problem, i.e. p(x,y, t) = 0.For the same simply supported boundary conditions we have a solution of the form:= e’°’ Umn cos(mtx/a)sin(nity/b) (A.2.60)m n= e’°t Vmn sin(mlcx/a) cos(nity/b) (A.2.61)m nw = e’° Wmn sin(m7tx/a) sin(nmy/b) (A.2.62)m nWhen substituted into the equations of motion, the solution yields the following:hii10)rnn L13 Umn 0112 L22 — Io) L23 Vmn = 0 (A.2.63)L13 L23 Lg3—phW Wmn 0where the components of the L are those defined by Eq. (A.2.57a-t).For each m,n pair in the series solution there are three eigenvalues, and three eigenvectors:0) mnj = (w mnl ‘ mn2’ mn3) (A.2.64)Umn Umni Umn2 Umn3Vmn = Vmni Vmn2 Vmn3 (A.2.65)Wmn i Wmni Wmn2 Wmn3Only two components of each eigenvector are independent, and can be normalized with respect to Wmnj.104The particular solution is assumed to be separable into functions of position and time:Nx = Umnj .cos(mitx/a)sin(nty/b).TjQ) (A.2.66a)m=1 n=1 j13= Vmnj.sin(mTtx/a)cos(n31y/b).Tmnj(t) (A.2.66b)m=1 n=1 j=13Wmnj sin(mitx/a)sin(niry/b).Tmnj(t) (A.2.66c)m1 nI j=1Substituting the above expressions into the equations of motion, and applying the homogeneous solution ofEq. (A.2.63), yields the following:, Umnjcos(m1tX/a)sin(n1ty/b)[W;inj Tmnj+Tmnjj=O (A.2.67a)m=1 n=1 j=1sin(m7cx/a)cos(nty/b)[coj Tmnj + Tmnj] = 0 (A.2.67b)m=1 n=1 j=1sin(mitx/a)sin(nicy/b)[oj Tmnj + Tmnj] = (A.2.67c)m=1 n=i j=1 phwhere Timn(t) is a time dependent function yet to be determined, and the dot subscript indicates a derivativewith respect to time.In order to solve for the time function Tmnj(t), Eq. (A.2.67a-c) must be orthogonalized. Love [19271presented a general method of orthoganalizing a three dimensional elasticity problem. Using this method,we apply Clebsch’s theorem to obtain the orthoganality condition.105The general conjugate property is:fS.0’i2 + + dz dy dx =0; r S (A.2.68)where the displacement fields are in the form:U=Ur(X,Y,Z)7;(t) (A.2.69a)v= (A.2.69b)w = Wr(X,y,Z)7(t) (A.2.69c)Examining the displacement field, Eq. (A.2.46a-c) and the particular solution, Eq. (A.2.66a-c), thedisplacements, in Love’s notation are:= ZUmnj cos(mitx/a)sin(nny/b) (A.2.70a)Vr = ZVmnj sin(mtx/a)cos(nity/b) (A.2.70b)Wr = Wmnj sin(mirx/a)sin(niry/b) (A.2.70c)and Tr(t) Tmnj(t).Thus substituting are assumed forms foriir, r’ r’ and integrating through the thickness, the conjugateproperty becomes:I010 “efg,mnj dydx=0 e,fgm, n,j (A.2.71)whereefg, mnj = I Uej;c cos(eitx/a) sin(fity/b) Umnj cos(mtx/a) sin(rnty/b)+1 Vefg sin(etx/a) cos(ftty/b) Vmnj sin(mtx/a) cos(nity/b) (A.2.72)+ph Wefg sin(eix/a) sin(ficy/b) Wmnj sin (mitx/a) sin(rnty/b)The three equations of motion, Eq. (A.2.67a-c), must now be rearranged to match the above expression.This is achieved by multiplying Eq. (A.2.67a) by:I Uefg cos(etx/a) sin(ficy/b) (A.2.73a)106Eq. (A.2.67b) by:IVefg sin(eirx/a)cos(ftty/b) (A.2.73b)and Eq. (A.2.67c) by:phWefg sin(elrx/a) sin(fity/b) (A.2.73c)then summing the three results.The resulting single equation, in terms of the time dependent variable Tmnj, is:±[nnj efg,mnj Tmnj1efg,mnj Tmnj]_ PhWefg sin(eicx/a)sin(fity/b). = 0 (A.2.74)m=I n=1 j=1Next, Eq. (A.2.74) is integrated over the area of the plate, with the only non-zero result occurringwhen (e,Jg) = (m,nj).After performing the integration, each set of indices being equal, we can drop the summation, andEq. (A.2.74) becomes:ab ab.—-Wmnj.1josin(m7tX/a)sin(n1ty/b)pz dydxOnjTmnj+Tmnj=M(A.2.75)mnjwhereMmnj ‘1mnj,mnj dydx(A.2.76)= IU1 + IV, + phW,Note that Eq. (A.2.75) is the familiar single degree of freedom equation of motion, and that Wmnj 1,because of the eigenvector normalization.As with the Kirchoff type analysis, we consider a dynamic load in the form:p(x, y, t) = p(x, y). F(t) (A.2.77)107The dynamic solution then becomes:Tmnj = T, cos(wmnj t)+- -‘‘—sin(omnj t)+abq1F(t)O)mnj sin(t—t)dtmnj 0mnj mnjwhere Pmfl is p(x,y) expressed as a Fourier series.The solution for lateral deflection of the dynamically loaded laminate is:w = sin(mltx/a) sin(nity/b)m=1 n=1A.3 Figures31 toT, COS(Omnjt) +——sin(o,,t) +abJ FQc) sin0mnj (t — t) dt}mnj WnjMmnjFigure A.2: Natural co-ordinate system.108y(A.2.78)(A.2.79)Figure A. 1: Principal material co-ordinate system.x2zFigure A.3: Differential plate element - force equilibrium.Figure A.4: Differential plate element - moment equilibrium.AFigure A.5: Rectangular laminate with general loading.p(x,yt) phW yxzxyMs,,p(x,y,t)xy//109yFigure A.6: Patch loading nomenclature.Figure A.7: Shear deformation nomenclature..1xan110APPENDIX BUBClmpact© SoftwareEach of the impact models developed in this thesis have been implemented in a user friendly computerprogram called UBC Impact©.B.1 The Impact InterfaceB.1.1 What’s On The ScreenWhen you start Impact, a new session begins, and the Impact screen is displayed. The following illustrationidentifies each part of the screen.i— Open menus on the menu bar to display Impactcommands.Click buttons onthe too:rto-———[He dit ptions Help Ncommands-______________quickly._________________Maximize_______________________CascadeArrange1 ProjectileUse the mouse Z Targetor the keyboard 3 Grid Viewerto choose menu 4 Graph Viewercommands. Picture1Projectile 1Target 6ridjiewer 6raFVieer PicturetT LiChild windows allow you to enter data, and to viewresultsThe status bar displays information aboutthe current session111The ToolbarUsing the mouse, you can quickly choose commands from the toolbar.Copy (Edit menu)Print (File menu) Simulation controlsNew (File menu) \ / Context sensitive Help [\ \ . . \ Status indicator iSave Input File (File menu)Open Input File (File menu) Click here to selectsimulation modelCommand buttons may not always be available, depending on the simulation status. For example, while asimulation is running or paused, the New command will be disabled.The buttons on the toolbar will automatically adjust position when you resize the Impact screen.The status indicator provides visual information regarding the simulation status. See Simulation Statusbelow.The Status BarThe status bar at the bottom of the Impact screen displays information about the current session.Status of the input Status of the calculated Simulation statusparameters laminate properties112Input StatusInfonnation regarding the status of input parameters is shown in the segment on the left side of the statusbar.Input Status DescriptionNone An Impact session has just started, and no parameters have beenentered by the user.File An input file has been loaded into the current session, or theexisting parameters have been saved to a file.Modified Changes to input parameters have been made since the last openor save operation.Laminate StatusThe constitutive properties of the target laminate are only calculated when required, i.e.. when you ask toview the laminate properties or when a simulation begins. The status of these calculated laminate propertiesis shown in the middle segment of the status bar.Laminate Status DescriptionNot calculated The laminate properties have not been calculated.Using current values The laminate properties have been calculated, and no changes toparameters affecting these properties have been made.Using old values Changes have been made to parameters affecting the laminateproperties, but these properties have not been recalculated.This feature is useful when changing parameters during a simulation.For example, you may be simulating a non-penetrating impact, and decide that the target plate is too stiff.Pause the simulation, and change the thickness of the target laminate. The status bar will indicate that theold laminate properties are being used.Restart the simulation and choose to recalculate the laminate properties. The status bar will now show thatthe current values are being used, i.e.. that the properties have been recalculated using the new targetthickness,113Simulation StatusThe status of the numerical engine used to perform Impact simulations is shown in the right most segmentof the status bar, as well as by the status indicator on the toolbar (see above).Simulation Status Status Indicator DescriptionIdle Steady grey No input parameters are available.Ready Steady red Parameters are available, simulation has notbeen started.Running Flashing green Simulation is runningPaused Steady yellow Simulation has been pausedStopped Steady red Simulation was halted by the user.Done Steady red Impact event is complete.B.1.2 Choosing CommandsCommands are grouped in the four main menus of the Impact screen. Some commands carry out an actionimmediately; others display a dialog box so that you can select options.Menu bar- Edit__Qptions__Window_______ew Ctrl+NOpen Input File... Ctrl+Ohrn1TflfllSave Input File As... You can press shortcut keys toSave Output choose some commandsSave Output As...Dimmed commands are notPrint Setup... %% available because of the ImpactExit session status.j/i: C:MOPCONE.lMIZ: C:1iMOPNPEN.lMI \ -‘ An ellipsis means the command: C:1iMOPFLAT.IMI \ displays a dialog box.4: C:MOPZHU.IMI5: C:MOPCROUCH.MIA check mark (V) indicates thatthe command is “on”.114Choosing a command by using the mouse Click the name of a menu on the menu bar,and then click the command name. To close a menu without choosing a command, click outside the menuarea.Choosing a command by using the keyboard Press ALT or F 10 to make the menu bar active, and then pressthe key corresponding to the underlined letter in the menu name. To choose a command press the key forthe underlined letter or number in the command name. To close a menu without choosing a commandpress ESC.The arrow keys (ARROw UP, ARROW DOWN, etc.) can also be used to choose menus and commands.Using the Shortcut MenuWhen you point to any of the three output child windows in Impact, a shortcut menu is available, byholding down the right mouse button. This menu is identical to the Edit menu on the menu bar.Commands in the shortcut menu can be chosen by dragging with the right mouse button, and releasingwhen the command is selected, or a command can be chosen by clicking the left mouse button over thedesired command.Using Shortcut KeysYou can choose some commands by pressing the shortcut keys listed on the menus to right of thecommand. For example, to open an input file, press CTRL+O.B.1.3 Dialog BoxesThe Impact interface can display two types of dialog boxes. Child windows are a type of dialog box thatremain on the screen at all times, while other dialog boxes are displayed only when required.115Dialog boxes contain options, commands, and text boxes.Click a command button to carry out an actionor display another dialog boxIimiClick an arrow to see a list; click ordrag to select an option.Although it is usually easiest to use the mouse while you work in a dialog box, you can also select optionsor fill in information with the keyboard. To move among the different options, press TAB. To select thecurrent option or command, press enter. To move instantly to a specific option, use the ALT key and theletter underlined in the title of the desired option. For example, to move to the “File Name” text box in thedialog box above, press ALT+N.Child WindowsChild windows are a special type of dialog box because they remain on the screen at all times. Childwindows are used both to input parameters, and to display simulation results.Drag the title bar to move the dialog box.File Name: Directories:c:\temptempIDrives:C: drive_c116To invoke any changes press the Update command, orto undo any changes press Cancel._______— Click here to________ _ _minimize theImpact velocity:_____ __ __ __ _iimui dialog box.I 30.01 mi’s Conical_ _ __Mass: Friction Stress:.32kg lII4PaDimensions Elastic Piopertiesiameter: Tip Angle: M2duIus:I 7.51mm 17deq OIGPaTotal Length: Blunt Length: Eoissons Ratio:I 251mm I 21mm IParameters not required by thecurrently selected model appear innormal type. -Parameters needed by the selectedimpact model appear in bold type117Projectile Window The projectile window allows you to edit the dimensions and properties of theprojectile.Parameter Units DescriptionImpact Velocity rn/s Velocity of the projectile at the time of impact.Mass kg Mass of the projectile. It is assumed not to erode.Shape Shape of the projectile tip. Choose between conical,flat, or hemispherical.Friction Stress MPa Stress induced by sliding between the surface of theprojectile and the target material.Diameter mm Diameter of the projectile shaft. In the case of ahemispherical shape, the diameter of the tip isassumed to be equal to the shaft diameter.Total Length mm Length of the projectile from tip to toe.Tip Angle degrees Half angle of a conical tip.Blunt Length mm See UBC Conical model.Modulus GPa Elastic modulus of the projectile material.Poisson’s Ratio Poisson’s ratio of the projectile material.118Target Window The target window describes the dimensions, and material properties of the targetlaminate.Parameter Units DescriptionTarget Length mm Length of the target laminate in the major or 00direction.Target Width mm Length of the laminate in the 90° direction.Target Thickness mm Total thickness of the laminate.Average density kg/rn3 Density of the laminate.Bulge Delay mm See UBC Conical model.Flow Stress MPa Yield strength for a perfectly plastic material.Shear Strength MPa Yield strength, in shear of the target materialUltimate Strength MPa Ultimate strength of the target material.Failure StrainsFiber m/m Breaking strain of the laminate fibers.Matrix m/m Ultimate strain of the laminate matrix.Lateral Shear rn/rn See Awerbuch and Bodner model.Elastic Properties’E1 GPa Modulus of a lamina in the fiber direction.GPa Modulus of a lamina perpendicular to the fiberdirection.Poisson’s ratio in the plane of a lamina.V12G12 GPa Shear modulus in the plane of a lamina.G23 GPa Shear modulus in the 2-3 plane of a lamina.Other parameters are accessed using the command buttons which display dialog boxes.1 The lamina is assumed to be tranversely isotropic, thus E2 = E3, and G12 =G13.119View [A] [B] [D] Dialog BoxClicking the View [A] [B] [D] command button calculates the laminate properties and displays them in adialog box.[A] Matrix In plane extensional stiffnesses A11 .. .A33 The in plane shear stiffness, A and A55, arealso shown.[BI iViatrix Coupling stiffhesses B11.. .B33 These values are calculated by Impact but not used, as thelaminate is assumed to symmetric and orthotropic.[D] Matrix Bending stiffnessesD11.. .D33Laminate Dialog BoxThe laminate dialog box allows you to describe the lay-up of the target laminate.Type the orientation angle of theselected lamina (in degreees), orclick the arrows to adjust by anamount shown in step Size.Click here to set the orientation ofthe currently selected layer to zero.Select the number of layers in the laminate.Use the mouse to select the lamina you want to edit.120Stress Strain Dialog BoxThis dialog box allows you to input parameters used by the LAMP model.Static Indentation Dialog BoxThis dialog box allows you to input parameters used by the UBC Flat model.Picture Window This window displays a graphic of the impact event, including the damaged targetmaterial surrounding the projectile. The picture window operates only when the Zhu or UBC Conicalmodels are selected.Grid Viewer Results are shown in a spreadsheet type of display.Cells within the Grid Viewer can be selected and copied into the clipboard. The entire grid can also beselected, copied to the clipboard, or cleared. When a range of cells is copied to the clipboard, the columnlabels are automatically included.Results shown in the Grid Viewer include the following.Result Units DescriptionTime msec Time elapsed in impact event.Force N Contact force between projectile and target.Velocity m/s Velocity of projectile.Disp. Projectile mm Displacement of projectile. Displacement equalszero at start of the simulation.Defi. Target mm Deflection of the target mid-plane, at point ofimpact.(1 -D) Damage factor. See Zhu model and UBC Conicalmodel.Comments Information pertaining to the simulation. Forexample the perforation of the target will be notedin this column.121Graph Viewer Any of the numerical results shown in the Grid Viewer can be plotted in the GraphViewer.You can choose the results to be plotted, by double clicking over the Graph Viewer. The Graph Viewerdialog box will be displayed.— Choose one parameter for the X-Axis— Choose up to fiveparameters to plot.n flThJI!rX-Ax1s Y-AxesItime orce2 Inone II1. 3 none‘1’- .4 none5Ifl°’The Graph Viewer requires a large amount of processor time to update. Keep this window minimized whilenot in use, to speed up the simulation.Working With Child WindowsInitially the five child window icons will be shown at the bottom of the screen, double click these to inputdata or view output.Window ControlsIn the uppermost corner of each window are the window controls. Use the control box in the left corner formenu driven controls, and the control buttons in the right corner for mouse driven controls.122Using the Control Box Click on the Control Box to access the following menu.__________________________________This menu affects the status and appearance of the activewindow.MinimizeMaximizeClose AIt+F4Switch To... CtrI+EscCommand DescriptionRestore Sizes the window to the default size.Move Allows the user to size the Interface windowwhile displaying the double arrow pointer.Size The active window can be resized bydragging the double arrow mouse pointer ona window border.Minimize The active window is reduced to its icon.Maximize The window is resized to fit the entire screenarea. Child windows are sized to fit theImpact Interface window.Close Closing the Impact Interface will have thesame effect as using the Exit command.Closing a child window is equivalent tochoosing Minimize.Switch To The Switch To Dialog Box will appearallowing the user to transfer control toanother application present in the Windowsoperating environment.123Using the Mouse Controls You can invoke the Minimize command by clicking on theminimize button on the right side of the title bar. Click the maximize button to invoke the Maximizecommand. Click the restore button to invoke the Restore command.B.2 CommandsCommands are used to control the simulation process, open and save files, and to direct output.B.2.1 Simulation ControlBefore starting an Impact simulation, the appropriate parameters should be entered in the Target andProjectile child windows. As well, data controlling the numerical engine is required. Select the Solutionand Damage commands from the Options menu to display the appropriate dialog box.The appearance of warning dialog boxes can be controlled before or during a simulation by choosing theWarnings On command.Use the command buttons on the toolbar to start stop and pause the simulation.Solution Dialog BoxThis dialog box allows you to adjust the size of the time step, the number of modes, and the type of targetloading, all used by the numerical engine.rlime Step Loadin TypeI O (msec) Paint LoadNumb ol Modes[i 0 (18O)124Time Step The time increment at which results for the impact event are calculated. A large timeincrement will yield inaccurate results, and a very small time step will cause the simulation to proceed veryslowly.Number of Modes The number of modes, in each direction (x and y), used in the modal solutionwhen calculating the target deflection. Similar to the Time Step, an appropriate number of modes should bechosen to maximize efficiency.Loading Type The loading pattern used to calculate the target deflection.This option is only used by the Non-Penetrating model.Choices include:Loading Type DescriptionPoint Load A one dimensional force applied at the centerof the target laminate.Patch Load A patch, equal in length and width to thediameter of the projectile, and centered on thetarget laminate. The contact force is evenlydistributed over this patch.Distributed Load The contact force is evenly distributed overthe entire area of the target laminate.125Damage Dialog BoxThe Zhu model and the UBC Conical model calculate damage in the target laminate according to theparameters entered in this dialog box.Picture Window Update Determines how often, in relation to the Time Step, the PictureWindow is updated. Each time an update is performed a considerable delay to the simulation occurs.Minimizing the Picture Window prevents the window from being updated.Size of Damage Grid The size of the grid used to calculate the target damage. A grid of size Nwill require strain calculations to be performed at N2 points.Warnings On/OffUse this option to control the appearance of waming dialog boxes while a simulation is in progress. Thesedialog boxes require you to respond before the simulation can continue.To allow these waming dialog boxes to appear, select the Wamings On option from the Options menu. Acheck mark to the left of the menu item indicates that the option is on.Pictuie Window Update&Everyl Steps1.o EveiyStepso NoneDo not display damage ineiSize ol Damage Grid (N):(0< N < 100)126RunTo begin a simulation, click the Run command button from the toolbar.First, the Simulation dialog box will be displayed.Choose this option to add an informativeheader to the Grid Viewer.- Choose this option torecalculate the laminateproperties using thecurrent target parameters.Simulation dialog options, if grey in colour are not available, or not required. For example if the input filehad been saved prior to clicking the Run command, the Save Input option on the Simulation dialog box isnot required and will appear grey.Once a simulation is in progress the Run button will be replaced by the Halt button.HaltClick this button to completely stop the current simulation. Once halted, a simulation can not be resumed.After halting the simulation the Halt button will be replaced by the Run button.PauseClick this button to pause the current simulation. Once paused, the simulation can be resumed by clickingthe Resume button which replaces the Pause button.i— Choose this option to/ save the input file before/ starting a simulation.Click this button to begin asimulation.LI Save input data befae iunLI ffecalculate laminate pioperties.Add header to grid127You can edit target and projectile properties while a simulation is paused.ResumeClick this button to resume a paused simulation. The Pause button will replace the Resume button.B.2.2 Edit CommandsEdit commands are available to manipulate input and output data.CopyYou can copy images of the Graph Viewer or Picture Window to the clipboard. This is useful whenpreparing summaries of Impact sessions.Selected cells within the Grid Viewer can be copied to the clipboard in a form that is easily pasted into aWindows based spreadsheet.Select AllUse this command to select the entire contents of the Grid Viewer.Clear WindowYou can clear the contents of the active child window with this command.Clear InputsChoose this command to clear all the inputs. Data in the Target Window, Projectile Window, andassociated dialog boxes will be cleared.Clear OutputsUse this command to reset the Graph Viewer, and to clear the Grid Viewer and Picture Window.Clear AllChoosing this command is equivalent to choosing Clear Inputs and Clear Outputs.B.2.3 File CommandsTwo types of files are used by Impact: input files, and output files.128NewUse this command to start a new Impact session. All existing parameters in the input child windows will becleared.Open InputThe data contained in the input child windows can be stored in a file. Use this command to load an existinginput file.The standard Windows dialog box will be displayed.Save InputYou can save data in the input child window for later use with the Save Input command.The data is saved as unformatted text. You can edit the file with a standard text editor, such as Notepad, butdata is not in an easy to understand format. Editing an input file outside of the Impact interface is notrecommended.If a filename has not been given to the current Impact session, you will be prompted for a filename inwhich to save the data. Impact input files are normally given the “imi” extension (Impact Input).Save Input AsThis command allows you to save input data to a file with a filename you specify.Save OutputResults displayed in the Grid Viewer can be saved to a file with the Save Output command.The data is saved as formatted text, suitable for importing into Excel or other Windows based spreadsheets.To import into Excel, choose Windows(ANSI), tab delimited format, with no text qualifier.Save Output AsYou can establish a filename for the output data, or change the current filename with this command.129B.2.4 PrintingImages shown in either the Graph Viewer or the Picture Window can be sent to the printer via the PrintManager.Print SetupForm the File menu, select Print Setup to change or configure the default printer.Click here to see more setupoptions.The Impact interface sends data to the printer in a bitmap format. For best results setup your printer to suitimage printing. Avoid printing in high resolution Postscript mode.-Prin \*121øt?‘(currently HP LaseiJet 4141.1 PostScript on FILE:)C SpeciFic Eiintei: iIIIIIHP LaserJet 41414 PostScript on FILE:• Orientation PaperC Poitrait Size: Letter 8 1/2 x 11 in* andscape lource: lAuto SelectFor best results choose Landscape130PrintClick here to setup theprinter. See PrintSetup.The Damage option will only be available when the currently selected model is either the Zhu model or theUBC Conical model.B.3 Model ReferenceSix models are currently available to simulate impact events in Impact.Required input parameters, and available simulation results are summarized in this chapter. If any specialnotes pertaining to these parameters are required, they are listed. For detailed technical information refer tothe technical documnets included in the Impact documentation package.The Impact interface can display warnings during a simulation, indicating such things as perforation,excessive damage, and rebound. The types of warnings you can expect are listed for each model.B.3.1 LAMP ModelThis model, developed by Woodward and Crouch, calculates the ballistic limit, of a blunt projectileimpacting a layered homogeneous target.Choose the Print command from the File menu to display the Print dialog box.Select to print either the Graph or theDamage.The images will automatically be scaled to fit your printer.131This model differs from others, in that a series of “simulations” are performed by the numerical engine todetermine the ballistic limit. An initial velocity is assumed. Then using energy methods calculates theresidual velocity of the projectile, after penetration, is calculated. If the residual velocity is less than zero,i.e. the initial velocity was less than the ballistic limit, the initial velocity is incremented and the calculationrepeated. The initial velocity used in the first iteration with a residual velocity greater than zero, is chosenas the ballistic limit. Note that there is a non-zero residual velocity at the ballistic limit.The first iteration uses an initial velocity of 10 mIs, with subsequent iterations using an initial velocityincremented by 5 mIs. In some cases the residual velocity predicted can be substantial as the smallincremental velocity steps can result in a change in the penetration mechanics.The model results are presented in a dialog box, and not in the child windows.Ballistic Limit = 235.0 misWoik Done = 357.7 N mResidual Velocity = 41.5 misrn_IReferenceWoodward, R.L. and Crouch, I.G.“A Computational Model of the Perforation ofMulti-Layer Metallic Laminates,” MRL Research ReportMRL-RR-9-89, Materials Research Laboratory, DSTO, September, 1989.132
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Modelling the impact behaviour of fibre reinforced composite materials Pierson, Michael O. 1994
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Title | Modelling the impact behaviour of fibre reinforced composite materials |
Creator |
Pierson, Michael O. |
Date Issued | 1994 |
Description | Three analytical models describing the behaviour of composite targets under impact, and suitable for engineering applications are developed herein. Each model assumes that the impacting projectiles are rigid, and that the targets are fibre reinforced laminated plates. One model is concerned with the non-penetrating impact of hemispherical projectiles. A modal series solution which includes the effects of shear deformation and rotary inertia is developed to describe the target deformations in this model. Another model predicts the penetration behaviour of blunt projectiles. Results of static penetration tests are used by this model to characterize the damage caused by impacting blunt projectiles. The third model describes the penetration due to impacting conical shaped projectiles. The progression of damage as it is described by Zhu et al [1992] is used as a basis for the characterization of damage in this model. Each model is compared with experimental results obtained from low velocity instrumented impact tests, and high velocity ballistic tests. |
Extent | 2335380 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2009-03-04 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0099099 |
URI | http://hdl.handle.net/2429/5492 |
Degree |
Master of Applied Science - MASc |
Program |
Metals and Materials Engineering |
Affiliation |
Applied Science, Faculty of Materials Engineering, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 1994-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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