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Enhancing impact characterization and multi-criteria design optimization of glass fiber reinforced polypropylene… Alemiardakani, Mohammad 2014

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  ENHANCING IMPACT CHARACTERIZATION AND MULTI-CRITERIA DESIGN OPTIMIZATION OF GLASS FIBER REINFORCED POLYPROPYLENE LAMINATES  by Mohammad Alemiardakani  M.A.Sc., University of Tehran, 2007 B.Sc., Amirkabir University of Technology (Tehran Polytechnic), 2004  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY in THE COLLEGE OF GRADUATE STUDIES  (Mechanical Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Okanagan) June 2014 © Mohammad Alemiardakani, 2014 ii  Abstract Twintex®, a relatively new thermoplastic composite material system made of comingled polypropylene (PP) and glass fibers, has shown superior toughness in comparison with traditional glass-fiber epoxy composites. Along this improvement, this thesis aims at enhancing impact characterization of Twintex composites and developing a systematic approach for weave pattern selection and lay-up optimization of their laminates under impact. The research was conducted in experimental and numerical parts with a case study on potential application of Twintex in a highway guardrail. In the first part, a set of 180 drop-weight-impact and four-point-flexural experiments were performed for mechanical characterization of PP/glass laminates with different fiber architectures (balanced plain, balanced will, unbalanced plain, and unidirectional tape) before and after impact. Another set of the experiments was designed based on a Taguchi design of experiment (DOE) method and showed that this method has a good accuracy in predicting impact response of hybrid fiber reinforced plastic (FRP) composites such as Twintex. The X-ray microtomography and visual inspection techniques were also employed to investigate the interior and exterior damages induced to the specimens due to impact. These nondestructive evaluation techniques revealed that the impact damage mechanisms are highly dependent on the selected architecture of fibers. Also it is shown that, contrary to some previously published reports, impact resistance of FRP composites cannot be evaluated solely based on the extent of visible damages, and that inner damage along with associated damage modes must also be taken into account. Next, a case study was conducted to extract the criteria that may be used by engineers to assess impact-resistance of different Twintex FRP laminates for a potential guardrail application. Namely, a decision matrix with nine criteria was formed and three weighting techniques were developed under a multicriteria decision making environment to select the most appropriate fabric weave pattern for this specific application. The proposed multi-criteria approach is general and can assist designers to select optimum composite materials in other similar applications. The last part of this thesis deals with a simplified approach for numerical characterization (finite element analysis) of Twintex laminates under impact without a need for additional subroutine codes.    iii  Preface This PhD dissertation is written in publication format, i.e. chapters are based on published and submitted (under review) journal articles. As a result, the literature review, methodology, data analysis, discussion and conclusion on each sub-area of the research are extensively presented in each corresponding chapter.  List of publications: The following articles, which have been published or are under review in peer-reviewed journals, have formed the bulk of this thesis: 1) M. Alemi Ardakani, A. S. Milani, S. Yannacopoulos, L. Bichler, D. Trudel-Boucher, G. Shokouhi, H. Borazghi, “Micro-Tomographic Analysis of Impact Damage of FRP Composite Laminates: A Comparative Study”, Advances in Materials Science and Engineering, Volume 2013 , Article ID 521860, 10 pages, (2013). 2) M. Alemi Ardakani, A. S. Milani, S. Yannacopoulos, H. Borazghi “A Rapid Approach for Predication and Lay-up Optimization of Glass Fiber/Polypropylene Composite Laminates under Impact”, Submitted. 3) M. Alemi Ardakani, A. Milani, S. Yannacopoulos, G. Shokouhi, “An Intelligent Weave Pattern Selection Approach to Impact Optimization of Fiber Reinforced Composites under Multiple Criteria”, Submitted. 4) M. Alemi-Ardakani, A. S. Milani, S. Yannacopoulos (2014) “On Complexities of Impact Simulation of Fiber Reinforced Polymer Composites: A Simplified Modeling Framework”, Submitted.   Chapter 2 is based on article #1.   Chapter 3 is based on article #2.   Chapter 4 is based on article #3.   Chapter 5 is based on article #4.  iv  For all of the above publications, I was responsible for the literature review, method development, testing/ simulation, data analysis and writing the article. Dr. Milani and Dr. Yannacopoulos reviewed the articles and guided the research. In addition, our partners from industry and research institutes who cooperated in this research co-authored some of the articles as addressed in the above list. The contributions of these co-authors were:  Mr. Hossein Borazghi and Ms. Golnaz Shokouhi from AS Composite Inc. gave consultation on Twintex material selection and also provided the test samples.  Dr. David Trudel-Boucher from National Research Council Canada -Industrial Materials Institute (NRC-IMI) coordinated and supervised the impact and flexural tests.   Dr. Lucas Bichler from UBC Okanagan assisted in image processing during non-destructive damage inspections.  Besides the above journal articles, I have published and presented the following conference articles as part of my studies at UBC:  M. Alemi Ardakani, A. Afaghi Khatibi, A. Milani, H. Parsaiyan, “Low-Velocity Impact Behavior of Aluminum-Fiber/Epoxy Laminates – A Comprehensive Experimental Study”, ASME International Design Engineering Technical Conference, IDETC/CIE 2010, August 15-18, 2010: Montreal, QC, Canada  M. Alemi Ardakani, A. Milani, S. Yannacopoulos, “Impact Behaviour of Twintex Composite Sandwich Panels”, Abstract in 23rd Canadian Materials Science Conference, University of British Columbia- Okanagan Campus, June 22 - 24, 2011: Kelowna, BC, Canada  M. Alemi Ardakani, A. Milani, S. Yannacopoulos, D. Trudel-Boucherb, G. Shokouhi, “Flexural Toughness as an Attribute for Impact Damage Evaluation of Composite Laminates”, ASME International Mechanical Engineering Congress and Exposition, November 11-17, 2011: Denver, CO, USA   v  Table of Contents Abstract ……………………………………………………………………………………................................... ............................. ii Preface ………......................................................................................................... ....................................... iii Table of Contents .......................................................................................................................................... v List of Tables  ............................................................................................................................................. viii List of Figures ............................................................................................................................................... x List of Symbols .......................................................................................................................................... xiv List of Abbreviations  .................................................................................................................................. xv Acknowledgments  .................................................................................................................................... xvi Dedication ...….. ........................................................................................................................................ xvii Chapter 1: Introduction ................................................................................................................................ 1 1.1 Problem Definition and Motivation  ............................................................................................. 1 1.2 Twintex ® Composites  ................................................................................................................ 2 1.3 Research Objectives ..................................................................................................................... 3 Chapter 2: Microtomographic Analysis of Impact Damage in FRP Composite Laminates ......................... 4 2.1 Overview ...................................................................................................................................... 4 2.1.1 Historical Background  .......................................................................................................... 5 2.1.2 Example of XMT for Composites ......................................................................................... 6 2.2 Case Study .................................................................................................................................... 8 2.2.1 Sample Preparation  ............................................................................................................... 8 2.2.2 Impact and Post-impact Flexural Testing  ............................................................................. 8 2.2.3 Results of the Impact and Post-impact Bending Tests  ....................................................... 10 2.2.4 XMT Results ....................................................................................................................... 12 2.3 Summary of Findings for Microtomography Analysis of Composite Materials  ....................... 18 Chapter 3: Taguchi Predication and Lay-up Optimization of Composite Laminates under Impact  ......... 20 3.1 Overview .................................................................................................................................... 20 3.1.1 Motivation and Organization of this Chapter ...................................................................... 23 3.2 Experimentation ......................................................................................................................... 24 vi  3.2.1 Materials  ............................................................................................................................. 24 3.2.2 Taguchi Experimental Design for Impact Testing .............................................................. 24 3.3 A Discussion on Impact Test Data ............................................................................................. 27 3.4 Taguchi Prediction and Optimization  ........................................................................................ 29 3.4.1 Optimum Lay-up and Micro-tomography ........................................................................... 32 3.4.2 Contribution Percentage of Composite Layers  ................................................................... 35 3.5 Pre- and Post-Impact Four-Point Flexural Testing  .................................................................... 36 3.5.1 Why the UD laminate was found outlier in the current DOE model? ................................. 38 3.6 Summary of Findings for Taguchi Design of Experiment for Optimization of Composite Laminates under Impact  ........................................................................................................................ 39 Chapter 4: Multicriteria Weave Pattern Selection for Glass Fiber Reinforced Composites under Impact ............................... ..................................................................................................................................... 41 4.1 Overview .................................................................................................................................... 41 4.2 Experimental Procedure ............................................................................................................. 45 4.2.1 Sample Preparation and Selecting Performance Attributes ................................................. 45 4.2.2 Impact Testing  .................................................................................................................... 46 4.2.3 Pre- and Post-Impact Flexural Testing ................................................................................ 46 4.2.4 Non-destructive Damage Evaluation  .................................................................................. 47 4.2.4.1 Visual Inspection  ............................................................................................................ 47 4.2.4.2 Microtomographic Evaluation  ........................................................................................ 48 4.2.5 Experimental Results and Discussion  ................................................................................. 49 4.3 Multi-Criteria Decision Making (MCDM)  ................................................................................ 56 4.3.1 Why the use of MCDM is critical in this design application? ............................................. 57 4.3.2 TOPSIS MCDM method  .................................................................................................... 58 4.3.3 Weighting Methods  ............................................................................................................ 58 4.3.3.1 Adjustable Mean Bars (AMB) Direct Weighting Method  ............................................. 59 4.3.3.2 Modified Digital Logic (MDL) Method  ......................................................................... 64 4.3.3.3 Numeric Logic (NL) Method  ......................................................................................... 64 4.3.3.4 Entropy Method  .............................................................................................................. 65 4.3.3.5 Criteria Importance through Inter-criteria Correlation (CRITIC) Method ...................... 66 4.3.3.6 Modified Combinative Weighting (MCW) Method  ....................................................... 67 4.3.4 Multiple Criteria Material Selection Results and Discussions  ........................................... 68 vii  4.3.4.1 Adjustable Mean Bars (AMB) Weights  ......................................................................... 69 4.3.4.2 Modified Digital Logic (MDL) Weights ......................................................................... 70 4.3.4.3 Numeric Logic (NL) Weights  ......................................................................................... 71 4.3.4.4 Entropy Weights  ............................................................................................................. 72 4.3.4.5 Criteria Importance through Inter-criteria Correlation (CRITIC) Weights  .................... 73 4.3.4.6 Modified Combinative Weighting (MCW) Method: Different Practical Scenarios ........ 74 4.3.4.7 TOPSIS Ranking of Laminates ....................................................................................... 76 4.4 Summary of Findings for Multicriteria Weave Pattern Selection for Composites under Impact ....................... ......................................................................................................................................... 78 Chapter 5: On Complexities of Impact Simulation of Fiber Reinforced Polymer Composites: A Simplified Modeling Framework for Practitioners ...................................................................................................... 82 5.1 Overview .................................................................................................................................... 82 5.2 Case Study Experiments  ............................................................................................................ 84 5.3 Conventional Shell Finite Element Model and Limitations  ...................................................... 85 5.3.1 Effect of Strain Rate ............................................................................................................ 86 5.3.2 Effect of Bending ................................................................................................................ 89 5.3.3 Effect of Delamination ........................................................................................................ 89 5.3.4 Effect of fixture geometry and clamping condition  ............................................................ 91 5.4 Proposed Simplified Modeling  .................................................................................................. 93 5.4.1 Approach #1 ........................................................................................................................ 93 5.4.2 Approach #2 ........................................................................................................................ 97 5.5 Summary of Findings for Simplified Approaches for Impact Simulation of Composite Laminates ............................................................................................................................................... 98 Chapter 6: Overall Summary, Conclusions and Future Research Directions ........................................... 100 6.1 Overall Summary...................................................................................................................... 100 6.2 Conclusions  ............................................................................................................................. 102 6.3 Recommendations for Future Work  ........................................................................................ 106 References .................................................................................................................................................108 Appendix A: TOPSIS Method  ................................................................................................................. 122  viii  List of Tables Table 2.1      Tomography acquisition parameters used during imaging ..................................................... 12 Table 3.1      Specifications of the PP/glass composite plies used in the laminated plates for impact optimization ............................................................................................................................ 24 Table 3.2      Nine composite lay-up configurations used for the subsequent Taguchi L9 optimization     and four additional configurations employed for validation purposes ................................... 26 Table 3.3      Prediction of absorbed energy values by the Taguchi method ............................................... 31 Table 3.4      ANOVA results on the performed impact tests in Table 3.2 .................................................. 31 Table 4.1      Back face damage of the four tested PP/ glass laminates under 200J impact energy............. 54 Table 4.2      Comparison of internal damage area percentages from post-processed images in                Figure 4.8 ............................................................................................................................... 56 Table 4.3      Summary of results of the impact tests including, drop tower, visual inspection, XMT        and four-point flexural bending measurement; this is the basis of the decision matrix for subsequent MCDM/optimization calculations ....................................................................... 57 Table 4.4      Ranking of materials based on individual criteria (single objective optimizations)............... 58 Table 4.5      ROC weights for given number of attributes [133] ................................................................ 60 Table 4.6      Steps of the AMB weighting by the DM in the impact optimization case study ................... 69 Table 4.7      MDL weighting by the DM in the impact optimization case study ....................................... 71 Table 4.8      Continuation of table 4.7 ........................................................................................................ 71 Table 4.9      NL weighting by the DM in the impact optimization case study ........................................... 71 Table 4.10    Continuation of Table 4.9 ....................................................................................................... 72 Table 4.11    Normalized Decision making matrix (pij); note that the criteria become dimensionless ...... 72 Table 4.12    Calculated entropy (E), degrees of diversity (d) and weights of importance (w) for        different criteria according to the Entropy method ................................................................ 72 Table 4.13    Inter-criteria correlation factors (Rjk) according to the CRITIC method ............................... 73 Table 4.14    Summary of the four subjective (EW, AMB, MDL and NL) and the two objective       (Entropy & CRITIC) weighting methods ............................................................................... 73 Table 4.15    TOPSIS results based on the subjective and objective weights presented in Table 4.14 ....... 73 Table 4.16    Results of the four combinative weights based on four proposed scenarios to mimic the DM’s level of experience ....................................................................................................... 76 Table 4.17    Final MCDM results for the given laminate options; under the four different weighting scenarios in Table 4.16.......................................................................................................... 76 76 ix  Table 5.1      Quasi-static mechanical properties of twill weave Twintex ® composites (indices 1 and 2 refer to in-plane warp and weft directions, and 3 denotes the out-of-plane direction ..........855 Table 5.2      Correction factors applied to the mechanical properties of the Twintex sample ................. 95 Table 5.3      The ensuing modified material properties from Tables 5.1 and 5.2. .................................... 95 x  List of Figures Figure 1.1      Example applications of composite materials. [1] ..................................................................1 Figure 1.2      Damaged caused when a truck lost control in the freezing rain two miles south of the summit of Blewett Pass on US 97 on Feb 04, 2011. [4] .........................................................2 Figure 2.1      (a) Schematic of macro-tomography used in medical examinations; (b) helical body scanning [24]. .........................................................................................................................6 Figure 2.2      A high precision saw cutting the composite sample in the mid plane where the damage zone is present .................................................................................................................................7 Figure 2.3      Comparison between real impacted sample and the images obtained from nondestructive microtomography and destructive optical microscopy; red circles are to show comparable damage zones captured by the two methods. ..........................................................................7 Figure 2.4      Example of the histogram of XMT analysis in an impacted laminate; the varying severity of fiber disorientation, matrix cracking and delamination can be noticed depending on the distance from the impact center. .............................................................................................8 Figure 2.5      Set-up used during (a) the drop-weight impact testing and (b) post impact four-point bending ................................................................................................................................. 10 Figure 2.6      The average response curves of PW and UD samples subjected to 200 J impact loading; (a) impactor force vs. time, (b) force vs. displacement, (c) energy vs. time, and (d) the post-impact flextural test results. .................................................................................................. 11 Figure 2.7      Rear face of PW and UD samples subjected to 200J impact energy .................................... 12 Figure 2.8      Ultimate flexural strength comparisons before and after impact tests on UD and PW laminates ............................................................................................................................... 12 Figure 2.9      (a) XMT image of impacted PW laminate (top view at the impact center, left view at   10 mm from the impact center and front view close to the rear side of impact), (b) top view at the impact center, and (c) the processed image of top view at the impact center (for damage quantification purposes) ....................................................................................................... 14 Figure 2.10    (a) XMT image of impacted UD laminate (top view at the impact center, left view at 10 mm to the impact center and front view close to the rear side of impact), (b) left view at the impact center, and (c) the processed image of left view at the impact center. ..................... 15 Figure 2.11    XMT left view histogram of the impacted PW and UD laminates ....................................... 15 xi  Figure 2.12    XMT top view histogram of the impacted PW and UD laminates ....................................... 16 Figure 2.13    A cross-section of the impacted UD laminate showing a large through-thickness crack and clear delamination sites ........................................................................................................ 17 Figure 2.14    Area fraction of damaged zones in the impacted UD and PW samples, as function of distance from the impact center ............................................................................................ 18 Figure 3.1      Schematic of the laminates lay-up. ....................................................................................... 25 Figure 3.2      Clamping system used during the drop weight impact tests ................................................. 27 Figure 3.3      (a) Force-time, (b) velocity-time, (c) force-displacement and (d) energy-time diagrams of configuration #1 ([(PW)6]s) under 200J impact energy; ‘M’ refers to the same peak force point in different graphs ....................................................................................................... 28 Figure 3.4       Average absorbed energy (experimental) values for the 13 laminate configurations in Table 3.2 ......................................................................................................................................... 29 Figure 3.5       Interaction plots between (a) layers A&B, (b) layers B&C, and (c) layers C&A; using the L9 experimental data in Table 3.2. ....................................................................................... 32 Figure 3.6       Fiber bridging in a typical UD laminate [42] ....................................................................... 32 Figure 3.7      Taguchi model predictions via Eq (3.1); i.e., assuming no interactions ............................... 33 Figure 3.8      X-ray microtomographic images of configuration [TW,(PW)2,TW]s: predicted absorbed energy=41.0J ........................................................................................................................ 34 Figure 3.9      X-ray microtomographic images of configuration [UD0/90,TW,(PW)2]s: predicted absorbed energy=30.8J ........................................................................................................................ 34 Figure 3.10    Schematic of deformation distribution in a six-layer symmetric laminate due to flexural deformation ........................................................................................................................... 36 Figure 3.11    Four-point flexural bending test setup .................................................................................. 37 Figure 3.12    Flexural chord modulus before and after 200J impact (Due to impact, flexural chord modulus of UD, PW and TW laminates is reduced by 31%, 17% and 9%, respectively) .... 38 Figure 3.13    Microtomography cross-sections of an impacted UD laminate showing a large through-thickness crack and delamination. ........................................................................................ 39 Figure 4.1      Four different weave patterns examined in the case study for impact optimization; number of fibers in 22 bundles is twice the number of fibers in 11 bundles. .................................... 46 xii  Figure 4.2      (a) Drop-weight impact test tower, (b) Four-point flexural testing of samples;    support’s span (L) and loading span (L/2) ........................................................................................... 47 Fig. 4.3          Visual measurement of visible damage area in the rear face of an impacted sample (the blue scale bar is in cm) ................................................................................................................. 48 Figure 4.4      The x-ray micro-tomography system used for internal damage investigation of   impacted samples ................................................................................................................................. 48 Figure 4.5      (a) force-time, (b) deflection-time, (c) force-deflection, and (d) velocity-time diagrams of samples subjected to 200J impact energy; Images (e)  and (f), adopted from [30], show the front and side XMT slices of the UD laminate at 12 mm from the impact centre, respectively; notice a very large crack extended through half of the laminate thickness in image (e) and the long delamiation sites in image (f)........................................................... 51 Figure 4.6      (a) The front face of samples after four-point bending test (without impact), (b) after 200 J impact followed by four-point bending. Notice the circular trace of impactor in the middle of specimen in case (b). White arrows point to the traces of four point-bending loading fixture; for unimpacted specimen (case (a)) kinking occurred right below the loading bars while for the impacted specimen (case (b)) two parallel kinking occurred adjacent to the impactor. ............................................................................................................................... 52 Figure 4.7      Flexural stress-strain curves for (a) PW, (b) TW, (c) UW, and (d) UD samples before and after 200J impacts. The relative percentage loss of ultimate flexural strength (LUFS%) due to impact is 18.6%, 7.3%, 20.2% and 31.9%, for the PW, TW, UW, and UD laminate, respectively. Similarly, from areas under the stress-strain curves up to the maximum strength point, it was found that the relative percentage of fracture toughness loss (FTL%) due to impact is 17.8%, 11.1%, 14.7% and 29.8 %, for the PW, TW, UW, and UD laminate, respectively. .......................................................................................................... 53 Figure 4.8      X-ray microtomography slices at impact center for (a) PW, (b) TW, (c) UW, and (d) UD samples subjected to 200J impacts. ...................................................................................... 55 Figure 4.9      Processed images of the front and side sections of the x-ray microtomography images in Figure 4.7; quantitative values of the colored areas are given in Table 4.2. (thickness of all samples is 6 mm). ................................................................................................................. 56   xiii  Figure 4.10     Adjustable Mean Bars (AMB) weighing steps for a hypothetical example with 7 attributes: (a) initial state with equal weighing, (b) attributes 3 and 7 are weighted, (c) next attributes 1, 2 and 5 are weighted, and (d) finally attributes 4 and 6 are weighted; solid and hollow bars demonstrate the weighted and yet-to-be weighted attributes in each step, respectively). .............................................................................................................................................. 63 Figure 4.11    Prototype of a semi-rigid composite guardrail ...................................................................... 69 Figure 4.12    The AMB weighing procedure for the impact optimization case study with 9 attributes; solid and hollow bars demonstrate the weighted and yet-to-be weighted attributes, respectively. .......................................................................................................................... 70 Figure 5.1      (a) Impact test set up, and (b) FE model of the drop weight test using standard composite shell for the laminate. ........................................................................................................... 86 Figure 5.2      Comparison between experimental results and a conventional FEA using quasi-static properties and Hashin progressive damage criterion ............................................................ 86 Figure 5.3      Relationship between strain rate and (a) stiffness and (b) strength for the balanced twill weave Twintex  laminate (adapted from [37]) ..................................................................... 89 Figure 5.4      X-ray micro-tomography image of the impacted  Twintex laminate (sample delamination zones are shown with arrows) .............................................................................................. 91 Figure 5.5      Six different clamping configurations compared during impact testing [33] ....................... 92 Figure 5.6      A soft-core sandwich panel slipped out of the fixture under 200J impact ............................ 93 Figure 5.7      Comparison between experimental and two different numerical models; FEA with (i) quasi-static and (ii) modified material properties under Approach #1 ........................................... 96 Figure 5.8      Progression of the Hashin shear damage index during the impact event .............................. 96 Figure 5.9      Comparison between experimental and three different numerical methods; (i) FEA with quasi-static properties, (ii) FEA with modified material properties under approach #1, and (iii) FEA with modified material properties under approach #2 .......................................... 97 Fig. 6.1          An XMT slice of the Twintex plain woven laminate subjected to 200J impact showing shear deformation of  fiber bundles around the impact center. .................................................... 106  xiv  List of Symbols Symbol Definition Ai 𝐴∗ 𝐴− 𝐶𝑗 𝐶𝑖∗ 𝐶𝑗𝑘 𝑟𝑖𝑗 Si∗ Si− 𝜀̇ 𝑣𝑖𝑗 𝑤𝑗 𝑥𝑖𝑗 Alternatives or materials (𝑖 = 1, … ,𝑚) Positive-Ideal Solution Negative-Ideal Solution Criterion j or material properties (𝑗 = 1,… , 𝑛) Similarities to Positive-Ideal Solution  Comparative weight Normalized element of decision matrix  Separation from Positive-Ideal Solution Separation from Negative-Ideal Solution Strain rate Weighted normalized element of decision matrix Weight or importance of criteria j Elements of decision matrix, ith alternative or material, jth criterion   xv  List of Abbreviations Abbreviation Definition AE AMB ANOVA CRITIC CM DM EVD EW FEA FRP GFRP GA ID RLFT RLUFS MCD MCDM MDL NL FT UFS PW PM RF ROC RR RS TW UD UW WPM XMT Absorbed Energy  Adjustable Mean Bars weights Analysis of Variance  Criteria Importance through Inter-criteria Correlation Combinative Weights Decision Maker Exterior Visible Damage Equal Weights Finite element analysis Fiber Reinforced Polymer Glass Fiber Reinforced Polymer Genetic Algorithm Interior Damage Relative Loss of Flexural Toughness due to impact Relative Loss of Ultimate Flexural Strength due to impact Maximum Central Deflection Multi-Criteria Decision Making Modified Digital Logic method Numeric Logic weights Flexural Toughness (of healthy samples) Ultimate Flexural Strength (of healthy samples) Plain Woven Project Manager Reaction Force Rank Order Centroid weights Rank Reciprocal weights Rank Sum weights Twill Woven Unidirectional Unbalanced Woven Weighted Product Model X-ray Microtomography Technique   xvi  Acknowledgments First and above on all, my thanks and praises are to God, The One, The Most Merciful.  In addition, I would like to express my heartfelt appreciation to my supervisors Dr. Abbas S. Milani and Dr. Spiro Yannacopoulos. It was an honor for me to be their student during these years. The intimate atmosphere full of encouragement, technical and financial support and care provided by them helped me accomplish this thesis confidently and with so many good memories. Moreover, I would like to appreciate the insightful comments and guidance I received from the rest of my supervisory committee members, Dr. Lukas Bichler and Dr. Kian Mehravaran. I also wish to acknowledge the financial support of Natural Sciences and Engineering Research Council of Canada (NSERC). Many thanks goes to our industrial collaborators, Ms. Golnaz Shokouhi and Mr. Hossein Borazghi from AS Composite Inc., who initiated this research project, provided the test samples and gave valuable practical advices that only senior industrial fellows like them were capable of. I would also like to thank Dr. David Trudel-Boucher from the NRC-Industrial Materials Institute (IMI) for coordinating the experimental part of the work at their institute. In addition, I wish to thank all my colleagues at The University of British Columbia, Composites Research Network (CRN). At the end, I would like to show my deepest appreciation and love to the people that words cannot express how grateful I am to them: my beloved parents, sisters, my cousin, Hamid and last but not least, my beloved wonderful wife, Samira, for all her love, sacrifice and divine teachings. xvii  Dedication        1  Chapter 1: Introduction 1.1 Problem Definition and Motivation Over the past few decades, composite materials and structures have been playing significant roles in our daily life. Figure 1.1 shows example applications of such materials in different air, land, and marine industries. Occasionally, the shown structures in this figure, among many others, may experience damage over their lifetimes or be subjected to some type of impact loading. That is why a full understanding of impact behavior of composite materials is one of the necessities in today’s modern composite manufacturing industries to be able to design optimum and safe structures.   Figure 0.1: Example applications of composite materials. [1] It is also notable that due to high specific strength (strength-to-weight ratio) and other superior characteristics of composites, several manufacturers are progressively finding these materials as a good replacement for traditional materials [1]. For example, the most common material candidates for aircrafts fuselages are aluminum alloys series 2000 and 7000, but the fuselage of the most recent Boeing product (Boeing 787) is fully made of composite [2]. Some other industries, such as transportation, are considering composite materials in high risk products (such as roadside barriers). A fundamental question in these applications would be, ‘is there any composite material that can withstand impact energy more efficiently than its original metallic counterpart, namely steel, while offering several other advantages such as lower weight, easier manufacturing and installation, lower maintenance cost, etc?’ In order to 2  answer this question, the first step would be to fully identify the impact response of various candidate composite materials and configurations for specific applications. As science and technology advance, new and stronger fiber and resin systems are being introduced to the global market. One of these new material systems is the co-mingled glass-reinforced polypropylene fabrics known as “Twintex®”. The scope of this PhD research is impact characterization, optimization and design of Twintex composite laminates for their potential application in improving the impact performance of high risk structures such as highway guardrails. The US Department of Transportation reported 900 fatal crashes and 27,000 injurious crashes due to vehicle-guardrail crashes in 2009 [3]. Figure 1.2 [4]  shows a sample guardrail that was hit by a truck and was ruptured completely. Fortunately in this particular accident, the truck was stopped before falling into the steep shoulder.  Figure 0.2: Damaged caused when a truck lost control in the freezing rain two miles south of the summit of Blewett Pass on US 97 on Feb 04, 2011. [4] 1.2 Twintex ® Composites Among different fiber reinforced polymers (FRPs), Twintex® (with co-mingled polypropylene filaments and glass fibers) is a relatively new material system which was originally introduced by Saint-Gobain Inc. in France and as of November 1, 2007 was owned and manufactured by Owens Corning Inc. The commingling technology used in manufacturing of Twintex fabrics allows a high glass content of up to 80 percent, which in turn can facilitate the manufacturing of very strong and lightweight products [5]. In 3  comparison to a number of thermosetting matrices such as epoxy, the high toughness of thermoplastic polypropylene (PP) matrix in Twintex provides higher mode-I and -II interlaminar toughness and in general more superior impact properties [6]. Moreover, in contrast to the conventional woven composite preforms, Twintex products can be simultaneously consolidated and formed in continuous production lines. Twintex is also known to be environmentally friendly (in comparison to thermoset composites) because of its thermoplastic polypropylene matrix content [5] , [7].  1.3 Research Objectives The main objectives of this research can be summarized as follows. 1. Studying the effect of different fiber weave patterns on impact behaviour of Twintex laminates using mechanical testing and non-destructive inspection techniques including X-ray microtomography. 2. Investigating the applicability of a Taguchi design of experiment (DOE) approach on the layup optimization of FRP composite materials with a minimal number of test requirements. 3. Developing a systematic framework for multiple criteria decision making (MCDM) in material selection of Twintex laminates under impact loading. In doing so, new weighting techniques should be developed for more realistic applications of existing MCDM techniques for composite impact optimization. 4. Developing a simplified finite element modeling framework enabling general industrial practitioners to model FRP composites under impact, without a need to write intricate material subroutines.     4  2 Chapter 2: Microtomographic Analysis of Impact Damage in FRP Composite Laminates With the advancement of testing tools, the ability to characterize mechanical properties of fiber reinforced polymer (FRP) composites under  extreme loading scenarios has allowed designers to use these materials in high-level applications more confidently. Conventionally, impact characterization of composite materials is studied via nondestructive techniques such as ultrasonic C-scanning, infrared thermography, X-ray, and acoustography. None of these techniques, however, enable 3D micro-scale visualization of the damage at different layers of composite laminates. In this chapter, a 3D microtomographic technique has been employed to visualize and compare impact damage modes in a set of thermoplastic laminates. The test samples were made of commingled polypropylene (PP) and glass fibers with two different architectures, including the plain woven and unidirectional. Impact testing using a drop-weight tower, followed by post-impact four-point flexural testing and nondestructive tomographic analysis demonstrated a close relationship between the type of fibre architecture and the induced impact damage mechanisms and their extensions. 2.1 Overview During experimental analysis of impact behaviour of FRP composites, it is common to use either nondestructive or destructive detection methods to investigate the induced damage modes and their extension in test samples. Different nondestructive methods have been used in the literature, from simple visual methods [8]–[12] to more complex thermal- or electrical-based [13]–[16] methods, ultrasonic C-scanning [17]–[20], and X-ray imaging [21]–[23]. Each method has its own advantages and disadvantages and may be suitable for a particular application/material type. Nevertheless, a common limitation of these methods is that they are generally unable to give a full 3D image of the interior part of the material, hence making it difficult to provide complete information  regarding the location and extent of different damage modes such as matrix cracking, fiber breakage, fiber pull-out, fiber-matrix debonding, and delamination. On the other extreme, the destructive methods have been of less desire for sensitive applications as they 5  can be the source of additional damage in the impacted zone of structures such as fiber breakage, fiber pull-out, or delamination growth. The previous shortcomings can be well addressed by using today’s advanced X-ray microtomography techniques (XMTs), which is the main focus of this chapter. Namely, the present work aims at a detailed comparison of damage state in impacted woven fabric and unidirectional thermoplastic laminates via XMT, thereby arriving at a correlation between the observed damage distributions and the underlying reinforcement type.  2.1.1 Historical Background   X-ray microtomography technique (XMT) is known as a nondestructive technique for 3D microstructure reconstruction and visualization of the interior parts of objects with a resolution in the order of micrometers. Johann Radon, a Czech mathematician, was the first scientist who conceived a mathematical solution for the reconstruction of X-ray images in 1917 [24]. Allan Cormack, a South African physicist, continued the previous work and developed an algorithm for the geometrical reconstruction problem at Tufts University in 1964. Following this, Godfrey Hounsfield built the first CT (computed tomography) scanner at EMI Research Labs in the UK in 1972. It is worth mentioning that Cormack and Hounsfield received Nobel Prize in 1979 because of their contributions in building the first CT scanner and its effect on medical imaging applications [24]. Figure 2.1 shows a schematic of the macrotomography technique that today is used in medical examinations. Medical CT scanners use a point source X-ray and an array of detectors. The patient body is inserted into the machine chamber. At the same time X-ray source and detectors rotate around the body and collect the X-ray images, that is, helical body scanning.  Because of the sensitivity of human body to high radiation exposure, the energy and dosage of X-rays in these machines are set to be low and as a result the ensuing image resolutions are often low [25]. This limitation led Elliot and Dover in 1982 to build a more precise machine with higher exposure capability and image resolution (12 μm) for industrial applications and microanalyses [26]. Another difference between the industrial microtomography machines (XMT) and the medical CT scanners is that the X-ray source and detectors in XMT machines are stationary and the sample rotates. Depending on the need, one 6  can set the machine to take several thousands of scans in a complete rotation of the sample between 0ᵒ and 360ᵒ. Subsequently, post-processing software is used to reconstruct the 3D image of the sample which contains all geometrical information of the interior microstructure. (a)               (b) Figure 2.1: (a) Schematic of macro-tomography used in medical examinations; (b) helical body scanning [24]. 2.1.2 Example of XMT for Composites Before presenting the conducted case study, let us illustrate a general example of an XMT image (obtained by Xradia microXCT-400 machine) as compared to an image obtained from the same sample through a destructive method. Namely, an impacted composite sample was cut with a slow speed diamond saw (Figure 2.2) and the cross-section of the impacted zone was examined by an optical microscope (Figure 2.3). In the nondestructive counterpart of this analysis, the specimen needed no physical cutting and the XMT image (Figure 2.3) shows a slice (virtual cut) of the 3D image of the material microstructure in the mid-plane. The comparison of the two images shows that microtomography has captured the interior damages reasonably well. Slight differences between these images can be due to the damage induced during the cutting process in the destructive method (microscopy) such as fiber breakage, fiber pull-out, cracking, compressing or opening the delaminated layers, which in turn implies an advantage of using tomography as a nondestructive method.  Additionally, in the destructive method the cut sample may not be used for further investigations at different planes, whereas in the XMT the virtual cutting plane can be moved over the sample to scrutinize the microstructure in arbitrary sections. Figures 2.4(a) to 2.4(d) show the trend (histogram) of such interactive analysis for four different cutting X-ray Point Source  Detectors 7  planes (namely, at different distances from the specimen center). Each XMT slice has four sub-images (top, front, and left views). The cutting planes are shown by red, blue, and green lines. Figure 2.4(a) reveals the damaged cross-sections when the top and left cutting planes (blue and red lines) are far from the impact center as noted in the front view. Figure 2.4(b) shows the tomography slice when the top cutting plane (blue line) was placed near the impact center. In Figure 2.4(c), the left cutting plane (red line) was moved towards the impact center and in Figure 2.4(d) the left cutting plane was almost at the center where most of the damage is noticed from the side view.  Figure 2.2:  A high precision saw cutting the composite sample in the mid plane where the damage zone is present.  Figure 2.3: Comparison between real impacted sample and the images obtained from nondestructive microtomography and destructive optical microscopy; red circles are to show comparable damage zones captured by the two methods.  8      Figure 2.4: Example of the histogram of XMT analysis in an impacted laminate; the varying severity of fiber disorientation, matrix cracking and delamination can be noticed depending on the distance from the impact center. 2.2 Case Study 2.2.1 Sample Preparation Two sets of test samples were prepared using vacuum bagging to laminate 12 layers of polypropylene/E-glass preform (with a fiber volume fraction of 60%–70%) using two different reinforcement patterns including the plain woven (PW) and unidirectional (UD). The laminates’ size was chosen for impact testing based on ASTM D7136 [27] with a rectangular shape (150 × 100 mm) and the total thickness of 6mm (which was the maximum feasible thickness due to manufacturing limitations). 2.2.2 Impact and Post-impact Flexural Testing Each composite laminate type (PW or UD) was tested under a drop-weight tower (Figure 2.5(a)).  Impact tests were conducted using Dynatup Model 8200 impact machine with a 12.35-kg striker having a 1-inch Up Front Left (a) (d) (b) (c) 9  (in diameter) hemispherical tip. Each test was repeated twice and all four sides of the specimens were completely clamped during the impact event. The impact energy was kept  constant  at  200 J. Force history  during the  impact  event was collected by a  load  cell, a  quartz piezoelectric force sensor, mounted  on the impactor. The acceleration of impactor as a function of time, 𝑎(𝑡), was calculated by Newton’s second law of motion (Eq. (2.1)) from the collected force history 𝐹(t) and the impactor mass 𝑚:  𝑎(𝑡) =𝐹(𝑡)𝑚 (2.1) The velocity 𝑣(𝑡) and displacement of impactor 𝑥(𝑡) were found by numerical integration as: 𝑣(𝑡) = 𝑣0 + 𝑔𝑡 − ∫𝐹(𝑡)𝑚𝑑𝑡𝑡0 (2.2) 𝑥(𝑡) = ∫ 𝑣(𝑡)𝑑𝑡𝑡0 (2.3) where v0 is the velocity of impactor at the time of hitting the sample measured by an infrared velocity detector; see [28] for more details of drop-weight test kinematics. After impact testing, a post-impact four-point flexural experiment (Figure 2.5(b)) was conducted on each specimen. The motivation was to study the post-impact resistance of the impacted composite laminates for their potential application, for example, as a highway guardrail between inspection/repair intervals and also to find the deterioration of their effective mechanical properties due to the impact event. All the results presented in the next sections are normalised with respect to the fiber volume fraction. 10                                                            (a)                                                                  (b) Figure 2.5: Set-up used during (a) the drop-weight impact testing and (b) post impact four-point bending. 2.2.3 Results of the Impact and Post-impact Bending Tests Figures 2.6(a) and 2.6(b) show the average contact force and displacement of projectile from repeats of the test. In comparison to UD samples, Figure 2.6(a) suggests that PW has exerted more force to the impactor. The energy has been calculated via:  𝐸(𝑡) = ∫𝐹(𝑡). 𝑑𝑥 (2.4) where 𝐹  is the reaction (contact) force and 𝑥  is the impactor displacement. Figure 2.6(c) shows the average energy of impactor for the two experiments. Subtraction of the energy of impactor at the time of hitting the sample (200 J) from that at the rebounce indicates the dissipated energy due to permanent damage in the material. This energy is represented by the area trapped between the penetration and rebound curves in Figure 2.6(b) or the final flat energy level in Figure 2.6(c) after about 8 ms. According to these diagrams, UD laminates have absorbed more energy than PW laminates. Hence, it may be concluded that the absorbed energy has been decreased by increasing the reinforcement waviness from unidirectional to plain weave pattern, given comparable laminate thicknesses and fiber contents. 11                       (a)                                                      (b)                         (c)                                                         (d)     Figure 2.6: The average response curves of PW and UD samples subjected to 200 J impact loading; (a) impactor force vs. time, (b) force vs. displacement, (c) energy vs. time, and (d) the post-impact flextural test results. Figure 2.6(d) shows the average results of flexural testing for impacted samples. It confirms that the impacted plain woven composite has withstood post-impact bending forces much better than the impacted unidirectional composites. For comparison purposes, the four-point flexural testing was also performed on PW and UD healthy samples (i.e., before impact damage). Accordingly, Figure 2.8 indicated that the deterioration percent of ultimate flexural strength due to impact is 19% for the PW material and 32% for the UD material. This result is in agreement with the energy results in Figure 2.6(c): the more the absorbed energy by the material, the higher the deterioration of effective mechanical properties of the sample after the impact. Hence, we can conclude that UD samples have been damaged more severely than PW samples under impact. However a question would then be why is the visible (exterior) damage in PW samples much more apparent than UD samples as illustrated in Figure 2.7. XMT technique was employed to answer this question as it can illustrate the interior damage of the samples. 01020300 1 2 3 4 5 6 7 8 9 10PWUDContact Force(kN)Time (ms)200J-AVERAGE0102030-5 -3 -1 1 3 5 7 9 11 13 15PWUDForce (kN)Displacement (mm)200J-AVERAGE0501001502 00 1 2 3 4 5 6 7 8 9 10PWUDTime (ms)Energy (J)200J-AVERAGE0501001502002500 0.5 1 1.5 2 2.5 3Flexural Stress (MPa)Flexural Strain (%)Four-Point Flexural TestingPW_Post-Impact_200JUD_Post-Impact_200J12     Figure 2.7: Rear face of PW and UD samples subjected to 200J impact energy  Figure 2.8: Ultimate flexural strength comparisons before and after impact tests on UD and PW laminates  2.2.4 XMT Results As addressed earlier, X-ray microtomography tests were conducted using Xradia microXCT-400 machine with sample dimensions of 6×40×120 mm3. Table 2.1 shows the acquisition parameters and the test set-up used during tomography. Images obtained by this technique comprised 1024 × 1024 pixels of 33.57 µm. Table 2.1: Tomography acquisition parameters used during imaging X-ray Source Detector Tomography Setup Power=10 watt Voltage= 62 kV Current= 155 µA Magnification= 0.39 X Filter: No  No. of radiographs= 630 image Angle of rotation: -110˚ to 110˚ Illumination time= 1 s per radiograph Figures 2.9(b) and 2.10(b) show a 25 mm × 10 mm window cropped from the top and left cross-section views of PW and UD specimens. Fiber layers are also marked in these images. Figures 2.9(c) and 2.10(c) PW UD 13  represent Figures 2.9(b) and 2.10(b) after image processing using Buehler Omnimet 9.5 software. The image processing enabled measuring the damage areas quantitatively. The green and dark red regions in Figures 2.9(c) and 2.10(c) indicate the healthy and damaged regions, respectively. Figure 2.9(c) suggests the presence of several delamination sites, matrix crushing, and separations (branching) of fiber bundles within inner layers. This view also shows fiber breakage of two layers close to to the impact center as well as a large delamination between the third and fourth layers.  A set of virtual rulers were placed in the top and left views of both Figures 2.9(a) and 2.10(a) with the total lengths of 28, 24, ... ,4 mm. These rulers were used as indicators for subsequent image analyses to cut the 3D XMT images from −14mm to +14mm distance from the impact center with a spacing of 2mm. Images obtained from these cuts on one side of the impact center are presented in histogram forms (Figures 2.11 and 2.12). The useful length of field of view in collected tomography images was considered to be 20mm (to avoid edge effects that deteriorate the image resolution); hence the results in Figures 2.11 and 2.12  were included up to 10 mm (on one side) from the impact center.  Comparing results in Figures 2.11 and 2.12, it is first noticed that the states of damage at the top and left cross-sections are not generally identical, given the same cut distance from the impact  center (specially for the UD sample). This is most likely because of the non-symmetric impact boundary condition during the drop-weight tests due to the non-square shape of the fixture (130×80 mm). Expectedly, cracks and delaminations have been propagated longer in the direction with larger specimen dimension, which lays on the left view of tomography images. As the cutting plane goes farther from the impact center, we notice that the PW sample appears to be more and more undamaged (comparable to the healthy state). For UD laminates, from Figure 2.10(a), no severe local damage is observed under the impact center. There were, however, well-distributed small dark regions (dots) on the top view (see Figure 2.12 for results). Each of these dark regions would correspond to a delamination which can be traced in the corresponding left view in Figure 2.11. It was interesting that, in contrast to the PW laminate, if we go far from the impact center (up to 10 mm which was the maximum useful field of view), there still exits 14  evidence of some locally delaminated zones in the UD laminate and their intensity does not decrease rapidly. This means that the extent of damage in the UD laminate in the form of several micro-delaminations would be higher than that in the PW sample. Comparing the top view histograms of UD and PW laminates in Figure 2.12, another main difference between the two impacted materials is revealed: a very large through-thickness crack and fiber breakage have occurred in the UD laminate starting from the impacted face of the sample (marked with a white arrow in Figure 2.13). In fact, the calculated larger magnitude of absorbed energy in the UD laminate, 57.218 J versus 36.2 J for the PW sample, also shown in Figure 2.6(c), could be linked to this large through-thickness crack and fiber breakage in addition to the aforementioned distributed local delaminations across the sample. It should be added that a similar crack was visible in all test repeats of the UD material. Relating to the reinforcement architecture, the high waviness in the plain woven laminates would act as a barrier against impact pulse. On the other hand, flat UD fibers have allowed the impact wave to propagate from the center to the structure more easily without a large local damage under impactor.  (a)     12 layers of fibers          (b)      (c) Figure 2.9: (a) XMT image of impacted PW laminate (top view at the impact center, left view at 10 mm from the impact center and front view close to the rear side of impact), (b) top view at the impact center, and (c) the processed image of top view at the impact center (for damage quantification purposes). 3D Top Left Front 15   (a)             (b)                                                         (c) Figure 2.10: (a) XMT image of impacted UD laminate (top view at the impact center, left view at 10 mm to the impact center and front view close to the rear side of impact), (b) left view at the impact center, and (c) the processed image of left view at the impact center.  Distance from the impact center (mm) 0 2 4 6 8 10 PW       UD       Figure 2.11: XMT left view histogram of the impacted PW and UD laminates 16     PW UD Distance from the impact center (mm) 0   2   4   6   8   10                     Figure 2.12: XMT top view histogram of the impacted PW and UD laminates Figure 2.14 shows the area fraction of damaged regions (dark zones) obtained quantitatively from processed images in Figures 2.11 and 2.12. Each data point in Figure 2.14 has been calculated from the average response of the two cross- sections located symmetrically with respect to the impact center. According to the observed trends, the inner damaged area of PW samples decreases linearly by the distance from the impact center. Interestingly, in contrast to the PW sample, the damage faction of the UD sample has not varied notably by the distance from the impact center—it is nearly constant after 2 mm 17  across the sample within the given field of view. This result, in turn, confirms that the damage distribution has been more uniform in the UD sample. Also Figure 2.14 suggests that the damage fraction of UD samples has been overall lower than PW samples. On the other hand, as discussed in the previous sections, the deterioration of effective mechanical properties from the healthy to impacted samples has been more severe in the UD material (also this material has absorbed more energy as shown in Figure 2.6(c)). This means that for impact damage analysis and its linkage to the residual mechanical properties in the samples, next to the damaged area, one should look into other associated parameters. One of these key parameters is the corresponding damage mode to each damaged area. Although there is no evidence of severe local damage under impact center in UD samples, the very large through-thickness brittle crack and the associated fiber breakage mode, along with the distributed delaminations, have played a significant role in the absorption of impact energy in this material.  Figure 2.13: A cross-section of the impacted UD laminate showing a large through-thickness crack and clear delamination sites. Crack Delaminations 18   Figure 2.14: Area fraction of damaged zones in the impacted UD and PW samples, as function of distance from the impact center 2.3 Summary of Findings for Microtomography Analysis of Composite Materials PP/glass thermoplastic laminate samples were made using (i) unidirectional fibers (UD) and (ii) plain woven (PW) fabrics and subjected to 200 J impact energy as well as post-impact four-point bending. UD specimens absorbed (dissipated) more energy than the PW laminates. This was despite the fact the UD samples showed no or very little visible damage area in the outer faces. X-ray microtomography technique (XMT) was used to investigate the damage and its distribution inside the specimens. XMT analysis showed that the impact energy has been absorbed to create a severe local damage under the impact center of PW laminates, whereas well-distributed delamination zones were found across the unidirectional laminates even far from the impact center. The most likely reason would be that unidirectional fibers allow the impact wave to propagate more easily through the structure, whereas the waviness of woven fabrics can act as a barrier for damage propagation. A large through-thickness crack was also seen inside the UD sample, which has broken 6 out of 12 layers of the laminate. In summary, this case study suggests that the rear side visible damage in impacted FRP laminates cannot represent the entire damage extension and the associated loss of effective mechanical properties (here identified through post-impact flexural testing). Micro-cracks and distributed local delamination sites “inside” the samples can significantly contribute to the dissipation of impact energy. A powerful nondestructive 510152025300 2 4 6 8 10 12Damaged area fraction (%)Distance to the impact center (mm)UD - top viewUD- left viewPW- top viewPW- left view19  inspection method such as XMT can be used to visualize and quantify the damage state and its extent inside the specimens in 3D. Some clear differences were seen between damage states inside the impacted UD and  PW laminates and suggested that reinforcement selection should be made with ultimate care depending on the objectives of a given impact application.    20  3 Chapter 3: Taguchi Predication and Lay-up Optimization of Composite Laminates under Impact Despite numerous developments in numerical and analytical models of composites, experimentation is still considered among the most reliable techniques for impact characterization of fiber reinforced polymers (FRPs), and for the validation of associated models. This is mainly due to the immensely high level of complexity in impact response of FRPs. On the other hand, for the lay-up design and optimization purposes, fabrication and experimentation of a large number of FRP configurations can be often costly or infeasible. Taguchi is a known design of experiment (DOE) method used for cost reduction during experimental investigation and quality engineering of complex physical systems and processes. Efficiency, accuracy, strengths and limitations of this method in predicting and optimizing impact response of FRP composites studied in the present chapter, through a case study on polypropylene/E-glass laminates with unidirectional, plain woven, and twill weave fiber architecture options. In parallel, drop weight impact tests, X-ray micro-tomographic investigations, and pre- and post-impact four-point flexural bending tests are employed for two purposes: (a) gaining further knowledge on induced impact damage mechanisms, and (b) assessment and verification of the Taguchi results in the performed case study. In spite of the fact that impact events of these laminates are highly nonlinear and accompany high level of uncertainty, it was found that the Taguchi method is capable of predicting and optimizing the response of the FRPs with a minimal number of runs and a reasonable error. Finally, by correlating macro-level predictions to micro-level damage observations via X-ray tomography, the underlying assumptions of the method were scrutinized for future impact applications.      3.1 Overview Application of fiber reinforced polymer (FRP) composite materials is increasing rapidly owing to their high specific strengths (strength per weight ratios). Sufficient knowledge about mechanical properties of these materials is an indispensable need for designers to produce reliable composite products in high risk applications. Numerous investigations have been conducted on the characterization of FRP structures 21  under different types of loading. One branch of these studies is the response of FRP materials under impact loadings [29]–[33]. Cantwell et al. performed an investigation on high-velocity impact characterization of fiber reinforced metal laminates [34], [35]. Trudel-Boucher et al. [36] studied the induced damage in commingled glass polypropylene laminates (Twintex®) under impact events in the range of 1 to 10 J impact energy. Brown et al. [37] studied the effect of strain rate, ranging from the quasi-static to 100 s-1, on the mechanical properties of twill weaved Twintex® laminates. They reported a linear proportionality between mechanical properties and the logarithm of strain rate for this type of composites. In particular, they found that the tensile and compression moduli and strengths increase with increasing the strain rate, while the shear modulus and strength decrease. Naik et al. [38] performed an experimental study on the effect of laminate configuration (lay-up) on the low-velocity impact behavior of polymer based composites. They concluded that combining unidirectional plies and woven fabrics adds to the impact resistance of composite structures. They also found that laminates made of 3D woven fabrics show a better out-of-plane performance than 2D woven fabrics. González et al. [39] investigated the effect of ply clustering on the impact response of carbon-epoxy laminates. They used three configurations for this purpose: (a) [(45/0/-45/90)4]S, (b) [(452/02/-452/902)2]S, and (c) [454/04/-454/904]S. Their experimentation and developed numerical model showed that as clusters’ uniformity increases from the configuration (a) to (c), the number of interfaces available for delamination initiation and growth reduces. Thus, larger delaminations occur in more diversely clustered laminates and results in a lower mechanical resistance of the structure. Beside experimental works, several numerical research efforts have been dedicated to the impact simulation of FRP composite structures [40]–[48]. Specifically related to impact ‘optimization’ of composites, Yong et al. [49] used a genetic algorithm (GA) to choose the optimum fiber orientations that can minimize the central deflection and penetration of a slender laminated strip under low-velocity impact as well as that of a rectangular composite plate under high velocity impact. In their numerical tests, the GA showed a better performance in the high-velocity impact scenario where the search space was larger 22  and more non-linear. One main challenge in that study, as well as similar investigations on composite sandwich panels [50] and hybrid multilayered plates [51], has been the tuning of optimization parameters for better accuracy and robustness of the results, as well as the large number of FE evaluations required for optimization (in the order of hundreds). It has also been recommended that to improve confidence in the numerical modeling results, more accurate material constitutive (analytical) models are yet to be developed, especially when considering woven fabric reinforcements (in place of unidirectional fiber plies) and/or hybrid laminates [49]–[51]. The fact that a large number of factors affect the impact behavior of composite laminates makes the numerical simulation of these materials rather complicated. These factors can be due to anisotropic properties of laminates, non-linear behavior, uncertainties in manufacturing process, and interaction of micro-level damage mechanisms. As a result, a universal numerical approach is not yet developed to consider these factors simultaneously and predict the impact response of all FRP composite configurations. Most numerical models are application-specific and experimentation is still the most reliable methodology to understand the impact response of new FRP materials [52] and to validate the accuracy of numerical predictions [47], [48]. On the other hand, experimentation has its own challenges: testing is time consuming and costly. In this regard, some design of experiments (DOE) methods have been developed during the past few decades to reduce the number of required tests, time and cost by means of maximizing statistical information gained through minimum resources. One of the most effective methods of this kind is the Taguchi DOE approach, which was named after Genichi Taguchi [53]–[57]. The underlying theory is based on (a) quantifying the definition of quality using Karl Gauss’s quadratic loss functions, (b) introducing a set of orthogonal arrays (OAs) for designing experiments, and (c) combining mean and standard deviation measures for robust data analysis and design [57]. A part of orthogonal arrays (as fractional factorial designs) that are used in Taguchi’s models had also been developed by other researchers including Fisher [58], Tukey [59],Yates [60], Graybill [61], Kempthorne [62],Cochran and Cox [35], Scheffe [36].  23  The Taguchi method has been widely used to reduce the cost and improve quality of manufacturing products in several engineering fields. This is done by reducing the number of experiments, predicting the response of non-tested configurations, and finding optimum manufacturing parameters. For instance, El-Lahham et al. [37] used this method to optimize preform die geometries in a three-stage cold heading process of bolts. Kilickap [38] used the method to optimize a set of drilling parameters to reduce the delamination size in glass fiber-reinforced plastic (GFRP) parts. It was found that reducing the cutting speed and the feed rate can reduce the delamination notably. Maghsoodloo et al. [39] performed a thorough review and investigated advantages and disadvantages of Taguchi’s DOE methodology for quality, manufacturing and process engineering applications.  3.1.1 Motivation and Organization of this Chapter From the review above, it is perceived that a precise impact simulation of multi-layer hybrid composite structures (especially those including woven fabrics) is still a challenging task [40]–[51]. Alternatively, full factorial experimental methods are often time consuming and expensive, especially during composite lay-up optimization trials. The present work, hence, aimed to explore strengths and limitations of a ‘fractional’ factorial Taguchi approach in predicting and optimizing the impact energy absorption of FRP hybrid laminates (including both UD and woven layers) with a minimum number of trials. To this end, polypropylene/E-glass laminates with unidirectional, plain weave, and twill weave fibre architecture options are chosen as a case study. The Taguchi L-9 experimental design along with drop-weight impact tests are described in Section 3.2. Following, the impact test results are analyzed and discussed in Section 3.3. Section 3.4 is a demonstration of the proposed Taguchi prediction and optimization approach along with a discussion on impact characterization of the FRP composites using X-ray micro-tomography. The latter section, in particular, relates the DOE optimization results to the fundamental knowledge that was gained from the micro-tomographic observations on impacted samples. A post-impact quasi-static four-point flexural testing is employed in Section 3.5 to correlate the impact data to the bending behaviour of the composite samples. The same section also discusses a new concept on the assumption of absence of 24  outlier in the Taguchi approach, as related to underlying damage mechanisms. Finally, main results from performed experiments and the DOE analysis are summarized in Section 3.6. 3.2 Experimentation 3.2.1 Materials Composite plies used were made of commingled polypropylene filaments (PP) and glass fibers, known under the trade name of Twintex ®. Three different ply architecture options were considered: (1) plain weave, (2) twill weave, and (3) unidirectional (UD). Table 3.1 shows the material specifications used in subsequent lay-up optimization trials. Laminate specimens were fabricated with vacuum bagging at 200°C according to ASTM D7136 [40], in rectangular plate shapes of 10×15 cm2. Table 3.1. Specifications of the PP/glass composite plies used in the laminated plates for impact optimization Weave pattern Code %Wt Glass Fibers Ply thickness (mm) Plain Woven1 PW %60 0.5 Twill Woven2 TW %60 1 Unidirectional3 UD %70 0.5 1.      2.      3.  3.2.2 Taguchi Experimental Design for Impact Testing As addressed before, testing all possible composite configurations (stacking sequences) would be infeasible or very expensive. In the present case, the total number of possible configurations would exceed 300. As a result, the Taguchi method along with realistic manufacturing constraints was used to reduce the number of experiments. Namely, the Taguchi L9 design (with three factors, three levels each [55], [65]) was employed with the conditions that:  25   All laminate specimens should be balanced and symmetric.  Thickness of each cluster (layer) in the laminate should be at least 1mm (e.g., two PW plies can be used in one cluster to give a 1mm thickness; this strategy was well aligned with the findings of González et al. [39] on the effect of clustering strategies on impact resistance of composites as discussed in Section 3.1).    Total thickness of all specimens should be equal to 6 mm.  Figure 3.1 shows the schematic of specimens’ lay-up. Unidirectional layers are always used in the cross-ply form (UD0/90) to meet the balanced configuration constraint above. Accordingly, hereafter the DOE codes for factor levels are:  Level 1: (PW)2;  Level 2: TW Level 3:  UD0/90  Figure 3.1. Schematic of the laminates lay-up. Table 3.2 shows nine sample configurations resulting for the L9 design, four extra sample configurations for validation purposes, and two readings of test for each configuration. For the UD laminate (configuration #13) the measured energy values appeared to be much higher than the other laminate configurations, hence that particular test configuration was repeated five times to ensure there is no measurement error. All drop-weight impact tests were conducted similar to ASTM D7136, with four sides of the specimens clamped in a rectangular rig as shown in Figure 3.2. The tip diameter of the hemispherical stainless steel projectile was 1 inch. The 12.35 kg projectile hit the samples at 5.69 m/s, resulting in a 200J impact energy. 26  With three layers (three factors in the DOE terminology; see also Figure 3.1) and each three levels (Table 3.1), the total number of runs for a full factorial experimentation would be 33, multiplied by the number of repeats per configuration. Using the Taguchi L9, only 9 configurations needed to be tested (i.e., over 66% reduction in the optimization cost). Accordingly, impact response of the remaining 18 non-tested configurations is expected to be predicted by the Taguchi model and the optimum solution be found as will be shown in Section 3.4.  Table 3.2: Nine composite lay-up configurations used for the subsequent Taguchi L9 optimization and four additional configurations employed for validation purposes  Design Purpose Config. # Layer A Layer B Layer C Factor  D (not  Active here) Configuration Lay-up Energy (J) R1, R2** Standard Deviation Coefficient of Variation% Taguchi L9 1 1* 1 1 1 [(PW)6]s 35.66, 36.63 0.49 1.34 2 1 2 2 2 [(PW)2,(TW)2]s 31.41, 35.92 2.26 6.70 3 1 3 3 3 [(PW)2,(UD0/90)2]s 34.90, 36.63 0.90 2.42 4 2 1 2 3 [TW,(PW)2,TW]s 40.19, 41.11 0.46 1.13 5 2 2 3 1 [(TW)2,UD0/90]s 38.27, 38.93 0.33 0.85 6 2 3 1 2 [TW,UD0/90,(PW)2]s 38.90, 38.36 1.27 3.37 7 3 1 3 2 [UD0/90,(PW)2,UD0/90]s 36.88, 34.99 0.95 2.63 8 3 2 1 3 [UD0/90,TW,(PW)2]s 30.85, 30.26 0.30 0.97 9 3 3 2 1 [(UD0/90)2,TW]s 34.43, 36.45 1.01 2.85 For Validation 10 2 2 1  [(TW)2,(PW)2]s 41.26, 44.57 1.66 3.86 11 3 3 1  [(UD0/90)2,(PW)2]s 36.85, 36.95 0.05 0.14 12 2 2 2  [(TW)3]s 36.17, 38.25 1.04 2.79 13 3 3 3  [((UD)0/90)3]s*** 71.87, 62.69, 67.67, 68.45, 75.00 4.15 6.00 * Numbers 1, 2 and 3 assigned to layers A, B, and C corresponding to (PW)2, TW and UD0/90 respectively.  ** R1 and R2 are two readings from two impact tests repeats.  ***There were five repeats for this particular configuration.  27   Figure 3.2. Clamping system used during the drop weight impact tests 3.3 A Discussion on Impact Test Data The acceleration history of impactor was recorded by piezoelectric sensor of the drop weight impact tower. Multiplying the acquired acceleration values by the mass of impactor (12.35 kg), the force-time history of the impactor was extracted. Subsequently, the velocity and displacement of impactor were calculated by consecutive numerical integrations of acceleration-time history. Finally, the induced energy of the impactor was estimated by the integration of force-displacement diagram. The initial energy of impactor (200J) minus the energy of impactor at the time of final rebound (t=~8sec) represented the dissipated energy due the material damage during impact. Figure 3.3 shows sample results for configuration#1. Point M in this figure indicates the time when the projectile was bounced back by the specimen. A favorable repeatability of the test (denoted by R1 and R2 in Figure 3.3) is also observed and verified statistically from low coefficient of variation (CV) values in Table 3.2.   28            (a)                                                                                                     (b)               (c)                 (d) Figure 3.3. (a) Force-time, (b) velocity-time, (c) force-displacement and (d) energy-time diagrams of configuration #1 ([(PW)6]s) under 200J impact energy; ‘M’ refers to the same peak force point in different graphs.  The average values of the absorbed (damage) energy for tested configurations in Table 3.2 are presented in Figure 3.4. This figure suggests that the experimental result for configuration #13 which is uniformly made of UD layers may be considered an outlier (unusual) relative to other configurations. It must be emphasized that in statistics, outlier points are those that are distant from other observations and can be of two types: those indicating erroneous data/procedures, and those corresponding to test cases where a certain theory might not be valid. The outlier notion attributed to the UD samples herein falls within the second category. Consequently, it is not expected that, statistically, the Taguchi prediction model be able to predict the response of this configuration as accurately as other configurations. This check point is numerically assessed via model predications in the next section, along with micro-tomographic inspections to find its source at a material damage level.  M M M M 29   Figure 3.4. Average absorbed energy (experimental) values for the 13 laminate configurations in Table 3.2  3.4 Taguchi Prediction and Optimization  The following Taguchi predictive equation ([54], [67], [68]) with three factors was used to predict the response of laminate configurations:   ypredicted = y + (yA − y) + (yB − y) + (yC − y) (3.1) Where y is the total average of measured values in the L9 array (Table 3.2) and yx refers to the average absorbed energy in the corresponding layer x (i.e., A, B, or C). Table 3.3 shows that there is a slight difference between the experimental results and predicted responses, even for extra configurations that were included in Table 3.2 for validation purposes, expect for the outlier case (last row in Table 3.3). It is important to note that theoretically the predictive equation (3.1) assumes:   Experimental error is not significant compared to main effects;   There is no interaction between factors (composite layers); and   The underlying source of each factor effect (the state of damage in each type of fiber preform option) does not change from one laminate configuration to another. This assumption is very implicit and has not been received adequate attention in previous applications of the method.  Any disagreement between experimental and predicted results may be attributed to these assumptions. In order to statistically find the significance of each of the model factors against the interaction and random experimental errors (assumptions (i) and (ii) above), a standard ANOVA analysis (with α=0.5) was Outlier  30  applied on the measured data in L-9. ANOVA resulted in p-values shown in Table 3.4, indicating that the most significant factors in energy absorption capacity of the laminates are layers A and B. Also ANOVA showed that the factor interactions, which can be statistically represented by the missing factor D in the L-9 and is attributed to the lack of fit in Eq. (3.1), had a very insignificant effect compared to the main effects as seen in Table 3.4. This indicates that the Taguchi approach predicted the absorbed impact energy of the FRP laminates reasonably well within the experimental errors and also the assumption of no interaction between composite layers is fairly acceptable in the current application.  The concept of interaction between composite layers and its mechanical interpretation may be further dissected via interaction plots shown in Figure 3.5. These figures reveal that there are slight interactions between layers, especially when the assigned material to a factor (layer) is UD (notice, e.g., in Figure 3.5(a) that the UD curve slope is not the same as PW and TW curves). From a mechanical point of view, the presence of interaction can be linked, e.g., to the presence of fiber bridging between layers and/or deboudning of interface between adjacent layers without them being effectively damaged under induced energy. Figure 3.6 illustrates a sample image of a UD laminate with a clear fiber bridging between composite plies. In fact, from Table 3.3 it can be noticed that higher prediction errors are present when the composite layers (specially the first two layers, A and B, which have had the highest statistical influence in the impact response) take the same factor level (e.g., (2,2,1), or (3, 3, 1), or (1,1,1), etc).  The reason is most likely due to increase in delamination zones as described for clustering in stacking sequence in Section 3.1 and [39].      31  Table 3.3. Prediction of absorbed energy values by the Taguchi method Config. # Layer A Layer B Layer C Actual Absorbed Energy (J) (Experimental average) Predicted Absorbed Energy (J) (Taguchi Method) Prediction Error (%) 1 1 1 1 36.1 35.4 1.9 2 1 2 2 33.7 34.0 0.9 3 1 3 3 35.8 36.1 0.8 4 2 1 2 40.7 41.0 0.9 5 2 2 3 38.6 37.9 1.8 6 2 3 1 37.6 37.9 0.8 7 3 1 3 35.9 36.2 0.8 8 3 2 1 30.6 30.9 0.9 9 3 3 2 35.4 34.7 2.0 10 2 2 1 41.3 35.9 13.1 11 3 3 1 36.9 32.9 10.8 12 2 2 2 37.2 37.8 1.6 13 3 3 3 69.1 34.9 49.5 (outlier) 14 1 1 2  Not tested (-) 37.3 - 15 1 1 3 - 37.4 - 16 1 2 1 - 32.2 - 17 1 2 3 - 34.2 - 18 1 3 1  - 34.1  - 19 1 3 2  - 36.0  - 20 2 1 1 - 39.2 - 21 2 1 3  - 41.2  - 22 2 3 2  - 39.7  - 23 2 3 3  - 39.9  - 24 3 1 1  - 34.2  - 25 3 1 2  - 36.0  - 26 3 2 2  - 32.8  - 27 3 2 3  - 32.9  -  Table 3.4. ANOVA results on the performed impact tests in Table 3.2 Source Sum of squares DOF Mean Square F-value p-value A 93.13 2 46.57 25.08 < 0.0001 B 34.62 2 17.31 9.32 0.0043 C 9.91 2 4.95 2.67 0.1136 Residual 20.42 11 1.86            Due to interactions (D) 3.50 2 1.75 0.93 0.4293 Due to pure error 16.92 9 1.88   Total 158.09 17    32                                                            (a)                                                                                              (b)                                                                                                         (c) Figure 3.5. Interaction plots between (a) layers A&B, (b) layers B&C, and (c) layers C&A; using the L9 experimental data in Table 3.2.  Figure 3.6. Fiber bridging in a typical UD laminate [42] 3.4.1 Optimum Lay-up and Micro-tomography Figure 3.7 shows the summary of Taguchi prediction results from Table 3.3 values; According to the observed trends, and assuming that the response (absorbed energy) is of the ‘higher-the-better’ type, layer 33  A can maximally contribute to the laminate response when the assigned level is: TW; similarly when layer B takes (PW)2 ,and Layer C is assigned UD0/90, the response is maximized. This means that the optimum composite lay-up is predicted to be (2,1,3)s = [TW, (PW)2, UD0/90]s. In contrast, the configuration (3,2,1)s= [UD0/90, TW, PW)2]s is predicted to absorb the least damage energy. From Table 3.3, the corresponding predicted absorbed energy for these two (best and worst) configurations is 41.2 J and 30.9 J, respectively.  Next, nondestructive x-ray micro-tomography technique (XMT) was employed to investigate the accuracy of the above predictions from the viewpoint of induced internal damage. Among the existing tested configurations in the L-9 array, XMT was done on [UD0/90, TW, PW)2]s and [TW, (PW)2, TW]s laminates, which were predicted in Table 3.3 to give the lowest (30.9J) and the second highest (41.0 J) energy values, respectively. It is worth noting that from both Table 3.3 and Figure 3.7, it is clear that the performance of the configuration (2,1,2)s = [TW, (PW)2, TW]s is nearly identical to the aforementioned top ranked solution [TW, (PW)2, UD0/90]s (absorbed energy 41.0 J vs. 41.2 J).  Figure 3.7. Taguchi model predictions via Eq (3.1); i.e., assuming no interactions Theoretically, the higher the absorbed energy, the more severe the induced internal damage. Comparing Figures 3.8 and 3.9, much more severe local damages, fiber breakage and matrix cracks are seen in the configuration [TW,(PW)2,TW]s compared to the configuration [UD0/90,TW,(PW)2]s. In the latter laminate, mainly local delamination sites are observed, whereas in the former case with the high energy absorption 34  capacity a severe damage state is seen under the impactor, including matrix crushing and fiber pull-out. This shows that X-ray micro-tomography and the Taguchi prediction results support one another.    Figure 3.8. X-ray microtomographic images of configuration [TW,(PW)2,TW]s: predicted absorbed energy=41.0J  Figure 3.9. X-ray microtomographic images of configuration [UD0/90,TW,(PW)2]s: predicted absorbed energy=30.8J Layer A: UD Layer B: TW Layer C: PW Delaminations Delaminations Fiber breakage & Pull-out Local damage Matrix failure 35  3.4.2 Contribution Percentage of Composite Layers In order to reveal to which extent each design factor (layers A, B, or C in Figure 3.1) contributed to the impact damage of samples, a percentage contribution analysis with three factors was performed as follows [54], [70]: The total average response of the L9 array:  (𝑖 is the configuration#). 𝑦 =19∑𝑦𝑖9𝑖=1 (3.2) Total sum of squares: 𝑆𝑆𝑇𝑜𝑡𝑎𝑙 =∑(𝑦𝑖 − 𝑦)29𝑖=1 (3.3)  Sum of squares of factor A, B and C: (𝑖 = 1,2 and 3 means (PW)2, TW and UD0/90, and 𝑦𝑥,𝑖 is the ith-level average response): 𝑆𝑆 𝐴 = 3 ×∑(𝑦𝐴,𝑖 − 𝑦)23𝑖=1  (3.4)  𝑆𝑆𝐵 = 3 ×∑(𝑦𝐵,𝑖 − 𝑦)23𝑖=1 (3.5)  𝑆𝑆 𝐶 = 3 ×∑(𝑦𝐶,𝑖 − 𝑦)23𝑖=1 (3.6) % contribution of factor 𝑥: % 𝑐𝑜𝑛𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛 𝑥 =𝑆𝑆𝑥𝑆𝑆𝑇𝑜𝑡𝑎𝑙 (3.7) Results of this analysis indicated that layers A, B, and C have contributed to the damage response of the laminates by 58.90%, 21.89%, and 6.26%, respectively. This order of layers percentage contributions (A>B>C) can be further explainable in Figure 10 where the mechanism of deformation due to the out-of-plane impact may be analogous to a high speed flexural bending. The strain distribution during bending should suggest that the largest deformation occurs in the outer layers, which makes their contributions in the DOE analysis more significant than the inner layers.  36   Figure 3.10. Schematic of deformation distribution in a six-layer symmetric laminate due to flexural deformation Another noteworthy result in the same notion was that one could equivalently choose the best hybrid lay-up in the present study from a standard flexural bending attribute. It is known that a given impact energy can be absorbed by elastic and permanent (damage) deformations. For a given level of input energy, the larger the elastic deformation energy, the less the damage energy would be in the structure. Hence, in order to design a laminate with maximum damage energy (absorption) capacity, the elastic energy portion should be minimized. In other words, the hybrid laminate (specially the outer layers) should have the lowest possible flexural stiffness among given reinforcement options (here, (PW)2, TW, and UD0/90)s. This hypothesis was explored in the next section via four-point flexural bending tests. 3.5 Pre- and Post-Impact Four-Point Flexural Testing Quasi static four-point flexural tests (Figure 3.11) were conducted on three different laminate configurations; one uniformly UD: [(UD0/90)3]s, one uniformly twill woven: [(TW)3]s and one uniformly plain woven: [(PW)6]s, all with the same plate thickness of 6mm. Figure 3.12 shows the obtained flexural chord modulus values, along with the percentage loss of modulus due to damage (i.e., comparison of stiffness before and after impact). In order to minimize the elastic deformation portion and to maximize the permanent deformation (damage) in a hybrid laminate, TW which has the lowest modulus, should be assigned to the outermost layer A, followed by PW to layer B, and UD to layer C. This configuration could yield a more uniform stress distribution with minimum magnitude across the cross-section and 37  hence minimizing the elastic strain energy. Such functionally varying stiffness design would also reduce the potential of sliding between layers (fracture mode II) under flexural deformation. The combination of different weave patterns also increase the chance of delamination and as a result increase in energy absorption. The optimum material configuration [TW,(PW)2,UD0/90]s found in Section 3.1 is in complete agreement with the result of above bending stiffness optimization.  Results in Table 3.3 confirmed that there are a number of different hybrid laminate configurations that offer increased energy absorption capacity compared to the uniformly distributed lay-ups. The only exception from this is the laminate with uniformly unidirectional plies, [(UD0/90)3)s]. This sample showed a very high capacity in energy absorption (~69.1J) and also the Taguchi method was not able to predict its response properly. The XMT was employed in the next section to explore the reason behind this observation at a micro-structural level.    Figure 3.11. Four-point flexural bending test setup  38   Figure 3.12. Flexural chord modulus before and after 200J impact (Due to impact, flexural chord modulus of UD, PW and TW laminates is reduced by 31%, 17% and 9%, respectively) 3.5.1 Why the UD laminate was found outlier in the current DOE model? Figure 3.13 shows the XMT image of a UD sample subjected to the same impact energy level (200J).  Comparison between the acquired images in Figures 3.8, 3.9, 3.13 for [TW,(PW)2,TW]s, [UD0/90,TW,(PW)2]s and [(UD0/90)3]s reveals that damage mechanisms in the uniform UD laminate is completely different than the other laminates. Namely, the conventional 45 degree brittle crack observed in the UD laminate in Figure 3.13 was not present in any of the hybrid configurations. It is worth adding that the same through-thickness crack was observed in all five test repeats of this laminate configuration. This would suggest that the high modulus of unidirectional plies (see also Figure 3.12) has made the UD laminate to act more like a stiff homogenous material with a brittle behavior (the reason will be discussed in details in Section 4.2.5). A severe delamination zone is also visible in Figure 3.13. The above mentioned crack and delamination have dissipated a much larger amount of energy (on average ~69.1J) in the UD laminate compared to all other configurations in Table 3.3. The reason that the Taguchi method has not predicted the absorbed energy of the UD laminate (~50% error) is deemed to be mainly due to this major change of damage mechanism (i.e., violating assumption (iii) listed in Section 3.4). Hence, close attention should be paid to DOE model predictions for FRP impact applications since such statistical models can lose their predictability when the state of damage mechanisms governing the impact response of layers changes significantly from one composite configuration to another. 39   Figure 3.13. Microtomography cross-sections of an impacted UD laminate showing a large through-thickness crack and delamination. 3.6 Summary of Findings for Taguchi Design of Experiment for Optimization of Composite Laminates under Impact The Taguchi design of experiments (DOE) method was employed in this study to optimize the configuration of PP/glass composite laminates subject to impact loading. Drop-weight impact testing, pre- and post-impact four-point flexural bending, coupled with X-ray micro-tomography (XMT) were used to analyze the impact response of the fabricated homogenous and hybrid  FRP laminates using the plain woven, twill woven and unidirectional ply combinations. It was found that:  The Taguchi approach is an effective method to predict the impact response of the FRP laminates with a minimum number of runs (here a 66% reduction in the number of tests was realized using the L-9 array).  Slight differences between experimental and Taguchi predicted results were attributed to the interactions between composite layers, such as fiber bridging or debounding of interface especially in the presence of unidirectional layers. Crack Delamination 40   X-ray micro-tomography images closely correlated the measured absorbed energy values via drop weight impact tests to the Taguchi predictions and the interior damage state in the impacted laminates. It was seen that the optimized configuration with the highest energy absorption capacity was damaged most significantly in order to dissipate the induced energy.  Results of the optimization for the out-of-plane impact response coincided with the optimization of laminates lay-up via functionally variable stiffness criterion.  Combination of different type of reinforcement layers (e.g., the hybrid configuration [TW, (PW)2, UD0/90]s), provided the FRP structure with the capacity to absorb more impact energy as compared to the uniformly reinforced [(PW)6]s  and [(TW)3]s laminates.   The measured response of pure unidirectional (UD) laminate configuration was found to be unusually high relative to all other configurations (i.e., it showed approximately twice higher energy absorption capacity which could not be predicted by the Taguchi model). The reason was investigated through X-ray micro-tomography. It was found that the damage mechanism in impacted UD laminate was notably different from that of other configurations. Instead of severe matrix crushing and fiber pull-out seen on the back side of impacted specimens with any combination of woven and unidirectional FRPs, a brittle through-thickness crack was observed in the uniform UD laminate configuration.   This gave a new understanding that the Taguchi technique can be a very cost-effective and reliable method in prediction and optimization of impact response of FRP laminates, expect when the governing damage mechanisms in factor (layer) options are significantly different for some configurations. Further case studies and evaluation of the method in other impact applications may be worthwhile using other composite types and loadings.   41  4 Chapter 4: Multicriteria Weave Pattern Selection for Glass Fiber Reinforced Composites under Impact The focus of this Chapter is to recommend a systematic approach for selecting the architecture of fabric reinforcements in FRC composite products under impact events. Namely, through a case study, nine design criteria have been considered to evaluate and compare impact response of different polypropylene/glass laminates under a drop tower with 200J energy: (1) dynamic reaction force, (2) absorbed energy, (3) maximum central deflection, (4) interior damage intensity, (5) exterior damage intensity, (6) ultimate flexural strength, (7) the relative loss of strength due to impact, (8) flexural toughness, and (9) the relative loss of flexural toughness due to impact. The TOPSIS multiple criteria decision making (MCDM) method was implemented to rank four candidate laminates with different fiber architectures: (a) plain woven, (b) twill woven, (c) unbalanced woven, and (d) unidirectional fiber tapes. Five types of subjective and objective weighting methods were chosen within the MCDM framework to assign relative importance factors to the design criteria: the Entropy method, the modified digital logic (MDL) method, the criteria importance through inter-criteria correlation (CRITIC) method, and two new methods called ‘Numeric Logic, (NL) and ‘Adjustable Mean Bars’ (AMB). A combinative weighting scheme has been proposed to show how these subjective and objective weights can be combined and used in the TOPSIS model in order to find the final ranking of alternatives. The combinative method is specifically aimed to mimic practical situations where decision makers may have different levels of experience given an application. MCDM results for performed the case study on impact optimization of polypropylene/glass laminates are discussed and linked to composite damage analysis via mechanical test data and a set of X-ray microtomography evaluations. 4.1 Overview During last decades, numerous investigations have been conducted on material selection, design and optimization of fiber reinforced composites, and more specifically on polymer-based composites, using 42  different experimental, numerical and analytical methods. Among numerical methods, the finite element analysis (FEA) has been the most widely used technique [71]–[84]. In particular, the review articles by Ghiasi et al. [85], [86] show how different FEA techniques have been applied to stacking sequence optimization, constant stiffness design [87]–[98], and variable stiffness design [99]–[105] of composites. Other investigations have used these methods to optimize variable stiffness [106],  buckling capacity [107], [108], post-buckling progressive damage [109], thermomechanical [110] and elastic responses [111]. Regarding the impact design of composite, Yong et al. [49] used a genetic algorithm (GA) to choose optimal fiber directions that can minimize the central deflection and penetration of slender laminated strips as well as plates. In contrast to most of the above works that considered a single objective optimization, in most practical  applications, designers often require to consider multiple (usually conflicting) criteria simultaneously during their decision making. Hence, the present chapter was aimed at presenting a multiple criteria decision making (MCDM approach for material selection of composites under impact. In addition, due to a large number of inter-correlated damage mechanisms and uncertainties in modeling impact response of fiber reinforced polymers (FRPs), it was decided to merely use experimental data throughout the case study. Among different MCDM techniques such as Maximin, Maximax, Conjunctive, Disjunctive and Lexicographical [112], Elimination by Aspect [113], Simple Additive Weighting (SAW) [114], Weighted Product Method [115], Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) [116], ELECTRE [117], Median Ranking [118], PROMETHEE [119], Analytic Hierarchy Process (AHP) [120] and other methods, the TOPSIS method was selected.  TOPSIS was originally proposed by Hwang and Yoon [116]  and later modified by Yoon [121] and Hwang et al. [122]. The concept of this method is to find the optimum alternative in a given decision space that has the shortest distance from the so called ‘positive-ideal solution’ and the farthest distance from the ‘negative-ideal solution’, at the same time [116]. Positive- and negative-ideal solutions are often artificial (infeasible) and are merely hypothesized in TOPSIS for ensuring the best performance of the chosen alternative. It should also be noted that TOPSIS is among compensatory ranking methods of 43  MCDM where trade-offs between decision attributes (criteria) are allowed. More specifically, a good performance of a material candidate under one design attribute can compensate the poor performance of that material under some other attributes. This feature of the method can be suitable in near-the-end stages of a design process where short-listed material candidates have met the minimum requirements and the question remained is which material can maximize the overall performance of the structure under ‘all’ given criteria. Due to it simplicity and efficiency, TOPSIS has been widely used in the past in a diverse range of areas.  As a few examples, the technique was used by Davoodi et al. [123] for the design of a car bumper beam made of hybrid bio-composites, by Pakpour et al. [124] for optimum DNA extraction from agricultural soil samples, and by Jee and Kang [125] for the material selection of a flywheel. Behzadian et al. [126] reviewed 266 scholarly articles on the use of TOPSIS in other applications such as (a) supply chain management and logistics, (b) design, engineering and manufacturing systems, (c) business and marketing management, (d) health, safety and environment management, (e) human resources management, (f) energy management, (g) chemical engineering, (h) and water resources management. The review referred to 103 different journal publications between 2000 and 2012.  In order to successfully apply the standard TOPSIS method in engineering design, some critical notes should be taken into account: (1) the decision matrix must satisfy design requirements, (2) the desired attribute values are normally in a monotonically increasing or decreasing form (i.e., the higher the better, or the lower the better), (3) measured units should be commensurable [126] otherwise they should be normalized, (4) criteria weights should be assigned by the designer as closely as possible to the given application, and (5) in the case of changes in the decision matrix (e.g., adding/removing one material), the entire calculation process must be repeated as it can affect the final ranking. Perhaps the most critical input in TOPSIS similar to most other MCDM methods, is the assignment of criteria weights which can be based on subjective, objective, or combinative techniques. Examples of the developed subjective weighing techniques include Digital Logic and Modified Digital Logic [127]; Weighted Least-Square Method [128]; Delphi method [129]; Simple Multiattribute Rating Technique (SMART) [130] and its modified versions including  SMARTS [131] and SMARTER [132]–[134]; Simos’ procedure [135]; 44  Revised Simos’ procedure [136] for single decision making and the extended version for group decision making [137]. These techniques specify the weights solely based on the preferential judgments of the decision makers (DMs) and as the number of attributes increases, they can become very intricate. Due to several conditions such as the lack of experience, imprecise information, limited capability of DM for analyzing and correlating attributes and intangible nature of criteria, sometimes the DM may not be able to assign weights to criteria [138], [139]. In order to solve this problem, other objective weighting techniques such as Entropy [116] and Criteria Importance through Intercriteria Correlations (CRITIC) [140] have been proposed to extract statistical (unbiased) weights through analysis of a given decision matrix. Such objective methods, however, ignore the valuable input that can be gained from the DM’s expertise via subjective weighting. Accordingly, some researchers such as Jahan et al. [141], [142] developed combinative methods to account for both types of weighting and arrive at a single aggregated criteria weights.  In the present work, next to using the MDL, CRITIC, and Entropy methods, two new subjective techniques will be proposed with the goal of simplifying the judgment process for DMs with different levels of experience, especially when the number of criteria in the decision space becomes large. In addition, the combinative weighting method proposed in [141] has been slightly modified to be able to adjust to different levels of experience of DMs, or for group decision making processes. To show the proposed approach, a case study is performed on assessing the impact behaviour of four different polypropylene/glass composite laminates with plain weave, twill weave, unbalanced twill weave, and unidirectional fiber architectures. Drop tower impact testing, non-destructive damage evaluation, four point flexural testing have been used to obtain the necessary data for decision making process. Section 4.2 presents the entire experimental procedure along with discussions of material testing results. Section 4.3 then uses the data from Section 4.2 and discusses the MCDM methods and results. In the same section two new subjective weighting methods (namely, a Numeric Logic method and an Adjustable Mean Bars method) are introduced, along with a modified combinative weighting scheme to mimic different practical scenarios where decision makers may have different levels of expertise in the composite design and/or 45  MCDM fields. Finally, Section 4.4 highlights the main findings of the study and recommends a potential future work direction. 4.2 Experimental Procedure 4.2.1 Sample Preparation and Selecting Performance Attributes Four groups of test laminates were made from commingled polypropylene/glass fibers with four different weave patterns: (a) plain woven (PW), (b) twill woven (TW), (c) unbalanced twill woven (UW), and (d) unidirectional (UD) fiber tapes, as shown in Figure 4.1. In the MCDM framework, the resulting four groups of homogenous laminates will constitute the decision ‘alternatives’. Each laminate was made by vacuum bagging at 200°C and cut to a rectangular shape (150×100 mm) per ASTM D7136 [27]. In order to make a fair comparison in subsequent analyses, all laminates were fabricated at the same thickness (6 mm). Moreover, in order to attain balanced lay-out, unidirectional (UD) plies and unbalanced woven fabrics (UW) were always used in the cross-ply configuration; i.e. [UD0/90]n and [UW0/90]n, respectively. Performance criteria selected for the evaluation of impact response of the four laminate types were: the reaction force during the dynamic impact event (RF), absorbed impact energy (AE), the maximum central deflection of the laminate (MCD), areal fraction of induced interior damage (ID), the exterior visible damage area (EVD), ultimate flexural strength  of the healthy/unimpcated sample (UFS), the relative loss of ultimate flexural strength due to impact (RLUFS), flexural toughness of healthy sample (FT), and the relative loss of flexural toughness due to impact (RLFT). Accordingly, three categories of experimentations needed to be performed to obtain all the required numerical values under each criterion. These comprised of (a) drop tower impact testing, (b) four-point flexural testing, and (c) non-destructive damage evaluation testing. 46   PW (11,11) %60 glass fibers  TW (22,22) %60 glass fibers  UW (22,11) %60 glass fibers  UD (11,00) %70 glass fibers Figure 4.1: Four different weave patterns examined in the case study for impact optimization; number of fibers in 22 bundles is twice the number of fibers in 11 bundles.  4.2.2 Impact Testing Drop-weight Impact tests were carried similar to ASTM D7136 [27] using a Dynatub 8200 impact tester (Figure 4.2-a). The test machine was equipped with a mechanical mechanism to prevent impact repetitions due to the rebound. The only difference between these tests and the ASTM D7136 was the clamping system. Namely, because of the presence of relatively high impact energy (200J), it was decided to clamp all sides of the specimens rather than only clamping four points. All the tests were performed at the energy level of 200J and repeated twice using two samples per laminate configuration. A projectile with a hemispherical stainless steel tip with a diameter of 1-inch and the mass of 12.35kg struck each sample at a velocity of 5.69 m/s. 4.2.3 Pre- and Post-Impact Flexural Testing Four-point flexural tests, with two repeats per laminate type, were performed on both impacted and non-impacted specimens according to ASTM D7264 [143]. The goal was to measure both the post-impact mechanical properties of the laminates as well as their mechanical performance loss due to the impact. The latter can specially be important for specific applications where the structure (e.g., a guardrail) may experience multiple impacts during its service life and/or the maintenance schedule/repair interval times are too long. Figure 4.2-b shows the setup of this test, where the supports span (L) and the loading span (L/2) were set to be 100 mm and 50 mm, respectively.  47                                                                        (a)                                                 (b) Figure 4.2: (a) Drop-weight impact test tower, (b) Four-point flexural testing of samples; support’s span (L) and loading span (L/2) 4.2.4 Non-destructive Damage Evaluation In order to assess the extent of damage in the impacted specimens, two different non-destructive techniques were employed as follows. 4.2.4.1 Visual Inspection Visual inspection with the aid of a digital camera and a ruler was implemented to quantify the exterior visible damage areas on the rear side of impacted samples. Namely, as shown in Figure 4.3, a polygon was drawn around the boundary of each damaged zone and the corresponding area was calculated. Since the exact selection of the corner points of the polygon could be erroneous by the operator, the process was repeated six times by the same person and the data were averaged for each sample.    L/2 L 48   Fig. 4.3: Visual measurement of visible damage area in the rear face of an impacted sample (the blue scale bar is in cm) 4.2.4.2 Microtomographic Evaluation X-ray micro-tomography technique (XMT) was employed for the non-destructive evaluation (NDE) of the interior damage intensity in the impacted laminates. Figure 4.4 illustrates the features and operation of the employed Xradia microXCT-400 machine during the XMT analysis. The high power X-ray travels from the left to the right side of the set-up shown in Figure 4.4 by penetrating through the specimen mounted on the rotation stage and reaching to the scintillator detector. The adjusted lens then magnifies the projected image obtained by the detector and the CCD camera records the enlarged image and transfers it to a computer for further processing. The rotation stage was set to rotate between -110o to +110o and the detector was set to collect 630 images. The illumination time for each image was 1 second. The interior 3D microstructure of each sample was reconstructed from the 630 projections.   Figure 4.4: The x-ray micro-tomography system used for internal damage investigation of impacted samples 49  4.2.5 Experimental Results and Discussion Figure 4.5 shows the average force-time and energy-time histories obtained from the impact tests. According to Figure 4.5-a, TW and UD laminates have exerted the most and least forces to the impactor, respectively. According to Figure 4.5-b UW and PW have shown the fastest and slowest deflections, respectively. Figure 4.5-c shows the obtained force-deflections curve. The integration of the latter curve gives the energy history of impactor shown in Figure 4.5-d. Each flat part of the curve in Figure 4.5-d (with a value smaller than the original energy, 200J) indicates the portion of the striker’s energy that has been dissipated (absorbed) by the structure. More specifically, this energy has been dissipated through damage initiation and propagation in the laminates. Figure 4.5-d suggests that the UD samples have the highest capability to absorb energy while the lowest capability has been for the PW samples. This figure also shows that the absorbed energy is decreased by increasing the fiber waviness from UD, to UW, to TW, and to PW. The arrow in Figure 4.5-d points at the separation of the energy curve of the UD and UW laminates. This separation has occurred towards the end of the rebound stage, where normally no further significant damage is expected to be induced into the composite.  The root of higher energy absorption capacity of the UD laminate was further investigated with x-ray imaging in chapter 2 [30]. It was found that a through-thickness crack in the UD laminate (seen in all repeats of this configuration (Figure 4.6)) adjacent to the impactor perimeter (starting at ~6 mm from the impact centre) may have been responsible for this increase of energy absorption capacity. Interestingly, this crack was initiated and propagated on the impacted face of sample, where the corresponding plies would have been mainly under compression. The earlier report by Santulli et al. [144] also showed the presence of a long transverse crack starting on the impacted surface of twill composites. Since that crack is found adjacent to impactor on the top surface it can be inferred that it is likely produced due to intense local matrix shearing and associated fiber breakage and kinging in the compressive layers.  It is known that in contrast with the tensile loading conditions, fibers in FRP composites carry very low loads under compression [145]. Accordingly, depending on the geometrical and material conditions like 50  fiber architecture, different failure mechanisms such as fiber micro-buckling, kinking and fiber breakage  may occur in compressed FRP composites [146]. Micro-buckling as one of the main compression failure modes was first introduced by Dow and Gruntfest [147]. Rosen [148] developed a model for this failure mode, which later was revised by Sadowsky et al. [149]. Inspired from Argon’s work [150] on the combined effect of fiber misalignment and matrix yielding in composite fractures, Budiansky [151] found a sensitivity of micro-buckling to local fiber misalignments in the composites. Some researchers in the above cited works consider micro-buckling and kinking as one identical failure mode, while others differentiate between them. Several studies have been specifically conducted to find a correlation between fiber micro-buckling and kinking in UD  laminates [152]. According to [86], kinking in FRP composites under compression initiates from fiber breakage via micro-buckling of pre-existing misaligned fibers, followed by local matrix shearing and the formation of kink band. Since the critical strain required for fiber breakage was found to be much larger than the overall strain at which a kink band can exist, it was concluded that the presence of broken fibers is critical in the kink band formation and propagation. Our experimental results in Figure 4.6 support this hypothesis. In the healthy (unimpacted) UD samples under 4-point bending, no transverse cracking was observed whereas in all the impacted UD samples, the same type of cracking appeared on the edge of damaged area on the front face  of specimens. At the very beginning of collision, fibers closer to the free surface would have been crashed in the perimeter of impactor, hence facilitating the formation of fiber kinking and the subsequent strain-driven brittle fracture (transverse crack) during impact and post-impact bending. In the woven fabrics the extent of kinking would be much limited due to the geometrically interlocked yarns at each crossover point.   Interestingly, the transverse crack in the UD sample has advanced only through half of the laminate thickness (Figure 4.5-e & 4.5-f). A possible explanation may be that the lower part of the sample during rebound, or the post impact bending test, is mostly under tension where micro-buckling and kinking damage mechanisms are not active. Hence the through thickness crack path has transformed into a large delamination along the length of the sample, as seen from the side view in Figure 4.5-e. These 51  6 mm hypotheses, however, are yet to be further verified via further experimentation and advanced micro-level finite element simulations in the future.                                                                                                                      (e)                                                                                         (f) Figure 4.5: (a) force-time, (b) deflection-time, (c) force-deflection, and (d) velocity-time diagrams of samples subjected to 200J impact energy; Images (e)  and (f), adopted from [30], show the front and side XMT slices of the UD laminate at 12 mm from the impact centre, respectively; notice a very large crack extended through half of the laminate thickness in image (e) and the long delamiation sites in image (f).  (a) (b) (c) (d) 52                                                   (a)                                                                                               (b) Figure 4.6: (a) The front face of samples after four-point bending test (without impact), (b) after 200 J impact followed by four-point bending. Notice the circular trace of impactor in the middle of specimen in case (b). White arrows point to the traces of four point-bending loading fixture; for unimpacted specimen (case (a)) kinking occurred right below the loading bars while for the impacted specimen (case (b)) two parallel kinking occurred adjacent to the impactor.  Figure 4.7 shows the flexural stress-strain curves obtained by four-point quasi-static bending tests on healthy specimens before and after the impact. The area under the stress-strain curves up to the pick stress (ultimate strength) was measured and identified as “flexural toughness” of the laminates. Experimental results showed that ultimate flexural strength and flexural toughness of {PW, TW, UW, UD} samples reduced by {18.6%, 7.3%, 20.2%, 31.9%} and {17.8%, 11.1%, 14.7%, 29.8%} respectively. It can be concluded that the residual mechanical properties of TW and UD samples, respectively, have been affected the least and the most by the impact.    53             Figure 4.7: Flexural stress-strain curves for (a) PW, (b) TW, (c) UW, and (d) UD samples before and after 200J impacts. The relative percentage loss of ultimate flexural strength (LUFS%) due to impact is 18.6%, 7.3%, 20.2% and 31.9%, for the PW, TW, UW, and UD laminate, respectively. Similarly, from areas under the stress-strain curves up to the maximum strength point, it was found that the relative percentage of fracture toughness loss (FTL%) due to impact is 17.8%, 11.1%, 14.7% and 29.8 %, for the PW, TW, UW, and UD laminate, respectively. Table 4.1 shows the average of visual inspection trials for exterior damage of samples. Images in this table display a severe damage on the PW laminate. The general forms of the external damage on TW and UW laminates were similar to that of PW (i.e., a rectangular-shape damage area, localized within the warp and weft cell sizes of each weave pattern), with a trace of fiber breakage/delamination. No severe visible damage was found on any of the UD laminates’ rear side (only minor surface delaminations).   (a) (b) (c) (d) 54  Table 4.1: Back face damage of the four tested PP/ glass laminates under 200J impact energy Fabric Laminate Thickness Configuration Repeat #1 Repeat #2 Damaged Area (cm2) PW 6 mm [(PW)6]s   3.50 TW 6 mm [(TW)3]s   3.03 UW 6 mm [(UW0/90)2]s   3.07 UD 6 mm [(UD0/90)3]s   0.44  Figure 4.8 shows the x-ray microtomography (XMT) results including the top, front and left cross-sections at the impact center of PW, TW, UW and UD samples. It reveals that the interior damage state is highly dependent on the fiber weave pattern. For example, a very large localised fiber breakage, fiber pullout and matrix failure are observed in the PW and UW samples, while no severe local damage is seen at the impact center of UD laminates. Interestingly, ‘well-distributed’ matrix failure and delamination sites are observed in the TW sample. 55                                                (a) PW                                                                                      (b) TW                                              (c) UW                                                                                       (d) UD Figure 4.8. X-ray microtomography slices at impact center for (a) PW, (b) TW, (c) UW, and (d) UD samples subjected to 200J impacts. In order to quantify the damage extension in XMT slices, further image analysis was conducted. Figure 4.9 shows the color-coded processed images of the front and left sections of the XMT images at the impact center. The damage fraction areas obtained based on these images are presented in Table 4.2.  The average percentage of the damaged area between front and left views for each sample is used as one of the attributes in subsequent multi-criteria decision making.  Top Left 3D Front Top Left 3D Front Top Left 3D Front Top Left 3D Front 56   Top views Left views  PW  PW  TW  TW  UW  UW  UD  UD Figure 4.9. Processed images of the front and side sections of the x-ray microtomography images in Figure 4.7; quantitative values of the colored areas are given in Table 4.2. (thickness of all samples is 6 mm).  Table 4.2: Comparison of internal damage area percentages from post-processed images in Figure 4.8  Left view  Top view PW TW UW UD  PW TW UW UD Healthy area% (green zone) 0.596 0.684 0.493 0.834  0.578 0.671 0.500 0.821 Damaged area% (dark red zone) 0.404 0.316 0.507 0.166  0.422 0.329 0.500 0.179 4.3 Multi-Criteria Decision Making (MCDM) Table 4.3 summarizes the entire matrix of experimental data obtained in Section 2, which is now the basis of the decision making problem in this section. The nine design performance criteria are those described 57  in Section 4.2.1. For a general impact application, the criteria AE, UFS and FT would be benefit-like (i.e., the higher the better), while RF, MCD, ID, EVD, RLUFS and RLFT would be cost-like (i.e., the lower the better).  Table 4.3: Summary of results of the impact tests including, drop tower, visual inspection, XMT and four-point flexural bending measurement; this is the basis of the decision matrix for subsequent MCDM/optimization calculations  Materials  Impact Test (Dynamic Properties)  Nondestructive Evaluation  Pre- and Post-Impact Flexural Test (Quasi-Static Properties)  RF (N) AE (J) MCD (mm)  ID (%) EVD (mm2)  UFS (Mpa) RLUFS (%) FT (kN/m2) RLFT (%) PW  32,018 36.15 11.87  0.413 3.50  262.22 18.60 358.12 17.83 TW  34,121 37.21 12.43  0.322 3.03  222.67 7.34 294.41 11.08 UW  31,286 42.76 13.22  0.504 3.07  225.80 20.17 226.04 14.70 UD  29,514 69.14 12.37  0.173 0.44  206.59 31.95 148.25 29.79 4.3.1 Why the use of MCDM is critical in this design application? Table 4.4 shows the ranking of materials within each column of Table 4.3; i.e., based on single objectives related to each specific criterion (benefit or cost). This table reveals that each material can be strong in terms of some criteria and weak in terms of the others. For example, the UD laminate has shown very good response in terms of energy absorption with a minimal visible damage, but its residual strength has dropped significantly. Another example can be the PW laminate which has not absorbed the impact energy as highly as the other laminates but interestingly its interior and exterior damaged areas are significant. The strong point about this laminate configuration, however, is that its flexural strength and toughness have been better than all other configurations. In fact in Table 4.3, no two columns indicate the same ranking orders. These clear contradictions with respect to individual design criteria point to the critical need for a multiple criteria decision making model to find the final ranking of the material candidates.    58  Table 4.4: Ranking of materials based on individual criteria (single objective optimizations). Materials  Impact Test (Dynamic Properties)  Nondestructive Evaluation  Pre- and Post-Impact Flexural Test (Quasi-Static Properties)  RF AE MC  ID EVD  UFS RLUFS FT RLFT PW  3 4 1  3 4  1 2 1 3 TW  4 3 3  2 2  3 1 2 1 UW  2 2 4  4 3  2 3 3 2 UD  1 1 2  1 1  4 4 4 4  4.3.2 TOPSIS MCDM method As discussed in section 4.1, TOPSIS is known as an effective MCDM technique and has been widely used in diverse applications. In the present case, we employed TOPSIS to rank the four alternative materials and to choose the best laminate configuration for a low-velocity impact application such as semi-rigid guardrail where the internal damage can yield a significant reduction on the performance of the material. Mathematical derivations of the TOPSIS method can be found in other works (e.g. [116]) and its implementation steps have been summarized in Appendix A. 4.3.3 Weighting Methods One of the critical inputs in most MCDM methods, including TOPSIS, is to assign importance weights on the decision criteria. As reviewed in Section 1, there are several ways to define such weights [127]–[142]. Nevertheless, most of the developed techniques fall into two main categories: subjective and objective weighting. Subjective methods rely on the expert-opinion while the emphasis of the objective methods is on the statistical evaluation of data given in a decision matrix. Each these techniques has its own advantages and disadvantages. Potential uncertainty in expert judgment is the main disadvantage of the subjective methods, while the objective methods do not benefit from the expertise and experience of designers. In the current case study, a combination of these two techniques is used to comprise both expert’s opinion and the experimental facts. To this end, three subjective and two objective approaches were chosen: (a) a new (subjective) adjustable mean bars (AMB) method, (b) the (subjective) modified 59  digital logic (MDL) method [127], (c) a new (subjective) numeric logic (NL) method, (d) the (objective) Entropy method [116], and (f) the (objective) criteria importance through inter-criteria correlation (CRITIC) method [140]. Each of these techniques along with their combination is described below.  4.3.3.1 Adjustable Mean Bars (AMB) Direct Weighting Method The simplest method of objective weighting is the direct complete weight elicitation [116] where a highly experienced DM is able to assign the relative importance values of all criteria at once. This method is not advised for all the DMs because of its highly intuitive nature and potential inaccuracies in final ranking. Accordingly, less experienced DMs often opt to give an equal weighting (EW) to the criteria; however this approach can then become too conservative in some sensitive applications. Ordinal weighting techniques have been proposed in the literature to address this problem and assist the DMs; such as the rank sum (𝑅𝑆) weighting and rank reciprocal (𝑅𝑅) weighting [153], as well as the rank order centroid weighting (𝑅𝑂𝐶) [133]. In these techniques, the DM first ranks the attributes based on their priorities and then assigns weights in a descending order (𝑤1 > 𝑤2 > ⋯ > 𝑤𝑛), starting from the most important to the least important attribute. Barron and Barrett [134] compared the efficiency of the 𝐸𝑊, 𝑅𝑆, 𝑅𝑅 and 𝑅𝑂𝐶 techniques based on Eqs (4.1)-(4.4) using more than 10,000 test cases. They reported that 𝑅𝑂𝐶 outperforms the other techniques with the order of 𝑅𝑂𝐶 > 𝑅𝑅 > 𝑅𝑆 > 𝐸𝑊. 𝑤𝑗(𝐸𝑊) =1𝑛 (4.1) 𝑤𝑗(𝑅𝑆) =2(𝑛 + 1 − 𝑅𝑗)𝑛(𝑛 + 1) (4.2) 𝑤𝑗(𝑅𝑅) =1/𝑅𝑗∑ 1/𝑅𝑗𝑛𝑗=1 (4.3) 𝑤𝑗(𝑅𝑂𝐶) =1𝑛∑1𝑘𝑛𝑘=𝑗 (4.4) 60  Where n represents the number of attributes (j = 1,2, … , n) and Rj is the preferred rank of attribute j; Rj=1 represents the most important attribute. In Eq. (4.4), k indicates the number of alternatives to be found as optimum (usually k=1). In order to satisfy ∑ wj = 1nj=1 , the RS and RR weights should be normalized, e.g., with respect to the sum of weights. A problem with the above objective weighting methods is that they are merely based on criteria ranks and, thus, the distance between each consecutive criteria weights remains constant (i.e., the DM cannot give an extra emphasis to some specific criteria). For example, Table 4.4 shows the ROC weights for several cases with different number of attributes ranging from 2 to 9. As it can be seen these set of weights are constant and independent of the opinion of DM and data values. Table 4.5: ROC weights for given number of attributes [133]  Number of Attributes Rank j 2 3 4 5 6 7 8 9 1 0.7500 0.6111 0.5208 0.4567 0.4083 0.3704 0.3397 0.3143 2 0.2500 0.2778 0.2708 0.2567 0.2417 0.2276 0.2147 0.2032 3  0.1111 0.1458 0.1567 0.1583 0.1561 0.1522 0.1477 4   0.0625 0.0900 0.1028 0.1085 0.1106 0.1106 5    0.0400 0.0611 0.0728 0.0793 0.0828 6     0.0278 0.0442 0.0543 0.0606 7      0.0204 0.0335 0.0421 8       0.0156 0.0262 9        0.0123 ∑𝒘𝒋 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000  An ‘Adjustable Mean Bars’ (AMB) weighting method is proposed in this section with the goal of keeping the simple nature of direct weighting method, while allowing the DM to assign more emphasis on specific criteria of interest. Depending on the total number of attributes (n), this method can take up to n − 1 sub-weighting steps. In each step, the DM picks the most important attribute(s) and assigns a numeric weight between 0 and 1 according to Eq. (4.5): 61  𝑤𝑗,𝑎𝑑𝑗 =1 − ∑ 𝑤𝑖𝑚𝑖=1𝑛 − 𝑚+ 𝑘𝑗1𝑛2 (4.5) Where 𝑤i represents the weight of attribute i calculated in the previous steps and m is the total number of attributes weighted in the previous steps. 1𝑛2 is the minimum step size (unit vector) allowed for adjusting the weights, and 𝑘𝑗  is the emphasis factor of criteria j defined by the DM. This method is named adjustable mean bars (AMB) because, in each step, the DM sets the height of (𝑚 − 𝑛) number of mean bars equal to the average of the remaining weights (1−∑ 𝑤𝑖𝑚𝑖=1𝑛−𝑚) and then adjusts the weight of the most important attribute(s) by increasing their height(s) by (𝑘𝑗1𝑛2). It should be noted that the DM gives a positive integer value to kj according to the required emphasis of criteria j, with a constraint that the calculated weight in each step must be smaller than the assigned weights in the previous steps. For instance, 𝑘𝑗 =1 would mean that the DM would like to raise the relative importance of criteria j only one unit step (quantified as 1𝑛2) compred to the previous step. The second constraint is that the summation of all final weights should be equal to one ( ∑ 𝑤𝑗,𝐴𝑀𝐵 = 1 𝑛𝑗=1 ). Figure 4.10 visually illustrates how this method works, for an example where the total number of attributes is n =7. The solid and hollow bars in each step in Figure 4.10 demonstrate the weighted and yet-to-be-weighted attributes, respectively. The heights of the solid bars show the AMB weights.  For better clarity, the procedure of weighting in this example is further detailed below: Step 0 (default condition): All attributes are equally important. Thus, the default height of mean bars is w =17≅ 0.14. The step size is calculated to be 172≅ 0.02.  Step 1: the DM specifies that attributes 3 and 7 are the most important attributes with the relative emphasis factor of 3 (𝑘3 = 𝑘7 = 3). This means that the DM considers three unit step rise for attributes 3 and 7 compared to the other ones. As a result, the new weights for these attributes became: w3 = w7 =0.14 + (3)(0.02) = 0.20. 62  Step 2: The remaining weights with a total value of 0.6 (i.e., 1 − 2(0.2) = 0.6) should be distributed between unweighted attributes (1, 2, 4, 5, 6). So, the height of current hollow bars is set to 0.6 5⁄ = 0.12. Then, according to DM’s opinion, the second most important group of factors are attributes 1, 2 and 5 with the importance factor of one unit step more than attributes 4 and 6. Accordingly, the corresponding weights were adjusted based on Eq. (4.5) as: w1 = w2 = w5 =1−2(0.2)5+ 1(0.02) = 0.14 Step 3. The mean height of the two remaining bars (i.e., for attributes 4, 6) becomes 1−[2(0.2)+3(0.14)]2=0.09. Between these attributes, the DM gives a one unit higher importance to attribute 6; i.e., k6 = 1. Subsequently, the AMB weight for attribute 6 becomes: w6 =1 − [2(0.2) + 3(0.14)]2+ 1(0.02) = 0.11 Step 4. The weight of the remaining attribute (w4) is calculated as: ∑ wj = 1 7j=1   w4 = 1 − [2(0.2) + 3(0.14) + 0.11] = 0.07 It should be noted that after completing one round of weighting, the DM can compare the weight distribution to verify whether it is satisfactory based on his/her perception.  In the case of dissatisfaction, he/she can go back to any of the above steps and change the k values and repeat the procedure until a suitable distribution is found.    63                                          (a) initial step (default)                                                                                                                                        (b) step 1                                                                 (c) step 2                                                                                                                          (d) steps 3 and 4                                                                                          Figure 4.10. Adjustable Mean Bars (AMB) weighing steps for a hypothetical example with 7 attributes: (a) initial state with equal weighing, (b) attributes 3 and 7 are weighted, (c) next attributes 1, 2 and 5 are weighted, and (d) finally attributes 4 and 6 are weighted; solid and hollow bars demonstrate the weighted and yet-to-be weighted attributes in each step, respectively).  Adj. Mean=0.12 Mean=0.143 Adj. Mean=0.09 64  4.3.3.2 Modified Digital Logic (MDL) Method  For applications in which the number of design attributes is fairly large (similar to the current case; 9 criteria), assigning the importance weights among multiple criteria simultaneously may be very difficult for the decision maker (DM). The Digital Logic (DL) method has been developed to address this issue by suggesting pair-wise comparisons of criteria (which is essentially similar to the approach behind the Analytic Hierarchy Process/AHP [120]). The DL method has proven to be successful in increasing the reliability of decision results to a large extent, while providing a simple intuitive procedure for implementation purposes. In this method, two criteria are compared at a time and receive binary scores of 0 or 1 depending on their level of priority to the decision maker (1 for more important criterion, 0 for the less important one). Dehghan-Menshadi et al. [127] developed a Modified Digital Logic (MDL) method and others researchers including Diakoulaki et al. [140] and Chakraorty and Chatterjee [154] employed it to (a) give the possibility to the decision maker to assign equal weights to two attributes, and (b) not to eliminate the least important criterion from the decision matrix. These enhancements were achieved by changing the aforementioned binary scoring scheme from {0 and 1}, to a digital scoring scheme of {1, 2 and 3} to represent the less (1), equal (2), or more important (3) criteria. After all pair-wise comparisons are made, the MDL weights can be calculated as: 𝑤𝑗 =∑ 𝐶𝑗𝑘𝑛𝑘=1∑ ∑ 𝐶𝑗𝑘𝑛𝑘=1𝑛𝑗=1    ,      𝑗 𝑎𝑛𝑑 𝑘 = {1,… , 𝑛}  𝑎𝑛𝑑 𝑗 ≠ 𝑘 (4.6) If two criteria j and k are equally important, then 𝐶𝑗𝑘 = 𝐶𝑘𝑗 = 2, otherwise 𝐶𝑗𝑘 = 3 and 𝐶𝑘𝑗 = 1 if the attribute k is more important than the attribute j. If the attribute k is less important than the attribute j, then  𝐶𝑗𝑘 = 1 and 𝐶𝑘𝑗=3. 4.3.3.3 Numeric Logic (NL) Method A slight modification is applied to the MDL method in the present work to allow the more experienced DMs apply precise weighting among criteria. In the proposed Numeric Logic (NL) method, during each pair-wise comparison, the decision maker can assign any arbitrary numeric weight (𝑤1) between 0 and 1 65  to the first criterion, and 𝑤2 = 1 − 𝑤1 becomes the weight of the other criterion. In other words, in this case, weights during pair-wise comparisons are not limited to 0.25, 0.50 and 0.75 as in the MDL method. The final NL weights can be calculated again via the same Eq. (4.6) with the only difference being that 𝐶𝑗𝑘 this time represents the arbitrary numeric weights between criteria j and k (rather than digital scores). 4.3.3.4 Entropy Method The meaning of the term “entropy” depends on the field of its application. For example, in physics it implies the level of disorder in a system, and in transportation models it shows the dispersal of trips between two locations [155]. For some events or applications it quantifies the degree of randomness or fuzziness [156]. In MCDM, entropy relates to the degree of diversity within an attribute dataset [47]. The greater the degree of the diversity, the higher the weight of that attribute. In another words, the smaller the entropy within the data associated to an attribute, the greater the discrimination power of the attribute in changing ranks of alternatives. The steps for calculation of Entropy weights are as follows [116], [157]. Step 1. Normalization Since measured data under different criteria can be of different units or scales, a given decision matrix (e.g., Table 4.3) should be first transformed into a dimensionless space via: pij =xij∑ xijmi=1;  i=1,…,m  &  j=1,…,n (4.7) where xij is an element of the decision matrix corresponding to the ith alternative and the jth criterion. m is the total number of alternatives (here four; materials PW, TW, UW and UD), and n is the number of criteria (here the nine impact design attributes). Step 2. Calculation of the entropy (Ej) and the degree of diversity (dj) Entropy within the datasets of the normalized decision matrix for the jth criterion can be calculated via: 𝐸𝑗 = −1ln (𝑚)∑𝑝𝑖𝑗 𝑙𝑛 𝑝𝑖𝑗    𝑚𝑖=1 (4.8) 66  The degree of diversity (dj) is then calculated as: 𝑑𝑗 = 1 − 𝐸𝑗 (4.9) Step 3. Calculation of objective weights (wj)  The last step is the linear normalization of 𝑑𝑗 to find the relative weight of each criterion: 𝑤𝑗 =𝑑𝑗∑ 𝑑𝑘𝑛𝑘=1 (4.10) 4.3.3.5 Criteria Importance through Inter-criteria Correlation (CRITIC) Method In addition to the contrast intensity of attribute datasets in the decision matrix (the notion that was quantified by the Entropy method), there is another concept that is more recently taken into consideration by MCDM researchers. Diakoulaki et al. [140] noticed that the higher the level of interdependency between attributes, the larger the ranking outcome error. Criteria importance through inter-criteria correlation (CRITIC) was proposed in [140] as a new objective weighting method that can consider correlations between all given criteria. The method also included the contrast intensities (by means of standard deviations of criteria) and combined them with the weights from correlations. Alternatively, the modified model in [141] employed the Pearson product-moment correlation coefficients only and excluded the standard deviations of criteria in formulations. Since in the present work the contrast intensities will be taken into account by means of the Entropy method, we follow the same approach as in [141] to calculate the correlation weights as below. Step 1. Find the correlation coefficient Rjk calculated via the Pearson product-moments represents the correlation between the criteria j and k.   𝑅𝑗𝑘 =∑ (𝑥𝑖𝑗−?̅?𝑗)(𝑥𝑖𝑘−?̅?𝑘)𝑚𝑖=1√∑ (𝑥𝑖𝑗−?̅?𝑗)2𝑚𝑖=1 ∑ (𝑥𝑖𝑘−?̅?𝑘)2𝑚𝑖=1  ; (j & k=1,…n) (4.11) 67  Where m, ?̅?𝑗  and ?̅?𝑘  are the number of materials and the average values of criteria j and k, respectively.  𝑅𝑗𝑘 close to +1 or -1 indicates highly correlated criteria, while  𝑅𝑗𝑘 close to 0 indicates no correlation.  Step 2. Calculating the CRITIC weights  The next step is to calculate the weight of each criterion using its correlation to all other criteria: 𝑤𝑗,𝐶𝑅𝐼𝑇𝐼𝐶 =∑ (1 − 𝛽𝑅𝑗𝑘)𝑛𝑘=1∑ (∑ (1 − 𝛽𝑅𝑗𝑘)𝑛𝑘=1 )𝑛𝑗=1     ;  𝑗 and 𝑘 = 1, 2, … , 𝑛 (4.12) The sign function 𝛽 is equal to + 1 if the two objectives are of the same type (both the higher the better, or both the lower the better), otherwise 𝛽 = −1.  4.3.3.6 Modified Combinative Weighting (MCW) Method  Based on the above presented weighting methods, since there are more than one set of weights (some subjective and some objective), for a final decision making it is needed to congregate these weights into one single set.  Jahan et al. [141] proposed a combinative weighting (CW) method to aggregate three sets of weights corresponding to subjective, objective, and correlation weights  (𝑤𝑗,1, 𝑤𝑗,2 𝑎𝑛𝑑 𝑤𝑗,3) via the following formula. 𝑤𝑗,𝐶𝑊 =[(𝑤𝑗,1). (𝑤𝑗,2). (𝑤𝑗,3)]1/3∑ [(𝑤𝑗,1). (𝑤𝑗,2). (𝑤𝑗,3)]1/3𝑛𝑗=1  , 𝑗 = {1,2, … , 𝑛} (4.13) The assumption behind Eq. (4.13) is that the authority/power of all three sets of weights is equal. A Modified Combinative Weighting (MCW) method is proposed here to enable the analyst to assign different powers, 𝛼𝑝,  to different weighting systems. The MCW suggests:  𝑤𝑗,𝑀𝐶𝑊 =[(𝑤𝑗,1)𝛼1 . (𝑤𝑗,2)𝛼2 . … . (𝑤𝑗,𝑚)𝛼𝑚]1/(𝛼1+𝛼2+⋯+𝛼𝑚)∑ [(𝑤𝑗,1)𝛼1 . (𝑤𝑗,2)𝛼2 . … . (𝑤𝑗,𝑚)𝛼𝑚]1/(𝛼1+𝛼2+⋯+𝛼𝑚)𝑛𝑗=1  , 𝑗 = {1,2, … , 𝑛} (4.14) 68  where p and n are the number of weighting methods and attributes, respectively. This method also can be used in group decision making if, for example, the analyst is interested to give more power to the weights from a more experienced designer in the group. 4.3.4 Multiple Criteria Material Selection Results and Discussions In this section, the above described multi-criteria material selection approach (Sections 3.2 and 3.3) are applied on the decision matrix date of the laminate impact optimization problem (Table 4.3), with a potential application of the material candidates in a composite guardrail. In doing this, the five fundamental methods of AMB, MDL, NL, Entropy, and CRITIC are considered. For different objective methods (AMB, MDL, and NL), the same DM (decision maker) with the same level of experience was employed. In addition, three scenarios are introduced as to how a project manager/analysis (the final decision maker) would combine the above set of weights under the MCW formulation, depending on the level of experience of the DM and his/her experience in systems engineering. Each set of resulting combinative weights are then used as input in the TOPSIS method to find the best laminate among the given four options (UD, UW, PW, TW).   It is known that roadside barriers can be categorized into three different groups according to their performances: (1) rigid, (2) semi-rigid, and (3) flexible barriers [158]. A flexible barrier is allowed to deform up to 4 m off of the roadway to stop the vehicle. The maximum acceptable deflection for a semi-rigid guardrail system is 1m, whereas for a rigid system, no deflection is allowed and the errant vehicle should be redirected into traffic. The American Association of State Highway and Transportation Officials (AASHTO)—which is the institute in charge of developing standards on specifications, test protocols and guidelines for highway design and construction in the United States [159]- in the report AASHTO M-180 (entitled “Corrugated Sheet Steel Beams for Highway Guardrails”) recommends semi-rigid steel guardrail systems for highways [160]. Accordingly, criteria weightings by the DM in our case study are selected based on a semi-rigid design of composite guardrail (Figure 4.11). AASHTO in collaboration with Federal Highway Administration (FHWA), published NCHRP Report 350 [161] 69  entitled “Recommended Procedures for the Safety Performance Evaluation of Highway Features”. In that report the post-impact vehicular trajectory, the maximum velocity and ride down acceleration that occupants experience were mentioned among the main factors. Clearly these requirements can have roots in the impact reaction force, energy absorption, mechanical properties, and damage characteristics of the material used in the guardrail.  Figure 4.11. Prototype of a semi-rigid composite guardrail 4.3.4.1 Adjustable Mean Bars (AMB) Weights Considering the above guardrail design guides, the DM for AMB weighting ranked the importance of attributes in Table 4.3 as: 𝑤𝑃𝐼𝑈𝐹𝑆 = 𝑤𝐿𝑈𝐹𝑆 = 𝑤𝑃𝐼𝐹𝑇 = 𝑤𝐿𝐹𝑇 > 𝑤𝐴𝐸 > 𝑤𝑅𝐹 > 𝑤𝑀𝐶𝐷 > 𝑤𝐸𝑉𝐷 > 𝑤𝐼𝐷 with 𝑘𝑃𝐼𝑈𝐹𝑆 = 𝑘𝐿𝑈𝐹𝑆 = 𝑘𝑃𝐼𝐹𝑇 = 𝑘𝐿𝐹𝑇 = 𝑘𝐴𝐸 = 𝑘𝑅𝐹 = 3 &  𝑘𝑀𝐶𝐷 = 2 &  𝑘𝐸𝑉𝐷 = 𝑘𝐼𝐷 = . The subsequent AMB weighting procedure is summarized in Table 4.6 and also shown graphically in Figure 4.12.  Table 4.6. Steps of the AMB weighting by the DM in the impact optimization case study The steps in Table 4.6 are graphically shown in Figure 4.12. Step # Attribute(s) to be weighted Weighted attributed in previous steps (m) Emphasis factor (kj) AMB weights (wj) Height of remained mean bars (unweighted attributes) initial --- --- --- --- 0.111 1 UFS, RLUFS, FT, RLFT 0 3 0.147 0.082 2 AE 4 3 0.118 0.074 3 RF 5 3 0.110 0.061 4 MCD 6 2 0.085 0.050 5 EVD 7 1 0.062 0.037 6 ID 8 1 0.037 --- 70                                            (a) initial step                                                                           (b) step 1                                                (c) step 2                                                                               (d) step 3                                                (e) step 4                                                                              (f) step 5 Figure 4.12. The AMB weighing procedure for the impact optimization case study with 9 attributes; solid and hollow bars demonstrate the weighted and yet-to-be weighted attributes, respectively.  4.3.4.2 Modified Digital Logic (MDL) Weights For the MDL method, the DM compared each two attributes at a time and assigned digital scores of 1, 2, or 3 to them, based on his/her perception of their relative importance. The results, using Eq. (4.6), are shown in Tables 4.7& 4.8. 71  Table 4.7. MDL weighting by the DM in the impact optimization case study Attributes Relative Digital Weights (𝑪𝒋𝒌) RF  1 3 3 3 1 1 1 1              AE  3        3 3 3 2 2 2 1       MCD    1       1       3 3 1 1 1 1 ID     1        1      1      EVD      1        1       1     UFS       3        2       3    RLUFS        3        2       3   FT         3        2       3  RLFT                3             3           3 Table 4.8. Continuation of table 4.7 Attributes Relative Digital Weights (𝑪𝒋𝒌) #Positive Decisions Weighting Factors RF                 14 0.097 AE                 19 0.132 MCD                 12 0.083 ID  1 1 1 1 1           8 0.056 EVD 3     1 1 1 1       10 0.069 UFS    3    3    2 1 1    18 0.125 RLUFS     3     3   2    1 1  18 0.125 FT      3     3    3   3   2 22 0.153 RLFT          3       3     3   3 2 23 0.160 Remark: To complete the above table, every two criteria are compared relative to each other and receive digital scores; for example RF and AE have been compared and received scores of 1 (less important) and 3 (more important), respectively. Then, RF and MCD are compared and so on. If two criteria are equally important, each receives a score of 2. 4.3.4.3 Numeric Logic (NL) Weights In contrast to the MDL approach where the DM assigned digital scores to the attributes, in the numeric logic (NL) method, he/she could assign any relative weights during pair-wise comparisons. The results of this method are shown in Tables 4.9 and 4.10.  Table 4.9. NL weighting by the DM in the impact optimization case study Attributes Relative Numeric Weights (𝑪𝒋𝒌) RF  0.3 0.8 0.9 0.9 0.2 0.2 0.1 0.1              AE  0.7        0.8 0.9 0.9 0.5 0.5 0.5 0.4       MCD   0.2       0.2       0.8 0.6 0.2 0.2 0.1 0.1 ID    0.1       0.1      0.2      EVD    0.1       0.1      0.4     UFS      0.8       0.5      0.8    RLUFS       0.8       0.5      0.8   FT        0.9       0.5      0.9  RLFT         0.9       0.6      0.9 72  Table 4.10. Continuation of Table 4.9 Attributes Relative Numeric Weights (𝑪𝒋𝒌) Positive Decisions Weighting Factors RF                 3.5 0.097 AE                 5.2 0.144 MCD                 2.4 0.067 ID  0.4 0.1 0.1 0.1 0.1           1.2 0.033 EVD 0.6     0.2 0.2 0.1 0.1       1.8 0.050 UFS   0.9    0.8    0.5 0.4 0.4    5.1 0.142 RLUFS    0.9    0.8   0.5   0.4 0.4  5.1 0.142 FT     0.9    0.9   0.6  0.6  0.5 5.8 0.161 RLFT      0.9    0.9   0.6  0.6 0.5 5.9 0.164  4.3.4.4 Entropy Weights In order to find the entropy weights, first it was needed to normalize the decision matrix data as they were measured in different units/scales. Linearly normalized values using Eq. (4.7) are given in Table 4.11. Next, the entropy, the degree of diversity and the final objective weights (Table 4.12) were calculated according to Eqs. (4.8)-(4.10). Table 4.11: Normalized Decision making matrix (𝐩𝐢𝐣); note that the criteria become dimensionless. Materials Impact Testing (Dynamic Properties)  Nondestructive Evaluation  Pre- and Post-Impact Flexural Testing (Quasi-Static Properties)  RF AE MCD  ID EVD  UFS RLUFS FT RLFT  PW 0.25 0.20 0.24  0.29 0.35  0.29 0.24 0.35 0.24  TW 0.27 0.20 0.25  0.23 0.30  0.24 0.09 0.29 0.15  UW 0.25 0.23 0.26  0.36 0.31  0.25 0.26 0.22 0.20  UD 0.23 0.37 0.25  0.12 0.04  0.23 0.41 0.14 0.41  Next, entropy, degree of diversity and weights are calculated with Eqs. (4.8)-(4.10). Table 4.12 shows the results. Table 4.12: Calculated entropy (E), degrees of diversity (d) and weights of importance (w) for different criteria according to the Entropy method  Measures RF AE MCD ID EVD UFS RLUFS FT RLFT 𝑬𝒋 0.999 0.972 0.999 0.953 0.886 0.997 0.922 0.965 0.950 𝒅𝒋 0.001 0.028 0.001 0.047 0.114 0.003 0.078 0.035 0.050 𝒘𝑬𝒏𝒕𝒓𝒐𝒑𝒚 0.004 0.079 0.003 0.131 0.319 0.009 0.216 0.098 0.140  73  4.3.4.5 Criteria Importance through Inter-criteria Correlation (CRITIC) Weights Table 4.13 shows the inter-criteria Pearson correlations between the nine design criteria (𝑅𝑗𝑘) calculated via Eq. (4.11). The CRITIC weights were then calculated using Eq. (4.12) and results presented in Table 4.14. Table 4.15 shows the results of TOPSIS and ranks of alternative materials based on different sets of weights presented in Table 4.14.  Table 4.13: Inter-criteria correlation factors (𝐑𝐣𝐤) according to the CRITIC method  Attributes RF AE MCD ID EVD UFS RLUFS FT RLFT  RF  1 -0.83 -0.12 0.34 0.73 0.34 -0.99 0.72 -0.89  AE   1 0.06 -0.76 -0.99 -0.71 0.86 -0.89 0.92  MCD    1 0.41 0.01 -0.50 0.02 -0.49 -0.24  ID     1 0.85 0.58 -0.43 0.52 -0.70  EVD      1 0.75 -0.77 0.87 -0.88  UFS    Sym.   1 -0.33 0.89 -0.38  RLUFS        1 -0.70 0.93  FT         1 -0.66  RLFT          1  Table 4.14: Summary of the four subjective (EW, AMB, MDL and NL) and the two objective (Entropy & CRITIC) weighting methods  Weights RF (N) AE (J) MCD (mm) ID (%) EVD (mm2) UFS (Mpa) RLUFS (Mpa) FT (kN/m2) RLFT (kN/m2) 𝒘𝑬𝑾 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 𝒘𝑨𝑴𝑩 0.110 0.118 0.085 0.037 0.062 0.147 0.147 0.147 0.147 𝒘𝑴𝑫𝑳 0.097 0.132 0.083 0.056 0.069 0.125 0.125 0.153 0.160 𝒘𝑵𝑳 0.097 0.144 0.067 0.033 0.050 0.142 0.142 0.161 0.164 𝒘𝑬𝒏𝒕𝒓𝒐𝒑𝒚 0.004 0.079 0.003 0.131 0.319 0.009 0.216 0.098 0.140 𝒘𝑪𝑹𝑰𝑻𝑰𝑪 0.119 0.115 0.091 0.102 0.113 0.108 0.118 0.107 0.126  Table 4.15: TOPSIS results based on the subjective and objective weights presented in Table 4.14   TOPSIS Scores (C*)  Ranks   EW MDL NL AMB Entropy CRITIC  EW MDL NL AMB Entropy CRITIC PW  0.44 0.55 0.54 0.56 0.319 0.44  3 2 2 2 4 3 TW  0.57 0.68 0.66 0.67 0.494 0.58  1 1 1 1 2 1 UW  0.38 0.49 0.49 0.50 0.325 0.41  4 3 3 3 3 4 UD  0.48 0.31 0.35 0.31 0.560 0.46  2 4 4 4 1 2  74  4.3.4.6 Modified Combinative Weighting (MCW) Method: Different Practical Scenarios   Individual set of weights (wAMB , wMDL , wNL , wEntropy  and wCRITIC) calculated in the previous sub-sections, resonate differences between different weighting methods, given that the DM has been the same for all of them. The last step of weighting procedure is to aggregate these set of weights and arrive at one resultant set of weights to implement in the TOPSIS model and to rank the alternative materials. Based on Eq. (4.14), the following general formula can be written for the present case study: 𝑤𝑗,𝑀𝐶𝑊 =[(𝑤𝑗,𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑖𝑣𝑒)𝛼 . (𝑤𝑗,𝐸𝑛𝑡𝑟𝑜𝑝𝑦)𝛽 . (𝑤𝑗,𝐶𝑅𝐼𝑇𝐼𝐶)𝛾]1/(𝛼+𝛽+𝛾)∑ [(𝑤𝑗,𝑠𝑢𝑏𝑗𝑒𝑐𝑡𝑖𝑣𝑒)𝛼 . (𝑤𝑗,𝐸𝑛𝑡𝑟𝑜𝑝𝑦)𝛽 . (𝑤𝑗,𝐶𝑅𝐼𝑇𝐼𝐶)𝛾]1/(𝛼+𝛽+𝛾)𝑛𝑗=1  , 𝑗 = {1,2, … , 𝑛} (4.15) Four different scenarios were proposed to use Eq. (4.15) by the project manager (PM) based on the background and the level of expertise of the original DM (designer), as well as the level of complexity that the PW would like to include in a sensitive decision making process. These scenarios are as follows. Scenario I: The DM has a relatively low level of experience and as a result he/she is not fully confident in assigning subjective weights. Hence, the PM opts to select the most conservative approach and assign equal weights to all criteria (i.e., the EW method in Eq. (4.1)). In this case, in Eq. (4.15) we have 𝛼 =1 and 𝛽 = 𝛾 = 0, and the MCW weights are equal to the EW weights:  𝑤𝑗,𝑀𝐶𝑊−𝐼 = 𝑤𝑗,𝐸𝑊  , 𝑗 = {1,2, … , 𝑛} (4.16) Scenario II: The DM has a high level of experience and, therefore, is capable of assigning subjective/application-based weights. Depending on his/her level of confidence, from low to high, he/she might use the MDL, NL or AMB techniques. Still, the PM does not take into account the statistical/objective weights (Entropy and CRITIC) and merely relies on the designer’s input. Parameters in Eq. (4.15) for this case again are 𝛼 = 1 and 𝛽 = 𝛾 = 0, and the MCW weights are equal to one of subjective weights (MDL, NL or AMB):  75  𝑤𝑗,𝑀𝐶𝑊−𝐼𝐼 ={    𝑤𝑗,𝑀𝐷𝐿 𝑜𝑟𝑤𝑗,𝑁𝐿  𝑜𝑟𝑤𝑗,𝐴𝑀𝐵  }     , 𝑗 = {1,2, … , 𝑛} (4.17) Scenario III: The PM is interested to additionally include sets of objective weights into the final weighting scheme, with the aim of arriving at more accurate results. Hence, 𝛼 =  𝛽 = 𝛾 = 1, and the final weights are calculated as: 𝑤𝑗,𝑀𝐶𝑊−𝐼𝐼𝐼 =[(𝑤𝑗,𝑁𝐿). (𝑤𝑗,𝐸𝑛𝑡𝑟𝑜𝑝𝑦). (𝑤𝑗,𝐶𝑅𝐼𝑇𝐼𝐶)]1/3∑ [(𝑤𝑗,𝑀𝐷𝐿). (𝑤𝑗,𝐸𝑛𝑡𝑟𝑜𝑝𝑦). (𝑤𝑗,𝐶𝑅𝐼𝑇𝐼𝐶)]1/3𝑛𝑗=1  , 𝑗 = {1,2, … , 𝑛} (4.18) Note: Instead of NL, the results of MDL or AMB could be used.  Scenario IV: The original DM possesses a very high level of expertise in impact design of composite structures. Accordingly, because of his/her high confidence in all assigned subjective weights, the PM opts to give more power to the designer’s experience by increasing 𝛼 from one to two; i.e. 𝛼 = 2 and 𝛽 =𝛾 = 1. Alternatively, he/she can make a group decision making and employ two different methods of subjective weights by two different DMs (so that potential methodological inconsistencies during weighting are taken into account). In this case study, the well-experienced DM has enforced the effect of the NL weights by power of two via the following equation:  𝑤𝑗,𝑀𝐶𝑊−𝐼𝑉 =[(𝑤𝑗,𝑁𝐿)2. (𝑤𝑗,𝐸𝑛𝑡𝑟𝑜𝑝𝑦). (𝑤𝑗,𝐶𝑅𝐼𝑇𝐼𝐶)]1/4∑ [(𝑤𝑗,𝑁𝐿)2. (𝑤𝑗,𝐸𝑛𝑡𝑟𝑜𝑝𝑦). (𝑤𝑗,𝐶𝑅𝐼𝑇𝐼𝐶)]1/4𝑛𝑗=1  , 𝑗 = {1,2, … , 𝑛} (4.19) The final combinative weighting results of these four scenarios are presented in Table 4.16.     76  Table 4.16. Results of the four combinative weights based on four proposed scenarios to mimic the DM’s level of experience Weights RF (N) AE (J) MCD (mm) ID (%) EVD (mm2) UFS (Mpa) RLUFS (Mpa) FT (kN/m2) RLFT (kN/m2) 𝒘𝑴𝑪𝑾−𝑰 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 0.111 𝒘𝑴𝑪𝑾−𝑰𝑰 0.097 0.144 0.067 0.033 0.050 0.142 0.142 0.161 0.164 𝒘𝑴𝑪𝑾−𝑰𝑰𝑰 0.043 0.131 0.031 0.091 0.146 0.062 0.183 0.143 0.170 𝒘𝑴𝑪𝑾−𝑰𝑽 0.054 0.138 0.039 0.073 0.115 0.078 0.177 0.151 0.174  Table 4.17: Final MCDM results for the given laminate options; under the four different weighting scenarios in Table 4.16.   TOPSIS Scores (C*)  Ranks   Scenario I Scenario II Scenario III Scenario IV  Scenario I Scenario II Scenario III Scenario IV PW  0.44 0.56 0.46 0.50  3 2 2 2 TW  0.57 0.67 0.63 0.65  1 1 1 1 UW  0.38 0.50 0.44 0.46  4 3 3 3 UD  0.48 0.31 0.41 0.37  2 4 4 4  4.3.4.7 TOPSIS Ranking of Laminates The obtained sets of weights in Table 4.16 under different design scenarios were next applied to the TOPSIS steps in Appendix A to rank the four alternative laminates for the composite impact optimization under question. According to the results in Table 4.17, the degrees of similarity of each material option to the positive-ideal solution (i.e., Eq. (8.7) in Appendix A), also called the TOPSIS scores, indicate that the first rank material is the TWILL woven laminate, under all the four weighting methods. Interestingly, for this case study the results of Scenarios 2 to 4 are identical. The only observed difference was between Scenario 1 (i.e. equal weights) and other scenarios. According to scenario 1 the UD laminate is ranked second while scenarios 2 to 4 all ranked PW as second. One main reason is that the UD laminate has a superior rank with respect to the individual RF, AE, ID, and EVD criteria as seen in Table 4.4 (specially looking at the actual measured values in Table 4.3, this laminate in terms of minimized external visible damage/EVD, has been by far the best option). In contrast, UD has shown the poorest results for UFS, RLUFS, FT and RLFT. In scenarios 2 to 4, the subjective weights like NL and AMB have been part of 77  the weighting process. Namely in these subjective techniques, DM has given higher weights of importance to the criteria in which UD were weak (i.e. UFS, RLUFS, FT and RLFT) and lower importance to the criteria in which UD were strong, specially ID and EVD (see Table 4.16). Under scenario 2, which was purely based on designer’s opinion, he/she has perceived that both the external and internal damages (EVD and ID) would be somewhat automatically reflected in post impact (residual) mechanical properties and, hence, should not receive high weights. This notion on the DM’s perceptions can also be clearly seen from the lower number of positive decision that he/she has given during the MDL and NL method (Tables 4.8 and 4.10). Recalling the correlation coefficients in Table 4.13 (which are used in the CRITIC method), it becomes evident that as the DM had perceived, indeed the EVD has a high correlation (77%) with the relative loss of ultimate flexural strength (RLUFS), and 88% with the relative loss of flexural toughness (RLFT). However, the DM’s experience may have been less accurate regarding the correlation of the internal damage (ID) with other properties. From a micro-scale material behavior point of view, the mode of induced failure mechanism (e.g., fiber pull-out, fiber breakage, matrix cracking, kinking and delamination) would play much more important role in the residual mechanical properties than the internal damage area fraction. Table 4.13 has also revealed that reaction force and absorbed energy have the highest correlation with the loss of mechanical properties. Overall, from Table 4.3 it can be noted that the TW laminate performs comparably well under all criteria while showing the lowest loss of mechanical properties due to impact (i.e., both low values of RLUFS and RLFT), hence it has been chosen as the preferred weave pattern option within all the four weighting scenarios.  The above discussions in this example show on one hand the intelligence and usefulness that an MCDM model can provide to the decision makers for complex systems such as impact of fiber reinforced composites, and on the other hand, the critical need of trying different MCDM methods before making a final decision. Of course, in more realistic design cases, next to optimum material selection for a given structure, geometrical aspects/shape of the final structure, its manufacturing process, assembly stages, etc should be taken into account. 78  4.4 Summary of Findings for Multicriteria Weave Pattern Selection for Composites under Impact A broad range of mechanical and damage properties may be used as part of design criteria to judge the impact performance of composite laminates. These include the induced reaction force, absorbed (dissipated) energy, maximum deflection, interior and exterior damage intensities, ultimate flexural strength and its relative loss due to impact, flexural toughness and its relative loss during impact, etc. Destructive (impact and flexural testing) and non-destructive (e.g., visual inspection and x-ray microtomography) tests can be performed to obtain appropriate test data for given material candidates. In the presented case study, four different composite configurations (PW, TW, UW and UD) were considered with a hypothetical application in composite guardrails. Test results showed that each type of laminate is strong/agreeable in terms of some design criteria and somewhat weak in terms of others. As a result, a multiple criteria decision making method (namely, the TOPSIS method) was employed to select the laminate with the overall optimum performance. Weighting of criteria were performed based on both subjective and objective techniques, as well as their combinations. Two novel subjective weighting methods were proposed: adjustable mean bars (AMB) and numeric logic (NL) methods. The AMB enables the DM to adjust the weighting factors in a flexible step-by-step manner. It is also based on a simple graphical method, which may sometimes be easier to implement. Advantage of the NL method over the MDL method was deemed to be that the DM is not limited to three pre-defined digital levels of scoring during pair-wise comparisons. In the NL weighting, the DM can assign arbitrary numeric weights at each pair-wise comparison stage, hence increasing the accuracy of outcomed. Criteria Importance through Intercriteria Correlations (CRITIC) and Entropy methods were selected as objective approaches to quantify the intrinsic information of each attribute dataset. Entropy takes into account the degree of diversity of data under each attribute, while CRITIC accounts for correlations between attributes. In addition, a new combinative weighing method (MCW) was discusses. In comparison with the original combinative weighting (CW), the modified version (a) 79  allows for assigning more/less importance to specific objective and/or subjective weights, and (b) may be used in group decision making environments. Interestingly, the selected set of subjective, objective and combinative weighting methods ranked some criteria differently in the performed case study, and hence highlighted the importance of combinative weighting. Thus, a systematic methodology based on four practical scenarios was implemented to assist the material selection of composites based on the level of experience of the decision maker. Other findings specific to the performed example are:    In terms of dynamical responses such as absorbed energy, maximum reaction force and central deflection, the UD laminate showed a superior performance. In terms of damage areal intensity, the UD laminate also displayed a good performance. In terms of post-impact (residual) strength, however, the UD laminate performed very poor compared to the woven laminates (see Figure 4.7), confirming that for low velocity impact applications, where the post-impact/multiple impact properties along with associated reduction in mechanical properties may be of concern, woven composites should be used [52].  By changing the waviness of fabric from PW to UD, the capacity of structure to absorb impact energy was increased. On the other hand, the post-impact mechanical flexural strength and toughness declined.   From the x-ray tomographic images, it was understood that the type of woven fabric layers can play an important role as a barrier to the penetration of damage into the areas far from the impact center. In particular, PW and UW laminates showed very severe local damage while the TW laminate had a well-distributed small damage areas throughout the structure (with less fiber breakage and pull-out). The UD laminate, on the other hand, revealed completely different damage mechanism by producing a large through thickness crack farther from the impact center.   The observed transverse crack on the top surface of samples was associated to matrix shearing and fiber kinking under laminate compression. 80   The brittle like crack in the UD laminate, along with other observed delamination and kink zones, was suspected to be the source of the high energy absorption in the present case study. This was despite the fact that there was no notable visible damage in the back surface of the UD laminate (i.e., hidden damage state), whereas it was visible in all woven laminates. This trend is opposite to the results of drop tower tests on glass PW and UD laminates reported in [52], where a larger visible damage was seen in the UD laminate and also the energy absorption of the PW was higher than the UD laminate. The reason is presumed to be that in [52] the PW samples were in excess of ~10% fiber weight ratio over the UD samples, whereas in the current work this is reversed (see Figure 4.1). Similarly, from force-time diagrams (Figure 4.4-a), it is noted that the Hertzian force (which is a material property by means of the impact force limit at which the first kink/drop occurs in the response due to damage initiation) was higher for the woven laminates (<5kN) as compared to the UD laminate (>5kN).  These differences between the current results and those in [52] reemphasize that the impact behavior of FRP composites is a very complex phenomenon and in each design application (given specific fiber and resin materials, fabrication process, laminate thickness, fiber lay-up, level of applied impact energy, shape of the impactor, shape and condition of the test fixture/clamping [162], number of plies, etc), a separate detailed analysis is needed.   Considering the energy diagram in Figure 4.5-d, the response of all four composites have been in the rebounding zone and below the preformation limits (i.e., no full penetration of impactor occurred).  Also from the force-time response in Figure 4.5-a, it can realized that the UD laminate, compared to woven laminates with a cellular architecture, has resulted in less vibration (response oscillations) around the pick force where the first main composite damage occurs. This observation was also reported earlier by Evci and Gülgeç [52].   Single objective optimization was found to be an ineffective approach for material selection of composites which often involves multiple, conflicting design criteria. Accordingly, the application of MCDM models in the field is strongly recommended.  81  It was found that the MCDM (TOPSIS) ranking of materials is sensitive to the DM’s weighting. Thus, it is recommended to employ different individual and combinative weighting methods before making a final decision. The TOPSIS results based on the advanced weighting case (scenario 4 in section 4.4.6) in the present case study was: (𝑃𝑙𝑎𝑖𝑛 𝑊𝑜𝑣𝑒𝑛𝑇𝑤𝑖𝑙𝑙 𝑊𝑜𝑣𝑒𝑛𝑈𝑛𝑏𝑎𝑙𝑎𝑛𝑐𝑒𝑑 𝑊𝑜𝑣𝑒𝑛𝑈𝑛𝑖𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛𝑎𝑙) → (𝑅𝑎𝑛𝑘 2𝑅𝑎𝑛𝑘 1𝑅𝑎𝑛𝑘 3𝑅𝑎𝑛𝑘 4) ;  𝑆𝑐𝑜𝑟𝑒𝑠 = (0.490.650.460.38)  Using unbalanced (twill) woven fabrics, the laminate can still be symmetric using [0/90] layout. However, each single ply due to less decrimping is closer to the UD pattern and that is why the unbalanced twill woven (UW) laminate has absorbed more energy compared to the balanced twill in the current example. It is interesting that as compared to (balanced) PW, the UW laminate’s overall performance is only marginally weaker (compare the TOPSIS scores above, 0.46 vs. 0.49).   The MCDM methodology in this work was only applied to flat geometries (composite plates). Further application of the approach and comparison of composites using impact tests on more complex prototypes with actual 3D geometries may be worthwhile.    82  5 Chapter 5: On Complexities of Impact Simulation of Fiber Reinforced Polymer Composites: A Simplified Modeling Framework for Practitioners Impact modeling of fiber reinforced polymer composites is a complex and challenging task, in particular for practitioners with less experience in advanced coding and user-defined subroutines. Different numerical algorithms have been developed over the past decades for impact modeling of composites, yet a considerable gap often exists between predicted and experimental observations. In this Chapter, after a thorough review of reported sources of complexities in impact modeling of fiber reinforced polymer composites, two simplified approaches are presented for fast simulation of out-of-plane impact response of these materials considering four main effects: (a) strain rate dependency of the mechanical properties, (b) difference between tensile and flexural bending responses, (c) delamination, and (d) the geometry of fixture (clamping conditions). In the first approach, it is shown that by applying correction factors to the quasi-static material properties, which are often readily available from material datasheets, the role of these four sources in modeling impact response of a given composite may be accounted for. As a result a rough estimation of the dynamic force response of the composite can be attained without advanced coding. To show the application of the approach, a twill woven polypropylene/glass reinforced thermoplastic composite laminate has been tested under 200J impact energy and was modeled in Abaqus/Explicit via the built-in Hashin damage criteria. X-ray microtomography was used to investigate the presence of delamination inside the impacted sample. Finally, as a second and much simpler modeling approach, it is shown that by applying only a single correction factor over all material properties at once can still yield a reasonable prediction. Both advantages and limitations of the simplified modeling framework are addressed in the performed case study.  5.1 Overview Finite element analysis (FEA) has been employed in a large portion of past investigations on modeling and predicting the response of fiber reinforced composite materials. To give a few examples, a three-83  dimensional computational micro-mechanical model was developed for woven fabric composites by Ivanov et al. [163]. The impact response of unidirectional composite laminates was modeled by Aminjikaeai et al. [41] using a strain-rate dependent micro-mechanical model with a progressive damage behavior. Petrossian and Wisnom [164] developed interface elements to create a resin-rich area between plies to predict the onset and growth of delamination in composite laminates. Atas et al. [165] used cohesive zone elements with a bilinear traction-separation law to predict the delamination initiation and growth in pin-loaded composite laminates. A finite element model was developed by Komeili et al. [166] to consider the effect of meso-level uncertainties on the mechanical response of axially loaded woven composites. Next to modeling efforts, several impact and post-impact tests have also been conducted on composites (e.g., [6–8]) to understand damage mechanisms  experimentally and validate the associated finite element codes. Despite these efforts, a fully representative numerical model has not been developed to date to predict composites response under all different impact conditions, or it would be computationally very expensive. Fiber reinforced plastic (FRP) composites are known to be difficult materials to model numerically because of their nonlinear and occasionally non-repeatable mechanical responses under high velocity events. Sources of difference between their numerical modeling and experimental results are attributed to, on one hand, various uncertain parameters in the material (such as fiber misalignment/waviness, voids, non-uniform volume fraction distribution), and on the other hand, on modeling errors (such as assumptions made in fiber-matrix bonding behavior, rate- and deformation mode- dependency of the material parameters, etc). Factors contributing to the latter category (modeling complexities) will be reviewed in detail in Section 5.4. Multi-scale nature of composites and additional uncertainty during their manufacturing (such as curing time, evenness of applied pressure and temperature throughout the part, etc) can further add to the complexity in computational simulation of these materials. The present work is aimed at demonstrating simplified modeling frameworks to assist practitioners in using built-in options of FEA packages (e.g., the built-in Hashin model in Abaqus) for fast simulation of 84  high-speed impact response of composites (i.e., without a need for user-defined coding). Two approaches are suggested: (a) using different correction factors on different quasi-static properties of the material considering the effects from high strain rates, the bending mode, delamination, and clamping conditions; and (b) using only a single correction factor for all the material properties simultaneously. As a sample case study, the impact behavior of a glass fiber/polypropylene thermoplastic composite has been investigated against both approaches. 5.2 Case Study Experiments Composite coupons comprised of six layers of commingled glass fiber and polypropylene balanced twill weave (commercially known as Twintex®) were fabricated and tested under impact according to ASTM D7136 [27]. The size of samples was 6×100×150 mm. One centimeter width of all four sides of samples was fully clamped by the test fixture shown in Figure 5.1(a). Impact tests were done at 200J using the drop-weight impact test tower equipped with a one-inch diameter stainless steel hemispherical impactor. Piezoelectric sensors mounted on the impactor were used to record the reaction force exerted to the material during the collision period. The quasi-static material properties of the samples were taken from the Twintex® material datasheet shown in Table 5.1.   85  Table 5.1. Quasi-static mechanical properties of twill weave Twintex ® composites (indices 1 and 2 refer to in-plane warp and weft directions, and 3 denotes the out-of-plane direction)  Properties (𝒚𝒊)  Values Tensile Strength σ11  288 MPa  σ22  266 MPa Compression Strength σ11  155 MPa  σ22  150 MPa Shear Strength τ12  19 MPa  τ21  18 MPa Modulus of Elasticity E11  14 GPa  E22  13 GPa Shear Modulus G12  1.7 GPa  G13  1.8 GPa  G23  1.7 GPa Poisson’s Ratio υ12  0.1 --- Fracture toughness G  220 kJ/m2 Source: Adapted from http://www.ocvreinforcements.com/Pages/Mechanical_Properties.asp [168] 5.3 Conventional Shell Finite Element Model and Limitations The above mentioned glass fiber/PP laminates subjected to 200J drop weight impact was simulated in Abaqus/Explicit with deformable composite shell elements as shown in Figure 5.1(b). The model consists of a 3D analytical rigid projectile, a deformable laminate and an analytical rigid fixture (support). Similar to the actual test condition, the projectile was fully constrained, except in the vertical direction along which the velocity was defined at an initial value of 5.69 m/s (corresponding to a 200J energy given the initial height of the impactor and its mass). The assigned mesh type for the multilayer composite was S4R which is a 4-node doubly curved shell element with reduced integration. The hourglass control, finite membrane strains, second order accuracy and element deletion options were activated. The built-in Hashin progressive damage criterion was chosen to automatically decrease the mechanical properties of elements as a function of damage intensity during the impact event. 86    (a) (b) Figure 5.1. (a) Impact test set up, and (b) FE model of the drop weight test using standard composite shell for the laminate. Figure 5.2 compares the obtained numerical and experimental results. As addressed in Section 5.1, different factors can play a role in causing a deviation between these two responses. In particular, the effect of (a) strain rate, (b) bending mode during impact, (c) delamination, and (d) clamping system are discussed in details in the following sections.  Figure 5.2. Comparison between experimental results and a conventional FEA using quasi-static properties and Hashin progressive damage criterion  5.3.1 Effect of Strain Rate  The impact event modeled in the previous section relies on a dynamic deformation mode encompassing high strain rates. It is known that there is a significant difference between quasi-static and dynamic material properties of composites. Barre et al. [169] conducted a comprehensive review on relationships 87  between applied strain rate and effective mechanical properties of a wide range of fiber-reinforced thermoset matrix composites. Their review showed that the change in the composite effective properties is dependent on the type of fibers (glass/graphite/Kevlar), the resin type (epoxy/polyester), and the induced strain rate (ranging from a quasi-static slow test to 500 s-1).  With the exception of few cases, the general trend of increase in mechanical strength by increasing the strain rate was observed [169]. Barre et al. [169] also performed some experimental studies to investigate the effect of strain rate (10-1 to 10 s-1) on glass fiber-reinforced phenolic and polyester matrix composites. They found that the effective mechanical properties of the composites are additionally dependent on the architecture of fiber reinforcements as well as the test setup. Foroutan et al. [170] recently studied the effect of strain rate on mechanical properties of carbon fiber-reinforced epoxy and Bismalemide (BMI) matrix composites with three different weave patterns. In general, they found that maximum tensile and shear strengths increase with the strain rate, regardless of the type of fiber architecture. More specially, they reported up to 40% and 74% increase in the tensile and shear strength properties, respectively, under dynamic events. Although the majority of the current literature on impact response of FRPs has been devoted to thermosetting composites, some studies have been focused on thermoplastic composites. Todo et al. [171] investigated the high strain rate response of different types of fiber reinforced polyamides and found that, in general, polyamide matrix composites show an increased tensile strength and failure strain when subjected to high strain rates. Vashchenko et al. [172] reported a linear increase of tensile strength with respect to the log of strain rate in the range of 10-3–105 s-1 for polyamide matrix composites. Kawata et al. [173] researched on short graphite fiber reinforced nylons and also reported an increase in the tensile strength with increasing the strain rate. Bai et al. [174] tested the high strain rate properties of high density polyethylene (HDPE) composites. Their results showed a higher Young’s modulus as well as the tensile strength under high strain rate tensile loads.  Papadakis et al. [175], [176] focused on continuous glass fiber/ polypropylene composites (PlytronTM) and reported an increase in the elastic modulus and the shear strength values and a decrease in the failure strain and the shear modulus of the material as the 88  strain rate was increased. Surprisingly, as opposed to the previous findings on other composites, the tensile strength for this material was reported to be unaffected by strain rate. McKown and Cantwell [177] studied the behavior of self-reinforced polypropylene (PP) composites under high strain rate tensions at the range of 10-4–10 s-1. They noted that initial elastic stiffness, yield strength and tensile strength were increased with increasing the strain rate. They also reported a deterioration of the failure strain by increasing the strain rate.  A few earlier studies have been devoted to impact response of commingled woven glass/polypropylene thermoplastic composites. Bonnet [178] investigated the response of unbalanced woven (4:1) Twintex laminates subjected to a wide range of strain rates (10-1 to 100  s-1). Surprisingly, it was observed that with increasing the strain rate, the average tensile and shear moduli decreased for this material. However, the tensile and shear strengths increased with increasing the strain rate. Finally, Brown et al. [37] studied the effect of strain rate ranging from quasi-static to 100 s-1 on the mechanical properties of balanced twill woven PP/glass composites (Twintex ®), which is also the type of material used in the present study. According to [37], the tensile and compression moduli and strengths increase with increasing the strain rate, while the shear modulus and strength show a decrease at higher strain rates. Brown et al. [37] used the logarithmic function developed by Yen and Caiazzo [179], [180] for their curve fitting as follows: 𝐸𝑅𝑇 = 𝐸0(1 + 𝐴 𝑙𝑛𝜀̇𝐶)   (5.1) 𝑆𝑅𝑇 = 𝑆0(1 + 𝐵 𝑙𝑛𝜀̇𝐶) (5.2) Where 𝐸0 and 𝐸𝑅𝑇  are the quasi-static and adjusted (high strain rate) moduli, 𝑆0 and 𝑆𝑅𝑇 are the quasi-static and adjusted strengths, respectively. 𝐴, 𝐵 𝑎𝑛𝑑 𝐶  are constants obtained from experiments and 𝜀̇ represents strain rate.. The linear regression model between mechanical properties and the log of strain rate reported by Brown et al. [37] is summarized in Figure 5.3, and used in the present work during subsequent finite element models in Section 5.4.  89                                             (a)                                                                                                 (b) Figure 5.3. Relationship between strain rate and (a) stiffness and (b) strength for the balanced twill weave Twintex  laminate (adapted from [37]) 5.3.2 Effect of Bending Mechanical properties provided in material datasheets are often obtained via conventional uniaxial tension and compression tests at quasi-static rates. This is despite the fact that deformation during out-of-plane impacts is more similar to flexural bending modes than the axial tension or compression. Hallet [181] performed a statistical study on this topic and found that fiber reinforced composites show a notably higher tensile strength in the bending mode compared to standard axial tension. Further to [176], Santiuste et al. [182] were able to find a good match between their experimental and numerical results of composite laminated beams by increasing  the quais-static tensile strength by 40% in order to account for the effect of bending mode.  5.3.3 Effect of Delamination The only failure criterion built in Abaqus explicit 6.12 for progressive damage modeling of fiber reinforced composites was the Hashin criterion. This criterion considers fiber and matrix failure in tensile and compression with the following formulations [183], [184]: Fiber tension (σ̂11 ≥ 0):  Fft = (σ̂11XT)2+ α(τ̂12SL)2 (5.3) 90  Matrix tension (σ̂22 ≥ 0):   Fmt = (σ̂22YT)2+ (τ̂12SL)2 (5.4) Fiber compression (?̂?11 < 0):  𝐹𝑓𝑐 = (?̂?11𝑋𝐶)2 (5.5) Matrix compression (σ̂22 < 0):  Fmc = (σ̂222ST)2+ [(YC2ST)2− 1]σ̂22YC+ (τ̂12SL)2  (5.6) Where XT, XC, YT, YC, SL and ST are the longitudinal tensile strength, longitudinal compressive strength, transverse tensile strength, transverse compressive strength, longitudinal shear strength and transverse shear strength, respectively. σ̂11 , σ̂22 and τ̂12  refer to the in-plane normal and shear stresses (the 1-direction is aligned with fibers direction). The coefficient α defines the contribution of the shear stress to the fiber tensile failure initiation.  Equations (5.4)–(5.6) indicate that the conventional Hashin criterion does not take the delamination into account (there are modified versions of this criterion developed in the literature [182]; however they require user-defined coding to implement in the FE package at this point). This is despite the fact that damage detection tests have confirmed the presence of delamination in several composite structures under impact. Figure 5.3 shows an example of the image obtained via X-Ray Microtomography Technique (XMT) from the interior part of the impacted specimen in the present work. This image shows a cross-section 10 mm far from the impact center. The XMT reveals the presence of several delamination zones. Camanho et al. [39], [185]–[187] have also shown the presence of delamination in composites subjected to impact and explored the effects of stacking sequence and ply clustering on delamination initiation and growth.  Knowing that the conventional Hashin criterion cannot address the effect of delamination during numerical simulation, the difference between experimental and numerical results in Figure 5.2 can be partially explained. 91   Figure 5.4. X-ray micro-tomography image of the impacted  Twintex laminate (sample delamination zones are shown with arrows)  The next question would be, how much delamination can affect the global (effective) strengths of the composite. To this end, Soldatos and Shu [188] modeled perfectly and weakly bonded composite laminates. Their numerical results showed that the maximum capacity of weakly bonded laminates to carry tensile loads was declined. Colombo and Vergani [189] investigated the effect of delamination on the fatigue life of glass/epoxy composites and reported a 40% loss. Reis et al. [190] inserted Teflon layers at the middle of the plate thickness to make artificial delamination with different sizes (2 to 20 square mm) in carbon/epoxy laminates and revealed that the delamination can reduce the tensile strength up to 17%, but surprisingly it did not affect the Young’s modulus. It was also shown in [181] that both strength and Young’s modulus were independent of the size of delamination. The effect of delamination on the compressive behavior of glass fiber reinforced composites was investigated by Short et al. [191]. Artificial delaminated zones were made with PTFE (Polytetrafluoroethylene) films with different sizes (10 to 25 mm squares). Their work showed that the compression failure load decreases with increasing the size of delamination and also shifting the delamination zone towards the center of the laminate. A maximum strength reduction of 31% was reported for 25 mm2 delamination area placed at the center of the plate.  5.3.4 Effect of fixture geometry and clamping condition The current literature also shows that the impact response of composites can be highly dependent on the clamping condition. Daiyan et al. [192] investigated the effect of clamping on impact response of samples made of 20% mineral (talc) and 80% elastomer modified polypropylene compound (ISO code Delaminations 92  PP+EPDM-TD20), a material that is being used in automotive exterior parts. The samples were put on a stand with a circular hole (40 mm diameter) at the center. Samples were tested with and without a circular (40 mm inner diameter) clamp. The reaction force that unclamped samples exerted to the striker was up to 10% higher than that of the clamped samples. This behavior could be attributed to the higher internal damage in the clamped samples. Very recently, Nilakantan and Nutt [162] also reported a similar trend of composite behavior. They performed both experimental and numerical analyses on the effect of clamping geometry on the impact response of soft body armors made of plain weave Kevlar fabrics. The advantages and disadvantages of six tested designs (Figure 5.5) were discussed in [162]. corner point held  two-sides held  four-sides held  circular clamp  diamond clamp  corner plate held  Figure 5.5. Six different clamping configurations compared during impact testing [162] The comparison between the projectile deceleration after hitting the samples clamped in different configurations in Figure 5.5 showed that the maximum and minimum deceleration (which is proportional to the reaction force) were for the four-sides held and diamond clamps, respectively [162]. The velocity of projectile corresponding to the circular clamp stood between the velocities of the other two clamping configurations. This would suggest that the more severe/bigger area the sample is constrained by the clamping system, the sooner the failure occurrence would be likely, and the lower the peak force would be exerted to the projectile. If one compares the clamping condition of our experimental set-up (Figure 5.1(a)) to that of numerical simulation (Figure 5.1(b)), it is notable that the FE model assumes a fully clamped condition around the four sides of the sample, whereas in reality the tested samples would have had some degrees of freedom inside the fixture. This is mostly noticed when a sample slides out of the fixture under severe conditions (see Figure 5.6 for an example of such case when a soft-core sandwich panel was tested by the same test set-up and striker). In general, we could conclude that the numerical model, in contrast to reality, is modeled based on a fully clamped condition (with no degree of freedom in 93  tied nodes) and as a result the predicted peak reaction force should be expected to be lower than experimental results. This brings the idea of applying a fourth correction factor (𝜂𝑓𝑖𝑥 > 1) to mechanical properties to account for this clamping effect-- a notion that has been rarely taken into account during the impact modeling of composites.   Figure 5.6. A soft-core sandwich panel slipped out of the fixture under 200J impact 5.4 Proposed Simplified Modeling With the background presented above in Sections 5.3.1 to 5.3.3 regarding the effects of four main factors on accuracy of simulation of composites, the task is now to incorporate these effects into an easy-to-implement macro-level FE model to predict impact response of the tested Twintex composite, specially assuming this is done by a user with no advanced skills on writing material subroutines. 5.4.1 Approach #1 In order to make a numerical model able to capture experimental observations, the model input parameters should be carefully selected and, if needed, modified to reflect specific testing conditions and material behavior. This can be achieved by applying correction factors to the nominal (qausi-static) mechanical properties to bring in, e.g., the effects of high strain rate, flexural bending, delamination, and clamping system. The following equation can be proposed to apply these effects on input materials properties, i.e. elastic moduli, ultimate strengths and fracture toughness values:   94  (𝑦𝑚𝑜𝑑)𝑖 = (𝑦 × 𝜂?̇? × 𝜂𝑏𝑒𝑛 × 𝜂𝑑𝑒𝑙 × 𝜂𝑓𝑖𝑥)𝑖 =(𝑦 × 𝜂𝑡𝑜𝑡𝑎𝑙)𝑖 (5.7) Where 𝑦  and  𝑦𝑚𝑜𝑑  are the original and modified material properties; 𝜂?̇?  , 𝜂𝑏𝑒𝑛 , 𝜂𝑑𝑒𝑙 and 𝜂𝑓𝑖𝑥  are the correction factors for the strain rate, bending, delamination, and fixture clamping condition effects, respectively. These correction factors can be extracted from the literature if available. For the material under study, Twintex , the correction factors are extracted in Table 5.2 based on the available literature as follows.  According to the study by Santiuste [182] and their observation of 40% increase in composite strengths from tensile to bending mode, the correction factor for bending effect during impact simulation was set to be 1.4. Short et al. [191] reported a 17% and 31% deduction in tensile and compressive strengths, respectively, for completely delaminated laminates. Since such a severe large delaminaton was not observed in the x-ray micro-tomography of our samples (Figure 5.4), the corresponding correction factors were set at half of the effects reported in [191] as follows: 0.92 (i.e., 8% reduction) for the tensile strength, 0.85 (i.e., 15% reduction) for the compressive strength, and 0.70 for shear (i.e., 30% reduction, considering the shear as a more sensitive mode to delamination). Numerical simulation showed a wide range of strain rate on different elements during impact (0 to 1600 s-1). The average strain rate during whole impact event for all elements was found to be 63 s-1. Accordingly, this average strain rate was used to find the correction factor for strain rate effect from regression models in Figure 5.3 based on the work of Brown et al [37]. For instance, the regression model in Figure 5.3 suggests compression strength of 313.33 MPa at the strain rate of 63 s-1. Hence, the compression strength correction factor was found by dividing the high strain rate property (313.33 MPa) by the quasi-static one (178.14 MPa), yielding to 1.76. Also, a correction factor of 1.25 was set for the clamping condition effect (𝜂𝑓𝑖𝑥 = 1.25) due to potentially slight freedom of samples in the fixture. Here, according to Table 5.2, an example of final modified property is given for the tensile strength: XT𝑐𝑜𝑟𝑟𝑒𝑐𝑡𝑒𝑑 = XT𝑞𝑢𝑎𝑠𝑖−𝑠𝑡𝑎𝑡𝑖𝑐 × 𝜂?̇? × 𝜂𝑏𝑒𝑛 × 𝜂𝑑𝑒𝑙 ×95  𝜂𝑓𝑖𝑥 = 288 × 2.124 = 611.7 𝑀𝑃𝑎. Results of the modified material properties are summarized in Table 5.3. Table 5.2. Correction factors applied to the mechanical properties of the Twintex  sample Mechanical Properties  Correction Factors   Strain rate Bending Delamination Fixture Total 𝒚𝒊  𝜼?̇? 𝜼𝒃𝒆𝒏 𝜼𝒅𝒆𝒍 𝜼𝒇𝒊𝒙 𝜼𝒕𝒐𝒕𝒂𝒍 Tensile strength and fracture toughness  1.32 1.40 0.92 1.25 2.124 Compression strength and fracture toughness  1.76 1.40 0.85 1.25 2.616 Shear strength and fracture toughness  0.71 1.40 0.70 1.25 0.866 Tensile modulus  1.04 1.00 1.00 1.25 1.306 Compression modulus  2.15 1.00 1.00 1.25 2.688 Shear modulus  0.19 1.00 1.00 1.25 0.233  Table 5.3. The ensuing modified material properties from Tables 5.1 and 5.2. Modified Properties (𝒚𝒎𝒐𝒅,𝒊 = 𝒚𝒊 × 𝜼𝒕𝒐𝒕𝒂𝒍,𝒊)   Values Tensile Strength σ11  611.7 MPa  σ22  564.9 MPa Compression Strength σ11  405.5 MPa  σ22  392.5 MPa Shear Strength τ12  16.5 MPa  τ21  15.6 MPa Modulus of Elasticity E11  18.3 GPa  E22  17.0 GPa Shear Modulus G12  0.4 GPa  G13  0.4 GPa  G23  0.4 GPa Poisson’s Ratio υ12  0.10 --- Tensile Fracture Toughness G  467.2 kJ/m2 Compressive Fracture Toughness G  575.6 kJ/m2  96   Figure 5.7. Comparison between experimental and two different numerical models; FEA with (i) quasi-static and (ii) modified material properties under Approach #1 Figure 5.7 compares the results of the modified model with the original one in Section 5.3 as well as the experimental data. This figure reveals a considerable improvement in the prediction of force history after applying the correction factors in Table 5.2. Figure 5.8 shows the Hashin shear damage index, as an example of damage progression, during impact for an element close to the impact center. The observed trend shows that failure occurs at the beginning of the collision (in about 1 ms after the start of the impact). It also supports the notion of applying the correction factors and changing the material properties from the early stages of simulation for the tested material and configuration, especially under high energy impact.   Figure 5.8. Progression of the Hashin shear damage index during the impact event Initiation of shear damage 97  5.4.2 Approach #2 In general, the weak point about Approach #1 is that the user must estimate the average strain rate that the structure would experience under a given impact energy. Also the material properties at that specific strain rate should be accessible (via testing or published work on rate dependency of that particular material). Similarly, earlier estimations on bending mode, delamination, and clamping condition effects should be employed. However, if a material is new and only standard quasi-static data is available, a much simpler approach may be explored as follows.  In approach #2, one single correction factor is used for the entire set of quasi-static material properties (as opposed to Table 5.2 where different corrections factors were used for each property). This single correction factor for the twill weave Twintex laminate was found to be equal to 1.7 via trial and error. Figure 5.9 shows the FEA result of this approach versus the earlier approaches and the experimental data. Surprisingly, the result of approach #2 despite its simplicity by using only one single correction factor is close to approach #1, and both within a reasonable predictability of maximum impact force. It is believed that through similar future studies such correction factors on dynamic response of composites can be systematically explored and tabulated for different materials and test conditions, and eventually be used for simplified and fast simulations by industry practitioners.  Figure 5.9. Comparison between experimental and three different numerical methods; (i) FEA with quasi-static properties, (ii) FEA with modified material properties under approach #1, and (iii) FEA with modified material properties under approach #2 98  5.5 Summary of Findings for Simplified Approaches for Impact Simulation of Composite Laminates In spite of significant advances in numerical simulation of complex structures under different types of loadings, a fully representative model is not yet developed to predict the impact response of fiber reinforced plastic composites (FRPs) under different test conditions. The reason for this challenge may be found in different intrinsic and extrinsic characteristics of composites, from anisotropic properties to uncontrolled parameters during sample fabrication and/or testing. Different FRP composites show different and sometime unexpected mechanical responses, especially under dynamic loads, that make their numerical simulations a bigger challenge. For instance, contrary to many other conventional materials, the experimental study [37] shows that the shear modulus and strength of glass reinforced polypropylene composite decrease at higher strain rates. Or, according to [190], delamination insert films did not affect the Young’s modulus of carbon/epoxy laminates. Or, the study [182] shows that the tensile material properties under bending deformation can be notably different from those under normal tensile loading (up to 40% difference). Similarly, results in recent works [162], [192] imply that the different clamping conditions during drop weight testing can be a large source of variation in the material response. Considering all these four factors, detailed numerical modeling of an FRP can be an obstacle for practitioners, either due to the lack of experimental data, or the cost of advanced subroutines in conjunction to commercial FE packages. In this chapter, as a preliminary tool, two simplified modeling approaches were discussed to make it possible to predict the impact force of a FRP composite using built-in models in commercial FE packages (here Abaqus).  To show the application of the two approaches, a glass fiber reinforced polypropylene composite was subjected to impact at 200J and was simulated in Abaqus/Explicit. The built-in Hashin damage criterion was used for progressive damage of the material. The initial model using the material properties from quasi-static test data was far from the impact experimental results. To explain this deviation, X-ray micro-tomography was conducted on the samples and revealed delamination zones due to impact, which is not 99  considered in the conventional Hashin failure criterion. The other source of this deviation was due the difference between effective material properties in the tensile testing and the flexural testing (note that under out-of-plane impact, high speed bending is a dominant mode). Also, the quasi-static material properties needed to be corrected for the strain rate dependency effect during impact. Finally, further to earlier works [162], [192], it was deemed that the difference between the actual clamping system during the drop weight test and its modeled boundary condition in simulations can be another source of deviation, which was addressed by applying a fourth correction factor to the mechanical properties. Results of the modified model showed that by correcting the quasi-static material parameters using four factors, the predictability of the numerical model is notably enhanced (approach #1). In a second approach (approach #2), only one overall correction factor was applied to the entire set of material properties. The first approach was relatively more accurate compared to test data, however needed an estimation of average strain rate and the availability of literature on strain rate dependency data, clamping effect, delamination sensitivity, and bending mode effect for a given material. In contrast, the second method was found very easy to implement, and yet reasonably accurate, via iterating only a single constant correction factor.    100  6 Chapter 6: Overall Summary, Conclusions and Future Research Directions  6.1 Overall Summary  Fiber reinforced composite materials are increasingly employed in a wide variety of applications, from sports equipment to satellites. This popularity is mainly because of some desired properties of composites such as high specific strength (strength to weight ratio) and the ability to customize their material properties. Among different thermoplastic composites, commingled polypropylene/E-glass Twintex® fabrics (and similar commercial products such as Comfil®) have been proven to be suitable candidates in high impact resistant composite structures. In comparison to some other thermosetting counterparts, this type of composites can undergo higher-energy impact, thanks to their thermoplastic matrix and also the co-mingling technology used in their preforms. They also offer higher processing speeds during manufacturing and better recyclability options. Taking advantage of the above properties, this PhD research was conducted with the aim of developing a systematic approach for characterization, materials selection and multi-criteria optimization of Twintex laminates under impact, with a potential application in highway guardrails. The research was followed in two directions; (a) experimental (Chapters 2-4) and (b) numerical (Chapter 5). For the first part, two types of destructive tests (200J drop-weight impact, and pre- and post-impact four-point flexural bending) as well as two non-destructive tests (visual inspection and X-ray microtomography) were designed and performed to characterize the mechanical properties of different Twintex laminates with plain, twill, and UD architectures, under multiple criteria. Twintex samples were made using four different fabrics; (1) unidirectional fibers (UD), (2) balanced plain woven (PW), (3) balanced twill woven (TW) and (4) unbalanced twill woven (UW). The Taguchi design of experiments (DOE) method was employed to optimize the configuration of select Twintex laminates under the impact energy absorption criterion. The evaluation of the reliability/performance of the Taguchi statistical 101  prediction model in complex composite impact applications was of high interest. The method was found to be very effective and useful, especially for evaluating multiple hybrid composite laminates with a reduced number of experiments. However, the method accommodates one design factor (criterion) at a time (e.g., the absorbed energy), whereas in practice design of composites often necessitates the consideration of multiple design criteria. In order to address this issue, first, a broad range of mechanical properties were identified that would be used as criteria to judge the impact performance of composite laminates: (1) dynamic reaction force, (2) absorbed energy, (3) maximum central deflection, (4) interior damage intensity, (5) exterior damage intensity, (6) ultimate flexural strength, (7) the relative loss of strength due to impact, (8) flexural toughness, and (9) the relative loss of flexural toughness due to impact. Using the obtained test data from earlier chapters, the performance of four different composite configurations (PW, TW, UW and UD) were then judged via a multi-criteria decision making (MCDM) approach -- namely the TOPSIS technique. Among homogenous Twintex laminates, the optimum weave pattern found with this technique was the balanced twill weave. One critical stage of the MCDM procedure was assigning a weight of importance to each design criterion. In this work, next to standard methods such as Modified Digital Logic (MDL), two novel subjective weighting methods were proposed that can offer better flexibility to the decision makers. These new methods included: (a) the adjustable mean bars (AMB) method and (b) the numeric logic (NL) method. The Criteria Importance through Intercriteria Correlations (CRITIC) and the Entropy methods were also selected from the literature as two ‘objective’ approaches (i.e., without incorporating expert’s opinion) in order to quantify the intrinsic information under each attribute and assign a proper statistical weight of importance. In addition, a new combinative weighing method (MCW) was proposed and used to combine different ‘subjective’ and ‘objective’ weights. Different weighting schemes ranked the Twintex materials differently in the current case study and suggested a very high sensitivity of weighting in high risk applications of MCDM. Accordingly, a systematic procedure was proposed to better assist designers in composite materials selection. Namely, the methodology categorizes designers into three levels (basic, 102  intermediate and advanced levels) based on their experience in the field of impact design of composite materials and confidence in assigning weights. The method can give the decision makers options to select a weighting approach that would best fit their levels of experience. The proposed methodology was exemplified on the fabricated Twintex laminates as a preliminary case study to select the best weave pattern for a potential composite guardrail application.  In the second part of the thesis, two new preliminary approaches were proposed to simulate the impact response of FRP composite materials in commercial finite element packages, such as Abaqus, with no need for user-defined coding. Impact simulation of woven FRP composites structures is still known as a very challenging task in the composite research community and advanced FE users need to write their own subroutine codes to model specific material damage behaviours given different composite materials and fiber architectures. However, writing material subroutines is not always practical for general users of commercial FE packages, especially industrial manufacturers who look for fast and economic answers to their design problems.  6.2 Conclusions  The following specific conclusions were made in the experimental part of this thesis:   In some earlier reports on the impact behaviour of composite materials, the material impact resistance had been highly judged based on the visible damaged area on the rear face of impacted specimens. This research, within other recent works, revealed that neither the exterior visible damage nor the interior damage extent can represent the entire story about the impact resistance of the materials, or the loss of mechanical properties due to impact. There can be cases where the extent of damage area may be small, but depending on the associated failure mode(s), the ensuing energy absorption is high, and vice versa.   As a nondestructive inspection technique, X-ray microtomography (XMT) can be used to visualize and quantify the micro-cracks and distributed delamination sites “inside” the samples at 103  different cross sections. 3D images obtained by X-ray microtomography showed that damage modes and their propagations are highly dependent on the weave pattern of fibers in Twintex laminates.  UD specimens absorbed (dissipated) more energy than tested woven laminates. It is hypothesized that this is partially due to the ease of impact wave propagation in the unidirectional laminates. The other factor was found to be the presence of fiber kinking adjacent to the impactor boundary and the associated large transverse crack inside the UD samples.   The Taguchi design of experiment (DOE) approach was found to be a cost-effective and reliable method in predicting the impact energy absorption of FRP laminates and can be used for lay-up optimization purposes. An important assumption of the model, however, is that the governing damage mechanisms in different factor (layer) options are invariants among different laminates tested in the selected orthogonal arrays. Based on the present Taguchi results, the only composite configuration that did not meet this assumption was the homogenous UD laminate.     It was revealed that the laminate lay-up optimization for the out-of-plane impact response of Twintex composites can be equivalently, and much more conveniently, done via the optimization of lay-up based on a flexural stress distribution criterion.  It was found that some combinations of different types of reinforcements (i.e., hybrid configuration) such as [TW, (PW)2, UD0/90]s, increase the capacity of the FRP structure to absorb more impact energy as compared to the uniformly reinforced [(PW)6]s  and [(TW)3]s laminates.   For the multi-criteria impact optimization of FRPs, the proposed adjustable mean bars (AMB) weighting method was found superior to the previously reported basic subjective methods such as ROC. Advantages include: (a) enabling the DM to adjust the weighting factors more interactively, (b) providing visual aid during weighting iterations, and (c) more repeatability and re-adjustability.  104   Similarly, the advantage of proposed numerical logic (NL) method over the MDL method was that the DM is not limited to three pre-defined digital levels of importance to compare the criteria. Instead, the DM can assign any numeric weight between 0 and 1 during pairwise comparisons, which in turn increases the resolution/accuracy of ranking outcomes.   In comparison with earlier combinative weighting methods, the proposed modified combinative weighting (MCW) can accommodates DMs with different levels of experience, and as a result different level of reliability in the assigned weights. In the finite element analysis part of this thesis, two simplified modeling approaches were developed as a preliminary tool to make it possible to predict the impact force of FRP laminates using built-in models in commercial FE packages (herein Abaqus), without a need for advanced user-defined material subroutines. To show the application of the models, the previously experimented twill Twintex laminate was chosen as a sample FRP material. The specific conclusions made are as follows.  The initial FE model using the built-in Hashin damage criterion in Abaqus along with the quasi-static material properties was far from the real impact experimental results. Four possible sources can be identified as the main cause of this deviation:  i. Delamination zones are observed in the impacted Twintex laminates by X-ray micro-tomography while the conventional Hashin criterion does not consider delamination in its progressive damage model. ii. For composite laminates, there is a difference between effective material properties under normal tension and flexural bending loads. Knowing that the input properties in the FE model was obtained from normal tension tests, and also the fact that out-of-plane impact is more like a high speed bending than normal tension, the quasi-static material properties needed modifications. iii. The quasi-static material properties also needed to be varied for the strain rate dependency effect during impact.  105  iv. In most FE models it is assumed that samples were perfectly clamped in the fixture whereas in reality it is common that samples have some degrees of freedom to slide in the fixture. This could be another contributing factor to the deviation between simulation and experimental results; and accordingly an adjustment of effective composite properties may be needed.  In order to consider the above four effects in the composite impact simulation, two different approaches were proposed. In approach #1, for each of these factors, a correction factor can be multiplied to the original quasi-static mechanical properties. Hence, in total, four sets of correction factors were introduced to correct these four sources of errors. In approach #2, only one overall correction factor is applied to the entire set of material properties at once. The following sub-conclusions were  derived from these two approaches:  i. Results of both modified models showed that by correcting the quasi-static material parameters using correction factors, the predictability of the numerical model is considerably enhanced. ii. Approach #1 predicated a relatively more accurate force history during the impact—especially in predicting the first failure point (Hertzian force) as a material property. However, the second approach also gave a reasonably accurate prediction, especially on the impact peak force, when compared to the experimental data. iii. Approach #2 is much easier and faster to implement because no information is needed about the strain rate dependency, clamping effect, delamination sensitivity, and bending mode effect for a given material. It only requires finding one single correction factor via iteration. It should be added that despite of a significant improvement in prediction of contact force, these simplified approaches are not still capable of predicting the history of impactor energy accurately. This is 106  due to the fact that the predicted contact time/duration between impactor and samples was always less than actual data (i.e., the de-bounding occurred faster in the FE runs compared to experiments).  6.3 Recommendations for Future Work Most of the earlier FE models, including those presented in this thesis, on the impact response of woven composites have assumed an orthotropic material configuration for each layer of the given laminate. This is despite the fact that during static forming processes, it is already known that the angle between fiber bundles can largely change during deformation [193]. The X-ray tomography that was conducted in this research revealed that weft yarns around the impact center were severely sheared for some samples and yielded a non-orthogonal angle between warp and weft bundles (see, e.g., Figure 6.1).  To account for this local effect, a new rate-dependent nonlinear non-orthogonal user-defined FEM model can be developed in future, by means of correcting the material orientation of the composite layers due to potential large shear deformations during dynamic events such as impact.  Fig. 6.1: An XMT slice of the Twintex plain woven laminate subjected to 200J impact showing shear deformation of  fiber bundles around the impact center.  Other potential extensions of the work may include: 1. X-ray microtomography analysis of the hybrid Twintex laminates with varying thickness and reinforcement architectures subject to different levels of impact energy.  Impact Center 107  2. An investigation on the correlation between different impact factors and kink band formation in the laminates. 3. Validation of the proposed simplified models in Chapter 5 for other composite materials with different sizes and different impact test set-ups. In the case of successful validation, the list of materials and their corresponding correction factors can be formed in the future for fast implementations in a wide range of commercial FE packages.  4. Enhancing the detailed FEM modeling of Twintex laminates by adding a user-defined FORTRAN code to account for the deterioration of material properties as a function of damage state. 5. Validating the above FE model with the obtained drop weight impact test data. 6. Simulating a typical car accident with a full size 3D Twintex guardrail and subsequently performing shape optimization.  7. Final material recommendation for actual Twintex guardrail based on the MCDM tool developed in this thesis and the large scale model in Step 6. 8. A comparative study between the efficiency of the composite material found in step 7 and its original metallic counterpart, namely steel, in terms of impact resistance, durability, production and maintenance costs.  9. Making a full size Twintex guardrail prototype based on the obtained optimization results and running an actual car crash test. Of course the latter stage will necessitate a much large project scope where other components of the guardrail system such as posts and its joints should be analyzed next to the guardrail structure itself.      108  7 References [1] “http://www.sampe.org/events/LongBeach2013Prospectus.pdf.” . [2] “http://www.docstoc.com/docs/6320427/Boeing_787.” . 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Struct., vol. 86, no. 11–12, pp. 1164–1175, Jun. 2008.    122  8 Appendix A: TOPSIS Method This section presents a summary of ranking alternatives (material candidates) using the TOPSIS MCDM method [112].  Step 1. Normalization Since each attribute is measured on a different scale, normalization is required. The normalization is achieved by: 𝑟𝑖𝑗 =𝑥𝑖𝑗√∑ 𝑥𝑖𝑗2𝑚𝑖=1,    𝑖 = 1,… ,𝑚;   𝑗 = 1,… , 𝑛. (8.1) where i and j represent the corresponding row (material) and column (criterion)  in the given decision matrix. Step 2. Weighting Based on the importance of each given criterion, the weight (𝑤𝑗) from one of the subjective/objective/combinative methods should be applied to the corresponding normalized values of the decision making matrix:  𝑣𝑖𝑗 = 𝑤𝑗𝑟𝑖𝑗 (8.2) Note that summation of all weights must be equal to one (∑ 𝑤𝑗 = 1𝑛𝑗=1 ). Step 3. Identifying Positive-Ideal and Negative-Ideal Solution The positive-ideal (A*) and negative-ideal (A-) solutions are defined via the weighted normalized criteria values (𝑣𝑖𝑗) as follows: 𝐴∗ = {𝑚𝑎𝑥𝑖  𝑣𝑖𝑗|𝑗 ∈ 𝐽1, 𝑚𝑖𝑛𝑖  𝑣𝑖𝑗|𝑗 ∈ 𝐽2 , 𝑖 = 1,… ,𝑚} (8.3) And 123  𝐴− = {𝑚𝑖𝑛𝑖  𝑣𝑖𝑗|𝑗 ∈ 𝐽1, 𝑚𝑎𝑥𝑖  𝑣𝑖𝑗|𝑗 ∈ 𝐽2 , 𝑖 = 1,… ,𝑚} (8.4) Where J1 and J2 are the set of benefit and cost attributes, respectively. Note that the benefit and cost attributes refer to the higher the better and the lower the better type of attributes, respectively. Step 4. Calculation of Separations from Positive-Ideal and Negative-Ideal Solutions The separation among any two alternatives can be measured by an n-dimensional Euclidean distance. Accordingly, the separation of each alternative from the positive-ideal solution (A*) and negative-ideal solution (A-) is found as follows. 𝑆𝑖∗ = √∑(𝑣𝑖𝑗 − 𝑣𝑗∗)2𝑛𝑗=1    , 𝑖 = 1,… ,𝑚 (8.5) 𝑆𝑖− = √∑(𝑣𝑖𝑗 − 𝑣𝑗−)2𝑛𝑗=1    , 𝑖 = 1,… ,𝑚 (8.6) Step 5. Calculation of Similarities to Positive-Ideal Solution  The next step is to find the closeness of each material to the positive-ideal solution, also called the TOPSIS score.   𝐶𝑖∗ =𝑆𝑖−(𝑆𝑖∗ + 𝑆𝑖−) (8.7) 𝐶𝑖∗ is always between 0 and 1; 𝐶𝑖∗ = 0 when 𝐴𝑖 = 𝐴− and 𝐶𝑖∗ = 1 when 𝐴𝑖 = 𝐴∗. Step 6. Rank Preference Order The higher the 𝐶𝑖∗, the better the performance of the material. In another words, the descending order of 𝐶𝑖∗ shows the rank of materials. 

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