Bio-inspired composites : analysis of their fracture behavior using the XFEM

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POLITECNICO DI MILANO

School of Industrial and Information Engineering

Master in Biomedical Engineering

Bio-inspired composites: analysis of their

fracture behavior using the XFEM

Supervisor: Prof. Pasquale Vena Co-Supervisor: Ph.D. Flavia Libonati

Prof. Laura Vergani

Master of Science Thesis Andrés Eduardo Aguilar Coello Student number: 862591

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POLITECNICO DI MILANO

Scuola di Ingegneria Industriale e dell’Informazione

Laura Magistrale in Ingegneria Biomedica

Bio-inspired composites: analysis of their

fracture behavior using the XFEM

Relatore: Prof. Pasquale Vena Correlatori: Ph.D. Flavia Libonati

Prof. Laura Vergani

Tesi di Laurea Magistrale Andrés Eduardo Aguilar Coello Matricola: 862591

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Acknowledgements.

I would like to deeply thank to the Ph.D. Flavia Libonati for her guide and all the hours devoted to the discussions needed to bring this work to its end.

To my parents, my sister and my brother, who are always there, their support and love are invaluable tools for me.

To Carolina for her unconditional company along all the roads that life offers us.

In general, to my teachers and colleagues of the Politecnico for sharing their knowledge with me. Be assured that these teachings will be with me throughout my entire life.

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Abstract.

In the past few years, natural materials have been the spotlight in the pursue to develop damage-tolerant novel materials. Among these, nacre and bone-inspired materials are the focus of several studies which aim to exploit the outstanding properties that their natural counterparts exhibit. The mechano-biological interactions that take place at multiscale levels and contribute to obtain amplified mechanical properties (i.e. toughness) compared with their constituents have been deeply studied. Even tough, it is rather difficult to implement these multiscale features into man-made materials, the new designing and manufacturing resources available allow researchers to use different kinds of approaches to unravel these fracture’s characteristics in newly designed composites.

Through this work, simple bio-inspired topologies were modeled to mimic the particular toughening mechanisms observed in nacre and bone. By using stiff and soft polymers, we implemented the brick-and-mortar structure, characteristic of nacre and bone in the nanoscale, into 3D-printed composites. With the aid of numerical models, done with the commercial software Abaqus™ (Dassault Systems, 2016), and validated on previous experimental outcome, the main objective of this thesis work is to prove the validity in the use of the XFEM (eXtended Finite Elements Method) to analyze the fracture behavior of these bio-inspired composites. Moreover, the present work is intended to demonstrate the effects in the strength and toughness of the composites when changes in their topology are introduced.

Simplified geometries based in the unit cells that form the entire samples of the composites will be used. These models will be presented in two main groups, those with the basic brick-and-mortar topology, and those that included the mineral bridges (MBs) in the unit cell’s topology. The influence in the composites’ mechanical properties of the stiff material volume fraction (VF) is clearly reproduced by the Abaqus XFEM models. Additionally, the introduction of the MBs in their respective models, increased the strength and toughness of all the nacre-like composites modeled. However, the XFEM framework was not able to reproduce the failure mechanism shown by the nacre-like composites during the experimental testing. The complex nature in the mechanical properties of the base materials (specially the hyperelastic behavior of the soft polymer), and the still existent

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limitations of the XFEM framework in Abaqus were the main drawbacks in the fracture behavior analysis of these nacre-like composites.

Nevertheless, the FEM and XFEM frameworks are well positioned as valuable tools to be used in the design phases of these kind of new-materials. The results here obtained would help in the definition on a further methodology aim to improve, at least at some extent, the design stage of new bio-inspired composites with enhanced fracture behavior.

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Sommario.

Negli ultimi anni, i materiali naturali sono stati una fonte di ispirazione nel tentativo di sviluppare nuovi materiali resistenti al danneggiamento. Tra questi, la madreperla e i materiali di ispirazione ossea sono al centro di numerosi studi che mirano a sfruttare le proprietà eccezionali che le loro controparti naturali esibiscono. Le interazioni meccano-biologici che avvengono a diversi livelli di scala e contribuiscono ad ottenere proprietà meccaniche amplificate (i.e. tenacità) rispetto ai loro costituenti sono stati profondamente studiati. Anche se è piuttosto difficile implementare queste caratteristiche, che agiscono a diversi livelli di scala, in materiali artificiali, a causa delle attuali limitazioni nei processi produttivi. Recenti tecniche di produzione hanno permesso di studiare in modo più approfondito alcuni di questi meccanismi.

Attraverso questo lavoro di tesi, sono state modellate semplici topologie ispirate alla natura per imitare i particolari meccanismi di rafforzamento osservati nella madreperla e nell’osso. A livello di nano scala, entrambi i materiali naturali presentano un'architettura "brick-and-mortar", in cui le piastrine minerali a forma di mattone sono incorporate in una matrice di materiale polimerico organico. Qui, utilizzando polimeri rigidi e gommosi, abbiamo implementato questa topologia, in compositi stampati in 3D. Con l'aiuto di modelli numerici, realizzati con il software commerciale Abaqus ™ (Dassault Systems, 2016), e validati su precedenti risultati sperimentali, l'obiettivo principale di questa tesi è provare la validità nell’uso degli XFEM per analizzare il comportamento a frattura di questi compositi bio-ispirati. Inoltre, il presente lavoro ha lo scopo di dimostrare gli effetti nella resistenza e nella tenacità dei materiali compositi quando vengono introdotti cambiamenti nella loro topologia.

Verranno utilizzate geometrie semplificate basate sulle celle unitarie che formano l'intero campione dei compositi. Questi modelli sono presentati in due gruppi principali, quelli con la topologia di brick-and-mortar di base e quelli che includono i ponti minerali (MB) nella topologia della cella unitaria. L'influenza nelle proprietà meccaniche dei compositi della frazione di volume del materiale rigido (VF) è chiaramente riprodotta dai modelli XFEM di Abaqus. Inoltre, l'introduzione dei MB nei rispettivi modelli aumenta la resistenza e la tenacità di tutti i compositi nacre-simili modellati. Tuttavia, degli XFEM non sono stati in grado di riprodurre il meccanismo di fallimento mostrato dai compositi nacre-simili

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durante i test sperimentali. La natura complessa delle proprietà meccaniche dei materiali di base (specialmente il comportamento iperelastico del polimero gommoso) e le limitazioni ancora esistenti degli XFEM in Abaqus sono stati i principali inconvenienti nell'analisi del comportamento alla frattura di questi compositi nacre-simili.

Tuttavia, degli FEM e degli XFEM sono ben posizionati come strumenti preziosi da utilizzare nelle fasi di progettazione di questo tipo di nuovi materiali. I risultati qui ottenuti aiuterebbero nella definizione di un'ulteriore metodologia per migliorare, al meno in certa misura, la progettazione di nuovi compositi bio-ispirati con comportamento a frattura migliorato.

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Content

Acknowledgements. ... iii

Abstract. ... iv

Chapter 1 : Literature review ... 14

1.1 Introduction to bio-inspired composites ... 14

1.1.1 The bio-inspiration applied in di-novo composites ... 15

1.2 Biological materials ... 17

1.2.1 Nacre ... 18

1.2.2 Bone ... 22

Chapter 2 : Numerical background ... 29

2.1 Hyperelastic materials ... 29

2.1.1 Models for isotropic hyperelastic materials... 31

2.2 The eXtended Finite Elements Method ... 33

2.2.1 The enrichment functions ... 34

2.2.2 The cohesive segment method and phantom nodes... 35

2.2.3 The level set method ... 36

2.2.3 The enrichment procedure ... 36

2.2.4 The XFEM-based cohesive behavior ... 37

2.2.5 XFEM applied in finite strain fracture mechanics... 40

2.2.6 Limitations of the XFEM implementation in Abaqus ... 41

Chapter 3 : Materials and methods. ... 42

3.1 The composite material ... 42

3.1.1 The base materials ... 42

3.1.2 The bio-inspired composite architecture. ... 42

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3.3 XFEM models of the mechanical tests for the composite’s base materials. ... 44

3.3.1 The uniaxial tensile tests ... 45

3.3.2 The fracture tests ... 53

3.3.3 Final notes about the XFEM models in the base materials. ... 61

3.2 Experimental tests ... 63

3.2.1 Biaxial tensile test. ... 63

3.2.2. The compression tests. ... 72

3.4 Theoretical approach for a simplified model using unit cells. ... 74

3.4.1 Composites stiffness estimation ... 75

Chapter 4 : Abaqus XFEM models for the nacre-like 3D-printed materials ... 77

4.1 Unit cell models ... 77

4.1.1 Basic unit cell models. ... 78

4.1.2 Unit cell models with MBs ... 80

4.1.3 Nacre_70% model. ... 82

Chapter 5 : Results of the nacre-like composites Abaqus XFEM models... 84

5.1 Results in the basic unit cell models ... 84

5.1.1 Basic unit cell models uniaxial stress response ... 84

5.1.2 Basic unit cell models shear strain response. ... 85

5.1.3 Basic unit cell models crack status output. ... 88

5.2 Unit cell with mineral bridges models results ... 90

5.2.1 Unit cell with MB models uniaxial stress response ... 90

5.2.2 Unit cell with MBs models shear strain response. ... 91

5.2.3 Unit cell with MBs models crack status output ... 93

5.3 Nacre 70% model ... 95

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5.3.2 Nacre 70% model crack status output ... 98

Chapter 6 : Discussion and Conclusions ... 99

6.1 Discussion ... 99

6.2 Conclusions ... 103

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List of Figures

Figure 1-1. Schematic illustration of some features of nacre microstructure. (Fig. 1 in [10]) ... 19 Figure 1-2. Nanoscale mechanism controlling the shearing of the tablet's interfaces: a) bio-polymer acting as a viscoelastic glue. b) contact between the aragonite asperities. c) schematic view of mineral bridges in nacre. d) platelets waviness feature observed from scanning electron micrographs. (adapted form Fig. 8-9 in [7] ). ... 20 Figure 1-3. Scheme of the steady state crack in nacre. The J-integrals contours are also shown (Fig 17 in [11]) ... 22 Figure 1-4. Hierarchical multi-scale structure of bone, showing seven hierarchical levels, and their relation to the mechanisms of mechanical properties. The left panel shows the characteristic structural features; the right panel indicates associated deformation and toughening (fracture resistance) mechanisms. (Fig.17 in [7]) ... 23 Figure 1-5. Computational simulations comparing the stress-strain response of collagen fibril (blue line) and mineralized collagen fibril (red line). (Fig. 19 adapted from [7]) ... 24 Figure 1-6. Fracture model of bio composites. The stress-strain relation contributes to the fracture energy. The hard platelets remain intact as the soft matrix undergoes large shear deformation. The width of the fracture process is assumed to be in the order of the length of the HA minerals. (Fig. 7 in [6]) ... 26 Figure 2-1. Normal and tangential coordinates for a smooth crack. (Fig. 10.7.1-1 in [20]) 35 Figure 2-2. The phantom node method (Fig. 10.7.1-2 in [8]) ... 35 Figure 2-3. a) Propagating crack. b) Stationary crack. The red nodes are enriched with the Heaviside function; the green nodes are enriched with the asymptotic crack-tip singularity functions; the blue nodes are enriched with both functions (Fig.1.25 [15]) ... 36 Figure 2-4. a) Linear traction-separation response. b) Nonlinear traction-separation response (Fig. 10.7.1-6 in [20]). ... 38 Figure 2-5. Linear damage evolution. Here 𝛿𝑛0 stands for the initiation of the normal traction separation (at point A), while 𝛿𝑛𝑓 stands for the final magnitude of the traction separation (at point B). (Fig 32.5.6-3 [20]) ... 40

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Figure 3-1. TBP's material evaluation to define it hyperelastic behavior. Evaluation done using the available data from uniaxial tensile test. ... 45 Figure 3-2. BC applied in the dog-bone geometry used to model the TBP uniaxial tensile tests. ... 46 Figure 3-3. Mesh for the model of TBP uniaxial tensile test in the initial configuration. ... 47 Figure 3-4. Results for VM uniaxial tensile test model: a) Stress (S11) results contours at the first increment. b) Stress (S11) results contours at the last increment. c) Strain (E11) results contours at the first increment. d) Strain (E11) results contours at the last increment. ... 47 Figure 3-5. a) XFEM (crack) status for TBP under uniaxial tension. b) & c) XFEM status details; the red contours represent the complete broken elements. ... 48 Figure 3-6. TBP uniaxial tensile tests stress-strain response. Comparison between the experimental test and the Abaqus XFEM model. ... 49 Figure 3-7. Results for VM uniaxial tensile test model: a) Stress (S11) results contours first increment. b) Stress (S11) results contours last increment. c) Strain (E11) results contours first increment. d) Strain (E11) results contours last increment. ... 51 Figure 3-8. a) XFEM (crack) status for VM under uniaxial tension. b) XFEM status details; the red contours represent the complete broken elements. ... 51 Figure 3-9. VM uniaxial tensile tests stress-strain response. Comparison between the experimental test and the Abaqus XFEM model. ... 52 Figure 3-10. Results for TBP fracture test model: a) Stress (S22) results contours initial increment. b) Stress (E22) results contours intermediate increment. c) Stress (S22) results contours final increment. d) Strain (E22) results contours initial increment. e) Strain (E22) results contours intermediate increment. f) Strain (E22) results contours final increment. 55 Figure 3-11. Stress (S22) details to show the stress distribution in the crack tip for the TBP fracture test model towards the end of the simulation. ... 55 Figure 3-12. TBP fracture tests stress-strain response. Comparison between the experimental test and the Abaqus XFEM model ... 56

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Figure 3-13. a) Images from the experimental fracture test with the TBP samples. b) XFEM (crack) status for TBP during the fracture test. ... 57 Figure 3-14. a) XFEM status details; the red contours represent the complete broken elements. b) Stress (S22) details to show the stress peaks behind the crack tip. ... 57 Figure 3-15. Results for VM fracture test model: a) Stress (S22) results contours, initial increment. b) Stress (S22) results contours intermediate increment. c) Stress (S22) results contours, at higher stress value. d) Strain (E22) results contours, initial increment. e) Strain (E22) results contours, intermediate increment. f) Strain (E22) results contours, at higher strain value. ... 59 Figure 3-16. Stress (S22) details to show the stress distribution in the crack tip for the VM fracture test model. ... 59 Figure 3-17. a) XFEM (crack status) evolution for VM during the fracture test model. b) XFEM status details: shows the mesh shape, the crack propagation in green and the complete broken elements in red. Comparison with an image of the experimental output of the fracture test with the VM material. ... 60 Figure 3-18. VM fracture tests stress-strain response. Comparison between the experimental test and the Abaqus XFEM model. ... 60 Figure 3-19. a) Sketch: Square shape specimen for biaxial tests, all dimensions are reported in millimeters (mm). b)3D-printed square specimen for biaxial tests. c) Sketch: cruciform-fillet shape specimen for biaxial tests, all dimensions are reported in (mm) d) Final sample’s geometry. ... 64 Figure 3-20. Biaxial test rig. Image obtained by the authors during the execution of the experimental tests. ... 66 Figure 3-21. a) Cruciform-fillet TBP sample with the four white markers and the grips fixed. b) Cruciform-fillet TBP sample mounted in the biaxial test machine. ... 67 Figure 3-22. Results of biaxial test in the cruciform-fillet sample 1, stress-strain curve. ... 69 Figure 3-23. a) Cruciform-fillet sample initial configuration. b) Cruciform-fillet sample, stretching effect due to displacement-controlled application. c) Cruciform-fillet sample

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with a crack that has appeared and is starting to evolve through the material. d) The Cruciform-fillet sample fails. ... 70 Figure 3-24. TBP material evaluation for its hyperelastic behavior. a) Evaluation based in the available data of the uniaxial tensile test. b) Evaluation based in the available data of the biaxial tensile test. ... 71 Figure 3-25. Universal testing machine configured for compression tests. ... 72 Figure 3-26. Stress-strain results of the compression test in TBP material samples. The deflection in the stress-strain response can be identified around -50[MPa]. ... 73 Figure 3-27. a) Compression test, start. b) Compression test execution. c) Compression tests cylinder sample after reach failure in the tests. ... 74 Figure 3-28. a) The nanostructure of staggered hard plates in a soft matrix represents a convergent design of natural evolution. b) The primary load bearing zones of biological nanostructure showing mineral crystals primarily in tension and the soft matrix primary in shear. c) Forces acting on a single mineral plate diagram (Fig. in [6]). ... 75 Figure 4-1. Ucell model geometries for the different VF: Ucell_50% (top), Ucell_60% (middle), Ucell_70% (bottom). ... 77 Figure 4-2. Scheme of the BC applied in the Ucell_70%, the VM platelets are mainly subjected to uniaxial tension under lateral constraint from the TBP material of the adjacent Ucells. ... 77 Figure 4-3. Mesh sensitivity analysis in the Ucell models. The same seed size was used in the three models. ... 80 Figure 4-4. Scheme of the BC applied in the Ucell_70%-MB. ... 81 Figure 4-5. Mesh sensitivity analysis in the Ucell models. The same seed size was used in the three models. ... 82 Figure 4-6. Scheme of the BC applied in the 2D complete Nacre_70% XFEM model. ... 83 Figure 5-1. a) S11 contours of Ucell_50%. b) S11 contours of Ucell_60%. c) S11 contours of Ucell_70%. (deformation scale factor = 0.1) ... 85 Figure 5-2. a) E12 contours of Ucell_50%. b) E12 contours of Ucell_60%. c) E12 contours of Ucell_70%. (deformation scale factor = 0.1) ... 86

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Figure 5-3. a) Stress - strain curves obtained during the experimental testing in the nacre-like composites. b) Stress-strain curves obtained from the Abaqus XFEM Ucell models. . 86 Figure 5-4. Stress - strain curve related with the S11 contours output, showing the effect of the crack evolution in the stress-strain response. The small drop (red circle) is related with the energy released due to the propagation of the pre-existent crack. ... 87 Figure 5-5. a) XFEM status of Ucell_50%_MB and detail of the broken elements. b) XFEM status of Ucell_60%_MB and detail of the broken elements. c) XFEM status of Ucell_70%_MB and detail of the broken elements. Above each XFEM status, is placed the deformed status of the Ucell_MB model, showing the arrested crack before the VM platelet. ... 89 Figure 5-6. a) S11 contours of Ucell_50%-MB. b) S11 contours of Ucell_60%-MB. c) S11 contours of Ucell_70%-MB (deformation scale factor: 0.1). ... 90 Figure 5-7. a) E12 contours of Ucell_50%-MB. b) E12 contours of Ucell_60%-MB. c) E12 contours of Ucell_70%-MB (deformation scale factor: 0.1) ... 91 Figure 5-8. Stress-strain curves obtained during the experimental testing in the nacre-like composites with mineral bridges (MBs). b) Stress-strain curves obtained from the Abaqus XFEM Ucell_MBs models. ... 92 Figure 5-9. a) XFEM status of Ucell_50%_MB and detail of the broken elements. b) XFEM status of Ucell_60%_MB and detail of the broken elements. c) XFEM status of Ucell_70%_MB and detail of the broken elements. Above each XFEM status, is placed the undeformed status of the Ucell_MB model, showing the crack deflection found for each case, a grey color is used instead the black color to show the crack shape in the model’s geometry. ... 94 Figure 5-10. S11 contours of Nacre_70%. ... 95 Figure 5-11. E11 contours of Nacre_70%. ... 96 Figure 5-12. Comparison of the E11 output in the Nacre_70% Abaqus model with the experimental fracture test performed by Dimas et al., 2013. The interfaces where the strains are higher in the XFEM model, match the location where the TBP material start to fail in the 3-D printed nacre-like composite. In this experimental experience the authors

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used a similar combination of materials in a 3-D printed composite with a nacre-like topology. (adapted from Fig. 7 in [12]) ... 96 Figure 5-13. E12 contours of Nacre_70%. ... 97 Figure 5-14. a) Stress - strain curves comparison Nacre_70% Abaqus XFEM models. The solid lines represent the output obtained from the Abaqus XFEM models. The dashed lines are the experimental data. ... 97 Figure 5-15. Crack status (XFEM) contours of Nacre_70% ... 98

List of tables

Table 3-1. Mechanical properties of the acrylic-based polymers obtained from the uniaxial tensile tests (in [21]). ... 42 Table 3-2. Summary, Abaqus models’ configuration for the uniaxial tensile test in TBP and VM samples. ... 53 Table 3-3. Summary, Abaqus models’ configuration for the fracture test in TBP and VM samples. ... 61 Table 3-4. Time incrementation parameters found in Abaqus (first line) [20]. ... 62 Table 3-5. Results obtained from the experimental biaxial tests performed in two cruciform samples. The average and the standard deviation is calculated considering the four values of stress and strain obtained during the biaxial experimental tests. ... 69 Table 3-6. Resulting coefficients from the hyperelastic material evaluation using the available data from the uniaxial and the biaxial tensile tests. ... 71 Table 3-7. Elastic modulus values calculated with Equation 3-4 and compared with the experimental outcome reached by the different nacre-like composite samples. ... 76 Table 4-1. Materials definitions for the Ucell Abaqus XFEM models and general view of the modules configuration for the Ucell Abaqus XFEM models. ... 79 Table 4-2. General view of the modules configuration for the Ucell with MBs Abaqus XFEM models. ... 81

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Table 4-3. General view of the modules configuration for the Nacre_70% Abaqus XFEM model. ... 83 Table 5-1. Strength values comparison between Abaqus XFEM models and experimental data. ... 88 Table 5-2. Toughness values comparison between Abaqus XFEM models and experimental data. ... 88 Table 5-3. Strength values comparison between Abaqus XFEM models and experimental data. The error is calculated considering the experimental data and the Abaqus XFEM models’ output. ... 93 Table 5-4. Strength values comparison between Abaqus XFEM models and experimental data. The error is calculated considering the experimental data and the Abaqus XFEM models’ output. ... 93

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Introduction

Given the important developments achieved by the materials science, the engineering fields have been able to go further with the purpose of developing novel damage-tolerant materials. Similarly, the continuous and deep understanding of the biology, offers an invaluable opportunity for the development of new bio-inspired materials and technologies with a wide range of applications. Among these, nacre (i.e. ‘mother of pearl’) and bone-inspired materials are of special interest for the researchers, the designers, and in general for the diverse engineering fields.

Nacre and bone exhibit a complex architecture and the different hierarchical levels in their structures provide them with high strength and toughness. Clearly, if these values are compared with the properties showed by the typical engineering materials, their absolute value is lower. However, these natural composites far exceed the toughness of their constituents, being able to support elevated loads without reaching premature fracture. Additionally, these features are combined with a characteristic low-density, , reaching an extraordinary combination of lightness and resistance. The mechano-biological phenomena that takes place at these multiscale levels and contribute to obtain amplified toughness compared with their constituents, have been deeply studied. For instance, nacre’s fracture characteristics involve crack deflection, breaking of mineral bridges, bricks interlocking, and the organic biopolymer acting as a viscoelastic glue. On the other side, some bone’s fracture characteristics are similar and include additional features such as crack twist, constrained microcracking or sacrificial bonds.

Even though it is rather difficult to implement these multiscale features into man-made materials, owing to current manufacturing limitations, the new manufacturing resources available, allow researchers to use different kinds of approaches to unravel these fracture’s characteristics in newly designed composites. For instance, the additive manufacturing, commonly known as 3D-printing, is an emerging technology of extraordinary versatility that is proving its applicability to produce a wide range of materials, and among these, composites with bio-inspired topologies. Moreover, numerical simulations are extraordinary tools to be used in the design phase of all kinds of materials and devices. The combination of these technologies is an efficient alternative for this new material design.

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In a previous work introduced by Gu et al., 2017 simple bio-inspired topologies were used to fabricate new-composites using the 3-D printing technology [1]. The base materials choice was made to mimic, in a different order of magnitude, the mechanical properties and the interaction of the natural materials’ constituents. Therefore, a stiff polymer (VeroMagenta VM) was used to mimic the brick-shaped mineral platelets, whereas a soft polymer (TangoBlackPlus TBP) was used for the matrix mainly conformed by proteins in the natural materials. Fracture tests were executed in these composites, to analyze the toughening mechanisms that take place in these 3-D printed samples.

On the other side, through the current work, we modeled fracture tests executed in these bio-inspired composites using the XFEM framework available in Abaqus. To this purpose, first a literature review of nacre’s and bone’s characteristics was realized, additionally, the main topics related with the numerical background used for the models were included. The third chapter introduces information about the polymers used for the bio-inspired composite. Moreover, the Abaqus XFEM models of the uniaxial tension and the fracture test, were here implemented for the composite’s constituents in order to define the mechanical properties to be set in the composites’ models. Furthermore, additional experimental tests aim to increase the information available for the characterization of the TBP material were included. In the fourth chapter, the configuration used in the different composites Abaqus XFEM models is explained. Finally, the outcome obtained for the simulations performed are included in the chapter five.

Motivation

The XFEM arose as an extension of conventional finite element method (FEM) to overcome the drawbacks that this extensively used methodology presented when fracture mechanics problems were analyzed (e.g. the need of remeshing the regions close to the crack tip after each increment). Since it first introduction, the XFEM has been proved it applicability in several fracture mechanics problems. Therefore, when the fracture behavior of bio-inspired composited is analyzed, the XFEM can result in an effective and valuable alternative, for the numerical simulations to be performed in the models based on this kind of composites materials.

For this reason, we aim to prove the validity of the XFEM framework, to analyze the fracture behavior of the bio-inspired composites here presented. In case of obtaining the

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appropriate outcome, this may lead to improvements in the design process of new bio-inspired composites, that pursuit to exploit improved mechanical properties as strength and toughness.

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Chapter 1 : Literature review

1.1 Introduction to bio-inspired composites

To describe the characteristics and the materials properties of the bio-inspired samples used along this thesis work, it is necessary first, to make a brief description of the composite materials and their relationship with biomimetics.

The composites embrace a wide variety of materials, all of them based in a simple principle of combining two or more parts in a single fragment. The word composite means “consisting of two or more distinct parts”. It is worth to mention that the combination of base materials to be considered in the composite, must comply with the non-miscibility of its constituents. For engineering purposes, the composite materials can be seen at different scales from the nano to the macro-scale depending on the possible applications where these can be used. To this effect, one of the main advantages of composite materials is their possibility to be used in a wide range of fields.

Since different materials are used to create new composites, the designers are able to make any kind of combination that allows the new composite to exploit the main benefits from the properties that the base materials have. In this way, the composite’s final properties can easily overcome those characteristics, and at the same time reduce the base materials drawbacks. Common examples can be found in our daily activities, reinforced concrete used in the constructions field, plywood elements use in furniture or reinforced textiles used for hard duty fabrics.

In this context, we can appreciate how the nature offers almost infinite examples of biological materials that, through the years of the evolution, have created very specific combinations, which resulted in materials with outstanding characteristics. Wood, bone, teeth or seashells are a few examples of natural composites which are also of special interest in the engineering fields due to their strength, lightweight, and fracture resistance. Historically, based on simple observation, humanity has been able to copy or mimic many features found in nature. It is possible to imagine that tweezers were inspired in how the birds use their beaks for feeding, but also for nest building. Another example is fishing nets, which may be observed in nature in the spider’s webs and fulfill the same functionality; to catch the prey. Finally, shields commonly seen in many kinds of animals

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from insects as exoskeletons to big reptiles as the turtle’s carapace [2]. An interesting fact about the bio-composites is the fact that there is no distinction made by nature between the structural and the chemical features in the material, nature gives them equal importance, achieving in this way optimized properties for the materials at all scales.

Nowadays, thanks to the continuous improvements that have taken place in the materials’ science, it is possible to continue with this antique technique of developing new instruments, tools and technologies inspired by nature’s creation. However, this can be only achieved with a correct interdisciplinary effort. Biology, Chemistry or Medicine are scientific fields of high interest and deeply related with the development of new bio-inspired composites. These fields keep providing a continuous and deep understanding of nature. Therefore, an invaluable opportunity arises to increase the effort to design and create as many bio-inspired materials and technologies as the current or future world’s situation would demand.

Figure 1-1. Ashby plot for natural composites. (Fig.1 in [3])

1.1.1 The bio-inspiration applied in di-novo composites

Here is where the biomimetics approach takes place in a formal manner. The term was coined by the American engineer Otto Schmitt, back in the 1960s. He resumed the biomimetics concept as the process to transfer ideas from biology to technology. Some synonyms can be mentioned related with the same concept: ‘biomimicry’, ‘bionics’, or ‘biologically inspired design’.

One of the major advantages that the biomimetic approach has, is the vast palette of natural structures, mechanisms and living beings that can inspire new discoveries. Through this

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work we will concentrate in shells (the nacreous layer) and bone, which are natural materials with outstanding mechanical properties. A further explanation about these materials properties can be found in section 1.2.

At this point, it is important to introduce the difference between two well-known fabrication techniques that are commonly used in the production of new materials and devices. These techniques are specially highlighted in the research process to design new materials using the biomimetic approach [4], [5].

The first technique is called “top-down” approach. It is when the manufacturing process starts with an entire piece of raw material. Then, this part is stripped down to obtain the final part. A good example of this process are the old-fashioned fishing-boats that used to be carved from a tree trunk. When this approach is applied in biomimetics, engineers and biologist join efforts to deduce the functional principles that guide the solutions found in nature. Moreover, the materials and the manufacturing techniques are selected, based on the most convenient solutions available. The top-down technique can be also used as an optimization strategy for the existing biomimetic systems and can be resumed as a problem-solving approach.

For the second technique, the “bottom-up” approach is presented. In this case, the raw materials are assembled by parts to produce bigger structures. For instance, when a boat is fabricated using several wooden parts, such as strips or posts, reaching in this way a complex device ready for sail. In the bottom-up technique, biologists are the ones who led the research, while the engineers took an insight of this research to design new bio-inspired materials or devices. This process can be seen as a creative approach. It has become an integral step in the design and manufacturing of new solutions, especially with those related with biomimetics. Additionally, the bottom-up approach provides a deeper understanding of the materials and devices. Here, the efforts are focused on getting a clear description of it basic building blocks. In this way, the compatibility of the structural and the chemical mechanisms that take place in the material can be better understood.

Nowadays, the nature-inspired approaches have promoted significant advances at an interdisciplinary level. This has enabled the development of new advanced materials and complex technologies like scaffolds for tissue engineering, new adhesives, or new generation of composites used in aeronautics.

17

1.2 Biological materials

The use of natural materials has been well recognized since the beginning of humanity. In fact, many needs were fulfilled by the specific utilization of this kind of materials. These resources were used for several purposes passing from clothes, through house-building and food. They soon became compulsory goods for the wellbeing of the community. Therefore, it may be implied that, over time, continuous efforts have been made to look deeper through this enormous palette of materials to fulfill the new challenges that humanity faces.

Figure 1-2. Nanostructures of nacre and bone. a) Bone's nanostructure consists of plate-like mineral crystals 2-4 nm in thickness and up to 100nm in length embedded in a collagen-rich protein matrix. b) Nacre’s elementary structure of plate-like mineral crystals 200-500 nm in thickness and a few micrometers in length with small fraction of soft matrix in between. (Fig. 1 in [6])

In this context, we are going to present two biological materials that inspired the composites fabricated in previous works: nacre, also known as “mother of pearl” and bone. These materials have in common a structural pattern at the nanoscale level that consists of a brick-and-mortar topology. This topology is reproduced in the composite material analyzed in the present work.

18 1.2.1 Nacre

Nacre is well accepted as the gold standard for biomimetic materials due to it mechanical performance. More commonly known as “mother-of-peal”, this bio-composite is an iridescent material that can be found in the inner layers of many seashells. Nacre’s outstanding mechanical properties, mainly it stiffness and toughness, are the result of a complex and specific hierarchical organization of its constituents. This hierarchical organization means that the specific features that influence the material’s strength and toughness, can be found at different length scales. The final scope of the nacre’s structure is to maintain the integrity of the shell, acting together with the external layers, to keep the animal safe in case of being attacked by a predator.

On one hand, brick-shaped mineral platelets made of crystalline aragonite (CaCO3) occupy

the main volume fraction of the composite with about 95%. They have a polygonal flat shape and microscopic size 0.4-0.5 µm thick and 8-10 µm wide. Their main scope is to provide strength and a better stress distribution of the loads acting on the composite material. On the other hand, a biological polymer (5% volume fraction) mainly composed by chitin and proteins (20-30 nm thick), acts as the mortar between the mineral platelets [7], [8]. This bio-polymer supports large strains, being able to dissipate big quantities of energy when mechanical deformation is applied to it. This specific characteristic provides ductility to the nacre material. Moreover, it was observed that the geometrical arrangement of the mineral platelets is highly uniform throughout the entire nacreous layer. Finally, the complementary properties given by the characteristics found in the aragonite mineral platelets and the chitin-based bio-polymer, provides the nacre material with the capability to sustain the loads and dissipate the mechanical energy in an effectively manner.

1.2.1.1 The deformation of nacre

In previous studies, the deformation of nacre was analyzed using different mechanical testing configurations. The uniaxial tensile test in the platelets’ direction is the most representative experimental mode when the deformation in nacre needs to be interpreted. During these tests, the nacre’s platelets sliding was observed and identified as the starting mechanism for the local deformation. This behavior, found at the microscale, is the main source of nacre’s outstanding mechanical properties [7], [9]. When the sliding phenomenon spread over the specimen’s entire geometry, it drove to the pullout of the

19

platelets and eventually to the final failure of the specimen. Meanwhile, the damage caused by the crack propagation led to a stepwise fracture pattern in the nacreous layer.

Figure 1-1. Schematic illustration of some features of nacre microstructure. (Fig. 1 in [10])

The nacre constituents must fulfill some requirements to reach the before-mentioned behavior. For instance, the bio-polymer material located in the platelets’ interfaces should be weaker than the bio-mineral material of which these platelets are composed. Moreover, the platelets aspect ratio (the ratio between the length and wide) must be high enough to increase the cohesion within the material. Nonetheless, the aspect ratio (ρ) maximum value is constrained by a minimum thickness for the platelet. Under this value, the aragonite platelets would behave in a brittle manner, causing a premature failure in the nacre material. Thus, it is possible to say, that the nacre’s mechanical performance is mainly controlled by the mechanisms that take place in the interfaces between the aragonite’s platelets.

Furthermore, additional mechanisms acting at the nanoscale, increase the nacre’s response when it bears shear stresses, beginning with the unfolding of the macromolecules which compose the bio-polymer. This phenomenon does not produce a specific effect for the nacre’s tensile strength. However, regarding the shear strains in nacre, the polymer behavior causes a hardening effect on the material. Additionally, mechanisms such as the nano-asperities found on the nacre’s surface, or the mineral bridges (MBs) that make connection between the mineral platelets, contribute to increasing its toughness. A final feature may be described in the nanostructure of the nacreous material, which consists in

20

the waviness of the tablet surfaces. This increases the platelets interlocking and therefore their sliding resistance. The platelets waviness is in straight relation with the tablets sliding behavior, which are the most important features for the increased strength found in nacre. When acting together, the sliding of the platelets generates compressive stresses that interrupt the platelet separation. Additionally, the waviness also induces the interlocking of the platelets when the sample is under tension loads.

Figure 1-2. Nanoscale mechanism controlling the shearing of the tablet's interfaces: a) bio-polymer acting as a viscoelastic glue. b) contact between the aragonite asperities. c) schematic view of mineral bridges in nacre. d) platelets

waviness feature observed from scanning electron micrographs. (adapted form Fig. 8-9 in [7] ).

1.2.1.2 Fracture behavior of nacre

Regarding the fracture of nacre, it is important to remember that this material naturally has many flaws, like porosity and defects in growth. These flaws represent potential points where the cracks may appear and evolve, leading to the material’s catastrophic failure. There are no such mechanisms to avoid the flaws present in nacre. However, nature has developed a group of solutions to provide robustness to it. In this way, the cracks growing can be arrested and allow the material to resist a premature fracture (toughness). Among these mechanisms, the platelets sliding phenomenon and the appearance of interface voids seem to be the most important damage tolerant characteristics. It is worth to mention that the interaction of the mechanisms just mentioned, allows to nacre fracture toughness be 20 to 30 times higher than the mineral part. The intrinsic toughness value in nacre is about J0

= 0.3 kJ/m2, compared to the value found in pure aragonite J0 = 0.01 kJ/m2.

An interesting experimental approach was used by Barthelat and Espinosa, 2007 to determine the nacre’s crack resistance curve (JR). They found that with the advancing of

the crack, the slope of the JR curve rises. Therefore, the maximum toughness measured was

21

1.5 kJ/m2. This increment can be interpreted as the capacity of the material to arrests the crack propagation and resists the pre-existent flaws.

The increase of fracture toughness in a material like nacre, despite its high content of a bio-mineral with fragile behavior (aragonite), is probably associated with the formation of a process zone around the crack. This process zone consists in a inelastic region located ahead of the crack tip [11]. The main actors on this process zone are the crack face bridging and the appearance of new inelastic regions when the crack is evolving in nacre. These can be further explained using simple J-integrals calculations where JA = J0 + JB +

JW. Where JA is related with the loads application, JB is the crack bridging contribution and

JW the wake of the crack in the process zone [11]. Moreover, the JW act as extrinsic

toughening mechanisms, which shields the crack tip remotely. The following equations describe how these terms are obtained.

𝐽_𝐵 = ∫ 𝜎_𝑦𝑦(𝑢)𝑑(2𝑢), 2𝑢 0 𝐽𝑊 = 2 ∫ 𝑈(𝑦)𝑑𝑦, 𝑤 0

[Eq. 1, Bridging effect and process zone contribution in the J-integral (in [11])]

Regarding the JB term, the tablet pullout distance u introduces the stress that acts in the

interface σyy (u). This stress can be obtained from a nacre’s tensile stress-strain curve.

Furthermore, to obtain the JB value, it is only necessary to compute the area under this

curve. The resulting JB = 0.012 kJ/m2 shows that the nacre’s toughness is minimally

influenced by the crack bridging phenomenon.

Otherwise, when the JW is analyzed, w is half width of the wake, U(y) is the energy density

and y is the distance from the crack plane. In this analysis, U(y) comes from the energy dissipated by tension along the platelets. Finally, the value found was JB = 0.75 kJ/m2.

Therefore, the conclusion is that the inelastic zones that rise during the crack evolution are the main source of the nacre’s toughening characteristic. Moreover, it seems probable that the viscoelastic properties of the bio-polymer are a major player in the energy dissipation mechanisms. All these characteristics acting together, give nacre the capability of stabilizing the crack propagation under different circumstances.

22

Figure 1-3. Scheme of the steady state crack in nacre. The J-integrals contours are also shown (Fig 17 in [11])

Lastly, it is important to note that the accurate calculation of the energy released from the fracture process in nacre, requires an extensive understanding of the constituent’s material properties under multiaxial loading. This includes, for instance, the hysteresis behavior found in the bio-polymer [7].

All these remarkable characteristics have inspired in the last years a complete new generation of manmade composites materials [1], [12], [13]. Furthermore, the deep understanding of nacre’s fracture mechanisms and the several efforts made to develop this di-novo composites, will strength the importance of bio-inspired materials.

1.2.2 Bone

Bone is another example of a bio-composite that exhibits extraordinary mechanical properties, combining low weight with outstanding strength and toughness. Moreover, it must be highlighted that bone has a remarkable self-healing capability. Its mechanicals properties are mainly attributed to its complex hierarchical organization and the typical characteristics exhibited by its nature as a composite.

The properties of bone are the result of millions of years of evolution. The bone tissue provides structural support to a wide variety of animals. For this reason, bone and its mechanical properties are of special relevance to the physiological survival needs of these animals.

The basic constituents of bone (at nanoscale) are, on one hand, regular shaped platelets of hydroxyapatite (Ca10(PO4)6(OH)2) mineral crystals. These hydroxyapatite (HA) crystals

23

have a nanometer size of about 50 x 25 x 3 [nm]. On the other hand, collagen fibers are the second basic constituent. In this case, these collagen fibers are mainly collagen type-I molecules, with 300 [nm] in length and ~1.5 [nm] in diameter [3]. These fibers act as the soft matrix where the HA crystals are embedded, forming in this way the arrangement of mineralized collagen fibrils in a typical brick-and-mortar topology. Similarly, this typical shape can be found in nacre’s nanostructure.

Figure 1-4. Hierarchical multi-scale structure of bone, showing seven hierarchical levels, and their relation to the mechanisms of mechanical properties. The left panel shows the characteristic structural features; the right panel

indicates associated deformation and toughening (fracture resistance) mechanisms. (Fig.17 in [7])

The base hierarchical levels (level 0: ~ 1nm and level 1: ~ 300nm) are composed by the tropocollagen molecules, which consist in a triple helical arrangement of collagen fibers with nanometric inclusions of HA crystals. Bone is defined by the basic building blocks consisting of collagen proteins and the HA minerals. In level 2 (~1 µm), the mineralized collagen fibrils (MCFs) can be identified. Arrays of fibrils connected by a protein phase are in the third level (~10 µm) of bone’s hierarchical structure. The fourth level (~50 µm) is mainly done by fibril arrays configured in different kind of patterns: random, parallel, tilted or woven orientations. In the fifth level, (~100 µm) cylindrical structures called osteons are considered as the most relevant feature. These structures are also known as

24

Harversian systems. They have large vascular channels in the center and their boundaries are defined by the so called “cement lines”. These features play a key role in the bone’s growth and remodeling mechanism. The final levels in this hierarchical organization are the bigger features in size. While the sixth level (~50 cm) makes a distinction between the main macrostructures that can be found, trabecular(spongy) and cortical(compact) bone. Finally, the seventh level (~1 m) includes the overall shape of a bone in any animal [7]. In the present work, we have focused in the basic levels of this hierarchical structure, especially in the MCFs. The inclusion of the HA platelets in the collagen matrix introduced an important alteration in the response of this composite material. The mineralization of the collagen fibrils increases the material’s strength and it capability to dissipate energy when deformation loads are acting on it. Additionally, following the simplest rule of mixtures in composites, the stiffness in these fibrils is increased.

Previous studies reveal that the specific brick-and-mortar arrangement, and the materials properties of it building blocks, provide the MCFs special toughening mechanisms, leading to a damage-tolerant bio-composite. Following the same basic criteria of nacre’s nanostructure, the HA minerals (45% in volume fraction) here provide strength and stiffness, whereas the collagen phase (65% in volume fraction) is responsible for supporting large deformations. Among these mechanisms, we can name the formation of local yield regions, the breaking of sacrificial bonds, the microcracks formation, and crack bridging. The mechanisms that make bone an outstanding material are further analyzed in the next section.

Figure 1-5. Computational simulations comparing the stress-strain response of collagen fibril (blue line) and mineralized collagen fibril (red line). (Fig. 19 adapted from [7])

25 1.2.2.1 Stiffness in bone.

The dominant volume fraction in bone is occupied by the collagen soft matrix (about 60%). The collagen strength and stiffness properties are much lower than those found in bone and in the HA minerals. Despite this, thanks to the bone’s nanostructure properties, it can be stiffened with correctly dimensioned HA platelets (~ 40% in volume fraction). To this purpose, the platelets aspect ratio was introduced [6]. This value is obtained from the ratio between the length (~ 100 nm) and the thickness (~ 2 to 3 nm) of the HA crystals found in the MCFs. A simple scaling law (Equation 2) shows that the composite stiffness (E) rises when the shear modulus (Gp) of the polymer is increased, and/or when the aspect

ratio (ρ) rises (the latter in a quadratic relation).

𝐸 ∝ 𝐺_𝑝𝜌2 _{[Eq. 2 in [6]]}

Therefore, this simple relation shows the remarkable importance and influence of the aspect ratio in the stiffness of the MCFs. With a value of 30 to 40 in the HA crystals aspect ratio, the composite stiffness will reach an amplification of 3 orders of magnitude compared with the initial collagen fibrils stiffness. The entire stiffness model for bone proposed by Gao et al., 2003, is further explained in section 2.4.

1.2.2.2 Fracture behavior of bone.

The nanostructure of bone is very similar to that found in nacre. It consists in a staggered nanostructure (brick-and-mortar topology) where the hard HA minerals crystals are embedded in a soft matrix of collagen fibrils. At this level, these nanostructures are mainly subjected to uniaxial loads. Due to it specific geometry, these loads transfer in a tension-shear chain path. Therefore, the HA platelets are under tension, while the soft collagen matrix is primarily under shear stresses. An additional and similar characteristic with nacre is the main scope that the bone’s building blocks perform to increase the toughness of the material. On one hand, the collagen proteins dissipate energy, allowing the mineralized collagen fibrils to bear large deformations. On the other hand, the HA platelets give strength and stiffness to this composite. Moreover, these platelets consent a more even distribution of the loads acting in the MCFs.

The capability of bone to tolerate the presence of multiple flaws or cracks at different level of it hierarchical structure has been widely studied. This special characteristic gives bone

26

the capability of fail under a uniform rupture, instead of failing by the propagation of a preexisting crack. To this effect, nature has presented a clever solution in the nanostructure of bone, that has been achieved by size reduction. This means that under certain dimensions, the bone’s nanostructure does not fail by the influence of the propagation of a crack. An interesting analysis about the importance of nanoscale in biological materials is included by Gao et al., 2005. Using fracture mechanics concepts, they develop a simplified model to explain this issue. Finally, they found that when the mineral crystals were reduced to the nanometer size, these crystals became flaw tolerant.

Figure 1-6. Fracture model of bio composites. The stress-strain relation contributes to the fracture energy. The hard platelets remain intact as the soft matrix undergoes large shear deformation. The width of the fracture process is

assumed to be in the order of the length of the HA minerals. (Fig. 7 in [6])

To further describe the capacity of a bio composite like bone to dissipate fracture energy we use as a reference the model presented in [6]. They considered a crack growing in a composite with the typical staggered topology found in bone or nacre nanostructure. In the mentioned model they assume that the crack is confined in a specific strip of localized deformation where the evolution path of the crack is expected. Thus, the energy is calculated from the following integral (Equation 3), where w is the width of the strip and

σ(ε) is the stress-strain response of the MCFs under tension.

𝐽𝑐 = 𝑤 ∫ 𝜎 (𝜀)𝑑𝜀 [Eq. 3]

At these conditions, the failure of the composite results from pulling out of the HA crystals form the collagen matrix. Therefore, the crack propagation arises in the interfaces of the HA platelets, whose material is the collagen soft matrix. Furthermore, making a simple estimation where the width of the strip is equal to the length of the HA platelets, the

27

authors confirmed that the fracture energy is dissipated mainly by the role played by the soft matrix. Thus, the fracture energy (Equation 4) calculation changed to:

𝐽_𝑐 = (1 − 𝜙)𝐿 ∫ 𝜏𝑝𝑑 𝜀𝑝, 𝑤ℎ𝑒𝑟𝑒: ∫ 𝜏𝑝𝑑 𝜀𝑝= Θ𝑝min(𝑆𝑝, 𝑆𝑖𝑛𝑡, 𝑆𝑚/𝜌) [Eq. 4]

In the previous equation, the integral calculates the deformation energy dissipated by the protein matrix per unit of volume. In this case τp and εp are the protein shear stress and

shear strain respectively. While, S stands for the strength of the different components of the bio-composite. Consequently, Sp represents the protein strength, Sint the interface, and Sm

the mineral strength. Finally, Θp is the effective strain, and depends on the deformation

range reached by the protein.

Based in the equation’s interpretation, we can infer that, for bio-composites, the toughness could increase if the soft phase is increased in volume fraction. In this case, there is more protein available to dissipate fracture energy. Meanwhile the mineral crystals length act to avoid the strains to be concentrated in region where the crack tip is located. Thus, for larger HA platelets, the crack path in bone must be increased, resulting in a higher value of fracture energy needed to reach the failure of this bio-composite material.

Similar to what was described previously for nacre, proteins unfolding, and the slippage produced in the composite’s interface, are the main sources to increase the toughness of this kind of materials. However, in bone it seems to be an additional feature that probably enhances the strength of the protein-mineral interface. This mechanism consists in sacrificial bonds in which the calcium ions cross-link protein peptides with negative charges, increasing in this way the strength of the peptide backbone. Moreover, these sacrificial bonds introduce a solution raising the interface strength to similar values of the protein strength. The mechanism mentioned here, promotes the slippage o the mineral platelets and the interface deformation to occur under similar stress conditions. In this way, the deformation range of the collagen matrix can be optimized to increase the toughness of the MCFs. It is necessary to comply with the HA crystals nano-dimensions in the bone composite or otherwise the soft matrix deformation will cause the brittle fracture of these crystals, even before they reach the optimized deformation levels.

To summarize, the staggered topology of the MCFs plays an essential role in the fracture mechanism that take place in bone. Additionally, to this geometrical feature, the specific properties shown by the main components of the bone, lead to obtain this composite with

28

extraordinary fracture properties. First, the nano-size of the minerals crystals is highly important to obtain the flaw-tolerant composite. Moreover, these minerals provide additional strength and stiffness to the protein matrix. Besides, the collagen matrix undergoing large deformation values, while the composite does not suffer large strain effects, is able to dissipate high quantities of deformation energy in this manner. Additionally, the sacrificial bonds formed by the calcium ions, led to the optimization of the strength in the mineral-protein interface. The latter mechanism has an effect too in the higher stress repose of the soft matrix. All these mechanisms acting together provide bone its exceptional characteristics of strength and toughness.

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Chapter 2 : Numerical background

This chapter is focused on the core of the theoretical background. Here, the main concepts related to the materials and methods used throughout this work, such as hyperelasticity, and the eXtended Finite Elements Method (XFEM), will be addressed. Overall, these concepts are presented from the perspective of their implementation in Abaqus.

2.1 Hyperelastic materials

The correct configuration of the materials’ properties in Abaqus, is a key point to obtain valid results from the simulations implemented [14]. Through this work we dealt with two materials that were considered as the constituents for the realization of the nacre-like composite. However, between these materials, the hyperelastic polymer, known as TangoBlackPlus (TBP) was the one that required more attention during its configuration in the Abaqus environment. For this reason, through this thesis work, additional information about hyperelasticity will be presented. This section will bring to the reader deep understanding about the typical mechanical behavior of these kind of materials.

In continuum mechanics, the hyperelastic materials are analyzed as materials that sustain large deformations. To describe their behavior, a free-energy function is introduced. This function is defined per unit of reference of volume or mass. The strain energy function is known as the Helmoltz free-energy function (ψ). This function will only depend on the deformation gradient (F), when only homogeneous material is considered. Therefore, the function becomes the strain-energy function: ψ = ψ(F).

Since for the fabrication of this composite, the TBP, a rubber-like material, was used, we analyze here a specific kind of materials called isotropic hyperelastic materials. Typically, rubber is included as one of the most known materials in that class. The isotropy is a physical property that indicates the independence of the material’s stress-strain response in a specific direction. The following equation defines the condition that must be met, to define a material as isotropic:

𝜓(𝑭) = 𝜓(𝑭∗_{) = 𝜓(𝑭𝑸}𝑻_{) [Eq. 5 in [15]]}

Where, ψ(F) stands for the strain energy function, F and F* are the deformation gradients and Q refers to all the orthogonal vectors. Hence, the Equation 3, defines that the strain-energies are always the same for all the orthogonal vectors[15].

30

Moreover, in large deformation mechanics, it is possible to express the strain energy function in terms of its invariants (Equation 7), or, in terms of its principal stretches (Equation 8). This is only valid due to the isotropy condition (Equation 6), when its valid for all the symmetric tensors C (C: Cauchy-Green Tensor) and all the orthogonal tensors

Q.

𝜓(𝑪) = 𝜓(𝑭∗𝑇𝑭∗) = 𝜓(𝑭𝑇𝑸 𝑭𝑸𝑇) = 𝜓(𝑪∗_{) [Eq. 6]}

𝜓 = 𝜓 [𝐼₁(𝑪), 𝐼₂(𝑪), 𝐼₃(𝑪)] [Eq. 7] 𝜓 = 𝜓(𝐂) = 𝜓 [𝜆_1,𝜆_2,𝜆₃] [Eq. 8]

It results very common for practical applications, consider the hyperelastic material as incompressible. This is especially valid when these formulations are used for numerical models. The incompressibility of a material can be explained as its capability of not suffering volume changes, when is subjected to finite strains. Based on the nature of the TBP material, the incompressibility constraint (J = 1 or λ1 λ2 λ1 = 1) in the TBP material

was considered. A specific strain-energy function (Equation 9) arises when the incompressibility constraint is considered in the formulation:

𝜓 = 𝜓(𝑭) − 𝑝 (𝐽 − 1) [Eq. 9]

To obtain the material’s fundamental equations it is necessary to differentiate the strain-energy function with respect to the deformation gradient F. in terms of Piola-Kirchoff stresses, where P stands for the First Piola-Kirchoff stress tensor (Equation 10) and S for the Second Piola-Kirchoff stress tensor (Equation 11); and σ represents the Cauchy stress tensor (Equation 10). 𝑷 = −𝑝𝑷−𝑇₊ 𝜕𝜓(𝑭) 𝜕𝑭 [Eq. 10] 𝑺 = −𝑝𝑭−1𝑭−𝑇 + 𝑭−1 𝜕𝜓(𝑭) 𝜕𝑭 [Eq. 11] 𝜎 = −𝑝𝐼 + 𝜕𝜓(𝑭) 𝜕𝑭 𝑭 𝑇 _{[Eq. 12]}

Moreover, in the specific case that the isotropy and incompressibility conditions hold in the hyperelastic material, the strain energy function can be expressed as function of C and their invariants. Due to the before-mentioned conditions the third invariant is not anymore, an

31

independent variable. Thus, under these conditions the strain-energy function can be expressed as follows (Equation 11):

𝜓 = 𝜓 [𝐼₁(𝑪), 𝐼2(𝑪)] − 1 2⁄ 𝑝 (𝐼3− 1) [Eq. 13]

It is worth to mention, that the formulation here presented agrees with the general description for hyperelastic material behavior described in the Abaqus Theory Guide. Here, the definition of the constitutive behavior is based as total stress-strain relationship, rather than in the framework of history-dependent materials [16].

2.1.1 Models for isotropic hyperelastic materials.

During the configuration of the hyperelastic material in Abaqus, it is of remarkable importance, to count with the correct definition of the hyperelastic model that better describes the mechanical behavior of the TBP material. To this purpose, we described here the three models that better fitted the experimental tests data, which was previously obtained from the execution of uniaxial and biaxial tensile tests.

2.1.1.1 The Polynomial form.

The strain energy function at it polynomial form is represented in Abaqus as follows:

𝑈 = ∑

𝐶

(𝐼̅

− 3)

(𝐼̅

− 3)

+ ∑

(𝐽

− 1)

In the Equation 14, the polynomial representation can be used up to order 6th _{of N. Usually,}

the value of N is 2, when the first and the second invariants are considered in the hyperelastic model. In the previous equation, Jel stands for the elastic volume strain, while

Di is defined by the compressibility of the material. In case of incompressible material, this

value is zero. Finally, to obtain the bulk (k0) and the shear modulus (µ0) of the material, the

calculations are always made using N=1.

𝜇₀ = 2(𝐶₁₀+ 𝐶₀₁ ) [Eq. 15]

𝑘_{0 = 2 𝐷}

𝑖

⁄ [Eq. 16]

2.1.1.2 The Arruda-Boyce model

This model was proposed by Arruda and Boyce back in the 1993 [17]. They developed this relation on an eight-chain representation of springs, that went from the center of a representative cubic unit, towards its corners. In this physical representation all the chains

32

are subjected to the same general deformation state. The advantage that this model exhibits, is the ability to capture the deformation of the hyperelastic materials, requiring only two parameters to describe their behavior (the initial shear modulus and the limiting chain extensibility). Moreover, the Arruda-Boyce (AB) model can approximate the finite deformation, including the upturn effect typically found in rubber-like materials at higher strain levels. This feature is possible due to the non-Gaussian statistic theory used by the authors, to describe the individual behavior of the chains in the model.

𝑈 = 𝜇 ∑

(𝐼̅

_{− 3}

) + 1 𝐷

⁄ (

− ln 𝐽

)

𝑖=1 [Eq. 17 (Eq. from [16])]

The coefficients C1, C2, C3, C4, C5, [16] are well defined and came from the Langevin

function, which is approximated with a Taylor series expansion. Moreover, the term µ is the first key value in the AB expression, the initial shear modulus. Besides, λm stands for

the so called locking stretch. This coefficient represents the second key parameter in the AB equation. It represents the point where the spring (polymer) chain get locked[15]. When the locking stretch is reached the slope of the stress-strain curve rises in a remarkable way. The initial bulk modulus can be obtained as it was proposed for the Polynomial form in the Equation 14. Finally, the elongation of the eight chains is represented by the first invariant (

𝐼̅

In Abaqus, a simplified representation (it requires one material’s parameter) of the volumetric strain energy was added to the deviatoric part of the strain energy. In this way, the material’s parameters can be estimated in a simpler way.

2.1.1.3 The Ogden form

This strain-energy function arises from the vision of the author to provide a model which is able to reproduce the mechanical behavior of an hyperelastic material under a reasonable and amenable mathematical analysis [18].

In Abaqus the strain energy potential is expressed with the following formulation (Eq. 4.6.1.-14 in [16]):

𝑈 = ∑

(𝜆

+ 𝜆

_{+ 𝜆}

− 3) + ∑

(𝐽

− 1)