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Failure mechanisms of additively manufactured porous biomaterials: Effects of porosity and type of unit cell J. Kadkhodapour, H. Montazerian, A.Ch. Darabi, A.P. Anaraki, S.M. Ahmadi, A.A. Zadpoor, S. Schmauder

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S1751-6161(15)00205-2 http://dx.doi.org/10.1016/j.jmbbm.2015.06.012 JMBBM1506

To appear in: Journal of the Mechanical Behavior of Biomedical Materials

Received date:5 March 2015 Revised date: 7 June 2015 Accepted date: 13 June 2015 Cite this article as: J. Kadkhodapour, H. Montazerian, A.Ch. Darabi, A.P. Anaraki, S.M. Ahmadi, A.A. Zadpoor, S. Schmauder, Failure mechanisms of additively manufactured porous biomaterials: Effects of porosity and type of unit cell, Journal of the Mechanical Behavior of Biomedical Materials, http: //dx.doi.org/10.1016/j.jmbbm.2015.06.012 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Failure mechanisms of additively manufactured porous biomaterials: Effects of porosity and type of unit cell J. Kadkhodapoura,b*, H. Montazeriana, A. Ch. Darabic, A. P. Anarakia ,S. M. Ahmadid, A. A. Zadpoord, S. Schmauderb a b

Department of Mechanical Engineering, Shahid Rajaee Teacher Training University, Tehran, Iran

Institute for Materials Testing, Materials Science and Strength of Materials (IMWF), University of Stuttgart, Stuttgart, Germany c

d

Department of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran

Department of Biomechanical Engineering, Delft University of Technology, Mekelweg 2, Delft 2628CD, The Netherlands

*

Corresponding Author, Email: [email protected], Tel: +98-21-22970052 1

Abstract Since the advent of additive manufacturing techniques, regular porous biomaterials have emerged as promising candidates for tissue engineering scaffolds owing to their controllable pore architecture and feasibility in producing scaffolds from a variety of biomaterials. The architecture of scaffolds could be designed to achieve similar mechanical properties as in the host bone tissue, thereby avoiding issues such as stress shielding in bone replacement procedure. In this paper, the deformation and failure mechanisms of porous titanium (Ti6Al4V) biomaterials manufactured by selective laser melting from two different types of repeating unit cells, namely cubic and diamond lattice structures, with four different porosities are studied. The mechanical behavior of the above-mentioned porous biomaterials was studied using finite element models. The computational results were compared with the experimental findings from a previous study of ours. The Johnson-Cook plasticity and damage model was implemented in the finite element models to simulate the failure of the additively manufactured scaffolds under compression. The computationally predicted stress-strain curves were compared with the experimental ones. The computational models incorporating the Johnson-Cook damage model could predict the plateau stress and maximum stress at the first peak with less than 18% error. Moreover, the computationally predicted deformation modes were in good agreement with the results of scaling law analysis. A layer-by-layer failure mechanism was found for the stretch-dominated structures, i.e. structures made from the cubic unit cell, while the failure of the bending-dominated structures, i.e. structures made from the diamond unit cells, was accompanied by the shearing bands of 45°.

Keywords: Cellular biomaterials, Ti6Al4V, lattice structures, the Johnson-Cook damage model, selective laser melting, bone substitutes.

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1. Introduction The explorations of biomaterials have advanced tissue engineering scaffolds that can be replaced with damaged tissues while retaining biological activities in the absence of original living tissues. Currently, autograft and allograft are the most commonly used bone grafting techniques in orthopedic surgeries owing to their superior osteoconductive properties and ability to be integrated with the host bony tissue (Kuremsky, Schaller, Hall, Roehr, & Masonis, 2010; Poehling et al., 2005). Autologous bone grafts are osteogenic, osteoinductive, osteoconductive, and biocompatible (Keating & McQueen, 2001); however, they are not without their drawbacks as they are prone to postoperative pain and morbidity due to the extra procedures in donor sites. Furthermore, many treatments are needed to prevent disease transmission, graft failures, and laxity, especially in the case of allograft procedures (Mirelis et al.; Young Iii & Toth, 2006). Hence, many attempts have been made in tissue engineering to design substitutions as scaffolds made of biocompatible materials such as coralline hydroxyapatite (Mygind et al., 2007), collagen based materials (Offeddu, Ashworth, Cameron, & Oyen, 2015), calcium sulfate, and bioactive glasses to mechanically match the host tissue. Moreover, tissue engineering scaffolds are expected to provide suitable conditions for cell ingrowth, cell migration, and differentiation (Billström, Blom, Larsson, & Beswick, 2013). It is frequently claimed in the literature that the cell culture state of scaffolds can be well related to the fluid permeability of cellular architecture (Dias, Fernandes, Guedes, & Hollister, 2012; Syahrom, Abdul Kadir, Abdullah, & Öchsner, 2013; Truscello et al., 2012). Morphological parameters such as pore architecture, pore size, relative density, as well as mechanical properties of the base material have been found as the most effective factors for governing the mentioned requirements. Stochastic architectures have shown localized deformations owing to their internal imperfections (Cansizoglu, Harrysson, Cormier, West, & Mahale, 2008), while higher specific mechanical properties have been observed for non-stochastic geometries (Queheillalt & Wadley, 2005). Most of the studies on open cell non-stochastic cellular structures are focused on 3D lattice-based geometries such as diamond (S. M. Ahmadi et al., 2014), honeycomb (Ajdari, Jahromi, Papadopoulos, Nayeb-Hashemi, & Vaziri, 2012), octahedral (Sun, Yang, & Wang, 2012), rhombic dodecahedron (Horn 3

et al., 2014), tetrakaidecahedral (Zargarian, Esfahanian, Kadkhodapour, & Ziaei-Rad, 2014), and crystalline lattices (Karamooz Ravari, Kadkhodaei, Badrossamay, & Rezaei, 2014). Recently, many studies have utilized triply periodic minimal surfaces (TPMS) as a promising approach in scaffold designing owing to their unique mechanical and biological features in addition to their capacity in producing functionally gradient porous structures (Almeida & Bártolo, 2014; Yang, Quan, Zhang, & Tian, 2014; Yigil et al., 2013; Yoo, 2011). Melchels et al. (Melchels et al., 2010) compared cell seeding capability of stochastic pore architecture resulted from salt leaching with gyroid structure and reported more than 10-fold permeability improvement using gyroid structure. They observed better cell penetration in the center of the scaffold with gyroid structure, while the scaffolds in random architectures were covered with a cell sheet on the outside. Mechanical aspects of cellular materials with random and regular pore architectures were also assessed by Cansizoglu et al. (O. Cansizoglu, D. Cormier, O. Harrysson, H. West, & T. Mahale, 2006). They argued that irregular network geometries with randomized connectivity had a more similar mechanical behavior to the bone due to the decay of stiffness, especially when nodes were randomly disconnected. On the other hand, it is intuitively clear that increasing porosity leads to losing the strength of scaffolds, while it enhances situations for cell ingrowth and nutrient transformation in addition to helping prevent stress shielding which affects the longevity of the implant (Parthasarathy, Starly, Raman, & Christensen, 2010). Furthermore, increasing the cell size of porous network has been observed to decrease mechanical strength and stiffness (Yan et al., 2014; Yan, Hao, Hussein, & Raymont, 2012). Hence, a compromise is needed between strength, stiffness, permeability, and mechanical properties of the base material in the designing procedure of tissue engineering scaffold. Giving this scenario, additive manufacturing (AM) techniques are increasingly developed due to their controllability on material, internal architecture, and consequently mechanical and biological responses of scaffolds. Among the biocompatible materials such as tantalum, chrome, cobalt, and stainless steel (Parthasarathy et al., 2010), medical grade titanium alloys, namely Ti6Al4V, has been commercially advanced for fabricating implants by many methods such as melt processing, powder processing, and vapor deposition (Harrysson, Cansizoglu, Marcellin-Little, Cormier, & West Ii, 2008). Moreover, 4

producing titanium implants by additive manufacturing is progressively developed, since they are biologically inert and provide adequate osteointegration properties in addition to exhibiting high mechanical strength and good corrosion resistance (Jamshidinia, Wang, Tong, & Kovacevic, 2014; Wieding, Souffrant, Mittelmeier, & Bader, 2013). However, elastic modulus of titanium is almost 114GPa, while for cortical bone, it ranges from 0.5 to 20GPa (Wieding, Wolf, & Bader, 2014) which implies adequate morphological manipulation of pores to match the host tissue. Selective laser melting (SLM) is an additive manufacturing process in which powder particles with the sizes below 40mm are melted selectively to produce solid structures with the materials having melting point of less than the temperature provided by laser, such as titanium, stainless steel, and nickel-based super alloys (McKown et al., 2008). The quality of the produced materials can be controlled by powder, melting and processing parameters, and geometry itself (Gorny et al., 2011). Furthermore, it allows manufacturing fully dense metal parts without any need for post-processes such as infiltration, sintering, and hot isostatic pressing (Yan et al., 2012). Brenne et al. (Brenne, Niendorf, & Maier, 2013) revealed that post-SLM heat treatment can improve ductility in monotonic loading response of Ti6Al4V samples. Moreover, they reported enhanced energy absorption by 4 point bending test and increased fatigue life after the post-heat treatment of the samples. Smith et al. (Smith, Guan, & Cantwell, 2013) evaluated finite element models by comparing their results with the experimental data of compression test on the lattice materials produced by SLM technique. They modeled lattice models by beam and brick elements and concluded that key mechanical properties of large lattices composed of a huge number of elements can be well predicted by investigating their constitutive unit cell. Moreover, the differences between numerical and experimental results were reduced considering the diameter variations of struts by reverse engineering of unit cell. With this picture in mind, the cubic and diamond lattice structures produced by SLM technique were considered in this paper and their mechanical response was evaluated under quasi-static loading conditions and compared at the relative densities of 11, 22, 28, and 35% to discuss pore architecture effects on their compressive behavior. In order to get insight into mechanical predictability of Ti6Al4V 5

lattice structures, the Johnson-Cook damage model was studied through finite element analysis. Furthermore, the relationships between deformation and mechanical properties of cellular structures were investigated by scaling laws analysis of specific Young's modulus obtained in the experimental procedure and the numerical deformation mechanism was compared with the results of scaling laws analysis on the experimental Young's modulus and compressive strength.

2. Materials and Methods 2.1. Experimental Procedure In order to study the relationships between deformation mode and mechanical properties, the experimental data of a previous work (Seyed Mohammad Ahmadi et al., 2015) for the samples with different deformation mechanisms, namely cubic and diamond, were utilized in this paper. Selective laser machines (Layerwise, Kueven, Belgium) were used to build the porous structures. The laser processing parameters and manufacturing procedures were similar to the ones used in our previous studies (Amin Yavari, Ahmadi, et al., 2014; Amin Yavari, van der Stok, et al., 2014; Amin Yavari et al., 2013). As previously stated, Ti6Al4V ELI powder (grade 23 according to ASTM F136) was used to produce lattice structures by laser melting technology. CAD models of the constitutive unit cell for each lattice geometry was produced and the block structures were generated to achieve cellular structures with approximately 11, 22, 28, and 35% volume fractions. Geometry of unit cells by which the cubic and diamond lattices are produced, is illustrated in Fig. 1(a). Different values of porosity were generated by changing the diameter of struts in lattice structures. Geometrical characterization of the structures is presented in Table 1. CAD models were imported to the additive manufacturing machine and cylindrical lattice samples were constructed on top of a solid titanium substrate with the rough length of 15 mm and diameter of 10 mm. Then, the samples were removed from the substrate using electrical discharge machining (EDM). The quality and micro-structure of the parent material could potentially influence the mechanical properties of the resulting porous structures. In our previous studies (Amin Yavari et al., 2013), we found the microstructure of the parent material to be typical of the Ti-6Al-4V alloy. Moreover, the 6

microstructure was largely similar for porous structures with different porosities. The parent material was of high quality as indicated by the very small percentage of pores (see e.g. Figure 5 in (Amin Yavari, Wauthle, et al., 2014)) in the parent material. High density of the parent material is crucial when studying the mechanical properties of additively manufactured porous biomaterials and that is why, in our experimental studies, we accept the parent material to be of acceptable quality only when its density well exceeds 90% (typical values of density >99%)” To evaluate the mechanical response of lattice structures, INSTRON 5985 mechanical testing machine (100 kN load cell) was used to perform compressive tests under displacement control mode with the deformation rate of 1.8 mm/min. Compressive tests were repeated for 5 times for each of the samples and the average value of the resulted stress-strain curves was considered for interpreting compressive characteristics of the samples. All the tests were performed with standard methods for porous metallic materials (ISO, 2011). In order to idealize the stress-strain curves, modulus of elasticity was considered as the slope of linear fits to the experimental data up to 0.02 strain. Further, yield stress was computed by intersecting the curve with 0.002 offset line parallel to elastic region and plateau stress was calculated as the average value of stress-strain curve from the onset of plateau region. Stress at the first peak of the stress-strain curve was also attributed to σ max . More details of the experimental procedure and results were reported elsewhere (Seyed Mohammad Ahmadi et al., 2015). The porous structures were scanned using micro computed tomography (µ-CT). The µ-CT images were then segmented and used to evaluate the architecture of the porous structures using algorithms that calculated porosity, strut thickness, and pore size. The details of the scanning protocol, segmentation algorithms, and algorithms used for calculation of the structural parameters could be found in our previous studies (Seyed Mohammad Ahmadi et al., 2015; Amin Yavari, van der Stok, et al., 2014; Amin Yavari et al., 2013).

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2.2. Simulation and Modeling Procedure In order to investigate the ability of the Johnson-Cook model in terms of predicting the mechanical behavior of lattice structures, the 3D FE analysis quasi-static simulations were performed using FE software. First, the CAD models of cubic and diamond lattice structures were constructed for the analysis and imported into ABAQUS/CAE. Then, lattice structures were meshed with tetrahedral elements (C3Dd) automatically. It is worth mentioning that cell size effect (or sample size to cell size ratio) may lead to high deviations of numerical results from experimental data, especially at higher strains. However, since cellular materials generally produce a massive amount of output files, extremely high process cost is resulted during the early analysis. A previous work (Smith et al., 2013) studied cell size effect and stated that many properties could be well predicted by simulating lower amounts of unit cells of non-stochastic cellular materials, rather than the whole structure. Hence, in order to prevent high CPU times for solving numerical model, 125 unit cells (5×5×5 unit cells patterned along three global coordinates) of lattice structures were modeled and computationally analyzed. On the other hand, as far as failure behavior of the structure is concerned, mesh sensitivity plays a key role in the accuracy of finite element model due to the high sensitivity of results to damage parameters. Thus, mesh sensitivity analysis was carried out in order to find the suitable mesh size in finite element analysis. The samples with almost 22% volume fraction was considered for mesh sensitivity analysis and CAD models of cubic porous structures with 100µm composed of 5×5×5 unit cells were discretized by tetrahedral elements with the global seed sizes of 7, 3, and 2 µm, as illustrated in Fig. 2. For all of the models, an elastic-plastic model was considered according to the experimental results obtained by (Cain, Thijs, Van Humbeeck, Van Hooreweder, & Knutsen, 2015). The elastic modulus of 110Gpa and Poisson's ratio of 0.3 was set as elastic properties and plastic stresses and strains were defined as shown in Fig. 3. Finally, corresponding to each model, compressive displacement was defined as boundary conditions with the strain rate matching experimental procedure and ABAQUS explicit code was employed to solve the FEM problems.

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2.3. Computational Approach It is obvious that material properties play an important role in the numerical investigation of mechanical properties for lattice structures, especially at higher strains. In this paper, the material properties of Ti6Al4V were set according to those reported in (Garciandia, 2009). In order to evaluate damage in lattice structures, the Johnson-Cook model was employed with the same approach as in (Johnson & Cook, 1985). In this failure model, after an element exceeds the failure limit, it will be removed from the structure. The fracture strain-based Johnson-Cook damage model is described as:

ε failure = ( D1 + D2 exp ( D3σ

*

)


*   1 + D4ln(ε )  [1 + D5T ]  

where D1 , D2 , D3 , D4 , and D5 are five material constants in the Johnson-Cook damage model,

D1 , D2 , and D3 are damage parameters related to the relationships between failure strain and stress triaxiality, D4 and D5 depend on strain rate and temperature, respectively, and σ* is stress triaxiality defined as hydrostatic stress ( σ h ) to the equivalent stress ( σ q ) ratio:

σ* =

σh σq

Calculating material properties and damage constants of lattice structures (stress-strain curve) is accompanied by difficulties since the need to frequent and different tensile tests on structures with different diameters. Since tests were performed at room temperature with the same strain rates, D4 and

D5 were ignored. Moreover, the values of D1 , D2 , and D3 were selected in terms of their fit to experimental data; consequently, the values of -0.68, 0.73, and -0.25 were found to well agree with the experimental data, respectively.

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3. Results and Discussion 3.1. Mesh Sensitivity Analysis In order to ensure convergence in finite element results, a mesh study was performed considering three seed sizes of 2, 3, and 7 µm to mesh the cubic cellular structures with the length of 100µm. Stress-strain curves corresponding to the defined seed sizes are indicated in Fig. 4. As can be seen, convergence was found between the mesh sizes of 3 and 2µm. Hence, due to the time consuming run of the models with 2µm mesh size, 3µm was set for the seed size of the described lattice models; thereby, almost 270000, 320000, and 280000 mesh numbers were resulted corresponding to 3µm mesh size for 22% volume fraction of cubic and diamond structures, respectively.

3.2. Characterizing Deformation Modes by Scaling Law Analysis of Experimental Data In order to compare effect of lattice morphology on compressive behavior of scaffolds, stress-strain curves of experimental compression test for the cubic and diamond lattice structures corresponding to four volume fractions of almost 11, 22, 28, and 35% are separately represented in Fig. 5. Stress-strain curves for all the specimens followed the same trend as that of typical cellular materials, in which after elastic regime, energy absorption was accomplished during plateau region up to the onset of densification. Cellular materials may go along with strain hardening or strain softening at the onset of plateau regime (Li, 2006). In the case of lattice structures characterized in this paper, according to the stress fall-downs after the first peak of stress at the end of the elastic regime, post-yield strain softening behavior was observed for all the specimens with different relative densities and lattice patterns. The obtained values from the idealization of experimental stress-strain curves are presented in Table 2. It should be noted that densification was not seen for all the specimens. Occurrence of densification depends on deformation and failure mechanism during crushing at higher strains. Lack of densification and consequently terminal hardening are expected in the cases in which deformation is accompanied by material separation and highly brittle failure of relatively thin struts. Comparison of collapse behavior from stress-strain curves showed that the set of the samples with almost 11 and 22% relative densities did 10

not totally meet densification up to 0.5 strain, while the onset of densification can be observed for the cubic and diamond lattice structures with 28 and 35% volume fractions. Moreover, the variations of stress during plateau region could be attributed to more trend to brittle failure of internal struts. Hence, comparing the compressive behavior of specimens in plateau regime can provide an understanding of failure mechanism of the samples. Monotonicity of stress in plateau was seen to be higher at lower relative densities for both cubic and diamond samples. Generally, increasing relative density as a result of increasing the diameters of struts leads to more stress drop due to the internal failure of material and explains the increased stress variations at higher volume fractions. Following the same approach as Ashbey's (Ashby, 2006), specific mechanical properties of cellular materials can be described by scaling analysis relative to density. Variation of normalized Young's n

ρ  E = C   ). Exponential values modulus is reported to be exponentially in relation with density as ( Es  ρs  of power fits to data were found to be 0.904 and 1.658 for the cubic and diamond structures, correspondingly. The obtained power trend was in good agreement with the one attributed to the structures with stretch dominated and bending dominated deformation, since the exponential value for the structure with stretch dominated deformation, namely cubic structure, approached 1, while the ones in which bending was dominated on the deformation, it tended to 2, which can be also described by the geometry of lattices. The orientation of struts in cubic lattice was all parallel and perpendicular to loading direction, while in the case of diamond, bending of internal struts was expected owing to their inclination relative to loading direction.

3.3. Validating Simulation Results by Experimental Data Computational stress-strain curves for the cubic and diamond structures corresponding to four volume fractions with the Johnson-cook damage model were compared with the experimental data in Fig. 6. In all of the simulations, the same material property was assigned to each lattice structure. Finite element results in the elastic regime were found to be in good agreement with the experimental data. 11

Mechanical characteristics of the models obtained by finite element analysis of the Johnson-Cook plasticity model are presented in Table 1 and compared with the experimental results. As stated in previous studies, a considerable amount of discrepancy can be explained by variations in diameter during SLM processes. Although yield strength as well as first peak of stress was in good agreement with the experimental data for all the samples, computational yield strain was seen to be higher than the experimental results, which can be attributed to cell size effect as a higher number of unit cells was tested in the experimental procedure, while 125 unit cells were modeled for FE analysis. In the plastic regime, although the amplitude of stress variations in numerical results was higher than the experimental data, plateau stresses (average value of stress in plateau regime) were in good agreement with the one obtained in the experimental results. The results in Table 2 demonstrate that the Johnson-Cook plasticity model could properly predict maximum stress ( σ max ) and plateau stress with relative errors of less than 18%. Elastic modulus was predicted with relative errors of less than 22% for diamond structure, while in the case of cubic structure, it was in the range of 28 to 117%. In general, it can be observed by comparing the computational and experimental data that the Johnson-cook damage model could properly predict the mechanical characteristics of non-stochastic lattice structures at higher strains. In addition, those lattice structures with a huge number of unit cells can be predicted considering lower unit cells without losing accuracy. On the other hand, the nominal (design) values of the structural parameters of the porous structures were quite close to the experimental values of the structural parameters determined using µ-CT (Table 1). This shows the high degree of fidelity of the actual porous structure to the CAD files used for manufacturing. It is therefore expected that finite element models, which are based on the nominal values of the structural parameters, are reasonably good representations of the actual porous structure. There are, nevertheless, some deviations from the nominal values that could potentially influence the finite element results. It is therefore recommended that the actual porous structure based on reconstructions of µ-CT images be used in the future studies so as to improve the accuracy of finite element simulations.

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3.4. Interpreting Failure Mechanism Whereas different volume fractions of each lattice structure were deformed with the same failure mechanism, deformation procedure is presented by representing plastic strain contours for the cubic and diamond with almost 22% volume fraction at different strains in Fig. 7 and Fig. 8, respectively. In order to illustrate deformation mechanisms and show the relationships between deformation and stress-strain curves, figures are addressed at their corresponding strain in computational stress-strain curves. Failure mechanism of the cubic lattice structure in elastic region was accompanied by the uniform deformation of vertical struts (Fig. 7(a)). Then, deformation was observed to be followed by a layer-bylayer deformation up to the total collapse of layers such that the first upper layer was crushed initially (Fig. 7(b)) and the others were collapsed in order up to the end of deformation (Fig. 7(d)). Stress fluctuations are explained by failure of each layer thanks to the buckling of micro-struts such that each stress drop corresponds to failure at a specific stage of failure. Since struts withstand axial deformation owing to their parallel direction relative to loading, higher variations were seen owing to the high release of energy as each stage of collapse. It shows that higher load bearing capacity can be expected when deformation is accompanied by buckling of micro-struts. Scaling law analysis of normalized Young's modulus as well as yield strength was confirmed by the mechanism of failure, as can be seen for stretch dominated deformation structure. As previously stated in (Kadkhodapour, Montazerian, & Raeisi, 2014), Diamond lattice structure is formed of 45° oriented plates placed parallel to sides of an internal pyramid which are connected with 45° struts at their joints. At the onset of plastic deformation (Fig. 8(a)), tying struts initiated to bend and resulted in the layer-by-layer crushing of plates in which central plates, namely diagonal plates, were crushed initially and led to the development of a continuous shear band of 45° around the model (Fig. 8(b)). Subsequently, deformation went along with crushing the outer plates as a result of bending -45° tying struts (Fig. 8(c)). As shown in the stress-strain curve, stress drops corresponding to each collapse of layers were discernible. Failure in diamond lattice geometry was seen to be accompanied by shearing of the micro-struts leading the scaffold to tend to bending dominated deformation. It should be noted that the 13

amplitude of stress variation in plateau region was found to be lower than cubic structure. Hence, it is deduced that the less energy is released at each stage of failure when shearing failure mechanism as a result of strut inclination compared with that observed in scaffolds with buckling failure mechanism. State of deformations for the diamond structures was also found to be in good agreement with the exponential value of scaling laws, since bending of struts plays a key role in the deformation mechanism of such structures. Generally, it can be inferred from the results that the micro-strut orientation plays crucial role in deformation mechanism and consequently in stress strain behavior of scaffolds. When the struts are design to be placed parallel to loading direction, buckling is excepted resulting the structure to experience stretch dominated deformation behavior while the more inclination of micro-struts the more shearing failure is observed in the whole of the structure. In this case, bending deformation mode dominate the compressive behavior of scaffold. As well, when bending deformation is dominated on scaffolds, the less specific mechanical properties is expected compared with the stretch dominated structures.

4. Conclusions This study examined the Johnson-Cook plasticity and damage model in terms of predicting mechanical response and deformation procedure of Ti6Al4V lattice structures produced by selective laser melting. Lattice structures with cubic and diamond geometries were modelled with the diameters such that four volume fractions of 11, 22, 28, and 35% were achieved corresponding to each lattice structure. Finite element models were prepared by repeating 5 unit cells along three global coordinates and consequently the lattice structures composed of 125 unit cells were achieved. Subsequently, cellular structures with almost 22% volume fraction were selected to study their convergence in finite element analysis. In order to analyze the deformation mechanism of the samples, scaling analysis on specific Young's modulus of the experimental results was performed and compared with the deformation procedure achieved by finite element analysis of the Johnson-Cook damage model. Cubic lattice as a structure with stretch dominated deformation represented higher specific mechanical properties; hence, deformation 14

mechanism was found to illustrate the mechanical properties of materials. The exponential value of power fit to experimental Young's modulus for the cubic and diamond structure was found to be 0.904 and 1.658, which implied the domination of stretching and bending deformation, respectively. These results of power fits to data were in good agreement with the deformation mechanism of lattices, since for bending dominated structure (diamond), deformation was accompanied by the shearing band of 45°, while layer-by-layer failure was seen for the cubic structure with stretch dominated deformation. Approaches such as scaling laws analysis or Maxwell criterion have been previously introduced to examine the deformation state of lattices and characterize mechanical properties. However, they are developed to show if bending is dominated on the deformation of a specific architecture or stretching. Hence, development of methods in which more details regarding deformation mode are quantitatively provided can improve understanding the effect of deformation and failure mechanism on mechanical characteristics of cellular materials, since structures with stretch dominated deformation are shown to provide more specific stiffness and strength, making them appropriate for structural applications, while bending dominated structures are reported to be more applicable for energy absorption application. In this study, two different lattice structures were studied and the deformation mechanism of each one was investigated. The topology and shape of the structures as a source of difference were examined in detail. Although more detailed investigations have to be done on the material property and damage mechanism in terms of considering geometries, production process, etc. Comparing computational stress-strain curves with the experimental ones illustrated good ability of the Johnson-Cook damage model in predicting stress at the first peak as well as plateau stress (with less than 18% relative error). It can be concluded that there is a potential to predict mechanical behavior of structures with a huge number of unit cells by modeling their constitutive unit cells to prevent current restrictions for solving large models. Using simulation methods, a clearer picture was derived for the deformation pattern and failure mechanism of porous lattice structures of different geometries. It was also shown that the geometry as a source of difference in mechanical properties can play a role in the dominant deformation pattern of structure. 15

References Ahmadi, S. M., Campoli, G., Amin Yavari, S., Sajadi, B., Wauthle, R., Schrooten, J., . . . Zadpoor, A. A. (2014). Mechanical behavior of regular open-cell porous biomaterials made of diamond lattice unit cells. J Mech Behav Biomed Mater, 34(0), 106-115. doi: http://dx.doi.org/10.1016/j.jmbbm.2014.02.003 Ahmadi, S. M., Yavari, S. A., Wauthle, R., Pouran, B., Schrooten, J., Weinans, H., & Zadpoor, A. A. (2015). Additively Manufactured Open-Cell Porous Biomaterials Made from Six Different Space-Filling Unit Cells: The Mechanical and Morphological Properties. Materials, 8(4), 18711896. Ajdari, A., Jahromi, B. H., Papadopoulos, J., Nayeb-Hashemi, H., & Vaziri, A. (2012). Hierarchical honeycombs with tailorable properties. International Journal of Solids and Structures, 49(11–12), 1413-1419. doi: http://dx.doi.org/10.1016/j.ijsolstr.2012.02.029 Almeida, H. A., & Bártolo, P. J. (2014). Design of tissue engineering scaffolds based on hyperbolic surfaces: Structural numerical evaluation. Med Eng Phys, 36(8), 1033-1040. doi: http://dx.doi.org/10.1016/j.medengphy.2014.05.006 Amin Yavari, S., Ahmadi, S. M., van der Stok, J., Wauthle, R., Riemslag, A. C., Janssen, M., . . . Zadpoor, A. A. (2014). Effects of bio-functionalizing surface treatments on the mechanical behavior of open porous titanium biomaterials. J Mech Behav Biomed Mater, 36(0), 109-119. doi: http://dx.doi.org/10.1016/j.jmbbm.2014.04.010 Amin Yavari, S., van der Stok, J., Chai, Y. C., Wauthle, R., Tahmasebi Birgani, Z., Habibovic, P., . . . Zadpoor, A. A. (2014). Bone regeneration performance of surface-treated porous titanium. Biomaterials, 35(24), 6172-6181. doi: 10.1016/j.biomaterials.2014.04.054 Amin Yavari, S., Wauthle, R., Böttger, A. J., Schrooten, J., Weinans, H., & Zadpoor, A. A. (2014). Crystal structure and nanotopographical features on the surface of heat-treated and anodized porous titanium biomaterials produced using selective laser melting. Applied Surface Science, 290(0), 287-294. doi: http://dx.doi.org/10.1016/j.apsusc.2013.11.069 Amin Yavari, S., Wauthle, R., van der Stok, J., Riemslag, A. C., Janssen, M., Mulier, M., . . . Zadpoor, A. A. (2013). Fatigue behavior of porous biomaterials manufactured using selective laser melting. Materials Science and Engineering: C, 33(8), 4849-4858. doi: http://dx.doi.org/10.1016/j.msec.2013.08.006 Ashby, M. F. (2006). The properties of foams and lattices. Philos Trans A Math Phys Eng Sci, 364(1838), 15-30. doi: 10.1098/rsta.2005.1678 Billström, G. H., Blom, A. W., Larsson, S., & Beswick, A. D. (2013). Application of scaffolds for bone regeneration strategies: Current trends and future directions. Injury, 44, Supplement 1(0), S28S33. doi: http://dx.doi.org/10.1016/S0020-1383(13)70007-X Brenne, F., Niendorf, T., & Maier, H. J. (2013). Additively manufactured cellular structures: Impact of microstructure and local strains on the monotonic and cyclic behavior under uniaxial and bending load. Journal of Materials Processing Technology, 213(9), 1558-1564. doi: http://dx.doi.org/10.1016/j.jmatprotec.2013.03.013 Cain, V., Thijs, L., Van Humbeeck, J., Van Hooreweder, B., & Knutsen, R. (2015). Crack propagation and fracture toughness of Ti6Al4V alloy produced by selective laser melting. Additive Manufacturing, 5(0), 68-76. doi: http://dx.doi.org/10.1016/j.addma.2014.12.006 Cansizoglu, O., Harrysson, O., Cormier, D., West, H., & Mahale, T. (2008). Properties of Ti–6Al–4V non-stochastic lattice structures fabricated via electron beam melting. Materials Science and Engineering: A, 492(1–2), 468-474. doi: http://dx.doi.org/10.1016/j.msea.2008.04.002 Dias, M. R., Fernandes, P. R., Guedes, J. M., & Hollister, S. J. (2012). Permeability analysis of scaffolds for bone tissue engineering. J Biomech, 45(6), 938-944. doi: http://dx.doi.org/10.1016/j.jbiomech.2012.01.019 16

Garciandia, L. E. (2009). Characterization of rapid prototyped Ti6Al4V bone scaffolds by the combined use of micro-CT and in situ loading. Carlos III De Madrid, Leganés. Gorny, B., Niendorf, T., Lackmann, J., Thoene, M., Troester, T., & Maier, H. J. (2011). In situ characterization of the deformation and failure behavior of non-stochastic porous structures processed by selective laser melting. Materials Science and Engineering: A, 528(27), 7962-7967. doi: 10.1016/j.msea.2011.07.026 Harrysson, O. L. A., Cansizoglu, O., Marcellin-Little, D. J., Cormier, D. R., & West Ii, H. A. (2008). Direct metal fabrication of titanium implants with tailored materials and mechanical properties using electron beam melting technology. Materials Science and Engineering: C, 28(3), 366-373. doi: http://dx.doi.org/10.1016/j.msec.2007.04.022 Horn, T. J., Harrysson, O. L. A., Marcellin-Little, D. J., West, H. A., Lascelles, B. D. X., & Aman, R. (2014). Flexural properties of Ti6Al4V rhombic dodecahedron open cellular structures fabricated with electron beam melting. Additive Manufacturing, 1–4(0), 2-11. doi: http://dx.doi.org/10.1016/j.addma.2014.05.001 ISO. (2011). Mechanical testing of metals -- Ductility testing -- Compression test for porous and cellular metals (Vol. 13314). Jamshidinia, M., Wang, L., Tong, W., & Kovacevic, R. (2014). The bio-compatible dental implant designed by using non-stochastic porosity produced by Electron Beam Melting® (EBM). Journal of Materials Processing Technology, 214(8), 1728-1739. doi: 10.1016/j.jmatprotec.2014.02.025 Johnson, G. R., & Cook, W. H. (1985). Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Engineering Fracture Mechanics, 21(1), 31-48. doi: http://dx.doi.org/10.1016/0013-7944(85)90052-9 Kadkhodapour, J., Montazerian, H., & Raeisi, S. (2014). Investigating internal architecture effect in plastic deformation and failure for TPMS-based scaffolds using simulation methods and experimental procedure. Materials Science and Engineering: C, 43(0), 587-597. doi: http://dx.doi.org/10.1016/j.msec.2014.07.047 Karamooz Ravari, M. R., Kadkhodaei, M., Badrossamay, M., & Rezaei, R. (2014). Numerical investigation on mechanical properties of cellular lattice structures fabricated by fused deposition modeling. International Journal of Mechanical Sciences, 88(0), 154-161. doi: http://dx.doi.org/10.1016/j.ijmecsci.2014.08.009 Keating, J. F., & McQueen, M. M. (2001). Substitutes for autologous bone graft in orthopaedic trauma. J Bone Joint Surg Br, 83(1), 3-8. Kuremsky, M. A., Schaller, T. M., Hall, C. C., Roehr, B. A., & Masonis, J. L. (2010). Comparison of Autograft vs Allograft in Opening-Wedge High Tibial Osteotomy. The Journal of Arthroplasty, 25(6), 951-957. doi: http://dx.doi.org/10.1016/j.arth.2009.07.026 Li, Q. M. (2006). Compressive Strain at the Onset of Densification of Cellular Solids. Journal of Cellular Plastics, 42(5), 371-392. doi: 10.1177/0021955x06063519 McKown, S., Shen, Y., Brookes, W. K., Sutcliffe, C. J., Cantwell, W. J., Langdon, G. S., . . . Theobald, M. D. (2008). The quasi-static and blast loading response of lattice structures. International Journal of Impact Engineering, 35(8), 795-810. doi: 10.1016/j.ijimpeng.2007.10.005 Melchels, F. P. W., Barradas, A. M. C., van Blitterswijk, C. A., de Boer, J., Feijen, J., & Grijpma, D. W. (2010). Effects of the architecture of tissue engineering scaffolds on cell seeding and culturing. Acta Biomaterialia, 6(11), 4208-4217. doi: http://dx.doi.org/10.1016/j.actbio.2010.06.012 Mirelis, J. G., García-Pavía, P., Cavero, M. A., González-López, E., Echavarria-Pinto, M., Pastrana, M., . . . Escaned, J. Magnetic Resonance for Noninvasive Detection of Microcirculatory Disease Associated With Allograft Vasculopathy: Intracoronary Measurement Validation. Revista Española de Cardiología (English Edition)(0). doi: http://dx.doi.org/10.1016/j.rec.2014.07.030 Mygind, T., Stiehler, M., Baatrup, A., Li, H., Zou, X., Flyvbjerg, A., . . . Bünger, C. (2007). Mesenchymal stem cell ingrowth and differentiation on coralline hydroxyapatite scaffolds. Biomaterials, 28(6), 1036-1047. doi: http://dx.doi.org/10.1016/j.biomaterials.2006.10.003 17

O. Cansizoglu, D. Cormier, O. Harrysson, H. West, & T. Mahale. (2006). An Evaluation of NonStochastic Lattice Structures Fabricated Via Electron Beam Melting. Paper presented at the Proceedings of the Solid Freeform Fabrication Symposium, Austin, Texas. Offeddu, G. S., Ashworth, J. C., Cameron, R. E., & Oyen, M. L. (2015). Multi-scale mechanical response of freeze-dried collagen scaffolds for tissue engineering applications. J Mech Behav Biomed Mater, 42(0), 19-25. doi: http://dx.doi.org/10.1016/j.jmbbm.2014.10.015 Parthasarathy, J., Starly, B., Raman, S., & Christensen, A. (2010). Mechanical evaluation of porous titanium (Ti6Al4V) structures with electron beam melting (EBM). J Mech Behav Biomed Mater, 3(3), 249-259. doi: http://dx.doi.org/10.1016/j.jmbbm.2009.10.006 Poehling, G. G., Curl, W. W., Lee, C. A., Ginn, T. A., Rushing, J. T., Naughton, M. J., . . . Smith, B. P. (2005). Analysis of Outcomes of Anterior Cruciate Ligament Repair With 5-Year Follow-up: Allograft Versus Autograft. Arthroscopy: The Journal of Arthroscopic & Related Surgery, 21(7), 774.e771-774.e715. doi: http://dx.doi.org/10.1016/j.arthro.2005.04.112 Queheillalt, D. T., & Wadley, H. N. (2005). Cellular metal lattices with hollow trusses. Acta Materialia, 53(2), 303-313. Smith, M., Guan, Z., & Cantwell, W. J. (2013). Finite element modelling of the compressive response of lattice structures manufactured using the selective laser melting technique. International Journal of Mechanical Sciences, 67, 28-41. doi: 10.1016/j.ijmecsci.2012.12.004 Sun, J., Yang, Y., & Wang, D. (2012). Mechanical Properties of Ti-6Al-4V Octahedral Porous Material Unit Formed by Selective Laser Melting. Advances in Mechanical Engineering, 2012, 1-11. doi: 10.1155/2012/427386 Syahrom, A., Abdul Kadir, M. R., Abdullah, J., & Öchsner, A. (2013). Permeability studies of artificial and natural cancellous bone structures. Med Eng Phys, 35(6), 792-799. doi: http://dx.doi.org/10.1016/j.medengphy.2012.08.011 Truscello, S., Kerckhofs, G., Van Bael, S., Pyka, G., Schrooten, J., & Van Oosterwyck, H. (2012). Prediction of permeability of regular scaffolds for skeletal tissue engineering: A combined computational and experimental study. Acta Biomaterialia, 8(4), 1648-1658. doi: http://dx.doi.org/10.1016/j.actbio.2011.12.021 Wieding, J., Souffrant, R., Mittelmeier, W., & Bader, R. (2013). Finite element analysis on the biomechanical stability of open porous titanium scaffolds for large segmental bone defects under physiological load conditions. Med Eng Phys, 35(4), 422-432. doi: 10.1016/j.medengphy.2012.06.006 Wieding, J., Wolf, A., & Bader, R. (2014). Numerical optimization of open-porous bone scaffold structures to match the elastic properties of human cortical bone. J Mech Behav Biomed Mater, 37, 56-68. doi: 10.1016/j.jmbbm.2014.05.002 Yan, C., Hao, L., Hussein, A., Bubb, S. L., Young, P., & Raymont, D. (2014). Evaluation of light-weight AlSi10Mg periodic cellular lattice structures fabricated via direct metal laser sintering. Journal of Materials Processing Technology, 214(4), 856-864. doi: http://dx.doi.org/10.1016/j.jmatprotec.2013.12.004 Yan, C., Hao, L., Hussein, A., & Raymont, D. (2012). Evaluations of cellular lattice structures manufactured using selective laser melting. International Journal of Machine Tools and Manufacture, 62, 32-38. doi: 10.1016/j.ijmachtools.2012.06.002 Yang, N., Quan, Z., Zhang, D., & Tian, Y. (2014). Multi-morphology transition hybridization CAD design of minimal surface porous structures for use in tissue engineering. Computer-Aided Design, 56(0), 11-21. doi: http://dx.doi.org/10.1016/j.cad.2014.06.006 Yigil, C., Tae-Hong, A., Hoon-Hwe, C., Joong-Ho, S., Jun Hyuk, M., Shu, Y., . . . Ju, L. (2013). Study of architectural responses of 3D periodic cellular materials. Modelling and Simulation in Materials Science and Engineering, 21(6), 065018. Yoo, D.-J. (2011). Computer-aided porous scaffold design for tissue engineering using triply periodic minimal surfaces. International Journal of Precision Engineering and Manufacturing, 12(1), 6171. doi: 10.1007/s12541-011-0008-9 18

Young Iii, S. D., & Toth, A. P. (2006). Complications of Allograft Use in Anterior Cruciate Ligament Reconstruction. Operative Techniques in Sports Medicine, 14(1), 20-26. doi: http://dx.doi.org/10.1053/j.otsm.2006.02.009 Zargarian, A., Esfahanian, M., Kadkhodapour, J., & Ziaei-Rad, S. (2014). Effect of Solid Distribution on Elastic Properties of Open-cell Cellular Solids Using Numerical and Experimental Methods. J Mech Behav Biomed Mater. Figure Captions Fig. 1. Representing produced lattice structures by SLM technique: (a) cubic, and (b) diamond structures (Seyed Mohammad Ahmadi et al., 2015) Fig. 2. Overview of meshed cellular structures with the mesh seed sizes of (a) 7µm, (b) 3µm, and (c) 2 µm Fig. 3. Stress strain curve of Ti6Al4V utilized as the input for FE analysis. (Cain et al., 2015) Fig. 4. Convergence of stress-strain curves corresponding to the three mesh seed sizes for 22% volume fraction models of (a) cubic, and (b) diamond lattice structures Fig. 5. Experimental compressive stress-strain curves for the cubic and diamond models at relative densities of almost (a) 11%, (b) 22%, (c) 28%, and (d) 35%; Samples are named with the combination of the first letter of their names together with the corresponding relative density (Seyed Mohammad Ahmadi et al., 2015). Fig. 6. Comparing experimental stress-strain data with the numerical simulation data for different volume fractions of (a) cubic, and (b) diamond structures; Yield strains are found to be more than the experimental results. In plateau region, higher stress variations with more amplitudes lead to deviations from the experimental data. Nevertheless, plateau stress and consequently energy absorption capability are well predicted. Fig. 7. Deformation procedure for the cubic lattice structure; Layer-by-layer deformation mechanism is confirmed by stretch dominated deformation in scaling laws. Fig. 8. Failure mechanism of diamond lattice structure at 22% volume fraction; Continuous sharing band of 45°, owing to crushing diagonal layers, is observed. Shearing of layers is accompanied by the bending failure of tying struts perpendicular to the diagonal plates.

19

Table Captions Table 1. Morphological characterization of lattice structures (Seyed Mohammad Ahmadi et al., 2015) Table 2. Comparison of mechanical properties obtained from the Johnson-Cook damage model with the experimental data. Relative densities for the maximum stress at the first peak and plateau stress are predicted with high accuracies. Experimental data are extracted from (Seyed Mohammad Ahmadi et al., 2015). Table 1. Morphological characterization of lattice structures (S. Ahmadi et al., 2014) Unit cell type

Cubic

Diamond

Sample Label C-10 C-22 C-27 C-35 D-11 D-21 D-28 D-35

Apparent density (%) Dry weighing µ-CT 11±0.1 21±0.2 26±0.2 34±0.1 11±0.1 20±0.2 26±0.4 34±0.3

Strut diameter (µm) Nominal µ-CT

13 24 28 37 11 21 28 36

348 540 612 720 277 450 520 600

Pore size (µm) Nominal µ-CT

451±147 654±190 693±200 823±230 240±46 416±65 482±70 564±76

1452 1260 1188 1080 923 750 680 600

1413±366 1139±359 1155±354 1020±311 958±144 780±141 719±130 641±137

Table 2. Comparison of mechanical properties obtained from the Johnson-Cook damage model with the experimental data. Relative densities for the maximum stress at the first peak and plateau stress are predicted with high accuracies. Experimental data are extracted from (S. Ahmadi et al., 2014). Sample Name C-10 C-22 C-27 C-35 D-11 D-21 D-28 D-35

Young’s Modulus (MPa)

Yield Stress (MPa)

Maximum stress ߪ௠௔௫ (MPa)

Experiment

Simulation

%Error

Experiment

Simulation

%Error

Experiment

Simulation

%Error

1578 3157 3691 4836 511 1505 2178 3694

2016 3831 6913 10513 559 1468 2049 4466

28% 21% 87% 117% 9% 2% 6% 21%

29 63 66 113 7 29 32 71

29 48 67 118 12 33 40 78

25% 2% 5% 70% 14% 27% 11%

30 77 111 185 15 47 57 113

31 72 104 189 16 48 59 114

3% 6% 5% 2% 6% 2% 3% 1%

20

Plateau stress ߪ௣௟ (MPa) ExperimentSimulation

11 40 64 142 8 29 38 82

13 39 43 133 7 25 31 59

%Error

16% 10% 14% 1% 10% 13% 16% 18%

Table 3. Morphological characterization of lattice structures (S. Ahmadi et al., 2014) Unit cell type

Cubic

Diamond

Sample Label C-10 C-22 C-27 C-35 D-11 D-21 D-28 D-35

Apparent density (%) Dry weighing µ-CT 11±0.1 21±0.2 26±0.2 34±0.1 11±0.1 20±0.2 26±0.4 34±0.3

Strut diameter (µm) Nominal µ-CT

13 24 28 37 11 21 28 36

348 540 612 720 277 450 520 600

Pore size (µm) Nominal µ-CT

451±147 654±190 693±200 823±230 240±46 416±65 482±70 564±76

1452 1260 1188 1080 923 750 680 600

1413±366 1139±359 1155±354 1020±311 958±144 780±141 719±130 641±137

Table 4. Comparison of mechanical properties obtained from the Johnson-Cook damage model with the experimental data. Relative densities for the maximum stress at the first peak and plateau stress are predicted with high accuracies. Experimental data are extracted from (S. Ahmadi et al., 2014). Sample Name

Young’s Modulus (MPa)

Yield Stress (MPa)

Maximum

σ max C-10 C-22 C-27 C-35 D-11 D-21 D-28 D-35

stress

σ pl

(MPa)

Experiment

Simulation

%Error

Experiment

Simulation

%Error

Experiment

Simulation

%Error

1578 3157 3691 4836 511 1505 2178 3694

2016 3831 6913 10513 559 1468 2049 4466

28% 21% 87% 117% 9% 2% 6% 21%

29 63 66 113 7 29 32 71

29 48 67 118 12 33 40 78

25% 2% 5% 70% 14% 27% 11%

30 77 111 185 15 47 57 113

31 72 104 189 16 48 59 114

3% 6% 5% 2% 6% 2% 3% 1%

21

Plateau (MPa)

ExperimentSimulation

11 40 64 142 8 29 38 82

stress

13 39 43 133 7 25 31 59

%Error

16% 10% 14% 1% 10% 13% 16% 18%

(a) (b) Fig. 9. Representing produced lattice structures by SLM technique technique: (a) cubic, and (b) diamond structures structure (Ahmadi et al., 2015)

Cubic

Diamond

(a) (b) (c) Fig. 10. Overview w of meshed cellular structures with the mesh seed sizes of (a) 7µm, (b) 3µm, 3µm and (c) 2 µm

22

Fig. 11. Stress strain curve of Ti6Al4V utilized as the input for FE analysis. (Cain, Thijs, Van Humbeeck, Van Hooreweder, & Knutsen, 2015)

(a)

23

(b) Fig. 12. Convergence of stress-strain curves corresponding to the three mesh seed sizes for 22% volume fraction models of (a) cubic, and (b) diamond lattice structures

(a)

24

Compressive Stress (MPa)

(b)

Compressive Stress (MPa)

(c)

(d) Fig. 13. Experimental compressive stress-strain curves for the cubic and diamond models at relative densities of almost (a) 11%, (b) 22%, (c) 28%, and (d) 35%; Samples are named with the combination of the first letter of their names together with the corresponding relative density (Ahmadi et al., 2015).

25

26

Stress (MPa) (a) (b) Fig. 14. Comparing experimental stress-strain data with the numerical simulation data for different volume fractions of (a) cubic, and (b) diamond structures; Yield strains are found to be more than the experimental results. In plateau region, higher stress variations with more amplitudes lead to deviations from the experimental data. Nevertheless, plateau stress and consequently energy absorption capability are well predicted.

27

(a)

(b)

(c) (d) Fig. 15.. Deformation procedure for the cubic lattice structure; Layer-by-layer layer deformation mechanism is confirmed by stretch dominated deformation in scaling laws.

28

(a)

(b)

(c) (d) Fig. 16.. Failure mechanism of diamond lattice structure at 22% volume fraction; Continuous sharing band of 45°, owing to crushing diagonal layers layers, is observed. Shearing of layers is accompanied by the bending failure of tying struts perpendicular to the diagonal plates.

29