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Finite element analysis for mechanical response of Ti foams with regular structure obtained by selective laser melting
Acta Materialia 97 (2015) 199–206
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Acta Materialia journal homepage: www.elsevier.com/locate/actamat
Finite element analysis for mechanical response of Ti foams with regular structure obtained by selective laser melting Hoon-Hwe Cho a, Yigil Cho b, Heung Nam Han a,⇑ a b
Department of Materials Science & Engineering and Center for Iron & Steel Research, RIAM, Seoul National University, Seoul 151-744, Republic of Korea Department of Materials Science and Engineering, University of Pennsylvania, 3231 Walnut Street, Philadelphia, PA 19104, USA
a r t i c l e
i n f o
Article history: Received 24 June 2015 Accepted 2 July 2015 Available online 13 July 2015 Keywords: Titanium foam Regular structure Dual-level finite element analysis X-ray microtomography Stability Defect
a b s t r a c t Metal-based cellular materials with periodic structures are currently the preferred choice as bone/ cartilage implants under load-bearing conditions due to their controlled pore interconnectivity and porosities. This report presents a new methodology for the structural analysis of periodic cellular materials using X-ray microtomography and dual-level finite element modeling (FEM). A three-dimensional (3D) structure of periodic titanium foam produced using selective laser melting (SLM) is obtained using an X-ray microtomography. A dual-level FEM based on the 3D structure is used to simulate the deformation behavior of titanium foam with a regular structure under uniaxial compression, and the computed results are compared directly with the interrupted uniaxial compression experiments performed on the deformed 3D structures. The deformation behaviors of simplified structures with cylindrical and hexahedral struts are simulated, and the computed results demonstrate that unavoidable defects in the actual structure affect the mechanical stability significantly. Additionally, buckling-induced deformation behavior is analyzed by introducing an imperfection in the actual and simplified structures. The effects of certain selected process variables, such as internal angle and diameter, are also examined through a series of process simulations. Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
1. Introduction Cellular materials (including ceramics, polymers, metals, and their alloys) are widely used in structural applications due to their noteworthy physical and mechanical properties [1]. Thus, material science has given considerable attention to cellular materials that have structural function and robustness and generated remarkable innovations [2–10]. To design these materials, it is thus important to establish a method that can predict the structural behavior under any deformation conditions, especially compressive deformation. A few attempts to analyze deformation behavior of foams under load-bearing conditions have been reported. Youssef et al. [11] proposed a comprehensive modeling methodology for the mechanical behaviors of cellular materials. The actual microstructure of polyurethane foam was obtained by processing X-ray tomographic data, and the deformation behavior was analyzed under uniaxial compression using actual microstructure-based finite element modeling. Singh et al. [12] used a generic non-destructive methodology based on the X-ray microtomography and direct finite element modeling to determine the mechanical properties ⇑ Corresponding author. E-mail address: [email protected] (H.N. Han). http://dx.doi.org/10.1016/j.actamat.2015.07.003 1359-6454/Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
of titanium foams and characterize their deformation behavior. They demonstrated that the localized collapse of large and unfavorably oriented pores occurred at dispersed points throughout the volume, which is critical in understanding the failure mechanisms of metallic foams in general and porous implants in particular. This methodology has been used to investigate damage under compressive loading in bone [13], bioactive glass [14], polymer [15], titanium [16,17], and aluminum foams [18–20]. However, these numerical studies considering the actual structure have focused on only cellular materials without periodicity. Periodic cellular materials are of great interest as regenerative medicine for bone and cartilage [21]. Specifically, metallic periodic foams are currently evaluated and used as implants because they offer four main benefits: (1) proliferation of cells into the structure is facilitated; (2) vascularization is promoted; (3) structure can be tuned to match the modulus of bone to reduce stress-shielding effects; and (4) controlled interconnectivity, porosity, and periodicity are achieved [12,22–24]. However, considerably less attention has been given to a new methodology of structural analysis for periodic foam using actual three dimensional (3D) structure based finite element modeling (FEM) despite significant practical importance of the method, which can guide the design of cellular materials with structural function and robustness. Only a few
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recent studies have examined the deformation behaviors of periodic foams without considering actual 3D microstructure. Kim et al. [25] developed a noble concept of dual-level simulation with a spatially representative volume element (RVE) for a nanohole structure of a Cu electrode to understand the mechanical stress state around the nanohole during crack initiation and propagation. They concluded that the nanohole-structured Cu electrode had significantly improved electrical stability, which enabled the realization of fatigue-damage-free metal electrodes. Cho et al. [26] presented a method for the efficient structural analysis of periodic cellular materials under inhomogeneous strain. They reported that the method provides an efficient numerical solution for the inhomogeneous strain field by having an upper-level finite element (UFE) solver summon on a lower-level finite element (LFE) solver to transfer the homogenization information on the fly. In the current study, a new methodology for the structural analysis of periodic foams based on X-ray microtomography and dual-level FEM is presented. A commercially pure titanium (CP-Ti) foam with a periodic regular structure, which was developed using selective laser melting (SLM), is used as an example. This foam is useful for supporting rapid and sustained osteo-integration, and can be matched well to the interconnected network, which is observed in cancellous bone. The 3D structure of the periodic Ti foam is obtained using the X-ray microtomography. The dual-level FEM based on the 3D structure is used to simulate the deformation behavior of the Ti foam with a regular structure under uniaxial compression, and the computed results are compared directly with the interrupted uniaxial compression experiments performed on the deformed 3D structures. The deformation behaviors of the simplified structures with cylindrical and hexahedral struts are simulated to examine the effects of unavoidable defects on the structural stability. Additionally, buckling-induced deformation behavior of the actual and simplified structures is analyzed by introducing an imperfection. Furthermore, the effects of a few selected process variables, such as internal angle and diameter, are examined through a series of process simulations. 2. Materials and experiments 2.1. Sample preparation and X-ray microtomography characterization The sample was produced from CP-Ti (Grade 1) using the SLM method. The manufacturing process has been detailed by Mullen et al. [27,28]. The sample was scanned at a resolution of 6 lm per voxel using a commercial X-ray microtomography unit (Phoenix v|tome|x, GE Measurement and Control, USA). The X-ray tube voltage and filament current were fixed at 100 keV and 70 lA, respectively. A copper filter of 0.5 mm thickness was used to absorb low energy X-rays and reduce beam hardening [29]. A rotation step of 0.5° was set within an angular range of 360°. After acquiring the two-dimensional (2D) radiographic images, 3D volumes of 640 512 640 voxel of the sample were then reconstructed using the commercial reconstruction software datos|x (GE Measurement and Control, USA). The pre-processing of the 3D volume, including a 3 3 3 median filtering and removing islands within the volume space, was accomplished using the commercial image visualization/analysis software Avizo 6.3 (VSG, USA). Fig. 1 provides the (a) 3D reconstructed Ti foam and (b) unit cell structure. The Ti foam has a porosity of 65% and an octahedral structure with a strut of 600 lm length 180 lm thickness. 2.2. Mechanical property characterization Two types of tests were performed to characterize the mechanical properties: (i) static compression test to obtain the intrinsic
mechanical property of monolithic Ti; and (ii) interrupted compression test followed by the X-ray microtomography to relate the deformation behavior of the Ti foam to the deformed microstructure. The static and interrupted tests were performed on cylindrical specimens. The dimensions of the monolithic Ti and Ti foam were 10 mm diameter 20 mm and 6 mm diameter 7 mm, respectively. A 100 kN load cell servo-hydraulic universal testing machine (Zwick GmbH, Germany) was used to perform the static compression test. The test was conducted in displacement control mode with a strain rate of 0.001 s1. The interrupted compression test was performed in a screw-driven 10 kN Zwick machine using successive small deformation stages. The displacement was recorded by applying a clip gauge over the test platen’s each step, and the X-ray microtomography was performed after each step. Fig. 2a and b present the engineering stress–strain curves of the (a) monolithic Ti and (b) Ti foam, respectively. The data will be used as input parameters for the FE model. 3. Dual-level FE method It is beneficial to contrast dual-level FE with conventional single-scale continuum modeling [25,26,30]. In the single-scale modeling, macroscopic material characteristics are described typically using homogenized material parameters, such as homog homogenized Poisson’s ratio m , or enized Young’s modulus E, homogenized collapse strength. The macroscopic material behavior is then analyzed according to a constitutive relation, where a derivation or parameterization of the homogenized stress–strain relation accommodating the nonlinear nature of porous media essentially determines the appropriateness of this single-scale modeling. However, in most cases, the macroscopic nonlinear constitutive relationship is extremely sensitive to micro-scale states, such as unit cell shape, architectural bifurcation, or orientation, which change continuously during loading. These weak points could be overcome using the dual-scale FE method. The dual-scale FE consists of two steps: (i) upper-level FE (UFE) analysis and (ii) lower-level FE (LFE) analysis, so called lower-level simulation. A prerequisite simulation is performed to obtain the deformation history of the local region of interest under any mechanical test. The deformation gradient at the macroscopic UFE can be by the following equation, if the characteristic length scale of the deformation is much larger than the unit cell spacing [31,32]:
2 @ x1
@X 1
6 J ¼ @ x ¼ 6 @ x2 @ XR 4 @ X 1 @ x3 @X 1
@ x1 @X 2 @ x2 @X2 @ x3 @X 2
@ x1 @X 3 @ x2 @X3 @ x3 @X 3
3 7 7 5
ð1Þ
R is the reference coordinates of the element that occupies where X at a certain time. If we define the displacement vector in position x R [33], ¼x X upper space (with the overhead bar symbol) as u
R @X @u @u J ¼ @ ðX R R þ uÞ ¼ @ X R þ @X R ¼ I þ @X R @X
ð2Þ
In tensor notation,
J ij ¼ dij þ u ij
ð3Þ
The deformation gradient of the local region of interest describes the periodic boundary condition in the lower level space for a unit cell of the crystal. The displacement of the boundaries of the unit cell subjected to the known deformation gradient J can be expressed using the well-known Cauchy-Born rule [34] as follows:
uB uA ¼ ðJ IÞðxB xA Þ
ð4Þ
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Fig. 1. (a) 3D reconstructed Ti foam; and (b) unit cell structure.
Fig. 2. Engineering stress–strain curves of (a) monolithic Ti; and (b) Ti foam. Note: inset in (b) presents interrupted compression test.
where u is a displacement vector; x is a position vector in lower space (without the overhead bar symbol); and subscripts A and B denote two points located on the boundaries of the unit cell that are separated by an integer lattice vector. J I means the displacement gradient. In the LFE calculation, the internal continuum elements of the unit cell are deformed under the constraints of Eq. (4) according to the following convergence criterion.
jF UFE F LFE F re j 6 T re
ð5Þ UFE
LFE
In the above expression, F and F are the average forces of the local region of interest and unit cell, respectively. The force residual F re must be less than the force residual tolerance, T re , which is a value of 2% in this study. The dual FE model is implemented in the ABAQUS/Standard software [35] by incorporating a user subroutine called UMAT (for UFE) and DISP (for LFE). To obtain the macroscopic deformation gradient histories of the representative volume element (RVE) of interest, a macro-scale simulation was performed under uniaxial compression, as indicated in Fig. 3a. The simulation was performed in displacement control mode with a strain rate of 0.001 s1, which is the same as the experimental condition. The computed deformation gradient histories were used as boundary conditions of the RVE of interest, as indicated in Fig. 3b. The RVE has 123,418 tetrahedral elements based on the actual
microstructure obtained from the X-ray microtomography. The material properties used in this simulation are provided in Fig. 2. For the characteristics of the octahedral structure, the volumetric compression was simulated in the displacement mode with a strain rate of 0.001 s1, which is the same as the uniaxial compression. The deformation gradients can be obtained by introducing a new method to apply volumetric compression to the macroscopic geometry. The outer surfaces of the macroscopic geometry are pinned while the temperature increases without any change of mechanical properties, thus imposing compressive volumetric strain. The computed deformation gradient histories obtained from this method were also used as boundary conditions of the RVE of interest, as mentioned above.
4. Results and discussion 4.1. Validation of dual-level FE method against compression experiments The stress–strain curve of the monolithic Ti (Fig. 2a) was used as the flow stress for the strut of foam for input into the UFE. The measured stress–strain curve of the Ti foam is shown in Fig. 2b. Typically, three regimes in the course of deformation are clearly observed in the stress–strain curve of the Ti foam: (i) initial
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Fig. 3. (a) Calculated maximum principal strain distribution of Ti foam in uniaxial compression; and (b) mesh system of RVE.
linear elastic region; (ii) plastic collapse plateau region, so called transition zone; and (iii) densification region. In the ensuing dual-level FE simulation, the UFE deformation gradient of the center region of the Ti foam is imposed on the boundary condition of the unit cell in the LFE calculation, as shown in Fig. 3. The computed stress–strain curve predicts the typical three regions well, which is in agreement with the experimental data (Fig. 4). The Ti foam, which has ‘micro-geometry’, may undergo extremely large displacement and geometrical softening in the course of deformation. The dual-level FE method considers this geometrical nonlinearity effect well. Certainly, a few localized regions accumulate more stress than others during uniaxial compression (Fig. 5b) and therefore yield earlier, which causes the structure to be less stiff and eventually leads to the structural transition (Fig. 5c) that manifests as the softening transition zone in the stress–strain curve. As the accumulated stresses spread over the frames, hardening occurs, and the structure becomes stiff again (Fig. 5d). Fig. 5 illustrates that the sources of stress accumulation are distributed primarily on the frame ligaments. Particularly, the localized irregularities appear to be the primary sources of localized deformation because those could function as unavoidable defects. The actual microstructure typically has unavoidable defects, which are developed during the manufacturing process, such as the SLM method in this case. This indicates that controlling the surface roughness or
Fig. 4. Calculated and measured stress–strain curves upon uniaxial compression. The black line was obtained from the dual-level FE calculation, and the red symbol represents the measured data upon uniaxial compression. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
quality in the process would be extremely important in determining the mechanical properties of the foam. For further validation of the computational method, the computed and measured images of the Ti foam are compared at different positions in the stress–strain curve (marked in Fig. 4), as indicated in Fig. 6. The representative pictures (cross-sectional image) taken from the X-ray microtomography during the interrupted test are compared with the computed images. First, in the linear elastic region (Fig. 6a), the strut of the unit cell is hardly compressed, and the cross section of the unit cell remains in original form, i.e., a butterfly shape. The strut of the unit cell is rapidly compressed in the transition zone (Fig. 6b), and consequently the cross-sectional image in the densification region (Fig. 6c) becomes a ball-like shape. Simulating this experiment directly is clearly beyond the capability of the single-scale finite element analysis. These results indicate that the dual-level FE calculation predicts the deformed shapes in excellent agreement with the observations, which validates the implemented dual-level FE method.
4.2. Structural responses of actual and simplified structures It should be noted that unavoidable defects on the actual microstructure significantly affect the mechanical behavior, especially structural stability. Intuitively, the architecture in which the applied load spreads out more homogeneously may be more stable against that load. In other words, given a deformation condition, the accumulation of stress and strain on a localized region, such as an unavoidable defect, would lead to the earlier onset of concentrated deformation and load shedding, which would cause the architecture to be susceptible to be transformed by bifurcation instability. Thus, for actual and simplified structures, the structural stability is numerically investigated here using the dual-level FE method. The simplified structures with cylindrical and hexahedral struts are designed to have identical porosity (65%) to the actual structure and no unavoidable defects. Fig. 7 shows the computed maximum Mises stress profiles for the actual and simplified unit cells under uniaxial compression. The stress accumulation in the actual structure is the steepest, which indicates that the actual structure with unavoidable defects is more unstable than the other simplified structures when comparing uniaxial compression. This phenomenon is more prominent in the computed Mises stress distributions of the actual and simplified structures, as shown in Fig. 8. The stress accumulation occurs at symmetric local regions of the frame ligaments in the simplified structures whereas it develops at the localized irregularities without symmetry in the actual structure. These results demonstrate that unavoidable
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Fig. 5. Computed Mises stress distributions of unit cell of Ti foam during uniaxial compression in the dual-level FE model. Points O, P, Q and R denote each position in the stress–strain curve (marked in Fig. 4).
Fig. 6. Computed and measured images of Ti foam at different levels of applied strain. Points P, Q and R denote each position in the stress–strain curve (marked in Fig. 4).
defects in the actual structure that are developed necessarily during the actual process would affect mechanical stability significantly. Furthermore, it should be noted that if we assess the general structural stability of the porous material under any mechanical test, the unavoidable defects, which could lead to the earlier onset of concentrated deformation, should be considered. Certain aspects of the stress accumulation are also sensitive to the shape of the strut. The simplified structure with cylindrical struts is more unstable than that with hexahedral struts (Fig. 7), even though these structures have the same cross-sectional area
along the considered direction. It should be noted that the stress accumulation occurs at similar local regions of the frame ligaments in these two simplified structures, as shown in Fig. 8b and c. To confirm the effect of given a deformation condition on the structural stability, we also computed the mechanical behavior of the Ti foam under volumetric (isotropic) compression. Fig. 9 show the computed total and top 0.1% average Mises stress profiles of the actual unit cell under uniaxial and volumetric compression. For the volumetric compression of the actual unit cell, stress accumulation is less steep than the uniaxial compression in the initial
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Fig. 7. Predicted maximum Mises stress profiles for actual and simplified unit cells upon uniaxial compression. Red and blue lines indicate cylindrical and hexahedral struts, respectively. The maximum Mises stresses were obtained from the averaged stress in 0.1% of elements that indicated the highest Mises stress. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
elastic region, thus indicating that the bulk modulus is smaller than the Young’s modulus. From the elastic region, we can also calculate effective bulk modulus under volumetric compression, but it is noted that in the dual-level FE, the applied deformation condition may be not perfectly matched the volumetric compression since we used the deformation gradient histories of the region of interest. In the plastic region, the maximum stress histories are nearly similar which means there is no significant difference of structural stability, especially in top 0.1% average Mises stress profiles, on the given deformation conditions such as uniaxial and
Fig. 9. Predicted total and top 0.1% average Mises stress profiles of actual unit cell upon uniaxial and volumetric compression.
volumetric compression. This phenomenon might be caused by the stress saturation in the plastic region while the concentrated stress region is expanding. 4.3. Buckling analysis of actual and simplified structures Buckling is characterized by the sudden failure of a structural member subjective to high compressive stress. Thus, it is extremely important to investigate buckling of 3D periodic porous structures when assessing structural stability. An imperfection in the form of the first buckling mode (determined using a linear buckling analysis) was introduced in the initial structures, and the response of the RVE was considered under uniaxial compression. Fig. 10a presents
Fig. 8. Computed Mises stress distributions of (a) actual and simplified structures with (b) cylindrical and (c) hexahedral struts under uniaxial compression. Point S denotes a position in the maximum Mises stress profiles (marked in Fig. 7).
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Fig. 10. (a) Predicted maximum Mises stress profiles for actual and simplified unit cells with imperfections upon uniaxial compression. Computed Mises stress distributions of (b) actual and simplified structures with (c) cylinder and (d) hexahedron struts at point T.
Fig. 11. (a) Predicted maximum Mises stress profiles with an internal angle; and (b) maximum Mises stress values at point S (marked in Fig. 7) versus a diameter of strut for simplified unit cell having cylinder struts upon uniaxial compression.
the computed maximum Mises stress profiles for the actual and simplified unit cells under uniaxial compression. The computed profiles are nearly similar to each other because the stress accumulation occurs primarily at the imposed imperfection region. In other words, a given imperfection excludes the possibility of the simulation to follow an unstable deformation path, and consequently the structural stability of the actual structure would be less vulnerable to unavoidable defects. It should be that an unavoidable defect on an actual structure would act as the imperfection of the structure under typical mechanical tests. The deformed shape induced by the buckling varies with the geometry of the structure (Fig. 10b–d), which is in agreement with the literature [36,37]. The deformed shape is remarkably different from the typical behavior of the structure under uniaxial compression (Fig. 8a–c) due to the imposed imperfection. Additionally, local regions in which stress accumulation occurs are completely different from the above mentioned results (Fig. 8a–c) despite the identical boundary conditions, except for the artificial imperfection. 4.4. Effects of process variables on structural stability As a final application of the dual-level FE simulation in this report, we would like to investigate the effects of certain selected process variables on the structural stability under uniaxial compression. Since an analysis for the simplified structure is sufficient to predict the effects of process variables on the structural stability, only the simplified structure with cylindrical struts, which is closer
to the actual structure in terms of stability (Fig. 7), is used for efficiency of computation. First, the effect of an internal angle on the structural stability was analyzed. The internal angle signifies a half angle between each strut on the cross section, as indicated in the inset of Fig. 11a, and this variable can be controlled in the SLM process. As the internal angle increases, the maximum Mises stress increases more rapidly (Fig. 11a), and consequently the structure with the larger angle is more unstable. This phenomenon can be explained by the thickness of the center body of the unit cell. If the internal angle is smaller, the thickness of the center body is larger, which indicates a larger stiffness of the unit cell. Therefore, the structure with the larger stiffness is less vulnerable to the given deformation mode. Additionally, the sensitivity of the structural stability on a diameter of the strut was examined as indicated in Fig. 11b. The diameter is determined using laser properties, such as beam diameter, dwell time and energy. The maximum Mises stress decreases as the diameter increases, and this result indicates that the structure with the larger diameter is more stable. Since the thickness increases when the diameter increases, this phenomenon can be explained by the above mentioned reason. 5. Conclusions In this report, we presented a new methodology for the structural analysis of periodic foams based on the X-ray
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microtomography and dual-level FE methods. The 3D structure of the periodic Ti foam with an octahedral structure is obtained using the X-ray microtomography, and the dual-level FE method based on the 3D structure provides an efficient and accurate numerical solution using the UFE and LFE solvers. The computed results are compared directly with the interrupted uniaxial compression experiments performed on the Ti foam. The three regions typically observed in the stress–strain curve, which is created from the architectural transformation during uniaxial compression, are reproduced well using the method. Additionally, the method predicts the deformed shapes in excellent agreement with the observations. These results validate the implemented dual-level FE method. The structural stability of the periodic foam based on the existence of unavoidable defects and the shape of the strut is quantitatively examined by having the simplified structures with cylindrical and hexahedral struts. The computed results indicate that the unavoidable defects on the actual 3D structure affect the mechanical stability significantly, and certain aspects of the stress accumulation are sensitive to the shape of the strut. The buckling-induced deformation behavior of the actual and simplified structures is analyzed by introducing an imperfection. This indicates that an unavoidable defect on an actual structure would act as the imperfection of the structure under a typical mechanical test. The effects of certain selected process variables, such as internal angle and diameter, are examined through a series of process simulations. This process confirms that the stability of the structure is more stable when the internal angle is smaller and the diameter is larger. The presented studies demonstrate sufficient possibilities, which can improve the deformability of functional periodic porous materials only by using a rigorous numerical model. Furthermore, the insights gained by performing a numerical parametric exploration serve as an important design guideline in fabricating practical materials with excellent structural function and robustness. Acknowledgements This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (NRF-2013R1A2A2A01008806). H.H.C. would like to acknowledge Prof. Peter Lee and his colleagues at the Diamond-Manchester Collaboration for hosting me and helping with the XMT portion of this work. References [1] L.J. Gibson, M.F. Ashby, Cellular Solids: Structure and Properties, second ed., Cambridge University Press, Cambridge, New York, 1997. [2] J.B. Thompson, J.H. Kindt, B. Drake, H.G. Hansma, D.E. Morse, P.K. Hansma, Bone indentation recovery time correlates with bond reforming time, Nature 414 (2001) 773–776. [3] G.E. Fantner, T. Hassenkam, J.H. Kindt, J.C. Weaver, H. Birkedal, L. Pechenik, J.A. Cutroni, G.A.G. Cidade, G.D. Stucky, D.E. Morse, P.K. Hansma, Sacrificial bonds and hidden length dissipate energy as mineralized fibrils separate during bone fracture, Nat. Mater. 4 (2005) 612–616. [4] E. Munch, M.E. Launey, D.H. Alsem, E. Saiz, A.P. Tomsia, R.O. Ritchie, Tough, bio-inspired hybrid materials, Science 322 (2008) 1516–1520. [5] R.M. Erb, R. Libanori, N. Rothfuchs, A.R. Studart, Composites reinforced in three dimensions by using low magnetic fields, Science 335 (2012) 199–204. [6] K. Jin, Y. Tian, J.S. Erickson, J. Puthoff, K. Autumn, N.S. Pesika, Design and fabrication of gecko-inspired adhesives, Langmuir 28 (2012) 5737–5742. [7] M.A. Meyers, J. McKittrick, P.Y. Chen, Structural biological materials: critical mechanics-materials connections, Science 339 (2013) 773–779. [8] D.C. Dunand, Processing of titanium foams, Adv. Eng. Mater. 6 (2004) 369–376. [9] J.L. Fife, J.C. Li, D.C. Dunand, P.W. Voorhees, Morphological analysis of pores in directionally freeze-cast titanium foams, J. Mater. Res. 24 (2009) 117–124. [10] J.Y. Cho, K.Y. Kim, Structure and compressive strength of silicon open-cell foam obtained by a centrifugal separation method, Met. Mater. Int. 19 (2013) 361–365.
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Acta Materialia journal homepage: www.elsevier.com/locate/actamat
Finite element analysis for mechanical response of Ti foams with regular structure obtained by selective laser melting Hoon-Hwe Cho a, Yigil Cho b, Heung Nam Han a,⇑ a b
Department of Materials Science & Engineering and Center for Iron & Steel Research, RIAM, Seoul National University, Seoul 151-744, Republic of Korea Department of Materials Science and Engineering, University of Pennsylvania, 3231 Walnut Street, Philadelphia, PA 19104, USA
a r t i c l e
i n f o
Article history: Received 24 June 2015 Accepted 2 July 2015 Available online 13 July 2015 Keywords: Titanium foam Regular structure Dual-level finite element analysis X-ray microtomography Stability Defect
a b s t r a c t Metal-based cellular materials with periodic structures are currently the preferred choice as bone/ cartilage implants under load-bearing conditions due to their controlled pore interconnectivity and porosities. This report presents a new methodology for the structural analysis of periodic cellular materials using X-ray microtomography and dual-level finite element modeling (FEM). A three-dimensional (3D) structure of periodic titanium foam produced using selective laser melting (SLM) is obtained using an X-ray microtomography. A dual-level FEM based on the 3D structure is used to simulate the deformation behavior of titanium foam with a regular structure under uniaxial compression, and the computed results are compared directly with the interrupted uniaxial compression experiments performed on the deformed 3D structures. The deformation behaviors of simplified structures with cylindrical and hexahedral struts are simulated, and the computed results demonstrate that unavoidable defects in the actual structure affect the mechanical stability significantly. Additionally, buckling-induced deformation behavior is analyzed by introducing an imperfection in the actual and simplified structures. The effects of certain selected process variables, such as internal angle and diameter, are also examined through a series of process simulations. Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
1. Introduction Cellular materials (including ceramics, polymers, metals, and their alloys) are widely used in structural applications due to their noteworthy physical and mechanical properties [1]. Thus, material science has given considerable attention to cellular materials that have structural function and robustness and generated remarkable innovations [2–10]. To design these materials, it is thus important to establish a method that can predict the structural behavior under any deformation conditions, especially compressive deformation. A few attempts to analyze deformation behavior of foams under load-bearing conditions have been reported. Youssef et al. [11] proposed a comprehensive modeling methodology for the mechanical behaviors of cellular materials. The actual microstructure of polyurethane foam was obtained by processing X-ray tomographic data, and the deformation behavior was analyzed under uniaxial compression using actual microstructure-based finite element modeling. Singh et al. [12] used a generic non-destructive methodology based on the X-ray microtomography and direct finite element modeling to determine the mechanical properties ⇑ Corresponding author. E-mail address: [email protected] (H.N. Han). http://dx.doi.org/10.1016/j.actamat.2015.07.003 1359-6454/Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
of titanium foams and characterize their deformation behavior. They demonstrated that the localized collapse of large and unfavorably oriented pores occurred at dispersed points throughout the volume, which is critical in understanding the failure mechanisms of metallic foams in general and porous implants in particular. This methodology has been used to investigate damage under compressive loading in bone [13], bioactive glass [14], polymer [15], titanium [16,17], and aluminum foams [18–20]. However, these numerical studies considering the actual structure have focused on only cellular materials without periodicity. Periodic cellular materials are of great interest as regenerative medicine for bone and cartilage [21]. Specifically, metallic periodic foams are currently evaluated and used as implants because they offer four main benefits: (1) proliferation of cells into the structure is facilitated; (2) vascularization is promoted; (3) structure can be tuned to match the modulus of bone to reduce stress-shielding effects; and (4) controlled interconnectivity, porosity, and periodicity are achieved [12,22–24]. However, considerably less attention has been given to a new methodology of structural analysis for periodic foam using actual three dimensional (3D) structure based finite element modeling (FEM) despite significant practical importance of the method, which can guide the design of cellular materials with structural function and robustness. Only a few
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recent studies have examined the deformation behaviors of periodic foams without considering actual 3D microstructure. Kim et al. [25] developed a noble concept of dual-level simulation with a spatially representative volume element (RVE) for a nanohole structure of a Cu electrode to understand the mechanical stress state around the nanohole during crack initiation and propagation. They concluded that the nanohole-structured Cu electrode had significantly improved electrical stability, which enabled the realization of fatigue-damage-free metal electrodes. Cho et al. [26] presented a method for the efficient structural analysis of periodic cellular materials under inhomogeneous strain. They reported that the method provides an efficient numerical solution for the inhomogeneous strain field by having an upper-level finite element (UFE) solver summon on a lower-level finite element (LFE) solver to transfer the homogenization information on the fly. In the current study, a new methodology for the structural analysis of periodic foams based on X-ray microtomography and dual-level FEM is presented. A commercially pure titanium (CP-Ti) foam with a periodic regular structure, which was developed using selective laser melting (SLM), is used as an example. This foam is useful for supporting rapid and sustained osteo-integration, and can be matched well to the interconnected network, which is observed in cancellous bone. The 3D structure of the periodic Ti foam is obtained using the X-ray microtomography. The dual-level FEM based on the 3D structure is used to simulate the deformation behavior of the Ti foam with a regular structure under uniaxial compression, and the computed results are compared directly with the interrupted uniaxial compression experiments performed on the deformed 3D structures. The deformation behaviors of the simplified structures with cylindrical and hexahedral struts are simulated to examine the effects of unavoidable defects on the structural stability. Additionally, buckling-induced deformation behavior of the actual and simplified structures is analyzed by introducing an imperfection. Furthermore, the effects of a few selected process variables, such as internal angle and diameter, are examined through a series of process simulations. 2. Materials and experiments 2.1. Sample preparation and X-ray microtomography characterization The sample was produced from CP-Ti (Grade 1) using the SLM method. The manufacturing process has been detailed by Mullen et al. [27,28]. The sample was scanned at a resolution of 6 lm per voxel using a commercial X-ray microtomography unit (Phoenix v|tome|x, GE Measurement and Control, USA). The X-ray tube voltage and filament current were fixed at 100 keV and 70 lA, respectively. A copper filter of 0.5 mm thickness was used to absorb low energy X-rays and reduce beam hardening [29]. A rotation step of 0.5° was set within an angular range of 360°. After acquiring the two-dimensional (2D) radiographic images, 3D volumes of 640 512 640 voxel of the sample were then reconstructed using the commercial reconstruction software datos|x (GE Measurement and Control, USA). The pre-processing of the 3D volume, including a 3 3 3 median filtering and removing islands within the volume space, was accomplished using the commercial image visualization/analysis software Avizo 6.3 (VSG, USA). Fig. 1 provides the (a) 3D reconstructed Ti foam and (b) unit cell structure. The Ti foam has a porosity of 65% and an octahedral structure with a strut of 600 lm length 180 lm thickness. 2.2. Mechanical property characterization Two types of tests were performed to characterize the mechanical properties: (i) static compression test to obtain the intrinsic
mechanical property of monolithic Ti; and (ii) interrupted compression test followed by the X-ray microtomography to relate the deformation behavior of the Ti foam to the deformed microstructure. The static and interrupted tests were performed on cylindrical specimens. The dimensions of the monolithic Ti and Ti foam were 10 mm diameter 20 mm and 6 mm diameter 7 mm, respectively. A 100 kN load cell servo-hydraulic universal testing machine (Zwick GmbH, Germany) was used to perform the static compression test. The test was conducted in displacement control mode with a strain rate of 0.001 s1. The interrupted compression test was performed in a screw-driven 10 kN Zwick machine using successive small deformation stages. The displacement was recorded by applying a clip gauge over the test platen’s each step, and the X-ray microtomography was performed after each step. Fig. 2a and b present the engineering stress–strain curves of the (a) monolithic Ti and (b) Ti foam, respectively. The data will be used as input parameters for the FE model. 3. Dual-level FE method It is beneficial to contrast dual-level FE with conventional single-scale continuum modeling [25,26,30]. In the single-scale modeling, macroscopic material characteristics are described typically using homogenized material parameters, such as homog homogenized Poisson’s ratio m , or enized Young’s modulus E, homogenized collapse strength. The macroscopic material behavior is then analyzed according to a constitutive relation, where a derivation or parameterization of the homogenized stress–strain relation accommodating the nonlinear nature of porous media essentially determines the appropriateness of this single-scale modeling. However, in most cases, the macroscopic nonlinear constitutive relationship is extremely sensitive to micro-scale states, such as unit cell shape, architectural bifurcation, or orientation, which change continuously during loading. These weak points could be overcome using the dual-scale FE method. The dual-scale FE consists of two steps: (i) upper-level FE (UFE) analysis and (ii) lower-level FE (LFE) analysis, so called lower-level simulation. A prerequisite simulation is performed to obtain the deformation history of the local region of interest under any mechanical test. The deformation gradient at the macroscopic UFE can be by the following equation, if the characteristic length scale of the deformation is much larger than the unit cell spacing [31,32]:
2 @ x1
@X 1
6 J ¼ @ x ¼ 6 @ x2 @ XR 4 @ X 1 @ x3 @X 1
@ x1 @X 2 @ x2 @X2 @ x3 @X 2
@ x1 @X 3 @ x2 @X3 @ x3 @X 3
3 7 7 5
ð1Þ
R is the reference coordinates of the element that occupies where X at a certain time. If we define the displacement vector in position x R [33], ¼x X upper space (with the overhead bar symbol) as u
R @X @u @u J ¼ @ ðX R R þ uÞ ¼ @ X R þ @X R ¼ I þ @X R @X
ð2Þ
In tensor notation,
J ij ¼ dij þ u ij
ð3Þ
The deformation gradient of the local region of interest describes the periodic boundary condition in the lower level space for a unit cell of the crystal. The displacement of the boundaries of the unit cell subjected to the known deformation gradient J can be expressed using the well-known Cauchy-Born rule [34] as follows:
uB uA ¼ ðJ IÞðxB xA Þ
ð4Þ
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Fig. 1. (a) 3D reconstructed Ti foam; and (b) unit cell structure.
Fig. 2. Engineering stress–strain curves of (a) monolithic Ti; and (b) Ti foam. Note: inset in (b) presents interrupted compression test.
where u is a displacement vector; x is a position vector in lower space (without the overhead bar symbol); and subscripts A and B denote two points located on the boundaries of the unit cell that are separated by an integer lattice vector. J I means the displacement gradient. In the LFE calculation, the internal continuum elements of the unit cell are deformed under the constraints of Eq. (4) according to the following convergence criterion.
jF UFE F LFE F re j 6 T re
ð5Þ UFE
LFE
In the above expression, F and F are the average forces of the local region of interest and unit cell, respectively. The force residual F re must be less than the force residual tolerance, T re , which is a value of 2% in this study. The dual FE model is implemented in the ABAQUS/Standard software [35] by incorporating a user subroutine called UMAT (for UFE) and DISP (for LFE). To obtain the macroscopic deformation gradient histories of the representative volume element (RVE) of interest, a macro-scale simulation was performed under uniaxial compression, as indicated in Fig. 3a. The simulation was performed in displacement control mode with a strain rate of 0.001 s1, which is the same as the experimental condition. The computed deformation gradient histories were used as boundary conditions of the RVE of interest, as indicated in Fig. 3b. The RVE has 123,418 tetrahedral elements based on the actual
microstructure obtained from the X-ray microtomography. The material properties used in this simulation are provided in Fig. 2. For the characteristics of the octahedral structure, the volumetric compression was simulated in the displacement mode with a strain rate of 0.001 s1, which is the same as the uniaxial compression. The deformation gradients can be obtained by introducing a new method to apply volumetric compression to the macroscopic geometry. The outer surfaces of the macroscopic geometry are pinned while the temperature increases without any change of mechanical properties, thus imposing compressive volumetric strain. The computed deformation gradient histories obtained from this method were also used as boundary conditions of the RVE of interest, as mentioned above.
4. Results and discussion 4.1. Validation of dual-level FE method against compression experiments The stress–strain curve of the monolithic Ti (Fig. 2a) was used as the flow stress for the strut of foam for input into the UFE. The measured stress–strain curve of the Ti foam is shown in Fig. 2b. Typically, three regimes in the course of deformation are clearly observed in the stress–strain curve of the Ti foam: (i) initial
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Fig. 3. (a) Calculated maximum principal strain distribution of Ti foam in uniaxial compression; and (b) mesh system of RVE.
linear elastic region; (ii) plastic collapse plateau region, so called transition zone; and (iii) densification region. In the ensuing dual-level FE simulation, the UFE deformation gradient of the center region of the Ti foam is imposed on the boundary condition of the unit cell in the LFE calculation, as shown in Fig. 3. The computed stress–strain curve predicts the typical three regions well, which is in agreement with the experimental data (Fig. 4). The Ti foam, which has ‘micro-geometry’, may undergo extremely large displacement and geometrical softening in the course of deformation. The dual-level FE method considers this geometrical nonlinearity effect well. Certainly, a few localized regions accumulate more stress than others during uniaxial compression (Fig. 5b) and therefore yield earlier, which causes the structure to be less stiff and eventually leads to the structural transition (Fig. 5c) that manifests as the softening transition zone in the stress–strain curve. As the accumulated stresses spread over the frames, hardening occurs, and the structure becomes stiff again (Fig. 5d). Fig. 5 illustrates that the sources of stress accumulation are distributed primarily on the frame ligaments. Particularly, the localized irregularities appear to be the primary sources of localized deformation because those could function as unavoidable defects. The actual microstructure typically has unavoidable defects, which are developed during the manufacturing process, such as the SLM method in this case. This indicates that controlling the surface roughness or
Fig. 4. Calculated and measured stress–strain curves upon uniaxial compression. The black line was obtained from the dual-level FE calculation, and the red symbol represents the measured data upon uniaxial compression. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
quality in the process would be extremely important in determining the mechanical properties of the foam. For further validation of the computational method, the computed and measured images of the Ti foam are compared at different positions in the stress–strain curve (marked in Fig. 4), as indicated in Fig. 6. The representative pictures (cross-sectional image) taken from the X-ray microtomography during the interrupted test are compared with the computed images. First, in the linear elastic region (Fig. 6a), the strut of the unit cell is hardly compressed, and the cross section of the unit cell remains in original form, i.e., a butterfly shape. The strut of the unit cell is rapidly compressed in the transition zone (Fig. 6b), and consequently the cross-sectional image in the densification region (Fig. 6c) becomes a ball-like shape. Simulating this experiment directly is clearly beyond the capability of the single-scale finite element analysis. These results indicate that the dual-level FE calculation predicts the deformed shapes in excellent agreement with the observations, which validates the implemented dual-level FE method.
4.2. Structural responses of actual and simplified structures It should be noted that unavoidable defects on the actual microstructure significantly affect the mechanical behavior, especially structural stability. Intuitively, the architecture in which the applied load spreads out more homogeneously may be more stable against that load. In other words, given a deformation condition, the accumulation of stress and strain on a localized region, such as an unavoidable defect, would lead to the earlier onset of concentrated deformation and load shedding, which would cause the architecture to be susceptible to be transformed by bifurcation instability. Thus, for actual and simplified structures, the structural stability is numerically investigated here using the dual-level FE method. The simplified structures with cylindrical and hexahedral struts are designed to have identical porosity (65%) to the actual structure and no unavoidable defects. Fig. 7 shows the computed maximum Mises stress profiles for the actual and simplified unit cells under uniaxial compression. The stress accumulation in the actual structure is the steepest, which indicates that the actual structure with unavoidable defects is more unstable than the other simplified structures when comparing uniaxial compression. This phenomenon is more prominent in the computed Mises stress distributions of the actual and simplified structures, as shown in Fig. 8. The stress accumulation occurs at symmetric local regions of the frame ligaments in the simplified structures whereas it develops at the localized irregularities without symmetry in the actual structure. These results demonstrate that unavoidable
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Fig. 5. Computed Mises stress distributions of unit cell of Ti foam during uniaxial compression in the dual-level FE model. Points O, P, Q and R denote each position in the stress–strain curve (marked in Fig. 4).
Fig. 6. Computed and measured images of Ti foam at different levels of applied strain. Points P, Q and R denote each position in the stress–strain curve (marked in Fig. 4).
defects in the actual structure that are developed necessarily during the actual process would affect mechanical stability significantly. Furthermore, it should be noted that if we assess the general structural stability of the porous material under any mechanical test, the unavoidable defects, which could lead to the earlier onset of concentrated deformation, should be considered. Certain aspects of the stress accumulation are also sensitive to the shape of the strut. The simplified structure with cylindrical struts is more unstable than that with hexahedral struts (Fig. 7), even though these structures have the same cross-sectional area
along the considered direction. It should be noted that the stress accumulation occurs at similar local regions of the frame ligaments in these two simplified structures, as shown in Fig. 8b and c. To confirm the effect of given a deformation condition on the structural stability, we also computed the mechanical behavior of the Ti foam under volumetric (isotropic) compression. Fig. 9 show the computed total and top 0.1% average Mises stress profiles of the actual unit cell under uniaxial and volumetric compression. For the volumetric compression of the actual unit cell, stress accumulation is less steep than the uniaxial compression in the initial
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Fig. 7. Predicted maximum Mises stress profiles for actual and simplified unit cells upon uniaxial compression. Red and blue lines indicate cylindrical and hexahedral struts, respectively. The maximum Mises stresses were obtained from the averaged stress in 0.1% of elements that indicated the highest Mises stress. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
elastic region, thus indicating that the bulk modulus is smaller than the Young’s modulus. From the elastic region, we can also calculate effective bulk modulus under volumetric compression, but it is noted that in the dual-level FE, the applied deformation condition may be not perfectly matched the volumetric compression since we used the deformation gradient histories of the region of interest. In the plastic region, the maximum stress histories are nearly similar which means there is no significant difference of structural stability, especially in top 0.1% average Mises stress profiles, on the given deformation conditions such as uniaxial and
Fig. 9. Predicted total and top 0.1% average Mises stress profiles of actual unit cell upon uniaxial and volumetric compression.
volumetric compression. This phenomenon might be caused by the stress saturation in the plastic region while the concentrated stress region is expanding. 4.3. Buckling analysis of actual and simplified structures Buckling is characterized by the sudden failure of a structural member subjective to high compressive stress. Thus, it is extremely important to investigate buckling of 3D periodic porous structures when assessing structural stability. An imperfection in the form of the first buckling mode (determined using a linear buckling analysis) was introduced in the initial structures, and the response of the RVE was considered under uniaxial compression. Fig. 10a presents
Fig. 8. Computed Mises stress distributions of (a) actual and simplified structures with (b) cylindrical and (c) hexahedral struts under uniaxial compression. Point S denotes a position in the maximum Mises stress profiles (marked in Fig. 7).
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Fig. 10. (a) Predicted maximum Mises stress profiles for actual and simplified unit cells with imperfections upon uniaxial compression. Computed Mises stress distributions of (b) actual and simplified structures with (c) cylinder and (d) hexahedron struts at point T.
Fig. 11. (a) Predicted maximum Mises stress profiles with an internal angle; and (b) maximum Mises stress values at point S (marked in Fig. 7) versus a diameter of strut for simplified unit cell having cylinder struts upon uniaxial compression.
the computed maximum Mises stress profiles for the actual and simplified unit cells under uniaxial compression. The computed profiles are nearly similar to each other because the stress accumulation occurs primarily at the imposed imperfection region. In other words, a given imperfection excludes the possibility of the simulation to follow an unstable deformation path, and consequently the structural stability of the actual structure would be less vulnerable to unavoidable defects. It should be that an unavoidable defect on an actual structure would act as the imperfection of the structure under typical mechanical tests. The deformed shape induced by the buckling varies with the geometry of the structure (Fig. 10b–d), which is in agreement with the literature [36,37]. The deformed shape is remarkably different from the typical behavior of the structure under uniaxial compression (Fig. 8a–c) due to the imposed imperfection. Additionally, local regions in which stress accumulation occurs are completely different from the above mentioned results (Fig. 8a–c) despite the identical boundary conditions, except for the artificial imperfection. 4.4. Effects of process variables on structural stability As a final application of the dual-level FE simulation in this report, we would like to investigate the effects of certain selected process variables on the structural stability under uniaxial compression. Since an analysis for the simplified structure is sufficient to predict the effects of process variables on the structural stability, only the simplified structure with cylindrical struts, which is closer
to the actual structure in terms of stability (Fig. 7), is used for efficiency of computation. First, the effect of an internal angle on the structural stability was analyzed. The internal angle signifies a half angle between each strut on the cross section, as indicated in the inset of Fig. 11a, and this variable can be controlled in the SLM process. As the internal angle increases, the maximum Mises stress increases more rapidly (Fig. 11a), and consequently the structure with the larger angle is more unstable. This phenomenon can be explained by the thickness of the center body of the unit cell. If the internal angle is smaller, the thickness of the center body is larger, which indicates a larger stiffness of the unit cell. Therefore, the structure with the larger stiffness is less vulnerable to the given deformation mode. Additionally, the sensitivity of the structural stability on a diameter of the strut was examined as indicated in Fig. 11b. The diameter is determined using laser properties, such as beam diameter, dwell time and energy. The maximum Mises stress decreases as the diameter increases, and this result indicates that the structure with the larger diameter is more stable. Since the thickness increases when the diameter increases, this phenomenon can be explained by the above mentioned reason. 5. Conclusions In this report, we presented a new methodology for the structural analysis of periodic foams based on the X-ray
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microtomography and dual-level FE methods. The 3D structure of the periodic Ti foam with an octahedral structure is obtained using the X-ray microtomography, and the dual-level FE method based on the 3D structure provides an efficient and accurate numerical solution using the UFE and LFE solvers. The computed results are compared directly with the interrupted uniaxial compression experiments performed on the Ti foam. The three regions typically observed in the stress–strain curve, which is created from the architectural transformation during uniaxial compression, are reproduced well using the method. Additionally, the method predicts the deformed shapes in excellent agreement with the observations. These results validate the implemented dual-level FE method. The structural stability of the periodic foam based on the existence of unavoidable defects and the shape of the strut is quantitatively examined by having the simplified structures with cylindrical and hexahedral struts. The computed results indicate that the unavoidable defects on the actual 3D structure affect the mechanical stability significantly, and certain aspects of the stress accumulation are sensitive to the shape of the strut. The buckling-induced deformation behavior of the actual and simplified structures is analyzed by introducing an imperfection. This indicates that an unavoidable defect on an actual structure would act as the imperfection of the structure under a typical mechanical test. The effects of certain selected process variables, such as internal angle and diameter, are examined through a series of process simulations. This process confirms that the stability of the structure is more stable when the internal angle is smaller and the diameter is larger. The presented studies demonstrate sufficient possibilities, which can improve the deformability of functional periodic porous materials only by using a rigorous numerical model. Furthermore, the insights gained by performing a numerical parametric exploration serve as an important design guideline in fabricating practical materials with excellent structural function and robustness. Acknowledgements This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (NRF-2013R1A2A2A01008806). H.H.C. would like to acknowledge Prof. Peter Lee and his colleagues at the Diamond-Manchester Collaboration for hosting me and helping with the XMT portion of this work. References [1] L.J. Gibson, M.F. Ashby, Cellular Solids: Structure and Properties, second ed., Cambridge University Press, Cambridge, New York, 1997. [2] J.B. Thompson, J.H. Kindt, B. Drake, H.G. Hansma, D.E. Morse, P.K. Hansma, Bone indentation recovery time correlates with bond reforming time, Nature 414 (2001) 773–776. [3] G.E. Fantner, T. Hassenkam, J.H. Kindt, J.C. Weaver, H. Birkedal, L. Pechenik, J.A. Cutroni, G.A.G. Cidade, G.D. Stucky, D.E. Morse, P.K. Hansma, Sacrificial bonds and hidden length dissipate energy as mineralized fibrils separate during bone fracture, Nat. Mater. 4 (2005) 612–616. [4] E. Munch, M.E. Launey, D.H. Alsem, E. Saiz, A.P. Tomsia, R.O. Ritchie, Tough, bio-inspired hybrid materials, Science 322 (2008) 1516–1520. [5] R.M. Erb, R. Libanori, N. Rothfuchs, A.R. Studart, Composites reinforced in three dimensions by using low magnetic fields, Science 335 (2012) 199–204. [6] K. Jin, Y. Tian, J.S. Erickson, J. Puthoff, K. Autumn, N.S. Pesika, Design and fabrication of gecko-inspired adhesives, Langmuir 28 (2012) 5737–5742. [7] M.A. Meyers, J. McKittrick, P.Y. Chen, Structural biological materials: critical mechanics-materials connections, Science 339 (2013) 773–779. [8] D.C. Dunand, Processing of titanium foams, Adv. Eng. Mater. 6 (2004) 369–376. [9] J.L. Fife, J.C. Li, D.C. Dunand, P.W. Voorhees, Morphological analysis of pores in directionally freeze-cast titanium foams, J. Mater. Res. 24 (2009) 117–124. [10] J.Y. Cho, K.Y. Kim, Structure and compressive strength of silicon open-cell foam obtained by a centrifugal separation method, Met. Mater. Int. 19 (2013) 361–365.
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