Defect-Tolerant Bioinspired Hierarchical Composites: Simulation and

Mar 3, 2015 - Laboratory for Atomistic and Molecular Mechanics, Department of Civil and Environmental ... Engineering Fracture Mechanics 2017 183, 79-...
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Defect-Tolerant Bioinspired Hierarchical Composites: Simulation and Experiment Reza Mirzaeifar,† Leon S. Dimas, Zhao Qin, and Markus J. Buehler* Laboratory for Atomistic and Molecular Mechanics, Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, United States S Supporting Information *

ABSTRACT: Defect tolerance, the capacity of a material to maintain strength even under the presence of cracks or flaws, is one of the essential demands in the design of composite materials, as manufacturing induced defects, or those generated during operation, can lead to catastrophic failure and dramatically reduce the mechanical performance. In this paper, we combine computational modeling and advanced multimaterial 3D printing to examine the mechanics of several different classes of defect-tolerant bioinspired hierarchical composites, built from two base materials with contrasting mechanical properties (stiff and soft). We find that in contrast to the brittle base constituents of the composites, the existence of a hierarchical architecture leads to superior defect-tolerant properties. We show that composites with more hierarchical levels dramatically improve the defect tolerance of the material. We also examine the effect of adding both self-similar and dissimilar hierarchical levels to the materials architecture, and show that the geometries with multiple hierarchical levels can retain a significant portion of their fracture strength in the presence of either large edge cracklike flaws or multiple small distributed defects in the material. We compare the stress distributions in materials with different numbers of hierarchies in both simulation and experiment and find a more uniform stress distribution in the uncracked region of materials with higher hierarchy levels. These results provide micromechanical insights into the origin of the higher defect tolerance observed in simulation and experiment. KEYWORDS: bioinspired design, defect tolerance, hierarchical composite, fracture toughness, 3D printing

1. INTRODUCTION A large class of biological materials with excellent mechanical properties such as spider silk, bone, nacre, and sea sponge exoskeletons are made of simple constituents with relatively weak mechanical properties.1−4 In recent years, the development of theoretical and computational bottom-up approaches for understanding the mechanical properties of hierarchical materials has been an active field of research.5−11 The enhancement in the mechanical properties in these materials is a consequence of the complex architecture used by nature to construct the biocomposite from some simple and weak building blocks. For example, experimental observations have shown that the fracture response of some mineralized biological materials, including bone and nacre, is strongly affected by the fracture mechanisms at the nanocomposite structure of these materials made of staggered hard platelike inclusions (mineral phase) in a soft matrix (organic phase).12−14 Depending on the specific functionality of the material (i.e., load-bearing, skeletal support, armor protection) the geometry is used as a design parameter and the biological materials response is improved for that specific application. Instead of forming complex irregular structures, natural materials are often composed of hierarchical structures, having building blocks on the same level assembled in the similar way and forming the building blocks for the next level. This design strategy simplifies the recipe but still retain © XXXX American Chemical Society

the design space to achieve complex architecture for different functions, specifically for superior mechanical functions. Here we focus on hierarchical structures and study how they affect the materials’ mechanical property in this work. The central hypothesis about the defect tolerance of bone and bonelike materials is that the nanostructure of these materials have been evolved to a specific hierarchical staggered arrangement of stiff and soft materials, which leads to a more uniform distribution of the stresses in the uncracked material region.14−17 In a hypothetical ideal state of evolved biological material under the presence of multiple cracklike flaws, the hierarchical material fails by a uniform rupture, rather than by crack propagation because of the stress delocalization at the crack tips and the uniform stress distribution facilitated by the nanostructural architecture. Such capability to retain the similar mechanical strength as their bulk material in spite of the existence of the crack is understood as defect tolerance, which is the focus of this study. We have recently used a bottom-up approach to systematically examine the effects of multiple hierarchical structures on fracture toughness of microscaled particle reinforced compoReceived: November 17, 2014 Accepted: March 3, 2015

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DOI: 10.1021/ab500120f ACS Biomater. Sci. Eng. XXXX, XXX, XXX−XXX

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ACS Biomaterials Science & Engineering

this definition, the defect tolerance is a measure of the percentage of the strength that is retained in the material with defects compared to the initial material without any defects.

sites by investigating how crack propagation is modified solely through the presence of hierarchical structures and without adding additional materials or mechanisms.18 In this work, we use a similar computational approach for studying the effect of multiple hierarchical levels on the defect tolerance of large scaled composites by examining several different self-similar (bonelike) and dissimilar (biocalcitelike) geometries. In the self-similar designs, all the hierarchical levels have the same pattern of stacking the stiff and soft phases repeated at different scales. Self-similarity in the geometry can improve the defect tolerance in the material by extending the stress delocalization into different length scales. In many biological materials such as bone, nacre, and sea sponge exoskeleton, the stacking of constituent phases at different hierarchical levels shows differences from the others. Aside from self-similar samples, we also investigate dissimilar hierarchical structures in this paper. The proposed designs in this paper do not aim at an exact duplication of biomaterials, but mostly focus on selecting some key features in each of the mentioned biomaterials and designing composites with architectures inspired by the key properties of these materials. Our focus in this work is on two major microstructural properties of mineralized biological materials, which are hierarchy and the staggered arrangement of constituents. The bioinspired designs also mimic some key properties of the biomaterial including, the high ratio of stiffness between the stiff and soft constituents, and the high aspect ratio of stiff platelets with an optimized thickness of platelets to prevent failure of the stiff phase before sliding happens between the two phases.8,19 In recent years, besides using theoretical and computational methods for understanding the fundamental design principals of biocomposites, using similar geometrical architectures to design and manufacture new bioinspired materials with superior mechanical properties has attracted much attention.12,18−23 With the recent developments in advanced manufacturing methods, some efforts have been reported on using these bioinspired designs and synthesizing composites with impressive mechanical characteristics from simple building blocks using self-assembly, layer-by-layer (LBL) templating, and 3D printing.19,21,24,25 Among these methods, 3D printers that are capable of simultaneously printing multiple materials of contrasting mechanical features represents an efficient tool for manufacturing cost-effective composites at large scales. In this work, we use multimaterial 3D printing to manufacture composites with several different bioinspired topologies with different levels of hierarchical structures. The 3D printed samples are then tested and the defect tolerance is studied for each design and number of hierarchy levels by cutting precracks with different sizes on the sample and comparing the strength with an original uncracked sample. The experimental results are consistent with the computational findings and the fabricated hierarchical composites exhibit structural properties far superior to their constituents. The major focus in this work is studying the improvement of defect tolerance in the material by increasing the number of hierarchical levels. The defect tolerance is directly related to resistance to fracture for a given amount of crack advance length. A theoretical method for studying this property is using J-integral and studying the Rcurve behavior of the material, which was reported in.18 In this work, we use the physical interpretation of defect tolerance by comparing the fracture strength of uncracked samples with the fracture strength in samples with different precrack sizes. By

2. METHODS In this work, we use both computational models and experimental methods for studying the mechanical properties of the bioinspired composites with different levels of hierarchy. The computational model is used for verifying the effect of hierarchical structure on the mechanical properties before fabricating the final samples and performing the experimental tests. 2.1. Computational Model. We have developed and implemented an efficient computational model for studying the mechanical properties of various hierarchical composites in a series of recent works.18−21,26 The same computational model is used in this paper to study the effect of multiple hierarchical structures on the mechanical response of bioinspired particle reinforced composites. The models is based on constructing a triangular spring-bead lattice. The neighbor beads are connected to each other by a linear spring that exerts equal forces to the beads by a Hookean force−extension law for out of equilibrium deformations. The spring constant depends on the location of beads, whether they are in the soft or stiff region, and the values of spring constants are obtained from the simple tension experimental tests on each of the phases individually. Because the computational model is mainly designed and employed for quantitatively studying the effect of hierarchical levels on the mechanical response, we do not attempt to precisely simulate the experimentally observed materials response in the model. We instead try to follow the general trend of the response and simplify the nonlinear response of the polymers in the experiments for constructing an efficient and simplified tool for predicting the overall response of the designed hierarchical composite before 3D printing the final samples. Mechanical tests are performed on both the stiff and soft base materials individually. A key property in constructing the computational model is the stiffness ratio between the soft and stiff phases, which is practically difficult to measure due to the mixing at the interfaces that happens during 3D printing the samples. We use a simplified model based on the Voigt’s rule of mixtures by assuming an approximate 3−4% mixing of the two base materials at the interface. The effective stiffness ratio is estimated to be between 45 and 60 by this method, and we select an intermediate value of 50 for our computational model. More details of this procedure are given in ref 19. To investigate the fracture mechanics in the simulations, we considered a strain-controlled failure in the bead−spring model. The experimental results show an extensibility of 10−25% for the stiff phase and 170−220% for the soft phase. However, our previous studies have shown that assigning such a large extensibility for the soft phase makes the model highly nonlinear and unstable. On the other hand, potential mixing at interfaces may also affect the fracture strain of each phase. To maximize the extensibility of the compliant phase without compromising the stability of the model, we adopted a 5% breaking strain for the stiff phase. The two phases are designed to have identical toughness modulus, which results in obtaining the extensibility of the soft phase as 35%. In the computational model, each spring depending on its assigned material type, is broken if the axial strain reaches these critical values during loading. This class of discrete network models has been used for studying the fracture mechanics of various materials particularly to study the influence of randomness and disorder on the fracture behavior of materials.27−30 Experimental measurements show the nonlinearities in the mechanical response of the photopolymers that have been used in the experimental part of this paper. Although the nonlinear response, particularly for the stiff phase with very high extensibility, can affect the particular crack propagation in the designed composite, we argue that the dominating fracture and deformation mechanisms mainly depend on the stiffness ratio and relative extensibilities of the composite constituents, and the qualitative computational results obtained from the simplified model can predict the mechanical response of the B

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Figure 1. (a) Schematic of the six different designs. The bonelike samples are composed of hierarchically ordered stiff platelets (shown with dark blue color in the figure), in a continuous soft phase (gray regions in the figure). The two-hierarchy (2H) biocalcitelike sample is constructed by arranging small particles of the soft phase (dark regions in the figure) in a continuous stiff phase (shown by the lighter color in biocalcitelike designs). The three- and four-hierarch level designs in the biocalcitelike samples are a combination of the bonelike and biocalcitelike designs with dissimilar building blocks. (b) Schematic of samples with edge cracks, and (c) the stress−strain response of stiff and soft phases in the computational model. The stress−strain response of some representative (d) bonelike and (e) biocalcitelike samples. The solid lines show the response of uncracked samples for different hierarchy levels and the dashed lines show the response for samples with an edge notch with 30% the width of sample. For bonelike samples, by increasing the number of hierarchy levels from two to three, the fracture strength decreases by almost 20%. However, increasing the hierarchy level from three to four does not have a significant effect on the fracture strength. For the biocalcitelike samples, the fracture strength of the two-hierarchy design is remarkably higher than all the other structures, mainly because this is the only design with the stiff material as the continuous phase. In both cases, the response of precracked sample is closer to the original uncracked sample at higher hierarchy levels. phase to 200 μm, remarkably larger than the printer’s resolution (of 16 μm). The volume ratio between the stiff and soft phases is kept constant in all the designs to make them comparable, and the thickness of soft regions in each design is calculated by this constraint. The dual material digital printing method is selected for printing all the samples. Four aluminum strips are attached to both faces of the samples in the sides using Loctite E-90FL epoxy in order to facilitate applying a uniform load to the end of each sample in the tensile machine. The aluminum strips were positioned such that all samples had the same effective length. Three samples are printed for each of the six designs (total of 18 samples are printed). One sample is tested without a pre crack. Two other tests are performed on samples with a precrack of 10 and 30% the total width of the specimen. The notches were cut with a 1/32″ (0.79 mm) thick carbide-slitting saw with a 60° included angle. For obtaining the experimental strain fields through digital image correlation (DIC), Speckle patterns are spray-painted on one side of all the samples with a range of gray tones to assist the DIC. The commercial software VIC-2D is used for analyzing the images. The printed specimens are tested in an Instron 5582 Universal Testing Machine with an Instron 100 kN static load cell. The uniaxial tensile tests are performed by applying displacement boundary conditions. The samples are clamped in place with serrated hardened steel grip faces attached to steel vice action grips. The load capacity of the grips is 100 kN and the spring stiffness of the entire testing device far exceeds the stiffness of our specimens. Depending on the precrack size, the uncracked length of the sample is centered with the force

printed systems with an acceptable accuracy. This argument is validated in our previous works, and by the experimental results in the present work, as will be further studied in the following. 2.2. Experimental Section. In this work we follow a procedure similar to the experiments in ref 19. An Objet Connex500 multimaterial 3D printer is used for manufacturing six different designs for bioinspired composite materials. Two polymers with distinct mechanical properties are selected for printing hierarchical samples with contrasting stiff and soft constituents. VeroWhitePlus and TangoBlackPlus polymers are used as the stiff and soft phases in manufacturing the composites, respectively. Both the selected materials are photopolymers, based upon proprietary acrylic-based photopolymer resins. A schematic of the six different designs is shown in Figure 1a. The bonelike samples are composed of hierarchically ordered stiff platelets (shown with dark blue color in the figure), in a continuous soft phase (gray regions in the figure). In the bonelike samples, the three- and four-hierarchy levels are modeled by constructing a hierarchical structure inside each building block as shown in Figure 1a. The two-hierarchy (2H) biocalcitelike sample is constructed by arranging small particles of the soft phase (dark regions in the figure) in a continuous stiff phase (shown by the lighter color in biocalcitelike designs). The three- and four-hierarchal level designs in the biocalcitelike samples are a combination of the bonelike and biocalcitelike designs with dissimilar building blocks as shown in Figure 1a. The out-of-plane thickness of all the geometries is 5 mm. To obtain high-quality interfaces, we set the smallest thickness of the soft C

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Figure 2. (a) Crack propagation for the six designs with a precrack 30% the total length, and (b) the fractured shape of same samples without the precrack. By considering the uncracked sample as the reference for each design, the relative decrease of fracture strength by creating a notch with different lengths on each sample is measured and compared for (c) bonelike and (d) biocalcitelike samples. In both cases, the results are also compared with the single stiff phase. The individual points are obtained directly from the computational model, and the continuous lines are polynomial curves fitted to the data points. The results for both the (c) bonelike and (d) biocalcite like designs shows the significant improvement of the defect tolerance by increasing the hierarchy level, while a sample made of the single stiff phase loses approximately 95% of its strength by having a notch with 20% the width of sample (the cracked sample maintains less than 5% of the original strength), the four-hierarchical bonelike and biocalcitelike designs maintain approximately 70 and 80% of their strength by having the same precrack size, respectively.

1d, e. The solid lines in these figures show the response of uncracked samples for different hierarchy levels. As shown in Figure 1e for bonelike samples, by increasing the number of hierarchy levels from two to three, the fracture strength decreases by almost 20%. However, increasing the hierarchy level from three to four does not have a significant effect on the fracture strength. For the biocalcitelike samples, the fracture strength of the two-hierarchy design is remarkably higher than all the other structures, mainly because this is the only design with the stiff material as the continuous phase. Having the continuous stiff phase in 2H biocalcitelike sample is also reflected in the brittle fracture of this sample with a sharp drop in the stress value at the fracture point. As will be shown in the following experimental results, this phenomenon is also observed in the experiments. The fracture strength of the biocalcitelike samples also slightly decreases by adding a hierarchy level to the 3H design. The loss of a fraction of the fracture strength by increasing the hierarchical levels in the structure have been observed in our previous computational studies on microscaled samples as well.18 However, as will be studied further studied in the following, the slight decrease in the fracture strength is associated with a remarkable improvement in the defect tolerance in all designs. It is worth noting that designing a hierarchical structure which maintains the fracture strength besides improving the defect tolerance remains and open research topic. A possibility to be explored in future communications is designing the materials based on a

applied through the vice action grips to ensure pure tension in the specimens prior to initial crack propagation. In calculating the reported stresses in all the samples, the nominal uncracked cross-section is considered. A displacement rate of 3 mm/min is selected in all the experiments to ensure the validity of quasi-static loading conditions. It is worth noting that the selected loading rate is considered a slow rate, because the constituents in the composite are highly extensible and fairly large deflections are required before the composite breaks. The stable crack propagation in all the experiments approves the validity of the selected loading rate.

3. RESULTS AND DISCUSSIONS 3.1. Computational Results. We study six different designs as shown in Figure 1a. Three different hierarchy levels are considered for each of the bonelike and biocalcitelike designs as shown in this figure. In order to study the defect tolerance in each design, we study the uncracked samples and compare their mechanical response with other samples with different sizes of an edge notch, schematically shown in Figure 1b. The mechanical loading is simulated by fixing one end and applying a quasi-static displacement controlled loading to the other end. As explained in section 3, the mechanical response of both the stiff and soft phases is assumed linear with fracture strains of 5 and 35%, respectively. The stress−strain responses of the stiff and soft phases are shown in Figure 1c. We have studied different sizes of notches up to 35% the total width of the sample (Lcr/L = 0.35 in Figure 1b). The stress−strain response of several representative samples are shown in Figure D

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Figure 3. (a) Crack propagation in a 2H bonelike sample with five small cracks distributed in the structure subjected to a uniaxial stress. It is shown that how the hierarchical structure impedes the crack propagation. These small cracks in a homogeneous single phase propagate rapidly and join each other to create larger defects which leads to a sudden fracture of the material and a remarkable decrease in the strength. (b) Stress−strain response for three different hierarchy levels in the presence of multiple cracks. Increasing the hierarchy level decreases the strength in each sample slightly. However, this increase is associated with a remarkable improvement in the defect tolerance, as shown in c. Although a homogeneous stiff phase can retain only 12% of its original strength in the presence of distributed small defects, the two-, three-, and four-hierarchical structures maintain approximately 50, 58, and 60% of the original strength in a sample with multiple cracks.

a sample made of the single stiff phase loses approximately 95% of its strength by having a notch with 20% the width of sample (the cracked sample maintains less than 5% of the original strength), the two-, three-, and four-hierarchical designs maintain approximately 50, 60, and 70% of the original strength in the presence of a precrack with the same length. This significant improvement is also observed for the biocalcitelike samples as shown in Figure 2d. The two-hierarchy level biocalcite-design has the least improvement and the general trend of the fracture strength reduction follows the single stiff phase, mainly because the stiff material is the continuous phase in the structure, and as previously shown in Figure 1, the fracture mechanism is very similar to the brittle fracture of the stiff material. However, both the three- and fourhierarchical biocalcitelike samples show a significant improvement in defect tolerance. Comparing the three- and fourhierarchical biocalcitelike samples with the same bonelike designs also shows a slight improvement in the defect tolerance in biocalcitelike designs. In practice, manufacturing each part is associated with a series of processing methods that can cause the creation of several small defects in the structure. The material used for manufacturing each part also contains several initial defects. These process-induced initial defects are a major source for initiation and propagation of microcracks in the structure which can significantly weaken the materials response in practice, compared to the ideal strength predicted in the design. To study the effect of having a hierarchical design on the fracture strength of samples with multiple small defects, we have considered a case study with different hierarchy levels and five small cracks distributed in the structure, as shown schematically in Figure 3a. The samples are subjected to a uniaxial stress, similar to the previous case study, and the stress−strain response and the crack propagation in each design is compared

more realistic architecture of bone. The stress−strain response of samples with a large precrack (30% the width of sample) are also shown in Figure 1d. e for the bonelike and biocalcitelike samples, respectively. To better understand the effect of hierarchical levels on the defect tolerance of different designs, we measure the fracture properties of all samples with various sizes of precracks and compared the stress−strain response with the case of having no hierarchy level on the sample (i.e., the samples made of a single soft phase). It is observed that the hierarchy on the structure impeded the propagation of the precrack in the structure and structures with higher hierarchy levels can maintain a larger portion of their strength even in the presence of large precracks in the structure. The crack propagation for all the six designs is shown in Figure 2a for samples with a precrack 30% the total length. The fractured shape of same samples without the precrack are also shown in Figure 2b. By considering the uncracked sample as the reference for each design, the relative decrease of fracture strength by creating a notch with different lengths on each sample is measured and compared in Figure 2. The results for both the bonelike and biocalcitelike samples are also compared with the single stiff phase in this figure. Figure 2c shows the relative decrease in fracture strength as a function of the precrack length up to 35% the total with of the sample. The individual points are obtained directly from the computational model, and the continuous lines are polynomial curves fitted to the data points. The minor fluctuations in the approximated curves can be mitigated by adding more data points to the graph. However, the presented fitted curves can show the general monotonic trend in the data points with an acceptable accuracy. The results for both the bonelike and biocalcite like designs shows the significant improvement of the defect tolerance by increasing the hierarchy level. As shown in Figure 2c, although E

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Figure 4. (a) Stress−strain response for the uncracked and precracked 3D printed bonelike samples for different hierarchy levels. As shown, the fracture in all the samples happens in multiple steps with two peaks in the stress−strain response (see the text for discussion), in contrast with the simplified computational simulations which was based on considering a linear response for the material up to the breaking point. This phenomenon is mostly caused by the nonlinear response of the actual printed material, the higher extension capacity of the soft phase and also the effect of mixing in the interfaces. (b) Top view of printed samples with three different bonelike hierarchical structures. (c) Relative decrease of strength for the two different precrack sizes for each of the designs. In each case, the relative decrease is calculated by comparing the strength of cracked sample with the strength of the original sample without a precrack. For comparison purposes, the same results are also shown for a homogeneous sample made of the stiff phase. As shown, although the homogeneous sample maintains less than 10% of its original strength by cutting a large precrack (30% the width) on the sample, the four-hierarchy sample maintains almost 80% of its strength with the same large precrack. (d) Snapshots of the crack propagation in the bonelike samples in a point near the final. In the lower hierarchy samples, the crack starts locally propagating from the notch tip in both small and large (10% and 30%) precracks, whereas the crack propagation is remarkably delocalized in the higher hierarchy levels.

the more significant defect tolerance associates with the higher hierarchical level of the composite material, we now focus on experimental analyses and compare the in silico observations with tensile test results on 3D printed models with the same architecture as used in the simulations. 3.2. Experimental Results. All the studied designs are 3D printed with the method explained in Section 2.2. For each of the designs (three hierarchy levels for bonelike and biocalcitelike composites), three different samples are prepared, one without any precrack and two samples with a notch that is 10 and 30% of the total width of the sample. The defect in each design is examined by applying a uniaxial tensile load to the samples and measuring the stress−strain responses. A summary of the experimental results for the bonelike samples is shown in Figure 4. The stress−strain response for the uncracked and precracked samples is shown in Figure 4a and the responses are compared for different hierarchy levels. As shown in this figure, the fracture in all the samples happens in multiple steps, in

to a homogeneous single phase sample. Figure 3a shows the crack propagation pattern in a sample with two hierarchy levels. It is clearly shown that how the hierarchical structure impedes the crack propagation. These small cracks in a homogeneous single phase propagate rapidly and join each other to create larger defects which leads to a sudden fracture of the material and a remarkable decrease in the strength. The stress−strain response for three different hierarchy levels in the presence of multiple cracks is shown in Figure 3b. Similar to the previous case study, increasing the hierarchy level decreases the strength in each sample slightly. However, this increase is associated with a remarkable improvement in the defect tolerance as shown in Figure 3c. As shown in this figure, although a homogeneous stiff phase can retain only 12% of its original strength in the presence of these five distributed cracks, the two-, three-, and four-hierarchical structures maintain approximately 50, 58, and 60% of the original strength in a sample with multiple cracks. To validate what we have shown here that F

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Figure 5. (a) Stress−strain response for the uncracked and precracked 3D printed biocalcitelike samples for different hierarchy levels. The stress− strain response of 2H samples is close to the brittle fracture, mainly because the stiff material forms the continuous phase in the sample. The fracture in 3H and 4H samples happens in multiple steps with two peaks in the stress−strain response (see the text for discussion), in contrast with the simplified computational simulations, which was based on considering a linear response for the material up to the breaking point. This phenomenon is mostly caused by the nonlinear response of the actual printed material, the higher extension capacity of the soft phase and also the effect of mixing in the interfaces. (b) The top view of printed samples with three different biocalcitelike hierarchical structures. (c) The relative decrease of strength for the two different precrack sizes for each of the designs. In each case the relative decrease is calculated by comparing the strength of cracked sample with the strength of the original sample without a precrack. For comparison purposes the same results are also shown for a homogeneous sample made of the stiff phase. As shown, while the homogeneous sample maintains only less than 10% of its original strength by cutting a large precrack (30% the width) on the sample the four-hierarchy sample maintains almost 70% of its strength with the same large precrack. (d) snapshots of the crack propagation in the biocalcitelike samples in a point near the final. In the 2H sample, a brittle fracture with sudden propagation of the crack is observed. In the 3H and 4H samples, the crack propagation is remarkably delocalized.

contrast with the simplified computational simulations which was based on considering a linear response for the material up the breaking point. This phenomenon is mostly caused by the nonlinear response of the actual printed material, the higher extension capacity of the soft phase and also the effect of mixing in the interfaces as explained in section 2.2. However, as will be shown in the following, the general findings in the computational section, particularly the effect of hierarchy on the defect tolerance is following the same trend as observed in the computational section. The top view of the printed samples with three different bonelike hierarchical structures is also shown in Figure 4b. Studying the curves in Figure 4a, two peak values are observed in the stress−strain response. The initial elastic linear response of the material is followed by a drop in the stress value which happens due to initiation of some small cracks in the structure. However, in contrast with a

homogeneous phase, these cracks are not propagated rapidly in the structure and the hierarchical structure blocks the initiated cracks, which is followed by stretching the material along with gradual propagation of cracked regions in the sample. At a specific point for each sample, either the propagated small cracks join to create a larger crack or the initial notch in the structure starts rapidly propagating and the sample breaks with a relatively sharp drop in the stress−strain curves. Comparing the stress−strain curves for different samples, it is observed that as predicted by the computational results, the strength is decreased at higher hierarchy levels. However, we observe that although the 2H samples lose a remarkable portion of their strength by cutting a large notch on the sample, higher hierarchy levels show a very low sensitivity even to large notches. Comparing the stress−strain responses for a 4H G

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hierarchical levels in the design, the sensitivity to the initial defect is remarkably reduced, and as shown for the 4H samples, the stress−strain response is almost not affected by cutting the 10 or even 30% notches. The relative change of strength for samples with the precrack compared to the original uncracked samples is shown in Figure 5c. It is obvious that all the three designs have significantly improved compared to the homogeneous single phase material, and also increasing the hierarchy level enhanced the defect tolerance capability. A snapshot of each sample after propagation of the crack in the material is shown in Figure 5d. Comparing the crack shape in different hierarchy levels shows the transformation of a localized crack propagation at low hierarchy levels into a distributed crack formation in the whole material at higher hierarchy levels. The localized crack propagation in the 2H biocalcitelike samples is consistent with the brittle fracture response of this design as discussed earlier. A more detailed history of crack propagation in each design is also shown in the Supporting Information for the biocalcitelike samples. To better understand the underlying micromechanics, and specifically the stress distribution in the higher hierarchy composites, we measure the strain field in the samples by digital image correlation (DIC). Figure 6 shows the strain fields

sample with different sizes of precracks clearly shows that the mechanical response is not affected by the notch on the sample, which hereby approves the significant improvement of defect tolerance property caused by higher level hierarchical structures. The relative decrease of strength for the two different precrack sizes for each of the designs is shown in Figure 4c. In each case, the relative decrease is calculated by comparing the strength of cracked sample with the strength of the original sample without a precrack. For comparison purposes, the same results are also shown for a homogeneous sample made of the stiff phase. As depicted in Figure 4c, the overall change of the relative decrease for different hierarchy levels is in a good agreement with the computational calculations. Compatible with the computational results, the homogeneous sample maintains only less than 10% of its original strength by cutting a large precrack (30% the width) on the sample. However, the defect tolerance is dramatically improved in the samples with higher hierarchy levels, such the four-hierarchy sample maintains almost 80% of its strength even with the large precrack, as demonstrated in Figure 4c. For further detail, Figure S1 depicts the crack propagation paths in all tested bonelike and biocalcitelike samples with and without the precracks. These analyses show that the localized crack initiation and propagation in low hierarchy levels change toward a distributed crack propagation mechanism at higher hierarchy levels, which consequently results in a significant improvement in the defect tolerance of these designs. The crack propagation in all bonelike samples is shown in Figure 4d. The snapshots shown in this figure correspond to a near final fracture situation when all the small cracks and also the initial notch (in precracked samples) is propagated in the sample. A series of more complete snapshots during the initiation and propagation of cracks in all the bonelike samples is given in the Supporting Information. As shown in Figure 4d and the figures in the Supporting Information, in the lower hierarchy samples, the crack starts locally propagating from the notch tip in both small and large (10 and 30%) precracks. However, crack propagation is remarkably delocalized in the higher hierarchy levels, and for the 4H sample, instead of a localized crack tip propagation, several smaller cracks initiate and propagate in the sample. This phenomenon is due to the ability of hierarchical structure in delocalizing the stress distribution in the material, which will be further studied by the DIC results and the strain distributions in different samples in the following. The stress delocalization in high hierarchical designs and the distributed crack propagation pattern, in contrast to the low hierarchy designs of the homogeneous phase, is directly associated with the defect tolerance improvement is samples with high hierarchical levels, as studied in the previous sections. A similar experimental study is performed on the biocalcitelike samples with three different hierarchy levels and different sizes of precracks as shown in Figure 5. The stress−strain responses of cracked and uncracked samples for different hierarchy levels are shown in Figure 5a and the top view of 2H, 3H, and 4H biocalcite like samples are shown in Figure 5b. Studying the stress−strain responses clearly shows the difference in the response of 2H samples compared to all the other designs, mainly because of the continuous stiff phase as discussed earlier. The two hierarchy level biocalcitelike design loses a large portion of its strength by cutting a large notch on the sample as shown in Figure 5a, and the material response is close to a brittle fracture. However, by increasing the

Figure 6. Measured strain field in the homogeneous stiff phase and three different bonelike hierarchical designs by digital image correlation (DIC). All the strain fields are normalized to the maximum value to qualitatively compare the strain distributions in different designs and the homogeneous material. As shown, the homogeneous stiff phase exhibits a strain concentration around the crack tip, which is directly related to the stress concentration at that region, whereas strain, and consequently the stress, distribution in the areas far from the crack tip is remarkably lower than the crack tip. The strain distributions in the 2H sample shows a slight delocalization in the material, although the concentration around the around the crack tip is not completely resolved. However, for higher hierarchy levels (3H and 4H), the strain distribution clearly shows the effect of hierarchical structure in delocalizing the strain and stress distribution in the material.

obtained from DIC in the homogeneous stiff phase and three different bonelike hierarchical designs for comparison. All strain fields are normalized to the maximum value to qualitatively compare the strain distributions in different designs, as well as the homogeneous material. Figure 6 shows that the homogeneous sample exhibits a significant strain concentration around the crack tip, which is directly related to the stress concentration at that region. In contrast, the strain (and consequently the stress) distribution in areas far from the crack tip are remarkably lower than the crack tip. This distribution is consistent with the localized crack propagation mechanism observed in the homogeneous materials. The strain distributions in the 2H sample shows a slight delocalization in the material, although the concentration around the around the crack tip is not completely resolved. However, for higher hierarchy levels (3H and 4H), the strain distribution clearly H

DOI: 10.1021/ab500120f ACS Biomater. Sci. Eng. XXXX, XXX, XXX−XXX

Article

ACS Biomaterials Science & Engineering Present Address

shows the effect of hierarchical structure in dramatically delocalizing the strain and stress distribution in the material. In the 3H and 4H designs, instead of a single and well-defined strain (or stress) concentration around the crack tip, strain is concentrated around the vertical portions of the soft matrix distributed throughout the whole material. This delocalization by the hierarchical structure is the origin of defect tolerance improvement by increasing the hierarchical levels, in agreement with the results reported above.

†

R.M. is currently at Mechanical Engineering Department, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by BASF-NORA. The authors also acknowledge support from ARO through a MURI Award W911NF-09−1-0541, as well as ARO/ISN and ONR-PECASE.

4. CONCLUSIONS In this paper, we studied the effect of using multiple hierarchical levels in designing defect-tolerant materials. It was shown that the microscopic structure of certain mineralized biological materials, including bone and nacre, can be used to design composite materials made of simple building blocks with superior material properties, by using geometry as the only design parameter. Several different designs with multiple selfsimilar and dissimilar hierarchical levels are studied by both computational simulations and tensile tests on 3D printed models. We find that in the designs with a high number of hierarchy levels, a large portion of the original strength is retained in the presence of larger flawlike cracks or several small defects in the material. We have shown that precise 3D printing provides an avenue to fast prototype samples that feature a consistent trend of defect tolerance of different hierarchical structures, in agreement with simulations. These findings demonstrated that the fabricated hierarchical composites exhibit significant improvement in the defect tolerance and structural properties, far superior to those of each of their constituents. Defects, which can be either intrinsic at interface or introduced during the manufacturing process, are common in all composite materials. It is critical to understand how to reduce their negative effect on the mechanical property of the material. The work reported here shows that by increasing the hierarchical level, inspired from biological materials, one can expect to obtain enhanced defect tolerance in the composite material, both in theory and experiment. Such a robust material will be useful to prevent catastrophic failure, which is essential to enhance the reliability that is critical in many engineering applications. Moreover, our study has demonstrated that the superior mechanics is primarily caused by the geometric arrangement of two materials with contrasting stiffness, but does not critically depend on the chemistry or other properties of the building blocks. This makes the knowledge applicable to many different engineering materials.

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ASSOCIATED CONTENT

S Supporting Information *

The following file is available free of charge on the ACS Publications website at DOI: 10.1021/ab500120f. Crack propagation paths in all tested bonelike and biocalcitelike samples with and without the precracks (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +1-617-452-2750. Fax: +1-617-324-4014. Lab URL: http://web.mit.edu/mbuehler/ www/. I

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