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Bioinspired toughening mechanism: lesson from dentin

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Bioinspir. Biomim. 10 (2015) 046010

doi:10.1088/1748-3190/10/4/046010

PAPER

Bioinspired toughening mechanism: lesson from dentin RECEIVED

13 April 2015 REVISED

6 June 2015 ACCEPTED FOR PUBLICATION

15 June 2015

Bingbing An1,2 and Dongsheng Zhang1,2 1 2

Department of Mechanics, Shanghai University, Shanghai 200444, People’s Republic of China Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai, 200072, People’s Republic of China

E-mail: [email protected]

PUBLISHED

10 July 2015

Keywords: dentin tubules, two-layered structure, toughening mechanism, fracture process zone

Abstract Inspired by the unique microstructure of dentin, in which the hard peritubular dentin surrounding the dentin tubules is embedded in the soft intertubular dentin, we explore the crack propagation in the bioinspired materials with fracture process zone possessing a dentin-like microstructure, i.e. the composite structure consisting of a soft matrix and hard reinforcements with cylindrical voids. A micromechanical model under small-scale yielding conditions is developed, and numerical simulations are performed, showing that the rising resistant curve (R-curve) is observed for crack propagation caused by the plastic collapse of the intervoid ligaments in the fracture process zone. The dentin-like microstructure in the fracture process zone exhibits enhanced fracture toughness, compared with the case of voids embedded in the homogeneous soft matrix. Further computational simulations show that the dentin-like microstructure can retard void growth, thereby promoting fracture toughness. The typical fracture mechanism of the bioinspired materials with fracture process zone possessing the dentin-like structure is void by void growth, while it is the multiple void interaction in the case of voids in the homogeneous matrix. Based on the results, we propose a bioinspired material design principle, which is that the combination of a hard inner material encompassing voids and a soft outer material in the fracture process zone can give rise to exceptional fracture toughness, achieving damage tolerance. It is expected that the proposed design principle could shed new light on the development of novel man-made engineering materials.

1. Introduction The typical fracture mechanisms of ductile materials, such as metals and polymer-rubber blends, are void growth and coalescence (Gurson 1977, Kim et al 1996, Seidenfuss et al 2011), and therefore enormous efforts were devoted to the investigation of the process of nucleation, growth and coalescence of voids. Employing the cell models, it is found that the void growth is strongly dependent on the stress triaxiality and the spatial arrangement of voids plays a critical role in the plastic collapse behavior of voided materials (Kuna and Sun 1996). The reduction in the inter-void spacing leads to the development of a strain concentration, thereby accelerating void coalescence (Bandstra and Koss 2008). In addition to the stress triaxiality, the Lode parameter also exhibits significant influence on void growth; with the increase in Lode parameter, the critical localization strain increases (Barsoum and Faleskog 2011). The effect of mechanical properties of © 2015 IOP Publishing Ltd

matrix materials on void growth also receives great attention. For void growth in anisotropic plastic solids, the effect of material anisotropy is subtle when the stress triaxiality is high. However, in the case of moderate stress triaxiality, void growth becomes strongly dependent on the material anisotropy (Keralavarma et al 2011). Liu et al (2012) investigated the void coalescence in the Face-centered Cubic (FCC) crystals and revealed that the voids in soft orientation grains showed the larger propensity of coalescence, compared with those in the hard orientation. The similar crystal orientation-dependent void growth behavior was observed in Body Centered Cubic (BCC) crystals (Yerra et al 2010). To obtain a good understanding of the role of void growth in fracture of ductile materials, the interaction of voids with a crack tip should also be elucidated. In this regard, two distinct mechanisms were taken into account. The mechanism of void by void, which is that the crack initiation and growth occur by the

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interaction of the blunting crack tip with the nearest single void, was adopted in early studies, and hence a single void in front of a blunting crack tip was considered in these studies (McMeeking 1977, Aravas and McMeeking 1985, Hom and McMeeking 1989). Later, the studies incorporated the mechanism of multiple void interaction, namely the interactions of multiple voids ahead of a crack tip during crack propagation, and an embedded fracture process zone represented by a single row of voids in front of a crack tip was modeled (Tvergaard and Hutchinson 1992, Gao et al 2005). The study by Tvergaard and Hutchinson (2002) revealed that the initial volume fraction of voids is a key factor dominating which mechanism is operative. For the low volume fraction of voids, the crack initiation and propagation are governed by the mechanism of void by void growth, while the mechanism of multiple void interaction plays a dominant role in the case of the high void volume fraction. Although the mechanisms of void growth and coalescence are well understood, suppression of void growth and improvement of fracture toughness of ductile materials are still not achieved, which calls for an efficient material design principle. Recently, the investigations of mechanical behaviors of biological materials revealed that bone-like materials (e.g. bone, dentin, enamel and nacre) possess a complex hierarchical structure, which leads to superior mechanical properties (Gao et al 2003, Bajaj and Arola 2009, Launey et al 2010). The secret of the extraordinary mechanical performance of biological materials composed of relatively weak components holds great potential for bioinspired materials designs. By mimicking the composite structure of bone-like materials, in which the hard mineral platelets are embedded in the soft organic matrix in terms of staggered arrangement, Munch et al (2008) fabricated a hybrid ceramic-based material using aluminum oxide and polymethyl methacrylate. This novel bio-inspired engineering material displays high strength and large fracture toughness, simultaneously, successfully replicating the designs of bone-like materials. The main characteristics, i.e. the combination of hard and soft materials and the staggered arrangement in nanostructure of bone-like materials, have been widely adopted in bioinspired materials design (Begley et al 2012, Bonderer et al 2008, Wang et al 2014). In spite of this progress, the previous studies on bioinspired materials are focused on emulating the nanostructure of bone-like materials and the potential of the novel designs of these materials in development of engineering materials are not fully exploited. Acting as the elastic base of enamel, dentin possesses high fracture toughness (Ivancik and Arola 2013). The remarkable feature of the microstructure of dentin is the dentin tubules existing in the mineral-rich hard peritubular dentin, which is surrounded by the protein-rich soft intertubular dentin, forming the unique two-layered structure (Bertassoni et al 2012, Gotliv 2

et al 2006, Ryou et al 2012). The role of the unique two-layered structure of dentin is still unknown. Whether such a unique structural design can be used to retard void growth and promote fracture toughness is unclear. So far, the designs of the microstructure analogous to dentin have not been employed in engineering materials. The primary purpose of this study is to provide a bioinspired design principle achieving damage-tolerance for voided materials, based on the special microstructure of dentin. A computational model with an embedded fracture process zone, in which the dentin-like two-layered structure is incorporated, is developed, and numerical simulations of crack propagation are performed. The role of the unique microstructure is discovered and the underlying toughening mechanisms caused by this structure are elucidated.

2. Method The typical microstructure of young dentin is presented in figure 1(a), where the dentin tubules exist in hard peritubular dentins, which are further embedded in the soft intertubular dentin. Inspired by the unique microstructure of dentin, the crack propagation in the material with a fracture process zone with voided structure under plane strain conditions is considered in this study. As shown in figure 1(b), a model of compact tension (CT) specimen is constructed, where a blunting crack tip with crack tip radius of R is introduced and the fracture process zone with voided structure is placed on the mid-plane of the CT specimen, which is taken as the crack path. The width of the CT specimen, W, should be much larger than the radius of the voids, r, in the fracture process zone to guarantee the small-scale yielding condition. Here, by preliminary numerical simulations, it is found that W = 260r can satisfy the small-scale yielding condition well, and hence this value of specimen width is adopted in all the following computational simulations. In order to investigate the microscopic deformation mechanisms, the discrete voids are explicitly introduced and the fracture process zone is represented by a row of cylindrical voids surrounded by a hard material (called the inner layer), which is further embedded in a soft matrix (called the outer layer). The inner layer has a radius of 2r and the void spacing is fixed at X0 = 6r. Tvergaard and Hutchinson (2002) pointed out that six voids in front of the crack tip were sufficient to study the deformation mechanisms of a ductile fracture, whereas the study by Hütter et al (2012) identified that in order to capture the mechanism of plastic collapse, the minimum number of voids involved in the process zone was 15. Therefore, in the present study, 15 discrete voids are involved in the process zone. Finite strains are considered in this study, and the outer layer and inner layer are assumed to be an elastic-plastic material with the true stress-logarithmic

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Figure 1. Microstructure of dentin and bioinspired materials. (a) The two-layered structure in dentin, (b) schematic of the compact tension specimen obtained from a bioinspired material.

strain relation in uniaxial tension given by

the second Piola–Kirchhoff stress and Lagrangian strain which can be expressed as E ij =

σ ⎧ , σ ⩽ σy ⎪ E ⎪ ε = ⎨ σ ⎛ ⎞1/ n ⎪ y⎜ σ ⎟ , σ>σ y ⎪ E ⎜⎝ σ y ⎟⎠ ⎩

1 2

(1)

where σ and ε are the true stress and logarithmic strain, respectively. σy is the initial yield stress and E is the elastic modulus. n denotes the strain hardening exponent. Owing to the symmetry of the CT specimen with respect to the crack plane, only half of the region is modeled, and the symmetric boundary conditions are imposed on the crack plane. A rigid cylindrical pin is introduced to apply the mode-I load, and the frictionless contact between the pin and the hole is taken into account. The numerical method based on incremental principle of virtual work is adopted to solve this problem. The incremental equation is written as

∫V ( Δτij δEij + τij Δu k,i δu k,j ) dV = ∫ ΔTi δu i dS − ∫ τij δE ij dV + ∫ S V S

(

)

ui, j + u j, i + u k, i u k, j with ui being the displacement field. Ti is the surface traction, and V and S are the volume and surface of the body in the reference configuration. Four-node isoparametric elements are used to approximate the displacement field and the volume integration in equation (2) is calculated employing Gauss integration strategy. With this numerical method, the stress and strain distributions in the model can be determined, which can be further used to calculate the crack-driving force in terms of stress intensity factor KI. Since the CT specimen model in this study stands for a standard specimen whose geometry conforms to the ASTM standard E1820, the following equation provided in this standard is utilized to calculate KI: KI =

P 2+α (0.886 + 4.64α B W (1 − α)3/2

)

− 13.32α 2 + 14.72α 3 − 5.6α 4 Ti δu i dS (2)

where Δ(·) and δ (⋅) denote the increment and variation of physical quantities, respectively; (·),i represents the differentiation of physical quantities in the reference configuration. τij and Eij are, respectively, 3

(3)

where P is the maximum opening load and B is the thickness of the specimen, which is equal to W/2 for a standard CT specimen. α represents the ratio of the crack length a to the specimen width W. For determining the crack length, it is necessary to define the crack tip, which depends on the criterion of

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crack growth. In the study by Tvergaard and Hutchinson (2002), the crack growth is considered to be associated with void coalescence and a critical reduction in width of the ligaments between voids is introduced to define void coalescence. Considering that such a criterion of defining crack growth may be too strict, Hütter et al (2012) took advantage of the view that the plastic collapse of the intervoid ligaments, i.e. geometrically softening of ligaments, is the dominant mechanism of crack growth in ductile materials, which has been substantiated by experiments, and hence a concept of effective crack growth was utilized to determine the crack length. Here, we also employ this method. The effective crack growth is given by

⎛ Wi ⎞ Δa = X 0 ∑ ⎜ 1 − ⎟ W0 ⎠ ⎝ i=1

Figure 2. Effect of σi/σo with Ei/Eo = 1 on the R-curve.

N

(4)

where Wi and W0 are the current width and initial width of intervoid ligaments, respectively; N is the number of ligaments. Using the dimensionless analysis, the crack driving force can be expressed as ⎛ Δa σi E i Eo ⎞ KI = Π⎜ , , , , n i , no⎟ σo X 0 ⎝ X 0 σo Eo σo ⎠

(5)

where σo and σi are, respectively, the initial yield stresses for the outer layer and inner layer; Eo and Ei are, respectively, the elastic moduli for the outer and inner layer. ni and no denote the strain hardening exponents for the inner layer and outer layer, respectively. In this study, we focus on the effect of elastic mismatch

( ) and of initial yield stress mismatch ( ), Ei Eo

σi σo

and hence in the numerical simulations, the other E material parameters are fixed. The values for σo , ni and o no are chosen as 500, 0.1 and 0.1, respectively, which correspond to the mechanical behavior of metallic materials (Tvergaard and Hutchinson 2002, Tvergaard 2009, Hütter et al 2012). It should be noted that the goal of this study is to propose a bioinspired material design principle for ductile materials, rather than to characterize the deformation mechanism of dentin. Therefore, we developed a sufficiently general numerical model such that the computational predictions based on this model can be applied to a wide range of ductile materials, such as metals and polymers.

3. Results The influence of initial yield stress mismatch, σi/σo, on the R-curve of the bioinspired material is shown in figure 2. It is found that the fracture toughness increases with crack propagation, i.e. rising R-curve. As the yield stress mismatch increases, the fracture toughness also increases. For the case of σi/σo = 2.5, the normalized fracture toughness for crack growth reaches 14.5 when the normalized crack growth attains 4

6. In contrast, the case of σi/σo = 1, which represents the classical engineering materials analyzed in previous studies (e.g. Tvergaard and Hutchinson 2002, Gao et al 2005), displays a normalized fracture toughness of 10 corresponding to the identical crack extension. The enhanced fracture toughness suggests a toughening mechanism caused by the yield stress mismatch. Interestingly, the slopes of R-curves, which are a measure of growth toughness (Nazari et al 2009, An et al 2011), are also promoted by increasing σi/σo. In order to further identify the role of yield stress mismatch on the ductile crack growth, we calculated the void growth rate (defined as the ratio of current volume of void to its initial volume) varying with crack-driving force. As shown in figure 3(a), in the case of σi/σo = 2, the void #1 closest to the crack tip first grows and then the void #2, which is adjacent to #1 starts to grow; the other voids grow in order, indicating that the mechanism of void by void growth is active. In the case of σi/σo = 1 (figure 3(b)), with the increase in crack-driving force, all the voids almost grow simultaneously, suggesting that the crack growth is dominated by the mechanism of multiple void interaction. These numerical results indicate that increasing the yield stress σi/σo leads to the transition of fracture mechanisms from multiple void interaction to void by void growth. The evolutions of plastic deformation in the fracture process zone for σi/σo = 1 and σi/σo = 2 are shown in figures 4 and 5. It can be found that in the case of σi/σo = 1, plastic deformation in the region in the vicinity of the crack tip first takes place; with increasing crack-driving force, the plastic deformation spreads to the intervoid ligaments and the height of the active plastic zone is nearly equal to the radius of voids (figure 4). For the case of σi/σo = 2, more diffusive plastic deformation develops in the region near the crack tip, in comparison to the case of σi/σo = 1; with the increase in crack-driving force, the ligament between the crack tip and the nearest void undergoes plastic deformation and other ligaments display negligible plastic deformation (figure 5). We further present a comparison of the growth rate of

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considered. Under this condition, the void spacing, X0, is taken as a random variable between 5 and 10r, and the effective crack growth can be calculated as N

(

Δa = ∑i = 1X 0 1 −

Wi W0

). An example for the R-curve

of such bioinspired material is shown in figures 8 and 9. It is found that the fracture toughness increases with the increase of both the yield stress mismatch and the elastic modulus mismatch.

4. Discussion

Figure 3. Growth of voids as a function of crack-driving force for (a) σi/σo = 2 and (b) σi/σo = 1. Note that the voids are numbered according to their position from crack tip, such that the nearest one is defined as no. 1.

void #1 between the two cases. As shown in figure 6, the normalized crack-driving force inducing void growth is 2.2 in the case of σi/σo = 1; while this critical crack-driving force reaches 7.6 in the case of σi/σo = 2. Comparing the void growth rates in the two cases, it is found that under the same crack-driving force, the case of σi/σo = 2 displays a smaller growth rate, compared with the case of σi/σo = 1. For example, at the crack-driving force of 9, the void growth rate in the case of σi/σo = 2 is 1.6, while it reaches up to 3 in the case of σi/σo = 1. This indicates that the case of σi/σo = 2 provides a strong resistance to crack propagation. Figure 7 shows the effect of elastic modulus mismatch on the R-curve of the bioinspired materials. The normalized fracture toughness in the case of Ei/ Eo = 5 reaches 11 when the normalized crack extension attains 6; while in the case of Ei/Eo = 1 (representing the case of voids in the homogeneous matrix), the fracture toughness is 10 corresponding to the same crack extension. As expected, the elastic modulus mismatch also affects growth toughness; increasing the elastic modulus of the inner layer gives rise to enhanced growth toughness. In order to investigate the influence of void arrangement in the fracture process zone, the R-curves for the bioinspired material with the fracture process zone with randomly distributed voids was also 5

In this study, the mechanical performance of the bioinspired materials was investigated by developing a computational model, in which the dentin-like microstructure is incorporated in the fracture process zone. For ductile materials, crack initiation and propagation are accompanied by void growth and coalescence (Gurson 1977, Hom and McMeeking 1989), and hence exploring the mechanisms governing void growth and coalescence is essential for developing a good understanding of ductile fracture. In this regard, computational modeling possesses prominent advantages over experiments, since the effect of varying a single material parameter can be assessed. However, for computational modeling, there are difficulties in calculating crack length for the voided materials. Tvergaard and Hutchinson (2002) and Gao et al (2005) defined a critical reduction in width of intervoid ligaments to determine crack extension. However, this method is difficult to implement, which may decrease the computational efficiency. Here, we adopt a concept of effective crack length to facilitate numerical simulations. This concept arises from the plastic collapse mechanism of intervoid ligaments. At a certain loading, the softening of intervoid ligaments induced by plastic collapse takes place and the active plastic zone further moves with increasing stress, implying an effective crack growth in the voided process zone. Such a mechanism is demonstrated by the experiments of ductile materials (Hütter et al 2012). Owing to the fact that nature creates biological materials with superior mechanical performance based on building blocks with inferior mechanical properties, the designs of biological materials inspire a new way to develop engineering materials with exceptional mechanical performance. The key feature of the nanostructure of bone-like materials, i.e. hard mineral platelets embedded in a soft protein matrix in terms of a staggered arrangement, is widely utilized in bioinspired materials design (e.g. Bonderer et al 2008, Munch et al 2008), which leads to the advent of novel man-made hybrid materials exhibiting a unique combination of strength and toughness. Despite these advances, the design principle of the microstructure of dentin has not been applied to the design of engineering materials up to now. The present work addresses this issue. Ryou et al (2012) experimentally reported that the intertubular dentin shows a lower elastic

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Figure 4. Equivalent plastic strain distribution in the fracture process zone for σi/σo = 1.

modulus and hardness, in comparison to peritubular dentin. Considering that hardness characterizes the resistance of materials to plastic deformation, it can be reasonably deduced that peritubular dentin has a greater yield stress compared with intertubular dentin. The previous experimental results indicate that dentin is a voided material with each void (dentin tubule) surrounded by a hard inner layer that is in turn embedded in a soft outer layer. The potential advantages of such a two-layered structure are explored in this study. As shown in figures 2 and 7, the fracture toughness of the bioinspired materials can be tuned by altering yield stress ratio and stiffness ratio. Increasing the yield stress ratio σi/σo enhances fracture toughness and the large stiffness ratio Ei/Eo also results in amplified fracture toughness. In addition, such a toughening mechanism caused by the unique two-layered structure is independent of the void arrangement in the fracture process zone. As shown in figures 8 and 9, in the case of randomly arranged void distribution, the same results are found, i.e. increasing the yield stress ratio and large stiffness ratio can benefit fracture 6

toughness. Based on the numerical predictions, we propose a bioinspired material design strategy, which is that employing the design of the combination of a hard inner layer and a soft outer layer in the fracture process zone of ductile materials will improve fracture toughness and achieve damage-tolerance. The underlying mechanism responsible for this design strategy is that the hard inner layer suppresses void growth and enhances the resistance to plastic collapse in intervoid ligaments (figures 3 and 5), thereby shielding the voids in the fracture process zone. In addition, adding a hard inner layer leads to the more diffused plastic zone around the crack tip (figure 5) and such diffused plastic deformation benefits energy dissipation, which shows a distinct advantage over the traditional homogeneous matrix containing voids, where the height of the active plastic zone is nearly equal to the size of voids. It should be noted that two major toughening mechanisms in biological materials exist, namely extrinsic the toughening mechanism and the intrinsic toughening mechanism (Bajaj and Arola 2009, Barthelat and Rabiei 2011). The extrinsic toughening

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Figure 5. Equivalent plastic strain distribution in the fracture process zone for σi/σo = 2.

Figure 6. Comparison of growth rate of the void nearest the crack tip (No. 1) in the cases of σi/σo = 1 and σi/σo = 2.

mechanism acts in the crack wake, for example crack bridging by uncracked ligaments in enamel (Bajaj and Arola 2009), while the intrinsic toughening mechanism emerges ahead of the crack tip, such as the dilation in front of the crack tip generated by sliding of tablets in nacre (Barthelat and Rabiei 2011). This study is focused on the intrinsic toughening and therefore did not account for the extrinsic toughening mechanisms. 7

Figure 7. Effect of Ei/Eo with σi/σo = 1 on the R-curve.

In reality, for crack propagation in dentin, the bridging caused by uncracked ligaments emerges, which reduces the stress intensity at the crack tip and increases the propensity of crack closure, thereby promoting fracture toughness (Nazari et al 2009). It is expected that combining the crack-bridging toughening and the proposed process zone toughening caused by the twolayered structure will considerably enhance the fracture toughness of bioinspired materials.

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Figure 8. Effect of σi/σo on the R-curve behavior of the bioinspired materials with the fracture process zone having randomly arranged void distribution. Note that the elastic modulus ratio, Ei/Eo, is fixed at 1.

perfectly bonded to the outer layer. Actually, the mechanical property mismatch between the two materials can lead to fracture at the interface (He and Hutchinson 1989), which is also observed in biological materials. For example, Bechtle et al (2010) experimentally reported that a crack tends to propagate along the dentin enamel junction (DEJ) that is the natural interface between dentin and enamel. Therefore, an interesting question arises: how is the proposed bioinspired toughening strategy active? The geometrically interlocking design inspired by the suture joints in biological materials (e.g. Jasinoski et al 2010) could provide the solution. Employing the bioinspired fractal-like interlockers, the strength of the interface between the inner layer and outer layer will be significantly enhanced (Zhang et al 2012), thereby increasing the fracture resistant capability of the interface and activating the toughening mechanism proposed in this study. More importantly, the bioinspired geometrically interlocking interfaces have been fabricated successfully using 3D printing techniques (Lin et al 2014).

5. Conclusions

Figure 9. Effect of Ei/Eo on the R-curve behavior of the bioinspired materials with the fracture process zone having randomly arranged void distribution. Note that the yield stress ratio, σi/σo, is fixed at 1.

The study by Tvergaard and Hutchinson (2002) identified that two fracture mechanisms contributing to crack propagation in ductile materials exists, namely void by void growth and multiple void interactions. The mechanism of void by void growth decreases the growth rate of voids and plays a critical role in retarding crack growth. Conversely, the mechanism of multiple void interactions leads to fast growth of voids. It is demonstrated that initial void volume fraction is the key factor controlling the activeness of the two mechanisms. Increasing the initial void volume fraction results in the transition of fracture mechanisms from void by void growth to multiple void interactions. The present study provides another method to accomplish this transition. Adopting the design of a dentin-like structure, the fracture mechanism of multiple void interactions induced by a high volume fraction of voids can be altered into void by void growth, which undoubtedly improves fracture resistance of ductile materials. In this study, the numerical simulations are performed based on the assumption that the inner layer is 8

This study explores the mechanics of crack propagation in the bioinspired materials with the fracture process zone having a dentin-like structure. The twolayered microstructure retards void growth in the process zone and enhances the capability of intervoid ligaments to resist plastic deformation. Compared with the traditional ductile materials with voids in a homogeneous matrix, this unique structural design gives rise to a more diffused plastic zone around the crack tip, thereby promoting energy dissipation during crack growth. In addition, employing the design of the microstructure of dentin enables the transition of fracture mechanisms from multiple void interaction to void by void growth. All these mechanisms caused by the dentin-like microstructure in the fracture process zone lead to amplified fracture toughness. Based on these results, we propose a bioinspired process zone toughening strategy, which is that adopting the design of the combination of a hard inner layer and a soft outer layer in the voided process zone enhances fracture toughness and achieves damagetolerance. This bioinspired design strategy can shed new light on the development of novel engineering materials with superior mechanical performance.

Acknowledgments The authors would like to thank the National Science Foundation of China #11172161, #11402141 and #11372173, the Innovation Program of Shanghai Municipal Education Commission #12ZZ092. Technical support from the Instrumental Analysis and

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Research Center at Shanghai University is also acknowledged.

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Bioinspired toughening mechanism: lesson from dentin.

Inspired by the unique microstructure of dentin, in which the hard peritubular dentin surrounding the dentin tubules is embedded in the soft intertubu...
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