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The effect of microcracking in the peritubular dentin on the fracture of dentin
Journal of Biomechanics xxx (2017) xxx–xxx
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The effect of microcracking in the peritubular dentin on the fracture of dentin Bingbing An a,⇑, H. Daniel Wagner b a b
Department of Mechanics, Shanghai University, Shanghai 200444, People’s Republic of China Department of Materials and Interfaces, Weizmann Institute of Science, Rehovot 76100, Israel
a r t i c l e
i n f o
Article history: Accepted 15 October 2017 Available online xxxx Keywords: Dentin Composite structure Fracture Microcracking
a b s t r a c t Dentin is a biocomposite possessing elegant hierarchical structure, which allows it to resist fracture effectively. Despite the considerable efforts to unravel the peculiar fracture behavior of dentin, the effect of microstructural features on the fracture process is largely unknown. In this study, we explore the interaction between the primary crack with crack tip located in intertubular dentin (ITD) and microcracking of peritubular dentin (PTD) ahead of the primary crack. A micromechanical model accounting for the unique composite structure of dentin is developed, and computational simulations are performed. It is found that the microcracking of PTD located in the crack plane in front of the primary crack tip can promote the propagation of the primary crack, increasing the propensity of coalescence of primary crack and microcracks nucleating in PTD. We show that the two-layer microstructure of dentin enables reduction in driving force of primary crack, potentially enhancing fracture toughness. The high stiffness of PTD plays a critical role in reducing the driving force of primary crack and activating microcracking of PTD. It is further identified that the microcracking of PTD arranged parallel to the crack plane with an offset could contribute to the shielding of primary crack. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Dentin is a mineralized hard tissue comprising the bulk of human teeth. The major function of dentin is to carry and transfer stresses from the outer enamel (Eltit et al., 2013; Waters, 1980). To fulfill this function, dentin needs to have the damage tolerance capability to mitigate fracture. Hence, there is increasing interest in unraveling the fracture behavior and underlying mechanisms of dentin. Imbeni et al (2003) conducted fracture toughness testing of dentin based on fracture mechanics, and determined the improved, lower-bound value of fracture toughness. The study by Kruzic et al. (2003) revealed that dentin displays a rising crack growth resistance curve (R-curve), indicating that the fracture toughness increases with crack extension. Such enhanced fracture toughness is attributed to the salient toughening mechanisms. It was found that uncracked ligaments develop in the crack wake, forming crack bridging which reduces the stress intensity at crack tip and leads to the propensity of crack closure (Bajaj et al., 2006; Kruzic et al., 2003; Nalla et al., 2004). In addition, the microcracking surrounding the main crack was observed, which gives rise to
⇑ Corresponding author.
the dilation in the region sustaining high stress and increases the compliance, thereby serving as an important toughening mechanism (Nalla et al., 2003a). These previous studies have identified the unique fracture behavior of dentin, and some important fracture mechanisms have been also revealed. However, a good understanding of how the mechanisms change the driving force necessary for fracture is not achieved. To gain deep insights into the fracture in dentin, it is essential to explore the structure-property relation. Dentin is a nanocomposite consisting of mineral crystals wrapped by soft protein matrix (Ten Cate, 2008). The prominent feature of dentin at the nanoscale is the staggered arrangement of mineral crystals, which is similar to bone and nacre. Such structural feature imparts the fascinating combination of high strength and superior toughness to dentin (Ji and Gao, 2004; Zhang et al., 2010). At the micoscale, dentin is a biocomposite comprised of protein-rich intertubular dentin (ITD) reinforced by mineral-rich peritubular dentin (PTD) containing dentin tubules (Bertassoni et al., 2012; Ryou et al., 2012). The composite microstructure plays a pivotal role in fracture of dentin. Nalla et al (2003b) showed experimentally that there exists a microdamage zone surrounding the main crack, in which the PTDs undergo microcracking, indicating that inelastic deformation takes place in the fracture process of dentin.
E-mail address: [email protected] (B. An). https://doi.org/10.1016/j.jbiomech.2017.10.022 0021-9290/Ó 2017 Elsevier Ltd. All rights reserved.
Please cite this article in press as: An, B., Daniel Wagner, H. The effect of microcracking in the peritubular dentin on the fracture of dentin. J. Biomech. (2017), https://doi.org/10.1016/j.jbiomech.2017.10.022
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Such inelastic deformation caused by microcracking of PTD is further investigated by Eltit et al (2013). In their experiments, it was found that the fracture process of dentin involves merging of the microcracks nucleating at dentin tubules. Jainaen et al (2009) experimentally investigated fracture properties of dentin, showing that crack growth in dentin microstructure is accompanied by the fracture of PTD/ITD interface. The study by Ivancik and Arola (2013) reported that the microstructure of dentin exhibits significant influence on the crack growth path, and that microcracking of PTD occurs ahead of the main crack tip. A substantial degree of microcracking in PTD takes place in the region containing high volume fraction of dentin tubules; whereas crack bridging is seldom observed in this region, leading to the low fracture toughness. The dependence of fracture toughness on the tubule density was further elucidated by Montoya et al. (2016). Using an empirical model for porous materials, they showed that the reduction in fracture toughness is associated with the increase in volume fraction of dentin tubules. However, such an empirical model only considers the effect of geometrical feature of tubules and neglects the important deformation mechanisms associated with the microstructure of dentin. The effects of microcracking and elastic modulus ratio between PTD and ITD on the fracture process in dentin are unknown. Recently, An and Wagner (2016) developed a micromechanical model of dentin accounting for the unique composite microstructure and the fracture mechanisms of PTD, ITD and the PTD/ITD interface, and revealed the competing fracture mechanisms in dentin. Subsequently, the interaction between crack and the composite microstructure of dentin was explored and it was revealed that the key factor controlling microcracking of PTD ahead of main crack and crack deflection along PTD/ITD interface is the tensile strength of PTD (An et al., 2017). The low strength of PTD could give rise to microcracking of PTD in front of the main crack. Despite the enormous progress, the role of microcracking of PTD in fracture of the microstructure of dentin is still largely unknown. On the one hand, dentin tubules act as stress concentrations, which could potentially induce microcracking of PTD. On the other hand, the stress intensity at the main crack tip located in ITD is high, which could give rise to crack propagation and fracture of ITD. Why is the microcracking of PTD, rather than fracture of ITD, observed in experiments? How does the microstructure of dentin affect the main crack with crack tip located in ITD? These questions call for a good understanding of the fracture mechanisms of dentin. The present study is motivated by the experiments conducted by Ivancik and Arola (2013) and Koester et al. (2008). The primary goal of this study is to reveal the effects of microcracking of PTD and of the composite microstructure on the fracture of dentin. A computational model is developed accounting for the unique composite microstructure of dentin. The interplay between fracture of ITD and microcracking of PTD ahead of the main crack tip is elucidated, and the role of high stiffness of PTD in reducing the crack driving force of the main crack is identified. In addition, the effect of microcracking of PTD arranged parallel to the crack plane with an offset on the driving force of main crack is also discussed.
2. Computational model We idealize the microstructure of dentin as a heterogeneous solid consisting of hollow cylindrical reinforcements embedded in soft matrix, as shown in Fig. 1a. Such type of idealized microstructure has been used in previous studies (An et al., 2017; An and Zhang, 2015), and it is demonstrated that the numerical simulations based on such idealization of microstructure can capture the important features of fracture in dentin. A block of dentin having length L and height 2h is considered, where three
rows of PTD containing tubules with radius r are included. The spacing between neighboring tubules is X0, and the thickness of PTD is tp. It is experimentally reported that crack growth in dentin with crack path in-plane with dentin tubules involves the merging of the microcracks nucleating in PTD with the crack located in ITD (Eltit et al., 2013; Ivancik and Arola, 2013). The fracture process in dentin is shown schematically in Fig. 1. Therefore, in this study the main crack is taken to be the crack which propagates through the PTD and the ITD, and the tip of the main crack is located in the ITD, as illustrated in Fig. 2. The plane strain conditions are assumed and the two edges of the dentin block are subjected to tensile strain ±e. Owing to the symmetry of dentin block and the loading with respect to crack plane, only half of the block is modeled. To study the influence of microcracking of PTD, we introduce several microcracks in the PTDs ahead of the main crack tip. As shown in Fig. 2, the PTD1, which is nearest to the main crack tip, encompasses one microcrack with length of a1, and the PTD2, which is arranged parallel to the crack plane with an offset of X0, contains two microcracks with length of a2. Similarly, two microcracks of length a3 are introduced in the PTD3, which is close to the PTD1. Both the PTD and the ITD are modeled as isotropic and linear elastic solids. The finite element method is employed to obtain the approximate solution of the boundary value problem. The model of dentin block is meshed with eight-node isoparametric elements, and the collapsed quadrilateral elements are used to model the crack tip. A total of approximately 15,000 elements are involved. To assess the fracture resistance of the microstructure of dentin, the crack driving force represented by the energy release rate at crack tip is calculated using the contour integral. Based on dimensional considerations, the driving force for the main crack is given by
Gi ¼ Ei e2 hf
ai a1 a2 a3 Ep L tp r X 0 ; ; ; ; ; v i; v p; ; ; ; h h h r L t p t p t p Ei
ð1Þ
where Ei ¼ Ei =ð1 v 2i Þ is the plane strain modulus of the ITD, with Ei and vi being the elastic modulus and Poisson ratio of the ITD, respectively. ai represents the length of the main crack, and Ep and vp are elastic modulus and Poisson ratio for the PTD, respectively. In the numerical simulations, we fix vi = 0.3, vp = 0.3, L/h = 2.3, tp/h = 0.125, r/h = 0.125 and X0/r = 5. These values of the parameters are comparable to that observed in experiments (e.g. Ivancik and Arola, 2013; Montoya et al., 2016) and representative of dentin microstructure (An and Wagner, 2016; An et al., 2017). As a consequence, the driving force Gi has the form
Gi ¼ Ei e2 hF
ai a1 a2 a3 Ep ; ; ; ; L t p t p t p Ei
ð2Þ
where F is a dimensionless function. Using the same numerical method, the driving force Gp of the microcrack in the PTD1 can be calculated, which has the form similar to Eq. (2). However, due to the difference in the mechanical properties of PTD and ITD, a distinct dimensionless function for Gp is expected. In the numerical simulations, the range of elastic modulus mismatch, Ep/Ei, is taken to be from 1.2 to 3, which is consistent with the experiments on stiffness of ITD and PTD (Kinney et al., 1996; Ziskind et al., 2011), and the effect of elastic modulus mismatch can be explored by such choice of the values. The numerical simulations are performed based on the fact that PTD exhibits larger elastic modulus than ITD (Kinney et al., 1999). 3. Results and discussion Fig. 3a shows the normalized driving force for main crack, Gi =ðEi e2 hÞ, as a function of crack length ai/h for three values of modulus mismatch (Ep/Ei). It is found that for large elastic modulus
Please cite this article in press as: An, B., Daniel Wagner, H. The effect of microcracking in the peritubular dentin on the fracture of dentin. J. Biomech. (2017), https://doi.org/10.1016/j.jbiomech.2017.10.022
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Fig. 1. Schematic illustration of fracture process in dentin. (a) The main crack approaching PTD. (b) Microcracking of PTD ahead of the main crack tip. (c) Merging of microcrack in PTD with the main crack.
Fig. 2. Schematics of the computational models of the microstructure of dentin. (a) Uniform arrangement of tubules. (b) Staggered arrangement of tubules.
mismatch, for example Ep/Ei = 2 and Ep/Ei = 3, the driving force for the main crack decreases as crack advances. The propagation of main crack is controlled by the competition between the driving force and fracture toughness of the ITD. Based on the results in Fig. 3a, it is identified that crack arrest may take place when the length of the main crack reaches a critical value corresponding to the circumstance where the driving force is lower than the fracture toughness. For the small modulus mismatch, Ep/Ei = 1.2, the normalized driving force shows weak dependence on crack length and a nearly constant driving force is observed. The effect of elastic modulus mismatch on crack driving force of main crack is complex. For short cracks, the driving force increases with the increase in modulus mismatch; while an opposite variation trend of driving force with modulus mismatch is observed for long cracks. The results shown in Fig. 3a suggest that the high stiffness of PTD plays
Fig. 3. Effect of the length of main crack on crack driving force. (a) Normalized driving force of the main crack as a function of the normalized length of main crack. (b) Normalized driving force of the microcrack in the PTD1 as a function of the normalized length of main crack. Note that the geometric parameters used in computational simulations are a1/tp = 0.4, a2/tp = 0 and a3/tp = 0.
a critical role in mitigating fracture. For a modest toughness of ITD, the main crack in the ITD could initiate in the case of high Ep/Ei due to the large driving force for short cracks. As the main crack propagates, the driving force drops. When the long cracks appear, the driving force could drop to a low value, indicating that crack arrest may take place. Although the low Ep/Ei could lead to low driving force for short cracks, the relatively high driving force for long cracks appears in this case. This is disadvantageous for mitigating fracture. Once cracks initiate, crack arrest is difficult to take place.
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Fig. 3b shows the variation of the normalized driving force for the microcrack in PTD1, Gp =ðEi e2 hÞ, with crack length ai/h for three values of modulus mismatch. It is found that as the main crack propagates in the ITD, the driving force for the microcrack in the PTD1 increases, indicating that crack growth in the ITD could promote fracture of the PTD ahead of the main crack tip, which is in a good agreement with experimental observations (Ivancik and Arola, 2013; Nalla et al., 2003b). The crack driving force Gp exhibits strong dependence on the elastic modulus mismatch; increasing Ep/Ei elevates the level of crack driving force Gp, suggesting that high stiffness of PTD could contribute to the microcracking of PTD in front of the main crack tip. It is important to note that the study by Ivancik and Arola (2013) is focused on the fracture toughness and toughening mechanisms of dentin, and it was revealed that crack bridging is the major source of toughening. The study by Koester et al. (2008) revealed that due to the small sizes of realistic cracks in human dentin and the relatively small crack-deflection angle, the origin of toughness of human dentin is more associated with crack branching compared with other toughening mechanisms. Koester et al. (2008) pointed out that crack branching requires the presence of microcracking of PTD ahead of the main crack tip, and the microcracking of PTD can also lead to the formation of crack bridging. Therefore, we confine our attention to the important mechanism of microcracking of PTD. The subsequent development of crack bridging and crack branching is not the focus of this study. The fracture toughness of dentin cannot be assessed based on our study, since it requires a model incorporating all the potential toughening mechanisms, which is beyond the scope of our study. To explore the interplay between crack growth in ITD and microcracking of PTD ahead of the main crack, we further analyze the effect of a1/tp. As shown in Fig. 4a, the driving force for the main crack increases as the microcrack advances in PTD1, indicating that microcracking of PTD ahead of the main crack facilitates crack growth in the ITD. We also show that the driving force for the main crack decreases with increasing modulus mismatch. This is consistent with the results shown in Fig. 3a, since we considered the case of long cracks. The normalized driving force for microcrack in PTD1, Gp =ðEi e2 hÞ, as a function of microcrack length is shown in Fig. 4b. It can be seen that the driving force Gp increases as microcrack advances, which increases the propensity of fracture of PTD. The elastic modulus mismatch has a significant influence on the driving force Gp; increasing the modulus mismatch leads to increased driving force. Nalla et al. (2003b) showed experimentally that crack growth in the microstructure of dentin is accompanied by the formation of a microdamage zone surrounding the main crack tip. The interaction between crack and the composite microstructure of dentin leads to the microcracking of PTD ahead of the main crack tip. Such fracture process is further confirmed by the experiments conducted by Ivancik and Arola (2013). From the mechanics point of view, the dentin tubules can be considered as voids which could give rise to stress concentration and lead to microcracking in PTD. The stress intensity at crack tip is so high that the main crack may propagate when the applied stress increases. The competition between the propagation of main crack and microcracking of PTD ahead of the main crack governs the fracture process in dentin. If the former dominates, the main crack can penetrate into PTD. Otherwise, microcracking of PTD occurs, which is followed by merging of microcracks with the main crack. The latter case is widely observed in experiments. However, the underlying mechanisms responsible for such behavior are still unknown. The present study has addressed this issue. We reveal that the propagation of main crack can increase the driving force for microcracks in PTD (Fig. 3b), promoting microcracking of PTD ahead of the main crack. And the
Fig. 4. Effect of the length of the microcrack in the PTD1 on crack driving force. (a) Normalized driving force of the main crack as a function of the normalized length of the microcrack in the PTD1. (b) Normalized driving force of the microcrack in the PTD1 as a function of the normalized length of the microcrack in the PTD1. The geometric parameters used in computational simulations are ai/L = 0.45, a2/tp = 0 and a3/tp = 0.
microcracking of PTD can elevate the level of driving force for the main crack (Fig. 4a), increasing the propensity of the growth of main crack. Hence, the propagation of main crack and microcracking of PTD facilitate each other and are two co-evolving fracture mechanisms. It is further revealed that the stiffness of PTD plays the critical role in determining which fracture mechanism dominates. The high stiffness of PTD can give rise to the increased driving force for microcracks in PTD and meanwhile enables the reduction in the driving force for main crack. This indicates that high stiffness of PTD activates the fracture mechanism of microcracking of PTD ahead of the main crack, promoting the merging of microcracks with the main crack. The normalized driving force for the main crack as a function of the length of the microcracks in the PTD2, which is arranged parallel to the crack plane with an offset, is shown in Fig. 5a. It can be seen that the driving force for the main crack decreases as microcracks advance in the PTD2, indicating that microcracking in the PTD2 can shield the tip of main crack, which potentially enhances fracture toughness. The similar shielding effect on the microcrack in the PTD1 is also found. As shown in Fig. 5b, the normalized driving force, Gp =ðEi e2 hÞ, decreases with the increase in
Please cite this article in press as: An, B., Daniel Wagner, H. The effect of microcracking in the peritubular dentin on the fracture of dentin. J. Biomech. (2017), https://doi.org/10.1016/j.jbiomech.2017.10.022
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Fig. 5. Effect of the length of the microcrack in the PTD2 on crack driving force. (a) Normalized driving force of the main crack as a function of the normalized length of the microcrack in the PTD2. (b) Normalized driving force of the microcrack in the PTD1 as a function of the normalized length of the microcrack in the PTD2. The parameters used in computational simulations are Ep/Ei = 3, ai/L = 0.45, a1/tp = 0.4 and a3/tp = 0.
Fig. 6. Effect of the length of the microcrack in the PTD3 on crack driving force. (a) Normalized driving force of the main crack as a function of the normalized length of the microcrack in the PTD3. (b) Normalized driving force of the microcrack in the PTD1 as a function of the normalized length of the microcrack in the PTD3. The parameters used in computational simulations are Ep/Ei = 3, ai/L = 0.45, a1/tp = 0.4 and a2/tp = 0.
the length of microcracks in the PTD2. We also investigated the effect of microcracking of the PTD3 which is close to the PTD1 but far from the tip of main crack. As shown in Fig. 6a, the driving force of the main crack slightly decreases as microcracks advance in the PTD3. Also, the driving force for the microcrack in the PTD1 drops slightly with increasing the length of the microcracks in the PTD3 (Fig. 6b). Comparing the effects of microcracking of the PTD2 and microcracking of the PTD3, it is found that the microcracking in the PTD2 plays even more important role in reducing crack driving force and shielding crack tip. It was identified that microcracking enables reduction in stiffness and release of residual stress, acting as an important toughening mechanism in brittle solids (Hutchinson, 1987). Such toughening mechanism is also active in biological hard tissues (Nalla et al., 2004). In dentin, the microdamage zone surrounding the main crack could contribute to the fracture toughness (Nalla et al., 2003a). However, the roles of the microcracks in distinct regions of the microdamage zone are unclear. The present study reveals that microcracks in the PTD aligned parallel to the crack plane with an offset play the dominant role in shielding the tip of main crack.
To investigate the influence of the tubule arrangement, the crack driving forces of the main crack and of the microcrack in the PTD1 are calculated in the case of staggered arrangement of tubules (illustrated in Fig. 2). As shown in Fig. 7a, the normalized driving forces, Gi =ðEi e2 hÞ, of the main crack in the cases of uniform arrangement and staggered arrangement of tubules are nearly identical, indicating that the driving force of main crack is independent of the tubule arrangement. Fig. 7b shows the normalized driving force of the microcrack in the PTD1 varying with the arrangement of tubules. It is found that staggered arrangement slightly reduces the driving force, whereas such a reduction in driving force is insignificant. It is important to note that the present study is focused on the interplay between the growth of main crack and the microcracking of PTD in front of the main crack, and the effect of the tubule geometry is not taken into account. In reality, dentin exhibits spatial variations in tubule density and radius (Ten Cate, 2008). It is valuable to assess the effect of tubule geometry changes on the fracture in dentin. In a recent study, it was found that the toughness of dentin increases with the decrease in tubule radius, and low tubule density can give rise to enhanced toughness of dentin (An, 2017).
Please cite this article in press as: An, B., Daniel Wagner, H. The effect of microcracking in the peritubular dentin on the fracture of dentin. J. Biomech. (2017), https://doi.org/10.1016/j.jbiomech.2017.10.022
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Conflict of interest statement The authors have no conflict of interest. Acknowledgments B.A. acknowledges the support from National Science Foundation of China #11402141. H. D. W. is recipient of the Livio Norzi Professional Chair at the Weizmann Institute. References
Fig. 7. Effect of tubule arrangement on (a) the driving force of the main crack and (b) the driving force of the microcrack in the PTD1. The parameters used in numerical simulations are Ep/Ei = 3, a1/tp = 0.4, a2/tp = 0 and a3/tp = 0.
4. Conclusions This study explores the effect of microcracking of PTD on the fracture in the microstructure of dentin. We reveal that the propagation of the main crack in ITD and microcracking of PTD ahead of the main crack are two co-evolving mechanisms involved in the fracture process of dentin. Crack growth in ITD promotes microcracking of PTD, and microcracking of PTD in front of the main crack, in turn, enables increased driving force for the main crack, facilitating crack propagation in ITD. The high stiffness of PTD plays the dominant role in promoting microcracking of PTD and suppressing the growth of the main crack in ITD, which leads to the experimentally observed fracture process in dentin, i.e. microcracking of PTD ahead of the main crack and subsequent merging of microcracks with the main crack. We further identified the role of microcracking of PTD in the microdamage zone surrounding the main crack. The microcracking of the PTD aligned parallel to the crack plane with an offset provides greater contribution to crack tip shielding, compared with microcracking of the PTD arranged along the initial crack line.
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Please cite this article in press as: An, B., Daniel Wagner, H. The effect of microcracking in the peritubular dentin on the fracture of dentin. J. Biomech. (2017), https://doi.org/10.1016/j.jbiomech.2017.10.022
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Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
The effect of microcracking in the peritubular dentin on the fracture of dentin Bingbing An a,⇑, H. Daniel Wagner b a b
Department of Mechanics, Shanghai University, Shanghai 200444, People’s Republic of China Department of Materials and Interfaces, Weizmann Institute of Science, Rehovot 76100, Israel
a r t i c l e
i n f o
Article history: Accepted 15 October 2017 Available online xxxx Keywords: Dentin Composite structure Fracture Microcracking
a b s t r a c t Dentin is a biocomposite possessing elegant hierarchical structure, which allows it to resist fracture effectively. Despite the considerable efforts to unravel the peculiar fracture behavior of dentin, the effect of microstructural features on the fracture process is largely unknown. In this study, we explore the interaction between the primary crack with crack tip located in intertubular dentin (ITD) and microcracking of peritubular dentin (PTD) ahead of the primary crack. A micromechanical model accounting for the unique composite structure of dentin is developed, and computational simulations are performed. It is found that the microcracking of PTD located in the crack plane in front of the primary crack tip can promote the propagation of the primary crack, increasing the propensity of coalescence of primary crack and microcracks nucleating in PTD. We show that the two-layer microstructure of dentin enables reduction in driving force of primary crack, potentially enhancing fracture toughness. The high stiffness of PTD plays a critical role in reducing the driving force of primary crack and activating microcracking of PTD. It is further identified that the microcracking of PTD arranged parallel to the crack plane with an offset could contribute to the shielding of primary crack. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Dentin is a mineralized hard tissue comprising the bulk of human teeth. The major function of dentin is to carry and transfer stresses from the outer enamel (Eltit et al., 2013; Waters, 1980). To fulfill this function, dentin needs to have the damage tolerance capability to mitigate fracture. Hence, there is increasing interest in unraveling the fracture behavior and underlying mechanisms of dentin. Imbeni et al (2003) conducted fracture toughness testing of dentin based on fracture mechanics, and determined the improved, lower-bound value of fracture toughness. The study by Kruzic et al. (2003) revealed that dentin displays a rising crack growth resistance curve (R-curve), indicating that the fracture toughness increases with crack extension. Such enhanced fracture toughness is attributed to the salient toughening mechanisms. It was found that uncracked ligaments develop in the crack wake, forming crack bridging which reduces the stress intensity at crack tip and leads to the propensity of crack closure (Bajaj et al., 2006; Kruzic et al., 2003; Nalla et al., 2004). In addition, the microcracking surrounding the main crack was observed, which gives rise to
⇑ Corresponding author.
the dilation in the region sustaining high stress and increases the compliance, thereby serving as an important toughening mechanism (Nalla et al., 2003a). These previous studies have identified the unique fracture behavior of dentin, and some important fracture mechanisms have been also revealed. However, a good understanding of how the mechanisms change the driving force necessary for fracture is not achieved. To gain deep insights into the fracture in dentin, it is essential to explore the structure-property relation. Dentin is a nanocomposite consisting of mineral crystals wrapped by soft protein matrix (Ten Cate, 2008). The prominent feature of dentin at the nanoscale is the staggered arrangement of mineral crystals, which is similar to bone and nacre. Such structural feature imparts the fascinating combination of high strength and superior toughness to dentin (Ji and Gao, 2004; Zhang et al., 2010). At the micoscale, dentin is a biocomposite comprised of protein-rich intertubular dentin (ITD) reinforced by mineral-rich peritubular dentin (PTD) containing dentin tubules (Bertassoni et al., 2012; Ryou et al., 2012). The composite microstructure plays a pivotal role in fracture of dentin. Nalla et al (2003b) showed experimentally that there exists a microdamage zone surrounding the main crack, in which the PTDs undergo microcracking, indicating that inelastic deformation takes place in the fracture process of dentin.
E-mail address: [email protected] (B. An). https://doi.org/10.1016/j.jbiomech.2017.10.022 0021-9290/Ó 2017 Elsevier Ltd. All rights reserved.
Please cite this article in press as: An, B., Daniel Wagner, H. The effect of microcracking in the peritubular dentin on the fracture of dentin. J. Biomech. (2017), https://doi.org/10.1016/j.jbiomech.2017.10.022
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Such inelastic deformation caused by microcracking of PTD is further investigated by Eltit et al (2013). In their experiments, it was found that the fracture process of dentin involves merging of the microcracks nucleating at dentin tubules. Jainaen et al (2009) experimentally investigated fracture properties of dentin, showing that crack growth in dentin microstructure is accompanied by the fracture of PTD/ITD interface. The study by Ivancik and Arola (2013) reported that the microstructure of dentin exhibits significant influence on the crack growth path, and that microcracking of PTD occurs ahead of the main crack tip. A substantial degree of microcracking in PTD takes place in the region containing high volume fraction of dentin tubules; whereas crack bridging is seldom observed in this region, leading to the low fracture toughness. The dependence of fracture toughness on the tubule density was further elucidated by Montoya et al. (2016). Using an empirical model for porous materials, they showed that the reduction in fracture toughness is associated with the increase in volume fraction of dentin tubules. However, such an empirical model only considers the effect of geometrical feature of tubules and neglects the important deformation mechanisms associated with the microstructure of dentin. The effects of microcracking and elastic modulus ratio between PTD and ITD on the fracture process in dentin are unknown. Recently, An and Wagner (2016) developed a micromechanical model of dentin accounting for the unique composite microstructure and the fracture mechanisms of PTD, ITD and the PTD/ITD interface, and revealed the competing fracture mechanisms in dentin. Subsequently, the interaction between crack and the composite microstructure of dentin was explored and it was revealed that the key factor controlling microcracking of PTD ahead of main crack and crack deflection along PTD/ITD interface is the tensile strength of PTD (An et al., 2017). The low strength of PTD could give rise to microcracking of PTD in front of the main crack. Despite the enormous progress, the role of microcracking of PTD in fracture of the microstructure of dentin is still largely unknown. On the one hand, dentin tubules act as stress concentrations, which could potentially induce microcracking of PTD. On the other hand, the stress intensity at the main crack tip located in ITD is high, which could give rise to crack propagation and fracture of ITD. Why is the microcracking of PTD, rather than fracture of ITD, observed in experiments? How does the microstructure of dentin affect the main crack with crack tip located in ITD? These questions call for a good understanding of the fracture mechanisms of dentin. The present study is motivated by the experiments conducted by Ivancik and Arola (2013) and Koester et al. (2008). The primary goal of this study is to reveal the effects of microcracking of PTD and of the composite microstructure on the fracture of dentin. A computational model is developed accounting for the unique composite microstructure of dentin. The interplay between fracture of ITD and microcracking of PTD ahead of the main crack tip is elucidated, and the role of high stiffness of PTD in reducing the crack driving force of the main crack is identified. In addition, the effect of microcracking of PTD arranged parallel to the crack plane with an offset on the driving force of main crack is also discussed.
2. Computational model We idealize the microstructure of dentin as a heterogeneous solid consisting of hollow cylindrical reinforcements embedded in soft matrix, as shown in Fig. 1a. Such type of idealized microstructure has been used in previous studies (An et al., 2017; An and Zhang, 2015), and it is demonstrated that the numerical simulations based on such idealization of microstructure can capture the important features of fracture in dentin. A block of dentin having length L and height 2h is considered, where three
rows of PTD containing tubules with radius r are included. The spacing between neighboring tubules is X0, and the thickness of PTD is tp. It is experimentally reported that crack growth in dentin with crack path in-plane with dentin tubules involves the merging of the microcracks nucleating in PTD with the crack located in ITD (Eltit et al., 2013; Ivancik and Arola, 2013). The fracture process in dentin is shown schematically in Fig. 1. Therefore, in this study the main crack is taken to be the crack which propagates through the PTD and the ITD, and the tip of the main crack is located in the ITD, as illustrated in Fig. 2. The plane strain conditions are assumed and the two edges of the dentin block are subjected to tensile strain ±e. Owing to the symmetry of dentin block and the loading with respect to crack plane, only half of the block is modeled. To study the influence of microcracking of PTD, we introduce several microcracks in the PTDs ahead of the main crack tip. As shown in Fig. 2, the PTD1, which is nearest to the main crack tip, encompasses one microcrack with length of a1, and the PTD2, which is arranged parallel to the crack plane with an offset of X0, contains two microcracks with length of a2. Similarly, two microcracks of length a3 are introduced in the PTD3, which is close to the PTD1. Both the PTD and the ITD are modeled as isotropic and linear elastic solids. The finite element method is employed to obtain the approximate solution of the boundary value problem. The model of dentin block is meshed with eight-node isoparametric elements, and the collapsed quadrilateral elements are used to model the crack tip. A total of approximately 15,000 elements are involved. To assess the fracture resistance of the microstructure of dentin, the crack driving force represented by the energy release rate at crack tip is calculated using the contour integral. Based on dimensional considerations, the driving force for the main crack is given by
Gi ¼ Ei e2 hf
ai a1 a2 a3 Ep L tp r X 0 ; ; ; ; ; v i; v p; ; ; ; h h h r L t p t p t p Ei
ð1Þ
where Ei ¼ Ei =ð1 v 2i Þ is the plane strain modulus of the ITD, with Ei and vi being the elastic modulus and Poisson ratio of the ITD, respectively. ai represents the length of the main crack, and Ep and vp are elastic modulus and Poisson ratio for the PTD, respectively. In the numerical simulations, we fix vi = 0.3, vp = 0.3, L/h = 2.3, tp/h = 0.125, r/h = 0.125 and X0/r = 5. These values of the parameters are comparable to that observed in experiments (e.g. Ivancik and Arola, 2013; Montoya et al., 2016) and representative of dentin microstructure (An and Wagner, 2016; An et al., 2017). As a consequence, the driving force Gi has the form
Gi ¼ Ei e2 hF
ai a1 a2 a3 Ep ; ; ; ; L t p t p t p Ei
ð2Þ
where F is a dimensionless function. Using the same numerical method, the driving force Gp of the microcrack in the PTD1 can be calculated, which has the form similar to Eq. (2). However, due to the difference in the mechanical properties of PTD and ITD, a distinct dimensionless function for Gp is expected. In the numerical simulations, the range of elastic modulus mismatch, Ep/Ei, is taken to be from 1.2 to 3, which is consistent with the experiments on stiffness of ITD and PTD (Kinney et al., 1996; Ziskind et al., 2011), and the effect of elastic modulus mismatch can be explored by such choice of the values. The numerical simulations are performed based on the fact that PTD exhibits larger elastic modulus than ITD (Kinney et al., 1999). 3. Results and discussion Fig. 3a shows the normalized driving force for main crack, Gi =ðEi e2 hÞ, as a function of crack length ai/h for three values of modulus mismatch (Ep/Ei). It is found that for large elastic modulus
Please cite this article in press as: An, B., Daniel Wagner, H. The effect of microcracking in the peritubular dentin on the fracture of dentin. J. Biomech. (2017), https://doi.org/10.1016/j.jbiomech.2017.10.022
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Fig. 1. Schematic illustration of fracture process in dentin. (a) The main crack approaching PTD. (b) Microcracking of PTD ahead of the main crack tip. (c) Merging of microcrack in PTD with the main crack.
Fig. 2. Schematics of the computational models of the microstructure of dentin. (a) Uniform arrangement of tubules. (b) Staggered arrangement of tubules.
mismatch, for example Ep/Ei = 2 and Ep/Ei = 3, the driving force for the main crack decreases as crack advances. The propagation of main crack is controlled by the competition between the driving force and fracture toughness of the ITD. Based on the results in Fig. 3a, it is identified that crack arrest may take place when the length of the main crack reaches a critical value corresponding to the circumstance where the driving force is lower than the fracture toughness. For the small modulus mismatch, Ep/Ei = 1.2, the normalized driving force shows weak dependence on crack length and a nearly constant driving force is observed. The effect of elastic modulus mismatch on crack driving force of main crack is complex. For short cracks, the driving force increases with the increase in modulus mismatch; while an opposite variation trend of driving force with modulus mismatch is observed for long cracks. The results shown in Fig. 3a suggest that the high stiffness of PTD plays
Fig. 3. Effect of the length of main crack on crack driving force. (a) Normalized driving force of the main crack as a function of the normalized length of main crack. (b) Normalized driving force of the microcrack in the PTD1 as a function of the normalized length of main crack. Note that the geometric parameters used in computational simulations are a1/tp = 0.4, a2/tp = 0 and a3/tp = 0.
a critical role in mitigating fracture. For a modest toughness of ITD, the main crack in the ITD could initiate in the case of high Ep/Ei due to the large driving force for short cracks. As the main crack propagates, the driving force drops. When the long cracks appear, the driving force could drop to a low value, indicating that crack arrest may take place. Although the low Ep/Ei could lead to low driving force for short cracks, the relatively high driving force for long cracks appears in this case. This is disadvantageous for mitigating fracture. Once cracks initiate, crack arrest is difficult to take place.
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Fig. 3b shows the variation of the normalized driving force for the microcrack in PTD1, Gp =ðEi e2 hÞ, with crack length ai/h for three values of modulus mismatch. It is found that as the main crack propagates in the ITD, the driving force for the microcrack in the PTD1 increases, indicating that crack growth in the ITD could promote fracture of the PTD ahead of the main crack tip, which is in a good agreement with experimental observations (Ivancik and Arola, 2013; Nalla et al., 2003b). The crack driving force Gp exhibits strong dependence on the elastic modulus mismatch; increasing Ep/Ei elevates the level of crack driving force Gp, suggesting that high stiffness of PTD could contribute to the microcracking of PTD in front of the main crack tip. It is important to note that the study by Ivancik and Arola (2013) is focused on the fracture toughness and toughening mechanisms of dentin, and it was revealed that crack bridging is the major source of toughening. The study by Koester et al. (2008) revealed that due to the small sizes of realistic cracks in human dentin and the relatively small crack-deflection angle, the origin of toughness of human dentin is more associated with crack branching compared with other toughening mechanisms. Koester et al. (2008) pointed out that crack branching requires the presence of microcracking of PTD ahead of the main crack tip, and the microcracking of PTD can also lead to the formation of crack bridging. Therefore, we confine our attention to the important mechanism of microcracking of PTD. The subsequent development of crack bridging and crack branching is not the focus of this study. The fracture toughness of dentin cannot be assessed based on our study, since it requires a model incorporating all the potential toughening mechanisms, which is beyond the scope of our study. To explore the interplay between crack growth in ITD and microcracking of PTD ahead of the main crack, we further analyze the effect of a1/tp. As shown in Fig. 4a, the driving force for the main crack increases as the microcrack advances in PTD1, indicating that microcracking of PTD ahead of the main crack facilitates crack growth in the ITD. We also show that the driving force for the main crack decreases with increasing modulus mismatch. This is consistent with the results shown in Fig. 3a, since we considered the case of long cracks. The normalized driving force for microcrack in PTD1, Gp =ðEi e2 hÞ, as a function of microcrack length is shown in Fig. 4b. It can be seen that the driving force Gp increases as microcrack advances, which increases the propensity of fracture of PTD. The elastic modulus mismatch has a significant influence on the driving force Gp; increasing the modulus mismatch leads to increased driving force. Nalla et al. (2003b) showed experimentally that crack growth in the microstructure of dentin is accompanied by the formation of a microdamage zone surrounding the main crack tip. The interaction between crack and the composite microstructure of dentin leads to the microcracking of PTD ahead of the main crack tip. Such fracture process is further confirmed by the experiments conducted by Ivancik and Arola (2013). From the mechanics point of view, the dentin tubules can be considered as voids which could give rise to stress concentration and lead to microcracking in PTD. The stress intensity at crack tip is so high that the main crack may propagate when the applied stress increases. The competition between the propagation of main crack and microcracking of PTD ahead of the main crack governs the fracture process in dentin. If the former dominates, the main crack can penetrate into PTD. Otherwise, microcracking of PTD occurs, which is followed by merging of microcracks with the main crack. The latter case is widely observed in experiments. However, the underlying mechanisms responsible for such behavior are still unknown. The present study has addressed this issue. We reveal that the propagation of main crack can increase the driving force for microcracks in PTD (Fig. 3b), promoting microcracking of PTD ahead of the main crack. And the
Fig. 4. Effect of the length of the microcrack in the PTD1 on crack driving force. (a) Normalized driving force of the main crack as a function of the normalized length of the microcrack in the PTD1. (b) Normalized driving force of the microcrack in the PTD1 as a function of the normalized length of the microcrack in the PTD1. The geometric parameters used in computational simulations are ai/L = 0.45, a2/tp = 0 and a3/tp = 0.
microcracking of PTD can elevate the level of driving force for the main crack (Fig. 4a), increasing the propensity of the growth of main crack. Hence, the propagation of main crack and microcracking of PTD facilitate each other and are two co-evolving fracture mechanisms. It is further revealed that the stiffness of PTD plays the critical role in determining which fracture mechanism dominates. The high stiffness of PTD can give rise to the increased driving force for microcracks in PTD and meanwhile enables the reduction in the driving force for main crack. This indicates that high stiffness of PTD activates the fracture mechanism of microcracking of PTD ahead of the main crack, promoting the merging of microcracks with the main crack. The normalized driving force for the main crack as a function of the length of the microcracks in the PTD2, which is arranged parallel to the crack plane with an offset, is shown in Fig. 5a. It can be seen that the driving force for the main crack decreases as microcracks advance in the PTD2, indicating that microcracking in the PTD2 can shield the tip of main crack, which potentially enhances fracture toughness. The similar shielding effect on the microcrack in the PTD1 is also found. As shown in Fig. 5b, the normalized driving force, Gp =ðEi e2 hÞ, decreases with the increase in
Please cite this article in press as: An, B., Daniel Wagner, H. The effect of microcracking in the peritubular dentin on the fracture of dentin. J. Biomech. (2017), https://doi.org/10.1016/j.jbiomech.2017.10.022
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Fig. 5. Effect of the length of the microcrack in the PTD2 on crack driving force. (a) Normalized driving force of the main crack as a function of the normalized length of the microcrack in the PTD2. (b) Normalized driving force of the microcrack in the PTD1 as a function of the normalized length of the microcrack in the PTD2. The parameters used in computational simulations are Ep/Ei = 3, ai/L = 0.45, a1/tp = 0.4 and a3/tp = 0.
Fig. 6. Effect of the length of the microcrack in the PTD3 on crack driving force. (a) Normalized driving force of the main crack as a function of the normalized length of the microcrack in the PTD3. (b) Normalized driving force of the microcrack in the PTD1 as a function of the normalized length of the microcrack in the PTD3. The parameters used in computational simulations are Ep/Ei = 3, ai/L = 0.45, a1/tp = 0.4 and a2/tp = 0.
the length of microcracks in the PTD2. We also investigated the effect of microcracking of the PTD3 which is close to the PTD1 but far from the tip of main crack. As shown in Fig. 6a, the driving force of the main crack slightly decreases as microcracks advance in the PTD3. Also, the driving force for the microcrack in the PTD1 drops slightly with increasing the length of the microcracks in the PTD3 (Fig. 6b). Comparing the effects of microcracking of the PTD2 and microcracking of the PTD3, it is found that the microcracking in the PTD2 plays even more important role in reducing crack driving force and shielding crack tip. It was identified that microcracking enables reduction in stiffness and release of residual stress, acting as an important toughening mechanism in brittle solids (Hutchinson, 1987). Such toughening mechanism is also active in biological hard tissues (Nalla et al., 2004). In dentin, the microdamage zone surrounding the main crack could contribute to the fracture toughness (Nalla et al., 2003a). However, the roles of the microcracks in distinct regions of the microdamage zone are unclear. The present study reveals that microcracks in the PTD aligned parallel to the crack plane with an offset play the dominant role in shielding the tip of main crack.
To investigate the influence of the tubule arrangement, the crack driving forces of the main crack and of the microcrack in the PTD1 are calculated in the case of staggered arrangement of tubules (illustrated in Fig. 2). As shown in Fig. 7a, the normalized driving forces, Gi =ðEi e2 hÞ, of the main crack in the cases of uniform arrangement and staggered arrangement of tubules are nearly identical, indicating that the driving force of main crack is independent of the tubule arrangement. Fig. 7b shows the normalized driving force of the microcrack in the PTD1 varying with the arrangement of tubules. It is found that staggered arrangement slightly reduces the driving force, whereas such a reduction in driving force is insignificant. It is important to note that the present study is focused on the interplay between the growth of main crack and the microcracking of PTD in front of the main crack, and the effect of the tubule geometry is not taken into account. In reality, dentin exhibits spatial variations in tubule density and radius (Ten Cate, 2008). It is valuable to assess the effect of tubule geometry changes on the fracture in dentin. In a recent study, it was found that the toughness of dentin increases with the decrease in tubule radius, and low tubule density can give rise to enhanced toughness of dentin (An, 2017).
Please cite this article in press as: An, B., Daniel Wagner, H. The effect of microcracking in the peritubular dentin on the fracture of dentin. J. Biomech. (2017), https://doi.org/10.1016/j.jbiomech.2017.10.022
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Conflict of interest statement The authors have no conflict of interest. Acknowledgments B.A. acknowledges the support from National Science Foundation of China #11402141. H. D. W. is recipient of the Livio Norzi Professional Chair at the Weizmann Institute. References
Fig. 7. Effect of tubule arrangement on (a) the driving force of the main crack and (b) the driving force of the microcrack in the PTD1. The parameters used in numerical simulations are Ep/Ei = 3, a1/tp = 0.4, a2/tp = 0 and a3/tp = 0.
4. Conclusions This study explores the effect of microcracking of PTD on the fracture in the microstructure of dentin. We reveal that the propagation of the main crack in ITD and microcracking of PTD ahead of the main crack are two co-evolving mechanisms involved in the fracture process of dentin. Crack growth in ITD promotes microcracking of PTD, and microcracking of PTD in front of the main crack, in turn, enables increased driving force for the main crack, facilitating crack propagation in ITD. The high stiffness of PTD plays the dominant role in promoting microcracking of PTD and suppressing the growth of the main crack in ITD, which leads to the experimentally observed fracture process in dentin, i.e. microcracking of PTD ahead of the main crack and subsequent merging of microcracks with the main crack. We further identified the role of microcracking of PTD in the microdamage zone surrounding the main crack. The microcracking of the PTD aligned parallel to the crack plane with an offset provides greater contribution to crack tip shielding, compared with microcracking of the PTD arranged along the initial crack line.
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Please cite this article in press as: An, B., Daniel Wagner, H. The effect of microcracking in the peritubular dentin on the fracture of dentin. J. Biomech. (2017), https://doi.org/10.1016/j.jbiomech.2017.10.022