Region dependent fracture resistance behavior of human dentin based on numerical simulation

$Page 1: Region dependent fracture resistance behavior of human dentin based on numerical simulation$
Appl. Math. Mech. -Engl. Ed., 35(3), 277–284 (2014)DOI 10.1007/s10483-014-1790-8c©Shanghai University and Springer-Verlag

Berlin Heidelberg 2014

Applied Mathematicsand Mechanics(English Edition)

Region dependent fracture resistance behavior of human dentin basedon numerical simulation∗

Yuan-zhi XU (��)1, Bing-bing AN (��)2,Dong-sheng ZHANG (��)3,4, Rao-rao WANG (��)1

(1. The Tenth People’s Hospital of Tongji University, Shanghai 200072, P. R. China;

2. Department of Physics, Shanghai University, Shanghai 200444, P. R. China;

3. Department of Mechanics, Shanghai University, Shanghai 200444, P. R. China;

4. Shanghai Key Laboratory of Mechanics in Energy Engineering,

Shanghai University, Shanghai 200072, P. R. China)

Abstract Dentin has a hierarchical structure and is composed of numerous tubuleswhose diameters and densities vary with the distances to the dentin-enamel junction.The unique structure determines the mechanical performance of dentin. In this study, amultiscale model, which is based on the combination of the virtual multidimensional in-ternal bond (VMIB) theory and the Monte Carlo method, is used to simulate the fracturebehavior of human dentin. Numerical simulations reveal that human dentin exhibits agraded resistance curve (R-curve). Among the three regions of dentin, superficial dentinshows the strongest resistance to crack propagation, and deep dentin has the weakestresistance. In addition, the predictions of fracture toughness of middle dentin agree wellwith the experimentally reported values, suggesting that the proposed model can be usedto characterize the fracture behavior of human dentin comprehensively and properly.

Key words virtual multidimensional internal bond (VMIB), Monte Carlo simulation,resistance curve (R-curve), crack growth, dentin

Chinese Library Classification Q66, R780.22010 Mathematics Subject Classification 74E05

1 Introduction

In recent years, the health and esthetics of teeth become increasingly important and attractmore and more concern from people. However, tooth fracture is always a threat to the healthof human teeth. Therefore, it is of great need to understand the crack propagation and fracturebehavior of human teeth comprehensively so as to help to prevent tooth disease and maintaintooth health. Human teeth are composed of enamel, dentin, and pulp. Dentin is a highlymineralized tissue and occupies most of the volumes of teeth. It locates between enamel andpulp with the function of protecting pulp. Therefore, it is imperative to establish a goodunderstanding on the fracture properties of dentin.

∗ Received Apr. 1, 2013 / Revised Sept. 27, 2013Project supported by the National Natural Science Foundation of China (No. 11172161), the Scienceand Technology Commission of Shanghai Municipality (No. 12ZR1423500), the Innovation Programof Shanghai Municipal Education Commission (No. 12ZZ092), and the Shanghai Leading AcademicDiscipline Project (No. S30106)Corresponding author Rao-rao WANG, Professor, Ph.D., E-mail: [email protected]

278 Yuan-zhi XU, Bing-bing AN, Dong-sheng ZHANG, and Rao-rao WANG

Conventional strength tests cannot provide enough insight into the fracture mechanism ofdentin[1] due to the complexity of fracture in dentin. The fracture of dentin is composed ofcrack initiation and crack propagation which cannot be properly characterized by strength.Recently, fracture mechanics has been introduced to study the fracture behavior of dentin[2–9].With the help of fracture mechanics, it is found that the fracture toughness of dentin dependson the orientation[2–3]. However, these studies are based on the assumption that the fracturetoughness of dentin is constant throughout the whole dentin. Kruzic et al.[6] found that thefracture toughness of dentin increases with crack propagation, i.e., dentin possesses a risingresistance curve (R-curve). Multiple toughening mechanisms had been proposed to accountfor the rising R-curve property, e.g., crack bridging by uncracked ligaments at the crack wakeand microcracking[6]. It is also found that hydration and aging have great influence on thefracture toughness[6–8]. Although many work has been conducted on the fracture mechanism ofdentin from different aspects, the critical fact is ignored that dentin is a typical heterogeneousmaterial. The mechanical behavior is not only related to its orientation but also the distancefrom the dentin-enamel junction (DEJ). However, the region dependent fracture behavior hasnot been reported in the literature yet.

Dentin has a hierarchical structure and is composed of mineral platelet and protein matrixat nano-scale[10−11]. At micro-scale, dentin consists of numerous tubules. The deformation andfracture behaviors of dentin closely depend on the structure and the corresponding properties atdifferent scales. Recently, an experimental study indicated that the fracture toughness of humandentin varied with the distance to the dentin-enamel junction[12]. Therefore, it is necessary toemploy the multiscale numerical simulation to study the mechanism of fracture of dentin. Toour knowledge, the fracture behavior of human dentin has also not been simulated by using themultiscale numerical model so far.

The virtual internal bond (VIB) model, which is developed by Gao and Klein[13], is a typicalmultiscale method and had been successfully applied to various fracture simulations[14–18]. Inthe VIB model, the macro-mechanical properties are derived from the properties of bondsbetween the material particles. The advantage of the VIB model is that it does not requirean extra fracture criterion. However, the Poisson’ ratio is fixed in the VIB model, restrictingthe application of the VIB model. Zhang and Ge[19] developed the virtual multidimensionalinternal bond (VMIB) model to solve this limitation. The VMIB model inherits the advantagesof the VIB model, while at the same time, introduces a variable Poisson’s ratio. It has beensuccessfully employed to some crack growth simulations[20–22].

In order to take the effect of the hierarchy and heterogeneity of human dentin into account,the present work simulates crack propagation in human dentin by using the VMIB model andMonte Carlo method to study the region effect on the R-curve to provide understanding to thefracture behavior of human dentin. The primary objective is to study the fracture behaviors ofdentin at different distances to the DEJ.

2 MethodsIn the VMIB model, the material is regarded as the assembly of numerous material

particles[19–20]. In the present study, dentin is modeled as a material consisting of numerousdentin particles. Virtual bonds exist between adjacent dentin particles. The macroscopicconstitutive relationship can be derived from the stiffness of the microscopic bond. A detaileddescription of the VMIB model is provided here. The elastic tensor in the VMIB model can becalculated as follows[20]:

Cijkl =∫ 2π

0

∫ π

0

kξiξjξkξlD(ϑ, ϕ) sin ϑdϑdϕ

+∫ 2π

0

∫ π

0

r(ξiη′jξkη′

l + ξiηj′′ξkηl

′′ + ξiηj′′′ξkηl

′′′)D(ϑ, ϕ) sin ϑdϑdϕ, (1)

Region dependent fracture resistance behavior of human dentin based on numerical simulation 279

where k is the normal stiffness, and r is the shear stiffness; ξ is the unit vector, and in thesphere coordinate system, ξ = (sin ϑ cosϕ, sin ϑ sin ϕ, cosϑ)T; η is the vector perpendicular toξ, η′ = ξ × (x1 × ξ), η′′ = ξ × (x2 × ξ), η′′′ = ξ × (x3 × ξ), and xi (i = 1, 2, 3) is the base vectorof the coordinate; D(ϑ, ϕ) is the bond density distribution function.

The bond stiffness function has great influence on the macroscopic properties of material.According to the previous studies[13–14,17], a two-parameter cohesive law, U ′(l) = Al exp(−l/B),can be used to model the fracture behavior. In the equation, U ′(l) is the force of bond. A isthe initial modulus of material, and B is the strain which corresponds to the ultimate tensilestrength. Zhang and Ge[19] also discussed the application of the two parameter cohesive law.Considering that the two-parameter cohesive law can describe the stress-strain response ofmaterials[19], the uniaxial tensile response of human dentin is written as

σ = Aε exp(−ε/B). (2)

In the present study, the plane strain condition is assumed and the following phenomenologicalstiffness functions are adopted:

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

k = k0 exp(− l

C1εt

),

r1 = r0 exp(− 2α1

C2(1 + μ)εt

),

r2 = r0 exp(− 2α2

C3(1 + μ)εt

),

where εt is the strain corresponding to the ultimate tensile strength; l is the stretch of bond givenby l = |ξiεijξj |, and εij is the strain tensor; α1, α2 are the rotation angles, and α1 = |ξkεklη

′l|,

α2 = |ξkεklηl′′|; μ is the Poission’s ratio; k0 = 2E

π(1+μ)(1−2μ) , and r0 = (1−4μ)Eπ(1+μ)(1−2μ)

[20]. C1, C2,

and C3 are model parameters, which can be determined by fitting the uniaxial tensile responseof materials. In the present study, C1 = 0.724 81, C2 = 0.709 53, and C3 = 1.465 20. The firststiffness function in (3) is used in [19]. Under plane strain condition, the elastic tensor can beobtained by substituting (3) into (1) as follows:

Cijkl =∫ π

0

(k0 exp

(− l

C1εt

)ξiξjξkξl + r0 exp

(− 2α1

C2(1 + μ)εt

)ξiη

′jξkη′

l

+ r0 exp(− 2α2

C3(1 + μ)εt

)ξiηj

′′ξkηl′′)D(ϑ)dϑ. (3)

Human dentin can be regarded as an isotropic material[23] and D(ϑ) = 1 for this case[19].The VMIB model can be used to model the fracture behaviors of different materials by assigningthe corresponding parameters of the materials. In this case, the strain corresponding to theultimate tensile strength of human dentin εt is introduced to delineate the mechanical propertiesof human dentin.

The bond stiffness characterizes the interactions between material particles. According to(3), the bond stiffness decreases with the increasing deformation. At the micro-scale, the de-crease of bond stiffness indicates that the distance between particles is increased. To an extremeextent, when the distance is large enough, the macroscopic cracking occurs[19]. According tothe study of Staninec et al.[24], the ultimate tensile strength of dentin can be characterized byusing the following Weibull distribution[24]:

σ = σ0(− ln(1 − P ))1m , (4)

where P is the fracture probability of dentin; σ is the ultimate tensile strength of humandentin; σ0 is the scale parameter, and m is the Weibull modulus. In the present study, the


strain corresponding to the ultimate tensile strength εt is considered to be a random variablewhich obeys the Weibull distribution.

In this study, three regions with different distances to DEJ, i.e., the superficial dentin, themiddle dentin, and the deep dentin, are considered. Considering the symmetry of the CTspecimen shown in Fig. 1(a), only half of the CT specimen is simulated. The correspondingWeibull parameters are listed in Table 1[24−25]. The simulation region is meshed with three-node triangular isoparametric elements. As shown in Fig. 1(b), the left-side region of the modelis meshed with ordinary linear elastic elements while the right-side of the model is meshed withVMIB elements. Totally, 11 342 elements are meshed in the model. A displacement-controlledload is applied at the hole to achieve the quasi-static loading. The effective crack length iscalculated by using the compliance method[26]. Accoding to the ASTM Standard[27], the R-curve of the material is determined[28]. The R-curve of the middle dentin is firstly calculatedto verify the employed method and determine the number of the random samples. Then, thisnumerical method is applied to predict the R-curves of the superficial dentin and the deepdentin, and the corresponding R-curves are also plotted to reveal the region dependent fracturebehavior.

Fig. 1 Compact tension specimen geometry (a) and finite element model of CT specimen (b)

Table 1 Weibull distribution parameters for three dentin locations[23–24]

Dentin location σ0/MPa m

Superficial 61.6 4.5Middle 48.7 4.5Deep 33.9 4.5


3 Results

Two hundred random samples are calculated in the middle dentin by using Monte Carlosimulations. The mean load-displacement curve is shown in Fig. 2(a), and the correspondingR-curve, which illustrates the fracture toughness with respect to the crack extension, is shownin Fig. 2(b).

Fig. 2 Mean load-displacement curve (a) and R-curve for crack propagation (b) in middle dentin

Nazari et al.[8] and Zhang et al.[9] have studied the aging effect on the fracture toughnessof human dentin by using an experimental approach, showing that the initial toughness, i.e.,the stress intensity factor at the onset of the crack extension, is in the range of 1MPa·m–1.4MPa·m 1

2 , and the plateau toughness, i.e., the maximum fracture toughness, is in the rangeof 1.08MPa·m–1.75MPa·m 1

2 . The initial and plateau toughnesses of the middle age group(35�age�55) are (1.22±0.06)MPa·m 1

2 and (1.43±0.1)MPa·m 12 , respectively. The present nu-

merical simulations indicate that the initial toughness of the middle dentin is 1.155MPa·m 12 ,

and the plateau toughnesses of the middle dentin is 1.46MPa·m 12 , consistent with the experi-

mental results for middle-aged patients.The results of all Monte Carlo simulations are shown in Fig. 3. The crack extension is ex-

pressed in the dimensionless form Δa/W , where W is the specimen width. Figure 3(a) showsthat the load-carrying capacity of dentin varies in the different regions. The mean values of

Fig. 3 Load-displacement curves for all Monte Carlo simulations where black curves are mean load-displacement curves for three dentin locations (a) and R-curves for crack propagation in threedentin locations (b)


the peak loads for crack growth in the superficial, the meddle, and the deep dentin are 12.83N,10.11N, and 7.12N, respectively.

Among the three dentin regions, the superficial dentin has the greatest load-carrying capacitywhile the deep dentin possesses the smallest one. Figure 3(b) shows that all the superficial, themiddle, and the deep dentin exhibit rising R-curves. The initial toughness of the superficial, themeddle, and the deep dentin are 1.4MPa·m 1

2 , 1.155MPa·m 12 , and 0.79MPa·m 1

2 , respectively.The plateau toughness are 1.86MPa·m 1

2 , 1.46MPa·m 12 , and 1.03MPa·m 1

2 , respectively.

4 Discussion

The numerical simulations show that the fracture toughness of human dentin increaseswith the crack growth, i.e., human dentin possesses a rising R-curve. The rising R-curvebehavior of dentin is attributed to various toughening mechanism. Kruzic et al.[6] found thatthe crack bridging by uncracked ligaments in the crack wake was responsible for the rising R-curve. Microcracking was also reported as a toughening mechanism[2]. In the VMIB model, theinteraction between material particles is modeled by the cohesive law. Since the cohesive lawrepresents the effects of the crack bridging and fracture processing zone near the crack tip[29],the rising R-curve behavior predicted by the present model is also capable of describing thecontributions of crack bridging and microcracking. For the consistency between the numericalsimulation and the experimental results, the fracture behaviors of the deep and superficialdentin are also analyzed by changing the parameters listed in Table 1.

The obtained results clearly indicate that the fracture toughness of human dentin closelydepends on its location. Superficial dentin has the strongest resistance to fracture, while deepdentin is the region most prone to crack. The previous study conducted by Bajaj and Arola[30]

identified that the fracture toughness of human enamel was higher than that of the middledentin. Imbeni et al.[31] observed a crack propagated from enamel to dentin, and found thatthe crack traversed the DEJ and stopped in the superficial dentin. It is reasonable to concludethat there should be a region which possesses very high fracture toughness and locates veryclosely to the DEJ. In such condition, the propagating crack can be arrested near the DEJ.From the numerical analysis, the maximum fracture toughness of superficial dentin reaches1.86MPa·m 1

2 , providing reliable resistance for the crack propagation from the outer enamel.The simulations provide more comprehensive understanding of fracture toughness of human

dentin. Due to the difficulties in preparing specimens, previous experimental studies werefocused on the fracture toughness of middle dentin. The numerical study provides a way tocharacterize the dependence of the fracture behavior of dentin on its location, i.e., the superficial,the middle, and the deep dentins, developing an understanding of the fracture behavior of dentinas a non-homogeneous material. The findings in this study also provide guidelines for the toothrestoration. It is found that tooth fracture is the major threat to the restored teeth. Basedon the present study, the materials with graded fracture toughness are required to reduce thepropensity of fracture of the restored teeth.

5 Conclusions

Numerical simulations of fracture of human dentin are carried out to study the fracturetoughness of dentin. The simulations show that human dentin exhibits rising R-curve behaviors.The R-curve property of human dentin depends on its location. Superficial dentin has the largestresistance to fracture, while deep dentin possesses the weakest fracture toughness. The presentstudy also demonstrates that the multiscale numerical simulations based on VMIB theory andMonte Carlo method could be used to characterize the fracture behaviors of human dentin aswell as other biological materials.


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Region dependent fracture resistance behavior of human dentin based on numerical simulation