Higher order Cauchy–Born rule based multiscale cohesive zone model and prediction of fracture toughness of silicon thin films

Abstract

In this work, we extend the multiscale cohesive zone model (MCZM) (Zeng and Li in Comput Methods Appl Mech Eng 199:547–556, 2010), in which interatomic potential is embedded into constitutive relation to express cohesive law in fracture process zone, to include the hierarchical Cauchy–Born rule in the process zone and to simulate three dimensional fracture in silicon thin films. The model has been applied to simulate fracture stress and fracture toughness of single-crystal silicon thin film by using the Tersoff potential. In this study, a new approach has been developed to capture inhomogeneous deformation inside the cohesive zone. For this purpose, we introduce higher order Cauchy–Born rules to construct constitutive relations for corresponding higher order process zone elements, and we introduce a sigmoidal function supported bubble mode in finite element shape function of those higher order cohesive zone elements to capture the nonlinear inhomogeneous deformation inside the cohesive zone elements. Benchmark tests with simple 3D models have confirmed that the present method can predict the fracture toughness of silicon thin films. Interestingly, this is accomplished without increasing of computational cost, because the present model does not require quadratic elements to represent heterogeneous deformation, which is the inherent weakness of the previous MCZM model. Quantitative comparisons with experimental results are performed by computing crack propagation in non-notched and initially notched silicon thin films, and it is found that our model can reproduce essential material properties, such as Young’s modulus, fracture stress, and fracture toughness of single-crystal silicon thin films.

Appendix

1.1 Appendix 1: Optimization of inner vector of diamond cubic lattice

First, we document the procedure on how to find the inner vector $\mathbf{v}$ position based on strain energy minimization.

In order to satisfy Eq. (40), we use Newton’s method,

$$\begin{aligned}&{} \mathbf{v}^{i+1} = \mathbf{v}^{i} + \delta \mathbf{v} \end{aligned}$$

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$$\begin{aligned}&\delta \mathbf{v} = \Bigl ( { \partial ^2 V \over \partial \mathbf{v}^2 } \Bigr )^{-1} \Bigl ( { \partial V \over \partial \mathbf{v} } \Bigr )~. \end{aligned}$$

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Then, similar to Eq. (41), one may find that

$$\begin{aligned}&{ \partial V \over \partial \mathbf{v} } = {1 \over 2 \varOmega _0} \sum _{j = 2}^5 \Bigl ( { \partial V_{1j} \over \partial \mathbf{v} } \Bigl ) \nonumber \\&\quad = {1 \over 2 \varOmega _0} \sum _{j = 2}^5 \Bigl [ { \partial V_{1j} \over \partial r_{1j} } { \partial r_{1j} \over \partial \mathbf{v} } \nonumber \\&\qquad + \sum _{k = 2, j \ne k}^5 \Bigl ( { \partial V_{1j} \over \partial r_{1k} } { \partial r_{1k} \over \partial \mathbf{v} } + { \partial V_{1j} \over \partial cos \theta _{1jk} } { \partial cos \theta _{1jk} \over \partial \mathbf{v} } \Bigr ) \Bigr ]\nonumber \\ \end{aligned}$$

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where

$$\begin{aligned}&{ \partial cos \theta _{1jk} \over \partial \mathbf{v} } = { \partial r_{1j} \over \partial \mathbf{v} } { \partial cos \theta _{1jk} \over \partial r_{1j} } + { \partial r_{1k} \over \partial \mathbf{v} } { \partial cos \theta _{1jk} \over \partial r_{1k} } \nonumber \\&\qquad +\, { \partial r_{jk} \over \partial \mathbf{v} } { \partial cos \theta _{1jk} \over \partial r_{jk} } \nonumber \\&\quad = \Bigl ( {1 \over r_{1k} } - {cos \theta _{1jk} \over r_{1j} } \Bigr ) { \partial r_{1j} \over \partial \mathbf{v} } +\Bigl ( {1 \over r_{1j} } - {cos \theta _{1jk} \over r_{1k} } \Bigr ) { \partial r_{1k} \over \partial \mathbf{v} }\nonumber \\&\qquad -\Bigl ( { r_{jk} \over r_{1j} r_{i1k} } \Bigr ) { \partial r_{jk} \over \partial \mathbf{v} } \end{aligned}$$

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and

$$\begin{aligned} \displaystyle { \partial r_{ij} \over \partial \mathbf{v} } = \left\{ \begin{array}{ll} \displaystyle { \partial r_{1j} \over \partial \mathbf{r}_{1j} } \displaystyle { \partial \mathbf{r}_{1j} \over \partial \mathbf{v} } = - \displaystyle { \mathbf{r}_{1j} \over r_{1j} } &{} i, j \in ~\alpha ,~ \mathrm{or } ~ i, j \in \beta \\ 0 &{} \mathrm{otherwise} \end{array} \right. \nonumber \\ \end{aligned}$$

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For the second derivative,

$$\begin{aligned}&{ \partial ^2 V \over \partial \mathbf{v}^2 } = {1 \over 2 \varOmega _0} \sum _{j = 2}^5 \Bigl ( { \partial ^2 V_{1j} \over \partial \mathbf{v}^2 } \Bigl ) \nonumber \\&\quad = {1 \over 2 \varOmega _0} \sum _{j = 2}^5 \biggl [ { \partial ^2 V_{1j} \over \partial r_{1j}^2 } { \partial r_{1j} \over \partial \mathbf{v} } { \partial r_{1j} \over \partial \mathbf{v} } + { \partial V_{1j} \over \partial r_{1j} } { \partial ^2 r_{1j} \over \partial \mathbf{v}^2 } \nonumber \\&\qquad + \sum _{k = 2, j \ne k}^5 \Bigl \{ { \partial V_{1j}^2 \over \partial r_{1k}^2 } { \partial r_{1k} \over \partial \mathbf{v} } { \partial r_{1k} \over \partial \mathbf{v} } + { \partial V_{1j} \over \partial r_{1k} } { \partial ^2 r_{1k} \over \partial \mathbf{v}^2 } \nonumber \\&\qquad + { \partial V_{1j}^2 \over \partial cos \theta _{1jk}^2 } { \partial cos \theta _{1jk} \over \partial \mathbf{v} } { \partial cos \theta _{1jk} \over \partial \mathbf{v} } + { \partial V_{1j} \over \partial cos \theta _{1jk} } { \partial ^2 cos \theta _{1jk} \over \partial \mathbf{v}^2 } \Bigr \} \biggr ]\nonumber \\ \end{aligned}$$

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$$\begin{aligned}&{\partial ^2 cos\theta _{1jk} \over \partial \mathbf{v}^2} = \Bigl ( {1 \over r_{1k} } - {cos\theta _{1jk} \over r_{1j} } \Bigr ) {\partial ^2 r_{1j} \over \partial \mathbf{v}^2 } + \Bigl ( {1 \over r_{1j} } - {cos\theta _{1jk} \over r_{1k} }\Bigr ) {\partial ^2 r_{1k} \over \partial \mathbf{v}^2 } \nonumber \\&\quad + \Bigl ( -{1 \over r_{1k}^2 } {\partial r_{1k} \over \partial \mathbf{v} } - {\partial cos\theta _{1jk} \over \partial \mathbf{v} } {1 \over r_{1j} } + {cos\theta _{1jk} \over r_{1j}^2 } {\partial r_{1j} \over \partial \mathbf{v} } \Bigr ) {\partial r_{1j} \over \partial \mathbf{v}} \nonumber \\&\quad + \Bigl ( -{1 \over r_{1j}^2 } {\partial r_{1j} \over \partial \mathbf{v} } - {\partial cos\theta _{1jk} \over \partial \mathbf{v} } {1 \over r_{1k} } + {cos\theta _{1jk} \over r_{1k}^2 } {\partial r_{1k} \over \partial \mathbf{v} } \Bigr ) {\partial r_{1k} \over \partial \mathbf{v}} \end{aligned}$$

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where the following relation,

$$\begin{aligned} \displaystyle { \partial ^2 r_{ij} \over \partial \mathbf{v}^2 } = \left\{ \begin{array}{lcl} \displaystyle {\partial \over \partial \mathbf{v} } \Bigl ( - \displaystyle { \mathbf{r}_{ij} \over r_{ij} } \Bigr ) = \displaystyle {1 \over r_{1j}} \Bigr ( \mathbf{I} - { \mathbf{r}_{ij} \displaystyle \mathbf{r}_{ij} \over r_{ij}^2 } \Bigl ), &{}&{} i, j \in \alpha ~{ \mathrm or }~,~ i, j \in \beta \\ 0 , &{}&{} \mathrm{otherwise} \end{array} \right. \nonumber \\ \end{aligned}$$

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and Eqs. (56)–(59) are applied.

1.2 Appendix 2: Divergence of the second order stress tensor

Second, we show how to find the divergence of the second order stress tensor based on the higher Cauchy–Born rule for the Tersoff potential.

According to Eqs. (50)–(53), we have,

$$\begin{aligned}&\nabla _X \cdot \mathbf{Q} = {1 \over 2 \varOmega _0} \sum _{j = 2}^5 \Biggl [ \nabla _X \Bigl ( { \partial V_{1j} \over \partial r_{1j} } { 1 \over 2r_{1j} } \Bigr ) \mathbf{r}_{1j} \otimes \mathbf{R}_{1j} \otimes \mathbf{R}_{1j} \nonumber \\&\quad + \Bigl ( { \partial V_{1j} \over \partial r_{1j} } { 1 \over 2r_{1j} } \Bigr ) \nabla _X \cdot ( \mathbf{r}_{1j} \otimes \mathbf{R}_{1j} \otimes \mathbf{R}_{1j} ) \nonumber \\&\quad + \sum _{k =2, k \ne j}^5 \biggl [ \nabla _X \Bigl ( { \partial V_{1j} \over \partial r_{1k} } { 1 \over 2r_{1k} } \Bigr ) \mathbf{r}_{1k} \otimes \mathbf{R}_{1k} \otimes \mathbf{R}_{1k} \nonumber \\&\quad + \Bigl ( { \partial V_{1j} \over \partial r_{1k} } { 1 \over 2r_{1k} } \Bigr ) \nabla _X \cdot ( \mathbf{r}_{1k} \otimes \mathbf{R}_{1k} \otimes \mathbf{R}_{1k} ) \nonumber \\&\quad + \nabla _X \Bigl \{ \Bigl ( {\partial V_{1j} \over \partial cos \theta } {1 \over 2r_{1j} } \Bigr ) \Bigl ( {1 \over r_{1k}} - {cos \theta \over r_{1j} } \Bigr ) \Bigr \} \mathbf{r}_{1j} \otimes \mathbf{R}_{1j} \otimes \mathbf{R}_{1j} \nonumber \\&\quad + \Bigl \{ \Bigl ( {\partial V_{1j} \over \partial cos \theta } {1 \over 2r_{1j} } \Bigr ) \Bigl ( {1 \over r_{1k}} - {cos \theta \over r_{1j} } \Bigr ) \Bigr \} \nabla _X \cdot ( \mathbf{r}_{1j} \otimes \mathbf{R}_{1j} \otimes \mathbf{R}_{1j} ) \nonumber \\&\quad + \nabla _X \Bigl \{ {\partial V_{1j} \over \partial cos \theta } {1 \over 2r_{1k} } \Bigl ( {1 \over r_{1j}} - {cos \theta \over r_{1k} } \Bigr ) \Bigr \} \mathbf{r}_{1k} \otimes \mathbf{R}_{1k} \otimes \mathbf{R}_{1k} \nonumber \\&\quad + \Bigl \{ {\partial V_{1j} \over \partial cos \theta } {1 \over 2r_{1k} } \Bigl ( {1 \over r_{1j}} - {cos \theta \over r_{1k} } \Bigr ) \Bigr \} \nabla _X \cdot ( \mathbf{r}_{1k} \otimes \mathbf{R}_{1k} \otimes \mathbf{R}_{1k} ) \nonumber \\&\quad - \nabla _X \Bigl \{ {\partial V_{1j} \over \partial cos \theta } {1 \over 2r_{jk} } \Bigl ( {r_{jk} \over r_{1j} r_{1k} } \Bigr ) \Bigr \} \mathbf{r}_{jk} \otimes \mathbf{R}_{jk} \otimes \mathbf{R}_{jk} \nonumber \\&\quad - \Bigl \{ {\partial V_{1j} \over \partial cos \theta } {1 \over 2r_{jk} } \Bigl ( {r_{jk} \over r_{1j} r_{1k} } \Bigr ) \Bigr \} \nabla _X \cdot ( \mathbf{r}_{jk} \otimes \mathbf{R}_{jk} \otimes \mathbf{R}_{jk} ) \biggr ] \Biggr ] \end{aligned}$$

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and based on the above equation, we can write the following expressions with abbreviating index,

$$\begin{aligned}&\nabla _X \Bigl ( { \partial V \over \partial r } { 1 \over 2r } \Bigr ) \mathbf{r} \otimes \mathbf{R} \otimes \mathbf{R} + \Bigl ( { \partial V \over \partial r } { 1 \over 2r } \Bigr ) \nabla _X \cdot ( \mathbf{r} \otimes \mathbf{R} \otimes \mathbf{R} ) \nonumber \\&\quad = \bigg [ {\partial \over \partial X_C} \Bigl ( { \partial V \over \partial r } { 1 \over 2r } \Bigr ) + \Bigl ( { \partial V \over \partial r } { 1 \over 2r } \Bigr ) {\partial \over \partial X_C} \biggr ] r_a R_A R_B (\mathbf{e}_a \otimes \mathbf{E}_A \otimes \mathbf{E}_B ) \cdot \mathbf{E}_C \nonumber \\&\quad = \bigg [ {\partial \over \partial r} {\partial r \over \partial F} {\partial F \over \partial X_B} \Bigl ( { \partial V \over \partial r } { 1 \over 2r } \Bigr ) \!+\! \Bigl ( { \partial V \over \partial r } { 1 \over 2r } \Bigr ) {\partial \over \partial r} {\partial r \over \partial F} {\partial F \over \partial X_B} \biggr ] r_a R_A R_B (\mathbf{e}_a \!\otimes \! \mathbf{E}_A ) \nonumber \\&\quad = { 1 \over 2 } { \partial ^2 V \over \partial r^2 } R_a G_B R_A R_B = { \partial ^2 V \over \partial r^2 } \mathbf{R} \otimes \mathbf{r}^{2nd} \end{aligned}$$

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Here, $\mathbf{r}^{2nd}$ is the additional term in atom coordination [see Eq. (27)] owing to second order deformation. And, an example for the angle term is,

$$\begin{aligned}&\nabla _X \biggl [ \Bigr ( {\partial V_{1j} \over \partial cos \theta } {1 \over 2r_{1j} } \Bigl ) \Bigr ( { 1 \over r_{1k}} - {cos \theta \over r_{1j} } \Bigl ) \biggr ] \mathbf{r}_{1j} \otimes \mathbf{R}_{1j} \otimes \mathbf{R}_{1j} \nonumber \\&\quad = {\partial \over \partial cos \theta } \biggl [ \Bigr ( {\partial V_{1j} \over \partial cos \theta } {1 \over 2r_{1j} } \Bigl ) \Bigr ( { 1 \over r_{1k}} - {cos \theta \over r_{1j} } \Bigl ) \biggr ] \nonumber \\&\qquad \times {\partial cos \theta \over \partial F } {\partial F \over \partial X_B} r_{1j}^a R_{1j}^A R_{1j}^B ( \mathbf{e}_a \otimes \mathbf{E}_A) \nonumber \\&\quad = \biggl [ \Bigr ( {\partial ^2 V_{1j} \over \partial cos^2 \theta } {1 \over 2r_{1j} } \Bigl ) \Bigr ( { 1 \over r_{1k}} - {cos \theta \over r_{1j} } \Bigl ) - \Bigr ( {\partial V_{1j} \over \partial cos \theta } {1 \over 2r_{1j} } \Bigl ) { 1 \over r_{1j} } \biggr ] \nonumber \\&\quad {\partial cos \theta \over \partial F } G_B r_{1j}^a R_{1j}^A R_{1j}^B ( \mathbf{e}_a \otimes \mathbf{E}_A ), \end{aligned}$$

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where $\partial cos \theta / \partial \mathbf{F}$ has shown in Eq. (43), and

$$\begin{aligned}&{\partial V_{1j} \over \partial cos \theta } {1 \over 2r_{1j} } \Bigl ( {1 \over r_{1k}} - {cos \theta \over r_{1j} } \Bigr ) \nabla _X \cdot ( \mathbf{r}_{1j} \otimes \mathbf{R}_{1j} \otimes \mathbf{R}_{1j} ) \nonumber \\&\quad = {\partial V_{1j} \over \partial cos \theta } {1 \over 2r_{1j} } \Bigl ( {1 \over r_{1k}} - {cos \theta \over r_{1j} } \Bigr ) {\partial \over \partial r} {\partial r \over \partial F} {\partial F \over \partial X_B} r_{1j}^a R_{1j}^A R_{1j}^B (\mathbf{e}_a \otimes \mathbf{E}_A ) \nonumber \\&\quad = {\partial V_{1j} \over \partial cos \theta } {1 \over 2r_{1j} } \Bigl ( {1 \over r_{1k}} - {cos \theta \over r_{1j} } \Bigr ) {r_{1j}^a R_{1j}^a \over r_{1j}} {r_{1j}^a \over r_{1j} } G_B R_{1j}^A R_{1j}^B (\mathbf{e}_a \otimes \mathbf{E}_A ) \nonumber \\&\quad = {\partial V_{1j} \over \partial cos \theta } {1 \over r_{1j} } \Bigl ( {1 \over r_{1k}} - {cos \theta \over r_{1j} } \Bigr ) \mathbf{R}_{1j} \otimes \mathbf{r}_{1j}^{2nd}~. \end{aligned}$$

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Fig. 23

Fifteen nodes of isoparametric triangular prism element and a bubble node. Red circle is the point of a bubble node

1.3 Appendix 3: Second order shape function with a bubble mode

Second order shape function for wedge element with a bubble node in the center of element. Node positions are illustrated in Fig. 23.

$$\begin{aligned} N_1= & {} -{1 \over 2} ( 1 - \xi - \eta ) ( 2 \xi + 2\eta + \zeta ) ( 1 - \zeta ) - {1 \over 15} N_{16} \nonumber \\ N_2= & {} {1 \over 2} \xi ( 2 \xi - \zeta - 2 ) ( 1 - \zeta ) - {1 \over 15} N_{16} \nonumber \\ N_3= & {} {1 \over 2} \eta ( 2 \xi - \zeta - 2 ) ( 1 - \zeta )- {1 \over 15} N_{16} \nonumber \\ N_4= & {} - {1 \over 2} ( 1 - \xi - \eta ) ( 2 \xi + 2 \eta - \zeta ) ( 1 + \zeta ) - {1 \over 15} N_{16} \nonumber \\ N_5= & {} {1 \over 2} \xi ( 2 \xi + \zeta - 2 ) ( 1 + \zeta ) - {1 \over 15} N_{16} \nonumber \\ N_6= & {} {1 \over 2} \eta ( 2 \eta + \zeta -2 ) ( 1 + \zeta ) - {1 \over 15} N_{16} \nonumber \\ N_7= & {} 2 \xi ( 1 - \xi - \eta ) ( 1 - \zeta ) - {1 \over 15} N_{16} \nonumber \\ N_8= & {} 2 \xi \eta ( 1 - \zeta ) - {1 \over 15} N_{16} \nonumber \\ N_9= & {} 2 \eta ( 1 - \xi - \eta ) ( 1 - \zeta ) - {1 \over 15} N_{16} \nonumber \\ N_{10}= & {} 2 \xi ( 1 - \xi - \eta ) ( 1 + \zeta ) - {1 \over 15} N_{16} \nonumber \\ N_{11}= & {} 2 \xi \eta ( 1 + \zeta ) - {1 \over 15} N_{16} \nonumber \\ N_{12}= & {} 2 \eta ( 1 - \xi - \eta ) ( 1 + \zeta ) - {1 \over 15} N_{16} \nonumber \\ N_{13}= & {} ( 1 - \xi - \eta ) ( 1 - \zeta ^2 ) - {1 \over 15} N_{16} \nonumber \\ N_{14}= & {} \xi ( 1 - \zeta ^2 ) - {1 \over 15} N_{16} \nonumber \\ N_{15}= & {} \eta ( 1 - \zeta ^2 ) - {1 \over 15} N_{16} \nonumber \\ N_{16}= & {} {19683 \over 4096} \xi \eta ( 1 - \xi - \eta )(2\xi + 2\eta \nonumber \\&+ \zeta )(2\xi - \zeta -2)(2\eta - \zeta -2)\nonumber \\&\quad (2\xi + 2\eta -\zeta ) \nonumber \\&\quad (2\xi + \zeta -2)(2\eta + \zeta -2) ( 1 - \zeta ^2 ) \end{aligned}$$

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