Material point method for crack propagation in anisotropic media: a phase field approach

Abstract

A novel phase field formulation implemented within a material point method setting is developed to address brittle fracture simulation in anisotropic media. The case of strong anisotropy in the crack surface energy is treated by considering an appropriate variational, i.e. phase field approach. Material point method is utilized to efficiently treat the resulting coupled governing equations. The brittle fracture governing equations are defined at a set of Lagrangian material points and subsequently interpolated at the nodes of a fixed Eulerian mesh where solution is performed. As a result, the quality of the solution does not depend on the quality of the underlying finite element mesh and is relieved from mesh distortion errors. The efficiency and validity of the proposed method are assessed through a set of benchmark problems.

Appendices

Appendix A: Variational approach of the anisotropic phase field model

In the energy balance equation (9) the rate of the kinetic energy is evaluated as

$$\begin{aligned} \dot{\mathcal {K}} \left( \dot{\mathbf {u}} \right) = \frac{\hbox {d}}{\hbox {d}t} \int \limits _{\varOmega }\frac{1}{2} \rho |\dot{\mathbf {u}}|^{2} \hbox {d}{\varOmega }= \int \limits _{\varOmega }\left( \left[ \rho \ddot{\mathbf {u}} \right] \cdot \dot{\mathbf {u}} \right) \hbox {d}{\varOmega }. \end{aligned}$$

(61)

Similarly, the rate of the external work is expressed as

$$\begin{aligned} \dot{\mathcal {W}}^{ext} \left( \dot{\mathbf {u}} \right) = \int _{\partial {\varOmega }_{\bar{t}}} ( \varvec{\bar{t}} \cdot \dot{\mathbf {u}}) \,\hbox {d} \partial {\varOmega }_{\bar{t}} + \int _{\varOmega }(\varvec{b} \cdot \dot{\mathbf {u}}) \,\hbox {d} {\varOmega } \end{aligned}$$

(62)

and the rate of the internal work is defined accordingly as

$$\begin{aligned} \dot{\mathcal {W}}^{int} \left( \dot{\mathbf {u}},\dot{c},\nabla \dot{c} \right) = \frac{\hbox {d} {\varPsi }_{pot}}{\hbox {d}t} = \frac{\hbox {d}}{\hbox {d}t} \int \limits _{\varOmega }\left( \psi _{el} + \bar{\mathcal {G}}_{c} \mathcal {Z}_{c,Anis} \right) \hbox {d}{\varOmega }. \end{aligned}$$

(63)

Applying the divergence theorem in Eq. (63), the rate of the internal work $\dot{\mathcal {W}}^{int} \left( \dot{\mathbf {u}},\dot{c},\nabla \dot{c} \right)$ assumes the following form

$$\begin{aligned} \dot{\mathcal {W}}^{int} \left( \dot{\mathbf {u}},\dot{c},\nabla \dot{c} \right) = \mathcal {B}_1 + \mathcal {B}_2 + \mathcal {B}_3 + \mathcal {B}_4 , \end{aligned}$$

(64)

where the components $B_i$, $i=1\dots 4$, assume the following expressions

$$\begin{aligned} \mathcal {B}_1= & {} \frac{\hbox {d}}{\hbox {d}t} \int \limits _{\varOmega }\psi _{el} \hbox {d}{\varOmega }\nonumber \\= & {} \int _{\partial {\varOmega }} \left( \left[ \varvec{\sigma } \varvec{n} \right] \cdot \dot{\mathbf {u}} \right) \,\hbox {d} \partial {\varOmega }- \int \limits _{\varOmega }\left( \left[ \nabla \cdot \varvec{\sigma } \right] \cdot \dot{\mathbf {u}} \right) \,\hbox {d} {\varOmega }+ \int \limits _{\varOmega }\left( \psi _{el_c} \dot{c} \right) \,\hbox {d} {\varOmega }; \end{aligned}$$

(65)

$$\begin{aligned} \mathcal {B}_2= & {} \frac{\hbox {d}}{\hbox {d}t} \int \limits _{\varOmega }\left( \bar{\mathcal {G}}_{c} \left[ \frac{{{{\left( {c - 1} \right) }^2}}}{{4{l_0}}} \right] \right) \hbox {d}{\varOmega }= \int \limits _{\varOmega }\left( \left[ \bar{\mathcal {G}}_{c} \frac{ \left( c-1 \right) }{2{l_0}} \right] \dot{c} \right) \,\hbox {d} {\varOmega }; \end{aligned}$$

(66)

$$\begin{aligned} \mathcal {B}_3= & {} \frac{\hbox {d}}{\hbox {d}t} \int \limits _{\varOmega }\left( \bar{\mathcal {G}}_{c} \left[ {l_0} | \nabla c |^2 \right] \right) \,\hbox {d} {\varOmega }\nonumber \\= & {} \int _{\partial {\varOmega }} \left( \left[ \bar{\mathcal {G}}_{c} l_0 2 \nabla c \right] \cdot \varvec{n} \dot{c} \right) \,\hbox {d} \partial {\varOmega }- \int \limits _{\varOmega }\left( \left[ \bar{\mathcal {G}}_{c} l_0 2 {\varDelta }c \right] \dot{c} \right) \,\hbox {d} {\varOmega }; \end{aligned}$$

(67)

and

$$\begin{aligned} \begin{aligned} \mathcal {B}_4&= \frac{\hbox {d}}{\hbox {d}t} \int \limits _{\varOmega }\left( \bar{\mathcal {G}}_{c} \left[ l_0^{3} \sum _{\begin{array}{c} ijkl \end{array}} \gamma _{ijkl} \frac{\partial ^2 c}{\partial x_{i} \partial x_{j}} \frac{\partial ^2 c}{\partial x_{i} \partial x_{j}} \right] \right) \,\hbox {d} {\varOmega }\\&= \int \limits _{\varOmega }\left( \bar{\mathcal {G}}_{c} \left[ l_0^{3} \sum _{\begin{array}{c} ijkl \end{array}} \gamma _{ijkl} \frac{\hbox {d}}{\hbox {d}t} \left( \frac{\partial ^2 c}{\partial x_{i} \partial x_{j}} \frac{\partial ^2 c}{\partial x_{k} \partial x_{l}} \right) \right] \right) \,\hbox {d} {\varOmega }\\&=\int \limits _{\varOmega }\left( \bar{\mathcal {G}}_{c} \left[ l_0^{3} \sum _{\begin{array}{c} ijkl \end{array}} \gamma _{ijkl} \left( \frac{\hbox {d}}{\hbox {d}t} \left( \frac{\partial ^2 c}{\partial x_{i} \partial x_{j}} \right) \frac{\partial ^2 c}{\partial x_{k} \partial x_{l}} + \frac{\partial ^2 c}{\partial x_{i} \partial x_{j}} \frac{\hbox {d}}{\hbox {d}t} \left( \frac{\partial ^2 c}{\partial x_{k} \partial x_{l}} \right) \right) \right] \right) \,\hbox {d} {\varOmega }\\ {}&= \mathcal {T}_1 + \mathcal {T}_2 , \end{aligned} \end{aligned}$$

(68)

respectively, where

$$\begin{aligned} \mathcal {T}_1=\int \limits _{\varOmega }\left( \bar{\mathcal {G}}_{c} \left[ l_0^{3} \sum _{\begin{array}{c} ijkl \end{array}} \gamma _{ijkl} \left( \frac{\hbox {d}}{\hbox {d}t} \left( \frac{\partial ^2 c}{\partial x_{i} \partial x_{j}} \right) \frac{\partial ^2 c}{\partial x_{k} \partial x_{l}} \right) \right] \right) \,\hbox {d} {\varOmega }\end{aligned}$$

(69)

and

$$\begin{aligned} \mathcal {T}_2=\int \limits _{\varOmega }\left( \bar{\mathcal {G}}_{c} \left[ l_0^{3} \sum _{\begin{array}{c} ijkl \end{array}} \gamma _{ijkl} \left( \frac{\partial ^2 c}{\partial x_{i} \partial x_{j}} \frac{\hbox {d}}{\hbox {d}t} \left( \frac{\partial ^2 c}{\partial x_{k} \partial x_{l}} \right) \right) \right] \right) \,\hbox {d} {\varOmega }\end{aligned}$$

(70)

Components $\mathcal {T}_1$ and $\mathcal {T}_2$ are further expanded employing the divergence theorem into

$$\begin{aligned} \begin{aligned} \mathcal {T}_1&= \int _{\partial {\varOmega }} \left( \left[ \bar{\mathcal {G}}_{c} l_0^{3} \sum _{\begin{array}{c} ijkl \end{array}} \gamma _{ijkl} \left( \frac{\hbox {d}}{\hbox {d}t} \left( \frac{\partial c}{\partial x_{i} } \right) \frac{\partial ^2 c}{\partial x_{k} \partial x_{l}} \right) \right] \cdot \varvec{n} \right) \,\hbox {d} \partial {\varOmega }- \int \limits _{\varOmega }\left( \left[ \bar{\mathcal {G}}_{c} l_0^{3} \sum _{\begin{array}{c} ijkl \end{array}} \gamma _{ijkl} \left( \frac{\hbox {d}}{\hbox {d}t} \left( \frac{\partial c}{\partial x_{i} } \right) \frac{\partial ^3 c}{\partial x_{j} \partial x_{k} \partial x_{l}} \right) \right] \right) \,\hbox {d} {\varOmega }\\ {}&= \int _{\partial {\varOmega }} \left( \left[ \bar{\mathcal {G}}_{c} l_0^{3} \sum _{\begin{array}{c} ijkl \end{array}} \gamma _{ijkl} \left( \frac{\hbox {d}}{\hbox {d}t} \left( \frac{\partial c}{\partial x_{i} } \right) \frac{\partial ^2 c}{\partial x_{k} \partial x_{l}} \right) \right] \cdot \varvec{n} \right) \,\hbox {d} \partial {\varOmega }\\ {}&\quad - \left( \int _{\partial {\varOmega }} \left( \left[ \bar{\mathcal {G}}_{c} l_0^{3} \sum _{\begin{array}{c} ijkl \end{array}} \gamma _{ijkl} \left( \frac{\hbox {d} c}{\hbox {d}t} \frac{\partial ^3 c}{\partial x_{j} \partial x_{k} \partial x_{l}} \right) \right] \cdot \varvec{n} \right) \,\hbox {d} \partial {\varOmega }- \int \limits _{\varOmega }\left( \left[ \bar{\mathcal {G}}_{c} l_0^{3} \sum _{\begin{array}{c} ijkl \end{array}} \gamma _{ijkl} \left( \frac{\hbox {d} c}{\hbox {d}t} \frac{\partial ^4 c}{\partial x_{i} \partial x_{j} \partial x_{k} \partial x_{l}} \right) \right] \right) \,\hbox {d} {\varOmega }\right) \\ {}&= \int _{\partial {\varOmega }} \left( \left[ \bar{\mathcal {G}}_{c} l_0^{3} \sum _{\begin{array}{c} ijkl \end{array}} \gamma _{ijkl} \left( \frac{\hbox {d}}{\hbox {d}t} \left( \frac{\partial c}{\partial x_{i} } \right) \frac{\partial ^2 c}{\partial x_{k} \partial x_{l}} \right) \right] \cdot \varvec{n} \right) \,\hbox {d} \partial {\varOmega }\\&\quad - \left( \int _{\partial {\varOmega }} \left( \left[ \bar{\mathcal {G}}_{c} l_0^{3} \sum _{\begin{array}{c} ijkl \end{array}} \gamma _{ijkl} \left( \frac{\partial ^3 c}{\partial x_{j} \partial x_{k} \partial x_{l}} \right) \right] \cdot \varvec{n} \dot{c} \right) \,\hbox {d} \partial {\varOmega }- \int \limits _{\varOmega }\left( \left[ \bar{\mathcal {G}}_{c} l_0^{3} \sum _{\begin{array}{c} ijkl \end{array}} \gamma _{ijkl} \left( \frac{\partial ^4 c}{\partial x_{i} \partial x_{j} \partial x_{k} \partial x_{l}} \right) \right] \dot{c} \right) \,\hbox {d} {\varOmega }\right) \end{aligned} \end{aligned}$$

(71)

and

$$\begin{aligned} \begin{aligned} \mathcal {T}_2&=\int _{\partial {\varOmega }} \left( \left[ \bar{\mathcal {G}}_{c} l_0^{3} \sum _{\begin{array}{c} ijkl \end{array}} \gamma _{ijkl} \left( \frac{\partial ^2 c}{\partial x_{i} \partial x_{j}} \frac{\hbox {d}}{\hbox {d}t} \left( \frac{\partial c}{\partial x_{l} } \right) \right) \right] \cdot \varvec{n} \right) \,\hbox {d} \partial {\varOmega }- \int \limits _{\varOmega }\left( \left[ \bar{\mathcal {G}}_{c} l_0^{3} \sum _{\begin{array}{c} ijkl \end{array}} \gamma _{ijkl} \left( \frac{\partial ^3 c}{\partial x_{i} \partial x_{j} \partial x_{k}} \frac{\hbox {d}}{\hbox {d}t} \left( \frac{\partial c}{\partial x_{l} } \right) \right) \right] \right) \,\hbox {d} {\varOmega }\\&= \int _{\partial {\varOmega }} \left( \left[ \bar{\mathcal {G}}_{c} l_0^{3} \sum _{\begin{array}{c} ijkl \end{array}} \gamma _{ijkl} \left( \frac{\partial ^2 c}{\partial x_{i} \partial x_{j}} \frac{\hbox {d}}{\hbox {d}t} \left( \frac{\partial c}{\partial x_{l} } \right) \right) \right] \cdot \varvec{n} \right) \,\hbox {d} \partial {\varOmega }\\&\quad - \left( \int _{\partial {\varOmega }} \left( \left[ \bar{\mathcal {G}}_{c} l_0^{3} \sum _{\begin{array}{c} ijkl \end{array}} \gamma _{ijkl} \left( \frac{\partial ^3 c}{\partial x_{i} \partial x_{j} \partial x_{k}} \frac{\hbox {d} c}{\hbox {d}t} \right) \right] \cdot \varvec{n} \right) \,\hbox {d} \partial {\varOmega }- \int \limits _{\varOmega }\left( \left[ \bar{\mathcal {G}}_{c} l_0^{3} \sum _{\begin{array}{c} ijkl \end{array}} \gamma _{ijkl} \left( \frac{\partial ^4 c}{\partial x_{i} \partial x_{j} \partial x_{k} \partial x_{l}} \frac{\hbox {d} c}{\hbox {d}t} \right) \right] \right) \,\hbox {d} {\varOmega }\right) \\&=\int _{\partial {\varOmega }} \left( \left[ \bar{\mathcal {G}}_{c} l_0^{3} \sum _{\begin{array}{c} ijkl \end{array}} \gamma _{ijkl} \left( \frac{\partial ^2 c}{\partial x_{i} \partial x_{j}} \frac{\hbox {d}}{\hbox {d}t} \left( \frac{\partial c}{\partial x_{l} } \right) \right) \right] \cdot \varvec{n} \right) \,\hbox {d} \partial {\varOmega }\\&\quad - \left( \int _{\partial {\varOmega }} \left( \left[ \bar{\mathcal {G}}_{c} l_0^{3} \sum _{\begin{array}{c} ijkl \end{array}} \gamma _{ijkl} \left( \frac{\partial ^3 c}{\partial x_{i} \partial x_{j} \partial x_{k}} \right) \right] \cdot \varvec{n}\dot{c} \right) \,\hbox {d} \partial {\varOmega }- \int \limits _{\varOmega }\left( \left[ \bar{\mathcal {G}}_{c} l_0^{3} \sum _{\begin{array}{c} ijkl \end{array}} \gamma _{ijkl} \left( \frac{\partial ^4 c}{\partial x_{i} \partial x_{j} \partial x_{k} \partial x_{l}} \right) \right] \dot{c} \right) \,\hbox {d} {\varOmega }\right) , \end{aligned} \end{aligned}$$

(72)

respectively.

Substituting Eqs. (71) and (72) in Eq. (68) the following expression is derived for $\mathcal {B}_4$

$$\begin{aligned} \begin{aligned} \mathcal {B}_4&= \int _{\partial {\varOmega }} \left( \left[ \bar{\mathcal {G}}_{c} l_0^{3} \sum _{\begin{array}{c} ijkl \end{array}} \gamma _{ijkl} \left( \frac{\hbox {d}}{\hbox {d}t} \left( \frac{\partial c}{\partial x_{i} } \right) \frac{\partial ^2 c}{\partial x_{k} \partial x_{l}} \right) + \bar{\mathcal {G}}_{c} l_0^{3} \sum _{\begin{array}{c} ijkl \end{array}} \gamma _{ijkl} \left( \frac{\partial ^2 c}{\partial x_{i} \partial x_{j}} \frac{\hbox {d}}{\hbox {d}t} \left( \frac{\partial c}{\partial x_{l} } \right) \right) \right] \cdot \varvec{n} \right) \,\hbox {d} \partial {\varOmega }\\&\quad - \int _{\partial {\varOmega }} \left( \left[ \bar{\mathcal {G}}_{c} l_0^{3} \sum _{\begin{array}{c} ijkl \end{array}} \gamma _{ijkl} \left( \frac{\partial ^3 c}{\partial x_{j} \partial x_{k} \partial x_{l}} \right) + \bar{\mathcal {G}}_{c} l_0^{3} \sum _{\begin{array}{c} ijkl \end{array}} \gamma _{ijkl} \left( \frac{\partial ^3 c}{\partial x_{i} \partial x_{j} \partial x_{k}} \right) \right] \cdot \varvec{n} \dot{c} \right) \,\hbox {d} \partial {\varOmega }\\&\quad + 2 \int \limits _{\varOmega }\left( \left[ \bar{\mathcal {G}}_{c} l_0^{3} \sum _{\begin{array}{c} ijkl \end{array}} \gamma _{ijkl} \left( \frac{\partial ^4 c}{\partial x_{i} \partial x_{j} \partial x_{k} \partial x_{l}} \right) \right] \dot{c} \right) \,\hbox {d} {\varOmega }. \end{aligned} \end{aligned}$$

(73)

Substituting Eq. (73) in the energy balance equation (9) expression (14) is finally established.

Appendix B: Transformation of surface energy density to polar coordinates

The surface energy density and their corresponding reciprocal expression polar plots are evaluated according to the methodology introduced in [21]. In this, the Cartesian coordinate system $\mathbf {x}\left( x_1 , x_2 \right)$ is transformed to $\mathbf {x}_{\theta } \left( {x}_{1_{\theta }} , {x}_{2_{\theta }} \right)$ where the ${x}_{1_{\theta }}$ axis is defined along the crack path ${\varGamma }$ and ${x}_{2_{\theta }}$ axis is the axis normal to the crack interface as shown in Fig. 20. Angle $\theta$ is the counterclockwise angle between ${x}_{1}$ axis and ${x}_{1_{\theta }}$.

Fig. 20

Coordinate system transformation

Thus, coordinate transformation from $\mathbf {x}\left( x_1 , x_2 \right)$ to $\mathbf {x}_{\theta } \left( {x}_{1_{\theta }}, {x}_{2_{\theta }} \right)$ is performed through the transformation equation (74)

$$\begin{aligned} \left\{ \begin{array}{r} x_{1_{\theta }} \\ x_{2_{\theta }} \\ \end{array} \right\} = \begin{bmatrix} \cos (\theta )&-\sin (\theta ) \\ \sin (\theta )&\cos (\theta ) \\ \end{bmatrix} \left\{ \begin{array}{r} x_{1} \\ x_{2} \\ \end{array} \right\} = \mathbf {x}_{\theta } = \mathbf {R}_{\theta } \mathbf {x} \end{aligned}$$

(74)

with the inverse transformation defined as

$$\begin{aligned} \left\{ \begin{array}{r} x_{1} \\ x_{2} \\ \end{array} \right\} = \begin{bmatrix} \cos (\theta )&\sin (\theta ) \\ -\sin (\theta )&\cos (\theta ) \\ \end{bmatrix} \left\{ \begin{array}{r} x_{1_{\theta }} \\ x_{2_{\theta }} \\ \end{array} \right\} = \mathbf {x} = \mathbf {R}_{\theta }^{T} \mathbf {x}_{\theta } . \end{aligned}$$

(75)

Assuming that $c \left( \mathbf {x} \left( \mathbf {x}_{\theta } \right) \right) \approx c \left( \mathbf {x} \left( x_{2_{\theta }} \right) \right)$ and applying the chain rule, the phase field first spatial derivatives are expressed as

$$\begin{aligned} \frac{\partial c}{\partial x_1} = \frac{\partial c}{\partial x_{1_{\theta }}} \frac{\partial x_{1_{\theta }}}{\partial x_1} + \frac{\partial c}{\partial x_{2_{\theta }}} \frac{\partial x_{2_{\theta }}}{\partial x_1} \approx \frac{\partial c}{\partial x_{2_{\theta }}} \frac{\partial x_{2_{\theta }}}{\partial x_1} = \frac{\partial c}{\partial x_{2_{\theta }}} \sin (\theta ) \end{aligned}$$

(76)

and

$$\begin{aligned} \frac{\partial c}{\partial x_2} = \frac{\partial c}{\partial x_{1_{\theta }}} \frac{\partial x_{1_{\theta }}}{\partial x_2} + \frac{\partial c}{\partial x_{2_{\theta }}} \frac{\partial x_{2_{\theta }}}{\partial x_2} \approx \frac{\partial c}{\partial x_{2_{\theta }}} \frac{\partial x_{2_{\theta }}}{\partial x_2} = \frac{\partial c}{\partial x_{2_{\theta }}} \cos (\theta ) , \end{aligned}$$

(77)

respectively. Similarly, the second spatial derivatives are expressed as

$$\begin{aligned} \frac{\partial ^2 c}{\partial {x_1}^2}= & {} \frac{\partial }{\partial {x_1}} \left( \frac{\partial c}{\partial {x_1}} \right) = \frac{\partial }{\partial x_{1_{\theta }}} \left( \frac{\partial c}{\partial {x_1}} \right) \frac{\partial x_{1_{\theta }}}{\partial x_1} + \frac{\partial }{\partial x_{2_{\theta }}} \left( \frac{\partial c}{\partial {x_1}} \right) \frac{\partial x_{2_{\theta }}}{\partial x_1} \nonumber \\\approx & {} \frac{\partial }{\partial x_{2_{\theta }}} \left( \frac{\partial c}{\partial {x_1}} \right) \frac{\partial x_{2_{\theta }}}{\partial x_1} = \frac{\partial ^2 c}{\partial x_{2_{\theta }}^{2}} \sin ^{2}(\theta ) \end{aligned}$$

(78)

$$\begin{aligned} \frac{\partial ^2 c}{\partial {x_2}^2}= & {} \frac{\partial }{\partial {x_2}} \left( \frac{\partial c}{\partial {x_2}} \right) = \frac{\partial }{\partial x_{1_{\theta }}} \left( \frac{\partial c}{\partial {x_2}} \right) \frac{\partial x_{1_{\theta }}}{\partial x_2} + \frac{\partial }{\partial x_{2_{\theta }}} \left( \frac{\partial c}{\partial {x_2}} \right) \frac{\partial x_{2_{\theta }}}{\partial x_2} \nonumber \\\approx & {} \frac{\partial }{\partial x_{2_{\theta }}} \left( \frac{\partial c}{\partial {x_2}} \right) \frac{\partial x_{2_{\theta }}}{\partial x_2} = \frac{\partial ^2 c}{\partial x_{2_{\theta }}^{2}} \cos ^{2}(\theta ) \end{aligned}$$

(79)

and

$$\begin{aligned} \begin{aligned} \frac{\partial ^2 c}{\partial {x_1} \partial {x_2}}&= \frac{\partial }{\partial {x_1}} \left( \frac{\partial c}{\partial {x_2}} \right) = \frac{\partial }{\partial x_{1_{\theta }}} \left( \frac{\partial c}{\partial {x_2}} \right) \frac{\partial x_{1_{\theta }}}{\partial x_1} + \frac{\partial }{\partial x_{2_{\theta }}} \left( \frac{\partial c}{\partial {x_2}} \right) \frac{\partial x_{2_{\theta }}}{\partial x_1} \\&\approx \frac{\partial }{\partial x_{2_{\theta }}} \left( \frac{\partial c}{\partial {x_2}} \right) \frac{\partial x_{2_{\theta }}}{\partial x_1} = \frac{\partial ^2 c}{\partial x_{2_{\theta }}^{2}} \cos (\theta ) \sin (\theta ) \approx \frac{\partial ^2 c}{\partial {x_2} \partial {x_1}}, \end{aligned} \end{aligned}$$

(80)

respectively. Higher-order spatial derivatives are defined accordingly as

$$\begin{aligned} \frac{\partial ^3 c}{\partial {x_1}^3} \approx \frac{\partial ^3 c}{\partial x_{2_{\theta }}^{3}} \sin ^{3}(\theta ) \qquad \hbox {and}\qquad \frac{\partial ^3 c}{\partial {x_2}^3} \approx \frac{\partial ^3 c}{\partial x_{2_{\theta }}^{3}} \cos ^{3}(\theta ) \end{aligned}$$

(81)

and

$$\begin{aligned} \begin{aligned}&\frac{\partial ^4 c}{\partial {x_1}^4} \approx \frac{\partial ^4 c}{\partial x_{2_{\theta }}^{4}} \sin ^{4}(\theta ){,}\quad \frac{\partial ^4 c}{\partial {x_2}^4} \approx \frac{\partial ^4 c}{\partial x_{2_{\theta }}^{4}} \cos ^{4}(\theta ){,}\quad \frac{\partial ^4 c}{\partial {x_1}^2 \partial {x_2}^2} \approx \frac{\partial ^4 c}{\partial x_{2_{\theta }}^{4}} \cos ^{2}(\theta ) \sin ^{2}(\theta ) \\&\frac{\partial ^4 c}{\partial {x_1} \partial {x_2}^3} \approx \frac{\partial ^4 c}{\partial x_{2_{\theta }}^{4}} \cos ^{3}(\theta ) \sin (\theta ){,}\quad \frac{\partial ^4 c}{\partial {x_2} \partial {x_1}^3} \approx \frac{\partial ^4 c}{\partial x_{2_{\theta }}^{4}} \sin ^{3}(\theta ) \cos (\theta ). \end{aligned} \end{aligned}$$

(82)

Employing Eqs. (76)–(82), the functional $\mathcal {Z}_{c,Anis}$ of Eq. (4) is expressed in polar coordinates as

$$\begin{aligned} \begin{aligned} \mathcal {Z}_{c,Anis}&= \left[ {\frac{{{{\left( {c - 1} \right) }^2}}}{{4{l_0}}} + {l_0} | \nabla c |^2 } + l_0^{3} \sum _{\begin{array}{c} ijkl \end{array}} \gamma _{ijkl} \frac{\partial ^2 c}{\partial x_{i} \partial x_{j}} \frac{\partial ^2 c}{\partial x_{k} \partial x_{l}} \right] \\&\approx \left[ \frac{{{{\left( {c - 1} \right) }^2}}}{{4{l_0}}} + {l_0} \left( \frac{\partial c}{\partial x_{2_{\theta }}} \right) ^{2} + l_0^{3} \gamma _{\theta } \left( \frac{\partial ^{2} c}{\partial x_{2_{\theta }}^{2}} \right) ^{2} \right] , \end{aligned} \end{aligned}$$

(83)

where

$$\begin{aligned} \begin{aligned} \gamma _{\theta }&= \gamma _{1111} \sin ^{4}(\theta ) + \gamma _{2222} \cos ^{4}(\theta ) + \gamma _{1212} \cos ^{2}(\theta ) \sin ^{2}(\theta ) \\&\quad +2 \gamma _{1122} \cos ^{2}(\theta ) \sin ^{2}(\theta ) + 2 \gamma _{1112} \cos (\theta ) \sin ^{3}(\theta ) \\&\quad + 2 \gamma _{2212} \sin (\theta ) \cos ^{3}(\theta ). \end{aligned} \end{aligned}$$

(84)

Furthermore, the Euler–Lagrange equation is rewritten in the form

$$\begin{aligned} c - 1 - 4l_0^{2} {\varDelta }c + 4 l_0^{4} \sum _{\begin{array}{c} ijkl \end{array}} \gamma _{ijkl} \frac{\partial ^4 c}{\partial x_{i} \partial x_{j} \partial x_{k} \partial x_{l}} = 0 \Rightarrow c - 1 - 4l_0^{2} \left( \frac{\partial ^2 c}{\partial x_{2_{\theta }}^{2}} \right) + 4 l_0^{4} \gamma _{\theta } \left( \frac{\partial ^4 c}{\partial x_{2_{\theta }}^{4}} \right) = 0. \end{aligned}$$

(85)

Equation (85) can then be numerically solved subject to the following boundary conditions, i.e.

$$\begin{aligned} \begin{aligned}&c \left( 0 \right) = 0\\&\frac{\partial c \left( 0 \right) }{\partial x_{2_{\theta }}} = 0\\&\frac{\partial c \left( \pm \infty \right) }{x_{2_{\theta }}} \left( \approx \frac{\partial c \left( \pm x_{lb} \right) }{\partial x_{2_{\theta }}} \right) = 0\\&\frac{\partial ^{2} c \left( \pm \infty \right) }{\partial x_{2_{\theta }}^{2}} \left( \approx \frac{\partial ^{2} c \left( \pm x_{lb} \right) }{\partial x_{2_{\theta }}^{2}} \right) = 0, \end{aligned} \end{aligned}$$

(86)

where $x_{lb}$ is the distance from the boundary, assuming that $x_{lb}=20 l_0$. Finally, the surface energy density is numerically evaluated as

$$\begin{aligned} \begin{aligned} \mathcal {G}_{c} \left( \theta \right) = \int _{-\infty }^{+\infty } \bar{\mathcal {G}}_{c} \mathcal {Z}_{c,Anis} \hbox {d}x_{2_{\theta }} \approx \int _{-x_{lb}}^{+x_{lb}} \bar{\mathcal {G}}_{c} \mathcal {Z}_{c,Anis} \hbox {d}x_{2_{\theta }}. \end{aligned} \end{aligned}$$

(87)

The maximum and minimum values of $\mathcal {G}_{c} \left( \theta \right)$ for $\theta \in \left[ 0,2\pi \right]$ define the $\mathcal {G}_{c_{\max }}$ and $\mathcal {G}_{c_{\min }}$, respectively. The polar plot of surface energy density $\mathcal {G}_{c} \left( \theta \right)$ can be rotated by angle $\phi$ through relation (88) below

$$\begin{aligned} \varvec{\gamma }_{\phi } = \varvec{Q}_{\phi } \varvec{\gamma } \varvec{Q}_{\phi }^{T}. \end{aligned}$$

(88)

The rotation matrix $\varvec{Q}_{\phi }$ is defined as

$$\begin{aligned} \varvec{Q}_{\phi } = \begin{bmatrix} c^2&s^2&-2 c s \\ s^2&c^2&2 c s \\ c s&-c s&c^2 - s^2 \\ \end{bmatrix} , \end{aligned}$$

(89)

where $c=\cos \left( \phi \right)$ and $s=\sin \left( \phi \right)$. The angle $\phi$ goes clockwise. In the cases of cubic and orthotropic symmetries the fourth-order tensor $\varvec{\gamma }$ is expressed, in global axes, as

$$\begin{aligned} \varvec{\gamma } = \begin{bmatrix} \gamma _{1111}&\gamma _{1122}&\gamma _{1112} \\ \gamma _{2211}&\gamma _{2222}&\gamma _{2212} \\ \gamma _{1211}&\gamma _{1222}&\gamma _{1212} \\ \end{bmatrix} = \begin{bmatrix} \gamma _{11}&\gamma _{12}&2\gamma _{14} \\ \gamma _{12}&\gamma _{22}&2\gamma _{24} \\ 2\gamma _{14}&2\gamma _{24}&4\gamma _{44} \\ \end{bmatrix} . \end{aligned}$$

(90)