The use of Powell-Sabin B-Splines in a higher-order phase-field model for crack kinking

Abstract

Phase-field models for brittle fracture in anisotropic materials result in a fourth-order partial differential equation for the damage evolution. This necessitates a $\mathcal {C}^1$ continuity of the basis functions. Here, Powell-Sabin B-splines, which are based on triangles, are used for the approximation of the field variables as well as for the the description of the geometry. The use of triangles makes adaptive mesh refinement and discrete crack insertion straightforward. Bézier extraction is used to cast the B-splines in a standard finite element format. A procedure to impose Dirichlet boundary condition is provided for these elements. The versatility and accuracy of the approach are assessed in two case studies, featuring crack kinking and zig-zag crack propagation. It is also shown that the adaptive refinement well captures the evolution of the phase field.

Appendices

Appendix A: crack tip displacement fields

For completeness, we present the formulations of asymptotic displacement fields under pure mode-I loading around a straight crack tip. We impose these displacement fields as boundary conditions to approximate singular stress fields (parameterised by the stress intensity factor $\text {K}_\text {I}$) around the crack tip. The asymptotic displacement fields are given as

$$\begin{aligned} \begin{aligned} u_x&= \frac{\text {K}_\mathrm {I}}{2\mu }\sqrt{\frac{r}{2\pi }}\cos \frac{\theta }{2}(\kappa -\cos \theta ) \\ u_y&= \frac{\text {K}_\mathrm {I}}{2\mu }\sqrt{\frac{r}{2\pi }}\sin \frac{\theta }{2}(\kappa -\cos \theta ) \end{aligned} \end{aligned}$$

(A.1)

where $\mu =E/ 2(1+\nu )$, $\kappa = 3-4\nu$ for plane strain and $\kappa = (3 -\nu )/(1+\nu )$ for plane stress, and $(r, \theta )$ are polar coordinates with origin positioned at the crack tip. The derivatives of displacement fields with respect to Cartesian coordinates (x, y) at the crack tip read:

$$\begin{aligned} \begin{aligned} \frac{\partial u_x}{\partial x}&= \frac{{\text {K}_{\text {I}}}}{4 \mu \sqrt{2 \pi r } }{\cos \frac{\theta }{2}}\left( -\cos \theta +\cos 2\theta +\kappa -1 \right) \\ \frac{\partial u_x}{\partial y}&= \frac{{\text {K}_{\text {I}}}}{4 \mu \sqrt{2 \pi r } }{\sin \frac{\theta }{2}}\left( \cos \theta +\cos 2\theta +\kappa +1 \right) \\ \frac{\partial u_y}{\partial x}&= \frac{{\text {K}_{\text {I}}}}{4 \mu \sqrt{2 \pi r } }{\sin \frac{\theta }{2}}\left( \cos \theta +\cos 2\theta -\kappa -1 \right) \\ \frac{\partial u_y}{\partial y}&= \frac{{\text {K}_{\text {I}}}}{4 \mu \sqrt{2 \pi r } }{\cos \frac{\theta }{2}}\left( \cos \theta -\cos 2\theta +\kappa -1 \right) \end{aligned} \end{aligned}$$

(A.2)

Appendix B: imposing Dirichlet boundary condition

We now show how to impose Dirichlet boundary conditions in the framework of Powell-Sabin elements. In Sect. 3, a special choice of Powell-Sabin (PS) triangle is defined along the boundary: (i) for vertex k with an angle $\gamma < \pi$, two sides of the Powell-Sabin triangle must be aligned with two boundary edges (Fig. 8(left)); (ii) for vertex k with an angle $\gamma = \pi$, one side of the Powell-Sabin triangle must be aligned with the boundary edge, see Fig. 8(right). Rewriting Eq. (12) with respect to the nodal degrees of freedom U yields

$$\begin{aligned} \begin{bmatrix}\eta _k^a &{} \eta _k^b &{}\eta _k^c\\ \beta _k^a &{} \beta _k^b &{}\beta _k^c\\ \gamma _k^a &{} \gamma _k^b &{}\gamma _k^c \end{bmatrix} \begin{bmatrix}U^{k,a}\\ U^{k,b}\\ U^{k,c}\end{bmatrix}=\begin{bmatrix}U^{k} \\ \frac{\partial U^k}{\partial x} \\ \frac{\partial U^k}{\partial y} \end{bmatrix} \end{aligned}$$

(B.1)

where $U^{k,i}$ is the nodal degrees of freedom of Powell-Sabin triangle corner $i\,\left( i=a,b,c \right)$, associated with vertex k; $U^{k}$ denotes field values at vertex k; for the example in Sect. 5.1, $U^{k}$ is given in Eq. (A.1). $\mathbf {\triangledown } U^k = \left[ \frac{\partial U^k}{\partial x}\,\frac{\partial U^k}{\partial y}\right]$ is the gradient of $U^k$; it is defined in Eq. (A.2) for the example in Sect. 5.1.

For the vertex k with an angle $\gamma < \pi$, the coefficients $\eta _k^i$, $\beta _k^i$ and $\gamma _k^i$ $\left( i=a,b,c \right)$ have the following conditions:

$$\begin{aligned} \begin{aligned} \eta _k^a = \eta _k^c&= 0, \quad \eta _k^b = 1 \\ \begin{bmatrix} \beta _k^a \\ \gamma _k^a \end{bmatrix}\cdot \mathbf {t}&= 0, \quad \begin{bmatrix} \beta _k^c \\ \gamma _k^c \end{bmatrix}\cdot \mathbf {v} = 0 \end{aligned} \end{aligned}$$

(B.2)

in which $\mathbf {t}$ and $\mathbf {v}$ are unit vectors along the boundary.

$$\begin{aligned} \mathbf {t} = \frac{\mathbf {x}_a-\mathbf {x}_b}{\left\| \mathbf {x}_a-\mathbf {x}_b \right\| },\qquad \mathbf {v} = \frac{\mathbf {x}_c-\mathbf {x}_b}{\left\| \mathbf {x}_c-\mathbf {x}_b \right\| } \end{aligned}$$

(B.3)

From Eqs. (B.1) and (B.2), we obtain:

$$\begin{aligned} U^{k,b} = U^k,\qquad U^{k,a} = U^k + \frac{\mathbf {\triangledown } U^k \cdot \mathbf {v}}{\begin{bmatrix} \beta _k^a \\ \gamma _k^a\end{bmatrix}\cdot \mathbf {v}} , \qquad U^{k,c} = U^k + \frac{\mathbf {\triangledown } U^k \cdot \mathbf {t}}{\begin{bmatrix} \beta _k^c \\ \gamma _k^c\end{bmatrix}\cdot \mathbf {t}} \end{aligned}$$

(B.4)

For the case of vertex k with an angle $\gamma = \pi$, the coefficients $\eta _k^i$, $\beta _k^i$ and $\gamma _k^i$ $\left( i=a,b,c \right)$ should satisfy

$$\begin{aligned} \begin{aligned} \eta _k^a \ne 0,\quad \eta _k^b \ne 0, \quad \eta _k^c = 0 \\ \begin{bmatrix} \beta _k^c \\ \gamma _k^c \end{bmatrix}\cdot \mathbf {v} = 0 \end{aligned} \end{aligned}$$

(B.5)

where $\mathbf {v}$ denotes the unit vector along the boundary given in Eq. (B.3).

Considering Eqs. (B.1) and (B.5), this leads to

$$\begin{aligned} U^{k,a} = \frac{U^k\varDelta _1 - \eta _k^b\mathbf {\triangledown } U^k\cdot \mathbf {v}}{\eta _k^a\varDelta _1-\eta _k^b\varDelta },\,\qquad U^{k,b} = \frac{-U^k\varDelta + \eta _k^a\mathbf {\triangledown } U^k\cdot \mathbf {v}}{\eta _k^a\varDelta _1-\eta _k^b\varDelta }, \end{aligned}$$

(B.6)

with $\varDelta = \begin{bmatrix} \beta _k^a \\ \gamma _k^a\end{bmatrix}\cdot \mathbf {v},\, \varDelta _1 = \begin{bmatrix} \beta _k^b \\ \gamma _k^b\end{bmatrix}\cdot \mathbf {v}$.