Multi-phase-field modeling of anisotropic crack propagation for polycrystalline materials

Abstract

A new multi-phase-field method is developed for modeling the fracture of polycrystals at the microstructural level. Inter and transgranular cracking, as well as anisotropic effects of both elasticity and preferential cleavage directions within each randomly oriented crystal are taken into account. For this purpose, the proposed phase field formulation includes: (a) a smeared description of grain boundaries as cohesive zones avoiding defining an additional phase for grains; (b) an anisotropic phase field model; (c) a multi-phase field formulation where each preferential cleavage direction is associated with a damage (phase field) variable. The obtained framework allows modeling interactions and competition between grains and grain boundary cracks, as well as their effects on the effective response of the material. The proposed model is illustrated through several numerical examples involving a full description of complex crack initiation and propagation within 2D and 3D models of polycrystals.

Appendices

Appendix 1: Smeared displacement jump approximation (JA2)

In [40] we proposed a method to approximate the displacement jump $\llbracket {\mathbf {u}}({\mathbf {x}})\rrbracket$ created by interface decohesion as a smooth transition without additional variable, defined as follows:

$$\begin{aligned} \llbracket {\mathbf {u}}({\mathbf {x}}) \rrbracket \simeq {\mathbf {w}}({\mathbf {x}}) = h \nabla {\mathbf {u}} ({\mathbf {x}}) \mathbf {n}^{\varGamma _B}, \end{aligned}$$

(66)

where $\mathbf {w}({\mathbf {x}})$ denotes the smoothed displacement jump approximation, h is a small scalar parameter and ${\mathbf {n}}^{\varGamma _B}$ the normal vector to $\varGamma ^B$ at the point ${\mathbf {x}}$.

The corresponding BVP can be rewritten as:

$$\begin{aligned} \left\{ \begin{array}{{lclll}} \nabla \cdot {\varvec{\sigma }}(\mathbf {u}, {\varvec{d}}) &{}=&{} 0 &{}&{} \forall \mathbf {x} \in \varOmega ,\\ \gamma _{\beta }\big [\mathbf {t}(\mathbf {w}, \kappa ) - \mathbf {n}^{\varGamma _{\beta }} \cdot {\varvec{\sigma }}(\mathbf {u}, {\varvec{d}})\big ] &{}=&{} 0&{}&{} \forall \mathbf {x} \in \varGamma _{\beta },\\ \mathbf {u} &{}=&{} \overline{\mathbf {u}} &{}&{} \forall \mathbf {x} \in \partial \varOmega _u,\\ \mathbf {n}_{t} \cdot {\varvec{\sigma }}&{}=&{} \overline{\mathbf {t}} &{}&{} \forall \mathbf {x} \in \partial \varOmega _t.\\ \end{array}\right. \end{aligned}$$

(67)

The Cauchy stress is here defined as follows:

$$\begin{aligned} \sigma (\mathbf {u}, {\varvec{d}})= & {} \dfrac{\partial W_u^e}{\partial \mathbf {\varepsilon }^e}\nonumber \\= & {} \bigg [ g({\varvec{d}}) \mathbb {C}^0 + k_0 \mathbf {1} \otimes \mathbf {1} \big [ 1 - g({\varvec{d}})\big ] \text {sign} ^- ( \mathop {\mathrm{tr }}\nolimits {{\varvec{\varepsilon }}^e}) \bigg ]: {\varvec{\varepsilon }}^e \nonumber \\ \end{aligned}$$

(68)

where the expression of elastic strain $\mathbf {\varepsilon }^e$ is given (45). Using the variation in the displacement for (67)$_1$, the variation in the displacement jump for (67)$_2$, we obtain the corresponding weak form:

(69)

The details of linearization and numerical implementation can be found in the work [40].

Appendix 2: Anisotropic elasticity

In a polycrystalline material, the elastic moduli mentioned in (29) depend on the relative orientation of different grains. Hence the rotation of the grains must be taken into account to determine the elastic stiffness tensor for each grain. This can be achieved by using the reference transformation tensor and grains coordinate system. Suppose that the components of the elastic stiffness tensor matrix $\mathbf {C}^0 _g$ (in Voigt notation) are given in grains coordinates, then the position-dependent elastic stiffness tensor with respect to the reference coordinate system is given by:

$$\begin{aligned} \mathbf {C}^0 = \mathbf {P}^{\text {T}} \mathbf {C}^0 _g \mathbf {P}, \end{aligned}$$

(70)

where $\mathbf {P}$ is the transformation tensor in Voigt’s notation. In the general 3D case, the transformation matrix $\mathbf {P}$ can be defined by the expression:

$$\begin{aligned} \mathbf {P}= & {} \left[ \begin{matrix} L_{11}^2&{} L_{21}^2&{} L_{31}^2&{} 2L_{21}L_{11} \\ L_{12}^2&{} L_{22}^2&{} L_{32}^2&{} 2L_{22}L_{12} \\ L_{13}^2&{} L_{23}^2&{} L_{33}^2&{} 2L_{23}L_{13} \\ L_{12}L_{11}&{} L_{21}L_{22}&{} L_{31}L_{32}&{} L_{11}L_{22}+L_{21}L_{12}\\ L_{11}L_{13}&{} L_{21}L_{23}&{} L_{31}L_{33}&{} L_{13}L_{21}+L_{23}L_{11}\\ L_{13}L_{12}&{} L_{23}L_{22}&{} L_{33}L_{32}&{} L_{13}L_{22}+L_{23}L_{12}\\ \end{matrix}\right. \nonumber \\&\left. \begin{matrix} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2L_{31}L_{11}&{} 2L_{21}L_{31}\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2L_{12}L_{32}&{} 2L_{32}L_{22}\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2L_{13}L_{33}&{} 2L_{32}L_{33}\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ L_{31}L_{12}+L_{11}L_{32}&{} L_{31}L_{22}+L_{21}L_{32}\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ L_{11}L_{33}+L_{31}L_{13}&{} L_{33}L_{21}+L_{23}L_{31}\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ L_{12}L_{33}+L_{13}L_{32}&{} L_{33}L_{22}+L_{23}L_{32}\\ \end{matrix}\right] \nonumber \\ \end{aligned}$$

(71)

where $L_{ij}$ are the components of the rotation matrix.

For most materials treated in this work we considered an anisotropic behavior with cubic symmetry, where the elastic stiffness tensor contains only three independent parameters, the corresponding matrix in 2D reads:

$$\begin{aligned} \mathbf {C}^0 _g = \left[ \begin{matrix} C_{11}&{} \quad C_{12}&{} \quad 0\\ C_{12}&{} \quad C_{11}&{} \quad 0\\ 0&{} \quad 0&{} \quad C_{44} \end{matrix}\right] . \end{aligned}$$

(72)

The corresponding bulk modulus is given as $k_0 = (C_{11} + 2C_{12})/3$ (see [22, 27]). The rotation of material orientation can be simply determined following [25]. The transformation matrix is here written as:

$$\begin{aligned} \mathbf {P} = \left[ \begin{matrix} c^2&{} \quad s^2&{} \quad 2cs\\ s^2&{} \quad c^2&{} \quad -2cs\\ -cs&{} \quad cs&{} \quad c^2-s^2 \end{matrix}\right] \end{aligned}$$

(73)

where $c = \text {cos} \varphi _1$, $s = \text {sin} \varphi _1$.

Appendix 3: Cohesive zone model for grain boundary

The Cohesive Zone Model (CZM) is used to model the grain boundary decohesion. An example of traction separation law is described in Fig. 33

Fig. 33

Separation law used to simulate grain boundary decohesion, where $t_u$ is fracture strength, $t(\llbracket \mathbf {u}\rrbracket , \kappa )$, and $\varPsi ^B$ are defined in (19)

In the case of polycrystalline materials, the fracture strength $t_u$ may depend on the misorientation between neighboring single crystals, defined by:

$$\begin{aligned} {\varDelta \theta = \theta _1^i - \theta _2^i,} \end{aligned}$$

(74)

where $\theta _1^i, \theta _2^i,$ are the orientations of two crystals along the considered grain boundary on plane i.

Following the work [26], the fracture strength is then expressed by the following:

$$\begin{aligned} {t_u(\varDelta \theta ^i) = t_u^{\text {avg}} + \dfrac{1}{3}\varDelta t_u \sum _i \text {cos} (4\varDelta \theta ^i),} \end{aligned}$$

(75)

where $t_u^{\text {avg}}$, $\varDelta t_u$ are the average and the maximal fracture strength deviation, respectively.

Fig. 34

2D description of crystal orientation between two grains

An illustration for the case of 2D plane stress is depicted in Fig. 34. In that case, the grain boundary fracture strength is written by:

$$\begin{aligned} {t_u(\varDelta \theta ) = t_u^{\text {avg}} + \varDelta t_u \text {cos} (4\varDelta \theta ).} \end{aligned}$$

(76)