An enhanced grain-boundary framework for computational homogenization and micro-cracking simulations of polycrystalline materials

Abstract

An enhanced three-dimensional (3D) framework for computational homogenization and intergranular cracking of polycrystalline materials is presented. The framework is aimed at reducing the computational cost of polycrystalline micro simulations, with an aim towards effective multiscale modelling. The scheme is based on a recently developed Voronoi cohesive-frictional grain-boundary formulation. A regularization scheme is used to avoid excessive mesh refinements often induced by the presence of small edges and surfaces in mathematically exact 3D Voronoi morphologies. For homogenization purposes, periodic boundary conditions are enforced on non-prismatic periodic micro representative volume elements ($\mu$RVEs), eliminating pathological grains generally induced by the procedures used to generate prismatic periodic $\mu$RVEs. An original meshing strategy is adopted to retain mesh effectiveness without inducing numerical complexities at grain edges and vertices. The proposed methodology offers remarkable computational savings and high robustness, both highly desirable in a multiscale perspective. The determination of the effective properties of several polycrystalline materials demonstrate the accuracy of the technique. Several microcracking simulations complete the study and confirm the performance of the method.

Appendices

Appendix 1: Anisotropic Green’s functions

The 3D Green’s functions for the anisotropic problem are the solutions of the following problem

$$\begin{aligned} C_{ijkl}G_{kq,jl}\left( \mathbf {\xi },\mathbf {\eta }\right) +\delta _{iq}\cdot \delta \left( \mathbf {\xi }-\mathbf {\eta }\right) =0 \end{aligned}$$

(12)

where $\mathbf {\xi }$ and $\mathbf {\eta }$ denote generic collocation and field points, $C_{ijkl}$ is the anisotropic fourth-order elasticity tensor, the commas followed by letters in the subscripts denote derivation with respect to the related components of $\eta$, $\delta _{iq}$ is the Kronecker delta and $\delta \left( \mathbf {\xi }-\mathbf {\eta }\right)$ is the Dirac delta function.

Defined the unit vector $\mathbf b =(\mathbf {\xi }-\mathbf {\eta })/\Vert \mathbf {\xi }-\mathbf {\eta }\Vert$ in the direction connecting $\mathbf {\xi }$ and $\mathbf {\eta }$, the anisotropic Green’s function can be written

$$\begin{aligned} G_{ij}=\frac{1}{8\pi ^2r}\int _0^{2\pi }M_{ij}^{-1}\left[ \mathbf z (\varphi )\right] d\varphi \end{aligned}$$

(13)

with

$$\begin{aligned} M_{ij}\left[ \mathbf z (\varphi )\right] =C_{ipjq}z_p(\varphi )z_q(\varphi ) \end{aligned}$$

(14)

where $r=\Vert \mathbf {\xi }-\mathbf {\eta }\Vert$, $\mathbf {z}$ is a unit vector lying on the plane $\Pi$ perpendicular to $\mathbf {b}$, $\mathbf {z}\cdot \mathbf {b}=0$, whose position is given in terms of the angle $\varphi$ with respect to a reference axis on $\Pi$, and $C_{ipjq}$ are the components of the stiffness tensor. The integration in Eq. (13) is the performed along a unit circle lying on the plane $\Pi$.

The kernels appearing in the formulation can all be expressed in terms of the Green’s functions $G_{ij}$ and their first derivatives, which are given by

$$\begin{aligned} G_{ij,q}=\frac{\partial G_{ij}}{\partial \eta _p}=\frac{1}{4\pi ^2 r^2}\int _{0}^{\pi }(-b_p M_{ij}^{-1}+z_p F_{ij})d\varphi \end{aligned}$$

(15)

where the integration is still performed on the plane $\Pi$ and

$$\begin{aligned} F_{ij}=C_{mnpq}M_{im}^{-1}M_{pj}^{-1}(z_nb_q+z_qb_n) \end{aligned}$$

(16)

For a more detailed introduction to the boundary element method for anisotropic elastic problems, the interested readers are referred to [56] and references therein.

Appendix 2: Interface micro-cracking modelling

The intergranular model used in the present study allows to analyze initiation, evolution and coalescence of microcracks within the aggregate. For the sake of completeness, the interface equations are briefly recalled here. For further details, the interested readers are referred to [11, 60].

When an interface is pristine, displacement continuity and traction equilibrium are enforced on it:

$$\begin{aligned} \begin{aligned}&\varPsi _i^{\,ab}=\widetilde{u}_i^{\,a}(\varvec{\eta }){+}\widetilde{u}_i^{\,b}(\varvec{\eta })=-\delta \widetilde{u}_i^{\,ab}(\varvec{\eta })\\&\varPhi _i^{\,ab}=\widetilde{t}_i^{\,a}(\varvec{\eta }){-}\widetilde{t}_i^{\,b}(\varvec{\eta }) \end{aligned} \quad i=1,2,3\quad \forall \varvec{\eta }\in \mathscr {I}^{\,ab} \end{aligned}$$

(17)

where all the quantities are expressed in local coordinates over each grain face (which motivates the signs in the above equations), see [10, 11].

Microcracking initiation and evolution is modelled replacing the traction equilibrium equations with cohesive laws embodying an irreversible damage parameter, upon fulfilment of the following intergranular threshold condition

$$\begin{aligned} t_e=\sqrt{\langle \widetilde{t}_n\rangle ^2+\left( \frac{\beta }{\alpha }\widetilde{t}_s\right) ^2}\ge T_\mathrm {max} \end{aligned}$$

(18)

where $t_e$ is an effective traction, $T_\mathrm {max}$ is the interface cohesive strength, and $\widetilde{t}_n$ and $\widetilde{t}_s$ are the normal and tangential traction components, respectively. In Eq. (18), $\langle \ \rangle$ are the McCauley’s brackets, and $\alpha$ and $\beta$ are suitable constants entering the cohesive law. When damage is introduced, the following irreversible extrinsic cohesive traction-separation laws are introduced

$$\begin{aligned} \begin{aligned}&\varPsi _i^{\,ab}= C_s(d^*)\delta \widetilde{u}_i^{\,ab}-\widetilde{t}_i,\qquad i=1,2\\&\varPsi _3^{\,ab}= C_n(d^*)\delta \widetilde{u}_3^{\,ab}-\widetilde{t}_3\\&\varPhi _i^{\,ab}=\widetilde{t}_i^{\,a}-\widetilde{t}_i^{\,b},\qquad i=1,2,3\\ \end{aligned} \end{aligned}$$

(19)

being

$$\begin{aligned} C_s(d^*)=\frac{\alpha T_\mathrm {max}}{\delta u_s^\mathrm {cr}}\frac{1-d^*}{d^*},\quad C_n(d^*)=\frac{T_\mathrm {max}}{\delta u_n^\mathrm {cr}}\frac{1-d^*}{d^*} \end{aligned}$$

where $\delta u_s^\mathrm {cr}$ and $\delta u_n^\mathrm {cr}$ are the critical displacement jumps in sliding mode and opening mode, respectively. $d^*$ is the irreversible damage parameter

$$\begin{aligned} d^*=\max _{\mathscr {H}_d}{d}\in [0,1], \quad \mathrm {with}\ d=\sqrt{\left\langle \frac{\delta \widetilde{u}_n}{\delta u_n^\mathrm {cr}}\right\rangle ^2+\beta ^2\left( \frac{\delta \widetilde{u}_s}{\delta u_s^\mathrm {cr}}\right) ^2} \end{aligned}$$

which is given by the maximum value reached by the effective displacement d during the loading history $\mathscr {H}_d$. It is worth noting that the above equations, in the case of very low levels of damage $d^*$, still provide consistent interface conditions: this can be realized considering that multiplying $\varPsi _j^{\,ab}$, $j=1,2,3$ by $d^*\approx 0$, in Eq. (10), corresponds to enforce $\delta u_j=0$, i.e. pristine interface conditions. The interested readers are refereed to [11] for further details, including a visual representation of the traction separation laws.

Finally, when the critical condition $d^*=1$ is reached, the interface fails and the cohesive laws are replaced by equations of separation, contact slip or contact stick [11].