A homogenization based yield criterion for a porous Tresca material with ellipsoidal voids

Abstract

This paper presents a rate-independent analytical model for porous Tresca ($J_{3}$-dependent) materials containing general ellipsoidal voids. The model is based on the nonlinear variational homogenization method which uses a linear comparison material to estimate the response of the nonlinear porous solid and is denoted as “MVAR”. Specifically, the model is derived by an original approach starting from a novel porous single crystal model (Mbiakop et al. in Int J Solids Struct 64–65:100–119, 2015b, J Mech Phys Solids 84:436–467, 2015c) by considering the limiting case of infinite slip systems which leads readily to the corresponding Tresca criterion. The MVAR yield surface is then validated using FEM on different unit-cells and various parameters including several porosity levels, several stress triaxiality ratios, different Lode angle and general void shapes and orientations. The MVAR model is found to be in good agreement with the finite element results for all cases considered in this study. Both the MVAR and the FEM computations reveal a strong sensitivity upon the microstructure anisotropy (void shape and orientation), and a dependence of the effective behavior on the third invariant of the applied stress. To the best knowledge of the authors, this is the first model in the literature that is able to deal with porous Tresca material and general void shapes and orientations. Moreover, the MVAR is used in a predictive manner to investigate the complex response of porous Tresca cases with strong coupling between the $J_{3}$-dependent matrix behavior and the (morphological) anisotropy induced by the shape and orientation of the voids. The simplicity of the present analytical study opens the possibility to adapt the present model to experimental results for various materials.

Appendix: Microstructural tensor in the limiting case of \(K \rightarrow \infty \) (isotropic Tresca matrix)

In the case of slip systems with identical CRSS $\tau _0$ and reference slip-rate $\dot{\gamma }_0$, the compliance tensor $\mathbf {S}_{K}$ of the linear comparison composite is given in Mbiakop et al. (2015b, c) by setting $\lambda ^{(s)}=\lambda$ and $\rho ^{(s)}=\rho$,

$$\begin{aligned} \mathbf {S}_{K}= & {} \frac{1}{2\lambda } \sum _{s=1}^{K} \mathbf {E}^{(s)} + \frac{1}{2\rho } \sum _{s=1}^{K} \mathbf {F}^{(s)} + \frac{1}{3\kappa }\mathbf {J}, \; \text {with} \nonumber \\ \mathbf {E}^{(s)}= & {} 2 \varvec{\mu }^{(s)} \otimes \varvec{\mu }^{(s)}, \, \mathbf {F}^{(s)}=\mathbf {K}-\mathbf {E}^{(s)}, \end{aligned}$$

(17)

$\forall s=1,K$, where $\lambda ,\,\rho ,\,\kappa$ serve to denote eigenvalues. While $\lambda$ is optimized in the context of the variational homogenization method, $\rho ,\,\kappa$ are set to infinity [see more details in Mbiakop et al. (2015c)].

In addition, the microstructural tensor can easily be computed in the case of an isotropic von Mises compliance matrix, whose compliance tensor of the corresponding linear comparison composite is $\mathbf {S}_{Mises} = (2 \mu _0)^{-1}\mathbf {K} + (3\kappa _0)^{-1}\mathbf {J}$), following for instance the numerical framework described in Aravas and Ponte Castañeda (2004) and Danas (2008).

Moreover, the numerical computation of the hydrostatic part of $\mathbf {S}^{*}/K$ in the infinite number of slip systems context (i.e. Tresca matrix) for spherical voids leads to a result (up to $2\,\%$) close to the microstructural tensor associated with the compliance tensor of a von Mises matrix. In addition, it was shown Benallal (2015) and we observe through FE simulations that a porous von Mises material and a porous Tresca material exhibit the same hydrostatic point. Hence, one can, as a first approximation, use the tensor $\mathbf {S}_{Mises}$ instead of $\mathbf {S}$ in the limiting case $K \rightarrow \infty$. Suitable values for $\mu _0$ and $\kappa _0$ must consequently be used. In order to achieve this goal, the deviatoric and hydrostatic projections of both tensors lead to

$$\begin{aligned} \mu _0=\frac{5}{\frac{K}{\lambda }+\frac{K}{\rho }}, \, \kappa _0=\kappa \end{aligned}$$

(18)

Thus, using the identity (18), one can readily show that in the limit $\rho \rightarrow \infty$, $\kappa \rightarrow \infty$ and $K \rightarrow \infty$, the microstructural tensor $\widehat{\mathbf {S}}^{*}_{K}$ leads to equation (7).