A homogenization-based constitutive model for isotropic viscoplastic porous media

Abstract

International audienceAn approximate model based on the "second-order" nonlinear homogenization method is proposed to estimate the effective behavior of isotropic, viscoplastic, porous materials. The model is constructed in such a way that it reproduces exactly the behavior of a "composite-sphere assemblage" in the limit of hydrostatic loadings, and therefore coincides with the hydrostatic limit of Gurson's criterion in the special case of ideal plasticity. As a consequence, the new model improves on earlier homogenization estimates, which have been found to be quite accurate for low triaxialities but overly stiff for sufficiently high triaxialities and nonlinearities. Additionally, the estimates delivered by the model exhibit a dependence on the third invariant of the macroscopic stress tensor, which has a nontrivial effect on the effective response of the material at moderate triaxialities. The proposed model is compared with exact results obtained for a special class of porous materials with sequentially laminated microstructures. The agreement is found to be quite good for the entire range of stress triaxialities, and all values of the porosity and nonlinearity considered

Numerical homogenization of the viscoplastic behavior of snow based on X-ray tomography images

“…In this particular case, for a given value of n, it can be shown (Abouaf, 1985;Geindreau et al, 1999b;Danas et al, 2008) that the macroscopic equivalent stress is written…”

“…This compressibility, characterized by E 1 , depends on the level of the mean pressure (S 1 /3) applied on the snow sample, as well as the mean shear stress (S 2 ). The third invariant S 3 characterizes the loading type and is linked to the Lode angle ϑ in the stress space (Lemaitre and Chaboche, 1985;Danas et al, 2008):…”

Static limit analysis and strength of porous solids with hill orthotropic matrix

“…The crucial step to obtain F(Σ) consists in finding a trial stress field σ satisfying both the equilibrium equations and the void boundary conditions and implementing it into (13). Once F(Σ) determined, the macroscopic flow rule is obtained by performing the derivation of the Lagrangian functional L(σ,λ) with respect to Σ.…”

“…1 Another class of models for porous materials has been proposed in the literature based on variational non linear homogenization techniques (see for instance [5], [6], [13] and [12]). …”

Aspects of Computational Homogenization at Finite Deformations: A Unifying Review From Reuss' to Voigt's Bound

“…In order to resolve this shortcoming, Ponte Castañeda [139,140] proposed an improvement of the method which was further extended in Refs. [141] and [142]. Later, deBotton and Hariton [143] and deBotton [144] obtained a general expression for the behavior of incompressible sequentially laminated composites in small deformation and finite elasticity and compared their results with Hashin-Shtrikman bounds and proposed estimates of Ponte Castañeda [136].…”