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

Abstract

International audienceThis paper presents a rate-independent ana- lytical model for porous Tresca ( J3 -dependent) mate- rials containing general ellipsoidal voids. The model is based on the nonlinear variational homogenization method which uses a linear comparison material to esti- mate 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 cor- responding 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 pre- dictive manner to investigate the complex response of porous Tresca cases with strong coupling between the J3-dependent matrix behavior and the (morphologi- cal) 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

Void growth yield criteria for intergranular ductile fracture

“…E.7, it can be shown that g n π 3 = g n (0), g ′ n (0) = g ′ n π 3 = 0 and g n (θ) + g ′′ n (θ) ≥ 0, which will be useful in what follows. Note that in the limit in which n → +∞, the single crystal becomes a perfectly plastic Tresca material, as already pointed out by other studies (Cailletaud, 2009;Mbiakop et al, 2016), which means that: Appendix E.2. Hollow sphere under hydrostatic loading It is recalled that Benallal (2018) demonstrated that a hollow sphere with an isotropic perfect-plastic matrix following a yield function expressed as:…”

A porous crystal plasticity constitutive model for ductile deformation and failure in porous single crystals

“…There has been significant research performed in this area in the recent past; these works have been thoroughly reviewed and discussed in the literature, for example by incorporating crystallographic aspect for anisotropic behaviour (Besson, 2010; Mbiakop et al., 2016; Han et al., 2013; Mbiakop et al., 2015; Paux et al., 2015; Song and Castañeda, 2017 and references therein) or anisotropy through phenomenological plasticity models (for details see Benzerga and Leblond, 2010; Benzerga et al., 2016; Besson, 2010). These works are based on Gurson-type approach (for details see Benzerga and Leblond, 2010; Benzerga et al., 2016; Besson, 2010; Paux et al., 2015 and references therein) and homogenisation approaches (Mbiakop et al., 2016; Han et al., 2013; Mbiakop et al., 2015; Song and Castañeda, 2017).…”

Fourier-based strength homogenization of porous media

“…These types of micro-mechanical estimates have been applied to various combinations of morphologies and local strength properties for instance by [15,29,22,13,1,31,2,49,14,53]. Most of these estimates do not however model third invariant effects at the macroscopic scale, except works by [44,3], who revisited the limit analysis problem on the Gurson hollow sphere, and by [10,32,33] using a "second-order" non-linear homogenization method.…”

“…Simple 3D representations of the porous medium have then been considered, such as pores in simple cubic, body centered cubic or face centered cubic periodic configurations [45,34] as well as the hollow sphere [41]. More representative microstructures involving multiple pores have only recently been considered [58,4,5,17,18,57,53,32,33], but are limited to approximately 50 pores. Most of the simulations where discretized using the Finite Elements Method (FEM) and solved either by solving an incremental elasto-plastic problem or directly by resorting to second-order programming solvers.…”

“…Exceptions to these are the FFT-based simulations carried out in [35,4,5,57] based on the efficient numerical method initially introduced in linear elasticity by [38,39] Most of these numerical studies are restricted to the exploration of loading modes comprising a hydrostatic part combined either to a pure shear mode or to an axi-symmetric shear mode, thus involving only two or three values of the Lode angle. Studies exploring complex stress states to reconstruct a 3D yield surface in the stress space are scarce [45,58,34,32,33] and often restricted to basic periodic cells or microstructures with a limited number of pores. The work of [58], involving FEM computations on actual aluminum foam microstructures obtained by micro-tomography, featuring both many pores and complex loading directions is an exception.…”