A model for porous single crystals with cylindrical voids of elliptical cross-section

Abstract

International audienceThis work presents a rate-dependent constitutive model for porous single crystals with arbitrary number of slip systems and orientations. The single crystal comprises cylindrical voids with elliptical cross-section at arbitrary orientations and is subjected to general plane-strain loadings. The proposed model, called modified variational model (MVAR), is based on the nonlinear variational homogenization method, which makes use of a linear comparison porous single crystal material to estimate the response of the nonlinear porous single crystal. The MVAR model is validated by periodic finite element simulations for a large number of parameters including general in-plane crystal anisotropy, general in-plane void shapes and orientations, various creep exponents (i.e., nonlinearity) and general plane strain loading conditions. The MVAR model, which at the present state involves no calibration parameters, is found to be in good agreement with the finite element results for all cases considered in this work. The model is then used in a predictive manner to investigate the complex response of porous single crystals in several cases with strong coupling between the anisotropy of the crystal and the (morphological) anisotropy induced by the shape and orientation of the voids

An analytical model for porous single crystals with ellipsoidal voids

“…To summarize, the previous cases of several HCP crystalline structures show that the MVAR model deals extremely well with the strong sensitivity of the porous single crystals behavior on the crystal anisotropy (i.e., number and orientation of slip systems), as already discussed in Mbiakop et al (2015b) in the context of two-dimensional microstructures. This implies that by default the present model is able to distinguish with high accuracy between different crystals (and effective loading).…”

“…In order to achieve this goal, the variational nonlinear homogenization method of Ponte Castañeda (1991a) has been used and modified (Danas and Aravas, 2012) to derive estimates but not bounds. The modified variational (MVAR) model presented in this study, which is an extension of the work of Mbiakop et al (2015b) in two-dimensions, has been validated by comparison with full field FE calculations of single-and multi-void periodic unit-cells. The MVAR model has been found to be in good agreement with the FE results for a wide range of parameters describing the number and orientation of the slip systems (i.e., crystal anisotropy), the creep exponent (i.e., nonlinearity) of the matrix crystal, the porosity and the void shapes and orientations.…”

“…This form is directly inspired by the viscoplastic stress potential of the matrix U in the special linear case of n = 1 in equation (10). Then, one can write (DeBotton and Ponte Mbiakop et al, 2015b)…”

“…Nonetheless, such an operation is beyond the scope of this study and will not be pursued here. It should be mentioned, however, that in the case of two-dimensional systems (2D crystal plasticity and cylindrical voids), the tensor S * can be evaluated analytically as discussed in Mbiakop et al (2015b).…”

“…The proposed model is based on appropriate extension of a former work of the authors (Mbiakop et al, 2015b) in two-dimensional porous single crystals. More specifically, in Sections 2 and 3, we use the variational linear comparison composite theory of Ponte Castañeda (1991a) to provide a fully analytical model, called the modified variational (MVAR) model (see Danas and Aravas (2012); Mbiakop et al (2015b)), in three-dimensions. Subsequently, in Section 4, we present in detail the finite element (FE) periodic unit-cell framework which will be used to assess the MVAR model as well as to visualize the underlying deformation fields in the context of porous single crystals.…”