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A constitutive model for elastic flexoelectric materials under small deformation based on second gradient continuum theory is developed, using a Toupin-like variational formulation to simultaneously obtain constitutive relations, balance equations and boundary conditions. The model includes three different electromechanical ''stresses'': a higher-order stress, an extended local electric force and a generalized Cauchy stress tensor. The constitutive equations of the model are obtained by postulating an internal energy density function which depends on both the strain and its gradient as well as the polarization. Finally, as an application of the model, we derive the explicit analytical expressions of the polarization and displacement vector fields for the problem of the polarization induced over a thin spherical shell subjected to hydrostatic loading conditions.

This paper presents some new applications of a model of ductile rupture of porous ductile materials proposed by Gologanu, Leblond, Perrin and Devaux (GLPD). The first application is concerned with the relationship between this model and the class of generalized standard materials. We show that the GLPD model fits in this class, provided that the porosity is not allowed to change and the hypothesis of linearized theory is adopted. The advantage of this property is that it automatically warrants the unicity of the solution for the 'projection' problem of the (supposedly) elastic stress predictor onto the GLPD's yield locus. The second application leads to some exact analytical solutions of the GLPD model constitutive equations for the problems of an elastic hollow sphere in the framework of linearized theory, and viscous in large deformations. Comparisons between the numerical predictions of the GLPD model and the analytical solutions confirm the robustness of the numerical scheme used to implement this model into SYSTUS © finite element (FE) code. Thus, these exact analytical solutions can be used to validate the implementation of the GLPD model in another finite element code.In the third application, comparisons between experimental and numerical load vs. displacement curves for an axisymmetric pre-cracked specimen made of a typical stainless steel are found to yield satisfactory results.