Elastoplastic nonlocal micromorphic formulation with damage and hardening gradients
Résumé
It is well known that the use of inelastic constitutive equations accounting for induced softening leads to pathological space (mesh) and time discretization dependency of the numerical solution of the Initial and Boundary value Problem (IBVP). To avoid this drawback a thermodynamically-consistent formulation incorporating first micromorphic damage and isotropic hardening gradients in the framework of micromorphic continuum is proposed. In this formulation the damage field is strongly coupled with a mixed isotropic and kinematic nonlinear hardening together with micromorphic damage and isotropic hardening variables. A thermodynamically-consistent formulation is used to introduce the two micromorphic scalar damage and isotropic hardening fields as a Degree of Freedom (DoF) and their first gradient as a state variable in the state potential. By using the principle of virtual power in the framework of the micromorphic continuum, additional PDEs together with appropriate boundary conditions are obtained. For the micromorphic damage the additional PDE is shown to be similar to the one widely used in the literature to regularize the IBVP exhibiting softening behaviours. A new finite element with two additional DoF is developed, and implemented into ABAQUS/EXPLICIT using the user’s subroutine VUEL while the nonlocal micromorphic constitutive equations where implemented through the user subroutine VUMAT. For the validation purpose, the 2D strain plane and 3D uniaxial tension of metallic sheet is performed using elastoplastic constitutive equations with mixed hardening fully coupled with the ductile isotropic damage. A detailed parametric study is conducted to show the influence of the micromorphic parameters on the mesh dependency of the F.E solution. For a given choice of the model parameters, the proposed micromorphic methodology is shown to be efficient in avoiding the pathological mesh dependency.