Advanced Nonlocal Elatoplastic Constitutive Equations in the Framework of the Micromorphic Continua
Résumé
The primary objective of this talk is to develop a straightforward formulation of gradient-based nonlocal constitutive equations accounting for the full coupling between the plastic flow with mixed isotropic and kinematic hardening and the isotropic ductile damage under large plastic strains, using the general framework of micromorphic continua ([1], [2]). In the present formulation, three micromorphic phenomena are taken into account namely, the isotropic damage, the isotropic hardening and the kinematic hardening. The principle of virtual power accounting for these three micromorphic phenomena leads to three additional balance or micromorphic momentum equations. These three additional PDEs (Partial Differential Equations) together with the classical equilibrium equations are used to define highly nonlinear and fully coupled initial and boundary value problem (IBVP) with four functionals. When expressed in terms of strain-like state variables, the additional PDE associated to the micromorphic damage leads to a widely used nonlocal Helmholtz equation with its appropriate boundary condition [3]. On the other hand, using the thermodynamics of irreversible processes for micromorphic continua, fully coupled constitutive equations are obtained in terms of the micromorphic variables and their first gradients. For this end, new micromorphic state variables are added to the classical local state variables and used in the appropriate state and dissipation potentials to derive both the stress-like variables (state relations) and the strain-like flux variables (evolution relations). If, for the sake of simplicity, the micromorphic dissipations are neglected, it is shown that, each state variable undergoing the micromorphic aspect is written under the additive form of local and nonlocal contributions [4]. From the four strong forms obtained in terms of strain-like variables, the associated weak forms are deduced using the classical weighted residual and Galerkin method. The associated discretized weak forms are implemented in the commercial FE code ABAQUS/Explicit via the construction of appropriate new elements having additional degrees of freedom, using the user-defined subroutine VUEL as well as the micromorphic constitutive equations through the VUMAT subroutine. The simulations of some simple and more or less complex mechanical structures are conducted for various local and micromorphic material parameters and different mesh sizes in order to show the efficiency of the proposed formulation in giving, at convergence, mesh independent solutions of the IBVP [4]. References [1] Eringen A. C., Nonlocal continuum field theories. Springer Verlag, New York, 2002. [2] Forest S., Micromorphic approach for gradient elasticity, viscoplasticity and damage, ASCE Journal of Engineering Mechanics, 135, 3, (2009), p.117-131. [3] Geers M.G.D., Ubachs R., Engelen R., Strongly non-local gradient-enhanced Finite strain elastoplasticity, Int Journal for Numerical Methods in Engineering, 56, (2003) p. 2039-2068. [4] Saanouni K., Hamed M., Micromorphic approach for finite gradient-elastoplasticity fully coupled with ductile damage. Formulation and computational aspects. Int. Jou. Solids and Structures, to appear (2013).