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Karel Matous



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Finite Element Formulation for Modeling Large Deformations in Elasto-viscoplastic Polycrystals


K. Matous and A.M. Maniatty

Department of Mechanical, Aerospace and Nuclear Engineering
Rensselaer Polytechnic Institute
110 8th Street, Troy, NY 12180

Abstract


Anisotropic, elasto-viscoplastic behavior in polycrystalline materials is modeled using a new, updated Lagrangian formulation based on a three-field form of the Hu-Washizu variational principle to create a stable finite element method in the context of nearly incompressible behavior. The meso-scale is characterized by a representative volume element, which contains grains governed by single crystal behavior. A new, fully implicit, two-level, backward Euler integration scheme together with an efficient finite element formulation, including consistent linearization, is presented. The proposed finite element model is capable of predicting non-homogeneous meso-fields, which, for example, may impact subsequent recrystallization. Finally, simple deformations involving an aluminum alloy are considered in order to demonstrate the algorithm.

Conclusions


The proposed computational model is shown to be effective in modeling elasto-viscoplastic behavior and texture evolution in a polycrystal subject to finite strains. The finite element framework, based on an updated Lagrangian formulation, adopts a kinematic split of the deformation gradient into volume preserving and volumetric parts together with a three-field form of the Hu-Washizu variational principle to create a stable finite element method. The consistent linearization of the resulting system of nonlinear equations is derived. The meso-scale is characterized by a representative volume element and is capable of predicting local non-homogeneous stress and deformation fields. The numerical analysis of plane strain compression and simple shear loading of a unit cell was compared to the widely used Taylor model. Such comparison is for information only, because the finite element analysis is influenced by the specific homogeneous boundary conditions resulting in non-homogeneous deformation on lateral boundaries. The present work is a first step toward linking the macro-scale to the meso-scale through computational homogenization, where a meso-structure is fully coupled with the deformation at a typical material point of a macro-continuum. In this work, the appropriate periodic boundary conditions have not yet been derived. Further, on-going work involves extending the present model to cover macro-meso transition including periodic fields.

Acknowledgment


This work has been supported by the National Science Foundation through grants CMS-0084987, DMI-0115330, and DMI-0115146.
2009 Notre Dame and Dr. Karel Matous