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.