## Multiscale Modeling of
Elasto-Viscoplastic Polycrystals

Subjected to Finite Deformations

K. Matous^{1} and A.M. Maniatty^{2}

^{1}Department of Aerospace and Mechanical Engineering

University of Notre Dame

Notre Dame, IN 46556, USA.

^{2}Department of Mechanical, Aerospace, and Nuclear
Engineering

Rensselaer Polytechnic Institute

Troy, NY 12180, USA.

^{
}
#### Abstract

In the present work, the elasto-viscoplastic
behavior, interactions between grains, and the texture evolution in
polycrystalline materials subjected to finite deformations are modeled
using a multiscale analysis procedure within a finite element
framework. Computational
homogenization is used to relate the grain (meso) scale to the
macroscale. Specifically, a polycrystal is modeled by a material
representative volume element (RVE) consisting of an aggregate of
grains, and a periodic distribution of such unit cells is considered to
describe material behavior locally on the macroscale. The elastic
behavior is defined by a hyperelastic potential, and the viscoplastic
response is modeled by a simple power law complemented by a work
hardening equation. The finite element framework is based on a
Lagrangian formulation, where 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 is adopted to
create a stable finite element method. Examples involving simple
deformations of an aluminum alloy are modeled to predict inhomogeneous
fields on the grain scale, and the macroscopic effective stress-strain
curve and texture evolution are compared to those obtained using both
upper and lower bound models.

#### Conclusions

Computational procedures for the analysis of a
homogenized macro-continuum with a locally attached periodic
mesostructure of elastic-viscoplastic crystals were presented. The
relationship between the behavior at the meso- and macroscales was
discussed, and an incrementally linearized form of the macroscopic
constitutive relations was derived.

The proposed multiscale formulation is shown to be
effective in modeling elasto-viscoplastic behavior and texture
evolution in a polycrystalline materials subject to finite strains. The
mesoscale is characterized by a representative volume element and is
capable of predicting local non-homogeneous stress and deformation
fields. A realistic grain structure, motivated by experimental
observations, is modeled with a displacement-based updated Lagrangian
finite element formulation using the Hu-Washizu variational principle
to create a stable method in the context of nearly incompressible
behavior. The elastic behavior is defined by a hyperelastic potential,
and the viscoplastic response is modeled by a simple power law
complemented by a work hardening equation. A fully implicit two-level
backward Euler integration scheme and the consistent linearization are
used to obtain an efficient algorithm, where large time steps can be
taken. The proposed multiscale analysis is capable of predicting
non-homogeneous meso-fields, which, for example, may impact subsequent
recrystallization.

Finally, examples are considered involving simple
deformations of an aluminum alloy to predict inhomogeneous fields on
the grain scale, and the macroscopic effective stress-strain curve and
texture evolution are compared to those obtained using both upper and
lower bound models.

Future work involves extending the method to 3D with
a parallel implementation and using the model for more detailed
studies. Further on-going studies are necessary to determine the
minimum number of grains needed for a representative statistical
sampling. The approach can be used to study the effect of local texture
on local deformation. In addition, other crystal plasticity models can
be implemented, and the approach can be used for supplying information
for and validation of macroscale constitutive models.

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© 2009 Notre Dame and Dr.
Karel
Matous