## 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.

Download the paper here
© 2009 Notre Dame and Dr.
Karel Matous