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Karel Matous





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Multiscale Modeling of Elasto-Viscoplastic Polycrystals
Subjected to Finite Deformations


K. Matous1 and A.M. Maniatty2

1Department of Aerospace and Mechanical Engineering
University of Notre Dame
Notre Dame, IN 46556, USA.

2Department 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