Variational modeling of microstructures in plasticity
The analysis and simulation of microstructures in solids has gained crucial importance, virtue of the influence of all microstructural characteristics on a material’s macroscopic, mechanical behavior. In particular, the arrangement of dislocations and oth
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, Ulrich Hoppe 1 , and Dennis M. Kochmann
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Ruhr-Universit¨ at Bochum, Institut f¨ ur Computational Engineering, Lehrstuhl f¨ ur Mechanik – Materialtheorie, D-44780 Bochum, Germany 2
California Institute of Technology, Graduate Aerospace Laboratories, 1200 E. California Blvd., MC 205-45, Pasadena, CA 91125, U.S.A. †
Corresponding author: [email protected]
Abstract The analysis and simulation of microstructures in solids has gained crucial importance, virtue of the influence of all microstructural characteristics on a material’s macroscopic, mechanical behavior. In particular, the arrangement of dislocations and other lattice defects to particular structures and patterns on the microscale as well as the resultant inhomogeneous distribution of localized strain results in a highly altered stress-strain response. Energetic models predicting the mechanical properties are commonly based on thermodynamic variational principles. Modeling the material response in finite-strain crystal plasticity very often results in a nonconvex variational problem so that the minimizing deformation fields are no longer continuous but exhibit small-scale fluctuations related to probability distributions of deformation gradients to be calculated via energy relaxation. This results in fine structures which can be interpreted as the observed microstructures. This manuscript is supposed to give an overview of the available methods and results in this field. We start by discussing the underlying variational principles for inelastic materials, derive evolution equations for internal variables, and introduce the concept of condensed energy. As a mathematical prerequisite we review the variational calculus of nonconvex potentials and the notion of relaxation. We use these instruments in order to study the initiation of plastic microstructures. Here we focus on a model of single-slip crystal plasticity. Afterward we move on to model the evolution of microstructures. We introduce the concept of essential microstructures and the corresponding relaxed energies and dissipation potentials, and derive evolution equations for microstructure parameters.
J. Schröder, K. Hackl (Eds.), Plasticity and Beyond, CISM International Centre for Mechanical Sciences, DOI 10.1007/978-3-7091-1625-8_2, © CISM, Udine 2014
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K. Hackl, U. Hoppe and D.M. Kochmann We then present a numerical scheme by means of which the microstructure development can be computed, and show numerical results for particular examples in single- and double-slip plasticity. We discuss the influence of hardening and of slip system orientations in the present model.
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Variational framework for inelastic materials Constitutive setting and variational principles
The macroscopic mechanical behavior of a material deforming inelastically under the action of external forces reflects the aggregate of all physical mechanisms occurring on smaller scales. The state of the microstructure is commonly described in terms of so-called internal or history variables. Thus, we describe the state of a g
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