Numerical treatment of cosserat based rate independent strain gradient plasticity theories
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Keywords
non–classical continuum theories, Cosserat continuum medium theory, couple stress theory, small scale inelastic response, finite element analysis, constitutive modeling, integration algorithms
Abstract
The current trend towards miniaturization in the microelectronics industryhas pushed for the development of theories intended to explain the behaviorof materials at small scales. In the particular case of metals, a class ofavailable non–classical continuum mechanics theories has been recently employedin order to explain the wide range of observed behavior at the micronscale. The practical use of the proposed theories remains limited due to issuesin its numerical implementation. First, in displacement–based finite elementformulations the need appears for higher orders of continuity in the interpolationshape functions in order to maintain the convergence rate upon meshrefinement. This limitation places strong restrictions in the geometries of theavailable elements. Second, the available inelastic constitutive models for smallscale applications have been cast into deformation theory formulations limitingthe set of problems to those exhibiting proportional loading only. In thisarticle two contributions are made for the particular case of a Cosserat couplestress continuum. First it describes a numerical scheme based on a penaltyfunction/reduced integration approach that allows for the proper treatment ofthe higher order terms present in Cosserat like theories. This scheme results in a new finite element that can be directly implemented into commercial finiteelement codes. Second, a flow theory of plasticity incorporating size effects isproposed for the case of rate independent materials overcoming the limitationsin the deformation theory formulations. The constitutive model and its correspondingtime–integration algorithm are coupled to the new proposed finiteelement and implemented in the form of a user element subroutine into thecommercial code ABAQUS. The validity of the approach is shown via numericalsimulations of the microbending experiment on thin Nickel foils reportedin the literature.
PACS: 81.40.Lm, 81.40.Jj, 46
MSC: 82B21, 65N30
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References
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