Computational micro-to-macro transitions of discretized microstructures undergoing small strains |
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Authors: | C Miehe A Koch |
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Institution: | Institut für Mechanik (Bauwesen) Lehrstuhl I, Universit?t Stuttgart, D-70550 Stuttgart, Pfaffenwaldring 7, Germany e-mail: cm@mechbau.uni-stuttgart.de, DE
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Abstract: | Summary The paper investigates algorithms for the computation of homogenized stresses and overall tangent moduli of microstructures
undergoing small strains. Typically, these microstructures define representative volumes of nonlinear heterogeneous materials
such as inelastic composites, polycrystalline aggregates or particle assemblies. We consider a priori given discretized microstructures, without focusing on details of specific discretization techniques in space and time. The key contribution of the paper is
the construction of a family of algorithms and matrix representations of the overall properties of discretized microstructures.
It is shown that the overall stresses and tangent moduli of a typical microstructure may exclusively be defined in terms of
discrete forces and stiffness properties on the boundary. We focus on deformation-driven microstructures, where the overall macroscopic deformation is controlled. In this context, three classical types of boundary
conditions are investigated: (i) linear displacements, (ii) constant tractions and (iii) periodic displacements and antiperiodic
tractions. Incorporated by the Lagrangian multiplier method, these constraints generate three classes of algorithms for the
computation of equilibrium states and the overall properties of microstructures. The proposed algorithms and matrix representations
of the overall properties are formally independent of the interior spatial structure and the local constitutive response of
the microstructure and are therefore applicable to a broad class of model problems. We demonstrate their performance for some
representative model problems including elastic–plastic deformations of composite materials.
Received 29 May 2001; accepted for publication 2 January 2002 |
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Keywords: | Micro-to-macro transition Microstructure Two-scale homogenization analysis Elastic– plastic composite |
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