zbMATH — the first resource for mathematics

Computational micro-to-macro transitions of discretized microstructures undergoing small strains. (English) Zbl 1032.74010
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 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 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.

74A60 Micromechanical theories
74A40 Random materials and composite materials
74Q15 Effective constitutive equations in solid mechanics
Full Text: DOI