Computational micro-to-macro transitions for discretized micro-structures of heterogeneous materials at finite strains based on the minimization of averaged incremental energy Original Research Article

The paper investigates algorithms for the computation of homogenized stresses and overall tangent moduli of micro-structures undergoing large-strain deformations. Typically, these micro-structures define representative volumes of nonlinear heterogeneous materials such as inelastic composites, polycrystalline aggregates or particle assemblies. We consider a priori given discretized micro-structures without focussing on details of specific discretization techniques for space and time. The key contribution of the paper is the construction of a family of algorithms and matrix representations of overall properties of discretized micro-structures which are motivated by a minimization of averaged incremental energy. It is shown that the overall stresses and tangent moduli of a typical micro-structure may exclusively be defined in terms of discrete forces and stiffness properties on the boundary. We focus on deformation-driven micro-structures where the overall macroscopic deformation is controlled. In this context, three classical types of boundary conditions are investigated: (i) linear deformation, (ii) uniform tractions and (iii) periodic deformation and antiperiodic tractions. Incorporated by Lagrangian multiplier methods, these conditions generate three classes of constrained minimization problems with associated solution algorithms for the computation of equilibrium states and overall properties of micro-structures. 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 micro-structure and are therefore applicable to a broad class of model problems. We demonstrate their performance for some representative model problems including finite elastic–plastic deformations of composites, texture developments in polycrystalline materials and equilibrium states of particle assemblies.