ABSTRACT
The computational subdomain level-0 (Ωl0) introduced in Chapter 13 invokes pure macroscopic analysis with homogenized material properties and constitutive relations. This subdomain assumes relatively uniform deformation with low gradients and “statistically” periodic local microstructure with periodically evolving variables. A class of hierarchical models known as the FE2 multiscale methods [120, 119] solve micro-mechanical RVE or unit cell models to obtain homogenized properties for macroscopic analysis. This method can be very expensive for problems with evolving properties. In an incremental analysis, it entails solving the RVE micromechanical problem for every element integration point in the computational domain. This can lead to prohibitively large computational overhead. To overcome this limitation, macroscopic constitutive laws of elastic-damage and elastic-plastic damage have been introduced by Ghosh et al. [335, 137] from homogenization of RVE response at microscopic scales. Parameters in these comprehensive constitutive models are calibrated to manifest the effect of morphology as well as evolving microstructural mechanisms. This results in anisotropic homogenized parameters that are functions of evolving overall microstructural variables such as plastic work or damage energy. This dependence is a departure from conventional continuum damage mechanics laws that have constant parameters, which are typically calibrated from limited macroscopic experiments. These reduced-order constitutive models are also significantly more efficient than the FE2 type models since they have limited information on microstructural morphology and do not have to solve the RVE problem in every step. This chapter discusses the development of homogenization-based continuum damage mechanics and ductile fracture models using VCFEM for micromechanical analyses. These models, developed in [335, 137, 209, 210, 141], explicitly incorporate the effects of microstructural variability, as well as anisotropy due to morphology and evolving plasticity and damage. Important tasks contributing to this overall objective includes (i) identification of a RVE or SERVE (statistically equivalent RVE); (ii) detailed micromechanical analyses by VCFEM including explicit mechanisms of plasticity and damage; (iii) asymptotic expansion-based homogenization with periodicity for reduced-order modeling; (iv) framework development for
anisotropic continuum plasticity and damage; and (v) calibration of the evolving model parameter functions. These are elaborated in the following sections for two classes of homogenized models, viz. a homogenization-based continuum damage mechanics (HCDM) model for brittle failure and a homogenizationbased continuum plasticity and damage (HCPD) model for ductile failure.
