| For the design of materials, it is important to faithfully model the physics due to interactions at the microstructural scales. While brute-force modeling of all the details of the microstructure is too costly, current homogenized continuum models suffer from their inability to capture the correct physics---especially where localization and failure are concerned. To overcome this limitation, a multi-scale continuum theory is proposed so that kinematic variables representing the deformation at various scales are incorporated. The method of virtual power is then used to derive a system of coupled governing equations, each equation representing a particular scale and its interactions with the macro-scale. A constitutive relation is then introduced to preserve the underlying physics associated with each scale. The inelastic behavior is represented by multiple yield functions, each representing a particular scale of microstructure, but collectively coupled through the same set of internal variables. The proposed theory has been successfully applied to model granular materials, strain gradient plasticity, porous metals, and high strength steel. In the case of high strength steel for which the microstructure of interest consists of two populations of inclusions at distinct scales, in an alloy matrix, the multi-scale framework is used to establish the relationship between micromechanical features and fracture toughness. |
