Multiresolution Mechanics for Nano/Micro-Structured Materials

To understand the mechanics 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 sufficiently 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. We present some applications of the theory in high strength steels, whose microstructure consists of two populations of inclusions at distinct scales, in an alloy matrix. We then give an overview of undergoing research including the modeling of cortical bone, and thermal-mechanical-chemical-electrical materials