Typical inhomogeneities to be evolved in this scale level are the deformation-induced structures, normally yielding lamellar or band-like morphologies accompanied by relatively large “misorientation” across the bands or the walls. Since the “misorientation” is introduced so as for the grain of interest to accommodate the imposed geometrical constraint from its surroundings, these substructures are roughly categorized in “geometrically-necessary” types of bands (GNBs), in contrast to the dislocation cells in Scale A (being “mechanically-necessary”), which mediates the other two scales, therefore, may be expressed as “absorber.” Also, the chapter discusses the inhomogeneity evolutions in Scale B based on FE-based simulations, which incorporates the incompatibility-tensor field model in its constitutive framework. Starting from showing preliminary simulation results, some advanced outcomes are presented, including modeling of metallurgical microstructures (e.g., martensite lath block and packet) as a further extension.

Descriptions of the inhomogeneity including dislocations and defects based on the differential geometry forms the basic core of FTMP. This chapter first provides the basic notions of differential geometry necessary for understanding “non-Riemannian plasticity.” The fundamental concepts and quantities are presented second, which is followed by some new features peculiar to the present field theory of multiscale plasticity.

The completion of the theory for MMMs (multiscale modeling of materials) is manifested itself partially as an identification of the right “flow-evolutionary” law explicitly, which describes generally the evolution of the inhomogeneous fields and the attendant local plastic flow accompanied by energy dissipation. The notion “duality” ought to be embodied by this law, although it still is a “working hypothesis,” deserving further investigations. Specifically, it represents the interrelationship between the locally stored strain energy and the local plastic flow as has been discussed in the context of polycrystalline plasticity in Chapter 12 for Scale C. In this final chapter, we will derive explicitly a candidate form of the flow-evolutionary law as a possible embodiment of the duality, which is followed by application examples.