One of the prominent advantages of the gauge formalism as a field theory is its sophisticated mathematical structure, being based on analytical mechanics. Everything about the system dynamics that we need can be derived by rote out of a Lagrangian density of the system, which itself can be determined uniquely based on the prescribed symmetry underlying in the physical phenomenon we want to describe. In our case, we can find how the dislocation and defect fields should be incorporated into the continuum theory of elasticity, with direct correspondences to the differential geometrical (DG) counterparts introduced in Chapter 6. Also the formalism can provide us with a bridge between the DG pictures and the method of quantum field theory (QFT) discussed in Chapter 8 via the Lagrangian density.

This chapter deals with the last keyword among the three in FTMP (see Chapt.5 for details), i.e., “Cooperation,” in relation to “stability.” First three topics provided in this chapter are (1) field equation and stability, (2) preliminary simulation for interaction field, and (3) stability of dislocation cell structure. The last one is about the “local versus global” nature of interscale cooperation, i.e., (4) global-local structure of stress and strain fields, which is also related with the topics taken up in Chapt.14.

This Chapter first presents a minimal set of basic concepts about “dislocations.” After giving a brief overview of the dislocation theory, specific notions such as “Lomer-Cottrell sessile junction” and “stacking fault energy” are detailed, which are exceptionally important for a comprehensive understanding of many of the characteristics, particularly, dislocation-dislocation interactions and their strengths. The second part provides a simple introduction to metallurgy, especially regarding crystallographic structures, placing a special emphasis on the substantial distinction between face-centered cubic (FCC) and body-centered cubic (BCC) structures, which is expected to greatly facilitate further understanding of the associated contrasting features between the two.

This chapter intends to provide a physically-sound foundation for constructing a constitutive framework applicable to a wide range of strain rate and a limited temperature rage around RT, by considering majorly the statistical mechanics-based dislocation dynamics, together with the phonon drag mechanism. In the statistical mechanics sense, dislocations in a crystal move most likely assisted by the thermal fluctuations at finite temperatures with the help of externally applied stress. Such a process can be described by the statistical mechanics-based dislocation dynamics. The probability to find the mean velocity of dislocations against obstacles under the external stress is given by the Arrhenius-type equation, which provides the basis of the constitutive modeling. The details of the formalism and its applicability are given in the following, after some important phenomenology and basic notions are presented.

Quantum field theory (QFT) provides us with one and almost only suitable language (or mathematical tool) for describing not only the motion and interaction of particles but also their “annihilation” and “creation” out of a field considered a priori in a sophisticated way, whose view seems to be suited for describing dislocations, as a particle or a string embedded within a crystalline ordered field. This chapter concisely overviews the method of QFT, emphasizing distinction from the quantum mechanics, conventionally used for a single and/or many particle problems, and its equivalence to the statistical mechanics. The alternative formalism based on Feynman path integral and its imaginary time representation are reviewed, as the foundation for our use in Chapter 10.

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.

This chapter identifies the critical scales for achieving successful multiscale modeling and simulations of metallic materials in plasticity. To that end, we will look into the hierarchical structure based on the viewpoints provided in Part I. The breaking down of the hierarchical scales in plasticity into finite numbers of representative scales of critical importance, in combination with the extensive use of the FTMP-based models, is expected to convert many of practically-important- but-difficult-to-solve problems much easier for us to tackle. As such examples, three research projects in progress of the author’s own are outlined next. The contents provided there will effectively lead us to step forward to the following three chapters in Part III, where FTMP-based approaches, models and perspectives are applied to the identified individual scales.

Metallic materials are known to be strongly rate and history dependent, where the history includes strain, strain rate, strain path, and temperature. The history effects manifest themselves as, for example, additional hardening and softening in terms of the stress response. Since the materials inevitably experience various histories both during forming/working processes into desired shapes or structures and during operations, modeling of these history effects has been one of the most challenging and difficult problems in the field of mechanics of materials. Most of these histories are brought about by the dislocation substructures with cellular morphology evolving during plastic deformation. In other words, the information about the loading “history” is stored in the evolved/evolving dislocation cells.

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.

In this chapter, roughly three perspectives are presented and discussed based on the author’s own views, which are at least conceptually informative and expected to greatly help enrich the understanding of “corporation” aspects of multiscale plasticity, either directly or indirectly. The topics chosen are “small world network,” “global analysis,” and “bio-inspired mechanics.” The first subject is related to an issue how the microscopic and macroscopic degrees of freedoms are organically interrelated in hierarchically constituted complex material systems. The last topic is about much more complex systems than the materials, that is, bio-systems, from which we expect to gain numerous insights for tackling our problems in more advanced fashion. Roughly two implications are picked up here, that is, “irreducible structure of life” and “tight coupling versus loose coupling” controversy in terms of the mechanism of bio-motors. The latter is evidently associated with the previously mentioned “weak ties.”

Like the previous question about “FCC versus BCC” posed in Chapter 1, here is another interesting question of a very fundamental kind that few might be able to answer clearly and appropriately: What is (are) the substantial distinction(s) between single crystals and polycrystals in terms of plasticity? Of course the secondary and tertiary factors such as the effect of “textures” should be excluded. This question is extensively discussed in this section.

This chapter intends to overview the field theory of multiscale plasticity (FTMP) in terms of the key concepts (keywords), the basic theories, and the fundamental hierarchical recognition (i.e., the identification of important scales). This will be followed by the introductions of several new features that the author himself has found and introduced afresh. Practically, the theory is applied via the crystal plasticity formalism-based framework as a tentative and convenient vehicle. So, the constitutive framework together with some detailed sets of modeling for the evolution equations therein is are also presented in the present chapter, i.e., strain gradient terms for the dislocation-density and the incompatibility tensors.