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 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.