Comprehensive description of deformation and fracture of solids as wave dynamics

Abstract

A field theory of deformation and fracture is presented. Applying the principle of local symmetry to the law of elasticity, this theory is capable of describing elastic deformation, plastic deformation, and fracture of solids based on the same theoretical basis. Using the Lagrangian formalism, the theory derives field equations analogous to the Maxwell equations of electrodynamics. The field equations yield wave solutions that represent the spatiotemporal behaviors of the velocity and rotation fields of solids under deformation. The dynamics of elastic deformation and plastic deformation are differentiated by the form of the longitudinal force acting on a unit volume. In the field equations, this longitudinal effect acts as the source term. In the elastic dynamics, the source term represents a restoring (energy-conservative) force proportional to the displacement from the equilibrium, and in the plastic dynamics it represents an energy-dissipative force proportional to the local velocity. Both effects are interpreted as the solid’s reaction to the external load. Fracture is characterized by the final stage of deformation, where the solid loses both energy-conservative and energy-dissipative reaction mechanisms. These behaviors are observed as different forms in the wave characteristics of the dynamics. Elastic deformation is characterized by longitudinal compression waves, while plastic deformation is characterized by transverse decaying waves. In the transitional stage from the elastic to the plastic regime, a solitary wave is generated if a certain condition is satisfied. Experimental observations of solids that exhibit these wave characteristics of the deformation field are presented.