Dynamic Processes in Structural Thermo-Viscoplasticity

A multiple field theory dates back to developments in linear thermo-elasticity where the thermal strain is considered to be imposed on the linear elastic background structure in an isothermal state. Load strain and thermal strain fields are coupled by the boundary conditions or exhibit volume coupling. Taking into account a proper auxiliary problem, eg, the structure loaded by a unit force under isothermal conditions, results in Maysel’s formula and thus represents the solution in an optimal manner. That approach was generalized to dynamic problems and its efficiency was enhanced by splitting the solution in the quasi-static part and in the dynamic part thus considering the inertia of mass in the latter portion. This paper reviews recent extensions of such a multiple field theory to nonlinear problems of structural thermo-viscoplasticity. Their basis is given in formulations of micromechanics. In an incremental setting, the inelastic strains enter the background material as a second imposed strain field. Considering the rate form of the generalized Hooke’s law such a three-field representation is identified by inspection. Geometric nonlinearity is not fully taken into account, only approximations based on a second order theory, and, for beams and plates with immovable boundaries, in Berger’s approximation, render a third “strain field” to be imposed on the linearized background structure. A novel derivation of the dynamic generalization of Maysel’s formula is given in the paper. An elastic-viscoplastic semi-infinite impacted rod serves as the illustrative example of the fully dynamic analysis in the multiple field approach. Ductile damage is taken into account and the dissipated energy is deposited under adiabatic conditions. Material parameters are considered temperature dependent. An extension to a finite element discretization of the background structure seems to be possible, when preserving the solely linear elastic solution technique. Quite naturally, the domain integral boundary element method of solution results from such a multiple field approach.