High-temperature deformation and recrystallization: A variational analysis and its application to olivine aggregates

We develop a framework for a variational analysis of microstructural evolution during inelastic high-temperature deformation accommodated by dislocation mechanisms and diffusive mass transport. A polycrystalline aggregate is represented by a distribution function characterizing the state of individual grains by three variables, dislocation density, grain size, and elastic strain. The aggregate's free energy comprises elastic energy and energies of lattice distortions due to dislocations and grain boundaries. The work performed by the external loading is consumed by changes in the number of defects and their migration leading to inelastic deformation. The variational approach minimizes the rate of change of free energy with the evolution of the state variables under constraints on the aggregate volume, on a relation between changes in plastic strain and dislocation density, and on the form of the dissipation functionals for defect processes. The constrained minimization results in four basic evolution equations, one each for the evolution in grain size and dislocation density and flow laws for dislocation and diffusion creep. Analytical steady state scaling relations between stress and dislocation density and grain size (piezometers) are derived for quasi-homogeneous materials characterized by a unique relation between grain size and dislocation density. Our model matches all currently available experimental observations regarding high-temperature deformation of olivine aggregates with plausible values for the involved micromechanical model parameters. The relation between strain rate and stress for olivine aggregates maintaining a steady state microstructure is distinctly nonlinear in stark contrast to the majority of geodynamical modeling relying on linear relations, i.e., Newtonian behavior.