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

Abstract

[1] 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.Citation: Hackl, K., and J. Renner (2013), High-temperature deformation and recrystallization: a variational analysis and its application to olivine aggregates,

Experimental deformation and recrystallization of olivine – processes and timescales of damage healing during postseismic relaxation at mantle depths

“…We focus on the grain growth law reported by Karato (1989), the range of mobility presented in the review by Evans et al (2001), and a recent quantitative estimate derived from analyzing piezometric relations of olivine aggregates (Hackl and Renner, 2013). For the temperature range of our experiments a spread of 10 −18 to 10 −11 m 4 J −1 s −1 is indicated (Fig.…”

“…10a), likely due to a lower nucleus density and/or predominant occurrence of fragments. Furthermore, variations in the driving force for growth during isostatic annealing have to be considered for the recrystallized grains that are almost free of defects and for the fragments that inherit the deformed microstructure (see for example Hackl and Renner, 2013, for an analytic expression of the dependence of a grain's size evolution on the relation between its energy state and that of its environment). The driving force for growth of new grains, i.e., the difference in their free energy and that of their environment, G, is here simply associated with two defect concentrations, the density of interfaces (surface area per volume for a single grain, here represented by the inverse of grain size, i.e., a grain-shape dependent geometrical factor is neglected Solid Earth, 4, 423-450, 2013 www.solid-earth.net/4/423/2013/ for simplicity but can be considered to be included in the uncertainty of the specific interface energy) and the density of dislocations (length of dislocations per volume),…”

“…The corresponding activation enthalpies are ∼ 160 kJ mol −1 and 335 kJ mol −1 from Karato (1989) and Hackl and Renner (2013), respectively. The latter is actually not a result of the analysis by Hackl and Renner (2013) but chosen relying on the consistency between diffusion creep rates and diffusion coefficients for Si-diffusion along grain boundaries in olaggregates (see Hirth and Kohlstedt, 2003). Toriumi (1982) reported 210 ± 20 kJ mol −1 for grain boundary migration at isostatic stress conditions.…”

A Variational Approach to Dynamic Recrystallization Using Probability Distributions

“…We investigate the mathematical model based on the work of Hackl and Renner [1] for the phenomenon of dynamic recrystallization in polycrystalline materials. In [1], a probability distribution function f (D, ρ) was introduced, where the state of individual grains is determined by grain size D and dislocation density ρ which we collect into the state variable x = (D, ρ).…”

“…In [1], a probability distribution function f (D, ρ) was introduced, where the state of individual grains is determined by grain size D and dislocation density ρ which we collect into the state variable x = (D, ρ). The distribution function has to satisfy the continuity equation:…”