Thermodynamically consistent phase field theory of phase transformations with anisotropic interface energies and stresses

The main focus of this paper is to introduce, in a thermodynamically consistent manner, an anisotropic interface energy into a phase field theory for phase transformations. Here we use a small strain formulation for simplicity, but we retain some geometric nonlinearities, which are necessary for introducing correct interface stresses. Previous theories have assumed the free energy density (i.e., gradient energy) is an anisotropic function of the gradient of the order parameters in the current (deformed) state, which yields a nonsymmetric Cauchy stress tensor. This violates two fundamental principles: the angular momentum equation and the principle of material objectivity. Here, it is justified that for a noncontradictory theory the gradient energy must be an isotropic function of the gradient of the order parameters in the current state, which also depends anisotropically on the direction of the gradient of the order parameters in the reference state. A complete system of thermodynamically consistent equations is presented. We find that the main contribution to the Ginzburg-Landau equation resulting from small strains arises from the anisotropy of the interface energy, which was neglected before. The explicit expression for the free energy is justified. An analytical solution for the nonequilibrium interface and critical nucleus has been found and a parametric study is performed for orientation dependence of the interface energy and width as well as the distribution of interface stresses. Disciplines Aerospace Engineering | Structures and Materials Comments This article is published as Levitas, Valery I., and James A. Warren. "Thermodynamically consistent phase field theory of phase transformations with anisotropic interface energies and stresses." Physical Review B 92, no. 14 (2015): 144106. doi:10.1103/PhysRevB.92.144106. Posted with permission. This article is available at Iowa State University Digital Repository: http://lib.dr.iastate.edu/aere_pubs/93 PHYSICAL REVIEW B 92, 144106 (2015) Thermodynamically consistent phase field theory of phase transformations with anisotropic interface energies and stresses Valery I. Levitas1,* and James A. Warren2 1Iowa State University, Departments of Aerospace Engineering, Mechanical Engineering, and Material Science and Engineering, Ames, Iowa 50011, USA 2Material Measurement Laboratory, National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA (Received 20 July 2015; revised manuscript received 21 September 2015; published 13 October 2015) The main focus of this paper is to introduce, in a thermodynamically consistent manner, an anisotropic interface energy into a phase field theory for phase transformations. Here we use a small strain formulation for simplicity, but we retain some geometric nonlinearities, which are necessary for introducing correct interface stresses. Previous theories have assumed the free energy density (i.e., gradient energy) is an anisotropic function of the gradient of the order parameters in the current (deformed) state, which yields a nonsymmetric Cauchy stress tensor. This violates two fundamental principles: the angular momentum equation and the principle of material objectivity. Here, it is justified that for a noncontradictory theory the gradient energy must be an isotropic function of the gradient of the order parameters in the current state, which also depends anisotropically on the direction of the gradient of the order parameters in the reference state. A complete system of thermodynamically consistent equations is presented. We find that the main contribution to the Ginzburg-Landau equation resulting from small strains arises from the anisotropy of the interface energy, which was neglected before. The explicit expression for the free energy is justified. An analytical solution for the nonequilibrium interface and critical nucleus has been found and a parametric study is performed for orientation dependence of the interface energy and width as well as the distribution of interface stresses. DOI: 10.1103/PhysRevB.92.144106 PACS number(s): 05.70.Np, 64.60.Bd, 68.35.Md