Phase field approach to martensitic phase transformations with large strains and interface stresses

Abstract

Thermodynamically consistent phase field theory for multivariant martensitic transformations, which includes large strains and interface stresses, is developed. Theory is formulated in a way that some geometrically nonlinear terms do not disappear in the geometrically linear limit, which in particular allowed us to introduce the expression for the interface stresses consistent with the sharp interface approach. Namely, for the propagating nonequilibrium interface, a structural part of the interface Cauchy stresses reduces to a biaxial tension with the magnitude equal to the temperature-dependent interface energy. Additional elastic and viscous contributions to the interface stresses do not require separate constitutive equations and are determined by solution of the coupled system of phase field and mechanics equations. Ginzburg-Landau equations are derived for the evolution of the order parameters and temperature evolution equation. Boundary conditions for the order parameters include variation of the surface energy during phase transformation. Because elastic energy is defined per unit volume of unloaded (intermediate) configuration, additional contributions to the Ginzburg-Landau equations and the expression for entropy appear, which are important even for small strains. A complete system of equations for fifth-and sixth-degree polynomials in terms of the order parameters is presented in the reference and actual configurations. An analytical solution for the propagating interface and critical martensitic nucleus which includes distribution of components of interface stresses has been found for the sixth-degree polynomial. This required resolving a fundamental problem in the interface and surface science: how to define the Gibbsian dividing surface, i.e., the sharp interface equivalent to the finite-width interface. An unexpected, simple solution was found utilizing the principle of static equivalence. In fact, even two equations for determination of the dividing surface follow from the equivalence of the resultant force and zero-moment condition. For the obtained analytical solution for the propagating interface, both conditions determine the same dividing surface, i.e., the theory is noncontradictory. A similar formalism can be developed for the phase field approach to diffusive phase transformations described by the Cahn-Hilliard equation, twinning, dislocations, fracture, and their interaction.Keywords dividing surface, large strains, phase field approach, phase transformation, surface stresses and energy Disciplines Aerospace EngineeringComments NOTICE: This is the author's version of a work that was accepted for publication in Journal of the Mechanics and Physics of Solids. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal ...

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

“…Thus, the anisotropic function ψ ∇ (∇ ∇ ∇η i ) results in a generally nonsymmetric contribution to the true (Cauchy) stress σ σ σ (i.e., force per current unit area) ∇ ∇ ∇η i ⊗ ∂ψ ∇ ∂∇ ∇ ∇η i [10,33,35,37].…”

“…Since, at the nano-and even microscales, interface stresses are important for solid-liquid and solid-solid interfaces, and have been broadly studied within sharp-interface approach [19][20][21][22][23][24][25][26][27] as well molecular dynamics [28,29], the interface stresses have been introduced in PFA as well, see [11,12,14,30] for melting and [31][32][33][34][35] for solid-solid PTs. Here we will follow the most advanced theories at small strains [33] and large strains [35], where a detailed literature review is presented with critical analysis of the previous approaches. As the sharp-interface counterpart, we will start with the Shuttleworth equation [20,27] for the magnitudeσ S of the interface stress,…”

“…Thus, even for melting, when shear modulus tends to zero across an interface, introducing extra elastic interface stresses would be harmful. That is why our main hypothesis in [14,30,[32][33][34][35] is that elastic interface stresses are completely defined from the solution of the Ginzburg-Landau and mechanics equations for a PT problem. Thus, the main problem reduces to introducing the structural contribution to the interface stresses σ st (see Fig.…”

“…This problem has been solved in [33,34,36] for small strain approximation and in [35] for general large strain formulation, but for the case of isotropic interface energy and, consequently, isotropic gradient energy ψ ∇ (∇ ∇ ∇η i ). It is important to note that in order to introduce interface stresses (i.e., physical phenomenon) that represent biaxial tension with a magnitude equal to the nonequilibrium interface energy, it is necessary to introduce some geometrically nonlinear features even when strains are infinitesimally small.…”

“…However, the anisotropic function ψ ∇ (∇ ∇ ∇η i ) results in a nonsymmetric contribution to the true (Cauchy) stress ∇ ∇ ∇η i ⊗ ∂ψ ∇ ∂∇ ∇ ∇η i [33,35,37]. As we will discuss in Section II this leads to a violation of the angular momentum balance, which requires symmetry of the true stress.…”