Unambiguous Gibbs dividing surface for nonequilibrium finite-width interface: Static equivalence approach

Abstract

The definition of all properties of the nonequilibrium interface depends on the choice of the position of the dividing surface. However, the definition of its position has been an unsolved problem for more than a century. A missing principle to unambiguously determine the position of the Gibbs' dividing surface is found: the principle of static equivalence. A sharp interface (dividing surface) is statically equivalent to a nonequilibrium finite-width interface with distributed tensile stresses if it possesses (a) the same resultant force, equal to the interface energy, and (b) the same moment, which is zero about the interface position. Each of these conditions determines the position of a sharp interface, which may be contradictory. This principle is applied to resolve another basic problem: the development of a phase field approach to an interface motion that includes an expression for interface stresses, which are thermodynamically consistent, and consistent with a sharp-interface limit. Using an analytical solution for a curved propagating interface, it is shown that both conditions determine the same dividing surface, i.e., the theory is self-consistent. The expression for the interface energy is also consistent with the expression for the velocity of the curved sharp interface. Applications to more complex interfaces that support elastic stresses are discussed. The definition of all properties of the nonequilibrium interface depends on the choice of the position of the dividing surface. However, the definition of its position has been an unsolved problem for more than a century. A missing principle to unambiguously determine the position of the Gibbs' dividing surface is found: the principle of static equivalence. A sharp interface (dividing surface) is statically equivalent to a nonequilibrium finite-width interface with distributed tensile stresses if it possesses (a) the same resultant force, equal to the interface energy, and (b) the same moment, which is zero about the interface position. Each of these conditions determines the position of a sharp interface, which may be contradictory. This principle is applied to resolve another basic problem: the development of a phase field approach to an interface motion that includes an expression for interface stresses, which are thermodynamically consistent, and consistent with a sharp-interface limit. Using an analytical solution for a curved propagating interface, it is shown that both conditions determine the same dividing surface, i.e., the theory is self-consistent. The expression for the interface energy is also consistent with the expression for the velocity of the curved sharp interface. Applications to more complex interfaces that support elastic stresses are discussed.

Interaction between phase transformations and dislocations at the nanoscale. Part 1. General phase field approach

“…This expression was derived in Levitas and Preston (2002b) and used in all our simulations (Levitas and Javanbakht (2012, 2014a). Here ΔG θ and Δs are the differences between the thermal part of free energy and entropy of M and A, respectively; A andĀ are the double-well barrier between A and M and between martensitic variants, respectively; parameters B and D control energy away from both the A and M i minima and do not affect the phase equilibrium or PT conditions; functions K ji are related to the jump in elastic moduli during PTs; θ e is the equilibrium temperature for stress-free A and M; A 0 is a parameter, and θ c is the critical temperature at which stress-free A loses its thermodynamic stability.…”

“…Complete system of coupled phase field and mechanics equations is presented for a general case and for small strain approximation. Simplified cases of the developed theory have already been used for the finite element solution of various problems on the interaction between PTs and plasticity (Levitas and Javanbakht (2012, 2014a). …”

“…In Javanbakht (2013, 2014a) a simplified system of equations for the interaction between a single martensitic variant and dislocation evolution was presented but without any derivations and justification. A number of important problems have been solved in Levitas and Javanbakht (2012, 2014a, which include revealing scaledependent athermal hysteresis for the semicoherent interface motion, pushing and inheriting dislocations by a moving interface, the generation of dislocations by a growing martensitic plate (including its arrest), and an order of magnitude reduction in PT pressure due to dislocations generated by applied shear stresses.…”

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

“…the nonequilibrium condition (see, e.g., [19]), it is equal to the excess energy with respect to H in the region with H phase x ≤ x i and with respect to L in the region with L phase x > x i : Here x i is the position of the Gibbs dividing surface (sharp interface), which was determined in [35,36] using the principle of static equivalence. For the chosen fourth-degree thermodynamic potential, it was determined that the dividing surface corresponds to η = 0.5, which remains true for anisotropic gradient energy.…”

“…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.…”