Phase-field gradient theory

Zeitschrift für angewandte Mathematik und Physik, vol. 72

Abstract

Abstract We propose a phase-field theory for enriched continua. To generalize classical phase-field models, we derive the phase-field gradient theory based on balances of microforces, microtorques, and mass. We focus on materials where second gradients of the phase field describe long-range interactions. By considering a nontrivial interaction inside the body, described by a boundary-edge microtraction, we characterize the existence of a hypermicrotraction field, a central aspect of this theory. On surfaces, we define the surface microtraction and the surface-couple microtraction emerging from internal surface interactions. We explicitly account for the lack of smoothness along a curve on surfaces enclosing arbitrary parts of the domain. In these rough areas, internal-edge microtractions appear. We begin our theory by characterizing these tractions. Next, in balancing microforces and microtorques, we arrive at the field equations. Subject to thermodynamic constraints, we develop a general set of constitutive relations for a phase-field model where its free-energy density depends on second gradients of the phase field. A priori, the balance equations are general and independent of constitutive equations, where the thermodynamics constrain the constitutive relations through the free-energy imbalance. To exemplify the usefulness of our theory, we generalize two commonly used phase-field equations. We propose a ‘generalized Swift–Hohenberg equation’—a second-grade phase-field equation—and its conserved version, the ‘generalized phase-field crystal equation’—a conserved second-grade phase-field equation. Furthermore, we derive the configurational fields arising in this theory. We conclude with the presentation of a comprehensive, thermodynamically consistent set of boundary conditions.

Authors 2

1. Luis Espath corresponding Aachen Department of Mathematics

   RWTH Aachen University

   Affiliation as printed

   Department of Mathematics, RWTH Aachen University, Pontdriesch 14-16, 52062, Aachen, Germany

   Department of Mathematics, RWTH-Aachen University, Aachen, Germany

2. Victor M. Calo

   Curtin University · Commonwealth Scientific and Industrial Research Organisation · Mineral Resources

   Affiliation as printed

   Curtin Institute for Computation, Curtin University, Kent Street, Bentley, Perth, WA, 6102, Australia

   Mineral Resources, Commonwealth Scientific and Industrial Research Organisation (CSIRO), 10 Kensington, Perth, WA, 6152, Australia

   School of Earth and Planetary Sciences, Curtin University, Kent Street, Bentley, Perth, WA, 6102, Australia

   Mineral Resources, Commonwealth Scientific and Industrial Research Organisation (CSIRO), Perth, Australia

   School of Earth and Planetary Sciences, Curtin University, Bentley, Perth, Australia