Phase-field gradient theory

• Published in: Zeitschrift für angewandte Mathematik und Physik Vol. 72; no. 2
• Main Authors: Espath, Luis, Calo, Victor
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.04.2021; Springer Nature B.V
• Subjects: Boundary conditions; Constitutive equations; Constitutive relationships; Engineering; Field theory; Flux density; Free energy; Mathematical Methods in Physics; Mathematical models; Smoothness; Theoretical and Applied Mechanics; 74N20; 82C26; 80A22; Gradient theories; 80A17; Swift–Hohenberg equation; Phase-field crystal equation; 35L65; Phase-field models
• ISSN: 0044-2275; 1420-9039
• DOI: 10.1007/s00033-020-01441-2

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.