Phase-field gradient theory

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