Phase-field gradient theory

Espath, Luis; Calo, Victor

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.

Citation

Espath, L., & Calo, V. (2021). Phase-field gradient theory. Zeitschrift für Angewandte Mathematik und Physik, 72(2), Article 45. https://doi.org/10.1007/s00033-020-01441-2

• Journal Article Type: Article
• Acceptance Date: Nov 23, 2020
• Online Publication Date: Feb 16, 2021
• Publication Date: 2021-04
• Deposit Date: Dec 6, 2022
• Publicly Available Date: Dec 8, 2022
• Journal: Zeitschrift für angewandte Mathematik und Physik
• Print ISSN: 0044-2275
• Electronic ISSN: 1420-9039
• Publisher: Springer Verlag
• Peer Reviewed: Peer Reviewed
• Volume: 72
• Issue: 2
• Article Number: 45
• DOI: https://doi.org/10.1007/s00033-020-01441-2
• Keywords: Applied Mathematics; General Physics and Astronomy; General Mathematics
• Public URL: https://nottingham-repository.worktribe.com/output/7711330
• Additional Information: Received: 17 January 2020; Revised: 11 November 2020; Accepted: 23 November 2020; First Online: 16 February 2021