Thermodynamic Restrictions on, and Eshelbian Forms of Interfacial Stress Under Mechanical, Thermal, and Chemical Driving Forces

Abstract

Abstract. In this article, the thermodynamic configurational force and velocity associated with a reaction-diffusion moving interface are studied to derive both the Cauchy stress, its Eshelbian form, and their Piola transformations to the reference configuration. The driving force on the interface is mathematically connected to mechanical, thermal, and chemical fields both in the bulk and on the interface, including anisotropic and inhomogeneous interface stress. Systematically applying a general interface transport theorem, we derive the balance laws, the thermodynamic principles, and the consequent thermodynamic restrictions on the interface stress under the driving forces. These forms are shown to mirror their bulk versions. Next, the velocity–force Eshelbian forms of momentum balance in the bulk are reviewed, followed by the derivation of Eshelbian forms for the analogous interface momentum balance equations. However, the interface momentum balance includes an additional curvature term that only vanishes on a planar surface. We illustrate the derived Eshelbian forms and the interface configurational force through three examples. In the first example, we show that the Griffith criterion of fracture mechanics naturally results from the second-law condition on a crack in two-dimensional plane. In the second and third examples, we calculate the configurational force on a planar interface as well as an interface at a constant radius of curvature. We show that when the curvature is constant, the interface configurational force in the current configuration vanishes but does not in the reference configuration. This indicates that all the inhomogeneities of the material are expressed through the bulk configurational force alone in the current configuration when curvature is constant.