A general theory of curved deformable interfaces in solids at equilibrium

Abstract

We discuss the deformation of a curved interface between solid phases, assuming small strains in the bulk phases and neglecting accretion at the interfaces. Such assumptions are relevant to the deformation of solid microstructures when atomic diffusion and the formation of defects such as dislocations are negligible. We base our theory on a constitutive equation giving the (excess) free energy ψ of the interface when the interfacial limits of the displacement fields in the abutting phases as well as the limits of the displacement gradients are known. Using general considerations of frame invariance, we show that ψ can depend on these quantities at most through: firstly the normal and tangential components of the jump in displacement at the interface (stretch and slip), secondly the average of the projected strain in the tangent plane (average tangential strain), thirdly the tangential component of the jump in the projected displacement gradient at the interface (relative tangential strain and relative twist) and fourthly the normal component of this jump in projected displacement gradient (relative normal shear and relative tilt). Conjugate to these deformational quantities are generalized interface stresses, two of which (those corresponding to average tangential strain and relative tangential strain) have been discussed in the literature. We derive local equilibrium conditions that relate these interface stresses to stresses in the bulk material. For an energy quadratic in the strains we determine interface elastic constants and discuss their qualitative form for simple limiting cases.