Chemomechanical equilibrium at the interface between a simple elastic solid and its liquid phase

Applying diffusion coupled deformation theory, we investigate how the elastic properties of a solid body are modified when forced to keep its chemical potential aligned with that of its melt. The theory is implemented at the classical level of continuum mechanics, treating materials as simple continua defined by uniform constitutive relations. A phase boundary is a sharp dividing surface separating two continua in mechanical and chemical equilibrium. We closely follow the continuum theory of the swelling of elastomers (gels) but now applied to a simple two phase one-component system. The liquid is modeled by a local free energy density defining a chemical potential and hydrostatic pressure as usual. The model is extended to a solid by adding a non-linear shear elastic energy term with an effective modulus depending on density. Imposing chemomechanical equilibrium with the liquid reservoir reduces the bulk modulus of the solid to zero. The shear modulus remains finite. The stability of the hyper-compressible solid is investigated in a thought experiment. A mechanical load is applied to a rectangular bar under the constraint of fixed lateral dimensions. The linear elastic modulus for axial loading is evaluated and found to be larger than zero, implying that the bar, despite the zero bulk modulus, can support a weight placed on its upper surface. The weight is stabilized by the induced shear stress. The density dependence of the shear modulus is found to be a second order effect reducing the density of the stressed solid (chemostriction).