Energy minimization for an elastic fluid

For a homogeneous elastic fluid, the specific internal energy function depends on the deformation gradient through the specific volume, which is essentially the determinant of the deformation gradient. Standard, elementary minimization problems are commonly stated in terms of the specific volume field and either the body of fluid is subject to a constant environmental pressure or the total volume is fixed. When the deformation field, itself, is of primary concern, the specific internal energy function is left expressed as a function of the deformation gradient and the standard basic minimization problem in elasticity is to minimize the total internal energy of the body subject to a given homogeneous deformation on its boundary. We discuss the relationship between these two basic problems. Finally, we provide a solution to the latter minimization problem when the specified boundary placement data is arbitrary but sufficiently regular. Our analysis excludes the possibility of cavitation.