On the Existence of Hydrostatic Pressure in Regular Finite Deformations of Incompressible Hyperelastic Solids

In this article, we establish conditions for the existence of hydrostatic pressures in a class of hyperelastic incompressible materials subjected to finite deformations. Although our approach is quite general, we restrict ourselves to materials for which the stored energy function is defined on an appropriate Sobolev space (W^{1, p}(Ω))ⁿ; these include, for example, the Mooney-Rivlin materials. Among the implications of our results is that hydrostatic pressures may exist only in a very weak sense if the minimizers ṵ of the total energy are irregular (e.g., ṵ ∊ (W^1,p(Ω))ⁿ – (W^2,p(Ω))ⁿ), but if ṵ is sufficiently smooth (e.g., ṵ ∊ (W^2,p(Ω))ⁿ), then pressures p exist which can be characterized so that (ṵ,p) is a solution of the weak equilibrium equations.