An implicit constitutive relation, in which the Hencky strain tensor is assumed to be a function of the Kirchhoff stress tensor, is applied to analyse isotropic compressible and quasi-incompressible nonlinear elastic solids. The implicit relation is based on the use of a Gibbs potential. Experimental data allow the determination of the determinant of the deformation gradient as a function of the spherical part of the stress, commonly referred to in the literature as 'pressure'. By substituting the expression for the Hencky strain tensor into the aforementioned relation, a first-order linear partial differential equation for the Gibbs potential is obtained. The solution of this equation defines a class of elastic body that can be used to fit experimental data for nonlinear compressible solids. Some boundary-value problems are solved, considering both homogeneous and non-homogeneous deformations (in this latter case the inflation of a cylindrical annulus). The implicit constitutive relation is applied to fit experimental data for a type of natural rubber and a class of polypropylene foam. Using these constitutive relations, the problem of inflation of a cylindrical annulus is further analysed numerically.