Plane stress problems for isotropic incompressible hyperelastic materials (Q6611157)

As commented in the paper under review (p.457): ``There is a rich tradition of exploring the consequences of assuming plane stress conditions in linear elasticity\dots However, the corresponding problem for nonlinear elasticity is rarely considered, complicated as it is by the two difficulties mentioned previously for general dual problems. (First, the stress-strain relation is not in general invertible. Secondly, even if invertibility can be achieved, the existence and uniqueness of a regular deformation generating a pre-assigned left strain-tensor field \(B\) is an open problem; see p.456). This is the main concern in this paper.''\N\NIn this paper, it is shown that if the problem is posed in terms of the Cauchy stress, a semi-inverse approach must be employed to obtain the displacement of a typical particle. If, however, the general plane stress problem is formulated in terms of the Piola-Kirchhoff stress, one needs to solve a nonlinear partial differential equation for both simple tension and simple shear, the trivial solution of which yields a homogeneous deformation. Explicit solution is attained for the special case of the neo-Hookean material.