The analysis of plane stress problems has long been a topic of interest in linear elasticity. The corresponding problem for non-linearly elastic materials is considered here within the context of homogeneous incompressible isotropic elasticity. It is shown that when 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, the deformation of a particle requires the solution of a non-linear partial differential equation for both simple tension and simple shear, the trivial solution of which yields a homogeneous deformation. It is also shown that the general plane stress problem can be solved for the special case of the neo-Hookean material.

平面应力问题的分析长期以来一直是线弹性领域感兴趣的话题。这里在均匀不可压缩各向同性弹性的背景下考虑非线性弹性材料的相应问题。结果表明，当问题涉及柯西应力时，必须采用半逆方法来获得典型粒子的位移。然而，如果用 Piola-Kirchhoff 应力来表述一般平面应力问题，则粒子的变形需要求解简单拉伸和简单剪切的非线性偏微分方程，其平凡解产生齐次形变。还表明，对于新胡克材料的特殊情况，一般平面应力问题可以得到解决。