Ericksen’s problem consists of determining all equilibrium deformations that can be sustained solely by the application of boundary tractions for an arbitrary incompressible isotropic hyperelastic material whose stress-free configuration is geometrically flat. We generalize this by first, using a geometric formulation of this problem to show that all the known universal solutions are symmetric with respect to Lie subgroups of the special Euclidean group. Second, we extend this problem to its anelastic version, where the stress-free configuration of the body is a Riemannian manifold. Physically, this situation corresponds to the case where nontrivial finite eigenstrains are present. We characterize explicitly the universal eigenstrains that share the symmetries present in the classical problem, and show that in the presence of eigenstrains, the six known classical families of universal solutions merge into three distinct anelastic families, distinguished by their particular symmetry group. Some generic solutions of these families correspond to well-known cases of anelastic eigenstrains. Additionally, we show that some of these families possess a branch of anomalous solutions, and demonstrate the unique features of these solutions and the equilibrium stress they generate.

中文翻译：

---

非弹性埃里克森问题：不可压缩非线性非弹性中的通用变形和通用本征应变

Ericksen 的问题包括确定所有平衡变形，这些变形可以仅通过应用边界牵引力来维持，对于任意不可压缩的各向同性超弹性材料，其无应力配置在几何上是平坦的。我们首先对此进行概括，使用该问题的几何公式​​来证明所有已知的通用解都是关于特殊欧几里得群的李子群对称的。其次，我们将此问题扩展到其无弹性版本，其中主体的无应力配置是黎曼流形。在物理上，这种情况对应于存在非平凡有限本征应变的情况。我们明确地描述了共享经典问题中存在的对称性的普遍特征应变，并表明在存在特征应变的情况下，通用解的六个已知经典族合并为三个不同的非弹性族，以它们特定的对称群为特征。这些族的一些通用解对应于众所周知的非弹性本征应变情况。此外，我们展示了其中一些家族拥有异常解的分支，并展示了这些解的独特特征以及它们产生的平衡应力。