Yavari Arash, Goriely Alain
School of Civil and Environmental Engineering and The George W. Woodruff School of Mechanical Engineering , Georgia Institute of Technology , Atlanta, GA 30332, USA.
OCCAM, Mathematical Institute , University of Oxford , Oxford OX1 3LB, UK.
Proc Math Phys Eng Sci. 2016 Dec;472(2196):20160690. doi: 10.1098/rspa.2016.0690.
The elastic Ericksen problem consists of finding deformations in isotropic hyperelastic solids that can be maintained for arbitrary strain-energy density functions. In the compressible case, Ericksen showed that only homogeneous deformations are possible. Here, we solve the anelastic version of the same problem, that is, we determine both the deformations and the eigenstrains such that a solution to the anelastic problem exists for arbitrary strain-energy density functions. Anelasticity is described by finite eigenstrains. In a nonlinear solid, these eigenstrains can be modelled by a Riemannian material manifold whose metric depends on their distribution. In this framework, we show that the natural generalization of the concept of homogeneous deformations is the notion of -deformations with covariantly constant deformation gradients. We prove that these deformations are the only universal deformations and that they put severe restrictions on possible . We show that, in a simply-connected body, for any distribution of universal eigenstrains the material manifold is a symmetric Riemannian manifold and that in dimensions 2 and 3 the universal eigenstrains are zero-stress.
弹性埃里克森问题包括在各向同性超弹性固体中寻找能对任意应变能密度函数保持的变形。在可压缩情况下，埃里克森表明只有均匀变形是可能的。在此，我们求解同一问题的滞弹性版本，即我们确定变形和本征应变，使得对于任意应变能密度函数都存在滞弹性问题的解。滞弹性由有限本征应变描述。在非线性固体中，这些本征应变可以由一个黎曼材料流形建模，其度量取决于它们的分布。在此框架下，我们表明均匀变形概念的自然推广是具有协变常变形梯度的 - 变形概念。我们证明这些变形是唯一的通用变形，并且它们对可能的 施加了严格限制。我们表明，在单连通体中，对于任何通用本征应变的分布，材料流形是一个对称黎曼流形，并且在二维和三维中通用本征应变是零应力。
