title: The anelastic Ericksen problem: universal eigenstrains and deformations in compressible isotropic elastic solids.
abstract: 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 ofcovariantly homogeneous deformations-deformations with covariantly constant deformation gradients. We prove that these deformations are the only universal deformations and that they put severe restrictions on possibleuniversal eigenstrains. 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.
authors.affiliation: School of Civil and Environmental Engineering and The George W. Woodruff School of Mechanical Engineering , Georgia Institute of Technology , Atlanta, GA 30332, USA.
authors.forename: Arash
authors.identifier: 0000-0002-7088-7984
authors.initials: A
authors.lastname: Yavari
authors.affiliation: OCCAM, Mathematical Institute , University of Oxford , Oxford OX1 3LB, UK.
authors.forename: Alain
authors.identifier: 0000-0002-6436-8483
authors.initials: A
authors.lastname: Goriely
keywords: eigenstrain; geometric mechanics; nonlinear elasticity; residual stress; universal deformation