The Anelastic Ericksen Problem: Universal Deformations and Universal Eigenstrains in Incompressible Nonlinear Anelastici
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The Anelastic Ericksen Problem: Universal Deformations and Universal Eigenstrains in Incompressible Nonlinear Anelasticity Christian Goodbrake1 · Arash Yavari2 Alain Goriely1
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Received: 17 March 2020 / Accepted: 19 September 2020 © The Author(s) 2020
Abstract 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. Keywords Universal deformation · Nonlinear elasticity · Eigenstrain · Residual stress · Lie group symmetry · Riemannian geometry · Material manifold · Commutative algebra Mathematics Subject Classification 74B20 · 70G45
B A. Goriely
A. Yavari [email protected]
1
Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK
2
School of Civil and Environmental Engineering & The George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA
C. Goodbrake et al.
1 Introduction Universal deformations in nonlinear elasticity are deformations that exist for all members of a particular class of materials in the absence of body forces. Given any member of a particular class of materials, any universal deformation for that class can be maintained by the application of surface tractions alone. For instance in unconstrained isotropic elastic materials, only homogeneous deformations are universal. However, adding material constraints, i.e., restricting the class under consideration, expands the set of universal solutions. In particular, under the imposition of incompressibility, there are five known families of universal deformations in addition to the universal homogeneous deformations, now restricted to isochoric homogeneous deformations in keeping with the material constraint. The process of obtaining and classifying all universal solutions is a highly nontrivial task. This line of research originates in the seminal work of Jerald Eri
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