Complementary Cauchy Elasticity

Abstract

We develop a general nonlinear theory of complementary Cauchy elasticity, in which the right Cauchy--Green strain is prescribed as a function of the second Piola--Kirchhoff stress, without assuming constitutive invertibility or the existence of a complementary strain-energy function. Complementary Cauchy elasticity is a distinct explicit constitutive representation within implicit elasticity, with its own stress-space geometry and integrability condition. The strain response defines a complementary stress-work $1$-form whose exterior derivative characterizes the obstruction to complementary hyperelasticity through the failure of major symmetry of the tangent compliance tensor. For general implicit elasticity, we show that the stress-work and complementary stress-work $1$-forms are simultaneously closed on the constitutive manifold, a condition characterized by its Lagrangian structure. We formulate complementary versions of classical work postulates, examine material symmetry, and use the Darboux classification of differential $1$-forms to obtain local canonical representations of anisotropic and isotropic complementary Cauchy elastic solids, including explicit non-hyperelastic isotropic representatives of the complementary Ericksen and Edelen classes.
We study kinematic and stress constraints, showing that stress constraints arise naturally in complementary Cauchy elasticity and provide a framework for extremal composite materials, including pentamode solids. For internally constrained materials, the constitutively relevant stress variable is an equivalence class modulo reactive stresses, rather than an individual total stress. This quotient structure provides the natural formulation of constrained implicit and complementary Cauchy elastic constitutive relations.
We formulate the balance laws and thermodynamics covariantly and establish a fundamental distinction in the derivation of the balance of angular momentum. In Cauchy elasticity, covariance of the energy balance yields the balance of angular momentum only when covariance of the generalized energy functions is additionally imposed. In complementary Cauchy elasticity, the balance of angular momentum follows directly from covariance of the energy balance, without requiring this additional assumption.
Finally, we linearize implicit, Cauchy, and complementary Cauchy elasticity about finitely deformed and prestressed configurations. We show that a nonlinear non-hyperelastic solid may have a hyperelastic linearized response at a given constitutive state, with the obstruction to hyperelasticity appearing only at higher orders. The resulting framework accommodates non-hyperelastic, internally constrained, and stress-constrained elastic responses within implicit elasticity.