Non-Conservative Elasticity

Abstract: Most theories and applications of elasticity rely on an energy function that depends on the strains from which the stresses can be derived. This is the traditional setting of Green elasticity, also known as hyperelasticity. However, in its original form, the theory of elasticity does not assume the existence of a strain-energy function. In this case, called Cauchy elasticity, stresses are directly related to strains. Since the emergence of modern elasticity in the 1940s, research on Cauchy elasticity has been relatively limited. One possible reason is that, for Cauchy materials, the net work performed by stress along a closed path in strain space may be nonzero, and such materials may therefore require access to both energy sources and sinks. Motivated by recent applications, including active solids, we revisit Cauchy elasticity and show that stress in a general anisotropic Cauchy elastic solid can be expressed in terms of six Edelen–Darboux potentials, reducing to three for isotropic and two for incompressible isotropic solids. We show that Noether’s theorem does not generally apply to Cauchy elasticity; in particular, unlike hyperelasticity, objectivity is not equivalent to the balance of angular momentum. The balance laws are formulated covariantly, and a generalized Doyle–Ericksen formula is derived. Material symmetry and work theorems are revisited in terms of the stress-work 1-form, which provides the basis for classifying Cauchy elastic solids through the Darboux classification of differential 1-forms and their generalized energy functions. We discuss the relevance of Carathéodory’s theorem to the accessibility property of Pfaffian equations and establish the existence of intrinsic geometric hysteresis, characterized by the net work of stress in cyclic deformations. Reversing the orientation of a cyclic deformation reverses the sign of the net work, demonstrating that the material response, although non-conservative, is non-dissipative. Finally, we discuss implicit elasticity, in which stress and strain are related through implicit constitutive equations, and focus on complementary Cauchy elasticity, where the right Cauchy–Green strain is prescribed as a function of the second Piola–Kirchhoff stress without assuming constitutive invertibility or a complementary strain-energy function. We discuss the complementary stress-work 1-form, its geometric structure, the characterization of complementary hyperelasticity, and the roles of kinematic and stress constraints in Cauchy and complementary Cauchy elasticity. We also highlight a fundamental asymmetry: covariance of the energy balance directly implies the balance of angular momentum in complementary Cauchy elasticity, whereas Cauchy elasticity requires an additional covariance assumption on the generalized energy functions.

MathematicsPhysics

Audience: researchers in the topic

Nečas Seminar on Continuum Mechanics

Series comments: This seminar was founded on December 14, 1966.

Faculty of Mathematics and Physics, Charles University, Sokolovská 83, Prague 8. Unless stated otherwise, we will meet on Mondays at 15:40 in the lecture hall K3 (2nd floor)