Complementary Cauchy Elasticity
Abstract
We develop a 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. We introduce the complementary stress-work 1-form and characterize the obstruction to complementary hyperelasticity through the failure of major symmetry of the tangent compliance tensor. For general implicit elasticity, we show that the simultaneous closure of the stress-work and complementary stress-work 1-forms is characterized by the Lagrangian structure of the constitutive manifold. Using the Darboux classification, we obtain local canonical representations of anisotropic and isotropic complementary Cauchy elastic solids, including explicit non-hyperelastic examples. We investigate kinematic and stress constraints, including applications to pentamode solids, and show that internally constrained constitutive relations are naturally formulated on equivalence classes of stresses modulo reactive stresses. We formulate the balance laws and thermodynamics covariantly and establish that covariance of the energy balance directly yields the balance of angular momentum in complementary Cauchy elasticity, unlike in conventional Cauchy elasticity. Finally, we linearize implicit, Cauchy, and complementary Cauchy elasticity about finitely deformed and prestressed configurations and show that nonlinear non-hyperelastic solids may possess hyperelastic linearized responses.
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BibTeX
@misc{yavari2026complementary,
title = {{Complementary Cauchy Elasticity}},
author = {Arash Yavari},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.25218},
eprint = {2610.25218},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.25218},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Complementary Cauchy Elasticity
AU - Arash Yavari
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2610.25218
UR - https://arxiv.org/abs/2610.25218
AB - We develop a 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. We introduce the complementary stress-work 1-form and characterize the obstruction to complementary hyperelasticity through the failure of major symmetry of the tangent compliance tensor. For general implicit elasticity, we show that the simultaneous closure of the stress-work and complementary stress-work 1-forms is characterized by the Lagrangian structure of the constitutive manifold. Using the Darboux classification, we obtain local canonical representations of anisotropic and isotropic complementary Cauchy elastic solids, including explicit non-hyperelastic examples. We investigate kinematic and stress constraints, including applications to pentamode solids, and show that internally constrained constitutive relations are naturally formulated on equivalence classes of stresses modulo reactive stresses. We formulate the balance laws and thermodynamics covariantly and establish that covariance of the energy balance directly yields the balance of angular momentum in complementary Cauchy elasticity, unlike in conventional Cauchy elasticity. Finally, we linearize implicit, Cauchy, and complementary Cauchy elasticity about finitely deformed and prestressed configurations and show that nonlinear non-hyperelastic solids may possess hyperelastic linearized responses.
ER -