[論文レビュー] Nonlinear Cauchy Elasticity

Arash Yavari, Alain Goriely|arXiv (Cornell University)|Dec 22, 2024

Advanced Mathematical Modeling in Engineering被引用数 5

ひとこと要約

This paper revisits Cauchy elasticity without assuming a strain-energy function, introducing Edelen-Darboux potentials, a covariant framework, and connections to active solids and geometric (Berry) phases.

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 hyper-elasticity. 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 the 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 the strain space may be nonzero. Therefore, such materials may require access to both energy sources and sinks. This characteristic has led some mechanicians to question the viability of Cauchy elasticity as a physically plausible theory of elasticity. In this paper, motivated by its relevance to recent applications, such as the modeling of active solids, we revisit Cauchy elasticity in a modern form.

研究の動機と目的

• Investigate Cauchy elasticity beyond hyperelasticity where no strain-energy function exists.
• Develop a covariant formulation and identify the stress-work 1-form as the fundamental object.
• Classify Cauchy elastic solids using generalized energy functions via exterior calculus and Darboux forms.
• Explore connections to active matter, geometric phase, and experimental characterization of non-hyperelastic constants.
• Characterize linear and anisotropic cases and discuss implications for anelasticity and Cosserat-Cauchy materials.

提案手法

• Define the stress-work 1-form as the central object in Cauchy elasticity and derive the generalized Doyle-Ericksen formula.
• Introduce Edelen-Darboux potentials to express stress in anisotropic Cauchy elasticity.
• Use exterior differential systems and Darboux classification to characterize generalized energy functions.
• Establish the covariant balance laws and discuss the relation between balance, objectivity, and angular momentum.
• Explore the Berry (geometric) phase as the net work of stress in cyclic deformations.
• Discuss implications for active solids, isotropic and anisotropic cases, and Cosserat-Cauchy materials.

実験結果

リサーチクエスチョン

• RQ1 How can stress be expressed in Cauchy elasticity without a strain-energy function?
• RQ2 What is the role of the stress-work 1-form and Edelen-Darboux potentials in classifying Cauchy elastic solids?
• RQ3 How do balance laws, objectivity, and angular momentum relate in Cauchy elasticity?
• RQ4 What are the generalized energy functions for different symmetry classes (isotropic, anisotropic, incompressible, compressible)?
• RQ5 How does Cauchy elasticity connect to active solids and what experimental implications arise for antisymmetric elastic constants?

主な発見

• Cauchy elasticity can be formulated covariantly with the stress-work 1-form as fundamental.
• In anisotropic Cauchy elasticity, six Edelen-Darboux potentials describe the stress; isotropic cases reduce to three (compressible) or two (incompressible).
• Noether’s theorem does not generally apply in Cauchy elasticity, and objectivity is not equivalent to angular-momentum balance.
• A generalized Doyle-Ericksen formula and a Darboux-based classification of generalized energy functions are established.
• A Berry-phase-like geometric phase exists in cyclic deformations, indicating non-dissipative yet non-conservative stress work.
• Connections to active solids and antisymmetric elastic constants are clarified, with specific notes on linear anisotropic cases and Cosserat-Cauchy extensions.