Nonlinear isotropic odd elasticity

Technical Summary: Nonlinear Isotropic Odd Elasticity

Problem Statement
The field of "odd" elasticity has recently emerged to describe the nonconservative elastic responses of active solids, particularly in two dimensions (2D). While linear odd elasticity introduces novel behaviors—such as coupling compression to rotation—that are absent in passive, energy-conserving materials, existing theories are restricted to the linear regime of small deformations. This limitation prevents the description of large deformations characteristic of active biological solids and metamaterials. Furthermore, a fundamental gap exists in three dimensions (3D): there is no isotropic linear odd elasticity in 3D, yet the physical reality of active matter suggests that nonlinear effects might still manifest. This paper addresses the need for a constitutive framework describing large, nonlinear deformations of isotropic Cauchy elastic solids, specifically investigating how "oddness" (nonconservative stress responses) alters classical nonlinear stability problems.

Methodology
The authors establish a constitutive framework for finite deformations of 2D and 3D isotropic Cauchy elastic solids. Unlike hyperelasticity, where stress is derived from a conserved strain energy function, Cauchy elasticity posits a direct, local relation between stress and the deformation gradient tensor ($F$).

1. 2D Framework: By enforcing objectivity and isotropy, the authors derive the general form of the Cauchy stress tensor $T$ for 2D isotropic materials. They identify that while the standard terms depend on the left Cauchy–Green tensor $B$, the 2D nature of rotations allows for additional terms involving the alternating symbol $\varepsilon$ and the commutator $[B, \varepsilon]$. These terms introduce the "odd" behavior. The authors propose a minimal nonlinear model by specializing the scalar coefficients of these terms, drawing analogies to neo-Hookean elasticity to define dependencies on the determinant $J$.
2. 3D Framework: The authors extend the theory to 3D incompressible isotropic Cauchy elasticity. They utilize the standard representation of stress in terms of $B$ and $B^2$ with arbitrary scalar functions of the principal invariants. They construct a minimal model by truncating these functions to linear order around the undeformed state, creating a six-dimensional space of Cauchy materials (with a four-dimensional hyperelastic subspace).
3. Application to the Rivlin Problem: To test these frameworks, the authors apply them to the classical Rivlin problem: the bifurcation of a square (in 2D) or cube (in 3D) under dead-load tractions. This problem is chosen because it lacks a linear analogue; bifurcations only occur in the nonlinear regime. The analysis involves solving for homogeneous deformations and, where necessary, analyzing inhomogeneous perturbations via incremental elasticity and acoustic tensors.

Key Contributions and Results

• 2D Compressible Odd Rivlin Square:

  • In the passive (even) case, the square branch bifurcates into rectangular branches at a critical load.
  • In the odd case, the authors find that oddness generally suppresses this bifurcation. The square does not bifurcate into anisotropic homogeneous deformations unless a specific parameter-matching condition ($\mu A + \kappa K = 0$) is met.
  • If this condition is met, the bifurcation occurs, but the resulting shape is a rotated parallelogram rather than a rectangle.
  • For "very odd" materials (where $AK > \kappa\mu$), inhomogeneous deformations can bifurcate, but this requires a threshold of oddness not present in linear theory.
  • Incompressible 2D Case: The bifurcation of the square branch is completely suppressed for any odd material ($K \neq 0$), and no inhomogeneous solutions exist.

• 3D Incompressible Odd Rivlin Cube:

  • Contrary to the 2D result, the authors discover that bifurcation behavior survives in 3D, despite the absence of linear isotropic odd elasticity in three dimensions.
  • The classical biaxial bifurcation of the Rivlin cube persists in the space of Cauchy elastic materials.
  • Furthermore, a triaxial bifurcation (where all three principal stretches differ) can emerge. However, this requires specific parameter matching conditions.
  • Crucially, the authors note that the conditions required for this additional triaxial bifurcation are less restrictive for odd materials than for even (hyperelastic) materials. This contrasts with the 2D result where oddness suppressed behavior.

Significance and Claims
The paper claims to establish the first description of large, nonlinear deformations for isotropic odd elastic solids. The results reveal unexpected nonlinear behaviors that arise even in minimal problems:

1. Dimensional Divergence: The effect of oddness is dimension-dependent. In 2D, oddness generally suppresses the rich bifurcation behavior seen in passive squares, whereas in 3D, oddness allows bifurcation behaviors to persist and potentially new ones to appear, despite the lack of a linear odd counterpart.
2. Nonconservative Mechanics: The work demonstrates that the lack of energy conservation (Cauchy elasticity) leads to distinct stability landscapes compared to hyperelasticity, particularly regarding the conditions under which symmetry-breaking occurs.
3. Foundational Step: The authors position this work as a first step in the program of nonlinear Cauchy elasticity, providing a basis for describing active biological solids. They explicitly state that their model is a "minimal" extension and that a full understanding of different nonlinearities, anisotropy, and specific biological realizations remains a challenge for future work. They do not propose specific experimental setups or applications beyond the theoretical framework and the Rivlin problem paradigm.