Odd elasticity (1902.07760v1)

Abstract: Hooke's law states that the forces or stresses experienced by an elastic object are proportional to the applied deformations or strains. The number of coefficients of proportionality between stress and strain, i.e., the elastic moduli, is constrained by energy conservation. In this Letter, we lift this restriction and generalize linear elasticity to active media with non-conservative microscopic interactions that violate mechanical reciprocity. This generalized framework, which we dub odd elasticity, reveals that two additional moduli can exist in a two-dimensional isotropic solid with active bonds. Such an odd-elastic solid can be regarded as a distributed engine: work is locally extracted, or injected, during quasi-static cycles of deformation. Using continuum equations, coarse-grained microscopic models, and numerical simulations, we uncover phenomena ranging from activity-induced auxetic behavior to wave propagation powered by self-sustained active elastic cycles. Besides providing insights beyond existing hydrodynamic theories of active solids, odd elasticity suggests design principles for emergent autonomous materials.

• The paper introduces odd elasticity by extending classical elasticity with two additional moduli (A and K⁰) that enable local work extraction during deformation cycles.
• It employs continuum mechanics, microscopic modeling, and simulations to reveal unique deformation behaviors like internal torque and activity-induced auxetic responses.
• The findings suggest novel design principles for autonomous metamaterials capable of energy harvesting and wave manipulation in active systems.

Analysis of Odd Elasticity in Active Media

The paper explores the concept of "odd elasticity" within the field of active media, challenging the traditional constraints imposed by energy conservation on the elastic coefficients that relate stress and strain, commonly known as the elastic moduli. Classical elasticity, as per Hooke's law, operates under the premise that these moduli are dictated by conservative forces, meaning stresses are derived from potential energy surfaces—an assumption fundamentally challenged by the authors.

Core Concepts and Methodologies

The investigation introduces the notion of odd elasticity in two-dimensional isotropic solids featuring non-conservative microscopic interactions, such as those encountered in active matter systems. Active matter systems comprise entities capable of converting stored or ambient energy into systematic movement, thus inherently exhibiting non-conservative behavior. Within this framework, the authors propose two additional elastic moduli: $A$ and $K^o$, expanding the classical model that typically includes the bulk modulus ($B$) and the shear modulus ($\mu$).

The paper employs continuum mechanics, coarse-grained microscopic modeling, and numerical simulations to underline its findings. A particularly noteworthy aspect is the assertion that odd elasticity enables the local extraction or injection of work in solid materials during deformation cycles—a property absent in traditional materials. The presented models demonstrate phenomena such as activity-induced auxetic behavior and wave propagation supported by self-sustaining active cycles.

Numerical and Theoretical Results

Numerically, the suggested existence of odd elasticity manifests through distinctive deformation behaviors not accounted for by traditional elasticity. For instance, when a material with a non-zero odd modulus $A$ undergoes compression, it generates a unique internal torque density without altering other mechanical characteristics significantly. The resulting deformation cycles are associated with net work generation, attributed to the odd moduli $A$ and $K^o$. This idea is visually evidenced in the authors' illustrative active elastic engine cycle diagrams.

The theoretical implications include a reframing of how autonomous materials can be engineered, suggesting principles for designing emergent materials capable of complex task performance, energy extraction, and transmission. These materials theoretically offer new pathways for fabricating self-sustaining and self-repairing structures, which are crucial for future innovations in material science and engineering applications.

Implications and Speculation on Future Directions

The implications of this research are manifold, both from a fundamental physics perspective and in practical applications. Theoretically, it expands the understanding of how elasticity can operate under non-conservative forces, paving the way for new theories in material science that incorporate active matter dynamics. Practically, odd elasticity could inspire the design of new metamaterials and autonomous systems, functioning as energy harvesters or novel wave manipulation devices. Possible future directions include the exploration of odd elasticity in biological systems, which are inherently active and might already exploit similar principles within cellular structures or tissue dynamics.

In conclusion, the paper offers a unified conceptual and mathematical framework for odd elasticity, opening up avenues for future explorations into active solids and the intersection of physics, materials science, and engineering. The work suggests a new class of materials that could reshape how mechanical and structural systems are conceived, designed, and utilized in both scientific and industrial contexts.