Odd Elasticity Once Removed

• Odd elasticity once removed is an effective constitutive behavior where antisymmetric stress–strain relations emerge through reduction and elimination of hidden variables.
• It is realized through dimensional reduction, coarse-graining of active systems, and integrating out auxiliary fields to yield nonconservative work cycles.
• This phenomenon underpins novel mechanical responses, influencing topological waves, self-propulsion, and defect dynamics in active and prestressed materials.

Odd elasticity, once removed, denotes a family of constructions in which the characteristic signature of odd elasticity—a constitutive response with a part antisymmetric under exchange of stress and strain channels—does not appear as a primitive microscopic law, but instead emerges in an effective description after coarse-graining, dimensional reduction, elimination of hidden variables, prestress expansion, or embedding in a generalized continuum with internal rotations (Fossati et al., 2022, Lin et al., 2022, Engstrom et al., 23 Sep 2025). In this sense, oddness is inherited rather than postulated: it can arise in a reduced plate theory of an active three-dimensional solid (Fossati et al., 2022), a continuum obtained from passive chiral metamaterial unit cells (Shaat et al., 2021), a structural Langevin theory inferred after coupling to a catalytic coordinate (Kobayashi et al., 2022), or a Cauchy-stress/Lagrangian-strain map around anisotropic prestress in an otherwise conservative medium (Engstrom et al., 23 Sep 2025).

1. Definition and conceptual scope

Scheibner, Souslov, Banerjee, Surowka, Irvine, and Vitelli introduced odd elasticity by lifting the equilibrium restriction that enforces the major symmetry of the elastic modulus tensor (Scheibner et al., 2019). In the standard linear constitutive form,

$$\sigma_{ij}=C_{ijk\ell}u_{k\ell}, \qquad u_{ij}=\frac12(\partial_i \xi_j+\partial_j \xi_i),$$

passive elasticity requires Maxwell–Betti reciprocity,

$C_{ijk\ell}=C_{k\ell ij},$

because stress derives from an elastic energy and quasistatic work over a closed strain cycle vanishes (Fossati et al., 2022). Odd elasticity is the failure of that major symmetry. A convenient decomposition is

$$C_{ijk\ell}=C^{\mathrm e}_{ijk\ell}+C^{\mathrm o}_{ijk\ell},$$

with

$$C^{\mathrm e}_{ijk\ell}=\frac12\left(C_{ijk\ell}+C_{k\ell ij}\right),\qquad C^{\mathrm o}_{ijk\ell}=\frac12\left(C_{ijk\ell}-C_{k\ell ij}\right).$$

The odd part $C^{\mathrm o}$ is the non-potential, antisymmetric part that allows the material to inject or extract work over quasistatic strain cycles (Fossati et al., 2022).

The review "Odd Viscosity and Odd Elasticity" places this in a broader continuum setting. It defines odd elasticity by the antisymmetric part of the Piola-Kirchhoff elastic tensor under exchange of index pairs and emphasizes that, unlike ordinary elasticity, odd elasticity is nonconservative: the work done depends on the path through deformation space, not just the endpoints (Fruchart et al., 2022). This suggests that the phrase "once removed" is best reserved for situations in which this same antisymmetric constitutive structure appears only after a secondary operation—reduction, elimination, reinterpretation, or homogenization.

2. Constitutive signatures and work cycles

In two dimensions, the standard isotropic odd-elastic form contains two ordinary moduli and two odd moduli. In the generalized basis of dilation, rotation, and two shears, the isotropic matrix can be written as

$$K^{\alpha\beta} = 2 \begin{pmatrix} B&0&0&0\ A&0&0&0\ 0&0&\mu&K^o\ 0&0&-K^o&\mu \end{pmatrix},$$

with $B$ the bulk modulus, $\mu$ the shear modulus, and $A$ and $K^o$ the odd moduli (Scheibner et al., 2019). The modulus $C_{ijk\ell}=C_{k\ell ij},$0 couples dilation to torque density, while $C_{ijk\ell}=C_{k\ell ij},$1 couples the two shear sectors antisymmetrically. In the language of the 2019 formulation, a pure shear stress can induce simple shear strain and vice versa with an antisymmetric relation (Scheibner et al., 2019).

The work density over a closed quasistatic cycle is the basic operational diagnostic. One form used in the literature is

$C_{ijk\ell}=C_{k\ell ij},$2

and the odd contribution is precisely the part that survives on a closed loop (Fossati et al., 2022). In the 2019 formulation, the work over a cycle is proportional to the area enclosed in the relevant strain subspace; in the isotropic case, the shear-sector contribution is proportional to $C_{ijk\ell}=C_{k\ell ij},$3 times the enclosed area (Scheibner et al., 2019). This is why an odd-elastic solid was described as a distributed engine: work is locally extracted, or injected, during quasistatic cycles of deformation (Scheibner et al., 2019).

Several later works preserve this constitutive signature but relocate its origin. A useful criterion for identifying odd elasticity once removed is therefore not the microscopic mechanism, but whether the reduced variables inherit a constitutive law with the same kind of antisymmetric channel and the same closed-cycle work structure. This suggests a unifying viewpoint across active plates, chiral metamaterials, catalytic micromachines, driven grains, and prestressed solids (Fossati et al., 2022, Shaat et al., 2021, Kobayashi et al., 2022, Huang et al., 2023, Engstrom et al., 23 Sep 2025).

3. Mechanisms of indirect emergence

One major route is dimensional reduction. In "Odd elasticity and topological waves in active surfaces" (Fossati et al., 2022), a free-standing, initially flat, moderately thick plate is treated in the Reissner–Mindlin approximation, with displacement field

$C_{ijk\ell}=C_{k\ell ij},$4

The reduced plate theory inherits two odd elastic moduli from a cylindrically symmetric three-dimensional active elastic tensor: $C_{ijk\ell}=C_{k\ell ij},$5 The paper explicitly states that these moduli are inherited from the three-dimensional constitutive relations and that the effective surface medium is not a generic 2D odd-elastic material written down from scratch (Fossati et al., 2022). This is a canonical example of odd elasticity emerging through dimensional reduction.

A second route is hidden microstructure and elimination of internal twist. "Mechanical activity and odd elasticity of passive, 2D chiral metamaterials" derives an asymmetric continuum elasticity tensor from a passive chiral unit cell made of rigid circles and ordinary elastic ligaments (Shaat et al., 2021). The continuum odd elasticity appears only after a discrete Newtonian model is coarse-grained and the rotational field $C_{ijk\ell}=C_{k\ell ij},$6 is eliminated. The odd elastic part depends on the chiral odd elastic moduli

$C_{ijk\ell}=C_{k\ell ij},$7

while the reversible chiral coupling $C_{ijk\ell}=C_{k\ell ij},$8 can be nonzero even when the solid is not odd elastic (Shaat et al., 2021). The odd response is therefore mediated by geometry and a hidden twist mechanism rather than inserted as a powered odd spring law.

A third route is elimination of an explicitly driven auxiliary variable. In "Onsager's variational principle for nonreciprocal systems with odd elasticity," a 2D displacement field $C_{ijk\ell}=C_{k\ell ij},$9 is coupled to an extra field $$C_{ijk\ell}=C^{\mathrm e}_{ijk\ell}+C^{\mathrm o}_{ijk\ell},$$0 conjugate to a constant active force density $$C_{ijk\ell}=C^{\mathrm e}_{ijk\ell}+C^{\mathrm o}_{ijk\ell},$$1 (Lin et al., 2022). After minimizing the Rayleighian and eliminating $$C_{ijk\ell}=C^{\mathrm e}_{ijk\ell}+C^{\mathrm o}_{ijk\ell},$$2, the reduced continuum dynamics acquires emergent odd elastic moduli

$$C_{ijk\ell}=C^{\mathrm e}_{ijk\ell}+C^{\mathrm o}_{ijk\ell},$$3

The paper presents this explicitly as a mechanism in which hidden driven variables plus dissipative coupling generate odd elasticity one elimination step later (Lin et al., 2022).

A closely related stochastic version appears in "Odd elasticity of a catalytic micromachine." There the microscopic model is not written with a bare antisymmetric elastic matrix between structural coordinates; rather, a reaction coordinate $$C_{ijk\ell}=C^{\mathrm e}_{ijk\ell}+C^{\mathrm o}_{ijk\ell},$$4 drives conformational coordinates $$C_{ijk\ell}=C^{\mathrm e}_{ijk\ell}+C^{\mathrm o}_{ijk\ell},$$5, and the structural sector is then fitted to an effective odd-Langevin theory with

$$C_{ijk\ell}=C^{\mathrm e}_{ijk\ell}+C^{\mathrm o}_{ijk\ell},$$6

after analyzing correlation functions (Kobayashi et al., 2022). The same paper defines a nonreciprocality

$$C_{ijk\ell}=C^{\mathrm e}_{ijk\ell}+C^{\mathrm o}_{ijk\ell},$$7

and reports that its behavior is similar to that of the odd elasticity (Kobayashi et al., 2022).

A control-theoretic and time-averaged version appears in "Odd elasticity in driven granular matter." There periodic shear plus ratchet-like friction generates persistent grain spinning, and only after averaging over the drive cycle does the material acquire a time-averaged constitutive law with odd shear modulus $$C_{ijk\ell}=C^{\mathrm e}_{ijk\ell}+C^{\mathrm o}_{ijk\ell},$$8 (Huang et al., 2023). In the approximately isotropic regime,

$$C_{ijk\ell}=C^{\mathrm e}_{ijk\ell}+C^{\mathrm o}_{ijk\ell},$$9

and the heuristic prediction is

$$C^{\mathrm e}_{ijk\ell}=\frac12\left(C_{ijk\ell}+C_{k\ell ij}\right),\qquad C^{\mathrm o}_{ijk\ell}=\frac12\left(C_{ijk\ell}-C_{k\ell ij}\right).$$0

with $$C^{\mathrm e}_{ijk\ell}=\frac12\left(C_{ijk\ell}+C_{k\ell ij}\right),\qquad C^{\mathrm o}_{ijk\ell}=\frac12\left(C_{ijk\ell}-C_{k\ell ij}\right).$$1 when $$C^{\mathrm e}_{ijk\ell}=\frac12\left(C_{ijk\ell}+C_{k\ell ij}\right),\qquad C^{\mathrm o}_{ijk\ell}=\frac12\left(C_{ijk\ell}-C_{k\ell ij}\right).$$2 (Huang et al., 2023). Here the odd modulus is a cycle-averaged output of a driven frictional state.

A disordered active-matter version is developed in "Odd elasticity in disordered chiral active materials." Starting from a Cosserat medium with internal rotations and local active torques, the fast internal angular momentum is eliminated, and the Cauchy stress acquires an odd elasticity tensor with

$$C^{\mathrm e}_{ijk\ell}=\frac12\left(C_{ijk\ell}+C_{k\ell ij}\right),\qquad C^{\mathrm o}_{ijk\ell}=\frac12\left(C_{ijk\ell}-C_{k\ell ij}\right).$$3

proportional to the active torque density $$C^{\mathrm e}_{ijk\ell}=\frac12\left(C_{ijk\ell}+C_{k\ell ij}\right),\qquad C^{\mathrm o}_{ijk\ell}=\frac12\left(C_{ijk\ell}-C_{k\ell ij}\right).$$4 (Lee et al., 6 Aug 2025). The paper explicitly states that odd elasticity naturally emerges as a nonlinear effect of internal particle rotations, not as a microscopic odd spring network.

4. Reduced and enriched continua

Odd elasticity once removed often appears in enriched continuum theories rather than ordinary Cauchy elasticity. "Realization of active metamaterials with odd micropolar elasticity" constructs a freestanding active metabeam with Timoshenko/Cosserat kinematics,

$$C^{\mathrm e}_{ijk\ell}=\frac12\left(C_{ijk\ell}+C_{k\ell ij}\right),\qquad C^{\mathrm o}_{ijk\ell}=\frac12\left(C_{ijk\ell}-C_{k\ell ij}\right).$$5

and constitutive law

$$C^{\mathrm e}_{ijk\ell}=\frac12\left(C_{ijk\ell}+C_{k\ell ij}\right),\qquad C^{\mathrm o}_{ijk\ell}=\frac12\left(C_{ijk\ell}-C_{k\ell ij}\right).$$6

The off-diagonal one-way coupling $$C^{\mathrm e}_{ijk\ell}=\frac12\left(C_{ijk\ell}+C_{k\ell ij}\right),\qquad C^{\mathrm o}_{ijk\ell}=\frac12\left(C_{ijk\ell}-C_{k\ell ij}\right).$$7 is identified as the odd micropolar modulus (Chen et al., 2020). The oddness is mediated by internal rotation, piezoelectric sensing and actuation, and local feed-forward control, and only then summarized as an effective constitutive coefficient (Chen et al., 2020).

"Odd Cosserat elasticity in active materials" similarly embeds odd elasticity in a 2D micropolar medium with displacement $$C^{\mathrm e}_{ijk\ell}=\frac12\left(C_{ijk\ell}+C_{k\ell ij}\right),\qquad C^{\mathrm o}_{ijk\ell}=\frac12\left(C_{ijk\ell}-C_{k\ell ij}\right).$$8 and microrotation $$C^{\mathrm e}_{ijk\ell}=\frac12\left(C_{ijk\ell}+C_{k\ell ij}\right),\qquad C^{\mathrm o}_{ijk\ell}=\frac12\left(C_{ijk\ell}-C_{k\ell ij}\right).$$9. Its stress law is

$C^{\mathrm o}$0

with $C^{\mathrm o}$1 the Cosserat modulus and $C^{\mathrm o}$2 the odd modulus (Surówka et al., 2022). In this formulation, odd elasticity is filtered through a rotational sector and competes with rotational relaxation.

A conceptually different reinterpretation appears in "Odd elasticity in Hamiltonian formalism." There the odd modulus is generated from an anisotropic effective mass in the kinetic term of a Hamiltonian field theory, with

$C^{\mathrm o}$3

and a corresponding nonlinear correction

$C^{\mathrm o}$4

at finite strain (Zhang et al., 2023). The paper argues that odd elasticity is not denied its non-conservative phenomenology, but reinterpreted as arising from a Hamiltonian curl force induced by anisotropic inertia. Within this mechanism, the odd elasticity is intrinsically anisotropic (Zhang et al., 2023).

A further reinterpretation is developed in "Conservative yet constitutively odd elasticity in prestressed metamaterials." For perturbations about an anisotropically prestressed equilibrium state, the incremental Cauchy-stress/Lagrangian-strain tensor is

$C^{\mathrm o}$5

with major antisymmetry

$C^{\mathrm o}$6

The paper emphasizes that this does not make the material energetically odd, because $C^{\mathrm o}$7 are not an energy-conjugate pair in this finite-strain prestressed setting (Engstrom et al., 23 Sep 2025). The oddness is therefore constitutive but not thermodynamic in the usual active sense.

5. Dynamical, topological, and transport consequences

Indirectly generated odd elasticity has consequences far beyond static constitutive asymmetry. In the active-plate problem, finite thickness and odd elasticity produce non-Hermitian topological wave dynamics. The out-of-plane sector carries a first Chern number, and the plate supports unidirectional shearing edge waves localized near the boundary (Fossati et al., 2022). The mechanism is explicitly two-step: thickness gaps the transverse shear sector, and odd elasticity endows it with chirality (Fossati et al., 2022).

In odd Cosserat media, the competition between the Cosserat modulus $C^{\mathrm o}$8 and the odd modulus $C^{\mathrm o}$9 produces exceptional points in the overdamped dispersion relation. In the reduced case,

$$K^{\alpha\beta} = 2 \begin{pmatrix} B&0&0&0\ A&0&0&0\ 0&0&\mu&K^o\ 0&0&-K^o&\mu \end{pmatrix},$$0

so the exceptional point occurs at

$$K^{\alpha\beta} = 2 \begin{pmatrix} B&0&0&0\ A&0&0&0\ 0&0&\mu&K^o\ 0&0&-K^o&\mu \end{pmatrix},$$1

The paper identifies a sharp boundary between a Cosserat-dominated regime of complete wave attenuation and an odd-elasticity-dominated regime of propagating waves (Surówka et al., 2022).

The prestressed conservative construction yields a different wave phenomenology. Along special directions, the tuned systems support a transverse string-like mode with $$K^{\alpha\beta} = 2 \begin{pmatrix} B&0&0&0\ A&0&0&0\ 0&0&\mu&K^o\ 0&0&-K^o&\mu \end{pmatrix},$$2 and an in-plane flexural soft mode with

$$K^{\alpha\beta} = 2 \begin{pmatrix} B&0&0&0\ A&0&0&0\ 0&0&\mu&K^o\ 0&0&-K^o&\mu \end{pmatrix},$$3

even though the underlying system is conservative (Engstrom et al., 23 Sep 2025). The soft mode has oscillating momentum density but constant energy current, as encoded by

$$K^{\alpha\beta} = 2 \begin{pmatrix} B&0&0&0\ A&0&0&0\ 0&0&\mu&K^o\ 0&0&-K^o&\mu \end{pmatrix},$$4

for $$K^{\alpha\beta} = 2 \begin{pmatrix} B&0&0&0\ A&0&0&0\ 0&0&\mu&K^o\ 0&0&-K^o&\mu \end{pmatrix},$$5 (Engstrom et al., 23 Sep 2025).

Odd elasticity once removed also reorganizes locomotion and path-space dynamics. "Self-organized swimming with odd elasticity" shows that an odd-elastic Purcell swimmer can develop a stable limit cycle and self-organized locomotion at low Reynolds number (Ishimoto et al., 2021). In "The Onsager-Machlup Integral for Non-reciprocal Systems with Odd Elasticity," odd elasticity enters the Euler–Lagrange equation for the most probable path as a velocity coupling,

$$K^{\alpha\beta} = 2 \begin{pmatrix} B&0&0&0\ A&0&0&0\ 0&0&\mu&K^o\ 0&0&-K^o&\mu \end{pmatrix},$$6

so the most probable transition path becomes non-reciprocal and entropy production over a closed cycle is

$$K^{\alpha\beta} = 2 \begin{pmatrix} B&0&0&0\ A&0&0&0\ 0&0&\mu&K^o\ 0&0&-K^o&\mu \end{pmatrix},$$7

(Yasuda et al., 2021). The learned three-sphere microswimmer provides a yet further reduction: after reinforcement learning has produced non-reciprocal cycles, an effective odd elasticity

$$K^{\alpha\beta} = 2 \begin{pmatrix} B&0&0&0\ A&0&0&0\ 0&0&\mu&K^o\ 0&0&-K^o&\mu \end{pmatrix},$$8

is extracted a posteriori from force-displacement correlations (Lin et al., 2023).

Defect physics is another major downstream manifestation. "Topological defects in solids with odd elasticity" shows that odd moduli modify defect strain fields, defect interactions, and the stability of bound dislocation pairs; isolated dislocations can self propel via microscopic work cycles active at their cores (Braverman et al., 2020). The same theme recurs in driven granular matter, where self-healing grain boundaries, chiral plastic vortices, and force-chain deflection appear in the presence of time-averaged odd elasticity (Huang et al., 2023).

At the lattice level, nonreciprocal elastic coupling supports periodic and quasiperiodic traveling waves. "Periodic and quasiperiodic traveling waves in nonlinear lattices with odd elasticity" studies a nonlinear ring lattice with asymmetric nearest-neighbor coupling,

$$K^{\alpha\beta} = 2 \begin{pmatrix} B&0&0&0\ A&0&0&0\ 0&0&\mu&K^o\ 0&0&-K^o&\mu \end{pmatrix},$$9

and reports periodic and quasiperiodic traveling waves, master-stability curves, and an Eckhaus instability identified from the curvature of the master stability curve (Giraldo et al., 18 May 2026). Here odd elasticity is most visible in the existence, directionality, and finite-size stability of coherent waves.

6. Distinctions, limitations, and unresolved boundaries

The literature repeatedly distinguishes odd elasticity from neighboring notions. Chirality alone does not guarantee odd elasticity: in the passive chiral metamaterial, $B$0 can be nonzero even when the odd moduli $B$1 and $B$2 vanish (Shaat et al., 2021). Odd elasticity is also distinct from odd viscosity. The review article states that odd elasticity is generally associated with microscopic non-conservative forces, whereas odd viscosity is usually associated with microscopic dynamics that do not obey time-reversal symmetry (Fruchart et al., 2022).

A recurring misconception concerns conservative systems that are only constitutively odd. The prestressed-metamaterial work stresses that a major-antisymmetric incremental tensor $B$3 does not imply an active or nonconservative solid, because work must be computed using an energy-conjugate stress–strain pair such as second Piola–Kirchhoff stress and Lagrangian strain (Engstrom et al., 23 Sep 2025). This suggests that "odd elasticity once removed" includes at least two logically distinct cases: genuinely nonconservative reduced elasticity, and conservative systems that reproduce the algebra of odd elasticity in a non-conjugate representation.

Several symmetry restrictions remain sharp. The active-plate paper notes that odd elasticity is incompatible with full spherical isotropy but allowed by cylindrical isotropy (Fossati et al., 2022). The Hamiltonian construction shows that its induced odd modulus is intrinsically anisotropic (Zhang et al., 2023). At the same time, "Nonlinear isotropic odd elasticity" shows that although isotropic linear odd elasticity is absent in three dimensions, nonlinear odd effects can survive in finite-deformation problems such as the 3D Rivlin cube (Zhao et al., 4 May 2026). This suggests that the domain of odd elasticity once removed extends beyond the linear 2D setting, but not without qualification.

Boundary conditions, validity ranges, and interpretation also matter. In the active-surface problem, the existence and count of edge modes depend subtly on boundary conditions, and the topological results are presented most explicitly in the purely active limit (Fossati et al., 2022). In the learned microswimmer and catalytic micromachine, the extracted odd elastic coefficients are effective, dynamical, and history-dependent; they are not static material constants of the underlying springs or molecular couplings (Lin et al., 2023, Kobayashi et al., 2022). In driven granular matter, the constitutive law is time-averaged and contingent on maintaining a periodically driven steady state (Huang et al., 2023).

Taken together, these works establish a consistent theme. Odd elasticity once removed is not a single mechanism but a mode of appearance: odd elasticity emerges as the constitutive fingerprint of hidden activity, hidden rotation, hidden prestress, hidden control, or hidden coarse-graining. Sometimes the removed layer is geometrical, as in plate reduction or passive chirality (Fossati et al., 2022, Shaat et al., 2021). Sometimes it is dynamical, as in Onsager elimination, catalytic coordinates, learned strokes, or driven spinning grains (Lin et al., 2022, Kobayashi et al., 2022, Lin et al., 2023, Huang et al., 2023). Sometimes it is representational, as in prestressed conservative media or Hamiltonian anisotropic inertia (Engstrom et al., 23 Sep 2025, Zhang et al., 2023). A plausible implication is that odd elasticity is less a narrowly defined material class than a recurrent effective structure that reappears whenever nonreciprocal mechanics is projected onto reduced elastic variables.