Odd Cosserat elasticity in active materials Piotr Sur owka1 2 3Anton Souslov4Frank J ulicher2 5and Debarghya Banerjee2 1Department of Theoretical Physics Wroc law University of Science and Technology 50-370 Wroc law Poland

2025-05-02 0 0 971.63KB 13 页 10玖币

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Odd Cosserat elasticity in active materials

Piotr Sur´owka,1, 2, 3 Anton Souslov,4Frank J¨ulicher,2, 5 and Debarghya Banerjee2, ∗

1Department of Theoretical Physics, Wroc law University of Science and Technology, 50-370 Wroc law, Poland

2Max Planck Institute for the Physics of Complex Systems,

N¨othnitzer Straße 38, 01187, Dresden, Germany

3W¨urzburg-Dresden Cluster of Excellence ct.qmat, Germany

4Department of Physics, University of Bath, Claverton Down, Bath, BA2 7AY, UK

5Cluster of Excellence Physics of Life, TU Dresden, 01062 Dresden, Germany

(Dated: January 16, 2023)

Stress-strain constitutive relations in solids with an internal angular degree of freedom can be

modelled using Cosserat (also called micropolar) elasticity. In this paper, we explore Cosserat

materials that include chiral active components and hence odd elasticity. We calculate static elastic

properties and show that the static response to rotational stresses leads to strains that depend on

both Cosserat and odd elasticity. We compute the dispersion relations in odd Cosserat materials

in the overdamped regime and ﬁnd the presence of exceptional points. These exceptional points

create a sharp boundary between a Cosserat-dominated regime of complete wave attenuation and

an odd-elasticity-dominated regime of propagating waves. We conclude by showing the eﬀect of

Cosserat and odd elastic terms on the polarization of Rayleigh surface waves.

The elastic behavior of an isotropic solid at equilibrium

can be characterized by two elastic constants, namely the

shear modulus and the bulk modulus [1]. This simple de-

scription of elastic properties using two coeﬃcients is pos-

sible because of symmetries such as: isotropy, parity, and

time reversal invariance. However, this simple deﬁnition

does not apply to a variety of other systems, for example

nematic solids [2] and Cosserat (or micropolar) solids [3].

Typical elastic solids can be microscopically modelled by

considering point masses connected by springs. By con-

trast, Cosserat elasticity is based on a more complex pic-

ture, and includes an angle (φ) describing the microscopic

orientational degree of freedom. Even for models consist-

ing of point particles, Cosserat-like elasticity can emerge

due to a geometry based on rotating elements [4]. Re-

cent advances in additive manufacturing (or 3D printing)

have led to rapid developments in the design of metama-

terials with Cosserat elasticity [5–11]. Cosserat elastic-

ity can also emerge in disordered solids [12–17], elastic

polymers [18, 19], and bio-membranes with viscoelastic

responses [20–22]. The Cosserat ﬁlament model has been

used to explore the eﬀect of microrotations in biological

ﬁlaments [23–27].

Active solids [28–33] are solids that are far from equi-

librium due to forcing at the microscopic scales [34–36].

To consider the eﬀect of activity and chirality, a situation

that can emerge when activity is present in the form of

an active torque, in an elastic Cosserat solid one must

include the eﬀect of odd elasticity. The odd elasticity is

connected to the breaking of two essential symmetries of

classical solids – parity invariance and time reversal in-

variance – and appears in the elasticity tensor as a term

breaking the major symmetry of the fourth rank elas-

tic tensor, i.e., κo

ijkl =−κo

klij . Recent literature [37–51]

∗banerjee@pks.mpg.de

has extensively studied the eﬀects of odd elasticity and

other forms of odd responses in solids and ﬂuids, and it is

therefore worthwhile to study the eﬀects of odd elasticity

in solids with active torques.

In this paper, we show how the simultaneous presence

of both odd elasticity and the Cosserat term aﬀects the

static and dynamic elastic response of chiral active solids

in the over-damped regime. We ﬁnd that the static re-

sponse to oﬀ-diagonal stresses is strongly dependent on

both Cosserat and odd elasticity. We also ﬁnd that dy-

namic modes have an exceptional point [52, 53] in the

dispersion relation due to the competition of Cosserat

and odd elasticity. In the overdamped regime, this excep-

tional point is characterised by a transition from damped

oscillations to diﬀusive (or attenuating) solutions. The

diﬀusive solutions near these exceptional points have

a diﬀusion coeﬃcient proportional to the square root

of the coeﬃcient of Cosserat elasticity, in contrast to

both equilibrium Cosserat solids which have no excep-

tional points and odd-elastic solids without Cosserat

terms. Furthermore, the edge of these solids exhibits edge

modes [1, 54, 55] whose polarization is aﬀected by the

combination of odd and Cosserat elasticity. The Cosserat

elastic coeﬃcient results in a renormalization of the usual

elastic terms for these edge waves, while the odd elastic

coeﬃcient mixes the longitudinal and transverse waves

at the edge.

Eﬀective theory with odd elasticity — We begin by

considering the stress tensor in a two-dimensional odd

Cosserat solid (Fig. 1). We can write the constitutive re-

lation between stress σij and strain uij in these materials

as:

σij =µuij +Bδij ukk +κc

2ij φ−1

2∇ × u

+κo∂iu∗

j+∂∗

iuj.(1)

Here, the vector uis the displacement ﬁeld. The stress

tensor σij depends on strain tensor uij , which is de-

arXiv:2210.13606v2 [cond-mat.soft] 13 Jan 2023

FIG. 1. Odd Cosserat materials. (a) The presence of extended objects at lattice points connected by springs necessitates

an additional angle variable to deﬁne the strain. These microrotations are modelled using the so-called Cosserat term. In

addition, the bonds can be chiral and active, which can be modelled (to leading order) using so-called odd elasticity. The active

bonds illustrated above can be thought of as springs that inject energy and angular momentum, i.e., possess some form of active

torque. (b) Rod-like elastic ﬁlaments (like actin ﬁlaments) have an angular degree of freedom and their elastic properties have

been modelled using Cosserat elasticity. (c) Granular material with active torque and (d) active biomembranes with chiral

processes are two other examples of odd Coserat materials.

ﬁned as the spatial gradients of the displacement uij =

1/2(∂iuj+∂jui), described by the elastic coeﬃcients µ

and B. The coeﬃcient of Cosserat elasticity κcdescribes

a coupling of the internal displacement gradients to an

orientational degree of freedom with angle φ. The two-

dimensional Levi-Civita symbol is denoted by ij . The

odd elastic coeﬃcient is denoted by κoand we deﬁne

u∗

i=ij uj. The odd-Cosserat model described in Eq.(1)

respects rotational invariance.

The dynamics of solids can depend not just on the re-

lation between the elastic stresses and strains, but also

on viscous stresses proportional to the strain rates i.e.,

σvis

ij =ηijkl ˙ukl. A combination of viscous and elastic

stresses in solids is described by the Kelvin-Voigt model

of viscoelasticity, which in turn can be generalised to in-

clude odd elastic terms [39]. However, in this paper we

focus on the elastic properties only and hence neglect for

simplicity the viscosus stress. In addition to the consti-

tutive relation given in Eq.(1), the equation of motion

for the displacement ﬁeld can be written as:

ρ∂2

tui+ Γ∂tui=∂jσij

I∂2

tφ+ Γφ∂tφ=α∇2φ−κcφ−1

2∇ × u+τa,(2)

where ρis the mass density and Iis the moment of inertia

density. The coeﬃcients Γ and Γφare friction coeﬃcients

that arise due to the damping of relative motion with

respect to a substrate. The coeﬃcient αis a diﬀusive

coeﬃcient. The term proportional to κcis required by

angular momentum conservation [56]. The active torque

is denoted by τa. The above equations (Eq. (2)) do not

take into account non-linear terms.

So far we have discussed the presence of odd elastic-

ity in the equation for uij . We now consider terms that

constitute active contributions in the equation of motion

for φ. The Cosserat term can be derived from a free en-

ergy F=Rdx(κc/2) (φ−(1/2)∇ × u)2. Model-A type

dynamics [57, 58] using this free energy gives us the equa-

tions of motion for Cosserat materials (see supplementary

section [59]). In order for a term to qualify as an active

contribution, the term has to be such that it cannot be

derived from an equilibrium free energy. Therefore, we

do not ﬁnd a linear active term in the equation for φ. The

leading order active contribution arising in this equation

is nonlinear and given by:

τa=λ|∇φ|2+. . . . (3)

Mathematically, these terms are similar to the type of

active terms that arise in active binary mixtures [60–

62] [63]. An important point to be noted here is that

because of the positive deﬁnite nature of the term, the

sign of λdecides the sense of the active torque, making

the system naturally chiral. However, in the presence

of either polar or nematic activity, it is possible to have

linear active terms. It is also possible to model active

torques in the form of torque dipoles [64–67].

Elastostatics of odd Cosserat materials — Using the

stress tensor in Eq.(1), we now discuss the static proper-

ties of odd Cosserat materials. Since we are considering

the static regime, all time derivatives drop out. In static

equilibrium, solids balance external stresses by the elas-

tic stresses, which are proportional to the strain. In the

case of Cosserat solids, there is an additional degree of

freedom, which is the angle φ. This force balance can be

written as a set of linear equations with the applied stress

on one side and the internal stress on the other side. If

we now additionally neglect the higher order spatial gra-

dients arising due to diﬀusion we then have a linear prob-

lem where we can choose a proﬁle of external stress and

from that obtain the strain in static equilibrium. Ma-

terial properties like the Young’s modulus (E), Poisson

ratio (ν), and odd ratio (νo, deﬁning the transverse tilt

of the solid under uniaxial compression, see Ref. [68]) can

be computed using this method. We ﬁnd the emergence

of auxetic properties (negative value of ν) in the limit

of large odd elasticity (2(κo)2> µB), which has been

previously reported for non-Cosserat odd elastic solids in

Ref. [68]. Details of this computation is provided in [59].

While the moduli (E,ν, and νo) remain largely un-

aﬀected under the application of a uniaxial pressure, we

can obtain generic expressions of strain in the presence

of applied stress. Let us consider a problem where we

have only applied rotational and transverse stresses, i.e.,

only σxy and σyx are non-zero. Under such an external

stress, components of the strain tensor have the form:

∂ux

∂x =−κo(v1+v2)

4κo2+µ2,

∂uy

∂y =κo(v1+v2)

4κo2+µ2,

∂uy

∂x =v2−v1

ε+v2−v1

κc+µ(v1+v2)

2(4κo2+µ2),

∂ux

∂y =−v2−v1

ε−v2−v1

κc+µ(v1+v2)

2(4κo2+µ2),(4)

where, v1=σxy,v2=σyx, and εis a regularizing term

added to the equation of motion of φto make the matrix

invertible. Physically, one can think of εas an eﬀective

renormalization of the higher order spatial gradients that

have been neglected and also from coupling to external

substrate.

Elastodynamics of odd Cosserat materials — We

now consider the dynamics of the system described

by Eq. (1) and Eq. (2). In the under-damped limit

(i.e., Γ →0, Γφ→0, and ignoring other viscous ef-

fects), we can obtain oscillatory solutions from the elas-

tic terms. We now consider propagating waves of the

form exp(i(k·x−ωt)), where xis the position coordi-

nate and kis the wavevector which gives us wavenumber

k=|k|. The frequency is given by ωand tis time. In the

limit of large odd elasticity, we obtain a dispersion rela-

tion: ω=eiπ/4kpκo/ρ, where, ωis the frequency and k

is the wavenumber. The above dispersion relation corre-

sponds to a propagating wave in displacement with speed

pκo/2ρand a damping constant pκo/2ρ. In the case of

a normal elastic solid without any Cosserat coupling or

odd elasticity, we have the usual elastic waves with dis-

persion relations ω=kpµ/ρ and ω=kp(B+µ)/ρ.

Therefore, unlike normal elastic waves, in odd elastic

solids, we have spontaneous damping and injection of

energy in the form of linear displacement ﬂuctuations

within the solid.

We now consider the over-damped regime, where we

can neglect inertial eﬀects and absorb the damping coef-

ﬁcients (Γ and Γφ) into the other coeﬃcients of the prob-

lem. In typical elastic materials, such an over-damped

limit gives rise to damped modes due to the eﬀects of the

shear modulus and bulk modulus. Odd elasticity gives

rise to propagating waves, which are analogous to the

Avron waves in odd-viscous ﬂuids [48, 69, 70]. However,

for an odd solid these are waves in displacement and not

in velocity. We now compute the dispersion relation for

a Cosserat solid in the presence of odd elasticity. For the

purpose of this calculation, we ignore the bulk modulus

and consider the limit where we can neglect the ﬂuc-

tuations in φ, i.e., φ=φ0. We obtain a closed form

expression of the dispersion relation given by:

ω=ik2

8−8µ−κc±pκc2−64κo2,(5)

which is consistent with our understanding that the

Cosserat term is predominantly a diﬀusive term in the

equations of elastodynamics, whereas odd elasticity gives

rise to wave solutions with speed proportional to √κo.

Note that we obtain an exceptional point at κc= 8κo,

where the eigenvalues are degenerate and have a square

root branch point [52, 53]. At the exceptional point, two

eigenvectors coalesce to a single one and the eigenvalues

are degenerate. At this exceptional point, we ﬁnd the

transition from a diﬀusive solution with diﬀusivity pro-

portional to √κcto a damped wave solution with speed

proportional to √κo. This is the key diﬀerentiating fea-

ture of the odd Cosserat elasticity from both equilibrium

elastic solids and active odd elastic solids.

For the usual Cosserat solids with κo= 0, the excep-

tional points are absent because of the Hermitian nature

of the matrix. The discriminant of the cubic equation

does not take negative values because it can be written

as a sum of positive numbers. However, in the more gen-

eral case, if we take into account the eﬀect of both κcand

κowe obtain ω=iΛ that are solutions to the cubic equa-

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OddCosseratelasticityinactivematerialsPiotrSurowka,1,2,3AntonSouslov,4FrankJulicher,2,5andDebarghyaBanerjee2,1DepartmentofTheoreticalPhysics,WroclawUniversityofScienceandTechnology,50-370Wroclaw,Poland2MaxPlanckInstituteforthePhysicsofComplexSystems,NothnitzerStrae38,01187,Dresden,Germany3Wurz...

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Odd Cosserat elasticity in active materials Piotr Sur owka1 2 3Anton Souslov4Frank J ulicher2 5and Debarghya Banerjee2 1Department of Theoretical Physics Wroc law University of Science and Technology 50-370 Wroc law Poland

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