Odd elasticity and topological waves in active surfaces Michele Fossati1Colin Scheibner2 3Michel Fruchart2 3and Vincenzo Vitelli2 3 4 1SISSA Trieste 34136 Italy

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Odd elasticity and topological waves in active surfaces

Michele Fossati,

Colin Scheibner,

2, 3

Michel Fruchart,

2, 3

and Vincenzo Vitelli

2, 3, 4, ∗

SISSA, Trieste 34136, Italy

James Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA

Department of Physics, The University of Chicago, Chicago, Illinois 60637, USA

Kadanoﬀ Center for Theoretical Physics, The University of Chicago, Chicago, Illinois 60637, USA

(Dated: October 10, 2022)

Odd elasticity encompasses active elastic systems whose stress-strain relationship is not compatible

with a potential energy. As the requirement of energy conservation is lifted from linear elasticity,

new anti-symmetric (odd) components appear in the elastic tensor. In this work, we study the

odd elasticity and non-Hermitian wave dynamics of active surfaces, speciﬁcally plates of moderate

thickness. We ﬁnd that a free-standing moderately thick, isotropic plate can exhibit two odd-elastic

moduli, both of which are related to shear deformations of the plate. These odd moduli can endow

the vibrational modes of the plate with a nonzero topological invariant known as the ﬁrst Chern

number. Within continuum elastic theory, we show that the Chern number is related to the presence

of unidirectional shearing waves that are hosted at the plate’s boundary. We show that the existence

of these chiral edge waves hinges on a distinctive two-step mechanism: the ﬁnite thickness of the

sample gaps the shear modes and the odd elasticity endows them with chirality.

I. INTRODUCTION

The elasticity of surfaces such as plates and shells plays

an important role from biological systems [

] to en-

gineered structures [

–

]. When modeling surfaces in

active and living systems, such as virus capsids [

–

] or

cell membranes [

], active forces are often added di-

rectly into the equations of motion, while the elasticity

itself is left unchanged [

]. Yet, the very relationship

between stress and strain can also be modiﬁed by internal

energy sources. In this situation, the elastic response can

include odd-elastic coeﬃcients [

–

], which describe

the part of the elastic response due to non-conservative

forces and therefore violate Maxwell-Betti reciprocity [

Odd elasticity has been engineered into robotic meta-

materials [

], and signatures have been reported in

collections of spinning magnetic colloids [

], starﬁsh em-

broys [

], and models of muscular hydraulics [

]. In

Ref. [

], the existence of these odd-elastic moduli in thin

active membranes has been predicted on the grounds of

symmetry.

Here we investigate the odd elasticity of thin plates

and how it aﬀects their vibrational dynamics, focusing

on the moderately thick regime in which the tilting of

the plate cross-section is independent from its midplane

deformation. First, a symmetry analysis reveals that an

isotropic, free-standing, moderately thick plate can ex-

hibit two independent odd-elastic moduli. By analyzing

the normal modes of vibration of odd-elastic plates, we

show that they can support edge modes in which waves

propagate in a unidirectional fashion at the border of

the plate. The waves propagating in these edge modes

do not backscatter when they encounter sharp edges or

defects. Since this robustness originates in the topologi-

cal nature of the edge modes, our ﬁndings may suggest

∗vitelli@uchicago.edu

strategies for designing desirable acoustic structures, such

as unidirectional waveguides [21–34].

II. THEORY

A. Odd elasticity

We start by reviewing the theory of odd elasticity,

which is the linear elasticity of solids that exert non-

conservative forces. Linear elasticity is the continuum

theory that describes the behaviour of solids under small

long-wavelength deformations. The deformation of the

solid is described by the displacement ﬁeld

ξi

(

) =

i−xi

giving the diﬀerence between the original position of a

point

of the elastic solid and its current position

We assume that only the variation of the internal relative

distances modiﬁes the physical state. The internal forces

then depend only on the strain tensor

uij

= 1

∂iξj

∂jξi

) at linear order. The forces between parcels of the

elastic continuum are described by the stress tensor

σij

In linear elasticity, one assumes a linear relation between

stress and strain

σij =Cijk`uk`.(1)

where

Cijk`

is the elastic tensor, assumed here to be homo-

geneous in space and frequency independent. Symmetry

constrains the structure of

Cijk`

. The strain tensor is

symmetric by construction and, if internal torques are

absent, the stress tensor is symmetric too. In this case,

the elastic tensor is symmetric under the exchanges

i↔j

and

k↔`

[

]. See Refs. [

] for cases in which the

displacement gradient tensor and the stress tensor are not

assumed to be symmetric.

If the system is conservative, another symmetry exists.

Suppose that the system undergoes a deformation

ξi

(

) in

time, with strains

uij

(

). The forces are conservative if the

arXiv:2210.03669v1 [cond-mat.soft] 7 Oct 2022

net work done is zero for every sequence of deformation

that begins and ends in the same conﬁguration. The work

per unit volume of an inﬁnitesimal deformation is given by

σij duij

and the work per unit volume under a ﬁnite cycle

of deformation is calculated as a line integral. We consider

a closed path

in the strain space, parameterized by

uij

(

). Let

σij

(

) =

Cijk`uk`

(

) be the associated stress

tensor. By applying Stokes’s theorem, we can express the

work Was the surface integral

W=IC

σij duij =ZS

∂σij

∂uk`

duij ∧duk` (2)

in which

∧

is the exterior product and

is a surface in

strain space such that

∂S

. Using the antisymmetry

of the exterior product, we can see that the forces are

conservative (W= 0 for all C) if and only if

∂σij

∂uk`

=∂σk`

∂uij

,(3)

or equivalently, if and only if

Cijk` =Ck`ij .(4)

This property is known as Maxwell-Betti reciprocity. A

system is said to be odd-elastic when Eq.

(4)

does not

hold, i.e. when its elastic tensor has components that

are odd under exchange

ij ↔k`

[

]. Note that we

have made no distinction between the Cauchy and Piola-

Kirchoﬀ stress tensors, because we have assumed that

there is no pre-stress in the system, see Ref. [

] for

details. Isotropy sets further constraints on the elastic

tensor. Odd elasticity is incompatible with spherical

isotropy (i.e. invariance under all rotations in 3D) but

nonetheless it is compatible with cylindrical isotropy (i.e.

invariance under all rotations preserving the

-axis) [

Here we employ the most general cylindrically symmetric

elastic tensor, see Appendix A for a derivation.

B. Surface description: thick plate

We consider a moderately thick and initially ﬂat surface,

i.e. a plate, whose midplane (the longitudinal plane that

cuts the plate’s thickness in half) lies at rest in the

x-y

plane, see Fig. 1. The plate has uniform thickness

along the

-axis at rest. In the Reissner-Mindlin theory

of moderately thick plates [

], the full three-dimensional

displacement ﬁeld of the plate

ξi

(

x, y, z

) is expressed in

terms of ﬁelds deﬁned on the horizontal midplane as

ξx(x, y, z) = ηx(x, y) + zφx(x, y)

ξy(x, y, z) = ηy(x, y) + zφy(x, y)

ξz(x, y, z) = w(x, y).

(5)

The ﬁeld

ηα

represents the horizontal displacement of the

midplane (

= 0) in the direction

α∈ {x, y}

, while

describes the vertical displacement. A line of points that

FIG. 1.

Kinematics of a moderately thick plate.

the Reissner-Mindlin theory of moderately thick plates, the

deformation of a plate (drawn in its undeformed reference state

in grey and in a deformed state in black) is parameterized

by ﬁve ﬁelds (

ηx

ηy

φx

φy

, and

) deﬁned on the midplane

(dash-dotted line). The ﬁelds

ηx

and

describe respectively

the horizontal and vertical displacements of the midplane. The

ﬁeld

φx

is the angle between a deformed normal line in the

direction and the

axis. The quantity

−∂xw

quantiﬁes the

slope of the midplane in the

direction. If

−∂xw

is diﬀerent

from φx, then uxz is non-zero.

lies vertical at rest is inclined of an angle

φα

between

the

-axis and the

-axis after the deformation (Fig. 1).

While the full displacement ﬁeld is deﬁned over a three-

dimensional space, the ﬁelds

ηα

φα

and

are deﬁned

over the midplane, which is a two-dimensional manifold.

The strains can be decomposed into two terms:

independent and

-linear

uij

zu1

. Here,

describes strains in which the top face and the bottom

face of the plate are deformed oppositely. Explicitly

2u0

αβ =∂αηβ+∂βηα

2u0

αz =φα+∂αw

zz = 0

2u1

αβ =∂αφβ+∂βφα

αβ = 0

zz = 0.

(6)

The bending of the midplane is quantiﬁed by

−∂αw

and

αz

is half of the angle between a deformed vertical

line and the normal to the deformed midplane, projected

in the

direction (Fig. 1). No

-linear term is present in

the transverse strain. The vertical strain

uzz

is identically

zero because the vertical displacement is independent from

. Since

ξα

ηα

zφα

, an originally vertical straight line

remains straight after the deformation, and since

uzz

= 0,

the line does not elongate.

C. Constitutive relations

In order to represent the tensorial constitutive relations

in matrix form, we choose a basis for the strains and the

stresses by deﬁning the basis tensors

Dij =1

√2



100

010

000



S1

ij =1

√2



100

0−1 0

000





ij =1

√2



010

100

000



Tx

ij =1

√2



001

000

100





ij =1

√2



000

001

010



.

(7)

The

entry is zero for all the ﬁve matrices because

uzz

= 0 by construction and

σzz

= 0 is further assumed

(see Appendix B for details). Here

describes cylin-

drical symmetric strains and stresses,

S1,2

describe the

planar shears and

Tx,y

the cross-section shears in

and

direction. This basis separates the irreducible represen-

tations of the group of rotations

(2) around the

axis:

is invariant,

Sα

transforms as a two-headed arrow in

the plane, and

Tα

transforms as an ordinary vector in

the plane. The constitutive relation Eq.

(1)

can then be

written in matrix form as

σa=Cabub(8)

where

σa

and

are the components of the stress and

strain in the basis.

We assume the plate to be made of an homogeneous

cylindrically isotropic material (see Appendix A). The

elastic tensor of the plate is obtained from the full three-

dimensional elastic tensor via a reduction procedure illus-

trated in Appendix B. In the basis

{D, S1, S2, T x, T y}

reads

Cab = 2 





B0 0 0 0

0µ1Ko

10 0

0−Ko

1µ10 0

0 0 0 µ2Ko

0 0 0 −Ko

2µ2







(9)

with

−6D2+6H2+9Bµ3

3B+4(√2D+µ3)

. All the moduli are inherited

from the three-dimensional constitutive relations. Here,

is the eﬀective 2D bulk modulus, which relates plane

isotropic dilations to the plane isotropic stress.

µ1

and

µ2

are passive shear moduli that couple respectively planar

shears (

uS1, uS2

) and cross-section shears (

uTx, uTy

). The

odd moduli

1, Ko

build the antisymmetric part of the

elastic tensor.

maps

uS1

−σS2

and

uS2

σS1

does the same on the basis elements Tx, T y.

The dynamical quantities that are relevant for the

plate’s dynamics are the net stress tensor

Nij

and the

moment tensor Mij deﬁned by

Nij =Zh/2

−h/2

dz σij

Mij =Zh/2

−h/2

dz zσij .

(10)

These are respectively the zeroth and ﬁrst moment of the

stress in z. Integration in the z-direction produces a net

stress that depends only on

and a moment tensor that

depends on the bending terms

. Using the constitutive

equations Eq.

(1)

with the elastic tensor in Eq.

(9)

, we

ﬁnd that the in-plane stresses are governed by





NS1

NS2



= 2h



B0 0

0µ1Ko

0−Ko

1µ1









(11)

while the bending moments are governed by





MS1

MS2



=h3

6



B0 0

0µ1Ko

0−Ko

1µ1









(12)

and the cross-sectional stresses by

NTx

NTy= 2hµ2Ko

−Ko

2µ2u0

Ty.(13)

A visual representation of the components of the strain,

net stress and moment tensor in the basis of Eq.

(7)

, is

given in Table I. We note that the constitutive equations

in Eqs. (11) to (13) assume that the plate arises as the

thin limit of a homogeneous three-dimensional solid. How-

ever, if the plate is not homogeneous along its thickness,

additional moduli can appear that couple the independent

equations in Eqs. (11) to (13), see Appendix B. Notice

that the moduli

µ2

and

set the stresses in response to

the cross section shearing. The matrix in Eq. (13) is pro-

portional to a rotation matrix whose chirality is set by the

modulus

. This will play a crucial role when we discuss

the specturm and topological modes in Sections III.

Having the constitutive relations, we can examine the

linearly independent cycles in strain space over which

work is extracted C=∂S. The work per unit surface is

W=Zh/2

−h/2

dz ZC

duaCabub

=hZC

du0

aCabu0

b+h3

12 ZC

du1

aCabu1

2ZS

Cab du0

a∧du0

b+h3

24 ZS

Cab du1

a∧du1

(14)

There are three independent ways to extract energy

with a cycle of deformations, represented in Fig. 2. Cy-

cling in the plane

, the energy density extracted

is equal to 2

hK0

times the area enclosed in the strain

space (Fig. 2a). A bending cycle that involves

and

extracts (

h3/

times the area enclosed (Fig. 2b).

With a cycle in the

plane, the density of work is

2hK0

1times the area enclosed (Fig. 2c).

D. Equations of motion

We now move on to the dynamics of the system. To

do so, let us assume that the elastic material evolves

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OddelasticityandtopologicalwavesinactivesurfacesMicheleFossati,1ColinScheibner,2,3MichelFruchart,2,3andVincenzoVitelli2,3,4,1SISSA,Trieste34136,Italy2JamesFranckInstitute,TheUniversityofChicago,Chicago,Illinois60637,USA3DepartmentofPhysics,TheUniversityofChicago,Chicago,Illinois60637,USA4KadanoCen...

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Odd elasticity and topological waves in active surfaces Michele Fossati1Colin Scheibner2 3Michel Fruchart2 3and Vincenzo Vitelli2 3 4 1SISSA Trieste 34136 Italy

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