Mechanics of Materials 107 (2017) 71–80

Contents lists available at ScienceDirect

Mechanics of Materials

journal homepage: www.elsevier.com/locate/mechmat

Research paper

Nonlinear large deformation of acoustomechanical soft materials

Fengxian Xin

a , b , ∗, Tian Jian Lu

a , b

a MOE Key Laboratory for Multifunctional Materials and Structures, Xi’an Jiaotong University, Xi’an 710049, PR China b State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an 710049, PR China

a r t i c l e i n f o

Article history:

Received 12 July 2016

Revised 17 December 2016

Available online 6 February 2017

a b s t r a c t

An acoustomechanical theory of soft materials is proposed to account for the nonlinear large deformation

of soft materials triggered by both the ultrasound waves and mechanical forces. This theory is formu-

lated by employing the nonlinear elasticity theory of the soft material and the theory of acoustic radia-

tion force, which takes into consideration of the combination loading of mechanical forces and acoustic

inputs. While the propagation of acoustic wave depends on material configuration, the radiation force

generated by wave propagation deforms the material configuration. This complex interaction reaches a

steady state when the mechanical stress and the acoustic radiation stress are able to balance with the

elastic deformation stress. The acoustomechanical theory is employed to characterize the acoustomechan-

ical behaviors of thin soft material layers under different boundary conditions (e.g., equal-biaxial forces,

uniaxial force, and uniaxial constraint). Prestretches arising from these boundary conditions are shown to

play significant roles in affecting the acoustomechanical repsonse of soft material: the same material ac-

tuated from different prestretches and boundary conditions exhibits different stretch-stress relations. This

novel functionality enables innovative design of acoustic sensors and actuators based on soft materials.

© 2017 Elsevier Ltd. All rights reserved.

1

r

d

i

(

A

i

t

fi

t

m

s

m

a

t

t

t

t

d

e

1

c

t

r

o

d

w

e

i

w

r

1

s

i

a

d

t

S

i

e

h

0

. Introduction

Soft active materials capable of undergoing large deformation in

esponse to various external stimuli are promising candidates for

esigning innovative sensors, actuators, medical devices, microflu-

dic manipulation devices, energy harvesters and adaptive robotics

Stark and Garton, 1955; Zhang et al., 1998; Zhang et al., 2002 ).

large body of existing works have been devoted to investigat-

ng the nonlinear large deformation and instabilities of soft ac-

ive materials, including dielectric elastomer actuated by electric

eld ( Suo et al., 2008; Zhao and Wang, 2014 ), magneto-active elas-

omers trigged by magnitude field ( Bustamante et al., 2006; Dorf-

ann and Ogden, 2004; Rudykh and Bertoldi, 2013 ), and those

ensitive to temperature ( Chester and Anand, 2011 ) and environ-

ent salinity ( Ohmine and Tanaka, 1982; Tanaka et al., 1980; Yeh

nd Alexeev, 2015 ). In sharp contrast, the acoustomechanical actua-

ion performance of soft active materials received much less atten-

ion. To address this deficiency, we develop an acoustomechanical

heory for the nonlinear deformation of soft materials subjected

o combined mechanical force and acoustic inputs. A variety of

isplacement/force boundary conditions are considered, including

qual-biaxial forces, uniaxial force, and uniaxial constraint.

∗ Corresponding author.

E-mail addresses: fengxian.xin@gmail.com (F. Xin), tjlu@mail.xjtu.edu.cn (T.J. Lu).

d

2

h

ttp://dx.doi.org/10.1016/j.mechmat.2017.02.001

167-6636/© 2017 Elsevier Ltd. All rights reserved.

The propagation of ultrasonic wave with a frequency beyond

MHz in medium can give rise to a steady time-averaged force

alled the acoustic radiation force, due mainly to acoustic momen-

um transfer between adjacent medium particles. This force can

each a magnitude of several times MPa in air when the amplitude

f input sound pressure is ∼1 MPa, which is sufficient large to in-

uce large nonlinear deformation in common soft active materials

ith a modulus of several times kPa ( Guo et al., 2016; Issenmann

t al., 2008; Xin and Lu, 2016a ).

Lord Rayleigh first formulated a theory of radiation pressure

nduced by compressional acoustic waves ( Rayleigh, 1902 , 1905 ),

hile Brillouin was apparently the first to point out the second-

ank tensor nature of radiation pressure ( Beyer, 1978; Brillouin,

925 ). Subsequently, acoustic radiation force has been extensively

tudied for calculating the acoustic radiation pressure on spheres

n the pathway of wave propagation ( Doinikov, 1994; Hasegawa

nd Yosioka, 1969; King, 1934; Yosioka and Kawasima, 1955 ), for

eveloping acoustical trapping and tweezers ( Caleap and Drinkwa-

er, 2014; Evander and Nilsson, 2012; Hu et al., 2007; Marx, 2015;

hi et al., 2009; Silva and Baggio, 2015 ), for achieving acoustic lev-

tation and contactless handling of matter ( Brandt, 2001; Foresti

t al., 2013; Foresti and Poulikakos, 2014; Xie et al., 2002 ), for

eforming fluid interface and biological tissue ( Issenmann et al.,

008; Mishra et al., 2014; Walker, 1999 ), for disrupting cross-linked

ydrogels for drug release ( Huebsch et al., 2014 ), and so on. These

72 F. Xin, T.J. Lu / Mechanics of Materials 107 (2017) 71–80

〈

w

m

i

m

c

t

i

e

fi

t

t

f

fi

u

n

d

m

r

s

∂

u

s

t

w

a

t

h

d

5

t

c

L

l

i

b

t

c

q

σ

w

f

m

o

d

t

s

p

u

t

w

t

s

w

s

investigations all take advantage of the large magnitude and non-

contact merits of acoustic radiation force.

Although the concept of acoustic radiation force has been ex-

plored in a variety of practical applications, at present there lacks

a comprehensive theoretical model to describe the large nonlinear

deformation of soft materials induced by a combination of acoustic

radiation forces and mechanical forces in the framework of con-

tinuum mechanics. The aim of this study is to formulate such an

acoustomechanical theory. To capture the main idea, we consider a

thin layer of nearly incompressible soft material subjected to com-

bined mechanical forces and acoustic inputs. Two opposing acous-

tic waves having identical frequency and amplitude are consid-

ered, so that the layer deforms but remains unmoved. The acous-

tic radiation force is calculated by determining the acoustic fields

in and out of the thin material layer (surrounded by fluid), with

the nonlinear elasticity of the material accounted for by adopt-

ing the Helmholtz free energy function. For illustration, the pro-

posed acoustomechanical theory is employed to analyze three spe-

cific cases (equal-biaxial force, uniaxial force, and uniaxial con-

straint), with special focus placed upon the effects of prestretches

caused by different boundary conditions on the acoustomechanical

response of the soft material.

2. Acoustomechanical theory of soft materials

Wave propagation in medium is essentially the transport of

energy density flux or momentum density flux, which gives rise

to radiation stresses in the medium when the propagation oc-

curs at nonlinear level or encounters discontinuous interfaces. For

instance, the forces induced by optical wave propagation are re-

ferred to as Maxwell electromagnetic radiation stresses and those

induced by acoustic wave propagation are called acoustic radiation

stresses. In particular, ultrasound wave propagating in soft materi-

als with a frequency as high as 1 MHz is able to generate a steady

time-averaged radiation force and hence cause material deforma-

tion, since the period of ultrasound wave is too short for material

response. This is the case considered in the present study.

Consider a homogenous isotropic soft material, whose bulk

modulus K = E/ 3( 1 − 2 v ) is generally much larger than its shear

modulus G = E/ 2( 1 + v ) so that the material is nearly incompress-

ible. As a result, the wave field in the soft material induced by nor-

mally incident acoustic waves is dominated by longitudinal waves.

In other words, as a good approximation, the nearly incompress-

ible soft material may be taken as fluid-like for wave propagation.

Correspondingly, acoustic radiation stress in the medium can be

derived by employing the second-order perturbation theory of the

Navier–Stokes equation, as ( Borgnis, 1953; Lee and Wang, 1993;

Livett et al., 1981; Xin and Lu, 2016b,c,d,e ):

〈 T 〉 =

[

ρa

2 c 2 a

⟨ (∂φ

∂t

)2 ⟩

−ρa

⟨( ∇φ)

2 ⟩

2

]

I + ρa 〈 ∇φ � ∇φ〉 (1)

where 〈 T 〉 is the second-rank acoustic radiation pressure (stress)

tensor, representing compression when positive and tension when

negative, 〈 · 〉 denotes time-average over an oscillation cycle, I is

the identity tensor, φ is the velocity potential, ρa is the medium

density, and c a = ( ∂ p / ∂ρ) s is the acoustic speed in the medium.

Wave propagation in the medium is governed by the Helmholtz

equation �φ − 1

c 2 a

∂ 2 φ∂ t 2

= 0 , which can be solved by combining the

corresponding boundary conditions in the Eulerian coordinates.

Once the wave field is determined, the acoustic radiation stresses

in and out of the material are calculated as:

⟨T

out ⟩=

[

ρ1

2 c 2 1

⟨ (∂ φ1

∂t

)2 ⟩

−ρ1

⟨( ∇ φ1 )

2 ⟩

2

]

I + ρ1 〈 ∇ φ1 � ∇ φ1 〉

(2)

T

in 〉 =

[

ρ2

2 c 2 2

⟨ (∂ φ2

∂t

)2 ⟩

− ρ2 〈 ( ∇ φ2 ) 2 〉

2

]

I + ρ2 〈 ∇ φ2 � ∇ φ2 〉

(3)

here the subscripts “1,2” represent the outside and inside

edium, respectively.

The radiation stress falls into the category of field concept as it

s induced by acoustic field, which can be treated as part of the

aterial law in Eulerian coordinates. To formulate the acoustome-

hanical theory, consider next the nonlinear elasticity of soft ma-

erials. To this end, the continuum material at a particular time

s assigned to the reference configuration. One can thence mark

ach material particle using its coordinate X in the reference con-

guration and mark each spatial point using its coordinate x in

he current configuration. Subsequently, the reference configura-

ion X is mapped to the current configuration x using the de-

ormation gradient F = ∂x / ∂X . The Cauchy stress is related to the

rst Piola-Kirchhoff stress via σ = s · F T / det (F ) . Let dV ( x ) be a vol-

me element with mass density ρ( x ) and body force f b ( x , t ). Let

( x ) dA ( x ) be a surface element with surface force f s ( x , t ), where

A ( x ) is the area of the element and n ( x ) is the unit vector nor-

al to the interface between two media (e.g., a thin soft mate-

ial layer and the surrounding fluid; Fig. 1 ) pointing outside the

oft material. Force balance of the volume element is described by

σ/ ∂x + f b = ρ∂ 2 u / ∂ t 2 , with force boundary condition σ · n = f s ,

( x , t ) being the displacement field.

To be specific, with reference to Fig. 1 , consider a thin layer of

oft material subjected to combined mechanical force and acous-

ic inputs. It is assumed that the two opposing incident acoustic

aves are time-harmonic plane waves having identical amplitude

nd frequency, which propagate along the thickness direction of

he layer. It is further assumed that the thin soft material layer

as a thickness comparable to acoustic wavelength and its in-plane

imensions are much larger. For instance, a layer of dimensions

mm × 100 mm × 100 mm satisfies this condition if an incident ul-

rasound wave traveling with a frequency on the order of MHz is of

oncern. The thin layer with initial undeformed dimensions ( L 1 , L 2 ,

3 ) in the reference configuration is deformed to dimensions ( l 1 , l 2 ,

3 ) in the current configuration. For the problem shown schemat-

cally in Fig. 1 , the acoustomechanical theory can be formulated

y incorporating the acoustic stress field into the nonlinear elas-

ic stress field, so that the soft material behaves as an acoustome-

hanical responsive material to external mechanical forces. Conse-

uently, its constitutive law can be rewritten as:

i j = F iK ∂W (F )

∂ F jK − 〈 T i j 〉 − p h δi j (4)

here W ( F ) is the Helmholtz free energy, which is a symmetric

unction of the principal stretches ( λ1 , λ2 , λ3 ) for an isotropic soft

aterial, and p h is the Lagrange multiplier to ensure the constraint

f nearly incompressibility ( λ1 λ2 λ3 ≈ 1), which pertains to the hy-

rostatic pressure. Note that the acoustic radiation stress is put on

he right side of Eq. (4) , implying that the acoustic input is con-

idered as an insider, i.e., the acoustic radiation stress behaves as

art of the material law since it is a field force. When a focused

ltrasound wave with high intensity is incident upon the soft ma-

erial, the induced acoustic radiation stress 〈 T ij 〉 can be comparable

ith mechanical stress, thus enabling large nonlinear deformation

o develop in the material.

In the Cartesian coordinates of Fig. 1 , the acoustic radiation

tress is a diagonal stress tensor for normally incident acoustic

aves and the principal stretches coincide with the coordinates,

o that the constitutive relation of the acoustomechanical soft

F. Xin, T.J. Lu / Mechanics of Materials 107 (2017) 71–80 73

Fig. 1. Schematic illustration of a layer of soft material subjected to combined mechanical force f and two counterpropagating acoustic inputs p = p 0 e jωt : (a) the layer has

undeformed dimensions ( L 1 , L 2 , L 3 ) in the reference configuration; (b) the layer is deformed to dimensions ( l 1 , l 2 , l 3 ) in the current configuration.

m

σ

σ

T

a

H

s

W

w

T

p

l

σ

σ

w

r

t

t

F

t

t

t

o

〈

d

c

σ

S

E

d

s

i

r

a

e

c

fi

t

t

e

d

s

b

3

t

a

t

f

m

a

a

s

e

n

o

s

l

F

a

o

a

l

e

s

t

t

p

aterial can be rewritten as:

1 − σ3 = λ1 ∂W ( λ1 , λ2 )

∂ λ1

− ( 〈 T 11 〉 − 〈 T 33 〉 ) (5)

2 − σ3 = λ2 ∂W ( λ1 , λ2 )

∂ λ2

− ( 〈 T 22 〉 − 〈 T 33 〉 ) (6)

he nonlinear elasticity of soft materials can be described by

dopting the Gent model ( Gent, 1996 ). In the Gent model, the

elmholtz free energy is expressed as a function of the principal

tretches:

( F ) = −μJ m

2

ln

(1 − λ2

1 + λ2 2 + λ2

3 − 3

J m

)(7)

here μ is the initial shear modulus and J m

is the extension limit.

he Gent model degrades to the neo-Hookean model when J m

ap-

roaches infinity. Inserting the Gent model into the constitutive re-

ation, one obtains:

1 + ( t 1 − t 3 ) =

μ(λ2

1 − λ2 3

)1 −

(λ2

1 + λ2

2 + λ2

3 − 3

)/ J m

(8)

2 + ( t 2 − t 3 ) =

μ(λ2

2 − λ2 3

)1 −

(λ2

1 + λ2

2 + λ2

3 − 3

)/ J m

(9)

here the homogenized acoustic stresses along the thickness di-

ection are given by:

1 =

1

l 3

∫ l 3

0 〈 T 11 ( z ) 〉 dz, t 2 =

1

l 3

∫ l 3

0 〈 T 22 ( z ) 〉 dz,

3 =

⟨T in 33 ( l 3 )

⟩−

⟨T out

33 ( l 3 ) ⟩

(10)

or a thin material layer, this spatial averaging can simplify the

heoretical calculation without significant loss of the nature and

he accuracy of the problem. Due to material mismatch at the in-

erface, the acoustic radiation stress jumps discontinuously from

ne side 〈 T in 33

( l 3 ) 〉 to the other side 〈 T out 33

( l 3 ) 〉 , the difference t 3 = T in

33 ( l 3 ) 〉 − 〈 T out

33 ( l 3 ) 〉 being the resultant stress that causes material

eformation.

Cauchy stresses induced by external mechanical forces f 1 and f 2 an be expressed as:

1 =

f 1 l 2 l 3

=

1

λ2 λ3

f 1 L 2 L 3

, σ2 =

f 2 l 1 l 3

=

1

λ1 λ3

f 2 L 1 L 3

(11)

ubstitution of (11) into (8) and (9) yields:

1

λ2 λ3

f 1 L 2 L 3

+ ( t 1 − t 3 ) =

μ(λ2

1 − λ2 3

)1 −

(λ2

1 + λ2

2 + λ2

3 − 3

)/ J m

(12)

1

λ1 λ3

f 2 L 1 L 3

+ ( t 2 − t 3 ) =

μ(λ2

2 − λ2 3

)1 −

(λ2

1 + λ2

2 + λ2

3 − 3

)/ J m

(13)

qs. (12) and (13) can be used to characterize the nonlinear large

eformation of soft materials actuated by acoustic wave at pre-

cribed force/displacement boundary conditions. The acoustic field

s solved in Eulerian coordinates, which highly depends on mate-

ial deformation. Specifically, at given material configuration and

coustic inputs, both the acoustic field and the acoustic stress gen-

rated by the acoustic field are calculated. This acoustic stress

auses material deformation. In turn, the deformed material con-

guration reshapes the acoustic field and acoustic stress distribu-

ion. Such acoustomechanical coupling reaches a steady state until

he acoustic stress and mechanical stress together balance with the

lastic deformation stress.

In the sections to follow, the effect of force/displacement con-

ition on the steady-state acoustical actuation response of a thin

oft material layer immersed in fluid is quantified, including equal-

iaxial force, uniaxial force and uniaxial constraint.

. Model validation

To verify the proposed acoustomechanical theory for soft ma-

erials, a numerical model is developed by using the commercially

vailable FE (finite-element) software COMSOL Multiphysics. Since

his problem includes acoustic wave propagation and material de-

ormation, the Pressure Acoustics module and the Solid Mechanics

odule are adopted to establish the numerical model. The bound-

ry conditions of wave propagation and mechanical deformation

re fully taken into consideration. The length of each element is

elected to be one-twenty of the acoustic wavelength at the high-

st frequency (10 MHz) of interest to ensure the accuracy of the

umerical model.

The comparison between the proposed acoustomechanical the-

ry and the developed FE model for the acoustomechanical re-

ponse and the equivalent acoustic stresses of/on the soft material

ayer are presented in Fig. 2 (a) and (b), respectively. As shown in

ig. 2 (b), the theory agrees well with the FE model for the equiv-

lent acoustic stresses. This is because at the given configuration

f the material layer (i.e., the given thickness of the layer l 3 ), the

coustic fields and acoustic radiation stresses can be readily calcu-

ated without overly depending on the material deformation. How-

ver, the FE calculation of the acoustomechanical response of the

oft material layer is relatively difficult. In the FE model, the ma-

erial deformation and acoustic fields are strongly coupled, and

hus this boundary-value problem should be dealt with by ap-

lying an incremental iterative scheme. We increased the acous-

74 F. Xin, T.J. Lu / Mechanics of Materials 107 (2017) 71–80

Fig. 2. (a) Comparison between the proposed theory and the numerical model. (a) the acoustomechanical response of soft material under two counterpropagating acoustic

inputs; (b) the corresponding equivalent acoustic stress (solid line: theoretical results; dot: numerical results). The initial thickness of the layer is considered to be L 3 = ,

here is the acoustic wavelength in the material.

Fig. 3. A thin layer of soft material subjected to equal-biaxial force f and acoustic

input p = p 0 e jωt .

t

t

λ

σ

〈

t

g

w

e

n

s

C

s

t

W

i

l

F

o

t

l

o

a

e

W

m

v

a

s

tic inputs p 2 0 / ( μρ0 c 2 0 ) step by step, and for each increment, the

FE model correspondingly calculated the acoustic radiation stress

and material deformation, and then updated the acoustic fields.

Repeating the above procedures, we can reach the series of steady

states at which the generated acoustic radiation stress balances

the deformation stress. Fig. 2 (a) just plots these series of steady

states, namely, the given acoustic inputs p 2 0 / ( μρ0 c 2 0 ) are related

to the corresponding material deformation λ3 = l 3 / L 3 . The pro-

posed acoustomechanical theory is able to give excellent predic-

tions, while the FE model can only reproduce parts of the acous-

tomechanical response due to the convergence problem of numer-

ical calculation. Of course, the FE model can be further refined if

one can take more efforts to overcome the convergence problem.

4. Acoustical actuation under equal-biaxial force

Consider the acoustical actuation of a thin soft material layer

subjected to equal-biaxial mechanical force, as shown in Fig. 3 .

The layer has undeformed dimensions ( L 1 , L 2 , L 3 ) in the ref-

erence configuration. Upon exerting the equal-biaxial mechani-

cal force f , the layer is deformed to dimensions ( λpre L 1 , λpre L 2 ,

( λpre ) −2 L 3 ) in the prestretch configuration. Further, when acoustic

inputs p = p 0 e jωt are applied through the layer thickness, the two

counterpropagating acoustic waves generate a symmetric acous-

ic field as well as symmetric distribution of acoustic stress along

he thickness direction. Hence, one has λ1 = λ2 = λ and λ3 =−2 due to material deformation, with mechanical stresses σ1 =2 = λ f / ( L 2 L 3 ) and acoustic stresses t 1 =

1 l 3

∫ l 3 0

〈 T 11 (z) 〉 dz and t 3 = T in

33 ( l 3 ) 〉 − 〈 T out

33 ( l 3 ) 〉 generated in the current actuated configura-

ion. Accordingly, the constitutive equations of (12) and (13) de-

rade to

λ f

L 2 L 3 + ( t 1 − t 3 ) =

μ( λ2 − λ−4 )

1 − ( 2 λ2 + λ−4 − 3 ) / J m

(14)

hich describes the stretch λ of the soft material layer caused by

qual-biaxial force f and acoustic inputs p = p 0 e jωt . At prescribed

ormalized mechanical force f /( μL 2 L 3 ), the normalized acoustic

tress is a function of material stretch λ, as

p 2 0

μρ0 c 2 0

=

1

( t 1 − t 3 )

p 2 0

ρ0 c 2 0

[

μ(λ2 − λ−4

)1 −

(2 λ2 + λ−4 − 3

)/ J m

− λ

μ

f

L 2 L 3

]

(15)

orrespondingly, for this fixed mechanical force f /( μL 2 L 3 ), the pre-

tretch λpre of the soft material layer can be obtained by solving

he following equation

f

μL 2 L 3 =

λpre − ( λpre ) −5

1 −[2 ( λpre )

2 + ( λpre ) −4 − 3

]/ J m

(16)

ith the mechanical force f /( μL 2 L 3 ) fixed, as the acoustic stress

s increased, the material layer deforms to a succession of equi-

ibrium states, as shown by the stretch-stress relation curve of

ig. 4 (a). The prestretch λpre induced by the mechanical force is

btained from the initially acoustic load point, namely, the interac-

ion between the stretch-stress curve and the zero acoustic stress

ine. At certain prestretch and acoustic stress, there exist at least

ne or multiple equilibrium states, demonstrating the nonlinear

coustomechanical behavior of the soft material layer. For differ-

nt prestretches, the stretch-stress curves exhibit different trends.

hen the prestretch is small, the curve varies significantly with

ultiple peaks and dips. When the prestretch is large, the curve

aries with fewer peaks and dips.

The trend of the curve can be interpreted by the variation of

coustic stress with stretch. As shown in Fig. 4 (b), as the in-plane

tretch increases, the initially intensive fluctuation of the acoustic

F. Xin, T.J. Lu / Mechanics of Materials 107 (2017) 71–80 75

Fig. 4. (a) Acoustomechanical response of soft material under equal-biaxial mechanical force; (b) the corresponding equivalent acoustic stress. The initial thickness of the

layer is considered to be L 3 / = 10 , here is the acoustic wavelength in the material.

Fig. 5. A layer of soft material subjected to uniaxial force f 2 and acoustic inputs

p = p 0 e jωt .

s

t

a

t

s

a

t

m

g

5

c

F

m

f

a

d

p

i

t

l

c

D

t

f

t

I

a(

I

s

U

g

i

F

t

F

c

m

c

a

i

f

s

s

t

s

t

tress becomes less intensive, which causes the variation trend of

he stretch-stress curve in Fig. 4 (a). When t 1 − t 3 approaches zero,

peak appears in the stretch-stress curve, while a dip occurs when

1 − t 3 reaches a local maximum. The fluctuation of the stretch-

tress curve implies multiple snap-through instabilities: when the

coustic stress reaches local maximum, any further ramping up of

he stress will induce a discontinuous jumping of material defor-

ation to a much larger deformation. This phenomenon enables

iant acoustical actuation at relatively small acoustic inputs.

. Acoustical actuation under uniaxial force

Consider a soft material layer subjected to a uniaxial mechani-

al force and two counterpropagating acoustic waves, as shown in

ig. 5 . In the reference configuration, the layer has undeformed di-

ensions ( L 1 , L 2 , L 3 ). In the prestretch configuration, the layer de-

orms to dimensions ( ( λpre ) −1 / 2 L 1 , λpre L 2 , ( λ

pre ) −1 / 2 L 3 ) when uni-

xial mechanical force f 2 is applied. When acoustic inputs p =p 0 e

jωt are applied through layer thickness, the layer deforms to

imensions ( λ1 L 1 , λ2 L 2 , λ3 L 3 ) in the actuation configuration. The

rincipal stretches are ( λ1 , λ2 , λ3 = ( λ1 λ2 ) −1 ) while the mechan-

cal stresses are σ1 = 0 and σ2 = λ2 f / ( L 1 L 3 ) . The acoustic stresses

1 =

1 l 3

∫ l 3 0

〈 T 11 (z) 〉 dz and t 3 = 〈 T in 33

( l 3 ) 〉 − 〈 T out 33

( l 3 ) 〉 can be calcu-

ated by employing the acoustic fields in the current (actuated)

onfiguration. Eqs. (12) and ( 13 ) thence degrade to:

( t 1 − t 3 ) =

μ(λ2

1 − λ2 3

)1 −

(λ2

1 + λ2

2 + λ2

3 − 3

)/ J m

(17)

1

λ1 λ3

f 2 L 1 L 3

+ ( t 2 − t 3 ) =

μ(λ2

2 − λ2 3

)1 −

(λ2

1 + λ2

2 + λ2

3 − 3

)/ J m

(18)

eformation of the thin soft layer at equilibrium states can be ob-

ained by solving the above two equations at given mechanical

orce and acoustic inputs.

When the layer is only subjected to uniaxial mechanical force,

he prestretch λpre 2

= λpre ( λpre 1

= λpre 3

= ( λpre ) −1 / 2 ) is obtained by:

f 2 μL 1 L 3

=

λpre − ( λpre ) −2

1 −[2 ( λpre )

−1 + ( λpre ) 2 − 3

]/ J m

(19)

n the current actuated configuration, the principal stretches λ1

nd λ2 are related to each other by:

1 − λ2

J m

f 2 μL 1 L 3

)λ4

1 +

[λ2

J m

f 2 μL 1 L 3

(J m

− λ2 2 + 3

)− λ2

2

]λ2

1

− λ2

J m

f 2 μL 1 L 3

λ−2 2 = 0 (20)

nserting Eq. (20) into Eq. (17) , one obtains the normalized acoustic

tress as:

p 2 0

μρ0 c 2 0

=

1

( t 1 − t 3 )

p 2 0

ρ0 c 2 0

(λ2

1 − λ2 3

)1 −

(λ2

1 + λ2

2 + λ2

3 − 3

)/ J m

(21)

nder fixed uniaxial mechanical force f 2 /( μL 1 L 3 ), the layer under-

oes a succession of equilibrium states when the acoustic stress

ncreases, as illustrated by the stretch-stress relation curve in

ig. 6 . The prestretch is obtained directly from the interaction be-

ween the stretch-stress curve and the zero acoustic stress line.

or any given mechanical force and acoustic stress, the acoustome-

hanical response exhibits nonlinear behavior as at least one or

ultiple equilibrium states exist. Similar to the equal-biaxial force

ase, the fluctuation trend can be explained by the variation of

coustic stress as a function of in-plane stretch. Different uniax-

al prestretches lead to different acoustically triggered material de-

ormations. Therefore, the corresponding acoustic stresses are pre-

ented in Fig. 7 for selected uniaxial prestretches. When the pre-

tretch is small, the fluctuation of the acoustic stress is firstly in-

ensive and then less intensive ( Fig. 7 (a) and (b)). When the pre-

tretch is large, the fluctuation changes from firstly less intensive

o intensive and eventually to less intensive ( Fig. 7 (c) and (d)).

76 F. Xin, T.J. Lu / Mechanics of Materials 107 (2017) 71–80

Fig. 6. Acoustomechanical response of a thin soft material layer under uniaxial me-

chanical force, with its initial layer thickness L 3 / = 10 , being acoustic wave-

length in the material.

Fig. 8. A thin layer of soft material constrained in x -direction and subjected to me-

chanical force f 2 and acoustic inputs p = p 0 e jωt .

l

w

6

j

c

i

Such variation of the acoustic stress affects the trend of stretch-

stress curve ( Fig. 6 ), reaching maximum when t 1 − t 3 approaches

zero and minimum when t 1 − t 3 approaches maximum. Such non-

Fig. 7. Equivalent acoustic stress of a thin soft material layer under uniaxial force at select

(b) λpre 2

= 2 . 07 ; (c) λpre 2

= 3 . 03 ; (d) λpre 2

= 4 . 0 .

inearity in acoustomechanical response of soft material offers a

ide space to design novel acoustic actuators.

. Acoustical actuation under uniaxial constraint

With reference to Fig. 8 , consider a thin soft material layer sub-

ected to uniaxial constraint in the x -direction and uniaxial me-

hanical force f 2 in the y -direction. Two opposing acoustic waves

mpinge upon the layer along its thickness direction. In the ref-

ed values of prestrecth λpre 2

, with its initial layer thickness L 3 / = 10 : (a) λpre 2

= 1 . 1 ;

F. Xin, T.J. Lu / Mechanics of Materials 107 (2017) 71–80 77

Fig. 9. Acoustomechanical response of a thin soft material layer under uniaxial constraint condition at selected prestretches: (a) λpre 1

= 0 . 8 ; (b) λpre 1

= 1 . 0 ; (c) λpre 1

= 2 . 0 ; (d)

λpre 1

= 3 . 0 . The layer has an initial thickness of L 3 / = 10 , being acoustic wavelength in the material.

e

L

t

x

s

r

s

t

σ

t

t

b

σ

I

c

t

t

t

b

S

s

T

u

f

F

s

r

t

s

t

rence configuration, the layer has undeformed dimensions ( L 1 , L 2 ,

3 ). In the prestretch configuration, the layer is uniaxially stretched

o have a stretch λpre 1

and then fixed onto a rigid substrate in the

-direction. Thereafter, the layer with prestretch λpre 2

is uniaxially

tretched by mechanical force f 2 in the y -direction. In the cur-

ent actuated configuration, the layer is deformed to have principal

tretches ( λ1 , λ2 , λ3 ) when acoustic inputs p = p 0 e jωt are applied

hrough its thickness. Clearly, the mechanical stresses σ 1 � = 0 and

2 = λ2 f 2 / ( L 1 L 3 ) . The acoustic stresses t 1 =

1 l 3

∫ l 3 0

〈 T 11 (z) 〉 dz and

3 = 〈 T in 33

( l 3 ) 〉 − 〈 T out 33

( l 3 ) 〉 are calculated using the acoustic fields in

he current actuated configuration. Therefore, Eqs. (12) and ( 13 )

ecome

1 + ( t 1 − t 3 ) =

μ(λ2

1 − λ2 3

)1 −

(λ2

1 + λ2

2 + λ2

3 − 3

)/ J m

(22)

λ2 f 2 L 1 L 3

+ ( t 2 − t 3 ) =

μ(λ2

2 − λ2 3

)1 −

(λ2

1 + λ2

2 + λ2

3 − 3

)/ J m

(23)

n the process of acoustically actuated deformation, given the me-

hanical force λ2 f 2 /( L 1 L 3 ) and acoustic inputs, the stretch λ2 of the

hin layer is determined by ( 23 ) and the constraint stress σ 1 is de-

ermined by ( 22 ).

Given the prestretch λpre 1

and the mechanical force f 2 /( μL 1 L 3 ),

he layer endures another prestretch λpre 2

that can be calculated

y:

f 2 μL 1 L 3

=

λpre 2

−(λpre

1

)−2 (λpre

2

)−3

1 −[ (

λpre 1

)2 +

(λpre

2

)2 +

(λpre

1 λpre

2

)−2 − 3

] / J m

(24)

ubstitution of Eq. (24) into ( 23 ) leads to the normalized acoustic

tress:

p 2 0

μρ0 c 2 0

=

1

( t 2 − t 3 )

p 2 0

ρ0 c 2 0

[ (λ2

2 − λ2 3

)1 −

(λ2

1 + λ2

2 + λ2

3 − 3

)/ J m

− λ2

μ

f 2 L 1 L 3

]

(25)

o investigate the acoustomechanical response of the thin layer

nder uniaxial constraint and uniaxial mechanical force condition,

our prestretches ( λpre 1

= 0.8, 1, 2 and 3) are selected as shown in

ig. 9 . As the acoustic stress increases, a succession of equilibrium

tates are achieved in the layer, as evidenced by the stretch-stress

elation curves of Fig. 9 . The value of λpre 2

is read from the in-

eraction between the stretch-stress curve and the zero acoustic

tress line. The fluctuation of the stretch-stress curve demonstrates

he nonlinear acoustomechanical response of the layer, which is

78 F. Xin, T.J. Lu / Mechanics of Materials 107 (2017) 71–80

Fig. 10. Equivalent acoustic stress of a thin soft material layer ( L 3 / = 10 ) under uniaxial constraint condition at selected prestretches: (a) λpre 1

= 0 . 8 ; (b) λpre 1

= 1 . 0 ; (c)

λpre 1

= 2 . 0 ; (d) λpre 1

= 3 . 0 .

fi

a

T

i

t

o

s

a

s

a

p

s

S

g

s

a

A

d

f

directly related to the variation of acoustic stress with increas-

ing in-plane stretch ( Fig. 10 ). The fluctuation of the acoustic stress

is firstly less intensive and then intensive, resulting in the vari-

ation trend of stretch-stress curve in Fig. 9 . The intensive fluc-

tuation when λpre 2

is small becomes less intensive when λpre 2

be-

comes large. To maintain the deformation, the required acoustic

stress reaches maximum when t 1 − t 3 approaches zero and mini-

mum when t 1 − t 3 approaches its maximum. The significant effect

of prestretch can be used to enhance acoustically actuated material

deformation.

7. Concluding remarks

We develop an acoustomechanical theory for soft materials sub-

jected to simultaneous mechanical forces and acoustic radiation

forces by combining the nonlinear elasticity theory of soft mate-

rials with the acoustic radiation stress theory. For illustration, two

counterpropagating ultrasonic waves impinging onto the surfaces

of a thin soft material layer immersed in fluid is considered. The

resulting acoustic field is capable of giving rise to acoustic radia-

tion stress in and out of the thin layer, particularly at the interface

between the soft material and the surrounding medium. While the

acoustic radiation stress deforms the material, the deformed con-

guration of the soft material in turn is able to reconstruct the

coustic field and the distribution of the acoustic radiation stress.

he material reaches a steady deformation state when the mechan-

cal stress and acoustic stress balance with the material deforma-

ion stress.

The developed theory is employed to investigate the sequence

f equilibrium state and the acoustomechanical response of thin

oft material layer under three different force/displacement bound-

ry conditions: equal-biaxial force, uniaxial force and uniaxial con-

traint. The results demonstrate the significant influence of bound-

ry condition on acoustomechanical response. Further, as wave

ropagation is sensitive to material configuration, prestretching the

oft material affects remarkably the acoustomechanical behavior.

pecifically, when the thin soft material layer is acoustically trig-

ered from different prestretches, it shows different stretch ver-

us stress relations, useful for designing novel acoustic sensors and

ctuators.

cknowledgements

This work was supported by the National Natural Science Foun-

ation of China ( 51528501 ) and the Fundamental Research Funds

or Central Universities ( 2014qngz12 ).

F. Xin, T.J. Lu / Mechanics of Materials 107 (2017) 71–80 79

A

r

a

B

t

i

i

l

H

i

a

t

fi

g

d

l

u

w

t

m

b

b

t

1 k 2 z

ρ2 2

k 21

u

k 1 z c2 2

k 2 1 z

2 1 k

2 2 z(

ρ2 1

k

u2 1 k

2 2 z

(ρ2

1 k

w

r

v

s

t

a

a

⟨

⟨

T

t

t

w

t

τ

τ

ς

ς

R

B

B

B

BB

C

C

D

D

E

F

ppendix. Acoustic fields of two counterpropagating waves

Although a real acoustic wave propagates through a soft mate-

ial at nonlinear level, the calculation of the induced acoustic radi-

tion stress only needs a linear level analysis for the acoustic fields.

ecause the acoustic radiation stress scales as the second-order of

he acoustic fields, it can be expressed using the first-order terms,

.e., the linear acoustic pressure and the linear velocity. Therefore,

n this appendix, acoustic wave propagation at linear level is ana-

yzed by rigorously considering boundary conditions.

Acoustic wave propagation in medium is governed by the

elmholtz equation �φ − 1

c 2 a

∂ 2 φ∂ t 2

= 0 , φ being the acoustic veloc-

ty potential and c a = ( ∂ p / ∂ρ) s the acoustic speed. Generally, the

coustic impedance of the soft material mismatches with that of

he outside surrounding medium so that ρ1 c 1 � = ρ2 c 2 . The acoustic

eld in each medium consists of both positive-going and negative-

oing waves. As illustrated in Fig. 1 (b), the acoustic boundary con-

itions need to ensure the continuity of acoustic pressure and ve-

ocity at the interfaces:

p 1 = p 2 | z=0 , p 2 = p 3 | z= l 3 , (A1)

1 z = u 2 z | z=0 , u 2 z = u 3 z | z= l 3 , (A2)

here ( p 1 , p 2 , p 3 ) are the acoustic pressures and ( u 1 z , u 2 z , u 3 z ) are

he components of the velocity along the z -direction in the left

edium, soft material and the right medium, respectively.

Obeying the governing equation of wave propagation and the

oundary conditions of ( A1 ) and ( A2 ), the acoustic fields generated

y two counterpropagating acoustic waves are the superposition of

hese two opposing fields, as:

p in =

2 ωI ρ1 ρ2 k 1 z e jωt [ ρ2 k 1 z sin ( k 2 z ( z − l 3 ) ) − ρ2 k 1 z sin ( k 2 z z ) + j ρ

2 ρ1 ρ2 k 1 z k 2 z cos ( k 2 z l 3 ) + j (ρ2

1 k 2

2 z +

in =

2 I ρ1 k 1 z k 2 z e jωt [ ρ1 k 2 z sin ( k 2 z ( z − l 3 ) ) + ρ1 k 2 z sin ( k 2 z z ) + j ρ2

2 ρ1 ρ2 k 1 z k 2 z cos ( k 2 z l 3 ) + j (ρ2

1 k 2

2 z + ρ

p out R = j ω ρ3 I e

jωt

[

e j k 3 z ( z−l 3 ) + e − j k 3 z ( z−l 3 ) j sin ( k 2 z l 3 )

(ρ2

2 k 2 1 z − ρ

2 ρ1 ρ2 k 1 z k 2 z cos ( k 2 z l 3 ) + j

out R = j k 3 z I e

jωt

(

−e j k 3 z ( z−l 3 ) + e − j k 3 z ( z−l 3 ) j sin ( k 2 z l 3 )

(ρ2

2 k 2 1 z − ρ

2 ρ1 ρ2 k 1 z k 2 z cos ( k 2 z l 3 ) + j

here p and u are the acoustic pressure field and velocity field,

espectively. The superscripts “in/out” represent the corresponding

ariables related to the inside and outside media, respectively. The

ubscripts 1, 2 and 3 denote the left medium, the soft material and

he right medium, respectively. ω is the angular frequency, I is the

mplitude of incident velocity potential, ρ is the medium density,

nd k is the acoustic wavenumber.

Under such conditions, the acoustic radiation stresses are:

T in 11

⟩=

⟨T in 22

⟩=

τ2 ς

∗2 e

−2 j k 2 z z + τ ∗2 ς 2 e

2 j k 2 z z

2 ρ2 c 2 2

, ⟨T in 33

⟩=

τ2 τ ∗2 + ς 2 ς

∗2

2 ρ2 c 2 2

(A7)

T out 11

⟩=

⟨T out

22

⟩=

τ1 ς

∗1 e

−2 j k 2 z z + τ ∗1 ς 1 e

2 j k 2 z z

2 ρ1 c 2 1

, ⟨T out

33

⟩=

τ1 τ ∗1 + ς 1 ς

∗1

2 ρ1 c 2 1

,

(A8)

he equivalent stresses are:

cos ( k 2 z ( z − l 3 ) ) + j ρ1 k 2 z cos ( k 2 z z ) ]

z

)sin ( k 2 z l 3 )

(A3)

os ( k 2 z ( z − l 3 ) ) − j ρ2 k 1 z cos ( k 2 z z ) ] )sin ( k 2 z l 3 )

(A4)

)+ 2 ρ1 ρ2 k 1 z k 2 z

2 2 z

+ ρ2 2

k 2 1 z

)sin ( k 2 z l 3 )

]

(A5)

)+ 2 I ρ1 ρ2 k 1 z k 2 z

2 2 z

+ ρ2 2

k 2 1 z

)sin ( k 2 z l 3 )

)

(A6)

1 = t 2 =

1

2 ρ2 c 2 2

{

1

2 j k 2 z l 3 [ τ ∗

2 ς 2 ( e 2 j k 2 z l 3 − 1 ) − τ2 ς

∗2 ( e

−2 j k 2 z l 3 − 1 ) ] }

(A9)

3 =

τ2 τ∗2 + ς 2 ς

∗2

2 ρ2 c 2 2

− τ1 τ∗1 + ς 1 ς

∗1

2 ρ1 c 2 1

(A10)

here the superscript asterisk ∗ means the complex conjugate of

he corresponding variable, and we have

1 = jω ρ1 I e j k 1 z l 3

×(

j (ρ2

2 k 2 1 z − ρ2

1 k 2 2 z

)sin ( k 2 z l 3 ) + 2 ρ1 ρ2 k 1 z k 2 z

2 ρ1 ρ2 k 1 z k 2 z cos ( k 2 z l 3 ) + j (ρ2

1 k 2

2 z + ρ2

2 k 2

1 z

)sin ( k 2 z l 3 )

)

(A11)

2 = jω ρ1 ρ2 k 1 z I

×(

e j k 2 z l 3 ( ρ2 k 1 z + ρ1 k 2 z ) + ( ρ1 k 2 z − ρ2 k 1 z )

2 ρ1 ρ2 k 1 z k 2 z cos ( k 2 z l 3 ) + j (ρ2

1 k 2

2 z + ρ2

2 k 2

1 z

)sin ( k 2 z l 3 )

)

(A12)

1 = jω ρ1 I e − j k 1 z l 3 (A13)

2 = jω ρ1 ρ2 k 1 z I

×(

e − j k 2 z l 3 ( ρ1 k 2 z − ρ2 k 1 z ) + ( ρ2 k 1 z + ρ1 k 2 z )

2 ρ1 ρ2 k 1 z k 2 z cos ( k 2 z l 3 ) + j (ρ2

1 k 2

2 z + ρ2

2 k 2

1 z

)sin ( k 2 z l 3 )

)

(A14)

eferences

eyer, R.T. , 1978. Radiation pressure-the history of a mislabeled tensor. J. Acoust.

Soc. Am. 63, 1025–1030 . orgnis, F.E. , 1953. Acoustic radiation pressure of plane compressional waves. Rev.

Modern Phys. 25, 653–664 . randt, E.H. , 2001. Acoustic physics: suspended by sound. Nature 413, 474–475 .

rillouin, L. , 1925. Ann. Phys. (Paris) 4, 528 . ustamante, R. , Dorfmann, A. , Ogden, R. , 2006. Universal relations in isotropic non-

linear magnetoelasticity. Quart. J. Mech. Appl. Math. 59, 435–450 .

aleap, M. , Drinkwater, B.W. , 2014. Acoustically trapped colloidal crystals that arereconfigurable in real time. Proc. Natl. Acad. Sci. 111, 6226–6230 .

hester, S.A. , Anand, L. , 2011. A thermo-mechanically coupled theory for fluid per-meation in elastomeric materials: application to thermally responsive gels. J.

Mech. Phys. Solids. 59, 1978–2006 . oinikov, A .A . , 1994. Acoustic radiation pressure on a rigid sphere in a viscous fluid.

P. Roy. Soc. A 447, 447–466 . orfmann, A. , Ogden, R.W. , 2004. Nonlinear magnetoelastic deformations. Q. J.

Mech. Appl. Math. 57, 599–622 .

vander, M. , Nilsson, J. , 2012. Acoustofluidics 20: applications in acoustic trapping.Lab Chip 12, 4667–4676 .

oresti, D. , Nabavi, M. , Klingauf, M. , Ferrari, A. , Poulikakos, D. , 2013. Acoustophoreticcontactless transport and handling of matter in air. Proc. Natl. Acad. Sci. 110,

12549–12554 .

80 F. Xin, T.J. Lu / Mechanics of Materials 107 (2017) 71–80

L

S

S

S

S

T

X

X

X

X

X

Y

Z

Z

Foresti, D. , Poulikakos, D. , 2014. Acoustophoretic contactless elevation, orbital trans-port and spinning of matter in air. Phys. Rev. Lett. 112, 024301 .

Gent, A.N. , 1996. A new constitutive relation for rubber. Rubber Chem. Technol. 69,59–61 .

Guo, F. , Mao, Z. , Chen, Y. , Xie, Z. , Lata, J.P. , Li, P. , Ren, L. , Liu, J. , Yang, J. , Dao, M. ,Suresh, S. , Huang, T.J. , 2016. Three-dimensional manipulation of single cells us-

ing surface acoustic waves. Proc. Natl. Acad. Sci. 113, 1522–1527 . Hasegawa, T. , Yosioka, K. , 1969. Acoustic-radiation force on a solid elastic sphere. J.

Acoust. Soc. Am. 46, 1139–1143 .

Hu, J.H. , Ong, L.B. , Yeo, C.H. , Liu, Y.Y. , 2007. Trapping, transportation and separationof small particles by an acoustic needle. Sens. Actuat. A 138, 187–193 .

Huebsch, N. , Kearney, C.J. , Zhao, X. , Kim, J. , Cezar, C.A. , Suo, Z. , Mooney, D.J. , 2014.Ultrasound-triggered disruption and self-healing of reversibly cross-linked hy-

drogels for drug delivery and enhanced chemotherapy. Proc. Natl. Acad. Sci. 111,9762–9767 .

Issenmann, B. , Nicolas, A. , Wunenburger, R. , Manneville, S. , Delville, J.P. , 2008. De-

formation of acoustically transparent fluid interfaces by the acoustic radiationpressure. EPL 83, 34002 .

King, L.V. , 1934. On the acoustic radiation pressure on spheres. P. Roy. Soc. A 147,212–240 .

ee, C.P. , Wang, T.G. , 1993. Acoustic radiation pressure. J. Acoust. Soc. Am. 94,1099–1109 .

Livett, A.J. , Emery, E.W. , Leeman, S. , 1981. Acoustic radiation pressure. J. Sound Vib.

76, 1–11 . Marx, V. , 2015. Biophysics: using sound to move cells. Nat. Meth. 12, 41–44 .

Mishra, P. , Hill, M. , Glynne-Jones, P. , 2014. Deformation of red blood cells usingacoustic radiation forces. Biomicrofluidics 8, 034109 .

Ohmine, I. , Tanaka, T. , 1982. Salt effects on the phase transition of ionic gels. J.Chem. Phys. 77, 5725–5729 .

Rayleigh, L. , 1902. On the pressure of vibrations. Philo. Mag. 3, 338–346 .

Rayleigh, L. , 1905. On the momentum and pressure of gaseous vibrations, and onthe connexion with the virial theorem. Philo. Mag. Ser. 6 10, 364–374 .

Rudykh, S. , Bertoldi, K. , 2013. Stability of anisotropic magnetorheological elastomersin finite deformations: a micromechanical approach. J. Mech. Phys. Solids. 61,

949–967 .

hi, J. , Ahmed, D. , Mao, X. , Lin, S.-C.S. , Lawit, A. , Huang, T.J. , 2009. Acoustic tweez-ers: patterning cells and microparticles using standing surface acoustic waves

(SSAW). Lab Chip 9, 2890–2895 . ilva, G.T. , Baggio, A.L. , 2015. Designing single-beam multitrapping acoustical tweez-

ers. Ultrasonics 56, 449–455 . tark, K.H. , Garton, C.G. , 1955. Electric strength of irradiated polythene. Nature 176,

1225–1226 . uo, Z. , Zhao, X. , Greene, W.H. , 2008. A nonlinear field theory of deformable di-

electrics. J. Mech. Phys. Solids. 56, 467–486 .

anaka, T. , Fillmore, D. , Sun, S.T. , Nishio, I. , Swislow, G. , Shah, A. , 1980. Phase transi-tions in ionic gels. Phys. Rev. Lett. 45, 1636–1639 .

Walker, W.F. , 1999. Internal deformation of a uniform elastic solid by acoustic radi-ation force. J. Acoust. Soc. Am. 105, 2508–2518 .

Xie, W.J. , Cao, C.D. , Lü, Y.J. , Wei, B. , 2002. Levitation of iridium and liquid mercuryby ultrasound. Phys. Rev. Lett. 89, 104304 .

in, F. , Lu, T. , 2016a. Acoustomechanics of semicrystalline polymers. Theore. Appl.

Mech. Lett. 6, 38–41 . in, F.X. , Lu, T.J. , 2016b. Acoustomechanical constitutive theory of soft materials.

Acta Mech. Sin. 32, 828–840 . in, F.X. , Lu, T.J. , 2016c. Acoustomechanical giant deformation of soft elastomers

with interpenetrating networks. Smart Mater. Strtuct. 25, 07LT02 . in, F.X. , Lu, T.J. , 2016d. Generalized method to analyze acoustomechanical stability

of soft materials. J. Appl. Mech. 83, 071004 .

in, F.X. , Lu, T.J. , 2016e. Tensional acoustomechanical soft metamaterials. ScientificReports 6, 27432 .

eh, P.D. , Alexeev, A. , 2015. Mesoscale modelling of environmentally responsive hy-drogels: emerging applications. Chem. Commun. 51, 10083–10095 .

Yosioka, K. , Kawasima, Y. , 1955. Acoustic radiation pressure on a compressiblesphere. Acta. Acust. Acust. 5, 167–173 .

hang, Q.M. , Bharti, V. , Zhao, X. , 1998. Giant electrostriction and relaxor ferroelec-

tric behavior in electron-irradiated poly(vinylidene fluoride-trifluoroethylene)copolymer. Science 280, 2101–2104 .

Zhang, Q.M. , Li, H. , Poh, M. , Xia, F. , Cheng, Z.Y. , Xu, H. , Huang, C. , 2002. An all-or-ganic composite actuator material with a high dielectric constant. Nature 419,

284–287 . hao, X. , Wang, Q. , 2014. Harnessing large deformation and instabilities of soft di-

electrics: theory, experiment, and application. Appl. Phys. Rev. 1, 021304 .