Content uploaded by William S. Oates
Author content
All content in this area was uploaded by William S. Oates on Jul 11, 2014
Content may be subject to copyright.
Unusual Field-Coupled Nonlinear Continuum
Mechanics of Smart Materials
William S. Oates 1, Hongbo Wang 2, and Robert L. Sierakowski 3
Florida Center for Advanced Aero Propulsion (FCAAP)
Department of Mechanical Engineering
Florida A & M and Florida State University
Tallahassee, FL 32310
Asian Office of Aerospace Research & Development
Air Force Office of Scientific Research
Tokyo, Japan
Abstract
A set of anisotropic constitutive relations are developed using nonlinear continuum mechan-
ics to model chemical and field-coupled material behavior in a relatively broad range of smart
materials. A Landau-based free energy function is formulated and numerically implemented
to elucidate how finite deformation and crystal anisotropy affect field-coupled deformation and
how deformation affects microstructure evolution. Rotationally invariant order parameters are
introduced within the Landau energy function to illustrate how field-coupled mechanics occurs
without introducing explicit phenomenological parameters. Spontaneous deformation due to
scalar, vector, and tensor order parameters is quantified using this approach. It is shown that
both the scalar and vector order parameters induce hydrostatic deformation while the second
order tensor order parameter induces a range of different anisotropic spontaneous deformation
states. Numerical simulations are given which illustrate unusual field-coupled microstructure
evolution for a select number of active materials in comparison with data given in the liter-
ature. The materials simulated range from chemically responsive glassy polymer networks,
1Email: woates@eng.fsu.edu, Telephone: (850) 410-6190
2Email: hongbo@eng.fsu.edu, Telephone: (850) 410-6172
3Email: Robert.Sierakowski.ctr@usafa.edu, Telephone: (850) 410-6172
soft polydomain liquid crystal elastomers, and tetragonal phase ferroelectric materials. The
results illustrate that finite deformation continuum mechanics can be useful in modeling many
unusual field-coupled, anisotropic constitutive relations without introducing explicit coupling
parameters.
1 Introduction
Smart materials are well known for their intrinsic field coupled material characteristics
which in many cases can provide simultaneous actuation and sensing for adaptive structure
applications. This unique multifunctionality has been the impetus for studying field-coupled
mechanics for several decades. Arguably, the most well known types of smart materials include
ferroelectric ceramics (Lines and Glass, 1977; Jaffe et al., 1971), magnetostrictive compounds
(Bertotti, 1998; Kittel, 1949), and shape memory alloys (Boyd and Lagoudas, 1996; Huang
and Brinson, 1998). Active polymers have also begun to emerge as viable artificial muscles
(Bar-Cohen and Zhang, 2008). These materials include dielectric elastomers (Kofod and
Sommer-Larsen, 2005), shape memory polymers (Lendlein and Kelch, 2002), ionic polymers
(Nemat-Nasser, 2002), and a relatively broad class of nanocomposites (Baur and Silverman,
2007) and liquid crystal elastomers (Warner and Terentjev, 2007).
A broad range of constitutive modeling frameworks have been developed to predict the
material behavior governing active materials. For a review of continuum methods applied to
active materials such as ferroelectric materials, magnetostrictive compounds, and shape mem-
ory alloys, see (Smith, 2005; Bertotti, 1998; Boyd and Lagoudas, 1996; Huang and Brinson,
1998; Seelecke and M¨uller, 2004; Huang and Brinson, 1998; J.E.Huber et al., 1999; J.E.Huber
and N.A.Fleck, 2001; Hwang and Arlt, 2000; Armstrong, 2000; Kamlah et al., 2008) and many
others. Efforts have also focused on extracting underlying material relations via configura-
tional forces or multiscale mechanics to understand how local defects may influence micro to
mesoscale behavior (Hildebrand and Abeyaratne, 2008; Kowalewsky, 2004; Su and Landis,
2007). Model development specific to liquid crystal elastomers can be found in (Warner and
2
Terentjev, 2007; Bladon et al., 1993; Lubensky et al., 2002; Adams et al., 2007).
Unified theories have also been developed with particular emphasis on fundamental electro-
magneto-elastics of solids which includes extensive analysis of the fundamental balance laws
and generalized constitutive relations (Maugin, 2011; Ericksen, 2008; Vu and Steinmann, 2010;
Toupin, 1963; Nelson, 1991; Hutter and Van, 1978; Trimarco, 2009). However, specific free en-
ergy functions and the corresponding constitutive relations that correlate polarization, magne-
tization, liquid crystal order, phase and deformation with experiments still present significant
challenges. One example of a unified constitutive model for ferroic materials (i.e., ferroelec-
tric, ferromagnetic, shape memory alloy) has been developed which includes nonlinear and
hysteretic rate-dependent material behavior for small strain problems under uniaxial loading
(Smith et al., 2006; Smith, 2005). A combination of continuum thermodynamics and statis-
tical mechanics was used to develop a homogenized energy model to predict the field-coupled
constitutive behavior.
Finite deformation electro-mechanics and magneto-mechanics modeling has also been stud-
ied in soft elastomeric materials (Zhao et al., 2007; Suo et al., 2008; McMeeking and Landis,
2005; Bustamante et al., 2008, 2009). For dielectric elastomers, it was shown that a super-
position of a mechanical energy function and an idealized dielectric energy function leads to
coupling between deformation of a polymer network and electrostatic fields. A set of electro-
static stresses were obtained based on fundamental thermodynamics and nonlinear continuum
mechanics. Whereas this concept has been shown to provide good predictions of electrostatic
stresses in dielectric elastomers, limited work has been conducted to correlate anisotropic
microstructure evolution in a broader class of active materials using nonlinear continuum
mechanics.
The concept of a microforce has been explored to quantify underlying microstructure
evolution in solids with emphasis on quantifying solid-solid phase evolution (Fried and Gurtin,
1994; Gurtin, 1996). This approach involves modeling an order parameter and its gradient
in combination with linear momentum to correlate the constitutive behavior during phase
transformations and microstructure evolution. The minimization of the free energy of the
3
order parameter results in an Euler-Lagrange equation that has been used extensively in
modeling a wide range of meso to microscale effects in active materials (Cao and Cross,
1991; Virga, 1994; Kittel, 1949; Chen, 2002). Whereas finite deformation was included in
the microforce balance that was described in (Gurtin, 1996), limited model development has
been conducted to correlate energy functions and constitutive relations for field-coupled and
chemically-coupled materials.
In this analysis, nonlinear mechanics is coupled with microstructure evolution and numeri-
cally implemented to quantify coupling between different order parameters (e.g., polarization,
liquid crystal order, and water vapor concentration) based on finite deformation mechanics.
These different order parameters range from scalar, vector, and second order tensors which
are used to compare the modeling framework with experimental results given in the literature.
It is shown that the superposition of mechanical energy and a multi-well free energy that de-
scribes the underlying microstructure is sufficient to predict many field-coupled constitutive
relations without introducing explicit phenomenological coupling parameters. Moreover, the
methodology is used to illustrate that this coupling remains in the limit of infinitesimal strain
as described by a comparison to experiments on ferroelectric materials. This is done by fitting
the dielectric and elastic constants and predicting piezoelectric coupling in ferroelectric lead
zirconate titanate. Recently unusual piezoelectricity was measured using time-resolved x-ray
experiments (Jones et al., 2007; Pramanick et al., 2011). A tensor order parameter is proposed
to predict this behavior instead of the typical polarization order parameter.
The free energy function includes a chemical potential governing the phase, anisotropy
energy for long range crystal anisotropy in a three dimensional solid, and an isotropic energy
for lower order nematic phase that occurs in liquid crystal polymer networks. This leads to
predictions of anisotropic deformation or swelling that is driven by the underlying microstruc-
ture and chemical changes. A relatively small number of material parameters (six or less) is
able to predict the salient features governing anisotropic, field-coupled or chemically-coupled
deformation. Conversely, applied traction on the surface or applied fields also lead to defor-
mation and changes in the internal microstructure. Predictions of bending deformation of
4
chemically responsive elastomers, unusual soft elasticity of polydomain nematic phase liquid
crystal elastomers, and anisotropic piezoelectricity and polarization switching of ferroelectric
materials are simulated to illustrate how the model can be used to predict a relatively broad
range of field-coupled material characteristics. In particular, experimental results given in the
literature are used as motivation for comparisons to the model (Harris et al., 2005; Urayama
et al., 2009; Jones et al., 2007; Pramanick et al., 2011).
In the following sections, a general set of governing equations are briefly described and
followed by numerical examples. In Sections 2 and 3, a brief review of the balance equations
and thermodynamic relations are given. In Section 4, several numerical results describing
how the model is applicable to a relatively broad number of active materials are presented.
Discussion and concluding remarks are given in the final two sections.
2 Governing Equations
The model is first considered by starting with a free energy description per current volume
ψ=ψM(FiK ) + ψA(c, c,i, φi, φi,j , Qij , Qij,k) (1)
where the energy is decomposed into mechanical effects in ψMand active microstructure effects
in ψA. The order parameters range from scalar, vector, and tensor order parameters that are
defined in the spatial configuration. The scalar order parameter cand its gradient c,i, are
used to quantify chemically induced deformation as a function of changes in concentration.
The vector order parameter, φi={Pi, Mi, ni}, may include polarization Pi, magnetization
Mi, or liquid crystal order ni, for example. The gradient on the order parameter is denoted
by φi,j and is often included in meso to microscale energy descriptions to approximate a
multipole expansion on electrostatic or magnetostatic behavior that gives rise to twinned
domain structures within a grain or single crystal material (Lines and Glass, 1977; Mitsui
and Furuichi, 1953). A tensor order parameter and its gradient has also been introduced as
Qij and Qij,k; respectively. This tensor is normally used to quantify quadrupole effects. Such
5
effects are normally negligible in materials with strong polarization or magnetization, but
the quadrupole tensor is typically the primary order parameter used in modeling a class of
active materials called liquid crystal polymer networks (LCNs) (Warner and Terentjev, 2007;
P. de Gennes and Prost, 1993). Tensor order parameters have also been used to model a
strain-like order parameter in ferromagnetic shape memory alloys to accommodate relatively
large strain changes near twinned martensitic variants that otherwise exhibit significantly
large stress concentrations (Landis, 2008).
When modeling liquid crystal polymer networks, the second order tensor order parameter
is related to the vector order parameter or director by Qij =Q/2(3ninj−δij ). In this equation,
Qdefines the order of the nematic phase and niis constrained to a unit vector that defines
the orientation of a rod-like liquid crystal cluster of uniformly aligned molecules. This form
only includes the deviatoric component such that anisotropic dielectric properties or the light
absorption tensor can be defined to be proportional to the tensor order parameter. Different
approaches have been proposed to address how to incorporate Qand niinto a modeling
framework; see (Ericksen, 1991) for comparisons. One potential method is to directly model
Qij as the tensor order parameter since it is measurable from nuclear magnetic resonance
techniques or indirectly from optical measurements (P. de Gennes and Prost, 1993). Here, we
will numerically implement a reduced order method that simulates the director as the vector
order parameter using the quadrupole definition, Qij = 1/2(3ninj−n2
0δij ) where n0is the
equilibrium director value for a monodomain and 0 ≤ |n| ≤ 1 instead of a unit vector. This
accommodates both Qand the traditional unit vector representation of the director into a
single vector order parameter and allows for the introduction of a relative simple free energy
function with a phase field model. However, this can lead to an approximation of Qij when
the model is implemented numerically using phase field methods due to a diffuse domain wall,
point or line defect. This is because the trace of Qij may not be zero in regions of defects.
Despite this approximation, it provides a reduced order phase field modeling approach that
only requires simulating three vector components versus five tensor components with the
constraint Qkk = 0. The differences in direct implementation of the quadrupole tensor versus
6
ti
ϕ
traction
potential
electric
concentration
magnetization
polarization or
liquid crystal order
nI
PI
microstructure
order parameter
Flux: light, heat
or chemistry I
φ
MI
c
Figure 1: Description of the nonlinear continuum model that is embedded with an order parameter
that describes polarization, magnetization, liquid crystal order, or concentration.
the director vector order parameter will be discussed in a future analysis.
A number of mechanical energy functions can be introduced to quantify stress-strain con-
stitutive behavior such as an anisotropic elastic constitutive law, or in the case of elastomers,
a hyperelastic energy function; see (Holzapfel, 2000) for examples. Here, coupling between
a generalized mechanical energy function is first presented. Specific mechanical and order
parameter energy functions will be given in subsequent sections to first provide qualitative
estimates on spontaneous strain as a function of material parameters and second, more quan-
titative estimates for different active materials using numerical simulation.
It is initially assumed that these materials can undergo finite deformation without fracture.
The assumption of large deformation is used to quantify the material coupling based on
nonlinear geometric effects and thermodynamic parameters in the reference configuration,
but the model is also shown to exhibit interesting coupling in the limit of infinitesimal strain
as will be shown for the case of lead zirconate titanate piezoelectric materials. To this end,
the nonlinear continuum model is formulated to include finite deformation which requires
introducing the deformation gradient
7
FiK =∂xi
∂XK
(2)
where xiis the spatial point and XKis the reference or material point (Malvern, 1969).
A set of relations are introduced to provide rotational invariance in the reference configu-
ration. The scalar, vector, and tensor order parameter relations in the reference and current
configuration are related by
φi=J−1FiK e
φK
φi,j =J−1FiK FjL e
φK,L
Qij =J−1FiK FjL e
QKL
c=J−1ec
c,i =J−1FiK ec,K
(3)
where the tilde (
e·) denotes order parameters in the reference configuration. The Jacobian is
defined by J= det(FiK ). The relations (3)1,4can be obtained using geometric arguments and
conservation of mass, respectively (Malvern, 1969). The remaining terms can be determined
using the principle of virtual work by introducing work conjugate variables for the order
parameter and its gradient; see (Zhao et al., 2007; Oates and Wang, 2009) for examples on
dielectric elastomers and liquid crystal elastomers. A similar relation can be introduced for the
second order tensor gradient, although we will focus on the reduced order approach that relates
the gradient of Qij to the gradient of the vector order parameter in this analysis. It should
be noted that this is not the only form that satisfies rotational invariance. The rotation
tensor could also satisfy invariance; however, determination of the field-coupled stresses is
more complex; see (McMeeking and Landis, 2005). The introduction of these relations will be
introduced into (1) to quantify material coupling based on nonlinear geometric effects.
Minimization of the free energy and kinetic energy per reference volume requires satisfying
linear momentum and microscale force balances on the order parameters e
φIand ec. Note that in
the present analysis e
QKL is defined to depend on the vector order parameter, so a tensor-based
8
Euler-Lagrange equation is not introduced for this tensor. These equations are
∂siK
∂XK
+Bi= 0
∂e
ξJI
∂XJ
+eπI+eγI= 0
∂e
ξJ
∂XJ
+eπ+eγ= 0
(4)
in the reference volume Ω0where the first equation is the quasi-static form of linear momentum,
the second equation describes a microforce balance on the vector order parameter e
φIand the
third equation describes a chemical potential balance. The nominal stress is defined by siK
and the body force is denoted by Bi. The microforce balance is governed by the micro-stress
tensor e
ξJ I , an internal body force eπI, and an external body force eγI. This form of the governing
equations is useful in formulating a computational framework in the reference configuration.
More details on the microforce and chemical potential balance equations can be found in
(Gurtin, 1996). Additional details that define these work conjugate variables in terms of the
free energy function are given in the following section.
A set of boundary conditions are defined for (4)1,2. For linear momentum, traction is
defined by Ti=siK ˆ
NKon Γ0. The unit normal in the reference description is denoted by ˆ
NK.
The boundary conditions for the microforce balance include eτI=e
ξJI ˆ
NJon Γ0.
The chemical flux relations on the surface and within the volume require coupling (4)3
with the conservation of mass
˙
ec=−e
JI,I + ˙min Ω0(5)
where e
JIis the mass flux and ˙mis the external mass density supply rate. The mass flux is
assumed to be proportional to the gradient of the chemical potential
e
JI=−AIJ eµ,J (6)
where eµis the chemical potential and AI J is a mobility tensor.
9
Phase evolution is often described by combining the conservation of mass and the chemical
potential balance relations which leads to the Cahn-Hilliard equation (J.Cahn, 1961). This
is obtained by solving (4)3as eπ=−e
ξK,K −eγand writing the work conjugate relation for
the chemical potential as eµ=∂e
ψ
∂ec+eπ, as discussed in detail in (Gurtin, 1996; Govindjee and
Simo, 1993). Substituting these relations together with (6) into (5) gives the Cahn-Hilliard
equation
˙
ec=
AIJ ∂e
ψ
∂ec−eγ−e
ξK,K !,J
,I
+ ˙m. (7)
Numerically, this equation is more easily solved in the original form that contains two second
order equations that describe conservation of mass (5) and the chemical potential balance (4).
A dissipative force based on the time derivative of e
φIwill be introduced within the defi-
nition of eπI. To satisfy objective rates on this vector order parameter, the time derivative of
(3)1is taken. This leads to an objective rate governed by
˚
e
φK=JHiK ˙
φi(8)
where ˚
e
φ=˙
e
φ− ∇ · ve
φ+F−1LF e
φ. This objective rate includes the divergence of velocity ∇ · v
for compressible deformation processes (i.e., ˙
J6= 0) and the velocity gradient which is defined
by Lij =∂vi
∂xj
(Malvern, 1969). This form of the objective rate is similar to a convective time
derivative often used in electrodynamic problems (Eringen and Maugin, 1990; Hutter and Van,
1978). The relation described by (8) is used in formulating the dissipative energy relation.
This dissipative energy relation is implemented in (4)2within the term defined by eπIas a
function of the evolution rate of the vector order parameter e
φI. The dissipative energy is
defined as e
ΠD=−βIJ
2
˚
e
φJ˚
e
φJwhere βIJ is an inverse mobility tensor.
By taking the variation of the dissipative energy e
ΠD, the microscale body force eπIwill
include dissipative effects according to
eπI=−eηI−βIJ˚
e
φJ(9)
10
where a conservative force, eηI, has been included and will be defined by a thermodynamic
potential function in the following section. The second term defines the dissipative effect in
terms of the objective microstructure rate and inverse mobility tensor.
Prior to defining the set of work conjugate relations, it is illustrative to first present the
nominal stress that is typically defined by siK =∂e
ψ
∂FiK
at equilibrium (Holzapfel, 2000). This
free energy is defined in the reference frame based on (1), (3), and e
ψ=Jψ as
siK =∂e
ψM
∂FiK
+∂e
ψA
∂FiK
(10)
which defines the total nominal stress where the first term represents stresses from stretching
mechanical bonds and the second term represents stresses from the order parameters. In this
formulation, the total nominal stress is based on an objective free energy function which leads
to a symmetric, total Cauchy stress. It should also be emphasized that only the total stress
is measurable; however, several free energy components are introduced to quantify different
internal field-coupled effects.
Also note that the stress in (10) is defined at equilibrium. Additional stresses from rate
dependence may occur due to deformation gradient coupling contained within ˚
e
φI. In the
subsequent numerical simulations, quasi-static loading is considered which will neglect the
rate dependent term. The magnitude of this stress component was verified to be negligible
for the stretch rates considered on liquid crystal elastomers and ferroelectric microstructure
problems for the given inverse mobility coefficients.
In summary, the nominal stress relation forms the key result of the model which illustrates
how an order parameter (i.e., polarization, magnetization, liquid crystal order, or changes in
concentration) may influence the constitutive behavior of an active material. Special forms of
e
ψwill be given to illustrate how the model predicts anisotropic active material behavior for
a select number of active materials. First, a brief description of the thermodynamic relations
and work conjugate variables is given.
11
2.1 Work Conjugate Relations
According to the first and second laws of thermodynamics, the applied power must be equal
to or surpass the sum of kinetic energy and free energy rate of the material. By combining the
first and second laws and applying the result to an arbitrary representative volume element
in the deformed configuration, the balance of work rate to the internally stored free energy
rate is
ρ˙
ψ≤σjivi,j +ξji ˙
φi,j −πi˙
φi+ξi˙ρ,i + (eµ−eπ) ˙ρ−Jiµ,i (11)
where the free energy rate per mass is ˙
ψ; see (Gurtin, 1996; McMeeking and Landis, 2005) for
details. Here, the total Cauchy stress has been introduced as σj i.
The time rate of change of the deformation gradient, dFiK
dt =∂vi
∂xj
∂xj
∂XK
=FjK
∂vi
∂xj
, is used
to compare the time rate of change of the free energy in (11) using the relation
˙
ψ=∂ψ
∂FiK
˙
FiK +∂ψ
∂φi
˙
φi+∂ψ
∂φi,j
˙
φi,j +∂ψ
∂ρ,i
˙ρ,i +∂ψ
∂ρ ˙ρ. (12)
The work conjugate variables are defined by comparing (11) and (12). The free energy is
written in terms of energy per volume in terms of the reference density, e
ψ=eρψ(Holzapfel,
2000). This provides a definition of the total Cauchy stress
σji =J−1Fj K
∂e
ψ
∂FiK
(13)
where J−1=ρ
eρ. Note that the free energy is assumed to include the effect of mechanics, the
order parameters, and electromagnetic energy in free space. Details regarding the form of the
objective thermodynamic energy functions for e
ψare introduced in the following section.
The work conjugate body forces and microstresses associated with the vector order pa-
rameter are
ηi=J−1∂e
ψ
∂φi
and ξji =J−1∂e
ψ
∂φi,j
.(14)
12
Based on the definition of Qij in terms of φi, the work conjugate field is ηi=J−1∂e
ψ
∂Qkl
∂Qkl
∂φi
.
It should be noted that only the conservative force that is associated with the first term in
(9) is included in the definition of ηi.
The work conjugate variables associated with the phase are
µ−π=J−1∂e
ψ
∂c ,and ξi=J−1∂e
ψ
∂c,i
.(15)
A substitution of the conjugate variables given in this section plus the spatial form of the
relations (9) and (6), into (12) gives
βij ˙
φi˙
φj+Aij µ,iµ,j ≥0.(16)
Since φiand µare independent variables, both βij and Aij must be positive semi-definite to
satisfy the second law of thermodynamics.
3 Thermodynamic Energy Functions
The form of the thermodynamic potential used to predict active material coupling and
microstructure evolution is presented in this section. An emphasis is placed on simplified free
energy functions with a minimal number of parameters necessary to predict a relatively broad
range of active material constitutive relations. Whereas additional coupling terms are valid to
consider within a free energy formulation, we propose that coupling from rotational invariant
order parameters based on a spatial description of the free energy is sufficient to predict a
broad range of smart material constitutive relations. Moreover, the utilization of a spatial
form of the free energy may reduce the number of phenomenological constants necessary to
fit to data as will be illustrated through numerical simulation.
A description of the chemical free energy function is first given followed by the vector and
tensor order parameters. The coupled stresses from each energy function are determined and
material parameter dependence on spontaneous strain is quantified for the vector and tensor
order parameters. Numerical implementation is then given in Section 4.
13
3.1 Scalar Order Parameter
A chemical free energy is written as a function of the concentration to quantify changes in
chemically induced deformation using the relation
ψc(c) = g
2(c−c0)2+h
4(c−c0)4(17)
per current volume where a reference concentration has been defined by c0at some initial
time. The phenomenological parameters include gand hwhich define the magnitude of the
chemical potential for a given change in concentration.
Nonlinear coupling between the deformation gradient and the concentration is introduced
within this energy function using the relation previously defined by (3)3. This requires that
the free energy in the reference frame be coupled to the deformation gradient via the Jacobian
Jas described by
e
ψc=gJ
2(J−1ec−c0)2+hJ
4(J−1ec−c0)4.(18)
The stress associated with the change in concentration is obtained from sc
iK =∂e
ψc
∂FiK
which
gives
sc
iK =−g
2J−1HiK (ec2−J2c2
0)+
−h
4HiK (−3J−3ec4+ 8J−2ec3c0−6J−1ec2c2
0+Jc4
0)
(19)
and the Cauchy stress component associated with chemical diffusion is obtained using the
relations (3), (19), and
σc
ij =J−1FjK
∂e
ψA
∂FiK
(20)
which gives
σc
ij =−g
2(c2−c2
0) + h
4(3c4−8c3c0+ 6c2c2
0−c4
0)δij (21)
14
which leads to hydrostatic stresses relative to the ambient concentration c0. This is expected
since absorption of chemical constituents would lead to a volumetric shape change.
3.2 Vector Order Parameter
The free energy of the vector order parameter is decomposed into terms that describe
three dimensional crystal anisotropy and an isotropic energy function that only governs the
magnitude of the order parameter. The anisotropy energy is equivalent to free energy that
describes the easy axis in ferromagnetic materials (Kittel, 1949). A similar decomposition has
been introduced to study different phases in ferroelectric materials (Heitmann and Rossetti,
2005). While this decomposition is useful to describe crystal anisotropy, it is shown that the
rotational invariant vector order parameter plays no role in predicting anisotropic spontaneous
deformation.
The three dimensional crystal anisotropy energy focuses on tetragonal phase materials
using
ψφ1=g|δijδkl −δikδjl |
4φiφjφkφl(22)
where the Kronecker delta is denoted by δij . The absolute value of |δijδkl −δikδjl |guarantees
positive free energy values that are not along the (001) orientation for tetragonal phase crystal
symmetry if g > 0. This free energy function is plotted in Figure 2(a).
The following isotropic energy function only provides minima that govern the magnitude of
the order parameter at equilibrium. It is expanded into an eighth order Landau free function
of the form
ψφ2=aφ
2φiφi+bφ
4φiφiφjφj+cφ
6φiφiφjφjφkφk+dφ
8φiφiφjφjφkφkφlφl(23)
where the phenomenological constants include aφ, bφ, cφ, and dφ. The effect of each higher
order term on spontaneous strain will be quantified in Section 3.4. By substitution of the
rotationally invariant order parameter in (3)1and the relation e
ψφ2=Jψφ2
15
e
ψφ2=aφ
IJ
2e
φIe
φJ+bφ
IJ K L
4e
φIe
φJe
φKe
φL+cφ
IJ K LM N
6e
φIe
φJe
φKe
φLe
φMe
φN+
dφ
IJ K LMN P Q
8e
φIe
φJe
φKe
φLe
φMe
φNe
φPe
φQ
(24)
where
aφ
IJ =J−1CI J aφ
bφ
IJ K L =J−3CI J CKLbφ
cφ
IJ K LM N =J−5CI J CKLCM N cφ
dφ
IJ K LMN P Q =J−7CI J CKLCM N CP Qdφ
(25)
The field coupled stresses associated with this order parameter are determined from the
nominal stress definition previously given by (10) and (13). For brevity, only the Cauchy
stress is given
σφ
ij =aφφiφj−δij
2φkφk+bφφiφj−3δij
4φkφkφkφk+
cφφiφj−5δij
6φkφkφkφkφlφl+dφφiφj−7δij
8φkφkφkφkφlφlφmφm.
(26)
This symmetric Cauchy stress illustrates order parameter coupling due to the introduction of
an objective free energy in (24) and the rotational invariant order parameters in (25). Also
note that the crystal anisotropy within ψφ1does not contribute to the field-coupled stress,
as shown in the Appendix. In the case of ferroelectric or ferromagnetic materials where the
vector order parameter is polarization or magnetization, the isotropic free energy function is
the only direct contributing factor to field-coupled deformation. However, it will be shown
in Section 3.4 that this stress is hydrostatic at the equilibrium order parameter value and
therefore plays a negligible role in predicting anisotropic deformation exhibited by most types
of smart materials.
In the following section, the Cauchy stress due to a second order tensor will be quantified.
Comparisons between the vector and tensor order parameter stresses are then given to illus-
trate deformation predicted by the model under zero stress prior to presenting larger scale
16
computational simulations.
3.3 Tensor Order Parameter
The energy function for a tensor order parameter is written in terms of a linear quadrupole
that is often used to model liquid crystal materials. While this tensor exist in ferroelectric and
ferromagnetic materials, its contribution to the constitutive behavior is normally smaller than
the polarization and magnetization (Loudon, 2000). The implications of using this tensor to
describe ferroelectric or ferromagnetic materials will be discussed further in Section 5.
The quadrupole free energy function considered here is based on the deviatoric component
of a second order tensor. This describes the anisotropic part of a second order tensor such as
the dielectric tensor according to Qij =1
2(3κij −δij κkk ) where κij represents the total dielectric
tensor (Ericksen, 1991). The free energy function in terms of Qij is
ψQ=aQ
2QijQji +bQ
3Qij Qjk Qki +cQ
4Qij Qjk Qkl Qli (27)
where the phenomenological parameters aQ, bQ,and cQgovern the order of the crystal structure
at the microscopic scale. The functional form of this free energy is identical to the Landau-
de-Gennes free energy function used to model liquid crystals (P. de Gennes and Prost, 1993).
An example of this free energy is plotted in Figure 2(b) for aQ<0, bQ>0, and cQ>0.
Similar to the vector order parameter free energy, a transformation to the reference domain
is introduced to numerically implement the free energy function in the reference configuration
for solid mechanics modeling. In the reference frame, the free energy is
e
ψQ=1
2aQ
IJ K L e
QIJ e
QKL +1
3bQ
IJ K LM N e
QIJ e
QKL e
QMN +1
4cQ
IJ K LM N RS e
QIJ e
QKL e
QMN e
QRS (28)
where rotational invariance is satisfied by introducing (3)3and e
ψQ=JψQinto (27). These
tensors are
17
aQ
IJ K L =J−1CI LCJK aQ
bQ
IJ K LM N =J−2CI N CJ K CLM bQ
cQ
IJ K LM N RS =J−3CIS CJ K CLM CN R cQ
(29)
where the Green deformation tensor, CK L =FiK FiL , has been introduced (Malvern, 1969).
For brevity, the nominal stress components are given in the Appendix. The Cauchy stress
component induced by the tensor order parameter is determined from (10), (13), and (28) and
given by
σQ
ij =aQ2QikQj k −1
2QmnQmn δij +cQ2Qik Qkm Qmj −2
3QklQlmQmk δij +
+cQ2QikQkmQml Qlj −3
4QklQlmQmn Qnk δij .
(30)
It will be shown in the following section that this tensor order parameter can induce either
prolate or oblate material behavior meaning positive spontaneous strain may occur in the
direction of larger anisotropy (prolate) or negative spontaneous strain in the direction of larger
anisotropy (oblate). Furthermore, since the sign of spontaneous strain changes relative to the
anisotropy, it is shown in the following section that a certain ratio of the phenomenological
−1 01
−1
0
1
0
0.5
1
φ1
φ2
Free Energy (normalized)
−1 01
−1
0
1
−2
−1
0
1
φ1
φ2
Free Energy (normalized)
−1 01
−1
0
1
−1
0
1
φ1
φ2
Free Energy (normalized)
(a) (b) (c)
Figure 2: Free energy plots for the nonlinear continuum model. (a) Anisotropy energy described by
ψφ1in (22). (b) The second order tensor energy function described by ψQin (27). (c) The combined
free energy function ψ=ψφ1+ψQ.
18
parameters decouple spontaneous strain from the tensor order parameter.
3.4 Order Parameter Induced Spontaneous Deformation
To illustrate the deformation predictions from the order parameter and free energy rela-
tions, an isotropic elastic free energy function is added to the model such that “mechanical”
stresses are balanced with the stresses induced by the order parameters. It is clear from (21)
that changes in a scalar order parameter only induces hydrostatic stress so this stress is not
analyzed here. Instead, comparisons between the vector and tensor order parameter are given
to illustrate anisotropic deformation or lack thereof. We will also focus on the small strain
limit to simplify the analysis. Detailed numerical analyses, including finite deformation, are
given in subsequent sections using elastic and hyperelastic free energy functions and coupling
with the order parameters.
The isotropic elastic free energy is denoted by
ψM=1
2E
(1 + ν)(1 −2ν)εiiεkk +E
1 + νεijεij (31)
where Eand νare the elastic modulus and Poisson ratio. Infinitesimal strain is denoted by
εij. The “mechanical” Cauchy stress relation is then given by
σM
ij =Eν
(1 + ν)(1 −2ν)εkkδij +E
1 + νεij .(32)
where this stress is added to the field-coupled stress which defines the total Cauchy stress
based on the free energy. This total stress is set to zero to quantify spontaneous strain.
Anisotropic deformation is computed for a uniformly aligned microstructure vector and tensor
order parameter (i.e., monodomain) in the X3direction. This gives φ3=φ0and φ1=φ2= 0
for the vector order parameter and Q11 =Q22 =−Q
2and Q33 =Qfor the tensor order
parameter. In the case of equilibrium under zero applied loads, the total Cauchy stress is
zero (σij =σM
ij +σA
ij = 0) where σA
ij is the coupled stress based on either (26) or (30). By
solving for the spontaneous strain due to the microstructure coupling, we have
19
ε0
11 =ε0
22 =1
E(ν−1)σA
11 +νσA
33
ε0
33 =1
E2νσA
11 −σA
33.
(33)
To further illustrate this coupling behavior, specific examples are given for the spontaneous
strain as a function of the vector and tensor order parameter. The limit of incompressibility
(i.e., ν= 0.5) is considered which simplifies the form of the spontaneous strain to illustrate
salient features governing the coupling behavior.
For the vector order parameter, we have
εφ
11 =εφ
22 =φ2
0
2Eaφ+bφφ2
0+cφφ4
0+dφφ6
0
εφ
33 =−φ2
0
Eaφ+bφφ2
0+cφφ4
0+dφφ6
0(34)
where these spontaneous strain components are equal at the equilibrium value for φ0. In
comparison with the energy function in (23), the polynomial expression in (34) is zero at ther-
modynamic equilibrium. Therefore the spontaneous strain is always zero for incompressible
materials when the root of φ0is chosen at the global energy minimum. It is interesting to
note that when the material is compressible, the spontaneous strain is equal in each direction
and therefore does not contribute to any anisotropic deformation. It should be noted that a
rigorous proof for all values of the Landau coefficients has not been done.
For the tensor order parameter, we have
εQ
11 =εQ
22 =3Q2
8E4aQ+ 6bQQ+ 5cQQ2
εQ
33 =−3Q2
8E4aQ+ 6bQQ+ 5cQQ2(35)
therefore the function, 4aQ+ 6bQQ+ 5cQQ2, is important in quantifying the magnitude and
sign of the spontaneous deformation. Moreover, this function is zero for a certain set of
constants. This is obtained by solving for the equilibrium vector order parameter by equating
the work conjugate variable ηi, previously given by (14), to zero. This leads to the solution
of the equilibrium microstructure vector order parameter
20
Q=
−bQ±qb2
Q−4aQcQ
2dQ
(36)
Normally aQ<0 and cQ>0 for a non-zero, real order parameter value. This relation for Q
is substituted into (35) and the case of zero spontaneous strain is obtained. It can be shown
that if
cQ=−2b2
Q
aQ
(37)
the spontaneous deformation is always zero at the equilibrium, positive Qroot for incom-
pressible materials. If cQ<−2b2
Q
aQ
the spontaneous deformation is prolate and if cQ>−2b2
Q
aQ
the spontaneous deformation is oblate. Similar behavior is known to occur in liquid crystal
elastomers, however, prolate behavior predominately occurs (Warner and Terentjev, 2007).
This decoupling relation will be used in quantifying unusual soft elasticity or lack thereof in
polydomain liquid crystal elastomers in Section 4.2. Due to the lack of anisotropic deforma-
tion predicted by the vector order parameter, it will also be proposed in Section 4.3 that the
tensor order parameter gives a better prediction of electromechanical coupling in ferroelectric
materials at the domain length scale.
This form of the free energy function implements a relatively small number of material
parameters to explore their effect on predicting complex active material mechanics. Higher
order gradients may be necessary to quantify twinned microstructures and interface energetics
in materials with heterogeneous microstructure evolution. Whereas finite deformation effects
would introduce stresses associated with order parameter gradients, these stresses are typically
small due to the magnitude of the phenomenological parameters and are therefore neglected
in the numerical simulations.
4 Numerical Implementation
The material model is applied to a select number of active materials that exhibit complex
microstructure evolution and nonlinear coupling with deformation. First, deformation from
21
chemical diffusion is described and compared with data in the literature for large bending
deformation of a hygroscopic liquid crystal polymer network (Harris et al., 2005). This is
followed by large deformation of a soft polydomain liquid crystal elastomer to show how soft
elasticity may or may not occur during uniaxial stretching as a function of the cross-linked
reference state. This is motivated by experimental results of polydomain soft elasticity as a
function of the cross-linked state (Urayama et al., 2009). Microstructure evolution has shown
to be responsible for unusual soft deformation in monodomain specimens, but similar behavior
in random polydomains is still not well understood (Biggins et al., 2009). Lastly the model is
applied to ferroelectric materials with comparisons to recent unusual anisotropic piezoelectric
characterization of lead zirconate titanate (Jones et al., 2007; Pramanick et al., 2011).
4.1 Chemically Deforming Materials
The unified model is used to predict large bending deformation induced by water vapor
absorption in a hygroscopic liquid crystal polymer network (Harris et al., 2005). This pro-
vides an illustration of how volume changes from absorption due to environmental influences
affect material deformation. The chemical potential balance, as described by conservation of
mass (5) and linear momentum (4)1of a finite deforming shell, is used to quantify stress and
deformation during changes in concentration. By neglecting localized concentration gradients
of multi-phase materials, the chemical potential balance from (4)3is neglected and only diffu-
sion of mass and mechanical equilibrium are necessary to predict material deformation from
absorption of chemical species from the environment.
The material considered here deforms in the presence of an external water vapor source or
pH due to a change in the liquid crystalline structure as the external chemical constituents in
the environment are absorbed into the glassy polymer network. The liquid crystal polymer
structure initially consists of both covalent and secondary bonds. Humidity-controlled defor-
mation occurs after the network is converted to a salt. This provides a hygroscopic material
which begins to swell under humid conditions as water infiltrates the material. The response
is relatively fast where reversible deformation can occur on the order of seconds (Harris et al.,
22
2005). Whereas order-disorder effects of the liquid crystal rod-shaped molecules may also
contribute to deformation, the scalar order parameter is able to predict the key deformation
characteristics governing these materials.
Prediction of this deformation from water absorption is obtained by implementing the
stress coupling with concentration that was described in Section 3.1 into the linear momentum
balance equation. Since these materials are typically synthesized as films with thickness on the
order of tens of microns, a finite deforming shell model using FEAP is modified to include the
chemically induced stress components (Taylor, 2010; Simo and Fox, 1989). Isotropic elastic
properties, as a function of the liquid crystal orientation, are introduced based on data given
in (Harris et al., 2007). The material parameters used in the constitutive model are given in
Table 1. The liquid crystals are aligned along the short axis of the film as denoted by the X1
direction which is associated with a larger modulus. The linear chemical potential parameter
gis set to zero in the simulation to illustrate large nonlinear bending deformation. Since g
and hgive different sensitivities to concentration, these parameters can be fit to data based on
bending measurements (Hays et al., 2010, 2011). The model model parameter hwas used to
qualitatively fit bending simulations to supplemental videos provided in (Harris et al., 2005)
where the film rotates approximately 90◦as it approaches a water source.
The bending moment is computed by assuming a variation in the relative humidity (RH)
in the environment to range between typical ambient conditions of 35% to 100% as the film
Table 1: Water vapor deforming material properties. The orthotropic elastic properties are denoted
by the elastic moduli EIand Poisson ratio νI. The shear moduli are defined by GI=EI/(2(νI+ 1)).
The out-of-plane direction is defined as the X3direction.
h1.28×1027 Nm9/mol4
E12.4 GPa
E21.6 GPa
ν1=ν20.35
23
approaches a water source. In all simulations, the humidity on the side of the film not exposed
to water vapor was held fixed at 35%. The concentration is computed from the ideal gas law
and humidity tables assuming ambient temperature (25◦C) and atmospheric pressure. This
gives a change in water vapor concentration at ambient conditions c35%
0= 3.97 ×10−7mol/m3
to ec100% = 1.13 ×10−6mol/m3. Using these values for water concentration, a linear increase
in concentration is predicted using the boundary value problem for the one dimensional case
described in (Bird et al., 1960). For modeling purposes, the ambient conditions are taken
far from the water source such that approximately constant concentration is applied to the
material surface during large bending deformation.
As the humidity increases to 100%, large chemically induced bending is predicted by
computing the “chemically” induced moment using
e
mα(c)=1
jZh+
h−
ξσcgαjdξ (38)
where gαis the convected basis which is a function of the deformation gradient and the unit
vectors in the reference configuration of the shell model (Simo and Fox, 1989). Note that the
initial geometry is considered to be a flat plate. The chemically induced stress is denoted by
σcand is based on (21). Also note that the thickness of the film is h=h+−h−where the
coordinate along the thickness direction is ξ. The variable jis the mid-surface Jacobian and
(α= 1,2) spans the mid-surface of the shell.
The nonlinear finite element results are illustrated in Figure 3 which include displacement
and rotation constraints along the center line of the short axis. The humidity source is located
underneath the film. As the humidity is increased from the bottom, the film expands on this
surface from water vapor absorption and produces large monotonic increases in bending. Also
note that the curvature is largest along the ydirection because the modulus is lower in this
direction.
24
0246810 −2 0246
0
1
2
3
4
x (mm)
y (mm)
Displacement (mm) z
Figure 3: Finite element shell results of water vapor controlled deformation of the hygroscopic liquid
crystal elastomer. The film is fixed in the center along the short axis and the water vapor source
is underneath the film. Bending increases as the water vapor concentration on the bottom surface
increases to 100%.
4.2 Polydomain Liquid Crystal Elastomers
In comparison to the glassy hygroscopic liquid crystal polymer network, soft elastomers
synthesized with liquid crystals are known for their shape change induced by liquid crystal or-
der and conversely, liquid crystal reordering under mechanical loading (Warner and Terentjev,
2007). Due to the soft elastomer network, large liquid crystal reordering may occur during
large deformation. Whereas prior work has illustrated microstructure mechanics of these ma-
terials in idealized monodomains (Oates and Wang, 2009; Mbanga et al., 2010), it is shown
here using the nonlinear continuum model that complex polydomain materials exhibit similar
soft elastic behavior (i.e., nonlinear stress-strain response) and transitions from a polydomain
to monodomain under large stretch. The effect of the cross-linked reference state is modeled
by modifying the Landau coefficients based on the decoupling ratio given by (37).
Since nematic phase liquid crystals do not exhibit three dimensional long range crystal
25
order, the energy function given by (27) is used to simulate these materials. This corresponds
to the free energy plot shown in Figure 2(b). The order parameter for the liquid crystal rod
shaped molecule is the director (ni=φi) which defines a microscale homogenization of the
orientation of a mesogen unit. In this case, the second order tensor is defined by
Qn
ij =1
2(3ninj−n2
0δij ).(39)
where the director is limited to the range 0 ≤ |ni| ≤ 1 by assuming time-averaged thermal
fluctuations may reduce the equilibrium director to be n0≤1. The director at equilibrium is
found by setting ηi= 0 in (14)1which ensures that Qij is traceless under zero external loading
in regions of a uniform domain.
The experiments given in (Urayama et al., 2009) were conducted by cross-linking the
liquid crystal elastomer in either the nematic or isotropic state. It was found that large soft
elastic behavior (e.g., nonlinear stress-strain behavior) was observed when the material was
cross-linked in the isotropic state, but not the nematic state. When the material is cross-
linked in the isotropic state, spontaneous deformation of the host elastomer occurs as nematic
phase liquid crystals form during cooling. In the nematic-cross linking state, the nematic
phase liquid crystals form prior to formation of the cross-linked elastomer and therefore do
not induce spontaneous deformation on the host elastomer network. In this case, we define
the reference state to have zero spontaneous deformation for a nematic cross-linked material
which requires (37) to hold. For the isotropic cross linked state, cQ<−2b2
Q
aQ
is used to define
a prolate liquid crystal elastomer material.
Since polydomain structures are of particular interest, energy penalties on the director
gradient are introduced. This energy is described by Frank elastic behavior (P. de Gennes and
Prost, 1993). The Frank elastic energy is approximated by a scalar parameter and gradients
on the director
ψG=K
2ni,jni,j (40)
where Kequally penalizes splay, twist, and bend of the liquid crystal director; see (P. de Gennes
26
and Prost, 1993).
The liquid crystal energy in the spatial domain per current volume is therefore repre-
sented by ψA=ψQ+ψG. A transformation of the energy to the reference configuration and
minimization of the energy leads to an Euler-Lagrange equation of the form
∂e
ξJI
∂XJ
−eηI=β0˚
enI.(41)
This relation is equivalent to (4)2except external microforces are neglected. The objective
rate ˚
e
φI=˚
enIwas defined previously in (8). An inverse mobility factor is defined by βI J =β0δI J
which was also previously defined by the tensor introduced in (9). This mobility factor neglects
liquid crystal anisotropy by assuming comparable dissipation in all directions.
The liquid crystal model is coupled to an incompressible hyperelastic neo-Hookean free
energy function to predict liquid crystal domain structure evolution. Details of this free
energy can be found in (Holzapfel, 2000). The introduction of polymer mechanics requires
satisfying linear momentum in the reference configuration as defined by
∂siK
∂XK
= 0 (42)
where body forces and inertial effects have been neglected. It should be noted that the
total nominal stress includes components from the hyperelastic behavior of the host elastomer
and the liquid crystal order parameter, siK =sM
iK +∂e
ψQ
∂FiK
. The first term sM
iK is stress from
stretching the elastomer network.
Numerical results are given for a polydomain elastomer that undergoes uniaxial stretch-
ing. The equations are implemented numerically using Comsol 3.5 on a quasi-3D domain. A
two dimensional problem is simulated using plane stress mechanics but includes the director
components in all three directions. Such approximations neglect out-of-plane director gradi-
ents and out-of-plane shear coupling with the deformation gradient. The parameters used in
the model are based on nominal values for a nematic phase liquid crystal (P. de Gennes and
Prost, 1993) and a soft elastomer; see Table 2. The polydomain structure is obtained by using
random initial conditions and the model is then allowed to reach equilibrium under zero trac-
27
Table 2: Parameters used in the finite element model for the isotropic cross-linked case. The param-
eters include Kas the distortional energy constant, aQ, bQ,and cQas the Landau parameters, µas
the shear modulus, and κas the bulk modulus. For the nematic cross-linked state, the parameters
were modified as aQ=−93.4 kPa, bQ=−96.6 kPa, aQ=−2b2
Q
cQ= 200, and µ= 300 kPa.
Name Value Unit
K 2.00 ×10−10 N
aQ-1.43 kPa
bQ-96.6 kPa
cQ200 kPa
β0500 N-s/m2
κ0.5 GPa
µ300 kPa
tion boundary conditions and roller conditions along the bottom edge. Numerical validation
is conducted by using mesh refinement until each computational run gives equivalent results.
Once an equilibrium polydomain configuration is obtained, the model is stretched from the
top using a constant velocity, quasi-static displacement control and fully clamped conditions
along the bottom.
The results illustrate soft elasticity for the case of cQ<−2b2
Q
aQ
and hyperelastic behavior
for the decoupled case where cQ=−2b2
Q
aQ
. Under continued increases in the stretch ratio, the
random polydomain elastomer is stretched into an aligned domain structure. In Figure 4,
the director component in the vertical direction is plotted at different points along the stress
versus stretch curve. Note that the different colors in the plot refer to the director vectors
pointing up or down; however, this corresponds to the same energy and strain state. The total
nominal stress and stretch is plotted for the stress-stretch components aligned with the load.
During stretching from the initial random state, the volume fraction of domains aligned with
the load increases at the expense of domains in other directions. During this process, liquid
28
crystal reorientation leads to relaxation of the total stress for cQ<−2b2
Q
aQ
(Figure 4(a)) which
corresponds to the plateau region. Once all the liquid crystal domain structures align with the
external load, the material behaves like a conventional hyperelastic solid. Conversely, when
cQ=−2b2
Q
aQ
, minimal spontaneous strain occurs at equilibrium prior to stretching. This is not
exactly zero everywhere due to defects along domain walls. As the material is stretched, the
liquid crystals reorient and align with the external load, but since the spontaneous strain is
approximately zero everywhere, no soft elastic plateau is observed. Despite this spontaneous
strain decoupling, reorientation of the liquid crystals still occur in the nematic cross-linked
model. This is due to the indirect effect of distortion of the Landau-deGennes free energy
function in the reference configuration. As the deformation gradient changes, the global
energy wells of the free energy function change which leads to driving forces for liquid crystal
reorientation.
Similar nonlinear mechanics and increases in the liquid crystal order were observed ex-
perimentally in the isotropic and nematic cross-linked state (Urayama et al., 2009). In those
experiments, two different cross-link densities (3% and 10%) were tested and similar soft elas-
tic stress-strain curves were measured for isotropic cross-linking and hyperelastic stress-strain
curves for nematic cross-linking specimens.
4.3 Ferroelectric Materials
Ferroelectric materials are known for their deformation induced by changes in polariza-
tion from stress or applied electric fields (Lines and Glass, 1977; Jaffe et al., 1971). This
behavior becomes highly nonlinear at large fields and stresses. In this section, the nonlin-
ear continuum modeling framework is applied to tetragonal phase ferroelectric materials to
quantify anisotropic, field-induced deformation as a function of four Landau energy parame-
ters and two elastic material parameters without introducing piezoelectric or electrostrictive
coefficients.
The choice of the free energy function and order parameters is motivated by time-resolved
29
(a) (b)
Figure 4: Nonlinear deformation of polydomain nematic phase liquid crystal elastomers. The mag-
nitude of the director in the vertical direction is plotted on top of the corresponding stress-strain
curves. The bottom edge was fully constrained while stretched at a constant rate of 0.01 µm/s along
the top edge. The results in (a) correspond to an isotropic cross-linked state while (b) corresponds
to the nematic cross-linked model when cQ=−2b2
Q
aQ.
x-ray data that has shown unusual anisotropic piezoelectric coupling in tetragonal phase lead
zirconate titanate materials (Pramanick et al., 2011). In these experiments, large piezoelectric
coupling was found when the angle between the applied electric field and polarization was at an
angle of approximately ∼40◦. The piezoelectricity decreased to approximately zero when the
polarization and electric field were parallel or perpendicular. The results suggest polarization
rotation and lattice reorientation of the tetragonal crystal structure strongly influences the
electro-mechanical behavior. Given this observation, we introduce the second order tensor
order parameter to predict anisotropic spontaneous deformation as opposed to the polarization
vector order parameter. This is based on the prediction of hydrostatic deformation using a
vector order parameter as described in Section 3.4. Comments on this approach versus using
explicit coupling between polarization and strain is given in the discussion section.
The tensor order parameter is defined to represent the anisotropic component of the dielec-
30
tric tensor for tetragonal phase crystal structure. This tensor may be alternatively denoted
as a linear quadrupole which introduces additional coupling within Gauss’ law, but is beyond
the scope of the present analysis. This behavior will be discussed in a future analysis. Instead,
the second order tensor is defined to be proportional to the polarization according to
QP
ij =1
2(3PiPj−P2
0δij ) (43)
where P0is the equilibrium value of the polarization and Piis the polarization in the spatial
domain. This gives a set of eigenvectors for Qij that are orientated parallel or perpendicular
to the polarization and is qualitatively consistent with quadrupole calculations based on the
effective charge and atom locations using the center of the unit cell as the origin. In the
undeformed reference state, the trace of this tensor is zero. This becomes an approximation
for large deformation or near twinned domain structures where a direct correlation between Qij
and Pibecomes more difficult. For most ferroelectric crystals, the small strain approximation
is reasonable.
The free energy function is constructed by combining the anisotropic energy in (22) with
the second order tensor energy function in (27). The combination of these two energy functions
given minimima wells for a typical tetragonal phase ferroelectric as illustrated in Figure 2(c).
This form of the Landau free energy is combined with the elastic free energy previously given
by (31). For numerical implementation, the elastic Cauchy stress tensor in (32) is combined
with the second order tensor induced stress given by (30). Note that there is no field-coupled
Cauchy stress induced by (22). The Landau parameters, Young’s modulus, and Poisson ratio
used in the model for effective monodomain simulations are given in Table 3. Comparisons
of these material parameters with a set of parameters for a fully coupled electro-mechanical
phase field model are also given. The differences in these models are described in subsequent
paragraphs.
Energy minimization at different field increments is first computed to quantify anisotropic
piezoelectric behavior and ferroelectric switching behavior for different polarization orienta-
tions relative to the applied field. This is obtained through minimization of the electric Gibbs
31
Table 3: The ferroelectric material parameters used in the effective monodomain and phase field
models include the four Landau parameters, aQ, bQ, cQ, and gand the Young’s modulus and Poisson
ratio are denoted by Eand ν, respectively.
Effective Monodomain Phase Field Model
aQ−2.6×108N-m6
C4−7.43 ×109N-m6
C4
g5.0×107N-m6
C45.0×1011 N-m6
C4
bQ−6.24 ×109N-m10
C6−1.2×1011 N-m10
C6
cQ2.08 ×1010 N-m14
C86.9×1011 N-m14
C8
E 8.0×109Pa 8.0×109Pa
ν0.35 0.35
free energy that is defined by g=ψ−EiPiwhere Eiis the electric field in the spatial config-
uration. A gradient based minimization is applied by satisfying, ∂g
∂Pi
= 0 for i= 1, 2, and 3,
at each electric field increment.
Using the effective monodomain parameters in Table 3, numerical simulations are con-
ducted to first predict spontaneous strain, dielectric behavior, and piezoelectricity for differ-
ent crystal orientations. The spontaneous strain predictions for an effective monodomain are
Es
33 = 1%, Es
22 =Es
11 =−0.34% for a ferroelectric with polarization oriented in the X3direc-
tion. The spontaneous polarization for this case is Ps= 0.58 C/m2. The dielectric constant
and piezoelectric behavior is quantified at different angles by rotating the local coordinates
in the X2−X3plane for an applied field fixed in the X3direction. The slope of the applied
field versus strain and polarization is obtained at different angles and is plotted in Figure 5.
The orientation of the polarization is found to have a dramatic effect on the dielectric and
piezoelectric response. The relative dielectric constant is largest when the polarization is or-
thogonal to the field and approaches a small, positive number when the field and polarization
are aligned. The piezoelectric coefficient (d333) is predicted to be slightly negative when the
32
500
1000
1500
2000
30
210
60
240
90
270
120
300
150
330
180 0
κr
33
d333 (pm/V)
θ
Ps
Figure 5: Polar plot of the relative dielectric constant and piezoelectric coefficient d333 as a function
of the polarization orientation. The model is rotated in the X2−X3plane. The maximum dielectric
constant is 1676 and occurs at 0◦while the maximum piezoelectric coefficient is d333 = 1124 pm/V
and occurs at θ= 35obased on the parameters in Table 3.
polarization is aligned with the field (θ= 90o). As the angle is reduced from 90oto 0o, the
coefficient d333 increases until it reaches a maximum at θ= 35oand then approaches zero as
the angle approaches zero. The results compare well with recent time-resolved x-ray experi-
ments on tetragonal phase PZT (Jones et al., 2007; Pramanick et al., 2011). The predictions
in the linear regime suggests polarization rotation is the primary source of deformation in
poled ferroelectric materials. This is in contrast to most other models in the literature.
The linear piezoelectric predictions are compared with nonlinear ferroelectric switching to
illustrate how the model predicts major loop polarization-electric field hysteresis and strain-
electric field butterfly hysteresis. The same gradient based optimization scheme is applied
for large bi-polar fields to predict polarization switching and deformation. The results for
an initial angle between the field and polarization of θ= 15◦is illustrated in Figure 6.
The results show the conventional hysteresis response. Similar behavior is observed at other
angles except the piezoelectric response and the coercive field varies as a function of the
33
crystal orientation relative to the field. Moreover, the relative dielectric susceptibility, defined
in Pi=χr
ijε0Ejwhere ε0is permittivity of free space, changes for different polarization
orientations, but ranges from approximately χr
33 = 50 to 1000 similar to the perturbation
analysis in Figure 5. This value is comparable to many ferroelectric materials and therefore the
model reasonably predicts both dielectric and electromechanical behavior by only introducing
four Landau coefficients, a Young’s modulus and a Poisson ratio. Note that the coercive field is
strongly dependent on the anisotropic energy constant galthough this term does not directly
contribute to the field-coupled Landau stress. This term penalizes 90◦switching as illustrated
previously in Figure 2(a) and therefore affects the field-induced polarization rotation and
electric field coercivity to 90◦polarization switching. As this coefficient increases, larger
coercive energy barriers for 90◦switching occur while simultaneously reducing the piezoelectric
response.
These results are based on an idealized monodomain single crystal, whereas in realistic
materials, many domains and grains influence the constitutive behavior. To account for a
more realistic estimate of the material behavior, the free energy was implemented into the
same finite element framework discussed in Section 4.2 for liquid crystal elastomers except
the model used the ferroelectric Landau energy within a time-dependent Ginzburg-Landau
equation. This equationis coupled with both linear momentum and Gauss’ law, Di,i = (ǫ0Ei+
Pi),i = 0 where ǫ0is the permittivity of free space. Using this approach similar deformation
from an initial polydomain state was observed during polarization rotation induced by an
applied electric field. The results were obtained by poling a random polydomain model in
two dimensions. The main distinction between the idealized monodomain and the phase field
polydomain was the magnitude of the Landau parameters necessary to stabilize the domain
structure due to the internal fields due to the introduction of Gauss’ law, unlike in the effective
monodomain model. The Landau parameters were increased by two to three orders while
holding the ratios approximately fixed so the spontaneous polarization was comparable to the
idealized monodomain. This increase in the Landau parameters increased the spontaneous
strain from 1% to approximately 10% as well as the coercive field as illustrated in Figure 6.
34
−30 −20 −10 0 10 20 30
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
E3 (MV/m)
Polarization (C/m2)
P3
P2
P1
−30 −20 −10 0 10 20 30
−0.5
0
0.5
1
E3 (MV/m)
Strain (%ε)
E33
E22
E11
(a) (b)
Figure 6: Ferroelectric hysteresis loops for the angle (θ= 15◦) in relation to the piezoelectric response
in Figure 5).
While the magnitude of spontaneous strain is larger that the typical experimental value for
tetragonal phase ferroelectric materials, the effect of polarization rotation predicts positive
strain increments until the model is fully poled. Once this occurs, the strain increments are
slightly negative for any larger applied fields.
5 Discussion
An active material nonlinear continuum modeling framework has been developed to pre-
dict field-coupled nonlinear material behavior in a relatively broad range of active materials.
Through the use of nonlinear continuum mechanics and rotational invariant order parame-
ters, it was shown that scalar and vector order parameters induce hydrostatic spontaneous
deformation at equilibrium. This somewhat surprising result for the vector order parameter
shows that explicit coupling could be added into the material model to potentially improve the
qualitative predictions of unusual active material coupling presented here. To further explore
the effect of finite deformation and coupling with microstructure order parameters, the second
order tensor order parameter Qij was shown to induced anisotropic deformation for certain
35
0 100 200 300 400 500
0.0
0.1
0.2
0.3
0.4
0.5
P
3
Polar izatio n (C/ m
2
)
Strai n
E
3
(MV/m)
0.00
0.03
0.06
0.09
0.12
E
33
Figure 7: The corresponding change in magnitude of polarization P3and strain E33 relative to the
applied electric field magnitude (E3) is plotted to illustrate how domain wall motion leads to increases
in strain.
ratios of the Landau coefficients. The magnitude and sign of the spontaneous deformation
induced by Qij depended on three Landau parameters where decoupling was determined to
occur for the ratio cQ=−2b2
Q
aQ
. For values that were less than or greater than this ratio, ei-
ther prolate or oblate spontaneous deformation was predicted and used to model liquid crystal
elastomers and ferroelectric materials.
Specific attributes of the model were first validated on chemically deforming materials
by introducing a chemical potential balance to accommodate changes in deformation as a
function of chemical absorption from absorbing gases from the surrounding environment. This
was coupled with nonlinear mechanics to obtain stresses induced by changes in water vapor
concentration. A fourth order chemical potential parameter was implemented numerically to
36
predict deformation observed experimentally (Harris et al., 2005, 2007, 2006). As expected,
changes in concentration lead to swelling; however, large anisotropic bending deformation was
predicted when the model was coupled with an orthotropic elastic free energy function that
represented the glassy elastomer network. It was shown that bending away form the water
vapor source occurs due to a concentration gradient of water vapor through the thickness of
the film. It should be noted that the chemically coupled stress given by (21) always predicts
compressive stress which creates swelling under zero applied traction. Thus, the reference
concentration c0is important in cases where shrinking may occur such as from changes in
polarity or pH due to absorption of the chemicals in the environment. Order-disorder processes
due to the underlying liquid crystal director may also contribute to deformation; however,
the scalar concentration relation is found to provide reasonable predictions of concentration
induced deformation.
The second numerical example illustrates how the nonlinear continuum model predicts soft
elastic behavior or lack thereof in polydomain liquid crystal elastomers as a function of cross-
linking in the isotropic or nematic state. Recent experiments have shown that under certain
cross-linking conditions, polydomain specimens can exhibit soft elastic behavior similar to
the well-known stress-strain plateau characteristics in monodomain liquid crystal elastomer
specimens (Urayama et al., 2009). The polydomain structure at the onset of cross-linking
had a significant impact on the nonlinear stress-strain constitutive behavior. For example,
cross-linking in the nematic (lower temperature) state produced no soft elastic behavior. The
soft elastic behavior was recently modeled using a set of homogenized free energy functions
that depend on a residual deformation gradient; however, stress-strain behavior in the nematic
cross-linked state was not considered in the model (Biggins et al., 2009). Here, changes in
the soft elastic behavior are modeled as a function of cross-linking by introducing a set of
Landau coefficients that decoupled or coupled spontaneous deformation with time-dependent
microstructure evolution. The ratio of the Landau coefficients was found to have a dramatic
effect on the soft elastic stress strain curve. For typical prolate deformation (positive defor-
mation in the liquid crystal director orientation), soft elasticity was predicted for the isotropic
37
cross-linked state. In the nematic cross-linked state where no spontaneous deformation oc-
curs, the stress-strain behavior is completely hyperelastic with no soft elasticity. The result
provides a relatively simple methodology to predict significantly different constitutive behav-
ior. However, note that quantitative model fits have not been done. This requires matching
both changes in the order parameter and nonlinear deformation with the Landau coefficients.
These issues will be explored in a future analysis.
In the third example, the nonlinear constitutive model was applied to ferroelectric ma-
terials where the anisotropic energy function ψφ1was combined with the Landau-deGennes
free energy function ψQ. This free energy function was chosen over vector order parameter
free energy function ψφ2since it allowed for predictions of anisotropic deformation without
introducing explicit coupling between polarization and strain. As described in Section 3.4,
the vector order parameter (i.e., polarization in this case) only induces hydrostatic stress and
therefore would not produce any changes in deformation during polarization rotation. Al-
ternative to introducing the tensor order parameter, explicit coupling could be added into
the model to predict many of the observed coupling behavior in ferroelectric materials. It
is important to note however that if explicit coupling is used, positive spontaneous strain
in the direction of polarization will also lead to a positive d333 coefficient when the external
field is aligned with the polarization. Recent data (Pramanick et al., 2011) suggest negligi-
ble lattice strain for co-aligned fields and polarization where the majority of piezoelectricity
is due to heterogeneous microstructure and polarization rotation. The proposed model that
uses a second order tensor order parameter appears to fit this data more closely. This is
somewhat surprising if the second order tensor order parameter is considered analogous to
a linear quadrupole. This is because dipole energy is typically larger than the quadrupole
energy (Loudon, 2000). A more rigorous analysis of the quadropole tensor, its work conjugate
electric field gradient, and potential deformation gradient coupling should be further explored
to quantify these different effects on the electromechanical constitutive behavior.
38
6 Concluding Remarks
A nonlinear, field-coupled constitutive model has been developed and numerically im-
plemented to correlate microstructure evolution with deformation from external mechanical
loads, fields, and chemical absorption. The model illustrates when the effect of finite defor-
mation coupling with microstructure becomes important or can be neglected when modeling
microstructure induced deformation in active materials. By introducing a relatively small
number of material parameters in the spatial domain, relatively complex nonlinear mechanics
and microstructure evolution was predicted. This was achieved by transforming the spa-
tial description of free energy and field quantities to the reference configuration. Reasonable
correlation of complex liquid crystal elastomer nonlinear mechanics and chemically induced
deformation was predicted. The model also predicted both anisotropic piezoelectricity and
ferroelectric behavior using four Landau parameters and isotropic elastic coefficients. The
ferroelectric results compared qualitatively with time-resolved x-ray experiments given in the
literature which is different than the conventional tetragonal lattice distortion model for lead
zirconate titanate.
7 Acknowledgment
This material is based upon work supported by the DARPA YFA program (grant N66001-
09-1-2105) and an NSF CAREER award (grant 1054465). Additional support was provided
by a summer faculty ASEE program with Eglin Air Force Research Laboratory. W. S. Oates
also appreciates an insightful discussion with Chad Landis. Any opinions, findings, and con-
clusions or recommendations expressed in this publication are those of the authors and do not
necessarily reflect the views of the funding sponsors.
39
8 Appendix
8.1 Nominal Stress Relations
The nominal stress for the quadrupole tensor is obtain from sQ
iK =∂e
ψQ
∂FiK
and explicitly
given by
sQ
iK = 2 aJ−1FiM CN Q e
QQK e
QMN +cJ−2FiM CN QCRS e
QQR e
QMN e
QSK +
+dJ−3FiM CM QCRS CT A e
QAK e
QST e
QQR e
QM N
−HiK a
2CMRCN S e
QMN e
QRS +2c
3CNRCS ACMB e
QMN e
QRS e
QAB+
3d
4CNRCS ACBC CDM e
QMN e
QRS e
QAB e
QCD
(44)
where CNQ =FkN FkQ. This result is based on the free energy given by (28) which leads to
the Cauchy stress given by (30).
It was noted in Section 3.3 that the anisotropic free energy given in (22) does not contribute
to the field-coupled stress. This free energy is written in the reference configuration as
e
ψφ1=gJ −3
4|(CM N CP Q −CMP CN Q |e
φMe
φNe
φQe
φQ(45)
and the nominal stress associated with this free energy component is
sφ1
rS =gJ−3
2|CMN e
φMe
φN−CMN e
φMe
φN|e
φAe
φBCAB
|CSQ|FrQ −CAB
|CP S|Fr P (46)
using the identity ∂(C:C)1/2
∂C=C
|C|. Obviously, the term in the absolute value brackets is
zero.
References
Adams, J., Conti, S., DeSimone, A., 2007. Soft elasticity and microstructure in smectic C
elastomers. Continuum Mech. Thermodyn. 18, 319–334.
40
Armstrong, W., 2000. The non-linear deformation of magnetically dilute magnetostrictive
particulate composites. Matl. Sci. Eng. A285, 13–17.
Bar-Cohen, Y., Zhang, Q., 2008. Electroactive polymer actuators and sensors. MRS Bulletin
33 (3), 173–177.
Baur, J., Silverman, E., 2007. Challenges and opportunities in multifunctional nanocomposite
structures for aerospace applications. MRS Bulletin 32, 328–334.
Bertotti, G., 1998. Hysteresis in Magnetism: for Physicists, Materials Scientists, and Engi-
neers. Academic Press, San Diego, CA.
Biggins, J., Warner, M., Bhattacharya, K., 2009. Supersoft elasticity in polydomain nematic
elastomers. Phys. Rev. Lett. 103, 037802–1–037802–4.
Bird, R., Stewart, W., Lightfoot, E., 1960. Transport Phenomena. John Wiley and Sons, New
York, NY.
Bladon, P., Terentjev, E. M., Warner, M., 1993. Transitions and instabilities in liquid crystal
elastomers. Phys. Rev. E 47 (6), R3838–R3840.
Boyd, J., Lagoudas, D., 1996. A thermodynamical constitutive model for shape memory
materials. Part II. The SMA composite material. Int. J. Plasticity 12 (7), 843–873.
Bustamante, R., Dorfmann, A., Ogden, R., 2008. On variational formulations in nonlienar
magnetoelastostatics. Math. Mech. Solids 13, 725–745.
Bustamante, R., Dorfmann, A., Ogden, R., 2009. Nonlinear electroelastostatics: a variational
framework. Z. angew. Math. Phys. 60, 154–177.
Cao, W., Cross, L., 1991. Theory of tetragonal twin structures in ferroelectric perovskites
with a first-order phase transition. Phys. Rev. B 44 (1), 5–12.
Chen, L., 2002. Phase-field models for microstructure evolution. Annu. Rev. Mater. Res. 32,
113–140.
41
Ericksen, J., 1991. Liquid crystals with variable degree of orientation. Arch. Rational Mech.
Anal. 113, 97–120.
Ericksen, J., 2008. Magnetizable and polarizable elastic materials. Mathematics and Mechanics
of Solids 13, 38–54.
Eringen, A., Maugin, G., 1990. Electrodynamics of Continua I: Foundations and Solid Media.
Springer-Verlag, New York.
Fried, E., Gurtin, M., 1994. Dynamic solid-solid transitions with phase characterized by an
order parameter. Physica D 72, 287–308.
Govindjee, S., Simo, J., 1993. Coupled stress-diffusion: Case II. J. Mech. Phys. Solids 41 (5),
863–887.
Gurtin, M., 1996. Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a mi-
croforce balance. Physica D 92, 178–192.
Harris, K., Bastiaansen, C., Broer, D., 2006. A glassy bending-model polymeric actuator
which deforms in response to solvent polarity. Macromol. Rapid Commun. 27, 1323–1329.
Harris, K., Bastiaansen, C., Broer, D., 2007. Physical properties of anisotropically swelling
hydrogen-bonded liquid crystal polymer actuators. J. Microelectromech. S. 16 (2), 480–488.
Harris, K., Bastiaansen, C., Lub, J., Broer, D., 2005. Self-assembled polymer films for con-
trolled agent-driven motion. Nano Lett. 5, 1857–1860.
Hays, M., Wang, H., Oates, W., 2010. Force characterization of hygroscopic liquid crystal
elastomers. Proc. 2010 ASME Conf. Smart Materials, Adaptive Structures, and Intelligent
Systems, SMASIS.
Hays, M., Wang, H., Oates, W., 2011. Nonlinear bending mechanics of hygroscopic liquid
crystal polymer networks. ASME J. Appl. Mech. (accepted).
42
Heitmann, A., Rossetti, G., 2005. Thermodynamics of polar anisotropy in morphotropic fer-
roelectric solid solutions. Phil. Mag. 90(1-4), 71–87.
Hildebrand, F., Abeyaratne, R., 2008. An atomistic investigation of the kinetics of twinning.
J. Mech. Phys. Solids 56 (4), 1296–1319.
Holzapfel, G., 2000. Nonlinear Solid Mechanics. John Wiley & Sons, Inc., Chichester.
Huang, M., Brinson, L., 1998. A multivariant model for single crystal shape memory alloy
behavior. J. Mech. Phys. Solids 46 (8), 1379–1409.
Hutter, K., Van, A., 1978. Field matter interactions in thermoelastic solids: A unification of
existing theories of electro-magneto-mechanical interactions. Springer-Verlag, Berlin.
Hwang, S., Arlt, G., 2000. Switching in ferroelectric polycrystals. J. Appl. Phys. 87, 869–875.
Jaffe, B., Cook, W., Jaffe, H., 1971. Piezoelectric Ceramics. Academic Press, London.
J.Cahn, 1961. On spinodal decomposition. Acta. Metall. 9, 795–801.
J.E.Huber, N.A.Fleck, 2001. Multi-axial electrical switching of a ferroelectric: theory versus
experiment. J. Mech. Phys. Solids 49, 785–811.
J.E.Huber, N.A.Fleck, Landis, C., McMeeking, R., 1999. A constitutive model for ferroelectric
polycrystals. J. Mech. Phys. Solids 47, 1663–1697.
Jones, J., Pramanick, A., Nino, J., Daymond, M., Oliver, E., 2007. Time-resolved and
orientation-dependent electric-field-induced strains in lead zirconate titanate ceramics.
Appl. Phys. Lett. 90, 172909.
Kamlah, M., Zhou, D., Laskewitz, B., 2008. Constitutive behavior of ferroelectric piezoceram-
ics: Experiments and modeling. GAMM-Mitt. 31 (2), 166–175.
Kittel, C., 1949. Physical theory of ferromagnetic domains. Rev. Mod. Physics 21 (4), 541–583.
43
Kofod, G., Sommer-Larsen, P., 2005. Silicone dielectric elastomer actuators: Finite-elasticity
model of actuation. Sensors and Actuators A 122, 273–283.
Kowalewsky, O., 2004. Theory of complex lattice quasicontinuum and its application to fer-
roelectrics. Ph.D. thesis, California Institute of Technology, Pasadena, CA.
Landis, C., 2008. A continuum thermodynamics formulation for micro-magneto-mechanics
with applications to ferromagnetic shape memory alloys. J. Mech. Phys. Solids 56, 3059–
3076.
Lendlein, A., Kelch, S., 2002. Shape-memory polymers. Angew. Chem. Int. Ed. 41, 2034–2057.
Lines, M., Glass, A., 1977. Principles and Applications of Ferroelectrics and Related Materials.
Clarendon Press, Oxford.
Loudon, R., 2000. The Quantum Theory of Light. Oxford University Press, Oxford.
Lubensky, T., Mukhopadhyay, R., Radzihovsky, L., Xing, X., 2002. Symmetries and elasticity
of nematic gels. Phys. Rev. E 66, 011702–1–011702–22.
Malvern, L., 1969. Introduction to the Mechanics of a Continuous Medium. Prentice-Hall,
Inc., Englewood Cliffs, NJ.
Maugin, G., 2011. Configurational Forces: Thermomechanics, Physics, Mathematics, and
Numerics. CRC Press, New York.
Mbanga, B., Ye, F., Selinger, J., Selinger, R., 2010. Modeling elastic instabilities in nematic
elastomers. Phys. Rev. E 82, 051701–1–4.
P. de Gennes, Prost, J., 1993. The Physics of Liquid Crystals. Oxford Science Publications,
Oxford.
McMeeking, R., Landis, C., 2005. Electrostatic forces and stored energy for deformable di-
electric materials. J. Appl. Mech 72, 581–590.
44
Mitsui, T., Furuichi, J., 1953. Domain structure of rochelle salt and KH2PO4. Phys. Rev.
90 (2), 193–202.
Nelson, D., 1991. Momentum, pseudomomentum, and wave momentum:
Toward resolving the Minkowski-Abraham controversy. Phys. Rev. A 44 (6), 3985–3996.
Nemat-Nasser, S., 2002. Micromechanics of actuation of ionic polymer-metal composites. J.
Appl. Phys. 92 (5), 2899–2916.
Oates, W., Wang, H., 2009. A new approach to modeling liquid crystal elastomers using phase
field methods. Modelling Simul. Mater. Sci. Eng. 17, 064004.
Pramanick, A., Damjanovic, D., Daniels, J., Nino, J., Jones, J., 2011. Origins of electro-
mechanical coupling in polycrystalline ferroelectrics during subcoercive electrical loading.
J. Am. Ceram. Soc. 94 (2), 293–309.
Seelecke, S., M¨uller, I., 2004. Shape memory alloy actuators in smart structures: Modeling
and simulation. Appl. Mech. Rev. 57 (1), 23–46.
Simo, J., Fox, D., 1989. On a stress resultant geometrically exact shell model.
Part I: Formulation and optimal parameterization. Comput. Meth. Appl. Mech. Engrg. 72,
267–304.
Smith, R., 2005. Smart Material Systems: Model Development. SIAM, Philadelphia, PA.
Smith, R., Seelecke, S., Dapino, M., Ounaies, Z., 2006. A unified framework for modeling
hysteresis in ferroic materials. J. Mech. Phys. Solids 54 (1), 46–85.
Su, Y., Landis, C., 2007. Continuum thermodynamics of ferroelectric domain evolution: The-
ory, finite element implementation, and application to domain wall pinning. J. Mech. Phys.
Solids 55, 280–305.
Suo, Z., Zhao, X., Greene, W., 2008. A nonlinear field theory of deformable dielectrics. J.
Mech. Phys. Solids 56, 467–486.
45
Taylor, R., 2010. http://www.ce.berkeley.edu/feap/.
Toupin, R., 1963. A dynamical theory of elastic dielectrics. Int. J. Engng. Sci. 1, 101–126.
Trimarco, C., 2009. An extended notion of enthalpy. Electromagnetic solids. IMA J. Appl.
Math. 74, 416–426.
Urayama, K., Kohmon, E., Kojima, M., Takigawa, T., 2009. Polydomain-monodoamin tran-
sition of randomainly disordered nematic elastomers with different cross-linking histories.
Macromolecules 42, 4084–4089.
Virga, E., 1994. Variational theories for liquid crystals. Chapman & Hall, London.
Vu, D., Steinmann, P., 2010. Material and spatial motion problems in nonlinear electro- and
magneto-elastostatics. Math Mech Solids 15, 239–257.
Warner, M., Terentjev, E., 2007. Liquid Crystal Elastomers–Revised Edition. Oxford Science
Publications, Oxford.
Zhao, X., Hong, W., Suo, Z., 2007. Electromechanical hysteresis and coexistent states in
dielectric elastomers. Phys. Rev. B 76, 134113–1–134113–9.
46
A preview of this full-text is provided by SAGE Publications Inc.
Content available from Journal of Intelligent Material Systems and Structures
This content is subject to copyright.
