Mathematical homogenization theory for electroactive continuum

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng (2012)Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nme.4311

Mathematical homogenization theory for electroactive continuum

Sergey Kuznetsov*,† and Jacob Fish

Columbia University, Department of Civil Engineering and Engineering Mechanics, 610 Seeley W. Mudd,500 West 120th Street, New York, NY 10027

SUMMARY

Two-scale continuum equations are derived for heterogeneous continua with full nonlinear electromechanicalcoupling using nonlinear mathematical homogenization theory. The resulting coarse-scale electromechanicalcontinuum equations are free of coarse-scale constitutive equations. The unit cell (or representative volumeelement) is subjected to the overall mechanical and electric field extracted from the solution of the coarse-scale problem and is solved for arbitrary constitutive equations of fine-scale constituents. The proposedmethod can be applied to analyze the behavior of electroactive materials with heterogeneous fine-scale struc-ture and can pave the way forward for designing advanced electroactive materials and devices. Copyright© 2012 John Wiley & Sons, Ltd.

Received 18 January 2011; Revised 16 February 2012; Accepted 20 February 2012

KEY WORDS: electroactive; composite; multiscale; homogenization; electromechanical coupling;unit cell; electostriction; fine-scale structure

1. INTRODUCTION

All industrial materials are heterogeneous at some scale. The overall behavior of materials iscontrolled by their fine-scale structure. Materials with tunable fine-scale structure can exhibitsuperior performance at coarse level: multiphase fine-scale structure allows improving usefulproperties while reducing the effect of undesirable properties of fine-scale constituents; the overallproperties can be tailored by changing fine-scale structure (mixing different materials at differentscales, changing the size and shape of constituents, etc.) to meet the needs of specific applications.Attempts have even been made to state an optimization problem aimed at constructing an optimalmaterial architecture [1–8]. Because of the synergy of fine-scale constituents it is possible to obtainnew properties that are not available in fine-scale constituents alone. Metamaterials [9–13] representone fascinating example of how tuning fine-scale structure provides a completely different dynamicbehavior of materials.

Historically, the primary goal of composite materials with tunable fine-scale structure was toreduce the weight especially for materials with aerospace applications. More recently, emphasishas been placed on electroactive materials, which found their use in actuators, transducers, andstructures capable of changing their properties (behavior) depending on the environment. These areso-called ‘smart structures’ where feedback loop comprises of sensors and actuators. The presenceof biasing fields makes such materials apparently behave differently [14, 15], which creates anopportunity to control the response of so-called smart structures to various excitations [16].

Electroactive materials are used for actuators, sensors, and smart structures. Applications rangebut not limited to underwater acoustics, biomedical imaging [17], robotic manipulations, artificialmuscles [18–20], active damping, resonators and filters for frequency control and selection for

*Correspondence to: Sergey Kuznetsov, Columbia University, Department of Civil Engineering and EngineeringMechanics, 610 Seeley W. Mudd, 500 West 120th Street, New York, NY 10027.

†E-mail: [email protected]

Copyright © 2012 John Wiley & Sons, Ltd.

S. KUZNETSOV AND J. FISH

telecommunication, precise timing and synchronization [14], MEM and NEM devices [21–23], flowcontrol [24], precise positioning [25, 26], conformal control surfaces [27], power electronic devices[28], computer memory applications [22,23], tunable optics, generators for harvesting energy, tactilesensors [29–32], etc.

Composites containing electroactive materials in combination with other suitable materials canbe used to replace pure electroactive materials with inherent limitations such as low actuationcapabilities, high electric fields, fast decay, and instabilities. The overall properties of suchcomposites are extremely sensitive to fine-scale details. Various studies [20, 27, 33–36] have shownthat electromechanical coupling can be significantly improved combining flexible electromechanicalmaterial and high dielectric modulus (or even conductive) materials. The overall response of acomposite actuator can be vastly superior to that of its constituents. In particular, composites basedon electroactive polymers have shown tremendous promise because of their actuation capabilitiesresulting from the ability to produce strains of up to 40% [27,29], and more than 100% in elastomeractuation [37]. The attractiveness of polymers is not only because of their electromechanicalproperties, but also because of their lightweight, short response time, low noise, durability, and highspecific energy. Proper optimization of their fine-scale structure can lead to a vast improvement inelectromechanical coupling [27].

On the modeling front, there are numerous nonlinear models [14, 38–50] capable of describingelectromechanical behavior for large deformation problems. There are a number of constitutivephenomenological and micromechanical models for electromechanical materials, includingelectrostrictors [51,52], ferroelectric switching [53–57], and surface-dominated nanomaterials [58].Therefore, although the behavior of different phases is well understood and can be describedsufficiently well, the computational cost of resolving fine-scale details is very high and is oftenbeyond the capacity of modern computers. Thus, models capable of describing the overall behaviorof electroactive composites without resolving fine-scale details are needed. The phenomenologicalequations describing the behavior of electroactive composites are usually very complex and thusthere is increasing need for multiscale-multiphysics based methods.

Numerous works can be found in the literature that employ homogenization theory to modelthe electromechanical coupling in composites and polycrystalline materials, but mostly restricted tolinear piezoelectric coupling [1,8,59–67]. Shroeder [68] developed a homogenization framework fornonlinear coupling to account for ferroelectric switching in the presence of a strong electric field. Formechanical problems a number of homogenization methods accounting for large deformations andarbitrary nonlinear constitutive relations have been developed [69–79]. In the present manuscript weextend the nonlinear homogenization framework proposed in [70] to nonlinear electromechanicalmaterials with full electromechanical coupling. The mathematical formulation proposed herebycan be applied to arbitrary fine-scale structures and arbitrary constitutive models of phases. To theauthors’ knowledge this is first attempt to develop a general nonlinear mathematical homogenizationtheory for electromechanical materials.

2. MATHEMATICAL MODEL

First, we present the equations describing the behavior of a deformable and polarizable body,subjected to mechanical and electrical excitations. We will restrict ourselves to the quasistaticapproximation, when elastic wavelengths considered are much shorter than electromagnetic wavesat the same frequency [14, 80]. In which case the effect of electromagnetic waves is neglectedand the electric field depends on time through boundary conditions and coupling to the dynamicmechanical fields. The formulation follows the works of Tiersten [38] and Yang [14] and is basedon full electromechanical coupling in a solid via Maxwell stress, general nonlinear constitutiveequations, large deformations, and strong electric fields.

Consider a deformable and polarizable body occupying volume � with boundary @�. When thebody is placed in an electric field, differential material elements experience both body forces andcouples because of the electric field. There is a full coupling through the Maxwell stress and via theconstitutive equations [14, 51]. Following Tiersten [38], the electric body force FE, couple CE andpower wE are used to derive the balance equations

Copyright © 2012 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng (2012)DOI: 10.1002/nme

MATHEMATICAL HOMOGENIZATION THEORY FOR ELECTROACTIVE CONTINUUM

F Ej D �eEj CPiEj ,i , (1)

CEi D "ijkPjEk , (2)

wE D �mEi P�i , (3)

where Ej denotes electric field, �m the current mass density, �e the current free charge density and� D Pi=�x the polarization per unit mass. From (1) and (2), the governing equations can be derived[14, 38].

Gauss law

Di ,i � �e D 0, (4)

whereDi denotes the electric displacement,

Di D "0Ei CPi , (5)

Pi and "0 are polarization and electric constant, respectively. Because divergence of polarizationdetermines induced charge density, the total charge density is given by

�te D �e �Pi ,i . (6)

The electric field is vortex free, which can be written as

"ijkEk,j D 0, (7)

which implies that the electric field can be expressed in terms of the gradient of electric potential

Ei D��,i . (8)

The continuity equation is given by

P�mC �mvi ,i D 0 (9)

and the linear momentum balance equation is

�ij ,j C �mbi CFEi D

��ij ,j C �

Eij ,j

�C �mbi D �m Pvi , (10)

where �ij is Cauchy stress, bi is the body force per unit mass (excluding electric body force); �Eij is

electrostatic stress, which is related to electric force �Eij ,j D F

Ei .

The electrostatic stress �Eij can be expressed as

�Eij DDiEj �

1

2"0EkEkıij D PiEj C "0

�EiEj �

1

2EkEkıij

�. (11)

Divergence of �Eij is given by

�Eij ,j DDj ,jEi CPjEi ,j D �eEi CPjEi ,j D F

Ei . (12)

The sum of Cauchy stress �ij and the electrostatic stress �Eij gives the total stress �ij D �ij C �E

ij ,which is symmetric and can be decomposed into symmetric tensor

�Sij D �ij CPiEj D �

Sj i (13)

and Maxwell stress tensor

�Mij D "0

�EiEj �

1

2EkEkıij

�D �M

j i . (14)



Using the total stress, the balance of linear momentum (10) can be rewritten as

�ij ,j C �mbi D �m Pvi , (15)

where the dot stands for time derivative.The angular momentum balance equation is given by

"ijk�jk CCEi D "ijk

��jk CPjEk

�D "ijk

��jk C �

Ejk

�D 0, (16)

or

"ijk�jk D 0, (17)

which suggests that the total stress is symmetric.The conservation of energy is given by

�m Pe D �ij vi ,j C �mEi P�i . (18)

Free energy can be introduced through Legendre transform

D e �EiPi , (19)

and

�m P D �ij vi ,j �Pi PEi . (20)

The free energy provides the constitutive relations

Pi D Pi .Ej , "kj /, (21)

�ij D �ij .Ek , "kl /. (22)

In the present manuscript the weak form of governing equations will be expressed in the initialconfiguration. Therefore, we will transform the strong form of governing equations into the initialconfiguration. The balance laws in the so-called Lagrangian description are given by

DJ ,J � �E D 0, (23)

where DL D JF �1L,iDi and �E D �eJ are Lagrangian description of electric displacement andcharge density, Di D J�1Fi ,LDL.

The electric field is vortex free in the Lagrangian description and is given by

"IJKEK,J D 0, (24)

where EK D EiFi ,K D ��,iFi ,K D ��,K is the Lagrangian description of electric field,Ei DEKF

�1K,i . Equation (24) can be used implicitly to describe electric field as a gradient of electric

potential.

KkL,LCBk D �M Pvk , (25)

where KiL D JF �1L,j �ij and �M D �mJ are the first Piola–Kirchhoff stress and material den-sity, respectively,Bk D �Mbk ,and �ij D J�1Fj ,LKiL. KiLis nonsymmetrical tensor, satisfyingcondition

"kijFj ,LKiL D 0, (26)

which confirms that moments resulting from electric body force acting on the infinitesimal volumeare equilibrated by internal forces.

The energy balance equation is given by

�M P D TSKLPSKL �PK

PEK , (27)



where the second Piola–Kirchhoff stress T SKLis

T SKL D JF

�1K,kF

�1L,l �

Skl . (28)

The Green–Lagrange strain is given by

SKL D .uK,LC uL,K C uM ,KuM ,L/=2, (29)

and the Lagrangian description of polarization vector PK is

PK D JF�1K,kPk , Pi D J

�1Fi ,KPK , (30)

�Sij D �ij CPiEj D J

�1Fi ,KFi ,LTSKL. (31)

The energy balance determines constitutive equations, which can be written as

T SKL DT

SKL.SKL, EK/D �M

@

@SKL,

PK DPK.SKL, EK/D��M@

@EK.

(32)

KiL and DL may be calculated from

KjL DFj ,K�M@

@SKLC JFi ,L"0

�EiEj �

1

2EkEkıij

�,

DK D "0JC�1KLEL � �M

@

@EK.

(33)

To complete the strong form (23), (25), (32), (33) of the initial-boundary value problem is given by

ui D Nui on @�u,

� D N' on @�' ,

KkLNL D NTK on @�T ,NDKNK D� N! on @�! ,

ui D Nui .0/ at t D 0,

@ui

@tD Nvi .0/,

(34)

where N! is surface charge density applied on the portion of the surface @�! and NTK is mechanicaltraction applied on the portion of the surface @�T ; Nui and N' are displacements and electric potentialprescribed on the surfaces @�u and @�' , respectively, such that

@�! \ @�' D @�u \ @�T D @�

@�! [ @�' D @�u [ @�T D 0. (35)

Free energy, which determines constitutive relationships for nonlinear electroelastic materials, canbe written as power series of SAB and EA[14]

�M .SKL, EK/D1

2c2ABCDSABSSD� eABCEASBC�

1

2�2 ABEAEBC

1

6c3ABCDEF SABSCDSEFC

C1

2kABCDEEASBCSDE �

1

2bABCDEAEBSCD �

1

6�3ABCEAEBEC

C1

24c4ABCDEFGHSABSCDSEF SGHC

C1

6k2ABCDEFGEASBCSDESFG C

1

4a1ABCDEF EAEBSCDSEF

C1

6k3ABCDEEAEBECSDE �

1

4�4ABCDEAEBECED , (36)



where constants c2ABCD , eABC , �2 AB , c3ABCDEF , kABCDE , bABCD , �3ABC , c4ABCDEFGH ,k2ABCDEFG , a1ABCDEF , k3ABCDE�4ABCD are often referred to as the fundamental materialconstants. The structure of .SKL, EK/ depends on the symmetries of particular materials underconsideration [81, 82].

From the above strong form (34) it follows that the electric fields, E and P , are coupled to themechanical fields, F and K, at two levels [51]. First is the electrostatic coupling, when the electricfields directly generate distributed forces via Maxwell stress, which in turn affects the mechanicalequilibrium of a solid. The resulting stresses depend on the second-order terms of the electric field,and while typically very small (10�5 MPa for a l MV/m field), could have a significant effect if thefluctuations of the electric field are large (field singularities) as in the case of heterogeneous mediawith high contrast in the dielectric moduli or in electrodes and conducting crack tips [27, 83, 84].This nonlinear phenomenon is universal and common for both dielectrics and conductors [81],and has no converse effects, because the mechanical stress state does not directly influence theelectrostatic balance [51], but can affect it through the motion, when such a motion changes electricfield.

The second level of coupling occurs in the constitutive behavior of dielectric materials, expressedby Equations (21), (22) or (32), (33). Polarization induces strain, while mechanical stress affectspolarization, which introduces full coupling. Most common electromechanical materials arepiezoelectrics, ferroelectrics, and electrostrictors [85].

In piezoelectrics, there is a linear dependence between applied stress and induced electricdisplacement (direct piezoelectric effect)

Di D dijk�jk (37)

and between applied electric field and induced strain (inverse piezoelectric effect),

"ij D dkijEk , (38)

where dijk are piezoelectric coefficients. A positive electric field results in positive strain, and viceversa, negative electric field results in negative strain.

Ferroelectric materials have spontaneous polarization Ps in the absence of an external electricfield. The charge because of spontaneous polarization is usually masked by the charges fromthe surroundings of the material. The direction of spontaneous polarization can be switchedby external electric field. Thus, ferroelectric polycrystallines are characterized by hysteresis inpolarization-electric field P(E) and induced strain-electric field "(E).

In electrostrictive materials induced strain is proportional to the square of polarization

"ij DQijklPkPl , (39)

where Qijkl are electrostrictive coefficients. For mildly strong electric fields, the dependencebetween polarization and electric field is linear and (39) may be written as

"ij DMijklEkEl , (40)

where Mijkl D �km�lnQkmln and �ij is dielectric susceptibility. Formulation of ‘converse’electrostrictive effect is also possible [86]. Here, both positive and negative electric fields resultin positive longitudinal strain.

In both electrostriction and electrostatic coupling the overall behavior is determined by the aver-age of the fluctuation of the electric field (rather than its average). Therefore, one can obtain largecoupling with relatively small macroscopic field [20, 36]. Li et al. [36] described a three-phasepolymer-based actuator with more than 8% actuation strain attained with an activation field of20 MV/m.



3. MATHEMATICAL HOMOGENIZATION FOR NONLINEAR COUPLEDELECTROMECHANICAL PROBLEM

In this section we will introduce scale separation and proceed with deriving equations for differentscales from the strong form of governing equations in the Lagrangian description derived in theprevious section.

Consider a body with volume �� and surface @�� made of a heterogeneous solid with char-acteristic size of the heterogeneity l , which is much smaller than the body dimensions so thatheterogeneities are considered to be ‘invisible’. The body is effectively homogeneous at the coarse-scale; its fine-scale structure can be observed by zooming (Figure 1) at any coarse-scale point A. Itis assumed that elastic waves induced in this body by external loads are long enough to neglect thedispersion because of the material heterogeneity, that is, lengths of elastic waves are assumed to bemuch bigger than the size of heterogeneity l . We will denote the volume and boundary of the bodyon the coarse-scale by � and @�, respectively.

We assume that the body is composed of a periodic repetition of unit cells (UC) with volume‚Y . The periodicity may be global, when the whole body is a lattice consisting of unit cells ‚Y(Figure 2(a)) or local, when periodicity holds only in a small neighborhood of coarse-scale point(Figure 2(b)) [78].

To describe the dependence of various fields ‰�.X/ on the coarse-scale and fine-scale coor-dinates, we will introduce global coordinate system X and local (unit cell) coordinate system Yassociated with fine-scale structure placed at every coarse-scale point. The two coordinates arerelated by Y D X=; is a small parameter 0 < << 1, which can be introduced as a ratioof the unit cell size to the global size of the structure. Dependence of the field ‰�.X/ on the twoscale coordinates is denoted as

Figure 1. A body with fine-scale heterogeneities.

(a) (b)

Figure 2. Global (a) and local (b) periodicity.



‰�.X/D‰ .X, Y/ , (41)

where superscript denotes existence of fine-scale details. Unit cell coordinates Y in (41) aredefined with respect to the initial (undeformed) fine-scale configuration. All fields depend on time,but for simplicity time dependence is omitted in the notation.

To construct the weak form of the governing equations, we will need to integrate over the volume��. This can be carried out as long as the size of the unit cell is infinitesimally small, which yieldsthe following fundamental lemma of homogenization [87, 88]

lim�!0C

Z��

‰ .X, Y/ d�D lim�!0C

Z�

0B@ 1

j‚Y j

Z‚Y

‰.X, Y/d‚

1CAd�, (42)

which also implies that the fine-scale domain‚Y exists at every point in the coarse-scale domain(Figure 3).

To derive the coarse-scale equations we will start from the equilibrium equation and Gauss lawin the Lagrangian description (23), (25). As before, superscript denotes dependence of a quantityon fine-scale details

K�kL,L.F�, E �/CB�k .X/D �

�M Pv, (43)

D�J ,J .F

�, E �/� ��E.X/D 0 (44)

with initial and boundary conditions

ui D Nui on @�u,

� D N' on @�' ,

KkLNL D NTk on @�T ,NDKNK D� N! on @�! ,

u�i D Nui .0/ at t D 0,

@u�i@tD Nvi .0/ at t D 0,

(45)

@��! \ @��' D @��u \ @��T D @�,@��! [ @��' D @��u [ @��T D 0.

(46)

Utilizing the asymptotic expansion [89–91] of displacement and electric potential fields yields

u�i .X/D ui .X, Y/D u0i .X/C u1i .X, Y/C 2u2i .X, Y/CO.3/, (47)

'�.X/D ' .X, Y/D '0.X/C '1 .X, Y/C 2'2.X, Y/CO.3/. (48)

Figure 3. Transition from the heterogeneous to homogeneous domain as ! 0C.



This form of asymptotic expansions is typically used in classical homogenization theory, where itis assumed that the size of the unit cell is of the order , that is, infinitesimally small in the limit.Therefore, the leading order terms u0i .X/ and '0.X/ are considered to be constant over the unit celldomain, that is, do not depend on the fine-scale coordinate Y. However, for large unit cell distortionsleading order terms may no longer be constant over the unit cell domain and therefore Equations(47), (48) have to be modified as described below.

Assuming slow variation of every term in (47), (48) over the extent of a unit cell, every term in(47) and (48) can be expanded in Taylor series around the unit cell centroid XD OX

u0i .X/D u0i

�OX�C

@u0i@XJ

ˇ̌̌ˇOXYJ C

2 @2u0i@XJ @XK

ˇ̌̌ˇOXYJYK CO

�3�

, (49)

uni .X, Y/D uni . OX, Y/C @uni@XJ

ˇ̌̌ˇOXYJ C

2 @2uni@XJ @XK

ˇ̌̌ˇOXYJYK CO.

3/, (50)

'0.X/D '0�OX�C

@'0

@XJ

ˇ̌̌ˇOXYJ C

2 @2'0

@XJ @XK

ˇ̌̌ˇOXYJYK CO.

3/, (51)

'n.X, Y/D 'n. OX, Y/C @'n

@XJ

ˇ̌̌ˇOXYJ C

2 @2'n

@XJ @XK

ˇ̌̌ˇOXYJYK CO.

3/, (52)

where XJ � OXJ D YJ was used.Substituting the above into asymptotic expansions (47), (48) gives the modified asymptotic

expansion

u�i .X/D ui . OX, Y/D Ou0i . OX/C Ou1i .OX, Y/C 2 Ou2i . OX, Y/CO.3/, (53)

where

Ou0i .OX/D u0i

�OX�

, Ou1i .OX, Y/D u1i

�OX, Y

�C@u0i@XJ

ˇ̌̌ˇOXYJ , (54a)

Ou2i .OX, Y/D u2i . OX, Y/C

@u1i@XJ

ˇ̌̌ˇOXYJ C

1

2

@2u0i@XJ @XK

ˇ̌̌ˇOXYJYk (54b)

and similarly for potential

'�.X/D '. OX, Y/D O'0. OX/C O'1. OX, Y/C 2 O'2�OX, Y

�CO.3/, (55)

where

O'0. OX/D '0�OX�

, O'1. OX, Y/D '1�OX, Y

�C@'0

@XJ

ˇ̌̌ˇYJ , (56a)

O'2. OX, Y/D '2. OX, Y/C@'1

@XJ

ˇ̌̌ˇOXYJ C

1

2

@2'0

@XJ @Xk

ˇ̌̌ˇOXYJYK . (56b)

In the classical homogenization the spatial derivative is defined as @f �=@XI D .@f .X, Y/ =@XI /C.1=/.@f .X, Y/=@YI /. Because in the modified asymptotic expansions (53) and (55) all termsare expanded in the vicinity of the unit cell centroid OX, the dependence on X was eliminated bysubstitution X� OXD Y; therefore, the spatial derivative reduces to

@f �.X/@Xi

D1

@f . OX, Y/@Yi

. (57)



Consequently, the displacement gradient may be expressed as

@u�i@XK

D1

@ui . OX, Y/@YK

D@ Ou1i

�OX, Y

�@YK

C @ Ou2i

�OX, Y

�@YK

CO�2�

, (58)

where

@ Ou1i

�OX, Y

�@YK

D@u1i .

OX, Y/@YK

C@u0i .X/@XK

ˇ̌̌ˇOX

,

@ Ou2i

�OX, Y

�@YK

D@u2i .

OX, Y/@YK

C@u1i .X, Y/@XK

ˇ̌̌ˇOXC@2u1i .X, Y/@XJ @YK

ˇ̌̌ˇOXYJ C

@2u0i .X, Y/@XJ @XK

ˇ̌̌ˇOXYJ .

(59)

The asymptotic expansion of the deformation gradient is given as

F �iK D ıiK C@u�i@XK

D F 0iK.OX, Y/C F 1iK. OX, Y/CO

�2�

, (60)

where

F 0iK.OX, Y/D ıiKC

@ Ou1i

�OX, Y

�@YK

D ıiKC@u1i .

OX, Y/@YK

C@u0i .X/@XK

ˇ̌̌ˇOXD F CiK

�OX, Y

�CF �iK.

OX, Y/, (61)

F CiK.OX/D ıiK C

@u0i .X/@XK

ˇ̌̌ˇOX�

@xi

@XK

ˇ̌̌ˇOX

, F �iK. OX, Y/D@u1i .

OX, Y/@YK

, F 1iK�OX, Y

�D@ Ou2i .

OX, Y/@Yk

.

(62)Similarly, the electric field is given as negative derivatives of electric potential

E �J .X/D �@'�

@XJD�

1

@'. OX, Y/@YJ

D�@ O'1

�OX, Y

�@YJ

� @ O'2

�OX, Y

�@YJ

CO�2�D

D E 0J .OX, Y/C E 1J .

OX, Y/CO�2�

,

(63)

where

E 0J .OX, Y/D�

@ O'1

@YJD�

@'1. OX, Y/@YJ

�@'0.X/@XJ

ˇ̌̌ˇOXD E CJ .

OX/C E �J

�OX, Y

�, (64)

E CJ .OX/D�

@'0.X/@XJ

ˇ̌̌ˇOX

, E �J

�OX, Y

�D�

@'1�OX, Y

�@YJ

, E 1J .OX, Y/D�

@ O'2

@YJ. (65)

The coarse-scale deformation gradient F CiK and electric field E CJ are related to the average of the

leading order deformation gradient F 0iK

�OX, Y

�in (61) and electric field E 0J .

OX, Y/ in (64) over the

unit cell domain

F CiK.OX/D

1

‚Y

Z‚Y

F 0iK.OX, Y/d‚Y D ıiK C

@u0i .X/@XK

ˇ̌̌ˇOX

, (66)

E CJ .OX/D

1

j‚Y j

Z‚Y

E0J

�OX, Y

�d‚D�

@'0

@XJ

ˇ̌̌ˇOX

, (67)



where conditionsR‚Y

.@u1i =@YJ )d‚D 0 andR‚Y

.@'1=@YJ /d‚D 0 were employed. These con-ditions are satisfied when the periodic of weakly periodic boundary conditions for fine-scaledisplacements and potential is used. This will be discussed in the next section.

Expanding stress and electric displacement in Taylor series around the leading order deformationgradient F0.X, Y/ and electric field E0. OX, Y/ yields

K�iJ .F�,E�/DKiJ .F0,E0/C

@KiJ

@FmN

ˇ̌̌ˇF0,E0

F 1mN C @KiJ

@EM

ˇ̌̌ˇF0,E0

E 1M CO.2/ (68)

DK0iJ C K1iJ CO.

2/

D�J .F

�,E�/DDJ�F0,E0

�C

@DJ

@FmN

ˇ̌̌ˇF0,E0

F 1mN C @DJ

@EM

ˇ̌̌ˇF0,E0

E 1M CO.2/ (69)

DD0J C D1

J CO�2�

Taylor expansion of stress and electric displacement around the unit cell centroid OX yields

KiJ .F�,E�/DKiJ .X, Y/DK0iJ . OX, Y/C �@K0iJ@XK

ˇ̌̌ˇOXYK CK

1iJ .OX, Y/

�CO

�2�

, (70)

DJ .F�,E�/DDJ .X, Y/DD0J .OX, Y/C

�@D0

J

@XK

ˇ̌̌ˇOX

YK CD1J

�OX, Y

��CO.2/. (71)

Again, here the relation .Xi � OXi / D Yi has been utilized. Further substituting above and thesecond time derivative of (53) into equilibrium Equation (43) and Gauss law (44) and neglectingO./ terms yields

�1@K0iJ .

OX, Y/

@YJC@K0iJ@XJ

ˇ̌̌ˇOX

C@KiJ

1. OX, Y/@YJ

CBi . OX, Y/� �M. OX, Y/@2 Ou0i

�OX�

@t2D 0, (72)

�1@D0

J .OX, Y/

@YJC@D0

J

@XJ

ˇ̌̌ˇOX

C@D1

J

�OX, Y

�@YJ

� �E. OX, Y/D 0. (73)

Collecting terms of equal orders in (72), (73) yields the two-scale equilibrium equation andGauss law

O.�1/ W@K0iJ

�OX, Y

�@YJ

D 0,@D0

J

�OX, Y

�@YJ

D 0, (74)

O.0/ W@K0iJ@XJ

C@K1iJ

�OX, Y

�@YJ

CBi

�OX, Y

�D �M

�OX, Y

� @2 Ou0i � OX�@t2

,

@D0J

@XJC@D1

J .OX, Y/

@YJD �E. OX, Y/.

(75)

Integrating O.0/ equations over the unit cell domain and exploiting weak periodicity conditionsfor fine-scale stress and electric displacement

R‚.@K1iJ .

OX, Y/=@YJ /dY D 0 andR‚.@D1

J .OX, Y/=

@YJ /dY D 0, yields the coarse-scale equations

@KCiJ . OX/@XJ

ˇ̌̌ˇOXCBCi

�OX�D N�M. OX/

@2 Ou0i. OX/

@t2,

@DCJ . OX/@XJ

ˇ̌̌ˇOX

D �CE .OX/,

(76)



where

KCiJ .OX/D

1

‚Y

Z‚Y

K0iJ

�F0�OX, Y

�,E

�OX, Y

��d‚, BCi . OX/D

1

‚Y

Z‚Y

Bi . OX, Y/d‚, (77)

DCJ .OX/D

1

‚Y

Z‚Y

D0J

�F0�OX, Y

�,E. OX, Y/

�d‚, �CE .X/D

1

‚Y

Z‚Y

�E. OX, Y/d‚. (78)

As is commonly used in homogenization theory we make use of the relation OX D X � Y and theunit cell infinitesimality, then

lim�!0

. OX/D lim�!0

.X� Y/D X (79)

and the centroid OX can be replaced by an arbitrary point X. This type of assumption is made forinstance in separating equilibrium equation with multiple orders of into multiple scale equilibriumequations.

Equations (76) with boundary conditions (75) define the coarse-scale boundary value problem.The set of Equations (74) together with periodic (or weakly periodic) boundary conditions,discussed in the next section defines the fine-scale boundary value problem.

The resulting coarse-scale equations are of the same form as initial Equations (43), (44). Theassumption that the resulting coarse-scale equations have the same form as the original equa-tions is often used in homogenization frameworks [68, 74, 78, 79]. Even though we have similarequations, in mathematical homogenization we did not make any assumption about the form ofmacroscopic equations in advance. The form of coarse-scale equations is a consequence of assump-tions made during derivations such as the infinitesimality of the unit cell and long wave limit. This isan important difference of mathematical homogenization from alternative frameworks. Accountingfor additional terms in asymptotic expansion (such as accelerations, gradient terms) will result incoarse-scale equations being different from the fine-scale equations and can lead to strain gradient,nonlocal continuum [88] or in appearance of additional terms, such as second space derivative ofacceleration [92].

4. BOUNDARY CONDITIONS FOR THE UNIT CELL PROBLEM

In this section various boundary conditions imposed on the unit cell will be considered.The most common are the periodic boundary conditions, when the displacement and electric

potential perturbations on the opposite sides of the unit cell are assumed to be equal.Consider the asymptotic expansions for displacement and electric potential fields (53), (55)

u�i .X/D ui . OX, Y/D u0i�OX�C

h�F CiJ

�OX�� ıiJ

�YJ C u

1i

�OX, Y

�iCO.2/,

'�.X/D '. OX, Y/D '0�OX�C

h�ECJ .

OX/YJ C '1. OX, Y/iCO.2/,

where (65) and (66) were used (we keep here only terms up to O ./, because terms O.2/ do notenter unit cell and macroscopic equations). The leading order terms, which represent the rigid bodytranslation of the unit cell and constant potential, are independent of the unit cell coordinates Y.TheO./ terms describe the unit cell distortion and electric field in it. The terms .F CiJ . OX/� ıiJ /YJand E CJ .

OX/YJ represent a uniform coarse-scale deformation and uniform coarse-scale electric field,

while the terms u1i

�OX, Y

�and '1. OX, Y/ capture the deviations from the uniform fields induced by

material heterogeneity. These terms are important for electrostriction effect [27] and making theseterms larger increases the electrostriction effect.



Y1

Y2

1y2

M S

(a) Initial unit cell (b) Deformed unit cell

M′ M′′ S′ S′′

C1 2C′

C3C4

C2

1C′

3C′ 4C′y2

ΘYy

Θ2

Figure 4. Definition of periodic boundary conditions.

Figures 4(a) and (b) show the initial and deformed shape of the unit cell, respectively. The dotted

line in Figure 4(b) depicts the deformed shape of the unit cell because�F CiJ

�OX�� ıiJ

�YJ , whereas

the solid line shows the contribution of the u1i

�OX, Y

�term.

At the unit cell vertices @‚vertY , the deviations from the uniform fields, u1i . OX, Y/and '1. OX, Y/, are

assumed to vanish. For the remaining points on the boundary of the unit cell, the deviations fromthe average u1i . OX, Y/, '1i . OX, Y/ are prescribed to be periodic functions. Figure 4 depicts two pointsM and S on the opposite faces of the unit cell with M and S termed as the master and slave points,respectively. The displacements of the two points are given as

Ou1i .OX, YM /D

�F CiJ .OX/� ıiJ

�YMJ C u

1i .OX, YM /, (80)

Ou1i .OX, YS /D

�F CiJ .OX/� ıiJ

�Y SJ C u

1i .OX, YS /. (81)

Subtracting Equation (81) from (80) and accounting for periodicity, that is, u1i . OX, YM / Du1i .OX, YS /, gives a multipoint constraint equation

Ou1i

�OX, YM

�� Ou1i

�OX, YS

�D�F CiJ .OX/� ıiJ

�.YMJ � Y

SJ /, (82)

where YM and Y S represent the local coordinates of the master and slave nodes on the unit cellboundary, respectively.

Similarly, periodicity of electric potential perturbations '1�OX, YM

�D '1. OX, YS /gives the

following constraint equation:

O'�OX, YM

�� O'

�OX, YS

�D�E CJ .

OX/�YMJ � Y

SJ

�. (83)

Periodic boundary conditions can be used for periodic heterogeneous medium, when the unit celldistortion is not considerable, otherwise they are not physical. For nonperiodic medium or in thecase of large unit cell distortions more general boundary conditions have to be used instead. Notethat the only time when the periodicity condition was exercised was in deriving Equations (66), (67).For (66), (67) to hold the following conditions must be satisfied:

Z‚Y

@u1i .X ,Y /@YJ

d‚Y D 0,Z‚Y

@'1.X ,Y /@YJ

d‚Y D 0. (84)



Applying Green’s theorem and exploiting relations (54b), (56b), and (66), (67) the above reduces to

Z@‚Y

�Ou1i

�OX, Y

��F CiK

�OX�� ıiK

�YK

�NKdY D 0,

Z@‚Y

�O'1�OX, Y

��ECK

�OX��YK

�NKdY D 0,

(85)

where @‚Y is the boundary of ‚Y and NK are the components of the unit normal to the boundary@‚Y . Equation (85) represents the so-called weak periodicity condition.

Alternatively to Equations (82), (83) and (85) an essential boundary condition

Ou1i .OX, Y/�

�F CiK.

OX/� ıiK�YK D 0,

O'1. OX, Y/��E CK .

OX/�YK D 0

(86)

is often exercised in practice. It corresponds to zero perturbations from coarse-scale fields on theunit cell boundary.

The essential boundary conditions can be enforced either in the strong form (86) or in the weakform as

R@‚Y

�Ou1i

�OX, Y

��F CiK

�OX�� ıiK

�YK

��idY D 0,

R@‚Y

�O'1�OX, Y

��ECK

�OX��YK

��'dY D 0,

(87)

where �i ,�' are the Lagrange multipliers representing unknown tractions and surface charge den-sity on @‚Y . If we choose �i D KCiJNJ , �' D DC

J NJ where KCiJ , DCJ are constant over ‚Y and

require (87) to be satisfied for arbitrary KCiJ and DCJ , we then obtain Equation (85). Therefore,

Equation (85) will be referred to as a weak compatibility boundary condition, while Equation (86)as the strong compatibility condition. Equation (85) is in the spirit of the work of Mesarovic andPadbidri [93], who imposed the unit cell to satisfy average small strains. The natural boundary con-dition (85) is equivalent to uniform traction and allows large boundary fluctuations [79], which innot consistent with assumptions made in asymptotic expansions.

The essential (86) and natural (85) boundary conditions can be combined in the so-called mixedboundary conditions by requiring additionally to (85) condition

ˇ̌̌�Ou1i

�OX, Y

��F CiK.

OX/� ıiK�YK

�NKdY

ˇ̌̌6 "u,ˇ̌̌�

O'1�OX, Y

��ECK .

OX/�YK

�NKdY

ˇ̌̌6 "'

(88)

This condition is more restrictive than (85), but more compliant than (86), which is enforced up totolerances "u, "' . The mixed boundary condition can be implemented in various ways. For instance,by defining double nodes on the boundary and placing a linear or nonlinear spring between them[70] or by introducing a pad region surrounding unit cell [88].

The different boundary conditions can be denoted for generality

g

��Ou1

'1

�D 0 on @‚Y (89)



5. TWO-SCALE PROBLEM

The two-scale problem, consisting of the unit cell Equations (74) subjected to periodic (or other)boundary conditions (88) and the coarse-scale Equations (76), is two-way coupled. The link betweenthe two scales is schematically shown in Figure 5.

The fine-scale problem is driven by the overall (coarse-scale) deformation gradient F CiK. OX/ andelectric field E CK .

OX/. They are calculated for every material point of the coarse-scale problem(in practice only for integration points of the coarse mesh) and used together with (88) to for-mulate boundary conditions to be imposed on the unit cell. Once the unit cell problem is solved,it provides the coarse-scale problem with coarse-scale stress KCiJ and electric displacement DC

J via(77), (78). This effectively provides a coarse-scale constitutive relationship. Additionally, the localcoarse-scale consistent tangent is derived from the fine-scale stiffness. This framework is similar tocomputational homogenization frameworks [68, 74, 79].

This scale-bridging approach belongs to the category of information-passing (sometimes referredto as hierarchical or sequential) multiscale methods which evolve a coarse-scale model by advanc-ing a sequence of fine-scale models in small windows (representative volume or unit cell) placed atthe Gauss points of the discretized coarse-scale model.

The coupled two-scale problem is summarized below:

(a) Fine scale problem

(90)

Figure 5. Information transfer between the coarse-scale and fine-scale problems.



(b) Coarse-scale problem

(91)

6. FINITE ELEMENT DISCRETIZATION

Both the coarse-scale and fine-scale problems are discretized using finite elements. The displace-

ment and electric potential fields of the fine-scale problem Ou1i

�OX, Y

�, O'1

�OX, Y

�are approximated

as

Ou1i .OX, Y/DN 1

B.Y/d1iB.OX/,

O'1. OX, Y/DN 1B.Y/'

1B

�OX�

,(92)

where subscript B denotes the node number, N 1B.Y / are the unit cell shape functions and d1iB , '1B

are the nodal displacements and potentials in the unit cell mesh. Let dMiC . OX/, 'MC .OX/ be the master

(independent) degrees of freedom and express d1iB , '1B by a linear combination of dMjC . OX/and

'MC .OX/ defined by

"d1iB.

OX/

'1B.OX/

#D

"T diBkC

. OX/ 0

0 T'BC .

OX/

#"dMkC. OX/

'MC .OX/

#(93)

so that the constraint equation (89) in the discrete form is satisfied

g

N 1B.Y /

"T diBkC

. OX/ 0

0 T'BC .

OX/

#"dMkC. OX/

'MC .OX/

# !D 0 on @‚Y . (94)

Then writing the Galerkin weak form of (90) and discretizing it using (92) yields the discreteresidual equation

"rd1kC.mC1nC1�d

1,mC1nC1�'1/

r'1C .

mC1nC1�d

1,mC1nC1�'1/

#�

Z‚Y

"T diBkC

. OX/ 0

0 T'BC .

OX/

#@N 1

B

@YJ

mC1nC1

�KjJDJ

d‚Y D 0,

(95)where the left subscript and superscript denote the load increment and the iteration count (forimplicit method) at the coarse-scale, respectively; mC1nC1 r

d1iB and r'1C , mC1nC1�d

1iBand mC1

nC1�'1are the

residuals, displacement, and potential increments in the .m C 1/th iteration of the .n C 1/th loadincrement, respectively. If the constitutive equations are defined in terms of the Cauchy stress and



electric displacement in the current configuration it is convenient to restate the unit cell problem asfollows:

(96)

where we have exploited the relation between the first Piola–Kirchhoff stressKiK , electric displace-ment DK in the Lagrangian description and their current configuration counterparts, Cauchy stress�ij and electric displacement Di

J�ij D FjKKiK ,

JDi D FiKDK(97)

and J in (97) is the determinant of FjK .Similarly, the coarse-scale displacements and potential Ou0i . OX , t /, O'0. OX , t / are discretized as

Ou0i .OX/DNC

A .OX/dCiA,

O'0. OX/DNCA .OX/'CA ,

(98)

where NCA .OX/, dCiAand 'CA are the coarse-scale shape functions, nodal displacements, and potential,

respectively. Writing the weak form of (91) and using discretization (98) the discrete coarse-scaleequations can be expressed as

(99)

wherenC1rd CiA , nC1r' CiA and nC1�d

CiA, nC1�'CA are the coarse-scale residuals, displacement, and

potential increments in the .nC 1/th load increment, respectively, and

MijAB D ıij

Z�

�CMNCA N

CB d�, (100)

f intiA D

Z�

@NCA

@XJKCiJd�, (101)

f extiA D

Z�

NCA B

Ci d�C

Z@�T

NCANT di , (102)

f' intA D

Z�

@NCA

@XJDCJ d�, (103)



f' extA D

Z�

NCA �

CE d�C

Z@�!

N 0A N!d , (104)

whereMijAB is the mass, f intiA and f ext

iA the internal and external forces, respectively. It is againconvenient to express the coarse-scale governing equations in the deformed configuration

MijAB D ıij

Z�x

�CmNCA N

CB d�x , (105)

f intiA D

Z�x

@NCA

@xj�Cij d�x , (106)

f extiA D

Z�x

NCA b

Ci d�x C

Z@�tx

NCANtidx , (107)

f' intA D

Z�x

@NCA

@xjDCj d�x , (108)

f' extA D

Z�x

NCA �

Ce d�x C

Z@�!x

NCA N!

dc x , (109)

where

�Cm D �CM=J

C , bCi D BCi =J

C , �Ce D �CE=J

C , Ntidx D NTid , N!xdx D N!d , (110)

�Cij D FCjKK

CiK=J

C (111)

and JC is the determinant of F CiJ ;�x , d�x ,@�x ,dx denote volume, infinitesimal volume,boundary and surface element in the current configuration.

We now focus on deriving a closed form expression for the overall Cauchy stress �Cij and electric

displacementDCj . Substituting (97) into (77), (78) and recalling d‚y D Jd‚Y , we have

KCiK.OX/D

1

j‚Y j

Z‚y

F �1Km�imd‚y , DCK .OX/D

1

j‚Y j

Z‚y

F �1KmDmd‚y . (112)

Inserting (112) into (111) and denoting the volume of the coarse-scale (intermediate) configurationasˇ̌‚�yˇ̌D NJ j‚Y j, the overall quantities can be expressed as

(113)

where �F �1jm D F CjKF�1Km maps the fine-scale deformed configuration ‚y into the coarse-scale

deformed configuration ‚�y as illustrated in Figure 6.The coarse-scale problem may be solved using either explicit or implicit time integration [94].



Y1

Y2

(a) Initial configuration

C1

3C 4C

2C

1C′ y1

2C′

1C′

3C′ 4C′

*y1

*2y

(b) coarse-scale deformed configuration

2C′

3C′ 4C′y2

(c) fine-scale deformed configuration

F

ΔF

F

YΘ

*yΘyΘ

Figure 6. Unit cell configurations: (a) initial, (b) coarse-scale (intermediate), and (c) fine-scaledeformed (final)

7. NUMERICAL EXAMPLES

In this section we will consider a numerical example illustrating the ability of the mathematicalhomogenization to resolve the coarse-scale behavior of heterogeneous materials with nonlinear elec-tromechanical coupling subjected to electric field. The results of the mathematical homogenization(to be referred as MH) will be compared with the direct numeric simulation (DNS) where a very finemesh is employed to resolve fine-scale details. For simplicity, we will consider a two-dimensionalproblem (plane strain).

For the nonlinear electromechanical material we will use a simple relaxor ferroelectric mate-rial model proposed in [51], where polarization and strain are used as independent variables. It isassumed that the material is isotropic, where the stress depends linearly on total strain. The polariza-tion induced strain"E depends on the square of polarization, that is, it is an electrostrictive material,where polarizations saturates to Ps at high electric fields. The internal energy function for relaxorferroelectric was proposed by Hom and Shankar as

U D1

2."� "

E/ W C W ."� "

E/C

1

2k

"jPj ln

�PsC jPjPs � jPj

�CPs ln

1�

�jPjPs

�2!#, (114)

"E is polarization-induced strain defined as

"EDQ W .PP/D .Q11 �Q12/PPCQ12jPj2I, (115)

CandQ are an isotropic elastic stiffness matrix and an isotropic electrostrictive strain coefficientmatrix, respectively; k is material constant; coefficientsQ11 and Q12 are defined so that the longi-tudinal and transverse induced strains relative to polarization direction are Q11jPj2 and Q12jPj2,respectively.

Stress and electric field are calculated from the internal energy by differentiation

� D@U

@"D C W ."�Q W .PP// , (116)



ED@U

@PD�2 ."�Q W .PP// W C WQ � PC

1

karctan h

�jPjPs

�PjPj

. (117)

The first term in (117) represents the converse electrostrictive effect, while the second term rep-resents the stress-free dielectric behavior. Polarization can be written in more compact formas

PD Ps tanh .kjRj/RjRj

, (118)

where

RD EC 2 ."�Q W .PP// W C WQ � PD EC 2� WQ � P. (119)

Thus, electric displacement will be

DD "0EC PD "0ECPs tanh .kjRj/RjRj

. (120)

For a given strain and electric field (which are known from the previous iteration), one cansolve (118) for induced polarization and then use (116) to calculate stress. The Jacobian for theconstitutive model can be derived from (118) and (116) as�

d�dD

D

�C� 4C WQ � P �Z � .Q � P/T W C �2C WQ � P �Z

2Z � .Q � P/T W C "0ICZ

�

�d"dE

, (121)

where

ZDhI�H �

h2 ."�Q W .PP// W C WQ� 4 .Q � P/T W C WQ � P

ii�1�H, (122)

HD Ps tanh .kjRj/�

IjRj�

RRjRj3

C

kPsRR

cosh2.kjRj/ jRj2. (123)

Numerical values of model parameters for PMN-PT-BT at 5°C are given in Table I. Dependenceof polarization and induced strain on electric field is presented in Figure 7.

The above model is valid for small strains, so consequently we consider a problem where inducedstrains are small, in which case stress, strain, electric displacement, and electric field coincide ininitial and deformed configurations.

Table I. Model parameters for PMN-PT-BT at 5°C.

Y (GPa) Q11.m4/C 2/ Q12.m

4/C 2/ Ps(C/m2/ k(m/MV)

115 0.26 1.33�10�2 –6.06�10�3 0.2589 1.16

(a) Polarization as function ofelectric field

(b) Electric field-induced strain

Figure 7. Electrostrictive material model.



7.1. Actuator example

This example demonstrates the ability of mathematical homogenization to model the dependence ofcoarse-scale behavior on fine-scale details.

Consider a beam with the top half made of electroactive material and lower half made of materialwith no electromechanical coupling (Figure 8). The electroactive material is a periodic compositewith unit cell consisting of matrix material and horizontal electroactive fiber. Both materials havethe same mechanical properties (linear isotropic material with Young’s modulus E and Poisson’sratio /, with fiber additionally having electromechanical coupling — electrostriction with parame-ters given in Table I . Unit cell dimensions are 100 �m �100 �m.

The left side of the beam is mechanically fixed and grounded. A potential 36 kV (this potentialcorresponds to the saturation of polarization) is applied to the right-hand side of the beam.

When voltage is applied, the electrostrictive material in the upper half of the beam extendsbecause of electrostriction effect and the beam will bend as shown on Figure 8(c). This permitsusing the beam as an actuator.

We considered three cases with different fiber thicknesses a: a=100 �m (i.e., the whole unit cellconsists of electrostrictive material), a=60 �m and a=40 �m. Deflections of the point A as a func-tion of pseudotime (pseudotime parameterizes load) for different fiber thicknesses a found by DNSsimulation are shown in Figure 9(a). It can be seen that different fiber sizes result in different actu-ation capabilities for the same loading, that is, coarse-scale response is sensitive to the fine-scaledetails.

The purpose of homogenization is to capture this dependence. Figure 9 shows also the compar-ison of DNS simulation with MH for different values of a. The MH simulations were performedfor two different meshes with 96 elements and with 192 elements to study the convergence of MHsolution to DNS solution. For a=40 �m, the error between DNS and MH was 4.8% for 96 elementmesh and 1.3% for 192 element mesh. For a=60 �m, the error was 4.9% and 1.4%, respectively. Fora=100 �m, error was 5%. These results suggest that the mathematical homogenization is capableof reproducing the reference solution and captures the dependence of the coarse-scale response onfine-scale details.

The dependence of macroscopic response on the volume fraction of inclusion can be addressedalso by a rule of mixtures. In fact, the constant strain mixture rule will give very good result forhorizontal fiber inclusions, while constant stress mixture rule provides accurate results for verticalfiber inclusion. However, in general mixture rules are not capable of addressing the dependenceof macroscopic response on the shape of inclusion. This is especially problematic if the shape ofinclusion can change during loading, for example because of electromechanical actuation.

(a) undeformed beam

(b) unit cell

(c) deformed beam

Figure 8. (a) geometry of the beam and boundary conditions: the upper half is made of heterogeneous mate-rial, containing electroactive phase and phase without electromechanical coupling and consisting of periodicrepetition of unit cells, (b) a square unit cell consisting of matrix and electrostrictive horizontal fiber, and

(c) deformation of the beam when voltage is applied to the right side.



(a) (b)

(c) (d)

Figure 9. (a) displacements of point A, DNS simulations of actuation with different fractions of elec-trostrictive phase and Comparison of DNS (red line) and MH (blue line for 96 element and green linefor 192 elements) solutions, for different thicknesses of electroactive fiber (b) a=100 �m, (c) a=60 �m and

(d) a=40 �m.

To verify the performance of the mathematical homogenization, we considered the same problemas depicted in Figure 8(a) for different electroactive inclusions including horizontal, vertical, and cir-cular inclusions with the same volume fraction. Figure 10 demonstrates the ability of mathematicalhomogenization to address the dependence of the macroscopic response on the shape of inclusions.It can be seen that mathematical homogenization solution agrees well with DNS for different inclu-sions, while the constant strain mixture solution coincides with the mathematical homogenizationsolution for horizontal fibers only and is insensitive to the shape of the inclusion. This ability of themathematical homogenization to address shape dependence justifies increased computational cost.

7.2. Beam bending

Now consider the whole beam (with the same dimensions as previously) made of an electrostrictivematerial subjected to a linearly varying traction F D 0.25 � 10E C 5 � Y Pa, as shown in Figure 11.The left side of the beam is mechanically fixed and grounded. Electric potential ' is applied to theright-hand side of the beam. Again, the unit cell consists of matrix and fiber as depicted inFigure 8(b) with a=40 �m. The matrix is assumed to be a hyperelastic material with stressdepending exponentially on the first invariant of the strain I1 D "11C "22.

The application of electric field changes the mechanical properties of the beam, which results indifferent deflections under the same loading. We compare the results of DNS and MH to show theability of MH to capture the change of mechanical properties because of biasing fields.



Figure 10. Comparison of displacements at the pointA for different inclusion shapes as obtained with math-ematical homogenization and direct numerical simulation. In all cases the inclusions volume fraction is taken

to be the same. Constant strain rule of mixtures coincides with homogenization with horizontal fiber.

Figure 11. Beam with mechanical load applied to the left end.

Figure 12. Comparison the DNS and MH simulations for beam bending.

Figure 12 shows the deflection of point A as function of pseudotime for the two cases. In the firstcase the applied potential was '=36 kV. The result of DNS simulation is depicted by a red line andMH (192 elements) simulation is depicted with a green line. The difference between DNS and MHat time t=0.5 was 5.6% for 96 elements and 1.6% for 192 elements. In the second case the rightend of the beam was grounded, '=0. The result of DNS simulation is depicted by a blue line and



MH (192 elements) simulation is depicted by magenta line. The difference between DNS and MHsimulation at time t=0.5 was 5.2% for 96 elements and 1.5% for 192 elements.

It can be seen that the mathematical homogenization is capable of capturing the change ofmechanical properties because of biasing electric fields and reproduces the reference solution withgood approximation.

7.3. Wave propagation in a layered beam

Finally, consider wave propagation in a beam consisting of vertical layers of matrix material withYoung’s modulus Y = 8.96E+9 Pa and Poisson’s ratio D 0.3, and electroactive fibers with the sameparameters as in previous examples (Figure 13(a)). The unit cell size has the same dimensions as inprevious examples; however, the fiber is oriented in the vertical direction and its thickness a=19.6�m. The beam has length 8.0�10�3 m and thickness 4.0�10�4 m. The right end of the beam is con-strained; the top and bottom surfaces are constrained in the y-direction. Displacement compressiveimpulse of the form

u.t/D

´Nv�t �

Timp

2�sin�2�tTimp

��, t 6 Timp

Nv , t > Timp

is applied in horizontal direction to the left end of the beam (Figure 13(b)), whereTimp D 1.0�10�6s,Nv D 1.0m/s. This boundary condition induces longitudinal compression wave in the beam. Twocases are compared: when both ends of the beam are grounded and when potential '=16.0 kV isapplied to the left end of the beam (potential is applied before displacement). The application of the

(a) (b)

Figure 13. (a) Layered beam and (b) displacement boundary conditions u.t/(measured in �m) prescribedat the left end of the beam and corresponding velocity v.t/.

Figure 14. Longitudinal stress distribution along the beam; comparison of DNS and MH solutions at timet=3 �s for two cases: when both ends of the beam are grounded and when electric potential 36 kV is applied

to the left end on the beam.



electric field induces biasing stress field in the beam, which results in increasing of the peak of thestress wave. The comparisons of the stress wave profile at time t D 3.0�s for DNS and MH simu-lations are shown on Figure 14. Mathematical homogenization solution is in good correspondencewith reference solution for both cases.

8. CONCLUSIONS AND FUTURE WORK

In this manuscript we have extended the mathematical homogenization theory for the case of mate-rials with full nonlinear electromechanical coupling, including coupling through Maxwell stress,which is substantial in the presence of a strong electric field, and especially important in the caseof electrostrictive materials. The framework proposed in this manuscript is applicable to arbitraryfine-scale structures and arbitrary constitutive models of phases and allows large deformations, largeelectric fields, and large distortions of UC.

Macroscale equations were derived without a priori assumptions on their form using asymptoticexpansions. This is a unique feature of the proposed mathematical homogenization framework incomparison to the alternative frameworks that can be found in the literature. The form of the coarse-scale equations is a consequence of assumptions made during derivations such as the infinitesimalityof the unit cell and long wave limit. Accounting for additional terms in asymptotic expansion(such as accelerations, gradient terms) will result in change of coarse-scale equations being differ-ent from fine-scale equations. In terms of solution procedure, described in Section 5, this model issimilar to existing homogenization techniques except for more general mixed boundary conditions,which are more accurate than the classical periodic, essential, or natural boundary conditions.

The examples presented in this manuscript show that nonlinear mathematical homogenizationcaptures well the coarse-scale behavior of heterogeneous electroactive composite and its dependenceon the fine-scale details.

At the same time the method shares the shortcomings common to the first-order homogenizationmethods; it is insensitive to the absolute size of the UC because of assumption of UC infinitesimality[88]. This lack of accuracy increases with the unit cell size and the magnitude of strains and electricfields inhomogeneities.

In the future we plan to study mathematical homogenization for various dynamic problems. Inparticular are studies of wave propagation in the media with periodic resonant structures. It wouldbe interesting to explore if mathematical homogenization can capture the negative effective refrac-tive indexes as it was found in metamaterials. Another approach would be to ascertain whether thepresent formulation can be used to control band gaps by biasing fields and will it allow extendingeffective band gaps for use in various devices (for subwavelength imaging, wave attenuation, etc.).The other interesting phenomenon, which possibly may be studied using mathematical homoge-nization, is the controlled response of smart structures to various impacts. Depending on the impact,a control strategy may be developed to obtain the desired response from the smart structure. Thisphenomenon might be utilized in the development of smart armor and other structures. Fine-scalestructure optimization aimed at optimizing coarse-scale properties, in particular for electroactivepolymers containing composites, is another useful application of the method. The asymptotic expan-sion would allow to isolate the term '1. OX, Y, t /, responsible for electrostrictive and electrostaticcoupling. Making this term larger will permit increasing the actuation.

ACKNOWLEDGEMENT

The authors wish to thank Dr. Peter Chung from the Army Research Laboratory for supporting this work.

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Mathematical homogenization theory for electroactive continuum

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