Effect of flexoelectricity on the electromechanical responseof graphene nanocomposite beam

S. I. Kundalwal . K. B. Shingare . Ankit Rathi

Received: 7 September 2018 / Accepted: 17 September 2018

� Springer Nature B.V. 2018

Abstract Owing to its unique multifunctional and

scale-dependent physical properties, graphene is

emerged as promising reinforcement to enhance the

overall response of nanotailored composite materials.

Most recently, the piezoelectricity phenomena in

graphene sheets was found through interplay between

different non-centrosymmetric pores, curvature and

flexoelectricity phenomena. This has added new

multifunctionality to existing graphene and it seems

the use of piezoelectric graphene in composites has yet

to be fully explored. In this article, the mechanics of

materials and finite element models were developed to

predict the effective piezoelectric and elastic (piezoe-

lastic) properties of the graphene reinforced nanocom-

posite material (GRNC). An analytical model based on

the linear piezoelectricity and Euler beam theories was

also developed to investigate the electromechanical

response of GRNC cantilever beam under both

electrical and mechanical loads accounting the flex-

oelectric effect. Furthermore, molecular dynamics

simulations were carried out to determine the elastic

properties of graphene which were used to develop the

analytical and numerical models herein. The current

results reveal that the flexoelectric effect on the elastic

behavior of bending of nanocomposite beams is

significant. The electromechanical behavior of GRNC

cantilever beam can be tailored to achieve the desired

response via a number of ways such as by varying the

volume fraction of graphene layer and the application

of electrical load. Our fundamental study highlights

the possibility of developing lightweight and high

performance piezoelectric graphene based nanoelec-

tromechanical systems such as sensors, actuators,

switches and smart electronics as compared with the

existing heavy, brittle and toxic piezoelectric

materials.

Keywords Graphene � Flexoelectricity �Piezoelectricity � Nanocomposite � Micromechanics �Finite element

1 Introduction

Flexoelectricity phenomenon is the response of elec-

tric polarization to an applied strain gradient and is

developed as a consequence of crystal symmetry in all

materials. Recent advances in nanoscale technologies

have renewed the interest in flexoelectricity due to the

obvious existence of large strain gradients at the

nanoscale level that leads to strong electromechanical

S. I. Kundalwal (&) � K. B. Shingare � A. RathiApplied and Theoretical Mechanics (ATOM) Laboratory,

Discipline of Mechanical Engineering, Indian Institute of

Technology Indore, Indore 453552, India

e-mail: kundalwal@iiti.ac.in

K. B. Shingare

e-mail: phd1701203008@iiti.ac.in

A. Rathi

e-mail: phd1801103008@iiti.ac.in

123

Int J Mech Mater Des

https://doi.org/10.1007/s10999-018-9417-6(0123456789().,-volV)(0123456789().,-volV)

coupling. The symmetry breaking at surfaces and

interfaces in nonpolar materials allows new forms of

electromechanical coupling such as surface piezoelec-

tricity and flexoelectricity, which cannot be induced in

the bulk materials. Piezoelectricity–electrical polar-

ization induced by a uniform strain (or vice versa)—is

the most widely known and exploited forms of

electromechanical coupling that exists in non-cen-

trosymmetric crystals. In non-centrosymmetric crys-

tals, the absence of the center of inversion results in the

presence of bulk piezoelectric properties. In contrast to

piezoelectricity, flexoelectricity can be found in any

crystalline material, regardless of the atomic bonding

configuration (Kundalwal et al. 2017). Piezoelectric

materials are commonly used where precise and

repeatable controlled motion is required such as

atomic force microscopy probes, smart structures,

and sensors and actuators (Kundalwal et al. 2013). The

conventional piezoelectric materials in the form of

piezoceramics are brittle, bulky and also sensitive due

to the content of high lead which is hazardous to the

environment (Bernholc et al. 2004). On the other hand,

piezoelectric polymers are light weight and environ-

mentally friendly, but typically show weaker piezo-

electric response. As compared to the existing

piezoelectric materials, there is always search for

light weight, high performance and environmentally

benign new piezoelectric materials.

The successful synthesis of 2D single layer

graphene sheet byNovoselov et al. (2004) has attracted

considerable attention from both academia and indus-

try. This is due to its unique scale-dependent elec-

tronic, mechanical and thermal properties (Zhang et al.

2005; Balandin et al. 2008; Lee et al. 2008; Gupta and

Batra 2010; Shah and Batra 2014a; Verma et al. 2014;

Alian et al. 2017; Kundalwal et al. 2017). It is widely

considered to be one of the most attractive materials of

the twenty-first century. Most recently, the piezoelec-

tric effect in non-piezoelectric graphene layers is found

by Kundalwal et al. (2017) using flexoelectric phe-

nomenon via quantum mechanics calculations. This

study showed that the presence of strain gradient in

non-piezoelectric graphene sheet does not only affect

the ionic positions, but also the asymmetric redistri-

bution of the electron density, which induces strong

polarization in the graphene sheet. The resulting axial

and normal piezoelectric coefficients of the graphene

sheet were determined using two loading conditions:

(1) a graphene sheet containing non-centrosymmetric

pore subjected to an axial load and (2) a pristine

graphene sheet subjected to a bending moment. The

results further revealed that the respective axial and

normal electromechanical couplings in graphene can

be engineered by changing the size of non-centrosym-

metric pores and radii of curvature.

Unique 2D structure and properties, and as it is a

semiconductor with zero bandgap make graphene

particularly suitable for nanoelectromechanical sys-

tems (NEMS) applications. In recent advances, it is

widely used as an energy harvester and in the NEMS

applications. Beams and cantilevers are two important

types of NEMS structures, and the latter one has more

potential application due to its linear behavior and

high sensitivity. Graphene beams have been exten-

sively studied in the last decade (Li et al. 2012 and

references therein), however, to date, a very few

researchers studied graphene-based cantilever beam

due to its challenging structure and fabrication. For

instance, Conley et al. (2011) proposed practical

realization of the strain-engineering scheme to control

electron properties of graphene cantilevers subjected

to significant variation of strain. Free-standing carbon

nanomaterial hybrid sheets, consisting of carbon

nanotubes (CNTs), exfoliated graphite nanoplatelets

and nanographene platelets, have been prepared by

Hwang et al. (2013) using various material combina-

tions and compositions. When subjected to tensile

strains, these carbon nanomaterial sheets showed

piezoresistive behavior, characterized by a change in

electrical resistance with applied strain. Kvashnin

et al. (2015) reported that a graphene sheet does not

show any polarization until the instantaneous quantum

fluctuative is responsible for the van der Waals

interactions. Da Cunha Rodrigues et al. (2015) studied

the electromechanical properties of a single-layer

graphene transferred onto SiO2 calibration grating

substrates via piezoresponse force microscopy and

confocal Raman spectroscopy. The calculated vertical

piezocoefficient of graphene was found about

1.4 nm V-1, that is, much higher than that of

conventional piezoelectric materials such as lead

zirconate titanate and comparable to that of relaxor

single crystals. The observed piezoresponse and

achieved strain in graphene are associated with the

chemical interaction of graphene’s carbon atoms with

the oxygen from underlying SiO2. The results provide

a basis for future applications of graphene layers for

sensing, actuating and energy harvesting.

123

S. I. Kundalwal et al.

Recently, metal matrix composite fascinated the

attention of the researchers because of its high

mechanical properties and specific stiffness. To

achieve increasing demands of high energy efficiency

and mechanical strength of material, many researchers

experimentally introduced the CNT-reinforced alu-

minium composites (CNT/Al) over the last two

decades. It is very critical to mix or disperse CNTs

uniformly in the aluminium matrix without decaying

their properties. Therefore, the graphene in the form of

graphene nanosheets (GNs) and graphene nanoplate-

lets (GNPs) were incorporated into the aluminium

matrix in the literature. For example, graphene

reinforced aluminum matrix fabricated using different

methods such as spark plasma sintering, flake powder

metallurgy and semi-solid powder processing (Wang

et al. 2012; Bastwros and Kim 2016; Tian et al. 2016;

Chen et al. 2018). They have studied the various

synthesis methods and showed the excellent enhance-

ment of mechanical properties with mere 0.3 wt% over

the pure aluminium matrix. Saber et al. (2014)

developed hybrid PZT based piezoelectric composite

for sensory applications by incorporating the thin film

of multi-walled carbon nanotubes (MWCNTs) and

GNPs. The nanocomposite films showed significant

improvement in the effective piezoelectric and stiffnes

properties of resulting nancomposite by * 50% and

200%, respectively. They also reported that the use of

GNPs in composite is better than MWCNTs when the

dynamic response and poling behavior are considered.

Kandpal et al. (2017) elucidated the role of addition of

GNPs into the nanocomposite for the improvement of

piezopotential response such as output voltage of

nanogenerator which can be used as an energy

harvester. Graphene oxide (GO), a derivative of

graphene, is also found the application in NEMS

because of its chemical reduction shows promising

route towards the bulk production of graphene for

commercial applications (Cui et al. 2016). Bhavanasi

et al. (2016) introduced the bilayer film made of

PVDF-TrFE and GO which exhibited extraordinary

energy harvesting performance with efficient transfer

of electromechanical energy. In their extensive

research, they concluded that GO film enhanced

voltage output and power density approximately by

2 times and 2.5 times, respectively, over that of PVDF-

TrFE film. Dasari et al. (2018) experimentally inves-

tigated GO-reinforced aluminum composite using the

liquid phase mixing and powder metallurgy

techniques.

A few analytical studies have also been conducted

to incorporate the effect of flexoelectricity which

identified some of its unexpected manifestations in

structural elements. Gharbi et al. (2011) observed an

important role of flexoelectricity in the hardening of

ferroelectrics at nano-indentation. Morozovska et al.

(2011) reported that the flexoelectricity plays an

important role in the electromechanical responce of

moderate conductors. The influence of the flexoelec-

tric effect on the mechanical and electrical properties

of bending piezoelectric nanobeams with different

boundary conditions was investigated by Yan and

Jiang (2013). A size-dependent model of a three-layer

microbeam including a flexoelectric dielectric layer

was proposed by Li et al. (2014) based on the theory

presented by Hadjesfandiari (2013). Rupa and Ray

(2017) obtained the exact solutions for the static

response of simply supported flexoelectric nanobeam.

The beam was subjected to the applied mechanical

load on its top surface while it was activated with the

prescribed voltage at its top and bottom surfaces.

The ability to fabricate NEMS with engineered

properties, based on stacking of weakly van der Waals

(vdW) interacting atomically thin layers, is quickly

becoming reality, and this has stimulated a new

research on graphene based composites. To the best of

current authors’ knowledge, a flexible piezoelectric

graphene-based cantilever beam that offers various

possibilities for developing NEMS has not been

studied yet which may represent an alternative to the

conventional piezoelectric nanostructures. This is

indeed the motivation behind the current study. As a

strikingly novel research goal, we focus on the

development of (1) analytical and numerical microme-

chanics models to determine the piezoelastic proper-

ties of GRNC, (2) a theoretical piezoelasticity model

for the GRNC cantilever beam to study its electrome-

chanical response, and (3) molecular dynamics (MD)

simulations to determine the mechanical properties of

graphene sheet.

2 Effective piezoelastic properties of GRNC

In this section, the piezoelastic properties of GRNC

were obtained considering a graphene sheet as a

piezoelectric continuum medium. To simplify

123

Effect of flexoelectricity on the electromechanical response

computational efforts, several researchers modeled the

graphene sheet as a continuum plate in order to obtain

its bulk properties (Park et al. 2010; Roberts et al.

2010). Most of the existing studies on the straining of

graphene are based on the analytical as well as

numerical solutions using the concept of continuum

elasticity and hence the graphene can be used as a

continuum medium (Gupta and Batra 2010; Gradinar

et al. 2013; Verma et al. 2014; Bahamon et al. 2015).

This implies that, for a homogeneously deformed

graphene, the displacement of each atom in it is given

by the deformation of the continuum medium, on

which the atom is embedded.

GRNC is assumed to be comprised of piezoelectric

graphene sheets and aluminum matrix. Such

nanocomposite can be considered to be made of

rectangular representative volume elements (RVEs)

containing both graphene sheet and matrix as shown in

Fig. 1, and we confined our micromechanical analysis

to a single RVE. We assumed that the bonding

between the graphene and aluminum matrix is perfect

and the resulting nanocomposite is homogeneous (Gao

and Li 2005; Song and Youn 2006; Jiang et al. 2009;

Kundalwal and Ray 2011, 2013). The upper and lower

surfaces of the GRNC lamina are electroded and the

transverse electric field is applied across them. Hence,

the electric polarization exists in the piezoelectric

medium when it is kept in a parallel plate capacitor

with an electric potential applied across the lamina

even if these parallel plates are not in contact with the

lamina. This is associated with the phenomenon of

converse piezoelectric effect and thus proposed

GRNC lamina may be considered as a capacitor

having two parallel plates in which the graphene and

aluminium matrix act as the dielectric medium. By

using the analytical and numerical micromechanics

models, the effective piezoelastic properties of GRNC

lamina can be obtained.

2.1 Analytical micromechanics models

Figure 1 shows a schematic of RVE of GRNC in

which the graphene is reinforced along the 3–direc-

tion. Assuming the graphene as a solid fiber, the

micromechanical model developed by Kundalwal and

Ray (2011) was modified to determine the elastic

properties of GRNC.

Based on the material coordinate system (1–2–3),

the constitutive relations for the different phases of

GRNC are given by

rrf g ¼ Cr½ � erf g; r ¼ g,m and nc ð1Þ

where the states of stress and stain vectors, and the

elastic coefficient matrix of the rth phase are

rrf g ¼ rr1 rr

2 rr3 rr

23 rr13 rr

12½ �T;erf g ¼ er1 er2 er3 er23 er13 er12½ �T;

Cr½ � ¼

Cr11 Cr

12 Cr13 0 0 0

Cr12 Cr

22 Cr23 0 0 0

Cr13 Cr

23 Cr33 0 0 0

0 0 0 Cr44 0 0

0 0 0 0 Cr55 0

0 0 0 0 0 Cr66

2666666664

3777777775

ð2Þ

In Eqs. (1) and (2), the respective g, m, and nc denote

the graphene, aluminum matrix, and GRNC. The

superscript r is used to denote the respective con-

stituent phases; rr1, r

r2, and r

r3 are the normal stresses

in the direction 1, 2, and 3, respectively; er1, er2, and er3

are the corresponding normal strains;rr12, r

r13, and r

r23

are the shear stresses; er12, er13, and er23 are the

corresponding shear strains; and Crij are the elastic

coefficients of rth phase.

In order to satisfy the perfectly bonding condition

between a fiber and the matrix, researchers mainly

used the iso-field conditions and rules of mixture

(Smith and Auld 1991; Benveniste and Dvorak 1992;

Fig. 1 Cross sections of

RVE of GRNC

123

S. I. Kundalwal et al.

Kundalwal and Ray 2011; Kundalwal 2017). The iso-

field conditions allow us to assume that the normal

strains in a fiber, matrix, and homogenized composite

are equal along the fiber direction while transverse

stresses in the same phases are equal along the

direction transverse to the fiber length. The rules of

mixture allows us to express the normal stress along

the fiber direction and the transverse strains along the

normal to the fiber direction of the homogenized

composite in terms of that in the fiber and matrix as

well as their volume fractions. Such iso-field condi-

tions and rules of mixture for satisfying the perfect

bonding conditions between a graphene and the matrix

can be expressed as

rg1

rg2

eg3rg23

rg13

rg12

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

¼

rm1

rm2

em3rm23

rm13

rm12

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

¼

rnc1

rnc2

enc3rnc23

rnc13

rnc12

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

ð3Þ

and

Vg

eg1eg2rg3

eg23eg13eg12

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

þ vm

em1em2rm3

em23em13em12

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

¼

enc1enc2rnc3

enc23enc13enc12

8>>>>>><>>>>>>:

9>>>>>>=>>>>>>;

ð4Þ

where vg and vm are the volume fractions of a

graphene and the aluminum metal matrix with respect

to the total volume of RVE of GRNC, respectively;

and vm = 1 - vg. Using Eqs. (1)–(4), the stress and

strain vectors of GRNC can be written in terms of the

corresponding stress and strain vectors of constituent

phases as follows:

rncf g ¼ C1½ � egf g þ C2½ � emf g ð5Þ

encf g ¼ V1½ � egf g þ V2½ � emf g ð6Þ

Also, the relationship between the stresses and strains

in a graphene and the aluminum matrix given by

Eq. (3) can be written as

C3½ � egf g � C4½ � emf g ¼ 0 ð7Þ

The matrices obtained in Eqs. (5)–(7) are given by

C1½ � ¼ vg

0 0 0 0 0 0

0 0 0 0 0 0

Cg13 C

g23 C

g33 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

2666666664

3777777775;

C2½ � ¼

Cm11 Cm

12 Cm12 0 0 0

Cm12 Cm

11 Cm12 0 0 0

vmCm12 vmC

m12 vmC

m11 0 0 0

0 0 0 Cm44 0 0

0 0 0 0 Cm44 0

0 0 0 0 0 Cm44

2666666664

3777777775;

C3½ � ¼

Cg11 C

g12 C

g13 0 0 0

Cg12 C

g22 C

g23 0 0 0

0 0 1 0 0 0

0 0 0 Cg44 0 0

0 0 0 0 Cg55 0

0 0 0 0 0 Cg66

2666666664

3777777775;

C4½ � ¼

Cm11 Cm

12 Cm12 0 0 0

Cm12 Cm

11 Cm12 0 0 0

0 0 1 0 0 0

0 0 0 Cm44 0 0

0 0 0 0 Cm44 0

0 0 0 0 0 Cm44

2666666664

3777777775;

V1½ � ¼

vg 0 0 0 0 0

0 vg 0 0 0 0

0 0 0 0 0 0

0 0 0 vg 0 0

0 0 0 0 vg 0

0 0 0 0 0 vg

26666664

37777775

and

V2½ � ¼

vm 0 0 0 0 0

0 vm 0 0 0 0

0 0 1 0 0 0

0 0 0 vm 0 0

0 0 0 0 vm 0

0 0 0 0 0 vm

26666664

37777775

ð8Þ

in Eqs. (6) and (7), egf g and emf g are the strain vectorsof a graphene and the aluminum metal matrix,

respectively, and they can be expressed in terms of

GRNC strain vector encf g. Subsequently, using

Eqs. (5)–(7), the states of stresses and strains in

GRNC can obtained as follows:

123

Effect of flexoelectricity on the electromechanical response

rncf g ¼ Cnc½ � encf g ð9Þ

where the effective elastic coefficient matrix Cnc½ � ofthe GRNC is given by

Cnc½ � ¼ C1½ � V3½ ��1þ C2½ � V4½ ��1 ð10Þ

and

V3½ � ¼ V1½ � þ V2½ � C4½ ��1C3½ � ð11Þ

V4½ � ¼ V2½ � þ V1½ � C3½ ��1C4½ � ð12Þ

Using the piezoelastic coefficients of graphene and

matrix, the effective piezoelectric constants of GRNC

can be determined using the existing models (Smith

and Auld 1991; Ray and Pradhan, 2006; Kumar and

Chakraborty 2009). The values of piezoelectric coef-

ficients of a composite material obtained by Smith and

Auld (1991) are larger than that of values determined

by Ray and Pradhan (2006) because of consideration

of iso-strain conditions in both x and y-directions by

former researchers. The effective piezoelectric con-

stant e31 of the GRNC defines the normal stress

induced in transverse x-direction because of the

application of a unit electric field in longitudinal

z-direction. The expression for the effective piezo-

electric constant e31 can be written as follows (Smith

and Auld 1991; Kumar and Chakraborty 2009):

e31 = vge31gðC11m � C12mÞ=ða1 + a2Þ ð13Þ

Similarly, the effective piezoelectric constants e32 and

e33 define the normal stresses induced in respective y-

and z-directions because of the application of a unit

electric field in the longitudinal z-direction. While the

piezoelectric constant e15 quantifies the induced shear

stress about y-direction per unit electric field applied

in the x-direction. The closed form expressions of

various effective piezoelectric constants of GRNC are

given below:

e32 = e32g � ðb1C22g + b2C23gÞ=Q ð14Þ

e33 ¼ vgðe33g � 2vme31gðC13g � C12mÞ=ða1 þ a2ÞÞð15Þ

e15 = e15gð1� vmC55g=ðvgC55m + vmC55gÞÞ ð16Þ

e24 ¼ e24gð1� vmC44g=ðvgC44m þ vmC44gÞÞ ð17Þ

Note that the GRNC is the transversely isotropic

material with the 3-axis as the axis of symmetry;

therefore, e31 ¼ e32 and e24 = e15. Hence, only the

three independent piezoelectric constants (e31; e33; and

e15) are required to study the piezoelectric behaviour

of GRNC. Subsequently, the effective permittivity

constant (233) of GRNC is determined by using the

following expression (Ray and Pradhan 2006):

233¼ vg 233g þvm233m þe31gvgvm=ðvmC11g þ vgC11mÞ ð18Þ

where

Q ¼ vmC22g þ vgC22m

� �vmC33g þ vgC33m

� �

� vmC23g þ vgC23m

� �2 ð19Þ

a1 ¼ vgðC11m þ C12mÞ ð20Þ

a2 ¼ vmðC11g þ C12gÞ ð21Þ

b1 ¼ vme31gðvmC33g þ vgC33mÞ � vme33gðvmC23g

þ vgC23mÞð22Þ

b2 ¼ �vme31gðvmC23g þ vgC23mÞ þ vme33gðvmC22g

þ vgC22mÞð23Þ

2.2 FE modeling of piezoelastic properties

of GRNC

It is important to mention that the micromechanics

models developed in the preceding section are based on

the rules of mixture and iso-field assumptions. Numer-

ical or experimental investigations may be carried out

to verify these assumptions because both the analyses

do not require any such approximations (Kundalwal

and Ray 2012). Therefore, in the current study, FE

models were developed to validate the assumptions

used in Sect. 2.1 by employing the commercial

software ANSYS 15.0. The FE analysis was catego-

rized into the fully coupled electromechanical analysis

in which the modeling of three-dimensional RVE or

unit cellwas carriedwith 20 noded brick element ‘‘solid

226’’ which has both displacement and voltage degrees

of freedom (DOF). Figures 2 and 3 show the RVEs of

123

S. I. Kundalwal et al.

homogenized andFEmeshofGRNC, respectively. The

RVE of GRNC was homogenized and analyzed under

various boundary conditions. FE model of the RVE of

homogenous transversely isotropic GRNCwith its axis

of transverse isotropy being along the graphene was

developed for evaluating the eight independent piezoe-

lastic coefficients: Ceff11 , C

eff12 , C

eff13 , C

eff33 , C

eff44 ; e

eff13 , e

eff15

and eeff33 . The effective piezoelastic coefficients of the

GRNC can be determined by applying the appropriate

boundary conditions to the RVE to be discussed in

subsequent sections.

The combined electromechanical relations of

GRNC are given by Eq. (24) in which Ceffij , e

effij , and

2effij are the effective elastic constants, piezoelectric

constants and permittivity constants of GRNC,

respectively.

Under the conditions of an imposed electrome-

chanical loads on the RVE of GRNC, the average

stress rij

� �, strain eij

� �, electrical displacement Di

� �

and electric field Ei

� �are defined as below

rij

� �¼ 1

V

Z

v

rij

� �dV; eij

� �¼ 1

V

Z

v

eij� �

dV

Di

� �¼ 1

V

Z

v

Dif gdV; Ei

� �¼ 1

V

Z

v

Eif gdV ð25Þ

where v represents the volume of the GRNC RVE and

the quantity with an overbar represents the volume

averaged quantity. It is evident from Eq. (24) that if at

any point in the GRNC only one normal strain is

present while the other strain components are zero

then three normal stresses exists. The ratio between

any one of these three normal stresses and the normal

strain yields an effective elastic coefficient. Thus,

three such elastic coefficients at the point can be

calculated with one numerical experiment. Hence, the

determination of a particular effective piezoelastic

coefficient from the FE model needs to prescribe the

appropriate boundary conditions on the faces of the

GRNC RVE.

2.2.1 Effective elastic coefficients Ceff13 and C

eff33

In order to compute the effective elastic constants

Ceff13 and Ceff

33 of the GRNC, the RVE shown in Fig. 4

can be deformed in such a way that the normal strain

e33 is only present in it while all other strain

components are zero. In order to achieve such states

of strain, displacements at the five boundary surfaces

ðx1 ¼ 0 and a; x2 ¼ 0 and b; x3 ¼ 0Þ need to pre-

scribed to zero (e11 ¼ e22 ¼ e23 ¼ e13 ¼ e12 ¼ 0Þ. Itshould be noted that x1; x2; and x3 denote the coor-

dinates corresponding to 1, 2, and 3-axes, respectively.

A uniform normal displacement (u3) along the 3-

direction needs to be applied on the surface (x3 ¼ l) of

the RVE such that it is subjected to e33 only. Likewise,the electric fields (voltage DOF) at all faces are

required to constrain to zero (E1 ¼ E2 ¼ E3 ¼ 0).

Using Eq. (25), the average stresses and strains

(r11;r33; and e33) can be obtained. Then, the values

of effective elastic coefficients textCeff33 ð¼ r33=e33Þ

and Ceff13 ð¼ r11=e33Þ can be determined using Eq. (24)

for different volume fractions of graphene. Figure 4

shows the stress and strain distributions in the RVE

obtained along the graphene direction.

r11

r22

r33

r23

r13

r12

D1

D2

D3

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>;

¼

Ceff11 Ceff

12 Ceff13 0 0 0 0 0 � eeff31

Ceff12 Ceff

22 Ceff13 0 0 0 0 0 � eeff31

Ceff13 Ceff

13 Ceff33 0 0 0 0 0 � eeff33

0 0 0 Ceff44 0 0 0 � eeff15 0

0 0 0 0 Ceff55 0 � eeff15 0 0

0 0 0 0 0 Ceff66 0 0 0

0 0 0 0 eeff15 0 2eff11 0 0

0 0 0 eeff15 0 0 0 2eff22 0

eeff31 eeff31 eeff33 0 0 0 0 0 2eff33

266666666666666664

377777777777777775

e11e22e33e23e13e12E1

E2

E3

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

9>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>;

ð24Þ

123

Effect of flexoelectricity on the electromechanical response

2.2.2 Effective elastic coefficients Ceff11 and Ceff

12

In order to determine the effective elastic coefficients

Ceff11 and Ceff

12 of GRNC, the RVE needs to be imposed

to the states of strain such that only normal strain e11 ispresent while all other strain components are zero

(e22 ¼ e33 ¼ e23 ¼ e13 ¼ e12 ¼ 0Þ: Such states of

strain can be attained by constraining the surfaces of

RVE in the following manner:

u1 ¼ 0 at x1 ¼ 0; u2 ¼ 0 at x2 ¼ 0 and b; u3 ¼ 0 at

x3 ¼ 0 and 1.

In the same way, the electric field at all surfaces of

RVE are required to constrain to zero

(E1 ¼ E2 ¼ E3 ¼ 0). Due to the essential boundary

conditions, the uniform normal displacement (u1) is

required to apply on the surface x1 ¼ a of the RVE

such that it is under the application of e11 only. Using

Eqs. (24) and (25), the average stresses and strain

(r11;r22 and e11) can be obtained to calculate the

values of Ceff11 ð¼ r11=e11Þ and Ceff

12 ð¼ r22=e11Þ. Thedistributions of stresses and strains are shown in Fig. 5

when the deformation is applied to the RVE in the

transverse direction to the length of graphene.

2.2.3 Effective piezoelastic coefficients

Ceff44 ; e

eff15 and Ceff

66

To estimate the effective elastic coefficient Ceff44 of

GRNC, an out-of-plane shear in the x1–x3 plane of the

RVE is required to subject the pure shear deformation

in such a way that the shear strain e23 is non-zero whilethe remaining strain components are zero. Such states

of strain can be achieved by prescribing the surface

given by x3 ¼ 0 of the RVE and imposing the uniform

distributed tangential force on the surface given by

x3 ¼ l. In the same way, the electric potential at all

surfaces of the RVE are required to constrain to zero

(E1 ¼ E2 ¼ E3 ¼ 0). Subsequently, the average shear

stress and strain (r23 and e23Þ induced in the RVE can

be determined using Eq. (25). Finally, the effective

elastic coefficient Ceff44 can be estimated using relation:

r23=e23. Note that the effective elastic coefficients Ceff66

is not independent elastic coefficient and it can be

computed directly from the relation Ceff11 � Ceff

12

� �=2.

In order to calculate the effective piezoelectric

coefficient eeff15 , the boundary conditions similar to

obtain Ceff44 can be used such that the RVEwas imposed

to shear strain e23 only ðe11 ¼ e22 ¼ e33 ¼ e13 ¼e12 ¼ 0Þ and electric potential needs to be constrainedto zero on the two surfaces of RVE (E1 ¼ E2 ¼ 0).

Using Eq. (24), the values of eeff15 can be calculated

using the ratio eeff15 ¼ D2=e23 for different volume

fractions of graphene. Figure 6 depicts the distribu-

tions of out-of-plane shear stresses and strains in the

RVE of GRNC.

2.2.4 Effective piezoelectric and permittivity

coefficients eeff33 ; e

eff13 and ε

eff33

The effective piezoelectric coefficients eeff13 and eeff33 are

proportional to the in-plane and out-of-plane actua-

tions of piezoelectic materials, respectively. In order

to determine the values of eeff13 ; eeff33 ; and permittivity

Fig. 2 Homogenized RVE of GRNC

Fig. 3 FE mesh of the RVE of GRNC

123

S. I. Kundalwal et al.

constant εeff33 , converse to the boundary conditions of

Ceff33 and Ceff

13 , all the surfaces of RVE are required to

constrain so that their normal displacements are zero

ðe11 ¼ e22 ¼ e33 ¼ e23 ¼ e13 ¼ e12 ¼ 0Þ. The

uniform electric potential needs to apply to the RVE

in 3-direction and its remaining surfaces are required

to constrain to zero electric potentials (E1 ¼ E2 ¼ 0).

Figure 7 illustrates the distributions of electric

Fig. 4 FE simulations

showing distributions of astrain e33 and b stress r33 in

the RVE

Fig. 5 FE simulations

showing distributions of astrain e11 and b stress r11 in

the GRNC RVE

Fig. 6 FE simulations

showing distributions of ashear strain ðc23Þ and bshear stress ðs23Þ in the

GRNC RVE

123

Effect of flexoelectricity on the electromechanical response

displacement and electric potential in the RVE.

Subsequently, the average values of r11, r33, D3 and

E3 can be computed using Eq. (25). Then, using

Eq. (24), the effective values of eeff33 , eeff13 and ε

eff33 can be

evaluated using the respective ratios �r33=E3,

�r11=E3, and �D3=E3.

3 Piezoelasticity model of GRNC cantilever beam

Tremendous research on nanocomposite structures has

been carried out in the last decade with the aim of

developing NEMS. Among all structural elements,

piezoelectric composite cantilever beams have found

wide applications such as sensors, transducers and

actuators in NEMS. Therefore, here an attempt was

made to show the potential of cantilever beammade of

flexoelectric GRNC in which strain gradient is incor-

porated using its effective piezoelastic properties

obtained in previous sections. The generalized equa-

tion for the internal energy density function can be

written as follows (Shen and Hu 2010):

U ¼ 1

2bklPkPl þ

1

2Cijkleijekl þ dijkeijPk þ fijklui;jkPl

þ rijklmeijuk;lm þ 1

2gijklmnui;jkul;mn

ð26Þ

where Pi and ui are the components for polarization

and displacement vector, while ei are the components

for strain vectors. Cijkl, dijk, and bkl are the fourth orderelastic coefficient, third order of piezoelectric

coefficient and second order tensor of reciprocal

dielectric susceptibility, respectively. fijkl is the fourth

order flexoelectric coefficient tensor, while gijklmn are

the elements for purely nonlocal elastic effects and

related to strain gradient elastic theories and rijklmrepresent strain and strain gradient coupling tensors.

The strain vector is defined as

eij ¼1

2

oui

oxjþ ouj

oxi

� �ð27Þ

For the sake of simplicity, the elements gijklmn and

rijklm are assumed to be zero. Hence, the constitutive

equations can be expressed as

rij ¼oU

oeij¼ Cijklekl þ dijkPk ð28aÞ

sijm ¼ oU

oui;jm¼ fijmkPk ð28bÞ

Ei ¼oU

oPi¼ bijPj þ djkiejk þ fjkliuj;kl ð28cÞ

where rij, Ei, and sijm are the Cauchy stress tensor, the

electric field, and the moment stress or the higher order

stress, respectively. In higher order stress, sijm is

induced by the flexoelectric effect while it does not

present in the classical theory of piezoelectricity. The

various tensorial terms in Eq. (28) are given by

Fig. 7 FE simulations

showing the distributions of

a electric displacement ðD3Þand b electric potential ðE3Þin the GRNC RVE

123

S. I. Kundalwal et al.

rx

ry

rz

ryz

rxz

rxy

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

¼

C11 C12 C13 0 0 0

C12 C22 C23 0 0 0

C13 C23 C33 0 0 0

0 0 0 C44 0 0

0 0 0 0 C55 0

0 0 0 0 0 C66

2666666664

3777777775

exeyez2eyz2exz2exy

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

þ

d11 d21 d31

d12 d22 d32

d13 d23 d33

d14 d24 d34

d15 d25 d35

d16 d26 d36

2666666664

3777777775

Px

Py

Pz

8><>:

9>=>;

ð29aÞ

Ex

Ey

Ez

8><>:

9>=>;

¼ �d11 d12 d13 d14 d15 d16

d21 d22 d23 d24 d25 d26

d31 d32 d33 d34 d35 d36

264

375

exeyez2eyz2exz2exy

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

þb11 0 0

0 b22 0

0 0 b33

264

375

Px

Py

Pz

8><>:

9>=>;

ð29bÞ

A schematic of GRNC cantilever beam is shown in

Fig. 8 which has width b, length L and height h,

referred to the Cartesian coordinate system x–z.

A GRNC cantilever beam is considered to be made

of graphene and aluminum metal matrix. The arrange-

ment of graphene and aluminum matrix layers can be

varied to obtain a different volume fraction of graphene

for the piezoelectric analysis. Here, the thickness of

each single graphene layer was taken as 0.34 nm (Alian

et al. 2015a, b, 2017; Kundalwal and Meguid 2017).

The piezoelectric GRNC cantilever beam is polar-

ized along the z-axis. A constant concentrated load is

applied at the free end (x ¼ L) of cantilever beam and

a constant voltage V is applied between the upper and

lower surfaces of beam i.e., at z ¼ þ h2and z ¼ � h

2,

respectively. If the w xð Þ is the transverse displacement

of bending of the beam then using Euler beam

hypotheses the axial displacement at any point in it

is given as (Yan and Jiang 2013)

u x; zð Þ ¼ u0 xð Þ � zdw xð Þdx

ð30Þ

where u0 xð Þ is the axial displacement along the

longitudinal axis of the beamwhich may be introduced

by mechanical and electric loads due to the elec-

tromechanical coupling or flexoelectric effect. Using

Eq. (27), the non-zero strain can be written as

ex ¼du0

dx� z

d2w

dx2ð31Þ

Here, the thickness of a cantilever beam is considered

as very small as compared to its length ðh � LÞ.Hence one can assume that transverse displacement of

a cantilever beam is greater as compared to its

longitudinal displacement. Therefore, strain gradient

ex;x ¼ d2u0

dx2 � z d3w

dx3 can be omitted because it is negli-

gible as compared to ex;z ¼ � d2w

dx2. Therefore for

further formulation, we only considered strain gradi-

ent � d2w

dx2, which is induced by the flexoelectricity.

Fig. 8 schematic of GRNC

cantilever beam

123

Effect of flexoelectricity on the electromechanical response

We can formulate the electric field (EzÞ consideringit is only present in the z-direction ðEx ¼ Ey ¼ 0Þ(Ying and Zhifei 2005) and it can be calculated against

the variation of the thickness of beam corresponding to

the constant electrical potential. From Eqs. (28c) and

(30), the electric field in thickness direction is given by

Ez ¼ b33Pz þ d31ex þ f13ex;z ð32Þ

The extra term f13ex;z in the above equation is differentfrom the theory of linear piezoelectricity which

contributes to the flexoelectric effect. For a complete

formulation of the problem, in the absence of body

charges, the Gauss’ law is expressed as

ε0o2uoz2

þ oPz

oz¼ 0 ð33Þ

The relationship between the internal energy density

function and electric enthalpy density function can be

defined as (Shen and Hu 2010)

H ¼ U� 1

220 u;zu;z þ u;zPz ð34aÞ

where ε0 ¼ 8:85� 10�12 C=V m is the permittivity of

free space or vacuum. The relation between an electric

potential (u) and the electric field (Ez) is given by

Ez ¼ �u;z ð34bÞ

With consideration of the electric boundary condition

uðh=2Þ ¼ V and uð�h=2Þ ¼ 0, the polarization (Pz)

and electric field (Ez) from Eqs. (32) and (33) can be

written in terms of u0; and w as follows

Pz ¼20 d31

20 b33 þ 1zd2w

dx2� d31

b33

du0

dxþ f13

b33

d2w

dx2� V

b33h

ð35Þ

Ez ¼ � d31

20 b33 þ 1zd2w

dx2þ V

h

� �ð36Þ

By substituting Eq. (35) into Eq. (28a), the axial stress

(rx) can be determined as

rx ¼ C11 �d231b33

� �du0

dx� C11 �

20 d231

20 b33 þ 1

� �zd2w

dx2

þ d31f13

b33

d2w

dx2� d31V

b33h

ð37Þ

Using Eq. (37), the axial force ðFxÞ can be determined

as follows

Fx ¼ r

h2

�h2

rxdz

¼ bh C11 �d231b33

� �du0

dxþ d31f13

b33

d2w

dx2� d31V

b33h

�

ð38Þ

By simplifying Eq. (38), we can obtain

bh C11 �d231b33

� �du0

dxþ d31f13

b33

d2w

dx2� d31V

b33h

� ¼ 0

ð39aÞ

� C11 �d231b33

� �du0

dx¼ d31f13

b33

d2w

dx2� d31V

b33hð39bÞ

Because of the absence of external body forces and

mechanical loads in the axial direction, the relaxation

strain in the cantilever beam is present which plays

important role in its softer and stiffer behavior. The

relaxation strain can be defined as

du0

dx¼

� C11 � d231b33

�

d31f13b33

d2w

dx2 � d31V

b33h

� ð40Þ

The governing equation of piezoelectric nanobeams

including the effect of flexoelectricity is obtained

using the energy method. Using Eqs. (26), (28), and

(34b), the internal energy density function can be

written as

U ¼ 1

2rxex þ sxxzex;z � u;zPz� �

ð41Þ

in which,

sxxz ¼20 d31f13

20 b33 þ 1zþ f 213

b33

� �d2w

dx2� d31f13

b33

du0

dx� Vf13

b33h

�

ð42Þ

123

S. I. Kundalwal et al.

Next, variational principle can be formulated over

the entire volume of the GRNC piezoelectric beam and

can be written as (Mindlin 1968):

�dZ

V

HdVþ dW ¼ 0 ð43Þ

Subsequently, the governing equations of GRNC

piezoelectric cantilever beam can be obtained as

follows

C11 �d231b33

� �bh

d2u0

dx2þ d31f13bh

b33

� �d3w

dx3¼ 0 ð44aÞ

C11 �20 d

231

20 b33 þ 1

� �bh3

12� f213bh

b33

� �d4w

dx4

þ d31f13bh

b33

� �d3u0

dx3

¼ 0 ð44bÞ

For the cantilever beam, the transverse displacement

and slope can be obtained using the following

boundary conditions

wjx ¼ 0;dw

dx

����x ¼ 0 ð45Þ

The moment and force at x = L become:

C11 �20 d

231

20 b33 þ 1

� �bh3

12� f213bh

b33

� �d2w

dx2

þ d31f13bh

b33

� �du0

dxþ f13Vb

b33¼ 0 ð46aÞ

� C11 �20 d

231

20 b33 þ 1

� �bh3

12� f213bh

b33

� �d3w

dx3

� d31f13bh

b33

� �d2u0

dx2þ F

¼ 0 ð46bÞ

where the first terms in above equations are the

effective bending rigidity considering the flexoelectric

effect. Substituting the boundary conditions

0� x�Lð Þ in governing Eqs. (46a) and (46b) of

GRNC cantilever beam, the transverse deflection can

be obtained as follows

wcantilever ¼Px2

6 EIð Þcantileverx� 3Lð Þ

� C11f13Vbx2

2 b33C11 � d231� �

EIð Þcantileverð47aÞ

In absence of flexoelectricity effect (f13 ¼ 0),

Eq. (47a) gets reduced to

wcantilever ¼Px2

6 EIð Þcantileverx� 3Lð Þ ð47bÞ

where,

EIð Þcantilever ¼ C11 �20 d

231

20 b33 þ 1

� �bh3

12� f213bh

b33

� d231f213bh

b33ðb33C11 � d231Þ

ffi C11 �20 d

231

20 b33 þ 1

� �bh3

12� f213bh

b33

4 Elastic properties of graphene

In order to estimate the effective piezoelastic proper-

ties of the GRNC, the elastic properties of graphene

are required first. Therefore, we first determined the

elastic properties of graphene sheet using MD simu-

lations. It is the the most widely used tools in the

theoretical study of nanostructures and it calculates the

time-dependent behaviour of a molecular system and

permits accurate property predictions via accounting

interactions between atoms and molecules at the

atomic level (Shah and Batra 2014b; Alian et al.

2015a, b; Kundalwal and Meguid 2017; Kothari et al.

2018). This serves a complement to conventional

experiments with cheaper and faster simulations. All

MD simulations runs were conducted with large-scale

atomic/molecular massively parallel simulator

(LAMMPS) (Plimpton 1995), and the molecular

interactions in graphene were described in terms of

Adaptive Intermolecular Reactive Empirical Bond

Order (AIREBO) force fields (Stuart et al. 2000).

The systematic steps followed in the MD simula-

tions are as follows. First, the intial structure of

graphene was prepared. Then, simulations were

started with an energy minimization using a conjugate

gradient method to obtain the optimized structure of

graphene. A graphene was considered as optimized

123

Effect of flexoelectricity on the electromechanical response

structure when the total potential energy between

adjacent steps was less than 1.0 9 10-10 kcal/mol.

Simulation in each case was performed in the constant

temperature and volume conical (NVT) ensemble

using a time step of 0.5 fs with the total time of 50 ps

to equilibrate the graphene structure, where the

velocity Verlet algorithm was used to integrate

Newton’s classical equations of motion. In the present

work, the load was applied to a graphene sheet and

then the energy was computed as the result of

interatomic interactions of neighbouring atoms. The

stress of the graphene sheet was computed by aver-

aging the obtained stress of each carbon atom in it.

Then, the stress–strain curve during tensile loading

was obtained and Young’s modulus (E) and Poisson’s

ratio (l) of graphene sheet were determined. The

determination of values of E and l a graphene layer

was accomplished by using the strain energy density-

elastic constant relations (Kundalwal and Choyal

2018). The equivalent continuum graphene sheet

was assumed to be a flat plate considering its wall

thickness of 3.4 A (Alian et al. 2015a, b, 2017;

Kundalwal andMeguid 2017). A direct transformation

to continuum properties was then made by assuming

that the potential energy density of discrete atomic

interactions of neighbouring atoms is equal to the

strain energy density of the continuous substance

occupying a graphene volume. The atomic volume

was determined from the relaxed graphene sheet with

the thickness of 3.4 A.

5 Results and discussions

As a first endeavor, MD simulations were carried out

to determine the elastic properties of graphene sheet.

The predicted values of E and l are summarized in

Table 1, and they agree well with the results obtained

by other researchers using different modeling tech-

niques and potentials as well as experimental esti-

mates (Lee et al. 2008; Zhang et al. 2012; Dewapriya

et al. 2015).

Next, the effective piezoelastic coefficients of

GRNC were evaluated using two different models

developed in Sect. 2. Then, the obtained effective

piezoelastic coefficients of GRNC were used to

study the electromechanical response of GRNC

cantilever beam. A concentrated force P = 1 nN

was applied at the end of GRNC cantilever beam

having width, b ¼ h, and length, l ¼ 20h. We

considered permittivity of graphene same as that of

graphite, 211¼233¼ 1:106� 10�10 F=m (Munoz-

Hernandez et al. 2017). Usually, the fiber volume

fraction in the composite varies considerably accord-

ing to the manufacturing route, typically from 0.2 to

0.7. Therefore, we calculated the piezoelastic proper-

ties of GRNC for different volume fractions of

graphene, from 0.2 to 0.7. Since the GRNC is

transversely isotropic with 3-axis as the axis of

symmetry, only the five independent effective elastic

coefficients of GRNC (C11, C12, C13, C33 and C44) are

presented here. Figure 9 illustrates the variation of the

effective axial elastic coefficient C33 of the GRNC

with the graphene volume fraction. It may be observed

that the value of C33 increases almost linearly with the

graphene volume fraction. This result is coherent with

the previously reported results for graphene based

nanocomposites (Zhao et al. 2010; Khan et al. 2012; Ji

et al. 2016; Garcıa-Macıas et al. 2018). Almost 100%

agreement between the two sets of the values of C33

estimated by the MOM and FE models ensures the

validity of the assumptions and the rules of mixture for

deriving the MOMmodel. Note that the effective axial

elastic coefficient (C33) was estimated considering the

Table 1 Properties of graphene and aluminium matrix

Material E(GPa) l e31 (C/m2) e33 (C/m2) e15 (C/m2) 233 (F/m)

Graphene 985 (Present) 0.26 (Present) - 0.221 Kundalwal

et al. (2017)

0.221 Kundalwal

et al. (2017)

- 0.221 Kundalwal

et al. (2017)1.106 �10�10

Munoz-Hernandez

et al. (2017)

Aluminum 70 (Scari

et al. 2014)

0.33 (Scari

et al. 2014)

– – – 1.504 �10�11

Sundar et al.

(2016)

123

S. I. Kundalwal et al.

iso-strain condition along the graphene direction and

hence such estimation belongs to the Voigt-upper

bound.

Figure 10 demonstrates the variation of the effec-

tive elastic coefficient C13 of the GRNC with the

graphene volume fraction. The MOM model overes-

timates the value of C13 as compared to the FE

Fig. 9 Variation of the

effective elastic coefficient

C33 of the GRNC with the

graphene volume fraction

Fig. 10 Variation of the

effective elastic coefficient

C13 of the GRNC with the

graphene volume fraction

123

Effect of flexoelectricity on the electromechanical response

estimates. This is attributed to the fact that the

Poisson’s effect was modeled appropriately in the

FE simulations. Extension-extension coupling occurs

between the dissimilar normal stress (r33) and normal

strain (e11) due to the applied load along the axis of

symmetry (i.e., 3-axis). Since the GRNC is

Fig. 11 Variation of the

effective elastic coefficient

C11 of the GRNC with the

graphene volume fraction

Fig. 12 Variation of the

effective elastic coefficient

C12 of the GRNC with the

graphene volume fraction

123

S. I. Kundalwal et al.

transversely isotropic with 3-axis as the axis of

symmetry, the values of C23 of the GRNC are found

to be identical to those of C13 and for brevity are not

presented here.

Figures 11 and 12 demonstrate the variation of

effective elastic coefficients C11 and C12 with the

graphene volume fraction. It may be observed from

Figs. 11 and 12 that the MOM model overestimates

Fig. 13 Variation of the

effective elastic coefficient

C44 of the GRNC with the

graphene volume fraction

Fig. 14 Variation of the

effective permittivity

coefficient 233 of the GRNC

with the graphene volume

fraction

123

Effect of flexoelectricity on the electromechanical response

the respective values of C11 and C12 over that of FE

predictions, especially at larger volume fractions of

graphene. This is due to the fact that the transverse

elastic properties of composites are usually the

function of properties of matrix material and the

predictions of both models are well agreed at lower

volume fractions of graphene. As expected, the

predictions by both models differ significantly as the

graphene volume fraction increases. The values of C22

Fig. 15 Variation of the

effective piezoelectric

coefficient e33 of the GRNC

with the graphene volume

fraction

Fig. 16 Variation of the

effective piezoelectric

coefficient e31 of the GRNC

with the graphene volume

fraction

123

S. I. Kundalwal et al.

of the GRNC are found to be identical to those of C11

but they are shown here for brevity.

Figure 13 demonstrates the variation of effective

elastic coefficient C44 with the graphene volume

fraction. Once again, the MOM model overestimates

the values of C44 over that of FE predictions,

especially at larger volume fractions of graphene.

This finding is coherent with the previously reported

results by Pettermann and Suresh (2000). They

reported that the analytical predictions of only the

longitudinal shear modulus with respect to the elastic

behavior of piezoelectric composite can be altered

significantly compared to the experimental findings.

Figures 14 and 15 demonstrate the variation of

effective permittivity 233 and piezoelectric coefficient

e33 with the graphene volume fraction. Both figures re-

veal that the values of 233 and e33 increased linearly as

the graphene volume fraction increases. Moreover, it

may be observed from this comparison that the two

sets of results are in excellent agreement validating the

analytical micromechanics model used herein. This is

attributed to the imposition of electric potential (volt

DOF in case of FE simulations) in the fiber direction.

Figure 16 illustrates the variation of effective piezo-

electric coefficient e31 of GRNC with the graphene

volume fraction. It may be observed from this

figure that the magnitude of e31 increases with the

graphene volume fraction. Figure 16 clearly shows a

good correlation between the analytical and FE

predictions at wide range of volume fractions of

graphene. Figure 17 demonstrates the variation of

effective piezoelectric coefficient e15 with the gra-

phene volume fraction. This figure shows that that the

magnitude of e15 increases with the graphene volume

fraction as found in case of e31. It may also be observed

from Fig. 17 that the predictions of both models are

well agreed at lower volume fractions of graphene.

Compared to other piezoelectric coefficients, the

predictions of e15 are found to more sensitive to the

larger volume fractions of graphene and this is

attributed to the consideration of boundary conditions

and in-plane behavior of GRNC; the same were

attributed to the variation of values of C44 as shown in

Fig. 13. The research works by Odegard (2004) and

Moreno et al. (2009) also reported the discrepancy in

the predictions of values of e15. Comparison of the

predictions of analytical micromechanics approaches

with that by the FE simulations reveals that the former

models yield conservative estimates of the piezoelas-

tic constants of the GRNC. Hence, predictions by the

analytical micromechanics approaches were consid-

ered in the subsequent section for investigating the

Fig. 17 Variation of the

effective piezoelectric

coefficient e15 of the GRNC

with the graphene volume

fraction

123

Effect of flexoelectricity on the electromechanical response

Fig. 18 Deflection of

GRNC cantilever beam

along its length with

different graphene volume

fractions at V = – 1 V

Fig. 19 Deflection of

GRNC cantilever beam

along its length with

different graphene volume

fractions at V = – 5 V

123

S. I. Kundalwal et al.

effect of flexoelectricity on the electromechanical

response of GRNC cantilever beam.

So far, the effective piezoelastic properties of

GRNC have been determined. To investigate the

effect of flexoelectricity on the electromechanical

response of GRNC cantilever beam, three discrete

values of graphene volume fractions (vg) are consid-

ered as 0.4, 0.6, and 0.8, and the electrical potentials

are considered as - 1 V, - 5 V, and - 10 V.

Figures 18, 19 and 20 illustrate the effect of flexo-

electricity on the GRNC cantilever beam for different

graphene volume fractions and electric potentials. In

our simulations, negative electric potential was

applied over the surface of the GRNC cantilever

beam. These results reveal that the effective bending

rigidity of the GRNC cantilever beams with consid-

eration of the flexoelectricity (f13 6¼ 0) is higher than

that of neglecting flexoelectric effect (f13 = 0). Con-

versely, as positive electric potential (? V) was

applied to the cantilever beam, the softer elastic

behavior exhibited by it when the effect of flexoelec-

tricity (f13) was taken into account; then, induced

inhomogeneous boundary conditions to the bending

behaviour of GRNC cantilever beam were found

exactly opposite to its effective bending rigidity.

Inhomogeneous boundary condition is nothing but the

addition of positive and negative relaxation moments

according to the applied negative and positive electric

potentials, respectively. Thus, it can be concluded that

the stiffer and softer elastic behaviour of GRNC

cantilever beam can be altered by varying electric

potential (± V) and it may find wide applications in

NEMS. In may be observed from Fig. 18, 19 and 20

that the responses of GRNC cantilever beams improve

as the values of volume fraction of graphene and

applied electrical potential are increased. This is

attributed to the fact that the (1) effective piezoelastic

properties of GRNC increase with the increase in

graphene volume fraction in it as shown in Figs. 9, 10,

11, 12, 13, 14, 15, 16 and 17, and (2) flexoelectric

effect enhances as the applied electrical potential

increases. From the above discussion, it may be

observed that the deflections of GRNC cantilever

beam remarkably influenced by the electromechanical

loadings and strain gradients effects.

Fig. 20 Deflection of

GRNC cantilever beam

along its length with

different graphene volume

fractions at V = – 10 V

123

Effect of flexoelectricity on the electromechanical response

6 Conclusions

As the first of its kind, this study reports the effective

piezoelectric and elastic (piezoelastic) properties of

the graphene reinforced nanocomposite material

(GRNC). The mechanics of materials (MOM) and

finite element (FE) models were developed to predict

the effective piezoelastic properties of GRNC using

the piezoelectric graphene. A few MD simulations

were also carried out to determine elastic properties of

a graphene. The calculated piezoelastic properties

were then used to a case study of cantilever beam

made of GRNC for investigating its electromechanical

response. For this purpose, an analytical beam model

was derived using the extended linear piezoelectricity

and Euler beam theories incorporating the flexoelec-

tricity effect. Specific attention was given to investi-

gate the effect of graphene volume fraction and

electrical load. Our results demonstrate that the

flexoelectricity has significant effect on the electrome-

chanical response GRNC beam with mere use of 40%

volume fraction of graphene. It is revealed that the

electromechanical response GRNC cantilever beam is

improved with the increase in the graphene volume

fraction and it can be tuned via applying different

electric potentials. The current results are significant,

which revealed that the flexoelectric phenomena in

graphene induced due to the strain gradient can be

exploited to form next generation NEMS for various

multifarious applications.

Acknowledgements The authors gratefully acknowledge the

financial support provided by the Indian Institute of Technology

Indore and the Science Engineering Research Board (SERB),

Department of Science and Technology, Government of India.

S.I.K. acknowledges the generous support of the SERB Early

Career Research Award Grant (ECR/2017/001863).

References

Alian, A.R., Kundalwal, S.I., Meguid, S.A.: Interfacial and

mechanical properties of epoxy nanocomposites using

different multiscale modeling schemes. Compos. Struct.

131, 545–555 (2015a). https://doi.org/10.1016/j.

compstruct.2015.06.014

Alian, A.R., Kundalwal, S.I., Meguid, S.A.: Multiscale model-

ing of carbon nanotube epoxy composites. Polymer (U.K.)

70, 149–160 (2015b). https://doi.org/10.1016/j.polymer.

2015.06.004

Alian, A.R., Meguid, S.A., Kundalwal, S.I.: Unraveling the

influence of grain boundaries on the mechanical properties

of polycrystalline carbon nanotubes. Carbon 125, 180–188(2017). https://doi.org/10.1016/j.carbon.2017.09.056

Bahamon, D.A., Qi, Z., Park, H.S., Pereira, V.M., Campbell,

D.K.: Conductance signatures of electron confinement

induced by strained nanobubbles in graphene. Nanoscale

7(37), 15300–15309 (2015). https://doi.org/10.1039/

C5NR03393D

Balandin, A.A., Ghosh, S., Bao, W., Calizo, I., Teweldebrhan,

D., Miao, F., Lau, C.N.: Superior thermal conductivity of

single-layer graphene. Nano Lett. 8, 902–907 (2008).

https://doi.org/10.1021/nl0731872

Bastwros, M., Kim, G.Y.: Fabrication of custom pattern rein-

forced AZ31 multilayer composite using ultrasonic spray

deposition. In: ASME 2016 11th International Manufac-

turing Science and Engineering Conference, MSEC-2016,

vol. 1, pp. V001T02A016–V001T02A016 (2016). https://

doi.org/10.1115/MSEC20168605

Benveniste, Y., Dvorak, G.J.: Uniform fields and universal

relations in piezoelectric composites. J. Mech. Phys. Solids

40(6), 1295–1312 (1992). https://doi.org/10.1016/0022-

5096(92)90016-U

Bernholc, J., Nakhmanson, S.M., Nardelli, M.B., Meunier, V.:

Understanding and enhancing polarization in complex

materials. Comput. Sci. Eng. 6(6), 12–21 (2004). https://

doi.org/10.1109/MCSE.2004.78

Bhavanasi, V., Kumar, V., Parida, K., Wang, J., Lee, P.S.:

Enhanced piezoelectric energy harvesting performance of

flexible PVDF-TrFE bilayer films with graphene oxide.

ACS Appl. Mater. Interfaces 8(1), 521–529 (2016). https://doi.org/10.1021/acsami.5b09502

Chen, F., Gupta, N., Behera, R.K., Rohatgi, P.K.: Graphene-

reinforced aluminum matrix composites: a review of syn-

thesis methods and properties. JOM 70(6), 837–845

(2018). https://doi.org/10.1007/s11837-018-2810-7

Conley, H., Lavrik, N.V., Prasai, D., Bolotin, K.I.: Graphene

bimetallic-like cantilevers: probing graphene/substrate

interactions. Nano Lett. 11(11), 4748–4752 (2011). https://doi.org/10.1021/nl202562u

Cui, Y., Kundalwal, S.I., Kumar, S.: Gas barrier performance of

graphene/polymer nanocomposites. Carbon 98, 313–333(2016). https://doi.org/10.1016/j.carbon.2015.11.018

Da Cunha Rodrigues, G., Zelenovskiy, P., Romanyuk, K.,

Luchkin, S., Kopelevich, Y., Kholkin, A.: Strong piezo-

electricity in single-layer graphene deposited on SiO2

grating substrates. Nat. Commun. 6, 7572 (2015). https://

doi.org/10.1038/ncomms8572

Dasari, B.L., Morshed, M., Nouri, J.M., Brabazon, D., Naher, S.:

Mechanical properties of graphene oxide reinforced alu-

minium matrix composites. Compos. B Eng. 145, 136–144(2018). https://doi.org/10.1016/j.compositesb.2018.03.022

Dewapriya, M.A.N., Rajapakse, R.K.N.D., Nigam, N.: Influ-

ence of hydrogen functionalization on the fracture strength

of graphene and the interfacial properties of graphene-

polymer nanocomposite. Carbon 93, 830–842 (2015).

https://doi.org/10.1016/j.carbon.2015.05.101

Gao, X.L., Li, K.: A shear-lag model for carbon nanotube-re-

inforced polymer composites. Int. J. Solids Struct. 42(5–6),1649–1667 (2005). https://doi.org/10.1016/j.ijsolstr.2004.

08.020

Garcıa-Macıas, E., Rodrıguez-Tembleque, L., Saez, A.: Bend-

ing and free vibration analysis of functionally graded

123

S. I. Kundalwal et al.

graphene vs. carbon nanotube reinforced composite plates.

Compos. Struct. 186, 123–138 (2018). https://doi.org/10.

1016/j.compstruct.2017.11.076

Gharbi, M., Sun, Z.H., Sharma, P., White, K., El-Borgi, S.:

Flexoelectric properties of ferroelectrics and the nanoin-

dentation size-effect. Int. J. Solids Struct. 48(2), 249–256(2011). https://doi.org/10.1016/j.ijsolstr.2010.09.021

Gradinar, D.A., Mucha-Kruczynski, M., Schomerus, H.,

Fal’Ko, V.I.: Transport signatures of pseudomagnetic

landau levels in strained graphene ribbons. Phys. Rev. Lett.

110(26), 266801 (2013). https://doi.org/10.1103/

PhysRevLett.110.266801

Gupta, S.S., Batra, R.C.: Elastic properties and frequencies of

free vibrations of single-layer graphene sheets. J. Comput.

Theor. Nanosci. 7(10), 2151–2164 (2010). https://doi.org/

10.1166/jctn.2010.1598

Hadjesfandiari, A.R.: Size-dependent piezoelectricity. Int.

J. Solids Struct. 50(18), 2781–2791 (2013). https://doi.org/10.1016/j.ijsolstr.2013.04.020

Hwang, S.H., Park, H.W., Park, Y.B.: Piezoresistive behavior

and multi-directional strain sensing ability of carbon nan-

otube-graphene nanoplatelet hybrid sheets. Smart Mater.

Struct. 22(1), 015013 (2013). https://doi.org/10.1088/

0964-1726/22/1/015013

Ji, X., Xu, Y., Zhang, W., Cui, L., Liu, J.: Review of function-

alization, structure and properties of graphene/polymer

composite fibers. Compos. A Appl. Sci. Manuf. 87, 29–45(2016). https://doi.org/10.1016/j.compositesa.2016.04.011

Jiang, B., Liu, C., Zhang, C., Liang, R., Wang, B.: Maximum

nanotube volume fraction and its effect on overall elastic

properties of nanotube-reinforced composites. Compos.

B Eng. 40(3), 212–217 (2009). https://doi.org/10.1016/j.

compositesb.2008.11.003

Kandpal, M., Palaparthy, V., Tiwary, N., Rao, V.R.: Low cost,

large area, flexible graphene nanocomposite films for

energy harvesting applications. IEEE Trans. Nanotechnol.

16(2), 259–264 (2017). https://doi.org/10.1109/TNANO.

2017.2659383

Khan, U., Young, K., O’Neill, A., Coleman, J.N.: High strength

composite fibres frompolyester filled with nanotubes and

graphene. J. Mater. Chem. 22(25), 12907–12914 (2012).

https://doi.org/10.1039/c2jm31946b

Kothari, R., Kundalwal, S. I., Sahu, S.K.: Transversely isotropic

thermal properties of carbon nanotubes containing vacan-

cies. Acta Mech. 229, 2787–2800 (2018). https://doi.org/

10.1007/s00707-018-2145-z

Kumar, A., Chakraborty, D.: Effective properties of thermo-

electro-mechanically coupled piezoelectric fiber rein-

forced composites. Mater. Des. 30, 1216–1222 (2009).

https://doi.org/10.1016/j.matdes.2008.06.009

Kundalwal, S.I.: Review on micromechanics of nano- and

micro- fiber reinforced composites. PolymCompos (2017).

https://doi.org/10.1002/pc.24569

Kundalwal, S.I., Choyal, V.: Transversely isotropic elastic

properties of carbon nanotubes containing vacancy defects

using MD. Acta Mech. 229, 2571–2584 (2018). https://doi.org/10.1007/s00707-018-2123-5

Kundalwal, S.I., Meguid, S.A.: Multiscale modeling of regu-

larly staggered carbon fibers embedded in nano-reinforced

composites. Eur. J. Mech. A/Solids 64, 69–84 (2017).

https://doi.org/10.1016/j.euromechsol.2017.01.014

Kundalwal, S.I., Ray, M.C.: Micromechanical analysis of fuzzy

fiber reinforced composites. Int. J. Mech. Mater. Des. 7(2),149–166 (2011). https://doi.org/10.1007/s10999-011-

9156-4

Kundalwal, S.I., Ray, M.C.: Effective properties of a novel

composite reinforced with short carbon fibers and radially

aligned carbon nanotubes. Mech. Mater. 53, 47–60 (2012).https://doi.org/10.1016/j.mechmat.2012.05.008

Kundalwal, S.I., Ray, M.C.: Effect of carbon nanotube waviness

on the elastic properties of the fuzzy fiber reinforced

composites. J. Appl. Mech. 80(2), 21010 (2013). https://

doi.org/10.1115/1.4007722

Kundalwal, S.I., Meguid, S.A., Weng, G.J.: Strain gradient

polarization in graphene. Carbon 117, 462–472 (2017).

https://doi.org/10.1016/j.carbon.2017.03.013

Kundalwal, S.I., Suresh Kumar, R., Ray, M.C.: Smart damping

of laminated fuzzy fiber reinforced composite shells using

1-3 piezoelectric composites. Smart Mater. Struct. 22(10),105001 (2013). https://doi.org/10.1088/0964-1726/22/10/

105001

Kvashnin, A.G., Sorokin, P.B., Yakobson, B.I.: Flexoelectricity

in carbon nanostructures: nanotubes, fullerenes, and

nanocones. J. Phys. Chem. Lett. 6(14), 2740–2744 (2015).

https://doi.org/10.1021/acs.jpclett.5b01041

Lee, C., Wei, X., Kysar, J.W., Hone, J.: Measurement of the

elastic properties and intrinsic strength of monolayer gra-

phene. Science 321(5887), 385–388 (2008). https://doi.

org/10.1126/science.1157996

Li, P., You, Z., Cui, T.: Graphene cantilever beams for nano

switches. Appl. Phys. Lett. 101(9), 093111 (2012). https://

doi.org/10.1063/1.4738891

Li, A., Zhou, S., Zhou, S., Wang, B.: Size-dependent analysis of

a three-layer microbeam including electromechanical

coupling. Compos. Struct. 116(1), 120–127 (2014). https://doi.org/10.1016/j.compstruct.2014.05.009

Mindlin, R.D.: Polarization gradient in elastic dielectrics. Int.

J. Solids Struct. 4(6), 637–642 (1968). https://doi.org/10.

1016/0020-7683(68)90079-6

Moreno, M.E., Tita, V., Marques, F.D.: Finite element analysis

applied to evaluation of effective material coefficients for

piezoelectric fiber composites. In: Brazilian Symposium on

Aerospace Eng. & Applications, 2005 (2009)

Morozovska, A.N., Eliseev, E.A., Tagantsev, A.K., Bravina,

S.L., Chen, L.Q., Kalinin, S.V.: Thermodynamics of

electromechanically coupled mixed ionic-electronic con-

ductors: deformation potential, Vegard strains, and flexo-

electric effect. Phys. Rev. B Condens. Matter Mater. Phys.

83(19), 195313 (2011). https://doi.org/10.1103/PhysRevB.83.195313

Munoz-Hernandez, A., Diaz, G., Calderon-Munoz, W.R., Leal-

Quiros, E.: Thermal-electric modeling of graphite: analysis

of charge carrier densities and Joule heating of intrinsic

graphite rods. J. Appl. Phys. 122(24), 245107 (2017).

https://doi.org/10.1063/1.4997632

Novoselov, K.S., Geim, A.K., Morozov, S.V., Jiang, D., Zhang,

Y., Dubonos, S.V., Grigorieva, I.V., Firsov, A.A.: Electric

field effect in atomically thin carbon films. Science (New

York, N.Y.) 306(5696), 666–669 (2004). https://doi.org/

10.1126/science.1102896

Odegard, G.M.: Constitutive modeling of piezoelectric polymer

composites Constitutive modeling of piezoelectric

123

Effect of flexoelectricity on the electromechanical response

polymer composites constitutive modeling of piezoelectric

polymer composites. Acta Mater. 52(18), 5315–5330

(2004)

Park, J.Y., Park, C.H., Park, J.S., Kong, K.J., Chang, H., Im, S.:

Multiscale computations for carbon nanotubes based on a

hybrid QM/QC (quantummechanical and quasicontinuum)

approach. J. Mech. Phys. Solids 58(2), 86–102 (2010)

Pettermann, H.E., Suresh, S.: A comprehensive unit cell model:

a study of coupled effects in piezoelectric 1-3 composites.

Int. J. Solids Struct. 37(39), 5447–5464 (2000). https://doi.org/10.1016/S0020-7683(99)00224-3

Plimpton, S.: Fast parallel algorithms for short-range molecular

dynamics. J. Comput. Phys. 117(1), 1–19 (1995). https://

doi.org/10.1006/jcph.1995.1039

Ray, M.C., Pradhan, A.K.: The performance of vertically rein-

forced 1-3 piezoelectric composites in active damping of

smart structures. Smart Mater. Struct. 15(2), 631–641

(2006). https://doi.org/10.1088/0964-1726/15/2/047

Roberts, M.W., Clemons, C.B., Wilber, J.P., Young, G.W.,

Buldum, A., Quinn, D.D.: Continuum plate theory and

atomistic modeling to find the flexural rigidity of a gra-

phene sheet interacting with a substrate. J. Nanotechnol.

2010, 1–8 (2010)

Rupa, N.S., Ray, M.C.: Analysis of flexoelectric response in

nanobeams using nonlocal theory of elasticity. Int. J. Mech.

Mater. Des. 13(3), 453–467 (2017). https://doi.org/10.

1007/s10999-016-9347-0

Saber, N., Araby, S., Meng, Q., Hsu, H.-Y., Yan, C., Azari, S.,

Lee, S.-H., Xu, Y., Ma, J., Yu, S.: Superior piezoelectric

composite films: taking advantage of carbon nanomateri-

als. Nanotechnology 25(4), 045501 (2014). https://doi.org/10.1088/0957-4484/25/4/045501

Scari, A.S., Pockszevnicki, B.C., Junior, J.L., Junior, P.A.A.M.:

Stress-strain compression of AA6082-T6 aluminum alloy

at room temperature. J. Struct. (2014). https://doi.org/10.

1155/2014/387680

Shah, P. H., Batra, R. C.: Elastic moduli of covalently func-

tionalized single layer graphene sheets. Comput. Mater.

Sci. 95, 637–650 (2014a). https://doi.org/10.1016/j.

commatsci.2014.07.050

Shah, P. H., Batra, R. C.: In-plane elastic moduli of covalently

functionalized single-wall carbon nanotubes. Comput.

Mater. Sci. 83, 349–361 (2014b). https://doi.org/10.1016/j.commatsci.2013.11.018

Shen, S., Hu, S.: A theory of flexoelectricity with surface effect

for elastic dielectrics. J. Mech. Phys. Solids 58(5), 665–677(2010). https://doi.org/10.1016/j.jmps.2010.03.001

Smith, W.A., Auld, B.A.: Modeling 1–3 composite piezo-

electrics: thickness-mode oscillations. IEEE Trans.

Ultrason. Ferroelectr. Freq. Control 38(1), 40–47 (1991).

https://doi.org/10.1109/58.67833

Song, Y.S., Youn, J.R.: Modeling of effective elastic properties

for polymer based carbon nanotube composites. Polymer

47(5), 1741–1748 (2006). https://doi.org/10.1016/j.

polymer.2006.01.013

Stuart, S.J., Tutein, A.B., Harrison, J.A.: A reactive potential for

hydrocarbons with intermolecular interactions. J. Chem.

Phys. 112(14), 6472 (2000). https://doi.org/10.1063/1.

481208

Sundar, U., Cook-Chennault, K.A., Banerjee, S., Refour, E.:

Dielectric and piezoelectric properties of percolative three-

phase piezoelectric polymer composites. J. Vac. Sci.

Technol. B, Nanotechnol. Microelectron. Mater. Process.

Meas. Phenom. 34(4), 41232 (2016). https://doi.org/10.

1116/1.4955315

Tian, W., Li, S., Wang, B., Chen, X., Liu, J., Yu, M.: Graphene-

reinforced aluminum matrix composites prepared by spark

plasma sintering. Int. J. Miner. Metall. Mater. 23(6),723–729 (2016). https://doi.org/10.1007/s12613-016-

1286-0

Verma, D., Gupta, S.S., Batra, R.C.: Vibration mode localiza-

tion in single- and multi-layered graphene nanoribbons.

Comput. Mater. Sci. 95, 41–52 (2014). https://doi.org/10.

1016/j.commatsci.2014.07.005

Wang, J., Li, Z., Fan, G., Pan, H., Chen, Z., Zhang, D.: Rein-

forcement with graphene nanosheets in aluminum matrix

composites. Scr. Mater. 66(8), 594–597 (2012). https://doi.org/10.1016/j.scriptamat.2012.01.012

Yan, Z., Jiang, L.Y.: Flexoelectric effect on the electroelastic

responses of bending piezoelectric nanobeams flexoelec-

tric effect on the electroelastic responses of bending

piezoelectric nanobeams. J. Appl. Phys. 113(19), 194102(2013). https://doi.org/10.1063/1.4804949

Ying, C., Zhifei, S.: Exact solutions of functionally gradient

piezothermoelastic cantilevers and parameter identifica-

tion. J. Intell. Mater. Syst. Struct. 16(6), 531–539 (2005).

https://doi.org/10.1177/1045389X05053208

Zhang, Y.B., Tan, Y.W., Stormer, H.L., Kim, P.: Experimental

observation of the quantumHall effect and Berry’s phase in

graphene. Nature 438(7065), 201–204 (2005). https://doi.

org/10.1038/nature04235

Zhang, Y.Y., Pei, Q.X., Wang, C.M.: Mechanical properties of

graphynes under tension: a molecular dynamics study.

Appl. Phys. Lett. 101(8), 081909 (2012). https://doi.org/10.1063/1.4747719

Zhao, X., Zhang, Q., Chen, D., Lu, P.: Enhanced mechanical

properties of graphene-based poly(vinyl alcohol) com-

posites. Macromolecules 43, 2357–2363 (2010)

123

S. I. Kundalwal et al.