Composite Structures 106 (2013) 635–645

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Composite Structures

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

A nonlinear finite element model using a unified formulationfor dynamic analysis of multilayer composite plate embeddedwith SMA wires

0263-8223/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.compstruct.2013.07.006

⇑ Corresponding author. Address: Centre of Excellence for Research in AdvancedMaterials and Structures, Faculty of Mechanical Engineering, K.N. Toosi Universityof Technology, Pardis Street, Molasadra Avenue, Vanak Square, Tehran, Iran. Tel.:+98 2188674747; fax: +98 2188674748.

E-mail addresses: smrkhalili2005@gmail.com (S.M.R. Khalili), mbd_dehkordi@yahoo.com (M. Botshekanan Dehkordi), erasmo.carrera@polito.it (E. Carrera).

S.M.R. Khalili a,b,⇑, M. Botshekanan Dehkordi a, E. Carrera c

a Centre of Excellence for Research in Advanced Materials and Structures, Faculty of Mechanical Engineering, K.N. Toosi University of Technology, Tehran, Iranb Faculty of Engineering, Kingston University of Technology, London, UKc Department of Mechanics and Aerospace Engineering, Politecnico di Torino, Torino, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Available online 11 July 2013

Keywords:Vibration dampingShape memory alloysMaterial nonlinearityNonlinear finite elementComposite plateAdvanced plate theories

In this study, a new nonlinear finite element model is presented in the frame work of Carrera’s UnifiedFormulation (CUF) for the dynamic analysis of SMA hybrid composite considering the instantaneousphase transformation and material nonlinearity effects, for every point on the plate. The CUF unify manytheories in a unified form which can be differed by the order of expansion and definition of the variablesin the thickness direction. The Brinson’s SMA constitutive equation is used to model the behavior of SMAwires. The governing equations are derived using the Reissner Mixed Variational Theorem (RMVT) inorder to enforce the interlaminar continuity of transverse shear and normal stresses between two adja-cent layers. A transient finite-element-based method beside an iterative incremental procedure is pre-sented to study the dynamic response of multilayered composite plate embedded with SMA wires. Asuppressed vibration of the plate is observed, which is due to the energy dissipation of SMA wires. Theparametric effects like length-to-thickness ratio, plate aspect ratio and also the effect of different bound-ary conditions, upon the loss factors are investigated. Results show that as the length-to-thickness ratioand also the plate aspect ratio increases, the loss factor decreases.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Vibration damping consists of a challenging task to increase fa-tigue life and comfort of advanced structures. The variation of theproperties of the material composing the structure provides a valu-able approach to this task. Only, very limited options are availablefor such changes in the material properties. The present work pro-poses the use of NiTi shape memory alloys for this purpose. Pre-senting an exhaustive mathematical model for these structures issuffering from the nonlinearity behavior which is due to the phasetransformation. Therefore, in the most cases, the proposed modelsare accompanied with a lot of simplifying assumptions. In thestudy by Jafari and Ghiasvand [1] dynamic analysis of SMA beamunder a moving load was investigated. In their study, the effectof hysteretic loops was modeled by substitution of an equivalentdamping ratio in the equation of motion. The dynamic analysis of

SMA beam was studied by Hashemi and Khadem [2]. They consid-ered the effect of the phase transformation. But In their researchthey assumed that, the beam behaves like a one-degree-of freedomsystem. Zbiciak [3] investigated the response history of SMA beamunder impulse loading. He employed the rheological scheme formodeling the behavior of SMA material. He assumed that thematerial properties of SMA are constant.

Recently, many considerations have been focused on theimprovement in the properties of the composite structures byshape memory alloys. In the study by Rogers and Barker [4] SMAwires were used to control the frequency of a graphite/epoxy mul-tilayered beam. Upon the heating of SMA wires, an axial force wasgenerated in the beam because of the shape memory effect. Theyshowed that, the fundamental frequency of the beam was in-creased significantly by utilizing 15% volume fraction of SMAwires. Baz et al. [5] demonstrated that the SMA wires embeddedto composite beams have a capability to control their natural fre-quencies. The effect of pre-strain, and also the effect of tempera-ture on the SMA wires, was taken to account in their study. Theeffect of SMA wires on the controlling of the buckling and fre-quency analysis of composite beam was studied by Baz et al. [6].They found that the buckling load of a flexible composite beamwas increased up to three times the uncontrolled beam. Epps and

Fig. 1. Pseudoelastic behavior of shape memory alloys [35].

636 S.M.R. Khalili et al. / Composite Structures 106 (2013) 635–645

Chandra [7] implemented an experimental–analytical investiga-tion on the composite beams embedded with SMA wires. Theydemonstrated that, as the volume fraction of SMA wires increasesthe natural frequency of SMA composite beams increases. How-ever, it cannot be found any details on the damping properties ofSMA wires. Khalili et al. [8] studied the nonlinear dynamic re-sponse of sandwich beam with SMA hybrid composite skins underimpulse loading. In their research a new element was proposedusing the high order sandwich panel theory. They investigate theinfluence of the SMA wires on the vibration suppression of sand-wich beam considering the phase transformation effects. In the re-search conducted by Ostachowicz et al. [9], dynamic and bucklinganalysis of composite plates embedded with SMA wires was inves-tigated. They showed that the SMA wires have a significant effecton the natural frequencies and the thermal buckling of these struc-tures. Lee and Lee [10] studied the buckling and post-bucklinganalysis multilayered composite plates embedded with SMA wires.They showed that the critical load of the composite plates is in-creased by activation of the SMA wires. Cho and Rhee [11] pre-sented the nonlinear finite element model for static analysis ofshape memory alloy wire reinforced hybrid laminate compositeshells. They used the Brison’s constitutive equation based on theiterative method for modeling the behavior of SMA wires.

The use of multilayer composite structures has been continu-ously growing in the recent years. Multilayered composite struc-tures are utilized in many components of automotive, aerospace,and transportation vehicles. In recent years, many theories devotedto analyze the multilayer composite structures. Kirchhoff [12](CLT) and Reissner–Mindlin [13,14] (FSDT) plate theories are notsuitable to analysis the multilayered composite structures [15]. Be-cause they cannot satisfy the continuity of transverse stresses be-tween two adjacent laminate. In addition, these theories are notable to fulfill the zig-zag maner of the displacement distributionalong the thickness direction. These conditions are called C0

z -requirements in Ref. [16]. In this regard, a lot of theories have beenpresented for modification the FSDT [17–23]. These theories areknown as higher-order shear deformation theories (HSDT). Khaliliet al. [24] modified high-order theory for sandwich panels HSAPT,by applying first-order shear deformation theory for face-sheetsand used the improved HSPAT to study the free vibration andlow velocity response of sandwich panels. A lot of finite-elementmodels are proposed using the HSDT models [17–23,25]. It shouldbe mentioned that, a closed-form solutions can be found only insome few cases, especially for the linear problems with specificboundary conditions [26]. Two-dimensional theories are dividedto some categories, based on the unknown variables. If the dis-placement field is only unknown, the corresponding theories areknown as classical models and the governing equations are derivedusing the Principle of Virtual Displacements (PVD). If the trans-verse stresses are also assumed as unknowns, the correspondingtheories are known as mixed theories [27,28]. Carrera et al.[16,29–33] presented a unified formulation (UF) of multilayeredtheories, for both the PVD and RMVT formulations. This can be re-ferred as Equivalent-Single-Layer (ESL), if the unknown variablesare considered for the whole plate, or (LW), if the unknown vari-ables are considered for each layer, individually.

In this study, the nonlinear dynamic analysis of composite mul-tilayered plate embedded with SMA wires is investigated based onthe Carrera’s unified formulation. The instantaneous phase trans-formation effects are considered for all the points on the platefor the first time. The Brinson’s SMA constitutive equation is usedto model the pseudoelastic behavior of shape memory alloys wires.In the present study, the (RMVT) is utilized to derivation of thegoverning equations. The governing equations of motion and thekinetic relations of phase transformation are coupled with eachother. Therefore, a transient finite-element-based method beside

an iterative incremental procedure is presented to study the dy-namic response of multilayered composite plate embedded withSMA wires. Finally, a new program code is written in MATLAB soft-ware in order to dynamic analysis of composite plate embeddedwith SMA wires.

2. Constitutive equation of the SMA wires

In this study, the constitutive equation of shape memory alloyshas been proposed by Brinson [34] is utilized. This constitutiveequation presents the relation between the stress (r), strain (e),temperature (T) and martensite fraction (n) as follows:

r� r0 ¼ EðnÞðeÞ � Eðn0Þðe0Þ þXðnÞðnsÞ �Xðn0Þðns0Þ þ hðT � T0Þð1Þ

where E(n) and h are the Young’s modulus and the thermoelasticcoefficient, respectively. The subscript 0 implies the initial condi-tions of the corresponding term. E(n) can be expressed as follows[34]:

EðnÞ ¼ EA þ nðEM � EAÞ ð2Þ

where EM and EA are the Young’s modulus of the shape memory al-loys in the martensite and austenite phases, respectively.

In addition, X(n) is the transformation tensor and can be writ-ten in terms of Young’s modulus as follows:

XðnÞ ¼ �eLEðnÞ ð3Þ

where eL is the maximum strain that can be recovered completely.In the model of Brinson, the martensite volume fraction is separatedinto two parts as follows:

n ¼ ns þ nT ð4Þ

where ns indicates the fraction of the martensite that is induced bystress and nT indicates the fraction of the martensite that is inducedby temperature. Kinetic relations of the phase transformation (seeFig. 1) are expressed as follows [34]:

For conversion to martensite:For T > Ms and rcr

s þ CMðT �MsÞ < r < rcrf þ CMðT �MsÞ

ns ¼1� ns0

2cos

prcr

s � rcrf

r� rcrf � CMðT �MsÞ

� � !þ 1þ ns0

2

nT ¼ nT0 �nT0

1� ns0ðns � ns0Þ

ð5Þ

For T < Ms and rcrs < r < rcr

f

S.M.R. Khalili et al. / Composite Structures 106 (2013) 635–645 637

ns ¼1� ns0

2cos

prcr

s � rcrf

r� rcrf

� � !þ 1þ ns0

2

nT ¼ nT0 �nT0

1� ns0ðns � ns0Þ þ DTn

ð6Þ

For Mf < T < Ms and T < T0

DTn ¼1� nT0

2ðcosðaMðT �Mf ÞÞ þ 1Þ

Else DTn ¼ 0

For conversion to austenite:For T > As and CA(T � Af) < r < CA(T � As)

n ¼ ns0

2cos aA T � As �

rCA

� �þ 1

� �ns ¼ ns0 �

ns0

n0ðn0 � nÞ

nT ¼ nT0 �nT0

n0ðn0 � nÞ

ð7Þ

where rcrs and rcr

f are the critical stresses for the start and end of thephase transformation, respectively.

3. Unified formulation and finite element analysis

The Unified Formulation (UF) allows to unify a lot of twodimensional theories for plates, by separation of the unknown vari-ables into functions which depend only on the thickness coordi-nate z, and the ones coincide with the in-plane coordinates (x, y)[16,33]. In the Carrera’s unified formulation (CUF) the global un-known a(x, y, z) and its variation da(x, y, z) can be expressed as fol-lows [16]:

aðx; y; zÞ ¼ FsðzÞasðx; yÞ; daðx; y; zÞ ¼ FsðzÞdasðx; yÞwith s; s ¼ t; b; r and r ¼ 2; . . . ;N

ð8Þ

Bold letters stand for arrays and the summing convention is ex-pressed by repeated indices s and s. Subscripts t, b and r representthe top, the bottom and the higher order terms of the expansion,respectively. The expansion of N can be written from first to fourthorder. In this study, Reissner’s mixed variational theorem (RMVT) isutilized to enforce the interlaminar continuity of the shear and nor-mal stresses between the two adjacent layers a priori. Most of theavailable two dimensional theories are divided into two categories;equivalent single layer models (ESL) and layer-wise models (LW). Inthe ESL models, the z power expansion is utilized for displacementfield as follows:

u ¼ u0 þ zrur ; r ¼ 1;2; . . . ;N ð9Þ

The index 0 represents the displacement values related to the refer-ence surface of the plate. Linear and higher-order expansions in thez-direction are described using the r-polynomials. N is a free param-eter by which the demanded model can be obtained. To write all themodels in a unified formulation, the relation (8) is represented asfollows:

u ¼ Ftut þ Fbub þ Frur ¼ Fsus s ¼ t; b; r; r ¼ 2; . . . ;N � 1 ð10Þ

By Comparison of the above relation with Eq. (8), it can be seen thatindex b presents the values related to (ub = u0), while index t standsfor the highest-order term (ut = uN). Therefore, Fs polynomials canbe defined as follows:

Fb ¼ 1; Ft ¼ zN ; Fr ¼ zr ; r ¼ 2; . . . ;N � 1 ð11Þ

The above expansions can be used for all the multilayer plate that iscoincided with the ESL models. For achieving the LW models, theexpansion in Eq. (8) must be utilized for every layer, individually.

The Taylor expansion is not appropriate for description of the LWmodels, since, in the Taylor expansion, the displacements at theinterfaces are not as unknowns to enforce the continuity betweenthe two adjacent layers. For this purpose, a combination of Legendrepolynomials [36–38] can be utilized as the thickness functions[16]::

uk ¼ Ftukt þ Fbuk

b þ Frukr ¼ Fsuk

s s ¼ t; b; r;

r ¼ 2; . . . ;N; k ¼ 1;2; . . . ;Nl ð12Þ

where Nl is the number of the layers constituting the plate. The indi-ces t and b stand for the values corresponding to the top layer andthe bottom layer, respectively. The thickness functions Fs(fk) are ex-pressed for the kth layer. The Legendre polynomials Pj(fk) are:

P0 ¼ 1; P1 ¼ fk; P2 ¼ð3f2

k � 1Þ2

; P3 ¼5f3

k

2� 3fk

2;

P4 ¼35f4

k

8� 15f2

k

4þ 3

8ð13Þ

where fk = 2zk/hk, while zk and hk are the local coordinate and thethickness of kth layer, respectively. Therefore, �1 6 fk 6 1, in thisregard see Fig. 2. The thickness functions are obtained using theLegendre polynomials as follows [16]:

Ft ¼P0 þ P1

2; Ft ¼

P0 � P1

2; Fr ¼ Pr � Pr�2; r ¼ 2;3; . . . ;N

ð14Þ

Therefore, these functions have the following characteristic:

fk ¼1; Ft ¼ 1; Fb ¼ 0; Fr ¼ 0�1; Ft ¼ 0; Fb ¼ 1; Fr ¼ 0

�ð15Þ

Consequently, the continuity of the displacement at the two adja-cent layers is fulfilled as follows:

ukt ¼ ukþ1

b ; k ¼ 1; . . . ;Nl � 1 ð16Þ

The transverse stresses are considered for every layer, indepen-dently, regardless of the ESL and LW models. For this purpose, theLW explanation that is used for displacements is utilized for thetransverse stresses to enforce the interlaminar continuity for boththe ESL and LW models:

rknM ¼ Ftr

knt þ Fbr

knb þ Frr

kns ¼ Fsr

knr; s ¼ t; b; r;

r ¼ 2; . . . ;N; k ¼ 1;2; . . . ;Nl ð17Þ

Therefore, the continuity of the transverse stresses at the two adja-cent layers is fulfilled as follows:

rknt ¼ rkþ1

nb k ¼ 1; . . . ;Nl � 1 ð18Þ

For abbreviating the name of the models, the new acronym is intro-duced. To achieve this, the first parameter of the acronym is corre-sponding to the type of the model; therefore, letter L means LW andE means ESL. The second parameter identifies the type of the vari-ational statements which is used in derivation of the governingequations; in this regard, M means (RMVT). The third parametershows the number N which identifies the order of model. For exam-ple LM3 means LW theory using RMVT with third order of displace-ment and stress fields in the layer.

In this study, a quadratic nine-nodes finite element is used toapproximate the displacements and the transverse stresses asfollows:

uk ¼ FsNiqksi rk

nM ¼ FsNigksi ði ¼ 1; . . . ;9Þ ð19Þ

where Ni are the Lagrange quadratic shape functions, also:

qksi¼½qk

uxsi qkuysi qk

uzsi�T

gksi¼½gk

uxsi gkuysi gk

uzsi�T ði¼1; . . . ;NnÞ ð20Þ

where qksi and gk

si are the nodal unknown of the element for the kthlayer.

Fig. 2. Dimensions and coordinate systems of the SMA multilayered composite plate.

638 S.M.R. Khalili et al. / Composite Structures 106 (2013) 635–645

4. Theoretical formulation and methods

Using the Hamilton’s principle, it can be found that:

dLint � dLFin� dLext ¼ 0 ð21Þ

where, Lint is the internal work, LFinis the work done by the inertial

force and Lext is the work done by the external force. The total inter-nal work is given by the mechanical work and the work due to thephase transformation.

Lint ¼ LintM þ LintSMA ð22Þ

where, LintM is the work done by the mechanical stresses and LintSMA

is the work due to the phase transformation of SMA wires.In the present study, the (RMVT) is utilized in derivation of the

governing equations [27,28]. For the study of SMA hybrid multilay-ered plate, the RMVT is explained as follows:

dLintM ¼XNl

k¼1

ZXk

ZAk

fdekpG

TðnÞrkpCðnÞ þ dek

nGTðnÞrk

nMðnÞ

þ drknM

TðnÞðeknGðnÞ � ek

nCðnÞÞgdXdzk ð23Þ

The statement drknM

Tek

nG � eknC

� �� �is Lagrange’s Multiplier which

fulfils the compatibility of the transverse strains en.X and Ak, implythe in-plane and the thickness domains of the lamina, respectively.Index G means that the corresponding terms are obtained using thegeometrical relations, while C denotes the corresponding terms thatare obtained by the constitutive equations. Also, index M impliesthat the stress components are considered a priori. In the frame-work of CUF, the strains are explained as follows:

ekpGðnÞ ¼ fexxðnÞ; eyyðnÞ; exyðnÞgkT ¼ Dp ukðnÞ ð24aÞ

eknGðnÞ ¼ fcxzðnÞ; cyzðnÞ; ezzðnÞgkT ¼ ðDnp þDnzÞ ukðnÞ ð24bÞ

where the indices p and n imply the in-plane and normal compo-nents, respectively. The operators Dp, Dnp and Dnz are the differen-tial matrices which are defined as follows:

Dp ¼@x 0 00 @y 0@y @x 0

264375; Dnp ¼

0 0 @x

0 0 @y

0 0 0

264375; Dnz ¼

@z 0 00 @z 00 0 @z

264375ð25Þ

Also, for the stress components, it can be written:

rkpðnÞ ¼ Ck

ppðnÞekpðnÞ þ Ck

pnðnÞeknðnÞ ð26aÞ

rknðnÞ ¼ Ck

npðnÞekpðnÞ þ Ck

nnðnÞeknðnÞ ð26bÞ

where rkp; rk

p and the matrices Ckpp; Ck

pn; Cknp and Ck

nn are:

rkpðnÞ ¼ rk

xxðnÞ;rkyyðnÞ;rk

xyðnÞn o

;

rknðnÞ ¼ rk

xzðnÞ;rkyzðnÞ;rk

zzðnÞn o

ð27aÞ

CkppðnÞ ¼

Ck11ðnÞ Ck

12ðnÞ Ck16ðnÞ

Ck12ðnÞ Ck

22ðnÞ Ck26ðnÞ

Ck16ðnÞ Ck

26ðnÞ Ck66ðnÞ

26643775; Ck

pnðnÞ ¼0 0 Ck

13ðnÞ0 0 Ck

23ðnÞ0 0 Ck

36ðnÞ

26643775

CknpðnÞ ¼

0 0 00 0 0

Ck13ðnÞ Ck

23ðnÞ Ck36ðnÞ

264375; Ck

nnðnÞ ¼Ck

55ðnÞ Ck45ðnÞ 0

Ck45ðnÞ Ck

44ðnÞ 0

0 0 Ck33ðnÞ

26643775ð27bÞ

where Ckij denotes the stiffness of the kth layer of the plate. The

Young’s modulus and other properties of a SMA hybrid compositelayer are obtained as follows [39]:

ElðnÞ ¼ Ecl kc þ EsðnÞks ð28aÞ

EtðnÞ ¼ Ect=ð1�

ffiffiffiffiffiks

pð1� Ec

t=EsðnÞÞÞ ð28bÞGltðnÞ ¼ Gc

ltGsðnÞ=ðkcGsðnÞ þ ksGcltÞ ð28cÞ

tlt ¼ tcltkc þ tsks ð28dÞ

where indices ‘s’ and ‘c’ imply the shape memory and the compositemedium, respectively. Since, the displacements and transversestresses are unknown in RMVT; therefore the constitutive equationsmust be expressed as follows:

rkpðnÞ ¼ eCk

ppðnÞekpðnÞ þ eCk

pnðnÞrknðnÞ ð29aÞ

eknðnÞ ¼ eCk

npðnÞekpðnÞ þ eCk

nnðnÞrknðnÞ ð29bÞ

The coefficients eCkpp;

eCkpn;

eCknp and eCk

nn in the above relations are de-fined as follows:eCk

ppðnÞ ¼ CkppðnÞ � Ck

pnðnÞCknn

�1ðnÞCk

npðnÞ eCkpnðnÞ ¼ Ck

pnðnÞCknn

�1ðnÞð30aÞeCk

npðnÞ ¼ �Cknn

�1ðnÞCk

npðnÞ eCknnðnÞ ¼ �Ck

nn

�1ðnÞ ð30bÞ

After substitution of Eqs. (19), (20), (24) and (29) in Eq. (23), onegets the following:

S.M.R. Khalili et al. / Composite Structures 106 (2013) 635–645 639

dLkintM ¼ / dqkT

si ðnÞ DTpðNiIÞeZkss

pp ðnÞDTpðNjIÞ

h iqk

sjðnÞn o

.X

þ / dqkTsi ðnÞ DT

pðNiIÞeZksspn ðnÞNj

h igk

sjðnÞn o

.X

þ / dqkTsi ðnÞ DT

nXðNiIÞEssNj þ Es;zsNiNjIh i

gksjðnÞ

n o.X

þ / dgkTsi ðnÞ NiEssDnXðNjIÞ þ Ess;z NiNjI

�qk

sjðnÞn o

.X

� / dgkTsi ðnÞ Ni

eZkssnp ðnÞDpðNjIÞ

h iqk

sjðnÞn o

.X

� / dgkTsi ðnÞ Ni

eZkssnn ðnÞNj

h igk

sjðnÞn o

.X

ð31Þ

where the following layer’s stiffness and compliance have beenintroduced:

eZksspp ðnÞ; eZkss

pn ðnÞ; eZkssnp ðnÞ; eZkss

nn ðnÞ� �¼ eCk

ppðnÞ; eCkpnðnÞ; eCk

npðnÞ; eCknnðnÞ

� �Ess ð32Þ

In the above relation, the symbols / . . . .X implies the integrals ondomain X and also:

Ess ¼Z

Ak

ðFsFsÞdz ð33Þ

Therefore, dLkintM can be expressed as follows:

dLkintM ¼ dqkT

si ðnÞ Kkssijuu ðnÞqk

sjðnÞ þ Kkssijur ðnÞgk

sjðnÞh i

þ dgkTsi ðnÞ Kkssij

ru ðnÞqksjðnÞ þ Kkssij

rr ðnÞgksjðnÞ

h ið34Þ

where

Kkssijuu ðnÞ ¼ / DT

pðNiIÞeZksspp ðnÞD

TpðNjIÞ

h i.X ð35aÞ

Kkssijur ðnÞ ¼ / DT

pðNiIÞeZksspn ðnÞNj þ DT

nXðNiIÞEssNj þ Es;zsNiNjIh i

.X ð35bÞ

Kkssijru ðnÞ ¼ / NiEssDnXðNjIÞ þ Ess;z NiNjI� Ni

eZkssnp ðnÞDpðNjIÞ

h i.X ð35cÞ

Kkssijrr ðnÞ ¼ / �Ni

eZkssnn ðnÞNj

h i.X ð35dÞ

Above relations are [3 � 3] ‘fundamental nuclei’ such that the wholestiffness structure can be obtained by expanding the indices andthen assembling the mentioned nucleus. The components of thisnucleus can be expressed as follows:

KkssijuuxxðnÞ ¼ /eZkss

pp11ðnÞ½Ni;xNj;x�.X þ /eZksspp16ðnÞ½Ni;yNj;x�.X

þ /eZksspp16ðnÞ½Ni;xNj;y�.X þ /eZkss

pp66ðnÞ½Ni;yNj;y�.X

KkssijuuxyðnÞ ¼ /eZkss

pp12ðnÞ½Ni;xNj;y�.X þ /eZksspp26ðnÞ½Ni;yNj;y�.X

þ /eZksspp16ðnÞ½Ni;xNj;x�.X þ /eZkss

pp66ðnÞ½Ni;yNj;x�.X

KkssijuuxzðnÞ ¼ 0

KkssijuuyxðnÞ ¼ /eZkss

pp12ðnÞ½Ni;yNj;x�.X þ /eZksspp16ðnÞ½Ni;xNj;x�.X

þ /eZksspp26ðnÞ½Ni;yNj;y�.X þ /eZkss

pp66ðnÞ½Ni;xNj;y�.X

KkssijuuyyðnÞ ¼ /eZkss

pp22ðnÞ½Ni;yNj;y�.X þ /eZksspp26ðnÞ½Ni;xNj;y�.X

þ /eZksspp26ðnÞ½Ni;yNj;x�.X þ /eZkss

pp66ðnÞ½Ni;xNj;x�.X

KkssijuuyzðnÞ ¼ 0

KkssijuuzxðnÞ ¼ 0

KkssijuuzyðnÞ ¼ 0

KkssijuuzzðnÞ ¼ 0

KkssijurxxðnÞ ¼ Es;zs / ½NiNj�.X

KkssijurxyðnÞ ¼ 0

KkssijurxzðnÞ ¼ /eZkss

pn13ðnÞ½Ni;xNj�.X þ /eZksspn36ðnÞ½Ni;yNj�.X

KkssijuryxðnÞ ¼ 0

KkssijuryyðnÞ ¼ Es;zs / ½NiNj�.X

KkssijuryzðnÞ ¼ /eZkss

pn23ðnÞ½Ni;yNj�.X þ /eZksspn36ðnÞ½Ni;xNj�.X

KkssijurzxðnÞ ¼ Ess / ½Ni;xNj�.X

KkssijurzyðnÞ ¼ Ess / ½Ni;yNj�.X

KkssijurzzðnÞ ¼ Es;zs / ½NiNj�.X

KkssijruxxðnÞ ¼ Ess;z / ½NiNj�.X

KkssijruxyðnÞ ¼ 0

KkssijruxzðnÞ ¼ Ess;z / ½NiNj�.X

KkssijruyxðnÞ ¼ 0

KkssijruyyðnÞ ¼ Ess;z / ½NiNj�.X

KkssijruyzðnÞ ¼ Ess / ½NiNj;y�.X

KkssijruzxðnÞ ¼ � / eZkss

np13ðnÞ½NiNj;x�.X � /eZkssnp36ðnÞ½NiNj;y�.X

KkssijruzyðnÞ ¼ � / eZkss

np23ðnÞ½NiNj;y�.X � /eZkssnp36ðnÞ½NiNj;x�.X

KkssijruzzðnÞ ¼ Ess;z / ½NiNj�.X

KkssijrrxxðnÞ ¼ � / eZkss

nn55ðnÞ½NiNj�.X

KkssijrrxyðnÞ ¼ � / eZkss

nn45ðnÞ½NiNj�.X

KkssijrrxzðnÞ ¼ 0

KkssijrryxðnÞ ¼ � / eZkss

nn45ðnÞ½NiNj�.X

KkssijrryyðnÞ ¼ � / eZkss

nn44ðnÞ½NiNj�.X

KkssijrryzðnÞ ¼ 0

KkssijrrzxðnÞ ¼ 0

KkssijrrzyðnÞ ¼ 0

KkssijrrzzðnÞ ¼ � / eZkss

nn33ðnÞ½NiNj�.X ð36Þ

It should be mentioned that due to the location dependency of thecoefficients eZkss

ij ðnÞði; j ¼ p;nÞ, they must be remain in the integraldomain. The mentioned integrals and also the integral in the thick-ness direction are computed by the Gaussian quadrature method.To overcome the shear locking problem, the selective reduced inte-gration technique is employed in this study [40].

dLkintSMA is obtained as follows:

dLkintSMA ¼

ZXk

ZAk

ðdekxxGðnÞrk

xxsmaðnÞ þ dekyyGðnÞrk

yysmaðnÞÞdzdX ð37Þ

where rkxxsma and rk

yysma are the stresses due to the phase transfor-mation in the x and y directions, respectively. These stresses canbe expressed as follows:

rkxxsmaðnÞ ¼ kk

SxvkSxðnÞ ð38aÞ

rkyysmaðnÞ ¼ kk

SyvkSyðnÞ ð38bÞ

where kkSx; kk

Sy imply the volume fraction of the SMA wires in the xand y directions, respectively. The new parameters vk

Siði ¼ x; yÞ,which are due to the phase transformation, are explained asfollows:

vkSxðnÞ ¼ �eLEk

SxðnkxÞn

kx ð39aÞ

vkSyðnÞ ¼ �eLEk

SyðnkyÞn

ky ð39bÞ

In the above relations nki ði ¼ x; yÞ and Ek

Siði ¼ x; yÞ are the martensitevolume fraction and the Young’s modulus of the SMA wires in the xand y directions, respectively. In the framework of CUF, dek

xxGðnÞ anddek

yyGðnÞ can be obtained by the following relations:

dekxxGðnÞ ¼ Ni;xFk

sdqkxsiðnÞ ð40aÞ

dekyyGðnÞ ¼ Ni;yFk

sdqkysiðnÞ ð40bÞ

Fig. 3. Assembling scheme related to Kuu and M in ESL models.

640 S.M.R. Khalili et al. / Composite Structures 106 (2013) 635–645

After substitution of Eqs. (38), (39) and (40) in Eq. (37), one gets thefollowing:

dLkintSMA ¼ dqkT

si ðnÞPksma siðnÞ ð41Þ

where

PksmasiðnÞ ¼ pk

sxsiðnÞ pksysiðnÞ 0

h ið42aÞ

pksxsiðnÞ ¼

ZXk

Ni;xUksxSMAðnÞdX pk

sysiðnÞ ¼Z

Xk

Ni;yUksySMAðnÞdX ð42bÞ

where

UksxSMAðnÞ ¼ �Eskk

SxeLEkSxn

kx Uk

sySMAðnÞ ¼ �EskkSyeLEk

Synky ð43Þ

and

Es ¼Z

Ak

Fsdz ð44Þ

dLkFin

is obtained as follows:

dLkFin¼ dqkT

si ðnÞMksij €qk

sjðnÞ ð45Þ

where

Mkssij ¼mk

ss / ½NiNj�.X 0 00 mk

ss / ½NiNj�.X 00 0 mk

ss / ½NiNj�.X

264375 ð46Þ

and

mkss ¼

ZAk

qkFsFsdz ð47Þ

In order to obtain the work done by the external loads, it is assumedthat a pressure is distributed on the layer k, having the distantfk ¼ f1

k from the reference surface. Therefore, the work done by thispressure in obtained as follows:

dLkext ¼

ZXk

dukTðx; y; fk1ÞP

kðx; y; fk1ÞdX ð48Þ

where

uk ¼ F1sNiqk

si ði ¼ 1;2; . . . ;NnÞ ð49Þ

and Pkðx; y; fk1Þ is the pressure and can be expressed as follows:

Pk ¼ F1t Pk

t þ F1r Pk

r þ F1bPk

b ¼ F1sPk

s ; s ¼ t; b; r;

r ¼ 2;3; . . . ;N; k ¼ 1;2; . . . ;Nl ð50Þ

where

Pks ¼ Nipk

si ði ¼ 1;2; . . . ;NnÞ ð51Þ

pksi can be expressed as:

pksi ¼ pk

xsi pkysi pk

zsi

h iTð52Þ

Therefore:

Pk ¼ F1sNipk

si ð53Þ

After substitution of Eqs. (49) and (53) in Eq. (48), one gets thefollowing:

dLkext ¼

ZXk

dqkTsi ðnÞ F1

sF1s

� �ðNiNjÞpk

sjdX ¼ dqkTsi ðnÞP

ksi ð54Þ

where:

Pksi ¼ F1

sF1s

� � /½NiNj�.X pkxsj

/½NiNj�.X pkysj

/½NiNj�.X pkzsj

26643775 ð55Þ

After substitution of Eqs. (34), (41), (45) and (54) in Eq. (21), onegets the governing equations of motion as follows:

dqkTsi ðnÞ : Kkssij

uu ðnÞqksjðnÞ þ Kkssij

ur ðnÞgksjðnÞ þMksij €qk

sjðnÞ

¼ Pksi � Pk

sma siðnÞ ð56aÞdgkT

si ðnÞ : Kkssijru ðnÞqk

sjðnÞ þ Kkssijrr ðnÞgk

sjðnÞ ¼ 0 ð56bÞ

According to the above equations, it can be observed that stiffness/compliance matrices and also some force vector are variable withthe martensite volume fraction and so, these stiffness/compliancematrices and force vector are instantaneous and dependent on theposition of every point on the plate. At the same time, the martens-ite volume fraction is dependent on the stress and consequently thedisplacement values, therefore, these stiffness/compliance matricesand force vector are unknown. In other words, the governing equa-tions of motion and the kinetic relations of the phase transforma-tion are coupled with each other which makes the governingequations physically nonlinear. Figs. 3–5 are presented the assem-bling scheme for the fundamental nucleus. It must be mentionedthat the transverse stress unknowns are eliminated using the staticcondensation method [33]. Therefore, Eqs. (56) can be expressed asfollows:

KðnÞqðnÞ þM€qðnÞ ¼ PðnÞ ð57Þ

where

KðnÞ ¼ ½KuuðnÞ � KurðnÞðKrrðnÞÞ�1KruðnÞ�;PðnÞ ¼ P� PsmaðnÞ ð58Þ

4.1. Time discretization

The stiffness matrix and the force vector in Eq. (57) are physi-cally nonlinear, and therefore, an incremental solution techniquebeside an iterative method is utilized for solving the coupledequations.

The Newmark scheme is employed for time discretization of Eq.(57) by the following form [41]:

qmþ1 ¼ qm þ Dt _qm þ 1=2Dt2 €qmþc ð59aÞ_qmþ1 ¼ _qm þ Dt€qmþa ð59bÞ

where q and _q are the displacement and velocity vectors, respec-tively. Also, one has:

Fig. 4. Assembling scheme related to Kuu and M in LW models, Krr in both LW andESL models and Kru in LW models.

Fig. 5. Assembling scheme related to Kru in ESL models.

S.M.R. Khalili et al. / Composite Structures 106 (2013) 635–645 641

€qmþa ¼ ð1� aÞ€qm þ a€qmþ1 ð59cÞ

where a = 1/2, c = 8/5, [41]. Using the Eqs. (59a)–(59c), the Eq. (57)could be compacted by the following form:bKmþ1qmþ1 ¼ bPm;mþ1 ð60Þ

wherebKmþ1 ¼ Kmþ1 þ a3M ð61aÞbPm;mþ1 ¼ Pm;mþ1 þMða3qm þ a4 _qm þ a5 _qmÞ ð61bÞ

and the coefficients a3, a4 and a5 are obtained as follows [41]:

a3 ¼ 2=ðcðDtÞ2Þ; a4 ¼ 2=ðcðDtÞÞ; a5 ¼ 1=c� 1 ð62Þ

An estimation of the initial acceleration can be obtained by:

€q0 ¼M�1ðP0 � Kq0Þ ð63Þ

where the subscript 0 denotes the initial condition of the corre-sponding vector. After every step, the velocity and acceleration vec-tors are calculated as follows:

€qmþ1 ¼ a3ðqmþ1 � qmÞ � a4 _qm � a5 €qm ð64aÞ_qmþ1 ¼ _qm þ a2 €qm þ a1 €qmþ1 ð64bÞ

with a1 = aDt and a2 = (1 � a)Dt.

4.2. The proposed iterative incremental approach

In order to solve the nonlinear coupled equations, the followingsteps are proposed:

1- At the first step, the initial values of q0; _q0 and fngk0 are

assumed (they are usually assumed as zero).2- Determine the initial acceleration €q0 by Eq. (63).3- Define the new time tm+1 = tm + Dtm+1 and putfnpgk

mþ1 ¼ fngkm.

4- Update the material properties and vkSiði ¼ x; yÞ using the

fnpgkmþ1.

5- Calculate the qm+1 by Eq. (60) based on the fnpgkmþ1.

6- Compute frgkmþ1 using qm+1and fnpgk

mþ1.7- Calculate fngk

mþ1 using the kinetic relations of the phasetransformation based on the frgk

mþ1. (It is explained inFig. 6).

8- This iterative scheme continues until a specified conver-gence criterion is obtained. For this aim the following crite-rion can be used:

maxnk

ij;mþ1 � nkpij;mþ1

��� ���nk

ij;mþ1

��� ���0B@

1CA < d ð65Þ

where the index p indicates the predictor, k indicates the layer k andnij, m+1(j = x, y) implies the martensite volume fraction of the ithgauss point in the x and y directions, respectively, for the element.Also, d is a small numeral. When the convergence criterion is ful-filled, _qmþ1 and €qmþ1 are determined by Eq. (64). Then, the next timeincrement is started and the foregoing steps are repeated from step3 to step 8. If the convergence criterion is not fulfilled, a newapproximation of fnpgk

mþ1 is computed using the relaxation methodas:

9- fnpgkmþ1 ¼ fng

kmþ1 þ 1 fngk

mþ1 � fnpgk

mþ1

� �and steps 4 to 8 are

repeated again, till the convergence criterion is fulfilled.

In order to identify the condition of the phase transformation onevery gauss point of the plate, the algorithm scheme in Fig. 6 is uti-lized at any time increment.

5. Numerical results and discussion

A new code is written in MATLAB software for derivation the re-sults based on the above formulations. This section consists of twoexamples. At the first example for verifications the present finitemodel, a particular problem is studied and compared with the pub-lished results. In the second example a nonlinear dynamic analysisof SMA multilayered plate is implemented. Some parametric stud-ies such as length-to-thickness ratio, plate aspect ratio and also theeffect of different boundary conditions, upon the loss factors areinvestigated.

Example 1. A static analysis of a simply supported composite platewith [0�/90�]s lay-up without SMA wires, is studied in order tovalidate the present two dimensional finite element model. In thispaper it is assumed that all of the layers of the plate have equalthickness. The material properties of the lamina are as follows:

E1

E2¼ 25;

G12

E2¼ G13

E2¼ 0:5;

G23

E2¼ 0:2; m12 ¼ m13 ¼ m23 ¼ 0:25

start

0,1, >−+ mimi εε

Detect loading or unloading

loadingunloading

No

Check the phase transformationCheck the phase transformation

Msmi σσ <+1,

mimi ,1, ξξ =+

Yes

No

Mfmi σσ >+1,

No

Asmi σσ >+1,

mimi ,1, ξξ =+

Yes

No

use the M A Phase transformation’s kinetic

equation

Afmi σσ <+1,

01, =+miξ

Yes

No

use the A M Phase transformation’s kinetic

equation

11, =+miξ

Yes

Yes

Fig. 6. Algorithm scheme for dynamic phase transformation.

Table 1Normalized deflections at the center of the composite plate with a/b = 1.

a/h Uða=2; b=2;0Þ

4 3D [42] 1.937LM4 1.931

10 3D [42] 0.737LM4 0.736

20 3D [42] 0.513LM4 0.513

50 3D [42] 0.446LM4 0.445

100 3D [42] 0.435LM4 0.435

642 S.M.R. Khalili et al. / Composite Structures 106 (2013) 635–645

The square composite plate is subjected to bi-sinusoidal loadover the top surface of the plate in z direction as follow:

Pz ¼ pz sinpxa

� �sin

pyb

� �In this example the deflection is normalized as follows:

Uðx; y; zÞ ¼ uz100h3E2

pza4

Normalized deflections at the center of the plate are presented inTable 1 for different a/h ratios. As can be seen, the results of thepresent model are in very good agreement with the 3D elasticitysolution obtained by Pagano [42]. As such, the maximum discrep-ancy is less than 0.3%. It should be mentioned that, the validationof the dynamical phase transformation algorithm scheme is doneby the authors in the Ref [43].

Example 2. This example deals with the dynamic analysis of thecomposite plate embedded with SMA wires. The composite platehas a [0�/90�/90�/0�]s lay-up. According to the Fig. 2 the layers arenumbered from the bottom layer to the top layer. The materialproperties of the layers are as follows [44]:

E1 ¼ 50 GPa; E2 ¼ E3 ¼ 10 GPa; G12 ¼ G13 ¼ G23 ¼ 5 GPa;m12 ¼ m13 ¼ m23 ¼ 0:25; q ¼ 1600 kg=m3

The geometrical dimensions of the plate are as follows:

a ¼ b ¼ 1 m; h ¼ 0:05 m

The SMA wires are embedded in the layers 1 and 8 in the x directionand the layers 2 and 7 in the y direction. The volume fraction of SMAwires is chosen 40% for each lamina. The material properties of theSMA wires are shown in Table 2. The boundary conditions are sim-ply supported and a step impulsive pressure is applied on the topsurface of the plate in the z direction. The amplitude of uniformpressure is P = 3 MPa. Fig. 7 shows the stress–strain history at themidpoint (a/2, b/2) of the top lamina considering the pseudoelasticeffect of the SMA wires. As can be seen, the stress–strain diagramexhibits the hysteresis loops. Fig. 8 shows the variation of the mar-tensite volume fraction (MVF) of the SMA wires with time for thementioned point. In Fig. 9, the time response of the deflection atthe center point of the plate uz(a/2, b/2, 0) is presented. From thisfigure, it can be observed that, the amplitude of vibration decreasesregularly, such that at the time t = 0.06 s, the amplitude is reducedto 68% of its value at the first peak (t = 0.0034 s). This phenomenonis because of the hysteresis loops which dissipate the energy grad-ually. In addition, from Fig. 7, it can be seen that as the plate vi-brates, the region of dissipated energy diminishes gradually andtherefore the rate of reduction in amplitudes decreases. The vibra-tion goes until the stress–strain curve arrives the linear-elastic stateand after this situation, the plate vibrates with constant amplitude

Table 2Material properties of Nitinol alloy [34].

Ea = 67 GPa T = 50 �C CM = 8 MPa/�CEM = 26.3 GPa Mf = 9 �C CA = 13.8 MPa/�Crcr

s ¼ 100 MPa Ms = 18.4 �C q ¼ 6500 kg=m3

rcrf ¼ 170 MPa As = 34.5 �C

eL = 0.067 Af = 49 �C

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.0180

0.5

1

1.5

2

2.5

3

3.5

4x 10

8

strain

stre

ss (

pa)

Fig. 7. Stress–strain curve for the center of the top layer.

0 0.01 0.02 0.03 0.04 0.05 0.060

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

time (sec.)

MV

F

Fig. 8. Variation of the MVF with time at the center of the top layer.

0 0.01 0.02 0.03 0.04 0.05 0.060

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

time (sec.)

defl

ecti

on (

m)

Fig. 9. Response history of deflection at the center of the plate with a/h = 20 anda/b = 1 and SSSS boundary conditions.

Table 3Deflection at the first pick of the response for different a/h ratios based on thedifferent models.

a/h IP (MPa) EM1 EM2 EM3 LM1 LM2 LM3 Suitable model

10 20 0.0716 0.0717 0.0734 0.0735 0.0737 0.0737 EM320 3 0.0788 0.0788 0.0790 0.0791 0.0791 0.0791 EM130 1 0.0855 0.0855 0.0856 0.0856 0.0856 0.0856 EM140 0.45 0.0904 0.0904 0.0904 0.0904 0.0904 0.0904 EM1

0 0.01 0.02 0.03 0.04 0.05 0.060

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

time (sec.)

defl

ecti

on (

m)

EM1 EM2 EM3 LM1 LM2 LM3

Fig. 10. Response history of deflection at the center of the plate for different modelswith a/h = 20 and a/b = 1.

5 10 15 20 25 30 35 40 450.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

a/h

Los

s Fa

ctor

%

Fig. 11. Variation of the loss factor with a/h ratio for a/b = 1.

S.M.R. Khalili et al. / Composite Structures 106 (2013) 635–645 643

and more reduction in vibration amplitude cannot be seen (it can beseen in Fig. 12). In other words, due to the dissipation of energy bythe SMA wires, the amplitude of the vibration goes to decrease, un-til it is arriving to the linear-elastic states of the wires and after thissituation, it vibrates with constant amplitude.

In this part the effect of a/h on the dynamic response of thesquare simply supported SMA hybrid composite plate is investi-gated. At the first step, for selecting a suitable model, a maximumdeflection at the first peak at the center of the plate, uz(a/2, b/2, 0),for different a/h ratios is presented in the Table 3 based on the dif-ferent models. Since, the problem in this study is nonlinear; there-fore, the run time of the program is very high. Hence, for saving thecomputational cost, the selected models in Table 3 are used for thenext analysis. For example, the dynamic response of the plate with

0 0.01 0.02 0.03 0.04 0.05 0.060

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

time (sec.)

defl

ecti

on (

m)

Fig. 12. Response history of deflection at the center of the plate with a/h = 20,a/b = 1 and CCCC boundary conditions.

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

time (sec.)

defl

ecti

on (

m)

Fig. 13. Response history of deflection at the center of the plate with a/h = 20,a/b = 1 and CSCS boundary conditions.

0.25 0.5 0.75 1 1.25 1.5 1.75

0.5

1

1.5

2

2.5

3

3.5

a/b

Los

s Fa

ctor

%

Fig. 14. Variation of the loss factor with a/b ratio for a/h = 20.

Table 4Intensity of pressure (IP) and suitable model for different a/b ratios with a/h = 20 andSSSS boundary conditions.

a/b 0.5 0.75 1 1.25 1.5

IP (MPa) 1.3 1.8 3 4.4 7Suitable model EM1 EM1 EM1 EM1 EM3

644 S.M.R. Khalili et al. / Composite Structures 106 (2013) 635–645

a/h = 20 is shown in Fig. 10 for different models. As can be seen, agood agreement is observed between the different models, so itcan be concluded that the accuracy of the model EM1 with lesscomputational cost is suitable for analyzing the plate witha/h = 20. It should be mentioned that the intensity of the pressure(IP) for different a/h ratios is such that the maximum deflection atthe first peak for different a/h ratios is in the same range. Loss fac-tor of the SMA multilayered plate is computed using the measure-ment of the vibration amplitude within the vibration and thefollowing relation:

f ¼ 12p

1n

lnx1 � xmean

xn�1 � xmean

� �ð66Þ

Fig. 11 shows the variation of the loss factor with a/h ratio. It can beseen that as the a/h ratio increases, the loss factor decreases. Inother words, as the thickness increases, the capacity of SMA wiresfor dissipating the energy increases. The reason for this phenome-non is that, as the thickness increases, the value of the stress inthe SMA wires increases, and therefore reaches its critical magni-tude for transformation.

It is known that the boundary conditions can have an importanteffect on the analysis of the composite multilayered plate. For thisaim, the boundary conditions SSSS, CCCC and CSCS are investi-gated, where S and C stand for the simply supported and clamped

boundary conditions, respectively. The results from the CCCC andCSCS boundary conditions are shown in Figs. 12 and 13, respec-tively. The intensity of pressure is 9, 6.5 and 3 MPa for CCCC, CSCSand SSSS, respectively. It can be seen that, the capability of CCCCboundary conditions in damping is higher than the other boundaryconditions, such that the loss factor is 2.67%, 2.48% and 1.58% forCCCC, CSCS and SSSS boundary conditions, respectively. Therefore,as the stiffness of the edges increases, the capacity of SMA wires fordissipating the energy increases.

Plate aspect ratio (a/b) has an important effect on the analysis ofthe SMA hybrid composite plate. In this regard, the composite platehas a [0�/90�/90�/0�]s lay-up and the SMA wires are embedded inthe layers 1 and 8 in the x direction, which the volume fractionof SMA wires is chosen 40% for each lamina. Fig. 14 shows the var-iation of the loss factor with different a/b ratios for a/h = 20 andSSSS boundary conditions. The IP and the suitable model are pre-sented in Table 4. It can be observed from Fig. 14 that, as a/b ratioincreases, the loss factor decreases. In other words, when thelength of the plate is shorter than the width of the plate, theSMA wires show more capacity for dissipating the energy.

6. Conclusion

In this research, a nonlinear dynamic analysis of the multilayercomposite plate embedded with SMA wires is implemented in theframe work of Carrera’s Unified Formulation (CUF), considering theinstantaneous phase transformation effects, for every point on theplate. The CUF has the capability to unify many theories in aunified form which can be differed by the order of expansion anddefinition of the variables in the thickness direction. This can be(ESL), if the unknown variables are considered for the whole plate,or (LW), if the unknown variables are considered for each layer,individually. The Brinson’s SMA constitutive equation is used tomodel the pseudoelastic effect of the SMA wires. The governingequations are derived from the Reissner Mixed Variational Theo-rem (RMVT) in order to enforce the interlaminar continuity oftransverse shear and normal stresses between two adjacent layers.

S.M.R. Khalili et al. / Composite Structures 106 (2013) 635–645 645

The governing equations of motion and the kinetic relations ofphase transformation are coupled with each other. Therefore, aniterative incremental finite-element-based scheme is proposedfor solving the coupled equations. The Newmark method is em-ployed to time discretization of the governing equations. The para-metric effects of length-to-thickness ratio, plate aspect ratio andalso the effect of different boundary conditions, upon the loss fac-tors are investigated. Results show that, as the length-to-thicknessratio and the plate aspect ratio increases, the loss factor decreases.Also, it can be concluded that, as the stiffness of edges increases,the capacity of SMA wires to dissipate the energy increases.

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