Accepted Manuscript

Thermo-Mechanical Behavior of Shape Adaptive Composite Plates with Sur-

face-Bonded Shape Memory Alloy Ribbons

M. Bodaghi, M. Shakeri, M.M. Aghdam

PII: S0263-8223(14)00417-6

DOI: http://dx.doi.org/10.1016/j.compstruct.2014.08.027

Reference: COST 5860

To appear in: Composite Structures

Please cite this article as: Bodaghi, M., Shakeri, M., Aghdam, M.M., Thermo-Mechanical Behavior of Shape

Adaptive Composite Plates with Surface-Bonded Shape Memory Alloy Ribbons, Composite Structures (2014), doi:

http://dx.doi.org/10.1016/j.compstruct.2014.08.027

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1

Thermo-Mechanical Behavior of Shape Adaptive Composite Plates

with Surface-Bonded Shape Memory Alloy Ribbons

M. Bodaghi, M. Shakeri†, M. M. Aghdam

Thermoelasticity Center of Excellence, Department of Mechanical Engineering,

Amirkabir University of Technology, Tehran, Iran

A B S T R A C T

In this paper, thermo-mechanical analysis of rectangular shape adaptive composite plates with surface-

bonded shape memory alloy (SMA) ribbons is introduced. A robust phenomenological constitutive model

is implemented to predict main features of SMA ribbons under dominant axial and transverse shear

stresses during non-proportional thermo-mechanical loadings. The model is capable of realistic

simulations of martensite transformation/orientation, reorientation of martensite variants, shape memory

effect, pseudo-elasticity and ferro-elasticity effects. A numerical process is addressed to solve the time-

discrete counterpart of the model using an elastic-predictor inelastic-corrector return mapping algorithm.

Considering small strains and moderately large rotations in the von Kármán sense, governing equations of

equilibrium are derived based on the first-order shear deformation theory. A Ritz-based finite element

method along with an iterative incremental strategy is developed to solve the governing equations of

equilibrium with both material and geometrical non-linearities. The capability of the material and

structural model is examined by a comparative study with numerical data available in the open literature

for laminated SMA beams. Effects of the pre-strain state, temperature, length and arrangement of the

SMA ribbon actuator are investigated, and their implications on the thermo-mechanical behavior of shape

adaptive composite plates are put into evidence, and pertinent conclusions are outlined.

Keywords: Shape memory alloy; Martensitic transformation; Smart composite plate; Thermo-mechanical

analysis; Finite element solution

† Corresponding Author. Tel.: +98-21-66405844; fax: +98-21-66419736. E-mail address: shakeri@aut.ac.ir (M. Shakeri)

2

1. Introduction

In recent years, a new class of smart materials known as shape memory alloys (SMAs) has

gained considerable attention in engineering community due to their unique attributes such as

pseudo-elasticity (PE) and shape memory effect (SME) [1]. The distinctive behavior of SMAs is

related to the martensitic phase transformation due to changes in the stress and/or temperature.

At high temperatures, SMA materials behave pseudo-elastically while producing hysteresis. On

the other hand, at low temperatures, SMAs exhibit shape memory effect and can recover their

original size and shape upon heating. SMAs can also attain their original shape when an

opposing force is applied to the material which is termed as ferro-elasticity (FE).

Over the last two decades, many macroscopic phenomenological models have been proposed

to simulated martensitic phase transformation in SMAs, see for instance [2-7]. One-dimensional

(1-D) models proposed by Tanaka [2], Liang and Rogers [3] and Brinson [4], and three-

dimensional (3-D) model developed by Boyd and Lagoudas [5] are the most notable SMA

models that researchers commonly exploited. However, it should be mentioned that the proposed

models [2-5] cannot reproduce the ferro-elasticity effect of SMAs when the stress sign is

changed in low temperatures. In the other words, these models are only able to simulate 1-D

SMA behavior under uniaxial tensile loading. Based on the experimental observations [6, 7],

SMA materials under multi-axial non-proportional thermo-mechanical loadings may experience

the so-called martensite variant reorientation according to loading direction. This phenomenon

was not also covered in the mentioned models [2-5]. In an attempt to model variant reorientation,

Panico and Brinson [6] proposed a 3-D phenomenological model with capability of the

simulation of martensitic transformation and reorientation of martensite in SMAs under multi-

axial non-proportional loading. Recently, Bodaghi et al. [7] developed a simple and robust model

3

to simulate main features of SMAs under two dominant normal and shear stresses including

martensitic transformation, orientation/reorientation of martensite variants, SME, PE and FE.

The shape memory effect can be employed to control the behavior of mechanical structures.

One way of achieving this goal is by attaching SMA actuators in the layer form to polymeric,

metallic or composite matrices while electrical current is normally employed to induce the

thermally driven transformation. In the hybrid composite with soft matrix and eccentrically

posed SMA components, shape change may be induced through SME strain recovery. On the

other hand, in the hybrid composite with stiff matrix and centered or eccentrically posed SMA

components, large stress may be produced through SME restrained recovery.

Recently, some research works have been dedicated to analyze active shape/stress/vibration

control of hybrid SMA laminates with embedded or surface-bonded SMA layers. For instance,

Ghomshei et al. [8] investigated thermo-mechanical deformation of a beam actuator consists of

two SMA layers bonded to the sides of a matrix layer, experimentally and numerically. They

developed a finite element solution for their analysis and used 1-D Brinson model [4] to predict

the thermo-mechanical response of SMA layers. Results revealed that the thermal actuation is

successfully achieved by applying electrical current to the SMA layers. Marfia et al. [9]

proposed a finite element model to study the static behavior of elastic beams with two integrated

SMA layers. They found that SMA layer actuators are able to produce large amounts of work

and recover 50% of the displacement as a result of the application of external forces by

performing temperature cycles on the SMA layers. Yang et al. [10] presented experimental

studies on the active shape control of composite structures with SMA wire actuators attached on

the surfaces of the structures using bolt-joint connectors. Using electric resistive heating, SMA

actuators were activated and quite large deformation of the SMA hybrid composite structures

4

was observed and discussed. Roh et al. [11] examined active shape control of plate and panel

structures with surface-bonded SMA layers. 3-D SMA model proposed by Boyd and Lagoudas

model [5] was employed to simulate the main characteristics of SMA layers. They utilized the

ABAQUS finite element program with an appropriate user-defined material (UMAT) subroutine

for modeling SMA layers and host elastic structure elements. Numerical results revealed that the

SMA actuator could generate enough recovery force to deform the host structure and sustain the

deformed shape subjected to large external load, simultaneously. Roh and Bae [12]

experimentally and numerically examined the thermo-mechanical behaviors of Ni-Ti SMA

ribbons, associated with stress and temperature-induced transformations. They modified 3-D

Boyd-Lagoudas model [5] into a plane stress condition and implemented the two-dimensional (2-

D) incremental formulation in the ABAQUS finite element program with the aid of a UMAT

subroutine. For application of the developed numerical 2-D SMA model, the feasibility of a

gripper actuator with surface-bonded SMA strips was numerically demonstrated. Results

revealed that, when the SMA strip is activated by raising its temperature, the strain recovered in

the activated SMA strip causes bending deformation due to the off-center placement of the SMA

strip. Transient response of a sandwich beam with SMA hybrid composite face sheets and

flexible core under dynamic loads was studied by Khalili et al. [13]. They used 1-D Brinson

model to predict pseudo-elastic behavior of SMA wires and developed a finite element model to

solve the non-linear governing equations. Recently, Bodaghi and his colleagues [14, 15]

investigated active shape/vibration control of thin homogeneous elastic beams under

static/dynamic loadings with integrated SMA layers using finite element method. They employed

the 3-D Panico-Brinson model [6] and reduced it to a 1-D tension-compression case to simulate

5

1-D behavior of SMA layers. They found that the SMA layers can be successfully used to

suppress static/dynamic deformation of smart beams under mechanical loads.

The literature survey reveals that most of researches have been devoted to study active

shape/vibration control of beam-like structures with integrated/embedded SMA layers. Also, it

can be found that researches employed generally 1-D or 3-D models [2-5] to simulate thermo-

mechanical behavior of SMA materials. As previously stated, these models are incapable of

simulating the ferro-elasticity effect revealed when the pre-strained SMA materials undergo

compressive stress at low temperatures. Moreover, these models do not take into account

martensite variant reorientation as an important phenomenon under non-proportional thermo-

mechanical loading conditions.

The aim of this work is to investigate thermo-mechanical behavior of shape adaptive

composite plates with surface-bonded SMA ribbon actuators using a robust SMA model with

capability of simulating ferro-elasticity and martensite variant reorientation. SMA ribbons pre-

strained in a combined axial-shear state are installed onto the top surface of the composite plate

to design shape adaptive composite structures. The robust phenomenological constitutive model

proposed by the authors [7] is implemented to characterize main aspects of the SMA ribbons

under dominant axial and transverse shear stresses during non-proportional thermo-mechanical

loadings. The SMA model is able to simulate martensite transformation/orientation, pseudo-

elasticity, shape memory effect and in particular reorientation of martensite and ferro-elasticity

features. The first-order shear deformation theory (FSDT) and von Kármán geometrical non-

linearity are assumed to describe displacement and strain fields of shape adaptive composite

plates. Based on the principle of minimum total potential energy, an SMA composite plate

element is first developed which is subsequently extended to the finite element equations of

6

equilibrium. An iterative incremental strategy is introduced to solve the coupled governing

equations of equilibrium with both material and geometrical non-linearities. Capabilities of the

material and structural formulations are first examined through comparative study with

numerical results available in the open literature for laminated SMA beams. A detailed analysis

of the influence of pre-strain state, temperature, length and arrangement of surface-bonded SMA

ribbons on the thermo-mechanical behavior of shape adaptive composite plates with clamped-

free and clamped-clamped boundary conditions is carried out. Due to the absence of similar

results in the specialized literature, this paper is likely to fill a gap in the state of the art of this

problem.

2. Materials and Methods

Consider a composite plate with length a, width b, and thickness ch , as depicted in Fig. 1a. In

order to control the structure deformation, shape memory alloys in ribbon form are perfectly

bonded to the top surface of the host plate while current heating is used to activate them. SMA

ribbons have rectangular cross section shd × with arbitrary length. For the sake of identification,

some quantities associated with the composite plate will be marked by a subscript “c”, while

those affiliated with the SMA ribbons by a subscript “s”, placed on the right of the respective

quantity. The middle plane of the substrate plate is considered as a reference plane. The origin of

the Cartesian coordinate system ),,( zyx is located at the upper-left corner of the host plate on

the reference plane.

During non-proportional thermo-mechanical loadings, the SMA ribbons experience two

dominant axial stress and transverse shear stress. The thermo-mechanical behavior of SMA

7

materials is first studied and then governing equations of equilibrium and solution strategy are

presented.

2.1. Time-continuous SMA constitutive model

In order to simulate main thermo-mechanical features of the SMA materials under combined

axial-shear non-proportional loadings, the simple and robust macroscopic phenomenological

model developed by the authors [7] is implemented and briefly presented here. The model in

particular takes into account effects of reorientation of martensite variants and ferro-elasticity

together with other factors such as martensite transformation/orientation, shape memory effect

and pseudo-elasticity.

The constitutive model is developed within the framework of continuum thermodynamics of

irreversible processes in the realm of a small-strain regime. The model considers the volume

fraction of self-accommodated martensite Tξ and the volume fraction of oriented martensite Sξ

as scalar internal variables and the preferred direction of oriented martensite variants θ as a

directional internal variable. Self-accommodated martensite is temperature-induced with no

associated shape change, while oriented martensite has a crystal structure that has been

preferentially oriented by the applied load and accompanied by an observable inelastic strain.

The internal variables ST ξξ , and θ play an important role in characterizing the martensitic

orientation/transformation and reorientation of martensite variants.

The total martensite volume fraction, ξ , is obtained as the sum of stress-induced and

temperature-induced parts:

10,10,10, ≤≤≤≤≤≤+= ξξξξξξ TSTS (1)

8

Assuming a small strain regime, justified by the fact that the approximation of large

displacements and small strains is valid for the present application, the total strain components

are additively decomposed into an elastic part and an inelastic part as:

ine

ine

γγγεεε

+=+=

(2)

where ε and γ denote axial strain and transverse shear strain, respectively. eε and eγ represent

the elastic strain components while inε and inγ are associated with the inelastic strains produced

during stress-induced transformation/orientation from austenite/self-accommodated martensite to

oriented martensite or reorientation of martensite variants.

The inelastic axial and transverse shear strains can be written as [7]:

)sin(3/

)cos(

θξεγ

θξεε

Suin

Suin

=

=(3)

where uε is the maximum uniaxial attainable transformation/orientation strain. As can be found

from Eq. (3), θ signifies the angle between the inelastic strain vector { }T

inin 3/γε and the

horizontal axis inε in the inelastic strain vector space 3/inin γε − . The angle θ can be derived

from Eq. (3) as:

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

===<=>><<<<≥>≥

−

+++

==

0,0

0,0

0,0

0,0

0,0

0,0

0,0

undefined

2/

2/

2)3/(atan

)3/(atan

)3/(atan

)3/(atan

),3/(2atan

inin

inin

inin

inin

inin

inin

inin

inin

inin

inin

inin

inin

εγεγεγεγεγεγεγ

ππ

πεγπεγπεγ

εγ

εγθ

(4)

9

where )atan(• and ),(2atan ∗• represent one- and two-argument arctangent functions whose

ranges are ]2/,2/[ ππ − and )2,0[ π , respectively. Unlike atan function that takes a single

argument, atan2 function considers two arguments and returns an angle in one of the four

quadrants of the two-dimensional coordinate system 3/inin γε − , depending upon the signs of

both inelastic strain components.

Taking time derivative of Eq. (3), one may obtain:

( )θθξθξεε ��

� )sin()cos( SSuin −= (5a)

( )θθξθξεγ ��

� )cos()sin(3/ SSuin += (5b)

Eq. (5) reveals the fact that the inelastic strain evolution is due to two contributions; i.e. from

pure transformation/orientation and from pure reorientation. The first term originates from

transformation/orientation of austenite/self-accommodated martensite )0( ≠Sξ� by applying

uniaxial or axial-shear proportional loadings )( cte=θ . On the other hand, the second term

represents the reorientation of previously developed oriented martensite )0( ≠Sξ when the load

direction changes through axial-shear non-proportional loadings )0( ≠θ� .

Introducing the Helmholtz free energy function and following standard thermo-dynamical

considerations [7, 16], the final form of the constitutive model can be derived as:

Stress components:

)( insess EE εεεσ −==

)( insess GG γγγτ −== ; )1/(21

sss EG ν+= (6)

Thermo-dynamical dissipative forces:

10

)cos(3)sin(

)(

~)sin(3)cos(

0

0

θτθσ

ξσθτθσ

θ ss

sT

SsssS

X

MTCX

MTCX

+−=

−−=

−−−+=

(7)

where ss GE , and sν are the isotropic extensional and shear moduli, and Poisson ratio assumed

to be the same for both martensite and austenite phases. sσ and sτ denote axial and transverse

shear stresses while SX , TX and θX are thermo-dynamic stress-like quantities associated

respectively to the internal variables TS ξξ , and θ . Also, σ~ denotes a relative kinetic stress while

C is the conventional slope of forward/reverse phase transformations lines. Furthermore, •

indicates the Macaulay bracket which calculates the positive part of the argument, i.e.

2/)( •+•=• .

In order to control the evolution of the internal variables, the following limit functions are

introduced:

)()sgn( /S

rfSSSS YXXF ξ−= (8a)

),,()sgn( /ssT

rfTTTT YXXF τσξ−= (8b)

θθθθ YXXF −= )sgn( (8c)

where )sgn(• denotes the signum function. Also, θY represents an internal energy that must be

overcome for the SMA microstructure to reorient, and is considered to be constant in its

phenomenological average. Furthermore, rfSY / and rf

TY / control the kinetics of Sξ and Tξ during

forward/reverse phase transformations which in the present SMA model take the following

forms:

11

7,0)ln()1()1ln(

0)ln()1()1ln()(

3210

3210/ =⎩⎨⎧

<−−+−++>+−+−−−

= −

−

nforYeYYY

foreYYYYY

SSSr

Sn

SSr

SSr

Sr

S

Sn

SSf

SSSf

SSf

Sf

SS

rfS ξξξξξξ

ξξξξξξξ�

�

(9a)

⎪⎩

⎪⎨⎧

<>

−−+++−

−=

0

0

))(1(3)(

)()(

002200

00/

T

T

sfTssss

fsT

Trf

Tfor

for

AACMAC

MMCY

ξξ

ξτσξ

ξ�

�

(9b)

In Eq. (9), rfS

rfS

rfS YYY /

2/

1/

0 ,, and rfSY /

3 are constant coefficients that describe the kinetics of stress-

induced forward/reverse phase transformation. Furthermore, 0sM , 0

fM , 0sA and 0

fA denote four

stress-free characteristic temperatures which are representative of martensite start and finish, and

austenite start and finish, respectively.

The model is finally completed by the classical Kuhn-Tucker conditions

0,0,0

0,0,0

0,0,0

=≠=≤

=≠=≤

=≠=≤

θξθξξξξξ

θθ��

��

��

SS

TTTT

SSSS

ForF

ForF

ForF

(10)

which reduce the problem to a constrained optimization problem.

A simple procedure for identifying the parameters of the SMA model is described now. The

parameter uε can be found from a uniaxial pseudo-elastic experimental test. The kinetic

parameters σ~,,,, /3

/2

/1

/0

rfS

rfS

rfS

rfS YYYY and the material parameter C are calibrated to fit uniaxial

stress-strain experimental curves for simple tension or pure torsion tests. Moreover, the

calibration of reorientation activation threshold, θY , can be achieved using combined axial-shear

loading tests.

2.1.1. Time-discrete frame

The purpose behind developing SMA constitutive models is to solve boundary value

problems and predict thermo-mechanical behavior of structural components composed of SMA

12

materials. To this goal, the numerical implementation of the SMA constitutive model is

presented in this section, while more details can be found elsewhere [7, 17]. From a

computational standpoint, the non-linear material behavior is often treated as a time-discrete

strain-driven problem. Accordingly, the time interval of interest ],0[ t is subdivided into sub-

increments and the evolution problem is solved over the generic interval ],[ 1+nn tt with nn tt >+1 .

In the following, the quantities with the superscript n are related to the preceding time step nt ,

whereas the ones with no superscript are referred to the current time step 1+nt . Knowing the total

axial and shear strains ),( γε and temperature )(T at time 1+nt and also all field and internal

variables at time nt (i.e., nnT

nS

nn θξξγε ,,,, ), the new values of TS ξξ , and θ can be calculated

from the following time-discrete system:

θθθθ

θ

ξθθξεγθθξεε

ξσθθξεγθθξεε

YXXF

YXXF

GEX

MTCGEX

Srf

SSSS

SusSus

SsSusSusS

−=−=

−+−−=

−−−−+−=

)sgn(

)()sgn(

)cos())sin(3/(3)sin())cos((

~)sin())sin(3/(3)cos())cos((

/

0

(11a)

)(1

),,,,()sgn(

)(/

0

nSS

nTTTS

STrf

TTTT

sT

if

YXXF

MTCX

ξξξξξξθξξγε

−−=⇒>+

−=

−−=

(11b)

along with the requirements:

0)(,0)(,0

0)(,0)(,0

0)(,0)(,0

=−≠=−≤

=−≠=−≤

=−≠=−≤

nTTT

nTTT

nnS

nnS

nSSS

nSSS

ForF

ForF

ForF

ξξξξ

θθξθθξξξξξ

θθ (12)

As can be found from Eq. (11), the evolution of internal variables θξ ,S from the system of

equations (11a) is uncoupled from Eq. (11b) associated to the evolution of Tξ . After

13

determination of Sξ and θ by solving the system of equations (11a), the new value of Tξ can be

computed through Eq. (11b). The solution of time-discrete constitutive model is performed by

means of an elastic-predictor inelastic-corrector return map procedure as in classical plasticity

problems [17]. First, a thermo-elastic prediction assumes that the internal variables remain

constant. Then, if the trial state is admissible, that is ),,(0 TSiFi θ=≤ , the step is elastic and it

represents the final state. However, once ),,(0 TSiFi θ=> the Kuhn-Tucker conditions are

violated which implies the trial state is inelastic and therefore, internal variables have to be

evaluated solving evolution equations (11). This can be accomplished by means of an iterative

strategy such as Newton-Raphson scheme. It should be mentioned that for the case of 0=nSξ , the

trial value of ),3/(atan2 nin

nin εγθ = becomes indefinite (see Eq. (4)) since the oriented

martensite and inelastic strains are absent. In this case, the trial value of θ is considered as

),3(atan2 εγ ss EG which allows the constitutive model to simulate martensite nucleation

process.

2.2. Governing equations

In the present work, SMA ribbons are employed to design shape adaptive composite

structures. SMA ribbons are first loaded in a combined axial-shear state and then unloaded to

achieve recoverable inelastic strains )cos( 000 θξεε Suin = and )sin(3 000 θξεγ Suin = . After

preparation process, the pre-strained SMA ribbons with a specific arrangement are installed onto

the top surface of the elastic composite plate. The SMA ribbons are activated by electrical

heating while they are thermally insulated from the rest of the host plate.

14

A two-layered plate element with length a and width b is considered consisting of a host

composite plate with a surface-bonded SMA layer, as illustrated in Fig. 1b. Since the main

purpose of the present study is to analyze global response of thin to moderately thick SMA

composite plates, equivalent single layer theory of the first-order shear deformation is used to

describe the kinematics of deformation of the smart plate. The FSDT provides a sufficiently

accurate description of global response for thin to moderately thick laminated structures with

complex constitutive behavior [18]. Based on the FSDT, axial displacements u and v along the

x and y -directions, respectively, and transverse displacement, w along the z -direction at any

material point within the SMA composite plate domain are given by:

),(),,(

),(),(),,(

),(),(),,(

yxwzyxw

yxzyxvzyxv

yxzyxuzyxu

y

x

=

+=+=

ψψ

(13)

where vu, and w are the generalized displacements of a reference point ),( yx on the reference

plane along yx, and z axes, respectively. Also, xψ and yψ denote the rotations of a transverse

normal to the reference plane about y and x axes, respectively.

Adopting the von Kármán concept of small strains and moderately large rotations, the non-

linear strain-displacement relations can be expressed as:

⎭⎬⎫

⎩⎨⎧

+⎭⎬⎫

⎩⎨⎧

+⎭⎬⎫

⎩⎨⎧

=0

ε

ε

ε

0

ε

εni

s

bm z (14)

where the total strain ε , membrane strain mε , bending strain bε , transverse shear strain sε and

non-linear in-plane strain niε in Eq. (14) are:

15

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=

⎭⎬⎫

⎩⎨⎧

++

=⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

+=

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

+=

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=

yx

y

x

nixx

yys

xyyx

yy

xx

b

xy

y

x

m

xz

yz

xy

y

x

ww

w

w

w

w

vu

v

u

,,

2,

2,

,

,

,,

,

,

,,

,

,

2/)(

2/)(

,,,, εεεεε

ψψ

ψψψψ

γγγεε

(15)

where the subscript )(, denotes the partial differentiation with respect to the spatial coordinate.

For further convenience, the total strain is rewritten in terms of the linear strain Lε and the

non-linear strain Nε as:

NL εεε += (16)

where

uDε LL = (17a)

( ))()( 432121 uDuDuDuDε NNNNN += (17b)

and

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

∂∂∂∂

∂∂∂∂∂∂∂∂∂∂∂∂

∂∂∂∂

=

y

x

L w

v

u

x

y

xzyzxy

yzy

xzx

ψψ

uD ;

01/00

10/00

//0//

/00/0

0/00/

(18a)

16

{ }

{ }00/00;

00000

00000

00/00

00/00

00000

00/00;

00000

00000

00/00

00000

00/00

43

21

yx

y

xy

x

NN

NN

∂∂=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

∂∂∂∂

=

∂∂=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

∂∂

∂∂

=

DD

DD

(18b)

The variation of the generalized strain components can be obtained as:

uDε δδ LL = (19a)

)()( 4321 uDuDuDuDε NNNNN δδδ += (19b)

where δ represents the variational symbol.

The host composite plate consists of multiple unidirectional fiber-reinforced laminae. The

elastic constitutive equations for each lamina oriented as an arbitrary angle with respect to the

reference coordinate system in a plane stress state can be expressed as:

εQσ cc = (20)

where

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=

5545

4544

662616

262212

161211

000

000

00

00

00

;

QQ

QQ

QQQ

QQQ

QQQ

c

xz

yz

xy

y

x

c Qσ

τττσσ

(21)

where ijQ ’s are the transformed plane stress-reduced stiffnesses, and can be expressed in terms

of orientation angle and engineering constants details of which can be found in [18].

17

The principle of minimum total potential energy is used to derive the governing equations of

equilibrium and the boundary conditions of the SMA composite plate element. This may be

stated as:

0=−=Π WU δδδ (22)

where Uδ and Wδ are the variations of the strain energy and of the work done by the external

loads, respectively, in case of a virtual displacement xwvu δψδδδ ,,, and yδψ of the entire SMA

composite structure.

The variation of the strain energy of the composite plate element coupled with the SMA layer

can be expressed as:

∫∫ ∫∫∫ ∫ −−+=

A

h

h ssT

A

h

h ccT s

s

c

c

dAdzdAdzU2/

2/

2/

2/σεσε δδδ (23)

where 2/)( sccs hhzz +−= denotes local thickness coordinate with respect to the mid-plane of

the SMA layer; A is the area of the smart plate element under consideration. It should be

mentioned that a shear correction factor, scfK is introduced in Eq. (23) to compensate the error

due to constant transverse shear stress assumption through the structure thickness [18]. This

factor is assumed to be 5/6 for the present static problem [18].

The present SMA composite plate element will be used to construct composite plate with

surface-bonded SMA ribbons. Since the width of SMA ribbon actuators is small in comparison

with their length, they can be considered as beam-like structures. This represents a system that

includes active SMA beams encased on the composite plate. During non-proportional thermo-

mechanical loadings, the SMA ribbons experience two dominant axial stress and transverse shear

stress. In the present work, it is assumed that the surface-bonded SMA ribbons are set parallel to

plate axes. When they are parallel to the x -direction, axial and transverse shear stress and strain

18

components xzx τσ , and xzx γε , are prominent and displacement components v and yψ and

differentiation with respect to the y variable are neglected. On the other hand, SMA ribbons in

the y -direction experience dominant stresses and strains yzy τσ , and yzy γε , and insignificant

displacement components u and xψ while differentiation with respect to the x variable is

ignored.

The variation of the work done by the distributed transverse load ),( yxF acting over the

surface of the SMA composite plate and concentrated generalized forces including axial forces

yx PP , , transverse shear force V and bending moments yx MM , in moving through their

respective virtual displacements is given by:

∑∫∫=

+++++=n

iyi

Tyixi

Txii

Tiyi

Tixi

Ti

A

T MMVwPvPudAFwW1

)( δψψδδδδδ (24)

where xiiii wvu ψ,,, and yiψ are generalized displacements of the i th local point of the SMA

composite plate element.

2.3. Finite element model

In this section, a Ritz-based finite element formulation is developed to analyze thermo-

mechanical behavior of the shape adaptive composite structures with surface-bonded SMA

ribbon actuators. In the present study, linear Lagrange shape functions iϕ )4,3,2,1( =i are used

to interpolate yxvu ψψ ,,, and quadratic serendipity interpolation functions iβ )8...1( =i are

used for w details of which can be found elsewhere [19]. The eight-node rectangular plate

element has 24 degrees of freedom. The generalized displacements can be expressed in terms of

nodal variables through shape functions as:

19

UΦu = (25)

where Φ and U are the element shape function matrix and the vector of generalized nodal

displacements which are defined as:

[ ] [ ]

{ } { } { } { }{ } { }

4141

814141

8141

...ˆ,...ˆ

...ˆ,...ˆ,...ˆ,ˆˆˆˆˆ

...,...,

yyxx

T wwvvuu

ψψψψ

ββϕϕ

==

====

==

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

yx

yx

ψψ

wvuψψwvuU

βφ

φ0000

0φ000

00β00

000φ0

0000φ

Φ

(26)

Substituting the displacement vector (25) into Eqs. (16), (17) and (19), the strain field and its

variation can be expressed in terms of nodal displacement vector, U , as:

( ))()()()()( 432121

ΦUDUΦDΦUDUΦDUΦDε NNNNL ++= (27a)

( ))()()()()( 4321 ΦUDUΦDΦUDUΦDUΦDε NNNNL δδδδ ++= (27b)

In the following, it should be recognized that )( 2ΦUDN and )( 4ΦUDN are scalar. These scalar

forms may be inserted in arbitrary locations within other matrix products.

It is worthy to mention that Eq. (27) presented for the rectangular plate element is also valid

for the SMA ribbon installed in the )or( yx -direction if the shape functions associated with in-

plane nodal displacements )ˆor(ˆ uv and nodal rotation functions )ˆor(ˆ xy ψψ in Φ are set zero

while the variable )or( xy in other shape functions are substituted with )2/or(2/ ab . Using Eq.

(27), the virtual strain energy of the SMA composite plate element (23) can be rewritten in the

discretized form as:

αδδδ inTTU fUKUU −= (28)

where

20

{

}{ }∫∫ ∫

∫∫ ∫

−

−

++=

=++

++

++

++=

+=

A

h

h sNinsT

NNinsT

NinsT

Lin

iNNiT

NNNNiT

N

NNNiT

NNNiT

N

NLiT

NNLiT

N

NNiT

LNNiT

L

A

h

h LiT

Li

sc

s

s

i

i

dAdz

scidAdz2/

2/ 4321

24332

142132

1

4231212

21121

4321

4321

2121

2/

2/

)()()()()(

,,))(()())()(()(

))()(()())(()(

))(()())(()(

))(()())(()()()(

ΦUDεQΦDΦUDεQΦDεQΦDf

ΦUDΦDQΦDΦUDΦUDΦDQΦD

ΦUDΦUDΦDQΦDΦUDΦDQΦD

ΦUDΦDQΦDΦUDΦDQΦD

ΦUDΦDQΦDΦUDΦDQΦDΦDQΦDK

KKK

ααααααα

αα

αα

αα

αααα

α

(29)

in which cQ and αsQ are the stiffness matrices of composite and isotropic SMA materials and

αinε denotes the inelastic strain vector for the SMA ribbon which are all defined by

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

−−

=

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

sscf

sscf

s

s

s

scfscf

scfscf

c

GK

GK

E

E

QKQK

QKQK

QQQ

QQQ

QQQ

)2(0000

0)1(000

00000

000)1(0

0000)2(

000

000

00

00

00

5545

4544

662616

262212

161211

αα

αα

αQ

Q

(30a)

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

−−−−

−−−−

=

))sin()sin((3)2(

))sin()sin((3)1(

0

))cos()cos(()1(

))cos()cos(()2(

00

00

00

00

θξθξεαθξθξεα

θξθξεαθξθξεα

α

SSu

SSu

SSu

SSu

inε (30b)

In Eq. (30), the parameter α is attributed to installation direction of the SMA ribbon. While this

parameter is disregarded for the host composite plate, it is set 1 or 2 for the SMA ribbon installed

in the x or y -direction, respectively. Considering the points mentioned above on performing

some manipulation to replicate elemental formulations for the SMA ribbon installed in the x or y-

21

direction, the three-dimensional integrals of 11 , ins fK and 22 , ins fK reduce to two-dimensional

integrals over szx, and szy, domain, respectively.

The potential energy of external loads (24) can be also expressed in terms of mechanical

nodal variables as:

mTW fUδδ = (31)

where

{ }Tyyxxyyxx

T

A

m

MMMMVVPPPP

dAF

4141814141 ...............+⎭⎬⎫

⎩⎨⎧

= ∫∫ 00β00f (32)

Finally, by substituting Eqs. (28) and (31) into the principle of minimum total potential

energy (22), the finite element governing equations of the composite plate element with a

surface-bonded SMA ribbon installed in the x or y-direction are derived as:

mSin wwww ffUK += ),...,,,(),...,( 8181 θξα (33)

Eqs. (29), (30b) and (33) reveal that any change of the oriented martensite volume fraction, Sξ ,

and/or the preferred direction of oriented martensite variants, θ , may induce force in the SMA

composite plate element. It should be mentioned that, since Sξ and θ are variable through the

SMA beam element, these quantities are held into the integral of αinf . In this work, these

parameters are assumed to be constant along the SMA element length, while the Gauss-Legendre

numerical integration rule is utilized to evaluate the integral of αinf through the thickness

direction.

Since Eq. (33) is a set of non-linear algebraic equations, it would be useful to introduce

tangent matrix defined as

22

URT ∂∂= / (34)

in which R is the residual vector and can be written as:

min ffKUR −−= α (35)

By differentiation of the residual vector R with respect to the nodal displacement vector U , the

tangent matrix can obtained in an explicit form as:

ααinsc KKKKT ˆˆˆ −++= (36)

where

{

}{ }∫∫ ∫

∫∫ ∫

−

−

+=

=+

++

++

+

++

+=

A

h

h sNinsT

NNinsT

Nin

iNNNiT

N

NNNNNiT

N

NNNNNiT

N

NNNiT

N

NLiT

NNLiT

N

A

h

h NNiT

LNNiT

Li

s

s

i

i

dAdz

scidAdz2/

2/ 4321

4433

24421321

24423121

2211

4321

2/

2/ 4321

2121

)()()()(ˆ

,,))(()()(

)))(())((()()(

)))(())((()()(

))(()()(

)()()()()()(

)()()()()()(ˆ

ΦDεQΦDΦDεQΦDK

ΦUDΦDUΦDQΦD

ΦUDΦDΦUDΦDUΦDQΦD

ΦUDΦDΦUDΦDUΦDQΦD

ΦUDΦDUΦDQΦD

ΦDUΦDQΦDΦDUΦDQΦD

ΦDUΦDQΦDΦDUΦDQΦDK

ααααα

α

α

α

α

αα

ααα

(37)

Finally, Eq. (33) can be used to generate global finite element governing equations of the

composite plate with surface-bonded SMA ribbons installed in the x and y-directions by

assembling and applying boundary conditions which results in:

mijSjini ww ffU)(K += ),,( θξ (38)

In a similar way, the global tangent matrix T can be also constructed. Eq. (38) is a non-linear

system of algebraic equations in terms of external nodal variables ),,,,( iyixiii wvu ψψ and internal

variables ),( jSj θξ which are distributed through the thickness of each SMA element. The

quantities of Sξ and θ at each local material (integration) point in each element sub-domain

depend on the unknown strain (or displacement) and temperature fields in the SMA beam

23

element through constitutive equations (11) constrained by (12). Thus, it can be found that the

governing equations of equilibrium (38) and martensitic transformation/ orientation/reorientation

equations (11) are coupled via the external and internal variables which makes the problem to be

more complicated. In order to solve the highly coupled non-linear system of equations, a solution

algorithm is introduced in the next section.

2.4. Solution procedure

In this section, a solution algorithm is proposed to solve the coupled governing equations of

the present problem described in previous sections in particular Eqs. (11) and (38) to analyze the

thermo-mechanical behavior of shape adaptive composite plates with surface-bonded SMA

ribbons. In solving the present problem that present both material and geometrical non-

linearities, the time history of the defined load-temperature applied in the finite element analysis

is partitioned into load steps. During each step of this incremental solution process, the global

finite element solver attempts to satisfy the balance of equilibrium in the global sense by

determining appropriate increment of total strain (or displacement) locally for each integration

point of each element. This is accomplished by using a Newton-Raphson iteration scheme. Given

the local inputs of total strain and temperature, the SMA constitutive model is employed to

compute the local internal variables increments which are returned to the global solver. In the

following, subscript p indicates function evaluations at time pt (similarly for 1+p ), assuming

time increment pp ttt −=Δ +1 . Furthermore, the superscript q indicates the quantity at the qth

iteration. In this respect, the converged value of U at time 1+pt is denoted by 1+pU . Moreover,

the parameter R signifies the global residual vector defined as min ffUKR −−= . The

algorithmic solution is summarized as:

24

a) Compute 11

11

111 / −

+−+

−++ −= q

pqp

qp

qp TRUU . IF 1=q THEN set pp UU =+

01 .

b) IF the relative residual defined as qp

qp

qp 1

111 / +

−++ − UUU is smaller than a predefined tolerance,

310− , and 1≠q THEN set qpp 11 ++ = UU and 1+→ pp , and GOTO step (a), ELSE set 1+→ qq

and GOTO step (c).

c) Check martensite transformation/orientation/reorientation per Eq. (11). IF ),,(0 TSiFi θ=≤

THEN GOTO step (a) ELSE compute internal variables using 11

−+

qpU and THEN GOTO step (a).

3. Numerical results and discussion

In order to examine capabilities of the present material and structural formulations, numerical

simulations of beams with a surface-bonded SMA layer under thermo-mechanical loadings are

carried out and compared with those available in the open literature. Afterward, a series of

parametric studies are performed to provide an insight into the influence of pre-strain state,

temperature, length and arrangement of SMA ribbons on the thermo-mechanical behavior of

shape adaptive composite plates. In all simulations discussed in this section, the SMA material is

considered to be initially in self-accommodated martensite and austenite phases for 0sMT < and

0sMT ≥ , respectively.

3.1. Thermo-mechanical analysis of laminated SMA beams

In order to explore and demonstrate efficiency of the developed formulations and achieve a

possible comparative study, thermo-mechanical response of thin homogeneous elastic beams

with a surface-bonded SMA layer is simulated. This is due to lack of any results in the

specialized literature related to the SMA composite plates. A pre-strained SMA layer with initial

25

values of 2.00 =Sξ and 4/,6/,00 ππθ = is installed on the top surface of the host beam. Smart

beams with clamped-clamped and clamped-free boundary conditions at edges 0=x and a

under loading-heating and heating-loading-heating paths are analyzed, respectively. Results of

clamped-clamped case with 00 =θ which is regarded to the SMA layer initially pre-strained in a

uniaxial tensile mode are compared with those reported by Bodaghi et al. [14]. The length and

width of the two-layered beam are considered to be 200 and 11.11 mm while local thickness of

each layer is 2.5 mm. It is assumed that the host elastic beam is made of Aluminum with

elasticity modulus of 70 GPa and Poisson ratio of 0.3 [14]. Similar material properties and

characteristic temperatures as reported in [6, 14] are used for the top SMA layer given in Table

1. Note that the reorientation activation threshold θY as presented in Table 1 was not considered

by Bodaghi et al. [14] who only modeled 1-D behavior of the SMA layer under uniaxial stress.

This parameter was adapted from the research work of Panico and Brinson [6]. The SMA layer is

activated by electrical heating which uniformly raised the temperature. Since the effect of

elevated SMA temperature on adjacent host layer is negligible, it is assumed that the temperature

of the host beam remains in the reference room temperature as a generally well-known

assumption [1, 14]. In all computations, )118( × mesh with 5 through-the-thickness Gauss points

per SMA element is used to achieve the converged results accurately to three significant digits.

In Fig. 2, thermo-mechanical response of the two-layered SMA beam with clamped-clamped

boundary conditions under loading-heating path is depicted. Included in the figure are also

predictions reported in [14] for the case of 00 =θ . The thermo-mechanical loading is denoted by

arrow symbols in Fig. 2. The beginning and end of each stage are also marked by solid circle

symbols. The laminated SMA beam firstly experiences an upward uniformly distributed load

with maximum magnitude of MPaF 8.00 = . In the next stage, the SMA layer initially at

26

reference temperature K310 is heated up to temperature K350 . The history of central

deflection during thermo-mechanical loading is illustrated in Fig. 2a. Also, the histories of

oriented martensite volume fraction and preferred direction of oriented martensite associated

with the mid-surface of the SMA layer at the clamped end and center points are shown in Fig.

2b-2c and 2d-2e, respectively.

As can be found from Fig. 2a and 2d, implementing )118( × mesh results in an acceptable

accuracy so that a good correlation between the present result and those of [14] for the case of

00 =θ is achieved. Regarding the effect of reorientation phenomenon )0( ≠θ� on the response of

fully clamped SMA beams, it is observed that initial state of inelastic transverse shear strain does

not affect significantly the central deflection history during thermo-mechanical loading. On the

other hand, however, it has generally a major effect on the histories of oriented martensite

volume fraction and preferred direction of oriented martensite during thermo-mechanical

loading. Fig. 2b and 2c shows that the mid-surface of the SMA layer with various values of 0θ

experiences simultaneous reverse martensitic phase transformation )0( <Sξ� and martensite

variants reorientation )0( >θ� at the clamped end due to the development of compressive stress at

mechanical loading stage. This phenomenon occurs under non-proportional loading and can be

interpreted using Eq. (5). In this case, inelastic axial strain is recovered while inelastic transverse

shear strain may have an increasing or decreasing trend. On the other hand, however, Fig. 2d and

2e reveals that variation of oriented martensite volume fraction and its direction is vice versa at

the center of SMA layer where it undergoes a tensile stress. During heating stage, the SMA will

regain its original shape by transforming back into the parent austenite phase via reverse

martensitic phase transformation, see Fig. 2b and 2d. However, since the SMA layer is coupled

with the elastic host beam, a large stress is induced by constrained strain recovery which

27

suppresses the structure deformation. At this step, oriented martensite variants of mid-surface of

the SMA layer at the clamped end reorient while they does not rotate at the beam center.

Variation of tip deflection and internal variables of the two-layered SMA beam with clamped-

free boundary conditions under heating-loading-heating path is illustrated in Fig. 3. The SMA

temperature is first raised from K310 to K320 . Maintaining SMA temperature, a downward

uniformly distributed load with maximum magnitude of KPaF 400 = is applied to the structure.

In the presence of the mechanical load, SMA temperature is raised up to K330 . The counterpart

of Fig. 2 for the present example is demonstrated in Fig. 3. As can be concluded from this figure,

the pre-strain in a shear state has a significant effect on the histories of both external and internal

variables during thermo-mechanical loading. Fig. 3a shows that the two-layered SMA beam with

6/,00 πθ = and 4/π bends upward when the SMA temperature rises above the austenite

starting temperature about K5.318 and it experiences always maximum, intermediate and

minimum deflection history, respectively. It implies that thermal SMA actuators with less initial

inelastic transverse shear strain result in more deformation. During heating stage, a large stress is

induced by constrained strain recovery through temperature driven reverse phase transformation

from oriented martensite to austenite since the SMA layer is coupled with the elastic host beam,

see Fig. 3b and 3d. On the other hand, Fig. 3c and 3e reveals that oriented martensite variants for

all three cases remain in their initial direction during the first heating stage. As can be observed

in Fig. 3a, application of mechanical loading reduces the tip deflection of the structure with

6/,00 πθ = and 4/π about 95, 196 and 452%, respectively. Regarding variation of internal

variables, Fig. 3d and 3e reveals that the mid-surface martensite of all three SMA layers at center

point does not transform and reorient during mechanical loading. At the clamped end point,

reorientation of martensite variants takes place at mid-surface of SMA layers with 6/0 πθ = and

28

4/π in the absence of any phase transformation, see Fig. 3b and 3c. It means that inelastic axial

strain increases while inelastic transverse shear strain decreases (cf. Eq. (5)). As emerges from

Fig. 3a, 3b and 3d, when the SMA is further heated up to K330 , reverse martensitic phase

transformation takes place at both clamped end and center points inducing recovery stress and

increasing the structure deformation. In this respect, it is worthy to mention that SMA actuators

with various values of 0θ result in a similar deformation difference. Fig. 3c and 3e also shows

that, during further heating stage, SMA materials with non-zero initial value of θ experience

martensite variants reorientation at the center point.

Finally, the presented results in Figs. 2 and 3 show that, unlike thin elastic beams which

experience a negligible elastic transverse shear strain [18], the inelastic transverse shear strain

has a significant variation within thin SMA layers via martensitic transformation and

reorientation mechanisms. This emphasizes that consideration of elastic-inelastic shear

deformation and modeling of reorientation of martensite variants are essential to accurately

predict thermo-mechanical behavior of pre-strained SMA materials even when they are very

thin.

3.2. Thermo-mechanical analysis of SMA composite plates

In this section, thermo-mechanical behavior of shape adaptive composite plates with surface-

bonded SMA ribbon actuators are investigated and discussed in detail. Consider a symmetric

laminated composite plate made of orthotropic laminae with fibers oriented in the x-direction.

The engineering parameters of the unidirectional fiber-reinforced lamina including longitudinal

Young modulus )( 1E , transverse Young modulus )( 2E , major Poisson ratio )( 12ν , in-plane

shear modulus )( 12G and transverse shear modulus )( 23G are defined as [18]:

29

GPaGGPaGGPaEGPaE 45.3,96.8,25.0,93.17,78.53 23121221 ===== ν

In all examples, unless otherwise stated, square composite plates with cantilever boundary

conditions; i.e. clamped at edge 0=x and other edges free with 200mm length and total

thickness of 5mm are assumed. Also, SMA ribbons with mm1.11 width, 5mm thickness and

various lengths are considered whose material parameters are given in Table 1. Initial values of

Sξ and θ are assumed to be 0.2 and 0, respectively. Numerical results are computed for six

different types of SMA ribbons arrangements (SRA) as depicted in Fig. 4. During uniform

temperature rise of SMA ribbons, it is assumed that the temperature of the host composite plate

remains in the reference room temperature [1, 14].

In all computations, )1818( × mesh with 5 through-the-thickness Gauss points per SMA

element was used to obtain the converged results up to three significant digits. It should be

mentioned that a convergence study was first conducted to ensure independency of finite element

results from the number of elements and then their accuracy was examined through solving

classical non-linear bending of the plate [19] which are not presented here for the sake of brevity.

Finally, for simplicity and generality of the results, the following non-dimensional parameters

are introduced:

0*0

*

**

**

**

/;/

;/;/

/;/

/;/

s

sc

MTTFFF

hhHHwwHzz

bvvbyy

auuaxx

==

+=====

==

(39)

Active shape control of the composite plate with SRA 1 is first studied results of which are

presented in Fig. 5. A thermo-mechanical loading path is applied to the SMA composite plate

which is ordered as heating-loading-heating. SMA ribbons at reference temperature

30

)310( 0sMK = are first heated up to mean temperature K330 which is equal to KAf 110 + .

Maintaining SMA ribbons temperature, the shape adaptive structure experiences a downward

uniformly distributed load with maximum magnitude of MPaF 1.00 = . Finally, SMA ribbons are

further heated to K370 . The histories of central tip deflection and oriented martensite volume

fraction associated with the mid-surface of the left SMA ribbon at the clamped end and center

points are depicted in Fig. 5a-5c, respectively. Moreover, the mid-surface configuration of the

SMA composite plate at the end of first heating, mechanical loading and second heating are

illustrated in Fig. 5d-5f, respectively.

As can be found from Fig. 5a-5c, the SMA composite plate is deformed when SMA ribbons

in free-stress state are heated above stress-free austenite start temperature 0317 fAK = . During

heating stage, the recovery stress induced by constrained strain recovery via temperature driven

reverse phase transformation increases and develops more deformation. It is observed that the

structure experiences positive deflection and in particular non-dimensional central tip deflection

becomes unit at the end of heating stage while oriented martensite volume fraction at the end and

center points of SMA ribbon reduces to 0.09. Fig. 5d shows that the shape adaptive plate with

SRA 1 experiences maximum deflection at free end of longitudinally attached SMA ribbons.

During mechanical loading stage, the structure bends downward as much as 1* −=w at central

tip point. As can be found from Fig. 5b, the induced stress is sufficiently large to start forward

phase transformation from austenite to oriented martensite during mechanical loading stage. Fig.

5b and 5c indicates that, when the transverse mechanical loading reaches approximately 045.0 F ,

volume fraction of oriented martensite at the clamped end of the mid-surface of the SMA ribbon

increases significantly up to 0.33. However, oriented martensite volume fraction has no

important change at center point due to the low stress level at this point. Since the nucleation of

31

oriented martensite corresponds to the plateau phenomenon, upon increasing oriented martensite

volume fraction, the response of shape adaptive composite plate softens so that the load-

deflection gradient is increased, see Fig. 5a. Also, Fig. 5e indicates that narrow free tip region of

the shape adaptive plate has minimum and maximum deflection at free corners of the plate and

free ends of SMA ribbons at the end of mechanical lodging stage due to low and high stiffness at

these points, respectively.

In the next step, SMA ribbons are heated up to K370 . As can be observed in Fig. 5c, oriented

martensite volume fraction at the mid-surface center of SMA ribbon reduces when the

temperature rises above the austenite starting temperature about K340 . However, Fig. 5b

reveals that oriented martensite at the clamped end of SMA ribbon transforms to austenite phase

at a higher temperature about K348 due to the higher stress level at this point. As can be seen in

Fig. 5a, austenitic phase transformation at the mid-surface of the clamped end of the SMA ribbon

affects significantly load-deflection curve so that its gradient increases at K348 . Regarding the

end of second heating stage, Fig. 5f illustrates that the structure undergoes a positive deflection

at a region near free edge in the y-direction, whereas the other regions have a negative deflection.

In a numerical sense, it is found that the non-dimensional deflection at central region of the plate

increases from 5.0− to 125.0− , while the boundary region near free edge in the y-direction with

initial non-dimensional deflection of 1− experiences approximately 125.0* =w at the end of

second heating stage. Therefore, it can be concluded that the second half of SMA ribbons has a

better performance in this stage compared to their first half. In conjunction with the mode shape

in the y-direction, it is observed that minimum deflection takes place at an interior region

between two SMA ribbons and at vicinity of free edges in the x-direction due to low stiffness and

constraint at these regions. Finally, it can be found from Fig. 5a and 5b, the thermo-mechanical

32

behavior of SMA composite plate in particular central tip deflection is in a direct correlation with

the change of oriented martensite volume fraction at the clamped end of the SMA ribbon mid-

surface.

Next, thermo-mechanical response of shape adaptive composite plate with three different

arrangements including SRAs 2-4 are demonstrated in Figs. 6-8, respectively. A similar thermo-

mechanical loading path as previously applied to the shape adaptive composite plate with SRA 1

is adapted here. The history of central tip deflection and the mid-surface configuration of the

SMA composite plate at the end of first heating, mechanical loading and second heating are

presented respectively in parts (a)-(d) of these figures.

Comparing the presented results in Figs. 5a and 6a reveals that, although bonding of two

additional SMA ribbons increases the structure stiffness, deflection of SRA 2 is always larger

than that of SRA 1 during the first heating stage. As can be seen, at the end of first heating stage,

the smart structure with SRA 2 experiences non-dimensional central tip deflection about 62.1

which is %62 more than that of SRA 1. These figures also indicate that the applied mechanical

loading can only reduce the central tip deflection of the structure with SRA 2 about 58% due to

the higher stiffness in comparison with SRA 1. In the second heating stage, SRA 2 demonstrates

a lower shape recovery compared to SRA 1 so that the central tip deflection of the structure with

SRA 2 does not change beyond KT 358= . Finally, Figs. 5 and 6 reveal that bonding of two

additional SMA ribbons affects the structure deformation in the y-direction and reduces it during

thermo-mechanical loading path by increasing the structure stiffness in the transverse direction.

Presented results in Figs. 7 and 8 are now analyzed. The preliminary conclusion drawn from

these figures is the fact that SMA ribbons set in the transverse direction may be successfully used

to deform cantilever composite plates in the y-direction.

33

As can be found from Figs. 5a and 7a, shape adaptive composite plates with SRAs 1 and 3

have a similar central tip deflection history during thermo-mechanical loadings. However, it is

observed that, although adding transverse SMA ribbons increases the structure stiffness, the

central tip deflection of SRA 3 is always slightly more than that of SRA 1 due to the existence of

longitudinal SMA ribbons at central region of SRA 3. The presented results in Figs. 5 and 7

reveal that the longitudinal mode shape of SRA 3 is similar to SRA 1, while their transverse

mode shapes are different at local regions where transverse SMA ribbons are installed on the

host plate. As can be seen in Fig. 7b, during the first heating step, SMA ribbons set in the

transverse y-direction bend the free corner region in a concave form. The applied mechanical

loading changes then the concave form to convex one, see Fig. 7c. However, Fig. 7d shows that

the induced recovery stress of transverse SMA ribbons due to the second heating stage is not

large enough to change the convex form of free corner regions.

Regarding Fig. 8, since shape adaptive composite plates with SRAs 2 and 4 have the same

arrangement of longitudinal SMA ribbons, it is expected that both structures have a similar

response in the longitudinal direction. As can be found from Figs. 6a and 8a, these structures

experience central tip deflection history in a similar scheme though that of SRA 2 has a larger

scale due to the longer SMA ribbons. It is observed that transverse SMA ribbons affect

significantly the structure mode shape in the y-direction. Influence of transverse SMA ribbon

actuators becomes more pronounced in the free end of the plate where they are installed. Fig. 8

displays that free corners of the plate have a maximum deflection during thermo-mechanical

loadings. In contrast to SRA 3, transverse SMA ribbons of SRA 4 produce a continuous concave

shape in the y-direction and they are able to sustain the concave shape during whole thermo-

mechanical loadings.

34

To examine the effect of SMA ribbon length on the active shape control performance, the

mid-surface configuration of the shape adaptive composite plate with SRA 2 and different SMA

ribbon lengths (i.e. SRAs 5 and 6) at the end of first heating, mechanical loading and second

heating are presented in Fig. 9. As can be seen in Figs. 6 and 9, structures with SRAs 2, 5 and 6

experience maximum, intermediate and minimum central tip deflection during thermo-

mechanical loadings. In a numerical sense, it can be found that SRAs 2, 5 and 6 lead to non-

dimensional central tip deflection of 53.1,62.1 and 97.0 at the end of first heating stage. Figs. 9b

and 6c show that SRA 5 like SRA 2 results a positive deflection at the end mechanical loading

stage. On the other hand, however, Fig. 9e indicates that the structure with SRA 6 undergoes

nearly zero deflection in initial one-third of the plate where reinforced by SMA ribbons, whereas

the remained part experiences negative deflection with a maximum around 7.0* −=w at the free

edge in the y-direction. Moreover, it is found that, unlike SRAs 2 and 5, the second heating of

SRA 6 has no significant effect on the structure deformation. Finally, presented results in Figs. 9

and 6 reveal that SRA 5 results responses similar to SRA 2 and can be considered to design

economical shape adaptive composite plate. In the following results, behavior of composite

plates with SRA 5 is further investigated.

Influence of initial value of oriented martensite volume fraction on the thermo-mechanical

behavior of shape adaptive composite plate with SRA 5 is investigated in Fig. 10. The histories

of central tip deflection and oriented martensite volume fraction of the left SMA ribbon at central

point of its clamped end are illustrated in Fig. 10 for 2.0,15.00 =Sξ and 0.3. Fig. 10a shows that

the shape adaptive composite plates with various initial values of Sξ have the same deflection

during the first heating up to KT 325= . By further hating, the temperature-deflection curves are

separated from one another. It is seen that the SMA ribbons with 2.0,3.00 =Sξ and 0.15 yield

35

maximum, intermediate and minimum deflection at the end of first heating stage. The reverse

phase transformation induces recovery stress in SMA ribbons. The more oriented martensite that

transforms to austenite, the larger the deflection becomes. It is interesting to mention that shape

adaptive composite plates with various 0Sξ have the same load-deflection gradient during

mechanical loadings. Fig. 10b reveals that forward martensitic phase transformation and

consequently plateau phenomenon occur in center of the clamped end of SMA ribbons with

various initial values of 0Sξ when the smart structure is loaded over 075.0 F . Due to this fact,

Fig. 10a indicates that the response of the structure slightly softens beyond 075.0 F . Finally, it is

found that SMA ribbons with more initial value of oriented martensite volume fraction produce

more deflection at the second heating stage.

In practical case, when the SMA ribbon is stretched to achieve a recoverable strain, both

inelastic axial and transverse shear strains are induced producing non-zero preferred direction of

oriented martensite variants, 00 ≠θ , due to presence of both axial and transverse shear stresses.

In Fig. 11, influence of initial preferred direction of oriented martensite on the active shape

control efficiency of the composite plate with SRA 5 is highlighted. The histories of central tip

deflection, oriented martensite volume fraction and preferred direction of oriented martensite

variants associated with the mid-surface of the left SMA ribbon at clamped end are shown in Fig.

11a-11c, respectively. As can be found form Fig. 11a, the initial direction of oriented martensite

variants has a significant effect on the deformation history of the shape adaptive composite plates

during thermo-mechanical loadings. A similar conclusion was achieved for cantilever laminated

SMA beams as presented in Fig. 3a. Also, comparing these two figures shows that deformation

history of cantilever smart beam and plate has a similar trend versus variation of initial value of

θ during thermo-mechanical loading. During the first heating stage, oriented martensite volume

36

fraction of all three cases always decreases through reverse martensitic transformation inducing

the recovery stress, see Fig. 11b. At the end of this stage, SMA ribbons with 6/,00 πθ = and

4/π have oriented martensite volume fraction of 0.03, 0.045 and 0.065, and experience

19.1,53.1* =w and 0.86 due to the expense of more oriented martensite, respectively. On the

other hand, Fig. 11c indicates that the oriented martensite of SMA ribbons with 6/0 πθ = and

4/π undergoes also reorientation ( 0≠θ� ) so that the preferred direction of oriented martensite

variants increases during the reverse phase transformation. In the present case dealing with

simultaneous reverse martensitic transformation and reorientation, inelastic axial strain is

recovered while inelastic transverse shear strain may have an increasing or decreasing tendency

(cf. Eq. (5)). Upon applying mechanical loading, the reorientation initiates as the mechanical

loading reaches approximately 014.0 F while no phase transformation occurs ( 0=Sξ� ). For the

case of )4/or(6/0 ππθ = , in )94.0or(86.014.0 000 FFFF ≤≤ where the reorientation takes

place in the absence of any phase transformation, inelastic axial strain increases while inelastic

transverse shear strain decreases (cf. Eq. (5)). Continuing the mechanical loading activates

forward martensitic transformation and volume fraction of oriented martensite increases up to

0.075. As can be found from Fig. 11b and 11c, when the SMA material with 6/0 πθ = and 4/π

is further heated above the austenite start temperature about K347 , the reverse martensitic

transformation initiates with no reorientation and the gradient of temperature-deflection path is

increased. Finally, the oriented martensite, its direction and consequently temperature-deflection

gradient become zero at austenite finish temperatures around K353 . Fig. 11 reveals that, SMA

ribbons pre-strained in a pure axial state, (i.e., 00 =θ ) experience no reorientation phenomenon

during thermo-mechanical loadings. However, it can be concluded that the modeling of

37

reorientation of martensite variants is an essential tool to accurately assess thermo-mechanical

behavior of SMA materials pre-strained in a combined axial-shear state.

In all previous examples of this section, SMA ribbon actuators experience a heating-loading-

heating path. In order to study influence of cooling and unloading phases, a thermo-mechanical

cycle including heating-loading-cooling-unloading-heating is applied to the shape adaptive

composite plate with SRA 5. SMA ribbons at the reference temperature are first heated up to

temperature )11(330 0 KAK f += followed by transverse mechanical loading up to MPa1.0 .

Maintaining the mechanical loading, the SMA ribbons are cooled back to K320 which is equal

to KAf 10 + . Next, the structure is fully unloaded and then SMA ribbons are heated up to K340 .

The histories of central tip deflection and oriented martensite volume fraction related to the

clamped end of the left SMA ribbon at the mid-surface are demonstrated in Fig. 12a and 12b,

respectively. It is obvious that the present results for the first heating and mechanical loadings

are similar to those depicted in Fig. 10 for 2.00 =Sξ . Upon cooling in the presence of the applied

load, austenite transforms to oriented martensite and consequently the previously developed

recovery stress decreases. The process is accompanied by the decrease of non-dimensional

central tip deflection up to 35.0− at the end of cooling. Unloading the SMA material at

temperature above 0fA causes the oriented martensite to revert to austenite. The stress is released

gradually by unloading and *w increases up to 84.0 . A subsequent heating of the SMA to K340

results in a reverse phase transformation from oriented martensite to austenite and produces

recovery stress leading to high non-dimensional central tip deflection of 92.1 .

The final example is dedicated to the investigation of thermo-mechanical response of shape

adaptive composite plate with SRA 2 and fully clamped and simply supported boundary

38

conditions at 0=x and a while the other edges are free. The material and geometrical

parameters of the plate and SMA ribbons are similar to the previous case but the width is

reduced with a scale of 1/3. Unlike previous examples, the structure is first loaded mechanically

in low temperatures and then SMA ribbons are heated to high temperatures. This thermo-

mechanical loading is applied with maximum magnitudes of MPa3 and K410 . The histories of

central deflection and oriented martensite volume fraction associated with the mid-surface of the

left SMA ribbon at supported end and center points are shown in Fig. 13a-13c, respectively. As

can be concluded from Fig. 13a, although both composite structures have a similar deflection

history, fully clamped plate always experiences a lower deflection during thermo-mechanical

loading since more constraints at the edges increase the structure stiffness. Regarding variation

of Sξ at supported edge of the clamped plate, Fig. 13b reveals that oriented martensite volume

fraction has a decreasing-increasing trend during mechanical loading. When the structure is

gradually loaded, oriented martensite of the clamped end of pre-strained SMA ribbon at the mid-

surface is first transformed to austenite by compressive stress developed at this point. As

mentioned before, the recovery of pre-strained SMA materials at low temperatures by

compressive stress refers to the ferro-elasticity. It can be concluded that the modeling of ferro-

elasticity is essential for accurate prediction of behavior of pre-stained SMA devices. As the load

further increases, the austenite is transformed to the oriented martensite due to the growth of high

tensile stress at this point. The change of stress state from compression to tension is due to the

fact that all ribbon section is stretched in the large deformation regime. In conjunction with the

variation of Sξ at simply supported edge, it can be found from Fig. 13b that oriented martensite

volume fraction slightly increases during mechanical loading. It can be attributed to the

monotonic development of low tensile stress at simply edge support. In a similar way, at the

39

center point of both structures, SMA materials experience high forward phase transformation

from oriented martensite to austenite during mechanical loading. Finally, Fig. 13 shows that the

temperature driven reverse phase transformation induces recovery stress which reduces the

central deflection of fully clamped and fully simply supported composite plates up to 40 and

30%, respectively. This difference is due to the fact that fully clamped composite plate has a

more pre-strain or oriented martensite at the beginning of thermal loading stage. Note that, since

clamped-clamped and simply supported-simply supported boundary conditions are more

constraint compared to clamped-free case, a larger temperature rise is required to suppress the

structure deformation.

4. Conclusion

A study on the thermo-mechanical behavior of rectangular shape adaptive composite plates

with surface-bonded shape memory alloy ribbons was presented. The main features of pre-

strained SMA materials under combined axial-shear non-proportional loadings were modeled

based on the phenomenological constitutive equations proposed by the authors [7]. The time-

discrete counterpart of the constitutive model was presented and the associated solution

procedure was described according to the return map algorithm. The structural model was based

on the first-order shear deformation theory and von Kármán geometrical non-linearity. The finite

element equations of equilibrium were derived via principle of minimum total potential energy.

An iterative incremental scheme was introduced to solve the governing equations of equilibrium

coupled with martensitic transformation/orientation/reorientation equations per external and

internal variables. Capabilities of material and structural model were first verified by a

comparison with numerical simulations of laminated SMA beams under thermo-mechanical

40

loadings available in the open literature. The implications of the pre-strain state, temperature,

length and arrangement of the SMA ribbon actuators, as well as of boundary conditions of the

plate on the thermo-mechanical behavior of shape adaptive composite plates were put into

evidence via a parametric study, and related conclusions were drawn. In this sense, as it has been

shown, consideration of elastic-inelastic shear deformation and modeling of ferro-elasticity and

reorientation of martensite variants are essential to accurately predict the thermo-mechanical

behavior of pre-strained SMA devices under proportional/non-proportional loadings.

The presented formulation and results are expected to contribute to a better understanding of

the behavior of SMA composite plates and to be instrumental toward an efficient design of shape

adaptive composite structures by conducting various optimization analyses on the position and

geometrical parameters of the SMA ribbons.

Acknowledgment

The authors would like to express their sincere gratitude to the anonymous reviewer who

made valuable comments and suggestions to improve the paper.

References

[1] Lagoudas DC. Shape memory alloys: modeling and engineering applications. Springer; 2008.

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[3] Liang C, Rogers CA. One-dimensional thermomechanical constitutive relations for shape memory

materials. J Intell Mater Syst Struct 1990;1:207-234.

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[4] Brinson LC. One-dimensional constitutive behavior of shape memory alloys: thermomechanical

derivation with non-constant material functions and redefined martensite internal variable. J Intell

Mater Syst Struct 1993;4:229-242.

[5] Boyd JG, Lagoudas DC. A thermodynamical constitutive model for shape memory materials. Part I:

the monolithic shape memory alloy. Int J Plast 1996;12(6):805-842.

[6] Panico M, Brinson LC. A three-dimensional phenomenological model for martensite reorientation in

shape memory alloys. J Mech Phys Solids 2007;55(11):2491-2511.

[7] Bodaghi M, Damanpack AR, Aghdam MM, Shakeri M. A phenomenological SMA model for

combined axial-torsional proportional/non-proportional loading conditions. Mater Sci Eng A

2013;587:12-26.

[8] Ghomshei MM, Khajepour A, Tabandeh N, Behdinan K. Finite element modeling of shape memory

alloy composite actuators: theory and experiment. J Intell Mater Syst Struct 2001;12:761-773.

[9] Marfia S, Sacco E, Reddy JN. Superelastic and shape memory effects in laminated shape-memory-

alloy beams. AIAA J 2003;41(1):100-109.

[10] Yang SM, Roh JH, Han JH, Lee I. Experimental studies on active shape control of composite

structures using SMA actuators. J Intell Mater Syst Struct 2006;17:767-777.

[11] Roh JH, Han JH, Lee I. Nonlinear finite element simulation of shape adaptive structures with SMA

strip actuator. J Intell Mater Syst Struct 2006;17:1007-1022.

[12] Roh JH, Bae JS. Thermomechanical behaviors of Ni-Ti shape memory alloy ribbons and their

numerical modeling. Mech Mater 2010;42:757-773.

[13] Khalili SMR, Botshekanan Dehkordi M, Carrera E, Shariyat M. Non-linear dynamic analysis of a

sandwich beam with pseudoelastic SMA hybrid composite faces based on higher order finite element

theory. Compos Struct 2013;96:243-255.

[14] Bodaghi M, Damanpack AR, Aghdam MM, Shakeri M. Active shape/stress control of shape memory

alloy laminated beams. Compos Part B 2014;56:889-899.

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[15] Damanpack AR, Bodaghi M, Aghdam MM, Shakeri M. On the vibration control capability of shape

memory alloy composite beams. Compos Struct 2014;110:325-334.

[16] Truesdell C, Noll W. The non-linear field theories of mechanics. Berlin: Springer; 1965.

[17] Simo JC, Hughes TJR. Computational inelasticity. New York: Springer; 1998.

[18] Reddy JN. Mechanics of laminated composite plates and shells: theory and analysis. Boca Raton:

CRC Press; 2004.

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Inc.; 2004.

43

List of Figures

Fig. 1. Schematic sketch of: (a) the shape adaptive composite plate with surface-bonded SMA ribbons; (b) an SMA composite plate element.

Fig. 2. Present simulation and possible comparison with those reported in [14]: the histories of the central deflection (a), oriented martensite volume fraction (b, d) and preferred direction of oriented martensite (c, e) associated with the mid-surface of the SMA layer of the clamped-clamped laminated SMA beam at the clamped end (b, c) and center (d, e) points for various initial values of θ .

Fig. 3. The histories of the tip deflection (a), oriented martensite volume fraction (b, d) and preferred direction of oriented martensite (c, e) associated with the mid-surface of the SMA layer of the clamped-free laminated SMA beam at the clamped end (b, c) and center (d, e) points for various initial values of θ.

Fig. 4. SMA ribbons arrangements.

Fig. 5. The histories of the central tip deflection (a), mid-surface oriented martensite volume fraction at the clamped end (b) and center (c) of the left SMA ribbon together with the mid-surface configuration of the cantilever SMA composite plate with SRA 1 at the end of first heating (d), mechanical loading (e) and second heating (f). Note that parts (e) and (f) are depicted in front view.

Fig. 6. Central tip deflection history (a) and mid-surface configuration of the cantilever SMA composite plate with SRA 2 at the end of first heating (b), mechanical loading (c) and second heating (d).

Fig. 7. The counterpart of Fig. 6 for the cantilever SMA composite plate with SRA 3. Note that part (c) is depicted in front view.

Fig. 8. The counterpart of Fig. 6 for the cantilever SMA composite plate with SRA 4.

Fig. 9. The mid-surface configuration of the cantilever SMA composite plates with SRAs 5 (a, b, c,) and 6 (d, e, f) at the end of first heating (a, d), mechanical loading (b, e) and second heating (c, f). Note that parts (e) and (f) are depicted in front view.

Fig. 10. The histories of the central tip deflection (a) and mid-surface oriented martensite volume fraction (b) at the clamped end of the left SMA ribbon of the cantilever SMA composite plate with SRA 5 for

various initial values of Sξ .

Fig. 11. The histories of the central tip deflection (a), oriented martensite volume fraction (b) and preferred direction of oriented martensite variants (c) at the clamped end mid-surface of the left SMA ribbon of the cantilever SMA composite plate with SRA 5 for various initial values of θ .

Fig. 12. The histories of the central tip deflection (a) and mid-surface oriented martensite volume fraction (b) at the clamped end of the left SMA ribbon of the cantilever SMA composite plate with SRA 5 under heating-loading-cooling-unloading-heating path.

Fig. 13. The histories of the central deflection (a) and oriented martensite volume fraction at the supported end (b) and center (c) points of the mid-surface of the left SMA ribbon of the SMA composite

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uratiatingpart

ion g (ats (e

of ta, d)e) an

the ), mnd (

canmech

(f) a

ntilehaniare

55

evericaldep

r SMl loapicte

MA adined i

comng (in fr

mpob, eront

osite) ant vie

e plnd sew.

lateseco

es wond

with d hea

SRatin

RAs ng (c

5 (ac, f)

a, b). N

b, c,Note

,) ane tha

nd at

(a)

Fig

(

g. 10

(b) a

0. T

at th

The

he c

ξ

his

clam

0Sξ

tori

mpe

0=

ies o

ed en

3.0

of th

nd o

he c

of t

cent

he l

tral

left

tip

SM

def

MA r

var

flec

ribb

riou

ction

bon

us in

ξ

n (a

of

nitia

56

0Sξ

(

a) an

the

al v

0=

(b)

nd m

can

alue

2.0

mid

ntile

es o

d-sur

ever

of ξ

rfac

r SM

Sξ .

ce o

MA

orien

A com

nted

mpo

d m

osit

marte

te p

ξ

ensi

late

0Sξ

ite v

e wi

0=

volu

ith S

15.0

ume

SRA

5

e fra

A 5

actio

for

on

r

(a)

p

Fipref

ig. 1ferre

ri

11. ed dibbo

Thedireon o

0θ

e hictioof th

0=

storon ohe c

0

riesof orcant

of rientilev

the ntedver

cend maSM

ntraarte

MA c

al tipensitcom

p dete v

mpo

eflevariasite

θ

ctioantse pla

57

0θ =

(

(c)

on (s (c)ate w

π=

(b)

a), ) at with

6/

orietheh SR

entee claRA

ed mamp

A 5 f

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site d miious

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θ

me frfacel va

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racte of alue

π=

tionthe s of

4/

n (b)lef

f θ

4

) anft SM.

nd MA

A

(a)

Fig(b

g. 12b) at

2. Tt the

The e cla

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toriped

ies od end

of thd of

he cf the

h

cente lehea

tral eft Sating

tip SMAg-lo

defA rioadi

flecibbong-

ctionon ocoo

n (aof tholing

58

(

a) anhe cg-un

(b)

nd mcantnloa

midtilevadin

d-surver Sng-h

rfacSMhea

ce oMA cating

oriencomg pa

ntedmposath.

d msite

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ensiate w

ite vwith

voluh SR

umeRA

e fra5 u

actiounde

on er

(a)

sup

pla

F

ppo

ate w

Fig.

orted

with

13.

d en

h SR

. Th

nd (

RA

he h

(b) a

2 u

isto

and

nde

ories

d cen

er lo

Cl

s of

nter

oadi

lam

f the

r (c)

ing-

mped

e ce

) po

-hea

d su

entra

oint

atin

uppo

al d

s of

g pa

ort

defle

f the

ath

a

ecti

e m

for

at x

on (

mid-s

ful

=x

59

(

(c)

(a)

surf

lly c

0 a

(b)

and

face

clam

and

d ori

e of

mpe

a .

ient

the

ed an

ted

e lef

nd s

S

ma

ft SM

sim

imp

rten

MA

mply

ply

nsite

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y sup

sup

e vo

bbon

ppo

ppor

olum

n of

orted

rt

me f

f th

d bo

frac

e SM

oun

ction

MA

dary

n at

A co

y co

t the

omp

ond

e

posi

ditio

te

ons

60

List of Table

Table 1. Material parameters used in the computations.

61

Table 1. Material parameters used in the computations.

Parameter Value Unit

0fM 306 K

0sM 310 K

0sA 317 K

0fA 319 K

ν 0.36 -

uε 3.8 %

E 68,400 MPa

C

10 1−MPaK

σ~ 21.05 MPa

fSY 0 88.92 MPa

fSY 1 0 MPa

fSY 2 12.25 MPa

fSY 3 0 MPa

rSY 0 88.92 MPa

rSY 1 0 MPa

rSY 2 0 MPa

rSY 3 0 MPa

θY 12.25 MPa