Analysis and control of large thermal deflection of composite platesusing shape memory alloy

Bin Duana, Mohammad Tawfika, Sylvain N. Goekb

Jeng-Jong Roa, Chuh Meia*

a Department of Aerospace Engineering, Old Dominion Univ., Norfolk, VA 23529b ES de Mecanique de Marseille, Marseille, France

ABSTRACT

A finite element method for predicting critical temperature and postbuckling deflection (large thermal deflection) is presentedfor composite plates embedded with prestrained shape memory alloy (SMA) wires and subjected to high temperatures. Thetemperature-dependent material properties of SMA and matrix, and the geometrical nonlinearities of large deflection areconsidered in the formulation. An incremental method consisting of small temperature increments and including the effect ofinitial deflection and initial stresses for material nonlinearities is presented. Within each temperature increment, the Newton-Raphson iteration method is used for calculating large thermal deflection. Results show that the critical buckling temperaturecan be raised high enough and the postbuckling deflection can be reduced and controlled for a given operating temperaturerange by the proper selection of SMA volume fraction, prestrain and alloy composition.

Keywords: Shape memory alloy, thermal buckling, thermal postbuckling

1. INTRODUCTION

SMA has a unique characteristic to recover a large prestrain (as large as 8-10%) completely when it is heated above theaustenite finish temperature Af. The austenite transformation start temperature As can be altered between -60oF (-50oC) and340 oF (170 oC) for Nitinol by varying the nickel content [1]. During the recovery process, a large tensile stress occurs if theSMA is restrained. To utilize this large recovery stress, the prestrained SMA is embedded in a fiber-reinforced laminatedcomposite plate, the SMA is thus restrained and large tensile in-plane forces are induced in the plate at temperatures higherthan Af. The large in-plane forces result in a stiffer plate and yield a higher critical temperature and a reduced thermalpostbuckling deflection.

Due to aerodynamic heating, the skin panels of high speed flight vehicles could potentially reach several hundred or thousanddegrees, for example, 350 oF (177oC) for the High Speed Civil Transport (HSCT) cruising at Mach 2.4. Large thermaldeflection of the skin panels may occur. This would alter the vehicle’s configuration, affect its aerodynamic characteristics,and lead to poor flight performance. The purpose of this paper is to investigate analytically the use of SMA recovery stress tocontrol the panel postbuckling deflection for a given temperature range.

The recovery stress for Nitinol shown in Fig. 1a [2] depends on temperature and prestrain value εr, the elastic modulus in Fig.1b [2] is also temperature-dependent (TD). The recovery stress and the elastic modulus both are highly nonlinear versustemperature. The properties of the matrix (aluminum or graphite-epoxy) are also TD and they can be approximated to belinear functions of temperature. The analysis for a SMA embedded composite plate thus has to consider the TD materialproperties of SMA and matrix and the geometrical nonlinearities of thermal postbuckling deflection.

Effects of TD material properties on buckling and postbuckling of plate and shell structures have been summarized in threeexcellent review papers [3-5]. Kamiya and Fukui [6] investigated the postbuckling of square isotropic plates with TDproperties. Finite difference method with iteration solutions was used and it obtained results for simply supported and

* Correspondence: Email: chmei@aero.odu.edu; Telephone: 757-683-3733; Fax: 757-683-3200

clamped edges with constrained in-plane displacements. The TD properties lower critical temperature, and reducepostbuckling stiffness for increasing temperature and deflection. Chen and Chen [7, 8] proposed a finite element method forthermal buckling and postbuckling behavior of laminated composite plates. The TD properties lower predictions of criticaltemperatures, and reduce thermal postbuckling strength. Noor and Burton [9] studied the effects of TD properties on theprebuckling stresses, critical temperatures and their sensitivity derivatives of antisymmetric angle-ply plates using the three-dimensional thermoelasticity solutions. Numerical results show that the TD properties reduced critical temperatures and theinfluence of prebuckling stresses on critical temperature is less significant for the TD than the temperature-independent (TI)property case. Lee et al. [10] recently studied the thermal effect for stiffened composite plates with the material degradationon the buckling, vibration and flutter characteristics using the finite element method. Their results indicate that the materialdegradation decreases the critical temperature and increases the postbuckling deflection. It is interesting to note that TDmaterial properties in references [6-10] are all assumed to be linear functions of the temperature, and only Noor and Burton[9] investigated the effect of prebuckling stresses on the critical temperature. In this paper, an incremental method formaterial nonlinearities with the Newton-Raphson iteration method for geometrical nonlinearities of large thermal deflectionis presented in the following for the analysis and control of postbuckling deflection of SMA embedded composite plates. Theeffects of prebuckling stresses and deformation on critical temperature as well as postbuckling deflection are considered inthe formulation.

Fig. 1a SMA recovery stress vs temperature Fig. 1b SMA modulus of elasticity vs with different prestrain [2] temperature [2]

2. FORMULATION

The present incremental method for the TD nonlinear properties of Nitinol in Fig.1 and the linearly varying TD properties ofmatrix (graphite-epoxy) consists of many small temperature increments and the effect of initial deflection and initial stresses.Within each temperature increment ∆T, the averaged material properties are computed and treated as constant and theincremental deflection and stress are determined. The first increment is from the reference or ambient temperature at whichthe plate is assumed in a state of free-stress. The incremental deflection and stresses are summed up at the end of eachincrement and they are the initial deflection and initial stresses for the next increment.

2.1 Constitutive Relation

Consider a thin SMA embedded composite lamina that the SMA and the graphite fibers have the same direction, the stress-strain relations of a general kth layer during a small temperature increment can be expressed as [11] as

0

10

20

30

40

50

60

70

80

90

-80 -40 0 40 80 120 160Temperature (C)

Mo

du

lus

Of

Ela

stic

ity

(GP

a)

Heating

Cooling

0

10

20

30

40

50

60

70

80

0 40 80 120 160 200 240 280 320

Temperature (F)

Rec

ove

ry S

tres

s (K

si)

εε r=1%

2%

3%

4%

5%

6%

7%

8% 9%

10

{ } [ ] { } { }( )

[ ] { } { } [ ] { }( ) skmmmskkrk

skkkok

ATTvQvQ

ATTQ

≥−+=

<−=−

,

,}{

∆ασε

∆αεσσ (1)

where {σo}, {α} and {σr} denote the initial stress, thermal expansion coefficient, and SMA recovery stress vectors,

respectively; νs and νm denote the volume fractions of SMA and matrix, respectively; [ ]Q and [ ]mQ denote the transformed

reduced stiffness matrices of the SMA embedded lamina (νs≠0) and the composite matrix (νs=0). Equation (1) is in anincremental form: the change of state of stress is thus {σ}-{σo}; and {σr} is the increment of SMA recovery stress during ∆T.The constitutive equations of the SMA embedded composite plate are given as

+

−

+

=

∆

∆

σ

σ

κε

M

N

M

N

M

N

DB

BA

M

N

T

T

r

ro

(2)

where the laminate stiffness [A], [B] and [D] matrices are TD, the incremental recovery stress resultants {Nr} and {Mr}depend on the temperature and prestrain (Figure 1a) and volume fraction of SMA, and the initial stress resultants {Nσ} and{Mσ} are due to the initial stress vector {σo}.

For large thermal deflections, the in-plane strain {εo} and curvature {κ} vectors are defined from the von Karman strain-displacement relations as

{ }ε

ε ε ε

o

x

y

y x

x

y

x y

x o x

y o y

x o y o x y

mo

bo

oo

u

v

u v

w

w

w w

w w

w w

w w w w

=+

+

++

= + +

,

,

, ,

,

,

, ,

, ,

, ,

, , , ,

/

/

{ } { } { }

2

2

2

2

(3)

{ }

−−

−

=

xy

yy

xx

w

w

w

,

,

,

2

κ (4)

{ } }{}{ κεε zo += (5)

where u, v and w are the in-plane and transverse displacements measured from the initial position (uo, vo, wo).

2.2 Finite Element Governing Equation

With the application of the variational principle and the kinematic boundary conditions, the governing equation for a SMAembedded composite plate undergoing large deflection subjected to a temperature increment ∆T can be written as

−

−

=

+

+

++

+

+

−

+

∆

∆

∆

mr

br

m

b

Tm

Tb

m

bb

m

bbo

mb

bmNBNm

m

brTN

mbo

bmobo

mmb

bmb

P

P

P

P

P

P

W

WN

W

WN

N

NNN

W

WKKK

K

KK

KK

KK

σ

σ

σ

00

0

3

1

00

0

02

1

00

0

00

0

00

0

0

21

1

111 (6)

or simply

[ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] { } { } { } { }rTorTNo PPPWNNNKKKKK −−=

+++++−+ ∆∆ σσ 211 3

1

2

1

2

1 (7)

where [K] is the system linear stiffness matrix; the geometrical stiffness matrices [KN∆T], [Kr], [Kσ] and [Ko] are due tothermal in-plane force {N∆T}, SMA incremental recovery stress {σr}, initial stress {σo}, and initial deflection {Wo},respectively; [N1] and [N2] are the first and second-order incremental stiffness matrices which depend linearly andquadratically upon system displacement vector {W}, respectively; [N1o] is linearly dependent on {W} and the known initialdeflection {Wo}. {P∆T}, {Pσ} and {Pr} are system load vectors which are due to thermal stress resultants, initial stress, andSMA incremental recovery stress, respectively. The subscripts b and m denote bending and in-plane displacements,respectively; subscripts mb (bm), Nm, NB indicate that the corresponding stiffness matrix or load vector is dependent onlaminate coupling stiffness [B], in-plane force components {Nm}= ([A]{εm

o}), and {NB}=([B]{κ}), respectively.

2.3 Solution Procedure

For the TD nonlinear material properties of Nitinol shown in Fig. 1 and TD properties of matrix, the thermal large deflectionat certain temperature is calculated using an incremental method. The temperature range between the reference temperatureand the final temperature is divided into many small increments ∆T. At reference temperature, the SMA embedded compositeplate is regarded as in a stress-free state (zero initial deflection and initial stresses). The first increment ∆T starts from thereference temperature. In each ∆T, the material properties are approximated as constant and the incremental deflection issolved by Eq. (7) using the Newton-Raphson iteration method. The total deflection is then obtained by adding theincremental deflection to the initial deflection obtained from previous calculation. The initial deflection and initial stressesare then updated and the next ∆T is introduced. For the ith iteration, Eq. (7) and the system displacement vector can be writtenas

[ ] { } { }iii PWK ∆=∆ +1tan (8)

{ } { } { }W W Wi i i+ += +1 1∆ (9)

where the tangent stiffness and the imbalance load vector are given by

[ ] [ ] [ ] [ ] [ ] iioilini NNNKK 211tan +++= (10)

{ } { } { } { } [ ] [ ] [ ] [ ] { }iiioilinrTi WNNNKPPPP

+++−−−=∆ ∆ 211 3

1

2

1

2

1σ (11)

with linear stiffness matrix as

[ ] [ ] [ ] [ ] [ ] [ ]rTNolin KKKKKK ++−+= ∆ σ (12)

The subscript i to the nonlinear incremental stiffness matrices in Eqs. (10) and (11) denotes that they are evaluated with{W}i. The solution scheme seeks to reduce the imbalance load {∆P}, and consequently {∆W}, to a specified small quantity(10-5 in this study).

The solution procedure discussed earlier is general for the prediction of large thermal deflection as well as the prediction ofcritical temperature and postbuckling deflection. The results given in the following is for a symmetrically laminatedcomposite plate with or without embedded Nitinol fibers subjected to a uniformly distributed temperature change. Therefore,both critical temperature and postbuckling deflection can be predicted.

3. RESULTS AND DISCUSSION

A 15×12×0.048 in. (38.1×30.5×0.12 cm) eight layered [0/45/-45/90]s graphite-epoxy plate with and without SMA (Nitinol) isstudied in detail. The material properties for Nitinol and graphite-epoxy are

Nitinolσr from Fig. 1a E from Fig. 1bAs 100oF (37.8oC) G 3.604 Msi (24.9 GPa), T<As

3.712 Msi (25.6 GPa), T≥As

ν 0.3 ρ 0.6067×10-3 lb-s2/in.4 (6450 kg/m3)α 5.7×10-6/oF (10.26×10-6/oC)

Graphite-Epoxy (∆T=T-Tref)E1 22.5×106 (1-3.53×10-4⋅∆T) psi E2 1.17×106 (1-4.27×10-4⋅∆T) psiG12 0.66×106 (1-6.06×10-4⋅∆T) psi ν12 0.22ρ 0.1458×10-3 (1550) α1 -0.04×10-6(1-1.25×10-3⋅∆T)/oFα2 16.7×10-6×(1+0.41×10-4⋅∆T)

One quarter of the plate is modeled with a 4×4 mesh or 16 C1 conforming rectangular plate finite elements [12]. The plateelement has 24 nodal displacements, 16 bending and 8 in-plane. The in-plane boundary conditions areu(0,y)=u(a,y)=v(x,0)=v(x,b)=0 at the four plate edges. Out-of-plane boundary conditions considered are simply supportedand clamped. The reference temperature of 70oF is used in the example.

3.1 Simply Supported Plate

The simply supported [0/45/-45/90]s composite plate with and without SMA is studies first. The nondimensional maximumdeflection versus temperature with SMA volume fraction vs=0, 10%, 15% and prestrain εr=3, 4 and 5% is shown in Fig. 2.The postbuckling deflections clearly indicate that the large in-plane forces induced in the plate from the Nitinol recoverystress result in stiffer plates and higher critical temperatures. The critical temperatures are given in Table 1. Skin panel ofHSCT could reach 350oF at cruise. Figure 2 and Table 1 show that the composite plate embedded with SMA (vs=10%,εr=5%) or (vs=15%, εr=3%) is suitable for HSCT application.

3.2 Clamped Plates

Figure 3 shows the maximum deflections for the clamped case with vs=10%, εr=3, 4 and 5% and vs=15%, εr=3%. The criticaltemperatures are also given in Table 1 and they showed that the clamped plate is stiffer than the simply supported. Thecomposite plate with SMA (vs=10%, εr=5%) or (vs=15%, εr=3%) is suitable for HSCT.

3.3 Weight Saving in Using SMA

The mass density of graphite-epoxy is 1550 kg/m3. Nitinol is much heavier which has a mass density of 6450 kg/m3. Thesaving in weight is based on critical temperature. The weight versus critical temperature of [0/45/-45/90]ns plate for n=1 to 5with no SMA is plotted in Figs. 4 and 5 for simply supported and clamped cases, respectively. The critical temperatures ofthe SMA embedded composite plates (n=1) and their corresponding weight are also given in Figs. 4 and 5. Theydemonstrated clearly the advantage in weight saving in using SMA for control of thermal buckling.

Table 1 Critical Temperature (Tcr, oF) of a [0/45/-45/90]s

Graphite-Epoxy 15×12×0.048 in. Plate

Volume fractionof Nitinol, vs %

Prestrain εr

%Critical Temp. Tcr

oFSimply Supported

0 0 92 10 3 271 4 300 5 371 15 3 417

Clamped 0 0 106 10 3 299 4 338 5 395 15 3 442

4. CONCLUSION

A finite element formulation is presented for analysis of critical temperature and postbuckling or large thermal deflection ofSMA embedded composite plates. Temperature dependent material properties of both SMA and matrix are considered in theformulation. Results demonstrated that it is feasible to control thermal deflections with proper percentages of SMA volumeand prestrain within a temperature range (maximum temperature of 350oF for HSCT). Future investigations include vibrationbehavior of SMA embedded composite plates and control of panel flutter and sonic fatigue of composite plates using SMA.

Fig. 2 Maximum deflection vs temperature Fig. 3 Maximum deflection vs temperature

Fig. 4 Plate weight vs critical temperature Fig. 5 Plate weight vs critical temperature

ACKNOWLEDGEMENTS

The first and fifth authors would like to acknowledge the support by grant F33615-91-C-3205, Air Force ResearchLaboratory.

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