COMPOSITES

www.elsevier.com/locate/compscitech

Composites Science and Technology 66 (2006) 2965–2970

SCIENCE ANDTECHNOLOGY

Thermal buckling behavior of cross-ply hybrid compositelaminates with inclined crack

Ahmet Avci a, Omer Sinan Sahin a,*, Necati Ataberk b

a Selcuk University, Department of Mechanical Engineering, Konya 42800, Turkeyb Selcuk University, Kadinhani Vocational School, Konya 42800, Turkey

Received 27 April 2005; received in revised form 4 January 2006; accepted 8 February 2006Available online 23 March 2006

Abstract

Thermal buckling analysis of symmetric and antisymmetric cross-ply laminated hybrid composite plates with an inclined crack sub-jected to a uniform temperature rise are presented in this paper. The first-order shear deformation theory in conjunction with variationalenergy method is employed in the mathematical formulation. The eight-node Lagrangian finite element technique is used for obtainingthe thermal buckling temperatures of hybrid composite laminates. The effects of crack size and lay-up sequences on the thermal bucklingtemperatures for symmetric and antisymmetric plates are investigated. The results are shown in graphical form for various boundaryconditions.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Hybrid composite plates; Thermal buckling; Finite element method

1. Introduction

Fiber reinforced structures are widely used in so manyengineering applications, because of their low weight andhigh strength. Stability of these structures is importantespecially at elevated temperatures.

The thermal buckling analyses of orthotropic platesincluding a crack were investigated by Avci et al. [1]. Ther-mal buckling analysis of symmetric and antisymmetriccross-ply laminated hybrid composite plates with a holesubjected to a uniform temperature rise for differentboundary conditions was studied by using finite elementsmethod by Avci et al. [2]. In that paper the effects of holesize, lay-up sequences and boundary conditions on thethermal buckling temperatures were investigated. Akbulutand Sayman [3] were studied the buckling behavior of lam-inated composite plates with central square openings forvarious boundary conditions and stacking sequences byusing finite element method.

0266-3538/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.compscitech.2006.02.009

* Corresponding author. Tel.: +90 332 223 18 98.E-mail address: ossahin@selcuk.edu.tr (O.S. Sahin).

The thermal buckling of isotropic and composite plateswith a hole by using both closed form solution and finiteelements method was investigated by Chang et al. [4]. Ther-mal buckling of antisymmetric cross-ply composite lami-nates was investigated by Mathew et al. [5]. Abramovich[6] investigated the thermal buckling behavior of cross-ply symmetric and nonsymmetrical laminated beamsemploying the first-order deformation theory. Murphyand Ferreira [7] studied theoretical and experimentalapproaches to obtain the buckling temperature and buck-ling mode for flat rectangular plates. Huang and Tauchert[8] studied the thermal buckling of clamped symmetricangle-ply laminated plates employing a Fourier seriesapproach and the finite element method. Prabhu andDhanaraj [9], Chandrashekhara [10], Thangaratnam andRamaohandran [11], and Chen et al. [12] also studied thethermal buckling of the laminates subjected to uniformtemperature rise or non uniform temperature fields usingthe finite element approach.

An extensive overview of the general buckling prob-lems of laminated composite plate was made by Liessa[13]. In that study, some complicated effects were

tk

h

Fig. 1. Geometry of the problem and coordinates.

2966 A. Avci et al. / Composites Science and Technology 66 (2006) 2965–2970

investigated such as shear deformation, hygrothermal fac-tors and post buckling behavior. The influence of temper-ature distribution on buckling modes has investigated byBednarczyk and Rihter [14]. The thermally induced buck-ling of antisymmetric angle-ply laminated plates withLevy-type boundary conditions was investigated by Chenand Liu [15]. Thermal buckling behavior of compositelaminated plates with transverse shear deformation wasstudied by Sun and Hsu [16]. Chockalingam et al. [17]investigated the thermal buckling of antisymmetriccross-ply hybrid laminates by using finite element tech-nique based on first-order shear deformation theory.Ramaohandran [18] and Chen et al. [19] also studied ther-mal buckling of laminates subjected to uniform tempera-ture rise or non uniform temperature fields using finiteelement approach. Local buckling of composite laminarplates was considered and the critical strains of laminatedplates with various shaped local delamination and differ-ent stacking patterns are obtained by making use of theenergy principle. by Wang et. al. [25]. Also non-linearthermal buckling for local delamination near the surfaceof laminated cylindrical shell problem was studied byWang et al. [26].

The present paper aims to determine the buckling tem-perature and buckling mode shapes for hybrid compositelaminates with different inclined crack by using the finiteelement method. The thermal buckling of symmetric andantisymmetric cross-ply laminates with cracks is investi-gated, based on the first-order deformation theory inconjunction with the variational energy method. Thefinite element approach is used for obtaining the thermalbuckling temperatures for boron/epoxy–glass/epoxyhybrid laminates. The effects of crack length, crack incli-nation angle and lay-up sequences on the thermal buck-ling temperatures are numerically solved. The bucklingbehavior of boron/epoxy–glass/epoxy hybrid compositeplates was compared with E-glass/epoxy plates. Theresults are presented in graphical form for variousboundary conditions.

2. Mathematical formulation

The laminated orthotropic construction of the plate isconsisted of N layers. Each layer is of thickness tk, so thath ¼

PNk¼1tk is the total thickness of the laminate.

The longitudinal and lateral dimensions of the laminateare a and b and subjected to uniform temperature differ-ence DT between ambient and laminated plate as shownin Fig. 1. The linear stress–strain relation for each layeris expressed with x, y-axes and has the form

rx

ry

sxy

8><>:

9>=>;

k

¼Q11 Q12 Q16

Q12 Q22 Q26

Q16 Q26 Q66

264

375

k

ex �axDT

ey �ayDT

cxy �axyDT

8><>:

9>=>;

k

syz

sxz

� �¼ Q44 Q45

Q45 Q55

" #cyz

cxz

� � ð1Þ

where rx, ry, sxy, syz and sxz are the stress components, Qij

are transformed reduced stiffnesses, which can be expressedin terms of the orientation angle and the engineering con-stant of the material [20]. DT is temperature difference, ax

and ay are the coefficients of thermal expansion in direc-tions of x and y-axes, respectively. axy is the apparent coef-ficient of thermal shear, such as:

ax ¼ a1 cos2 hþ a2 sin2 h

ay ¼ a2 cos2 hþ a1 sin2 h

axy ¼ 2ða1 � a2Þ sin h � cos h

ð2Þ

a1 and a2 are the thermal expansion coefficients of the lam-ina along the longitudinal and transverse directions of fi-bers, respectively.

In this study first-order shear deformation theory isused. The displacements u, v and w can be written asfollows:

uðx; y; zÞ ¼ u0ðx; yÞ þ zwxðx; yÞvðx; y; zÞ ¼ v0ðx; yÞ þ zwyðx; yÞwðx; y; zÞ ¼ wðx; yÞ

ð3Þ

where u0, v0, w are the displacements along to x, y, and z-axes, respectively, at any point of the middle surface, andwx, wy are the bending rotations of normals to the midplane about the x and y axes, respectively. The bendingstrains ex, ey and transverse shear strains cxy, cyz, cxz atany point of the laminate are

ex

ey

cxy

8><>:

9>=>; ¼

ou0

oxov0

oy

ou0

oy þov0

ox

��������

��������þ z

owx0

oxowy

oy

owxoy þ

owy

ox

��������

��������cyz

cxz

�������� ¼

owoy� wy

owox þ wx

����������

ð4Þ

Table 1Material properties

Material E1

(Gpa)E2

(GPa)G12

(GPa)m12 a1 (1/�C) a2 (1/�C)

E-glass/epoxy 15 6 3 0.3 7.0 · 10�6 2.30 · 10�5

Boron/epoxy 207 19 4.8 0.21 4.14 · 10�6 1.91 · 10�5

A. Avci et al. / Composites Science and Technology 66 (2006) 2965–2970 2967

The resultant forces Nx, Ny and Nxy, moments Mx, My

and Mxy and shearing forces Qx, Qy per unit length ofthe plate are given as

Nx Mx

Ny My

N xy Mxy

264

375 ¼ Z h=2

�h=2

rx

ry

sxy

8><>:

9>=>;ð1; zÞ dz

Qx

Qy

( )¼Z h=2

�h=2

sxz

syz

� �dz

ð5Þ

The total potential energy P of a laminated plate underthermal loading is equal to

P ¼ U b þ U s þ V ð6Þwhere Ub is the strain energy of bending, Us is the strainenergy of shear and V represents the potential energy ofin-plane loadings due to temperature change

U b ¼ 1=2

Z h=2

�h=2

Z ZRðrxex þ ryey þ sxycxyÞ dA

� �dz

U s ¼ 1=2

Z h=2

�h=2

Z ZRðsxzcxz þ syzcyzÞ dA

� �dz

V ¼ 1=2

Z ZR½N 1ðow=oxÞ2 þ N 2ðow=oyÞ2

þ 2N 12ðow=oxÞðow=oyÞ� dA�Z

oRðNb

nuon þ Nb

s uos Þ ds

ð7ÞHere dA = dxdy, R is the region of a plate excluding thecrack. Nb

n and Nbs are in-plane loads applied on the bound-

ary oR.For the equilibrium, the potential energy P must be sta-

tionary. The equilibrium equations of the cross-ply lami-nated plate subjected to temperature change can bederived from the variational principle through use ofstress–strain and strain–displacement relations. One mayobtain these equations by using dP = 0 [21,22].

2.1. Finite element formulations

In general, a closed form solution is difficult to obtain forbuckling problems [21,22]. Therefore numerical methods areusually used for obtaining an approximate solution.

In order to study the buckling of the plate, an eight-nodeLagrangian finite element analysis is applied in this study.The stiffness matrix of the plate is obtained by using theminimum potential energy principle. Bending stiffness[Kb], shear stiffness [Ks] and geometric stiffness [Kg] matri-ces can be expressed as

½Kb� ¼Z

A½Bb�T ½Db�½Bb� dA ð8Þ

½Ks� ¼Z

A½Bs�T ½Ds�½Bs� dA ð9Þ

and

½Kg� ¼Z

A½Bg�T ½Dg�½Bg� dA ð10Þ

where

½Db� ¼Aij Bij

Bij Dij

� �½Ds� ¼

k21A44 0

0 k22A55

" #

½Dg� ¼N 1 N 12

N 12 N 2

" #ð11Þ

ðAij;Bij;DijÞ ¼Z h=2

�h=2

Qijð1; z; z2Þ dz ði; j ¼ 1; 2; 6Þ ð12Þ

ðA44;A55Þ ¼Z h=2

�h=2

ðQ44;Q55Þ dz ð13Þ

A44 and A55, are the shear correction factors for rectangu-lar cross section are given by k2

1 ¼ k22 ¼ 5=6 [23].

The total potential energy principle for the plate satisfiesthe assembly of the element equations.

The element stiffness and the geometric stiffness matricesare assembled. The corresponding eigenvalue problem canbe solved using any standard eigenvalue extraction proce-dures [22,24]

½½K0� � kb½K0g��ui

vi

wi

8><>:

9>=>; ¼ 0 ð14Þ

where

½K0� ¼ ½Kb� þ ½Ks�; �kb½K0g� ¼ ½Kg� ð15ÞThe product of kb and the initial guess value DT is the

critical buckling temperature Tcr, that is

T cr ¼ kbDT ð16Þ

3. Numerical result and discussion

There are many techniques to solve eigenvalue prob-lems. In this study the Newton Raphson method is appliedto obtain numerical solutions of the problem. For thermalbuckling due to a DT temperature change in the plate, theuniaxial or biaxial in-plane loads are developed along therectangular edges, while the crack edges are free.

The E-glass/epoxy, and boron/epoxy are considered ascomponents of hybrid plate and their thermo-elastic prop-erties are given in Table 1. Here, E1 and E2 are elastic mod-uli in 1 and 2 directions, respectively, m12 is Poisson’s ratioand a1 and a2 are thermal expansion coefficients of thematerials used in the solution. The effect of a12 is neglected.

Stacking sequence of hybrid composite plates have beentaken both symmetric and antisymmetric. Stackingsequences have been represented below. Boron/epoxy andglass/epoxy layers are named B and G, respectively.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.2 0.4 0.6 0.8

2c/L

T/T

o

Φ=0

Φ=30

Φ=45

Φ=60

Φ=90

Fig. 3. Effect of crack size and position on buckling temperature ofantisymmetrically stacked hybrid composite plate.

2968 A. Avci et al. / Composites Science and Technology 66 (2006) 2965–2970

The sequence of 4 Layers symmetric lay up of Glass/epoxy–boron/epoxy is 0�G/90�B/90�B/0�G.

The sequence of 4 Layers antisymmetric lay up of glass/epoxy–boron/epoxy is 0�G/90�B/0�G/90�B.

Each layer has 0.25 mm thickness and the length of oneedge of square plate is 100 mm. 2c/L ratio, represents thecrack size to length of one side of composite plate and /represents the crack inclination angle as shown in Fig. 1.

A wide range of boundary conditions can be accommo-dated, but only one kind of boundary conditions is chosenas defined below:

Two edges clamped and two edges are free

At x ¼ � L2

andL2

u ¼ w ¼ wy ¼ wx ¼ 0

Fig. 2 shows the meshed plate. Four edges of plate havedivided into ten parts disregarding the crack size.

The buckling behavior of antisymmetrically stackedhybrid composite plate is shown in Fig. 3. In this figure,T/T0 ratio is used instead of buckling temperature and2c/L ratio is used instead of crack length c. Here, T0 repre-sents the buckling temperature of hybrid plate withoutcrack. Thus graph is plotted by using dimensionless axes.It can be seen that when the crack inclination angle /= 90� and / = 60� T/T0 ratio decreases while crack lengthincreases. The case of / = 30� and / = 0� T/T0 ratioincreases while crack length increases. These results showthat inclination angle effected to buckling temperature. Ifinclination angle decreases, the buckling temperature orT/T0 ratio decreases and converges the 1 while crack lengthincreases. For larger values of crack length, the differencebetween temperatures for different inclination anglesincrease. These results show that buckling temperature iseffected by both crack length and inclination angle.

Fig. 2. Typical mesh.

The buckling behavior of symmetrically stacked hybridplate is shown in Fig. 4. Fig. 3 shows that, the bucklingtemperature is depending on crack length and crack incli-nation angle. But T/T0 ratio in symmetrically stackedplate is greater than antisymmetrically stacked one forfixed crack length and inclination angle. This result showsthat the buckling temperature and consequently the buck-ling resistance of symmetrically stacked hybrid plate isgreater than that of antisymmetrically stacked hybridplates.

The relationship between crack size and buckling tem-perature is shown in Figs. 5 and 6. Buckling temperatureis plotted with respect to crack size for various inclinationangels of antisymmetrically stacked E-Glass/epoxy platesin Fig. 5. The smallest T/T0 ratios are obtained for crackinclination angle of / = 90�. On the other hand, it is con-cluded that, the effect of cracks upon buckling tempera-ture become clear as the crack length increases. Thoughthis behavior is similar for all crack inclination angles,for small inclination angles, this behavior can be seenclearly.

0

0.5

1

1.5

2

2.5

0.2 0.4 0.6 0.8

2c/L

T/T

o

Φ=0

Φ=30

Φ=45

Φ=60

Φ=90

Fig. 4. Effect of crack size and position on buckling temperatures ofsymmetrically stacked hybrid plate.

0

0.2

0.4

0.6

0.8

1

1.2

0.2 0.4 0.6 0.82c/L

T/T

o

Φ=0

Φ=30

Φ=45

Φ=60

Φ=90

Fig. 6. Variation of buckling temperatures of symmetrically stacked E-glass/epoxy plate.

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

0.2 0.4 0.6 0.82c/L

T/T

o

Φ=0

Φ=30

Φ=45

Φ=60

Φ=90

Fig. 5. Variation of buckling temperatures of antisymmetrically stackedE-glass/epoxy plate.

A. Avci et al. / Composites Science and Technology 66 (2006) 2965–2970 2969

The first buckled mode shapes generated glass/epoxycross-ply laminated plates with inclined crack are shownin Figs. 7 and 8. It is found that critical temperature forcrack angle of 0�is 26,284 �C, for crack angle of 60� is

Fig. 7. Buckled mode shape of hybrid plate with:

Fig. 8. Buckled mode shape of hybrid plate: (a)

20,851 �C, for crack angle of 90� is 17,148 �C and criticaltemperature for plate without crack is 24,572 �C, respec-tively. The mode shapes presented in Figs. 7 and 8 showconsiderable skewing for the laminated plates.

(a) crack angle of 0�; (b) crack angle of 60�.

with crack angle of 90�; (b) without crack.

2970 A. Avci et al. / Composites Science and Technology 66 (2006) 2965–2970

4. Conclusions

Thermal buckling behaviors of cross-ply laminatedhybrid plates with inclined crack have been examined byemploying the first-order shear deformation theory andfinite element technique. Both symmetric and antisymmet-ric lay-up sequence are considered.

Because of absence of bending-extension coupling, sym-metric cross-ply E-glass/epoxy laminates does not yield thehighest buckling resistance as expected.

Effect of crack upon thermal buckling is minimum whilecrack inclination angle is 90�. As the crack length increases,this effect becomes clear.

Effect of cracks upon thermal buckling for hybrid lami-nated composite plate and E-glass/epoxy laminates aredifferent.

Effect of crack upon thermal buckling for antisymmetri-cally stacked hybrid laminates is negative while crackscause positive effect on symmetrically stacked hybrid platesand effect of cracks upon thermal buckling for antisymmet-rically stacked E-glass/epoxy laminates is positive whilecracks cause negative effect on symmetrically stacked E-glass/epoxy plates.

The buckling temperature is affected the larger cracklength more than the small crack length as shown in Figs.3–6. For small crack length the high temperature or highthermal stresses can be supported by the imperforated sec-tion of the plate. This result can be seen for every perfora-tion angles in Figs. 3–6. T/T0 rates converge the ‘‘1’’ forsmall perforations at all plates.

Another result is that buckling temperature of the platefor every perforation angel is to increase while crack lengthis increasing. This result is expected. Because the highertemperature needs for reaching the same stress level forbuckling when the perforation is bigger. If the crack lengthis great , that means that stressed cross section is fewer sothis cross section must be heated much more than when theimperforated section is higher.

References

[1] Sahin OS, Avci A, Kaya S. Thermal buckling of orthotropic plateswith angle crack. J Reinf Plast Compos 2004;23(16):1707–16.

[2] Avci A, Sahin OS, Uyaner M. Thermal buckling of hybrid laminatedcomposite plates with a hole. J Compos Struct 2005;68:247–54.

[3] Akbulut H, Sayman O. An investigation on buckling of laminatedplates with central square hole. JRPC 2001;20:1112–24.

[4] Chang JS, Shiao FJ. Thermal buckling analysis of isotropic andcomposite plates with hole. J Therm Stress 1990;13:315–32.

[5] Mathew TC, Singh G, Rao GV. Thermal buckling of cross-plycomposite laminates. Comp Struct 1992;42(2):281–7.

[6] Abramovich H. Thermal buckling of cross-ply laminates using a firstorder deformation theory. Comp Struct 1994;28:201–13.

[7] Murphy KD, Ferreira D. Thermal buckling of rectangular plates. IntJ Solids Struct 2000;38:3979–94.

[8] Huang NN, Tauchert TR. Thermal buckling of clamped symmetriclaminated plates. J Thin-Walled Struct 1992;13:259–73.

[9] Prabhu MR, Dhanaraj R. Thermal buckling of laminated compositeplates. Comp Struct 1994:1193–204.

[10] Chandrashekhara MR. Buckling of multilayered composite platesunder uniform temperature field. In: Birman V, Hui D. Thermaleffects on structures and materials. ASME Pub., vol. 203, AMD, vol.110; 1990. p. 29–33.

[11] Thangaratnam KR, Ramaohandran J. Thermal buckling of compos-ite laminated plates. Comp Struct 1989;32:1117–24.

[12] Chen LW, Lin PD, Chen LY. Thermal buckling behavior of thickcomposite laminated plates under non-uniform temperature distribu-tion. Comp Struct 1991;41:637–45.

[13] Liessa AW. A review of laminated composite plate buckling. ApplMech Rec 1987;40:575–91.

[14] Bednarczyk H, Rihter M. Buckling of plated due to self-equilibratedthermal stresses. J Therm Stress 1985;8:139–52.

[15] Chen W, Liu WH. Thermal buckling of antisymmetric angle-plylaminated plates-an analytical levy-type solution. J Therm Stresses1993;16:401–19.

[16] Sun LX, Hsu TR. Thermal buckling of laminated compositeplates with transverse shear deformation. Comp Struct 1990;36(5):883–9.

[17] Chockalingam S, Mathew TC, Singh G, Rao GV. Critical temper-atures of hybrid laminates using finite elements. Comp Struct1992;43(5):995–8.

[18] Thangaratnam KR, Ramachandran J. Thermal buckling of compos-ite laminated plates. Comp Struct 1989;32:1117–24.

[19] Chen LW, Lin PD, Chen LY. Thermal buckling behavior of thickcomposite laminated plates under non-uniform temperature distribu-tion. Comp Struct 1991;41:637–45.

[20] Jones RM. Mechanics of composite materials. Tokyo: McGraw-HillKogagusha Ltd.; 1975.

[21] Lin CH, Kuo CS. Buckling of laminated plates with holes. J ComposMater 1989;23:536–53.

[22] Bathe KJ. Finite element procedures in engineering analysis. Engle-wood Cliffs (NJ): Prentice-Hall, Inc.; 1982.

[23] Whitney JM. Shear correction factors for orthotropic laminates understatic load. J Appl Mech 1973(march):302–4.

[24] Chen LW, Lin PD, Chen LY. Thermal buckling behavior of thickcomposite laminated plates under non-uniform temperature distribu-tion. Comp Struct 1991;41:637–45.

[25] Wang X, Lu G. Local buckling of composite laminar plates withvarious delaminated shapes. Thin-Walled Struct 2003;41:493–506.

[26] Wang X, Lu G, Xiao DG. Non-linear thermal buckling for localdelamination near the surface of laminated cylindrical shell. Int JMech Sci 2002;44:947–65.