Thermal postbuckling analysis of temperature dependent delaminated composite plates

Thin-Walled Structures 97 (2015) 296–307

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Thermal postbuckling analysis of temperature dependent delaminated composite plates S.F. Nikrad, H. Asadi n Thermoelasticity Center of Excellence, Department of Mechanical Engineering, Amirkabir University of Technology, Tehran 15875, Iran

art ic l e i nf o

a b s t r a c t

Article history: Received 23 September 2015 Received in revised form 25 September 2015 Accepted 25 September 2015

This paper deals with the nonlinear thermal stability of composite plate with embedded and throughthe-width delaminations under uniform temperature rise. The formulation is established within the framework of the higher order shear deformation theory by taking into account the von Karman geometrical nonlinearity. The thermomechanical properties of the laminates are assumed to be temperature-dependent. The nonlinear equilibrium equations derived by the minimum total potential energy principle, are solved using the Rayleigh–Ritz method along with the Newton–Raphson iterative procedure. For modeling the embedded and through-the-width delaminations, the plate is divided into a number of smaller regions. The proposed model is capable of analyzing both local buckling of the delaminated base laminate and sublaminate as well as the global buckling of the plate. Numerical results are presented to provide an insight into effects of delamination type, size of delamination and boundary condition on the critical buckling temperature difference, buckling mode and postbuckling behavior of the composite plate. It is found that presence of delamination leads to substantial reduction in the load carrying capacity of the composite plate. Furthermore, the results reveal that the buckling mode could be changed depending on the delamination area. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Thermal postbuckling Delaminated composite plates Embedded delamination Through-the-width delamination Local and global buckling

1. Introduction Fiber reinforced composite structures are extensively used as a structural member in various industries including aeronautics, aerospace and marine industries where a high ratio of strength/stiffness to weight are required. Apart from being lightweight, composite materials offer high specific strength, high-energy absorption capacity. Despite these merits, multilayered composite structures bring complication in analysis and design due to wide range of defects that may lead to drastic decrease of the stiffness and strength. Delamination is one of the most common failure modes in layered composite materials, which may result from manufacturing imperfections, layups, edge effects and various loadings. In particular, when composite structures are subjected to compressive forces, delamination becomes a limitation in the design process. Delamination has been a subject of foremost concern in engineering applications of composite structures due to the associated issues of structural stability, stiffness degradation, fracture and reduction in load carrying capacity. A brief literature review on the stability responses of delaminated composite structures is provided hereinafter. n

Corresponding author. E-mail addresses: [email protected] (S.F. Nikrad), [email protected] (H. Asadi). http://dx.doi.org/10.1016/j.tws.2015.09.027 0263-8231/& 2015 Elsevier Ltd. All rights reserved.

A model for the buckling analysis of delaminated composite beam and plate was introduced by Chai et al. [1]. It is found that the growth of the delamination could be stable, unstable, or an unstable growth followed by a stable growth. Bottega et al. [2] studied the stability behavior of circular plates with a circular delamination, which is situated in the center of the plate under assumption of asymmetric deformation. The buckling response of a thin elliptical delamination was presented by Shivakumar at al. [3] by means of the Rayleigh–Ritz method along with finite element method. Anastasidis et al. [4] analyzed the buckling and postbuckling behaviors of delaminated composite plates by simulating the contact of the delamination regions through the application of distributed springs of constant stiffness. Davidson [5] conducted theoretical and experimental studies to determine the strain and force at which delamination buckling occurred for a laminated composite with an elliptical shape delamination. The stability characteristic of delaminated composite beam-plate was investigated by Piao [6] using a consistent shear deformation theory. Suemasu [7,8] studied the buckling behavior of composite plate with a center delamination by employing classical plate theory (CPT) and first order shear deformation theory (FSDT). Adan et al. [9] conducted a study to examine the buckling load of laminated composite beam with multiple through-the-width delaminations. The buckling characteristic of the laminated composite plates with two centrally through-the-width delaminations

S.F. Nikrad, H. Asadi / Thin-Walled Structures 97 (2015) 296–307

was investigated by Shu [10] on the basis of classical plate theory. It was shown that the delaminated segments could buckle either together in a constrained mode, independently in a free mode, or in a mixed partially constrained mode, depending on the thickness of the delaminated layers. Wang and his co-authors [11,12] introduced spring simulation technique to determine the local buckling load and the strain energy release rate of laminated composite beams and plates with delamination. They found that symmetrically loaded delaminated plates results in a mode I fracture. The buckling response of a thin plate, which is bonded laterally to a thick plate, was presented by Shahwan et al. [13] using nonlinear spring distribution between two plates. The buckling characteristics of debonded sandwich panel under compression was obtained by Sleight et al. [14] using spring distribution between substrate and face sheet. Results of this work were obtained by utilizing finite element method (FEM) and Rayleigh– Ritz method, separately. Jane et al. [15] analyzed bifurcation buckling behavior of laminated composite plates by means of Rayleigh–Ritz method in conjunction with von Karman's nonlinear strain–displacement relations. In this study, the delaminated sublaminate was assumed to be thin and the local buckling of the plate was studied. Andrews et al. [16] introduced a technique by employing classical plate theory to investigate the elastic interaction of the multiple through-the-width delaminations in laminated composite plates under static out-of-plane loading. A discontinuity in the energy release rate is found when the delamination growth is considered. This discontinuity leads to the instantaneous shielding or amplification of the energy release rate, which may result in local snap-back and snap-through instabilities. Ovesy et al. [17] utilized spring simulated model to determine buckling load of composite plates with multiple through-the-width delaminations. In this study, the sublaminates were modeled as plates supported by an elastic foundation and the laminates contained multiple delaminations located through the thickness and length of the plate were analyzed by solving corresponding differential equations. Ovesy and his co-workers [18–20] investigated buckling and postbuckling behaviors of composite plates containing different types of delamination with arbitrary shape by using different plate theories. In their works, the formulations were developed based on the Rayleigh–Ritz technique by implementation of the polynomial series. Recently, a novel layerwise theory on the basis of FSDT was introduced by Ovesy and his colleagues [21,22] to determine the buckling and postbuckling behavior of delaminated composite plates containing multiple through-the-width delaminations. Results of

297

their studies indicate that postbuckling mode and the corresponding load carrying capacity strongly depend on the size of delaminations. Additional investigations on thermal stability of composite structures were also reported in the literature [23–33]. Despite many researches on the stability of delaminated composites under mechanical loads, there is no work supplied on the thermal instability of composite plates with embedded and through-the-width delaminations. Accordingly, further comprehensive studies are still indispensable for understanding the thermal postbuckling behavior of delaminated composite structures in severe environmental conditions. The present investigation aims to study the nonlinear thermal stability of composite plates containing through-the-width and embedded delaminations subjected to a uniform thermal loading. The thermomechanical properties of the plate are assumed to be temperature-dependent. Establishment of nonlinear equilibrium equations is performed within the framework of the higher order shear deformation theory along with the von Karman type of geometrical nonlinearity. The analytical method is accomplished based on the Rayleigh–Ritz approximate technique by the implementation of the complete polynomial series. The method is capable of handling both local buckling of the delaminated sublaminate and global buckling of the whole plate.

2. Governing equations Consider a rectangular composite plate with length L , width b , and total thickness h. A Cartesian coordinate system (x, y, z ) is defined, where x and y are located in the midplane and z defines the upward normal direction. The origin of the reference coordinate system is at the center of the midplane as illustrated in Fig. 1A. The displacement field of the composite plate is written based on the third order shear deformation theory (TSDT) as [34] u (x, y, z ) = u 0 (x, y ) + zϕx (x, y ) −

4z 3 ⎛⎜ ∂w0 ⎞⎟ ϕx (x, y ) + 2 ∂x ⎠ 3h ⎝

v (x, y, z ) = v0 (x, y ) + zϕy (x, y ) −

4z 3 ⎛ ∂w0 ⎞ ⎜ ϕy (x, y ) + ⎟ 2 ∂y ⎠ 3h ⎝

w (x, y, z ) = w0 (x, y )

(1)

where u0 , v0 and w0 represent the displacement components at midplane of the plate; ϕx and ϕy are the middle surface rotations about y and x axes, respectively. The von Karman nonlinear strain–displacement relation is adopted for small strain and moderate rotation deformation, [34]

Fig. 1. (a) A typical plate with an embedded delamination, and (b) the delamination front [20].

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{ε } + z{ε } + z {ε } { γ¯} = { γ } + z { γ } { ε¯} =

0 0

1

2

3

3

2

(2)

where ε¯ij and γ¯ij are components of the strain tensor which can be written as

⎧ ε3 ⎫ ⎧ ε1 ⎫ ⎧ 0⎫ xx xx ⎧ ε¯xx ⎫ ⎪ εxx ⎪ ⎪ ⎪ ⎪ ⎪ 3⎪ ⎪ ⎪ ⎪ ⎪ 0⎪ ⎪ 1⎪ ⎬ ⎨ ε¯yy ⎬ = ⎨ εyy ⎬ + z ⎨ εyy ⎬ + z 3 ⎨ εyy ⎪ 3⎪ ⎪ 1⎪ ⎪ γ¯ ⎪ ⎪ 0 ⎪ ⎩ xy ⎭ ⎪ γ ⎪ ⎪ ⎪ ⎭ ⎩ γxy ⎪ ⎭ ⎩ γxy ⎪ ⎩ xy ⎭

⎧ Mxx ⎫ ⎪ ⎪ ⎨ Myy ⎬ , ⎪ ⎪ ⎩ Mxy ⎭

(3)

⎛ ⎧ Q yz ⎫ ⎧ Ryz ⎫ ⎞ ⎜⎜ ⎨ ⎬, ⎨ ⎬ ⎟⎟ = ⎝ ⎩ Q xz ⎭ ⎩ Rxz ⎭ ⎠

⎫ ⎪ ⎪ ⎪ 2 ⎪ ∂v 0 1 ⎛ ∂w 0 ⎞ ⎬, + ⎜ ⎟ ⎪ ∂y 2 ⎝ ∂y ⎠ ⎪ ∂w 0 ∂w 0 ⎪ ∂v 0 ∂u 0 + + ∂x ∂y ⎪ ∂x ∂y ⎭ 2

⎧ ⎪ ⎧ 1 ⎫ ⎪ ε xx ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎨ εyy ⎬ = ⎨ ⎪ ⎪ ⎪ 1 ⎪ ⎩ γxy ⎪ ⎭ ⎪ ⎪ ⎪ ⎩

⎫ ⎧ ⎞ ⎛ 2 ⎪ ⎪ − 4 ⎜ ∂ϕx + ∂ w 0 ⎟ 2 ⎟ ⎪ ⎪ 3h2 ⎜⎝ ∂x ∂x ⎠ ⎧ 3⎫ ⎪ ⎪ ε ⎪ xx ⎪ ⎪ ⎞ ⎪ ⎛ 2 ⎪ ⎪ 3 ⎪ ⎪ − 4 ∂ϕy ∂ w0 ⎟ ⎜⎜ ⎬, ⎨ εyy ⎬ = ⎨ + ⎟ 2 ⎪ ⎪ ⎪ ⎪ 3h2 ⎝ ∂y ∂y ⎠ 3 ⎪ γxy ⎪ ⎪ ⎩ ⎭ ⎪ ⎞ 2 ∂ϕy ⎪ − 4 ⎛ ∂ϕx ∂ w0 ⎟ ⎪ ⎜⎜ + +2 ⎪ ⎪ ⎟ 2 ⎪ ∂x∂y ⎠ ⎪ ∂x ⎭ ⎩ 3h ⎝ ∂y ⎧ 2⎫ ⎧ 0⎫ ⎪ γ yz ⎪ ⎪ ⎪ γ −4 yz ⎨ ⎬= ⎨ ⎬ 0 2 ⎪ h2 ⎪ ⎩ γxz ⎪ ⎭ ⎩ γxz ⎪ ⎭

⎫ ⎪ ∂x ⎪ ⎪ ∂ϕy ⎪ ⎬, ∂y ⎪ ∂ϕy ⎪ ∂ϕx ⎪ + ∂y ∂x ⎪ ⎭ ∂ϕx

Q¯ 12 Q¯ 22 0 0 Q¯ 62

0

0

0 0 ¯ ¯ Q 44 Q 45 Q¯ 54 Q¯ 55 0

0

⎧ σ¯xx ⎫ ⎪ ⎪ ⎩ τ¯xy ⎭

⎧ τ¯ ⎫

∫−h /2 ⎨⎩ τ¯xzyz ⎬⎭ ( 1, z2) dz

(6)

{ ε0} − { εth}⎫⎪⎪ ⎪ ⎬ { ε1} ⎪ ⎪ 3 ⎪ {ε } ⎭ ⎧ 0 ⎫ γ ⎤ { } ⎪ ⎪ [Ds] ⎬ ⎥⎨ [Fs] ⎦ ⎪ γ 2 ⎪ ⎩ { }⎭ h /2

( Aij , Bij, Dij, E ij, Fij, Hij ) = ∫−h /2 Q¯ ij ( 1, z, z 2, z 3, z 4, z 6) dz (i, j = 1, 2, 6) h /2

( Asij, Dsij, Fsij ) = ∫−h /2 Q¯ ij ( 1, z 2, z 4) dz ⎧ ∂w 0 ⎫ ⎧ 0 ⎫ ⎪ ϕy + ⎪ ⎪ γ yz ⎪ ⎪ ∂y ⎪ ⎨ ⎬=⎨ ⎬, 0 ⎪ ⎪ ∂ w ⎪ 0⎪ ⎩ γxz ⎭ ⎪ ϕ + ⎪ ⎩ x ∂x ⎭

(i, j = 4, 5)

(8)

In addition, {εth} is thermal strain tensor and can be written as ⎧ ε xx ⎫th ⎪ ⎪ ⎨ ε yy ⎬ = ⎪ γ xy ⎭ ⎪ ⎩

(4)

Q¯ 16 ⎤ ⎛ ⎧ ε¯xx ⎫ ⎧ αxx ⎫ ⎞ ⎥⎜ ⎪ ⎪ ⎪ αyy ⎪ ⎟ Q¯ 26 ⎥ ⎜ ⎪ ε¯yy ⎪ ⎪ ⎪⎟ ⎪ ⎥⎜ ⎪ 0 ⎥ ⎨ γ¯yz ⎬ − ( T − T0 ) ⎨ 0 ⎬ ⎟ ⎜⎪ ⎪ ⎪ 0 ⎪⎟ 0 ⎥ ⎜ ⎪ γ¯xz ⎪ ⎪ ⎪⎟ ⎥ ⎜ ⎪ γ¯ ⎪ ⎩ 2αxy ⎭ ⎟⎠ Q¯ 66 ⎦ ⎝ ⎩ xy ⎭

(7)

The stiffness matrices in Eq. (7) are defined as

Considering T as the temperature distribution within the plate and T0 as the reference temperature, the transformed stress–strain relations of an orthotropic lamina in a plane state of stress are

⎡ ¯ ⎧ σ¯xx ⎫ ⎢ Q 11 ⎪ σ¯yy ⎪ ⎢ Q¯ ⎪ ⎪ ⎢ 21 ⎨ τ¯yz ⎬ = ⎢ 0 ⎪ τ¯ ⎪ ⎪ xz ⎪ ⎢ 0 ⎩ τ¯xy ⎭ ⎢ ¯ ⎣ Q 61

h /2

h /2

∫−h /2 ⎨⎪ σ¯yy ⎬⎪ ( 1, z, z 3) dz

⎧ ⎧ {N} ⎫ ⎡ [A] [B] [E] ⎤ ⎪ ⎪ ⎪ ⎢ ⎪ ⎥⎪ ⎨ {M}⎬ = ⎢ [B] [D] [F ] ⎥ ⎨ ⎪ ⎪ ⎢ ⎪ ⎥ ⎩ {P} ⎭ ⎣ [E] [F ] [H ]⎦ ⎪ ⎪ ⎩ ⎧ {Q }⎫ ⎡ [As] ⎨ ⎬=⎢ ⎩ {R} ⎭ ⎣ [Ds]

where ∂u 0 1 ⎛ ∂w 0 ⎟⎞ + ⎜ ∂x 2 ⎝ ∂x ⎠

⎧ Pxx ⎫ ⎞ ⎪ ⎪⎟ ⎨ Pyy ⎬ ⎟ = ⎪ ⎪⎟ ⎩ Pxy ⎭ ⎠

The stress resultants can be written in terms of the midplane displacements by using Eqs. (2)–(6)

0⎫ 2 ⎧ ⎧ γ¯yz ⎫ ⎧ ⎪ γyz ⎪ ⎪ γyz ⎫ ⎪ ⎨ ⎬ = ⎨ ⎬ + z2 ⎨ ⎬ 2⎪ 0⎪ ⎪ ⎩ γ¯xz ⎭ ⎪ ⎩ γxz ⎭ ⎩ γxz ⎭

⎧ ⎪ ⎧ 0⎫ ⎪ ε xx ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎨ εyy ⎬ = ⎨ ⎪ ⎪ ⎪ 0 ⎪ ⎩ γxy ⎪ ⎭ ⎪ ⎪ ⎪ ⎩

⎛ ⎧ Nxx ⎫ ⎜⎪ ⎪ ⎜ ⎨ Nyy ⎬ , ⎪ ⎜⎪ ⎝ ⎩ Nxy ⎭

where αij are the thermal expansion coefficients and Q¯ ij are the components of the transformed plane-stress reduced stiffness matrix. Based on the TSDT, the stress resultants Nij , Mij , Pij , Q ij and Rij are defined as

hk

( ) ( ) ( ) ( )

( ) ⎤⎥⎥ ⎥ α cos ( θ ) (T − T ) ⎥{ α } ⎥ − 2 cos ( θ ) sin ( θ )⎥ ⎦ sin2 θ k 2

11 22

k

k

k

0

(9)

where θ k is the ply angle in the kth lamina. The strain energy could be obtained by using Eqs. (3) and (5) as follows U=

=

(5)

⎡ cos2 θ k ⎢ ⎢ ⎢ sin2 θ k ∑∫ ⎢ h k =1 k −1 ⎢ k k ⎢⎣ 2 cos θ sin θ N

∫ ∫ ∫ 21 σ¯ T ε¯dV

b

L

2

2

1 ∫2∫2 2 −b −L

⎛ T T T ⎜ ε0 [A] ε0 + 2 ε0 [B] ε1 + ε1 [D] ε1 ⎜ T T T ⎜ 0 3 + 2 ε1 [F ] ε3 + ε 3 [H ] ε 3 ⎜ + 2 ε [E ] ε ⎜ T T T ⎜ 0 0 + 2 γ 0 [Ds] γ 2 + γ 2 [Fs] γ 2 ⎜ + γ [As] γ ⎜ ⎜ − 2 {ε}th [A] ε0 T − 2 {ε}th [B] ε1 T − 2 {ε}th [E] ε3 T ⎜ ⎜⎜ T ⎝ + {ε}th [A] {ε}th

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

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ dxdy ⎟ ⎟ ⎟ ⎟ ⎟⎟ ⎠ (10)

where L and b are length and width of the composite plate, respectively. The total potential energy of the plate is sum of the strain energies of each region

Fig. 2. A typical composite plate with a through-the-wide delamination.

S.F. Nikrad, H. Asadi / Thin-Walled Structures 97 (2015) 296–307

M

U total =

∑ U (j )

(11)

j=1

where U (j ) is the strain energy in the jth region (j = 1, 2, … M ). It is worth mentioning that since the plate is only subjected to thermal loading, the total potential energy Π is equal to the strain energy

(12)

Π = U total

rotations for the ith region. The current theoretical development offers no limitation in terms of application of different boundary conditions (i.e. clamped or simply-supported) on the edges of the plate. However, for the sake of brevity, the plate is assumed to be simply supported and immovable in the in-plane direction (SSSS). Mathematical expressions for this class of edge supports could be expressed as w (1 ) | u 0(1) | v0(1) |

3. Modeling of the embedded delamination In this section, it is assumed that embedded type of delamination exists between the layers of composite plate. The embedded delamination could take place at arbitrary locations and at arbitrary layers of the plate. To analyze a delaminated plate, shown in Fig. 1a, we introduce three domains with distinct displacement fields. The delamination divides the total thickness of the composite plate into a thicker part, called base laminate, and a thinner part, called sublaminate. It is worth mentioning that continuity requirements must be satisfied at the boundaries between the domains and between different plates defined in the thickness direction [20]. The necessary continuity conditions at the boundaries for displacement and rotation are written as [19] w (1) |x = L1 = w (2) |x = L1 = w (3) |x = L1,

∂x

| x = L2 =

∂w (2) ∂x

| x = L2 =

∂w (3) ∂x

y= b 2

=0

(14)

2

M1

w (1 ) =

w (2 ) =

m=0 n =0 M1 N1

L ⎞⎛ L ⎞⎛ b ⎞⎛ b⎞ (1 ) ⎛ ⎜ x + ⎟ ⎜ x − ⎟ ⎜ y + ⎟ ⎜ y − ⎟ x my n W mn ⎝ 2 ⎠⎝ 2 ⎠⎝ 2 ⎠⎝ 2⎠

L ⎞⎛ L ⎞⎛ b ⎞⎛ b⎞ (1 ) ⎛ ⎜ x + ⎟ ⎜ x − ⎟ ⎜ y + ⎟ ⎜ y − ⎟ x my n W mn ⎠⎝ ⎠⎝ ⎝ ⎠⎝ 2 2 2 2⎠ m=0 n =0

∑ ∑

M2

+

N1

∑ ∑

N2

∑ ∑

(2 ) W mn ( x − L1)

m=0 n =0 M1 N1

∑ ∑ ∑ ∑

2

2

2

( x − L2 ) ( y − b1) ( y − b 2 )

2 m n

x y

L ⎞⎛ L ⎞⎛ b ⎞⎛ b⎞ (1 ) ⎛ ⎜ x + ⎟ ⎜ x − ⎟ ⎜ y + ⎟ ⎜ y − ⎟ x my n W mn ⎝ 2 ⎠⎝ 2 ⎠⎝ 2 ⎠⎝ 2⎠

(3 ) W mn ( x − L1)

2

2

2

( x − L2 ) ( y − b1) ( y − b 2 )

2 m n

x y

m=0 n =0

(15)

In addition, the admissible functions for in-plane displacements are [19]

| x = L2

u 0(1) =

P1

Q1

∑ ∑ p =0 q=0

u (2) |x = L1 = u (1) |x = L1, z = h2 , u (2) |x = L2 = u (1) |x = L2, z = h2

u 0(2) =

(1 ) (2 ) (1 ) y = b1 = u | y = b1,z = h2 , u | y = b 2 = u | y = b 2, z = h2

P1

Q1

∑ ∑ p =0 q=0

L ⎞⎛ L⎞ (1 ) ⎛ ⎜ x + ⎟ ⎜ x − ⎟ x py q U pq ⎝ 2 ⎠⎝ 2⎠ L ⎞⎛ L⎞ (1 ) ⎛ ⎜ x + ⎟ ⎜ x − ⎟ x py q U pq ⎝ 2 ⎠⎝ 2⎠

P2 Q 2

u (3) |x = L1 = u (1) |x = L1, z = h3 , u (3) |x = L2 = u (1) |x = L2, z = h3

∑ ∑

+

u (3) | y = b1 = u (1) | y = b1,z = h3 , u (3) | y = b 2 = u (1) | y = b 2, z = h3

(2 ) U pq ( x − L1)( x − L2 )( y − b1)( y − b 2 ) x py q + h2 ϕx(1)

p =0 q=0

v (2) |x = L1 = v (1) |x = L1, z = h2 , v (2) |x = L2 = v (1) |x = L2, z = h2

−

v (2) | y = b1 = v (1) | y = b1,z = h2 , v (2) | y = b 2 = v (1) | y = b 2, z = h2 v (3) |x = L1 = v (1) |x = L1, z = h3 , v (3) |x = L2 = v (1) |x = L2, z = h3

u 0(3) =

(1 ) (3 ) (1 ) y = b1 = v | y = b1,z = h3 , v | y = b 2 = v | y = b 2, z = h3

4h23 ⎛ (1) ∂w (1) ⎞ ⎜ ϕx + ⎟ ∂x ⎠ 3h12 ⎝ P1

Q1

∑ ∑ p =0 q=0

L ⎞⎛ L⎞ (1 ) ⎛ ⎜ x + ⎟ ⎜ x − ⎟ x py q U pq ⎝ 2 ⎠⎝ 2⎠

P3 Q 3

⎛ ∂u (1) ⎞ ⎛ ∂u (1) ⎞ ⎟ |x = L1, z = h2 , ϕx(2) |x = L2 = ⎜ ⎟ |x = L2, z = h2 ϕx(2) |x = L1 = ⎜ ⎝ ∂z ⎠ ⎝ ∂z ⎠

+

⎛ ∂u (1) ⎞ ⎛ ∂u (1) ⎞ ⎟ | y = b 2, z = h2 ⎟ | y = b1, z = h2 , ϕx(2) | y = b 2 = ⎜ ϕx(2) | y = b1 = ⎜ ⎝ ∂z ⎠ ⎝ ∂z ⎠

−

∑ ∑

(3 ) U pq ( x − L1)( x − L2 )( y − b1)( y − b 2 ) x py q + h 3 ϕx(1)

p =0 q=0

⎛ ∂u (1) ⎞ ⎛ ∂u (1) ⎞ ⎟ |x = L , z = h 3 ⎟ |x = L , z = h3 , ϕx(3) |x = L2 = ⎜ ϕx(3) |x = L1 = ⎜ 1 2 ⎝ ∂z ⎠ ⎝ ∂z ⎠

v0(1) =

⎛ ∂u (1) ⎞ ⎛ ∂u (1) ⎞ ⎟ | y = b1, z = h3 , ϕx(3) | y = b 2 = ⎜ ⎟ | y = b 2, z = h3 ϕx(3) | y = b1 = ⎜ ⎝ ∂z ⎠ ⎝ ∂z ⎠

v0(2) =

⎛ ∂v (1) ⎞ ⎛ ∂v (1) ⎞ ⎟ |x = L , z = h2 , ϕy(2) |x = L2 = ⎜ ⎟ |x = L , z = h2 ϕy(2) |x = L1 = ⎜ 1 2 ⎝ ∂z ⎠ ⎝ ∂z ⎠

⎛ ∂v (1) ⎞ ⎛ ∂v (1) ⎞ ⎟ |x = L , z = h3 , ϕy(3) |x = L2 = ⎜ ⎟ |x = L , z = h 3 ϕy(3) |x = L1 = ⎜ 1 2 ⎝ ∂z ⎠ ⎝ ∂z ⎠

(13)

where h2 and h3 are the distance between the mid-planes of regions 2 and 3 to the mid-plane of the whole plate which are graphically shown in Fig. 1b. Moreover, the displacement field u(i), v (i), w (i), ϕx(i) and ϕy(i) are the corresponding displacements and

4h33 ⎛ (1) ∂w (1) ⎞ ⎟ ⎜ ϕx + ∂x ⎠ 3h12 ⎝

S1

R1

⎛

b ⎞⎛ b⎞ ⎟ ⎜ y − ⎟ x syr 2 ⎠⎝ 2⎠

⎛ ⎝

b ⎞⎛ b⎞ ⎟ ⎜ y − ⎟ x syr 2 ⎠⎝ 2⎠

∑ ∑ V sr(1) ⎜⎝ y +

s =0 r =0 S1 R1

∑ ∑ V sr(1) ⎜ y +

s =0 r =0 S2 R 2

∑ ∑ V sr(2) x − L1 x − L2 y − b1 s =0 r =0 4h23 ⎛ (1) ∂w (1) ⎞ ⎜ ϕy + ⎟ − ∂y ⎠ 3h12 ⎝ S1 R1 b ⎞⎛ b⎞ (1 ) ⎛ ⎜ y + ⎟ ⎜ y − ⎟ x syr v0(3) = ∑ ∑ V sr ⎝ 2 ⎠⎝ 2⎠ s =0 r =0 S3 R 3 (3 ) + ∑ ∑ V sr x − L1 x − L2 y − b1 s =0 r =0 4h33 ⎛ (1) ∂w (1) ⎞ ⎜ ϕy + ⎟ − ∂y ⎠ 3h12 ⎝ +

⎛ ∂v (1) ⎞ ⎛ ∂v (1) ⎞ ⎟ | y = b1, z = h2 , ϕy(2) | y = b 2 = ⎜ ⎟ | y = b 2, z = h2 ϕy(2) | y = b1 = ⎜ ⎝ ∂z ⎠ ⎝ ∂z ⎠

⎛ ∂v (1) ⎞ ⎛ ∂v (1) ⎞ ⎟ | y = b1, z = h3 , ϕy(3) | y = b 2 = ⎜ ⎟ | y = b 2, z = h3 ϕy(3) | y = b1 = ⎜ ⎝ ∂z ⎠ ⎝ ∂z ⎠

= w (1 ) |

The functions describing the displacement fields for the divided laminates should satisfy all of the boundary and continuity conditions. According to the continuity and boundary conditions specified in Eqs. (13) and (14), the displacement fields for the assumed domains are taken to be polynomials. The admissible functions for out-of-plane displacements are [19]

+

∂w (1) ∂w (2) ∂w (3) | y =b2 = | y =b2 = | y =b2 ∂y ∂y ∂y

v (3 ) |

y =− b 2

2

m=0 n =0 M3 N3

∂w (3) ∂w (2) ∂w (1) | y = b1 | y = b1 = | y = b1 = ∂y ∂y ∂y

u (2 ) |

w (1 ) |

= 0,

= v0(1) | b = 0 y=

y =− b 2

w (3 ) =

∂w (1) ∂w (2) ∂w (3) |x = L1 = |x = L1 = |x = L1 ∂x ∂x ∂x

x= L 2

= u 0(1) | L = 0, x=

x =− L 2

w (1 ) | y = b 2 = w (2 ) | y = b 2 = w (3 ) | y = b 2

∂w (1)

= w (1 ) |

x =− L 2

w (1 ) | x = L 2 = w (2 ) | x = L 2 = w (3 ) | x = L 2

w (1) | y = b1 = w (2) | y = b1 = w (3) | y = b1,

299

(

)(

)(

)( y − b 2 ) x syr

+ h2 ϕy(1)

(

)(

)(

)( y − b 2 ) x syr

+ h 3 ϕy(1)

Besides, the assumed rotational functions are [19]

(16)

300

S.F. Nikrad, H. Asadi / Thin-Walled Structures 97 (2015) 296–307

ϕx(1) =

J1

K1

∑ ∑ j =0 k =0

ϕx(2) =

J1

K1

∑ ∑ j =0 k =0

− ϕx(3) =

J1

K1

j =0 k =0

ϕy(1) =

2

1) j k xy + Φ (Xjk

J2

K2

∑ ∑ j =0 k =0

1) j k Φ (Xjk xy +

2) Φ (Xjk ( x − L1)( x − L2 )( y − b1)( y − b 2 ) x jy k

ϕx(1) |x =− L = ϕx(4) |x = L = 0, 2

J3

K3

∑ ∑ j =0 k =0

w (1) |x =− L = w (4) |x = L = 0, 2

u0(1) |x =− L = u0(4) |x = L = 0

(19)

u0(1) |x =− L = u0(4) |x = L = 0

(20)

2

2

2

2

2

T1

(1 ) i t ∑ ∑ ΦYit xy

i =0 t =0 I1 T1

I2

Wherein the out-of-plane displacement functions for SFSF boundary condition represent as [19]

T2

(1 ) i t (2 ) ∑ ∑ ΦYit x y + ∑ ∑ ΦYit x − L1 i =0 t =0 i =0 t =0 2 4h2 ⎛ (1) ∂w (1) ⎞ ⎜ ϕy + ⎟ − ∂y ⎠ h12 ⎝ I1 T1 I3 T 3 (1 ) i t (3 ) ϕy(3) = ∑ ∑ ΦYit x y + ∑ ∑ ΦYit x − L1 i =0 t =0 i =0 t =0 2 4h3 ⎛ (1) ∂w (1) ⎞ ⎟ ⎜ ϕy + − ∂y ⎠ h12 ⎝

ϕy(2) =

2

SFSF:

3) Φ (Xjk ( x − L1)( x − L2 )( y − b1)( y − b 2 ) x jy k

4h32 ⎛ (1) ∂w (1) ⎞ ⎜ ϕx + ⎟ ∂x ⎠ h12 ⎝

I1

2

∂w (1) ∂w (4) |x =− L = | L = 0 ∂x ∂x x =− 2 2

4h22 ⎛ (1) ∂w (1) ⎞ ⎟ ⎜ ϕx + ∂x ⎠ h12 ⎝

∑ ∑ −

w (1) |x =− L = w (4) |x = L = 0

1) j k xy Φ (Xjk

(

(

)( x − L2 )( y − b1)( y − b 2 ) xiyt M1

N1

m= 0 n= 0 M1 N1

)( x − L2 )( y − b1)( y − b 2 ) xiyt

⎛

L ⎞⎛ L⎞ ⎟ ⎜ x − ⎟ x m yn 2 ⎠⎝ 2⎠

⎛

L ⎞⎛ L⎞ ⎟ ⎜ x − ⎟ x m yn 2 ⎠⎝ 2⎠

(1) ⎜ x+ ∑ ∑ Wmn ⎝

w (1) =

(1) ⎜ x+ ∑ ∑ Wmn ⎝

w (2) =

m= 0 n= 0 M2

(17)

N2

(2) ∑ ∑ Wmn ( x − L1)2 ( x − L2 )2xmyn

+

m= 0 n= 0

It is noteworthy that in each function, the second series is defined to satisfy the continuity equations.

M1

N1

⎛

(1) ⎜ ∑ ∑ Wmn x+ ⎝

w (3) =

m= 0 n= 0 M3

N3

(3) ∑ ∑ Wmn ( x − L1)2 ( x − L2 )2xmyn

+

m= 0 n= 0

4. Modeling of the through-the-width delamination

M1

In this section, it is assumed that through-the-width type of delamination with arbitrary size exists between layers of the composite plate, which is located anywhere in the plate. In the case of through-the-width delamination, the composite plate is divided into four different regions as shown in Fig. 2. In a procedure similar to that followed in the case of the embedded delamination, in the case of a plate with a through-the-width delamination, the requirement of the continuity must be satisfied as follows [19]

N1

⎛

(1) ⎜ ∑ ∑ Wmn x+ ⎝

w (4) =

m= 0 n= 0

P1

Q1

⎛

L ⎞⎛ L⎞ ⎟ ⎜ x − ⎟ x py q 2 ⎠⎝ 2⎠

⎛

L ⎞⎛ L⎞ ⎟ ⎜ x − ⎟ x py q ⎠ ⎝ 2 2⎠

(1) ⎜x + ∑ ∑ Upq ⎝

u0(1) =

p= 0 q= 0 P1

Q1

(1) ⎜x + ∑ ∑ Upq ⎝

u0(2) =

p= 0 q= 0

x = L2

=

w (2 ) |

x = L2

=

w (3 ) |

P2

+

x = L2

Q2

(2) ∑ ∑ Upq ( x − L1)( x − L2 ) x pyq

⎞⎫ ⎛ L − x ⎞ ⎧ (1) 4h22 ⎛ (1) ∂w (1) + h2 ⎜ 2 |x = L1 ⎟ ⎬ ⎟ ⎨ ϕx |x = L1 − ⎜ ϕx |x = L1 + 2 ⎝ L2 − L1 ⎠ ⎩ ⎠⎭ ∂x 3h1 ⎝ ⎞⎫ ⎛ L − x ⎞ ⎧ (4) 4h22 ⎛ (4) ∂w (4) − h2 ⎜ 1 |x = L2 ⎟ ⎬ ⎟ ⎨ ϕx |x = L2 − ⎜ ϕx |x = L2 + 2 ⎝ L2 − L1 ⎠ ⎩ ⎠⎭ ∂x 3h 4 ⎝

u(3) |x = L1 = u(1) |x = L1, z = h3 , u(3) |x = L2 = u(4) |x = L2, z = h3

P1

Q1

⎛

(1) ⎜ x+ ∑ ∑ Upq ⎝

u0(3) =

v (2) |x = L1 = v (1) |x = L1, z = h2 , v (2) |x = L2 = v (4) |x = L2, z = h2

p= 0 q= 0

v (3) |x = L1 = v (1) |x = L1, z = h3 , v (3) |x = L2 = v (4) |x = L2, z = h3

P3

⎛ ∂u(1) ⎞ ⎛ ∂u(4) ⎞ ϕx(2) |x = L1 = ⎜ ⎟ |x = L1, z = h2 , ϕx(2) |x = L2 = ⎜ ⎟ |x = L2, z = h2 ⎝ ∂z ⎠ ⎝ ∂z ⎠

+

L ⎞⎛ L⎞ ⎟ ⎜ x − ⎟ x py q ⎠ ⎝ 2 2⎠

Q3

(3) ∑ ∑ Upq ( x − L1)( x − L2 ) x pyq p= 0 q= 0

⎛ ∂u(1) ⎞ ⎛ ∂u(4) ⎞ =⎜ ⎟ |x = L1, z = h3 , ϕx(3) |x = L2 = ⎜ ⎟ |x = L2, z = h3 ∂ z ⎝ ⎠ ⎝ ∂z ⎠

⎞⎫ ⎛ L − x ⎞ ⎧ (1) 4h32 ⎛ (1) ∂w (1) − h3 ⎜ 2 |x = L1 ⎟ ⎬ ⎟ ⎨ ϕx |x = L1 − ⎜ ϕx |x = L1 + 2 ⎝ L2 − L1 ⎠ ⎩ ⎠⎭ ∂x 3h1 ⎝

⎛ ∂v (1) ⎞ ⎛ ∂v (4) ⎞ ϕy(2) |x = L1 = ⎜ ⎟ |x = L1, z = h2 , ϕy(2) |x = L2 = ⎜ ⎟ |x = L2, z = h2 ⎝ ∂z ⎠ ⎝ ∂z ⎠ ⎛ ∂v (1) ⎞ ⎛ ∂v (4) ⎞ ϕy(3) |x = L1 = ⎜ ⎟ |x = L1, z = h3 , ϕy(3) |x = L2 = ⎜ ⎟ |x = L2, z = h3 ⎝ ∂z ⎠ ⎝ ∂z ⎠

(21)

p= 0 q= 0

∂w (1) ∂w (2) ∂w (3) |x = L1 = |x = L1 = |x = L1 ∂x ∂x ∂x 4 2 3 ( ) ( ) ( ) ∂w ∂w ∂w | x = L2 = | x = L2 = | x = L2 ∂x ∂x ∂x 2 1 2 ( ) ( ) ( ) u |x = L1 = u |x = L1, z = h2 , u |x = L2 = u(4) |x = L2, z = h2

ϕx(3) |x = L1

L ⎞⎛ L⎞ ⎟ ⎜ x − ⎟ x m yn 2 ⎠⎝ 2⎠

Moreover, in-plane displacement fields for the SFSF boundary condition are given as [19]

w (1) |x = L1 = w (2) |x = L1 = w (3) |x = L1 w (4 ) |

L ⎞⎛ L⎞ ⎟ ⎜ x − ⎟ x m yn ⎠ ⎝ 2 2⎠

⎞⎫ ⎛ L − x ⎞ ⎧ (4) 4h32 ⎛ (4) ∂w (4) + h3 ⎜ 1 |x = L2 ⎟ ⎬ ⎟ ⎨ ϕ |x = L2 − ⎜ ϕ |x = L2 + 2⎝ x ⎝ L2 − L1 ⎠ ⎩ x ⎠⎭ ∂x 3h 4

(18)

In this section, two different types of boundary conditions are considered: clamped-free (CFCF) and simply supported-free (SFSF). Mathematical expressions for these supports are CFCF:

P1

u0(4) =

Q1

p= 0 q= 0

and

⎛

(1) ⎜x + ∑ ∑ Upq ⎝

L ⎞⎛ L⎞ ⎟ ⎜ x − ⎟ x py q 2 ⎠⎝ 2⎠

(22)

S.F. Nikrad, H. Asadi / Thin-Walled Structures 97 (2015) 296–307 S1

v0(1) =

where h2, h3 and h4 are the distance between the mid-planes of regions 2, 3 and 4 to the mid-plane of the whole plate. Due to inplane geometrical symmetry, the displacement fields of plates located in region 4 are the same as those located in region 1. Furthermore, the in-plane displacement fields for the both SFSF and CFCF boundary conditions are the same, and out-of-plane and rotational functions for the CFCF boundary condition are defined in Appendix A and B. It is noteworthy that sufficient terms of series expansion should be taken into account to assure the convergence of the postbuckling paths. Since numerical analysis could be computationally expensive, hence developed method is of great significance to reduce the computational costs. The last step is the calculation of the unknown coefficients by using the minimum potential energy principle and solving the set of nonlinear equations. Substituting the admissible functions into the potential energy expression of each region (Eq. (10)), the total potential energy of the composite plate is calculated. Substituting the corresponding energy into the following expression, one can develop the nonlinear equilibrium equations.

R1

∑ ∑ V sr(1) x s yr s= 0 r= 0 S1

v0(2) =

R1

S2

∑ ∑ V sr(1) x s yr

+

s= 0 r= 0

R2

∑ ∑ V sr(2) ( x − L1)( x − L2 ) x s yr s= 0 r= 0

⎞⎫ ⎛ L − x ⎞ ⎧ (1) 4h22 ⎛ (1) ∂w (1) + h2 ⎜ 2 |x = L1 ⎟ ⎬ ⎟ ⎨ ϕ |x = L1 − ⎜ ϕ |x = L1 + ⎝ L2 − L1 ⎠ ⎩ y ⎠⎭ ∂y 3h12 ⎝ y ⎞⎫ ⎛ L − x ⎞ ⎧ (4) 4h22 ⎛ (4) ∂w (4) − h2 ⎜ 1 |x = L2 ⎟ ⎬ ⎟ ⎨ ϕy |x = L2 − ⎜ ϕy |x = L2 + 2 ⎝ L2 − L1 ⎠ ⎩ ⎠⎭ ∂y 3h 4 ⎝ S1

v0(3) =

R1

S3

∑ ∑ V sr(1) x s yr

+

s= 0 r= 0

R3

∑ ∑ V sr(3) ( x − L1)( x − L2 ) x s yr s= 0 r= 0

⎞⎫ ⎛ L − x ⎞ ⎧ (1) 4h32 ⎛ (1) ∂w (1) − h3 ⎜ 2 |x = L1 ⎟ ⎬ ⎟ ⎨ ϕy |x = L1 − ⎜ ϕy |x = L1 + 2 ⎝ L2 − L1 ⎠ ⎩ ⎠⎭ ∂y 3h1 ⎝ ⎞⎫ ⎛ L − x ⎞ ⎧ (4) 4h32 ⎛ (4) ∂w (4) + h3 ⎜ 1 |x = L2 ⎟ ⎬ ⎟ ⎨ ϕ |x = L2 − ⎜ ϕ |x = L2 + 2⎝ y ⎝ L2 − L1 ⎠ ⎩ y ⎠⎭ y ∂ 3h 4 S1

v0(4) =

R1

∑ ∑ V sr(1) x s yr

(23)

s= 0 r= 0

In addition, the assumed rotational functions for SFSF boundary condition define as [19] ϕx(1) =

J1

K1

1) j k xy ∑ ∑ Φ(Xjk

ϕx(2) =

J2

K1

j =0 k=0

−

2 ⎞ 4h2 ⎛ L2 − x ⎞ ⎛ (1) ∂w (1) |x = L1⎟ ⎜ ⎟ ⎜ ϕx |x = L1 + 2 ∂x ⎠ h1 ⎝ L2 − L1 ⎠ ⎝

+

2 ⎞ 4h2 ⎛ L1 − x ⎞ ⎛ (4) ∂w (1) | x = L2 ⎟ ⎜ ⎟ ⎜ ϕx |x = L2 + 2 ∂x ⎠ h 4 ⎝ L2 − L1 ⎠ ⎝ J1

J3

K1

K3

1) j k 3) ∑ ∑ Φ(Xjk x y + ∑ ∑ Φ(Xjk ( x − L1)( x − L2 ) x jy k j =0 k=0

j =0 k=0

2 ⎞ 4h ⎛ L − x ⎞ ⎛ (1) ∂w (1) − 23 ⎜ 2 |x = L1⎟ ⎟ ⎜ ϕx |x = L1 + ⎝ ⎠ L L x − ∂ ⎝ ⎠ 2 1 h1 2 ⎞ 4h3 ⎛ L1 − x ⎞ ⎛ (4) ∂w (1) | x = L2 ⎟ ⎜ ⎟ ⎜ ϕ | x = L2 + 2 ⎝ L − L ⎠⎝ x x ∂ ⎠ 2 1 h4

+ ϕx(4) =

J1

K1

1) j k xy ∑ ∑ Φ(Xjk

(24)

j =0 k=0

And I1

ϕy(1) =

T1

(1) i t xy ∑ ∑ ΦYit i=0 t=0 I1

ϕy(2) =

T1

I2

(1) i t xy ∑ ∑ ΦYit

+

i=0 t=0

T2

(2) ∑ ∑ ΦYit ( x − L1)( x − L2 ) xiyt i=0 t=0

⎞ 4h 2 ⎛ L − x ⎞ ⎛ (1) ∂w (1) − 22 ⎜ 2 |x = L1 ⎟ ⎟ ⎜ ϕy |x = L1 + ⎠ ∂y h1 ⎝ L2 − L1 ⎠ ⎝ ⎞ 4h22 ⎛ L1 − x ⎞ ⎛ (4) ∂w (1) + 2 ⎜ |x = L2 ⎟ ⎟ ⎜ ϕ |x = L2 + ⎠ ∂y h4 ⎝ L2 − L1 ⎠ ⎝ y I1

ϕy(3) =

T1

I3

(1) i t xy ∑ ∑ ΦYit i=0 t=0

− +

⎞ 4h32 ⎛ L1 − x ⎞ ⎛ (4) ∂w (1) |x = L2 ⎟ ⎟ ⎜ ϕy |x = L2 + ⎜ 2 ⎝L ⎠ ∂y h4 2 − L1 ⎠ ⎝ T1

i=0 t=0

(i ) W mn ,

i) where χ is the vector of unknowns χ = Φ(xmn , (i ) Φ ymn} . The Eq. (26) leads to the following matrix representation of the equilibrium equations

(27)

where K (T, χ ) , and F (T ) are the stiffness matrix and force vector, respectively. The generalized Newton–Raphson iterative method is employed to solve Eq. (27) and to trace the equilibrium path of delaminated composite plate. Moreover, for finding the critical buckling temperature difference, a trial and error procedure is adopted. According to this approach, an arbitrary temperature difference (ΔT ), is selected to perform the postbuckling analysis. If all out-of-plane degrees of freedom appear to be numerically zero, this will indicate that the selected temperature difference is less than the critical buckling temperature difference ΔTCr . In this case, the temperature difference is increased by small increments and the analysis is repeated. This process is continued until non-zero values are obtained for the out-of-plane degrees of freedom, when the corresponding temperature difference (ΔT ) designates the critical buckling temperature difference (ΔTCr ) of the structure [35]. It is noteworthy that the delaminated composite plates show complex buckling and postbuckling behaviors compared to perfect composite plate without delamination. In fact, when a delaminated composite plate is subjected to a uniform temperature rise, local buckling of the delaminated region or mixed buckling mode, which is a combination of local and global buckling, could occur before global buckling, as shown schematically in Fig. 3, [31]. The sublaminate buckles in the local mode while the base laminate remains its flatness. The local and mixed modes specifically are important for the composite design since they could result in the crack opening and crack propagation of the delaminated area [2].

5. Results and discussion

i=0 t=0

⎞ ⎛ L2 − x ⎞ ⎛ (1) ∂w (1) |x = L1 ⎟ ⎟ ⎜ ϕ |x = L1 + ⎜ ⎝ L2 − L1 ⎠ ⎝ y ⎠ ∂y

(1) i t xy ∑ ∑ ΦYit

(i ) V mn ,

T3

(3) ∑ ∑ ΦYit ( x − L1)( x − L2 ) xiyt

4h32 h12

I1

ϕy(4) =

+

(26) (i ) {Umn ,

K2

1) j k 2) x y + ∑ ∑ Φ(Xjk ∑ ∑ Φ(Xjk ( x − L1)( x − L2 ) x jy k j =0 k=0

ϕx(3) =

∂Π =0 ∂χi

[K (T , χ )]{χ} = {F (T )}

j =0 k =0 J1

301

(25)

The procedure outlined in the previous sections, is used herein to investigate the effect of delamination on the thermal postbuckling strength, end-shortening force, critical buckling temperature difference and buckling mode of composite plates. First a convergence and comparative studies are conducted to ensure the efficacy and validity of the developed solution procedure. Subsequently, a parametric study is performed to demonstrate the effects

302

S.F. Nikrad, H. Asadi / Thin-Walled Structures 97 (2015) 296–307

of the various involved parameters on the nonlinear thermal postbuckling response of the delaminated composite plate.

Table 1 Convergence study of the lateral deflection of delaminated composite plate in the postbuckling regime (LD/L = 0.7) .

5.1. Convergence study To obtain the sufficient terms needed to be involved in series expansion (Eqs. (15)–(17) and Eqs. (21)–(25)), a convergence study of the lateral deflection of the composite plate with through-the-width delamination in the postbuckling regime is carried out and presented in Table 1, for both clamped and simply supported boundary conditions. The material properties of the laminates are given in Table 2. The dimensions of the composite plate are 80 × 80 × 2 mm and stacking sequence of the sublaminate and base laminate are [0/90/90/0] and [0/90/90/0]3, respectively. It is observed from the Table 1, that the present results converge well as the number of terms in the series expansion increases. It is understood from the convergence study that Ni = Mi = ⋯ = Ii = Ti = 7, would be appropriate for further computation of the results. Therefore, in the present work, for all results, the number of degrees of freedom is equal to 960.

B.C.

1

5.2. Comparative studies To verify the accuracy of the present formulation and solution procedure, two comparative studies are addressed in Fig. 4 and 5. The thermal postbuckling equilibrium path of a perfect composite plate without delamination is compared with that obtained by Park et al. [32] and the comparison is depicted in Fig. 4. The results of Park at al. [32] were obtained based on first order shear deformation theory using finite element method. The material properties of the laminates are given in Table 2. The dimensions of the composite plate are 380 × 305 × 2 mm and the stacking sequence is [0/45/ − 45/90]s . The simply-supported boundary conditions are considered for all edges, which are immovable for the in-plane directions. As can be observed from this figure, the comparison is well-justified and the solution procedure seems to be accurate. The maximum discrepancy of less than 1% is observed between the current numerical results and those published in [32]. In another comparison study to verify the proposed method, a simply-supported laminated composite plate containing a rectangular embedded delamination is considered. The dimensions of the composite plate and rectangular embedded delamination are 100 × 50 × 2 mm and 80 × 40 mm (AD/A = 0.64), respectively. The stacking sequence of the sublaminate and base laminate are

2

Number of series expansion

ΔT ΔTCr

ΔT ΔTCr

= 0.6

=1

DOF 1 W sub h

W base h

2

W sub h

W base h

CFCF

Ni Ni Ni Ni Ni

= = = = =

⋯ ⋯ ⋯ ⋯ ⋯

= = = = =

Ti Ti Ti Ti Ti

= = = = =

3 4 5 6 7

240 375 540 735 960

0.75536 0.77381 0.77566 0.77566 0.77566

0.26344 0.26478 0.26498 0.26498 0.26498

1.0403 1.1197 1.1228 1.2290 1.2290

0.64762 0.65226 0.65299 0.65299 0.65299

SFSF

Ni Ni Ni Ni Ni

= = = = =

⋯ ⋯ ⋯ ⋯ ⋯

= = = = =

Ti Ti Ti Ti Ti

= = = = =

3 4 5 6 7

240 375 540 735 960

0.64141 0.65893 0.65991 0.66008 0.66008

0.30688 0.31032 0.31053 0.31062 0.31062

0.9387 0.9407 0.9419 0.9421 0.9421

0.4921 0.5072 0.5074 0.5074 0.5074

Sublaminate. Base laminate.

[0/90/90/0] and [0/90/90/0]3, respectively. The finite element analysis (FEA) is performed using ABAQUS 6.11 software. The layered 8-node solid elements with three degrees of freedom at each node are used. It is noteworthy that the mesh is further refined at the discontinuous delamination front in order to preserve the accuracy of the results (Fig. 5A). In the first step of buckling analysis, a linear static analysis is implemented. The second step is an eigenvalue analysis, which provides the results in terms of load factors and mode shapes. Subsequently, the mode shapes are used as postulated imperfections in order to perform an iterative nonlinear postbuckling analysis. The thermal postbuckling equilibrium path of the composite plate with rectangular embedded delamination is compared with the results of the finite element analysis. Fig. 5B gives a brief overview of the validity and accuracy of the proposed formulations and a close agreement with finite element method with maximum discrepancy of about 3% can be seen in the predictions. It should be noted that the number of degrees of freedom for the FEA model, which contains 12,000 solid elements is 288,000. Thus, it is understandable that the proposed method in this work is of great significance to reduce the computational costs. 5.3. Parametric studies In this section, to assess the effects of delamination on nonlinear thermal stability of composite plates, a composite plate made of graphite-epoxy is considered and thermomechanical properties of constituents are given in Table 2. In the rest of this paper, it is considered that L = b = 80 mm , h = 2 mm and other geometric parameters are given in each figure. It should be noted

Table 2 Thermomechanical properties of the Graphite/Epoxy. Graphite/Epoxy properties

E1m = 155 (1 − 3.53 × 10−4ΔT ) GPa E2m = 8.07 (1 − 4.27 × 10−4ΔT ) GPa G12m = G13m = 4.55 (1 − 6.06 × 10−4ΔT ) GPa G23m = 3.25 (1 − 6.06 × 10−4ΔT ) GPa α1m = − 0.07 × 10−6 (1 − 1.25 × 10−3ΔT )(1/°C)

Fig. 3. Buckling modes for delaminated composite plates.

α 2m = 30.1 × 10−6 (1 + 0.41 × 10−4ΔT )(1/°C) υ12m = 0.22

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303

that the effect of materials degradation is not taken into consideration in the temperature range used in the present analysis. Furthermore, for all results the temperatures are normalized by the critical buckling temperature difference ΔTCr of the composite plate without delamination.

Fig. 4. A comparison of postbuckling response of laminated composite plate without delamination.

Fig. 5. (A) A typical mesh for composite plates with a rectangular embedded delamination, and (B) comparison of the thermal postbuckling equilibrium path of the composite plate with an embedded delamination.

5.3.1. Embedded delamination It is assumed that the square embedded delamination takes place in the center of the plate, between the fourth and fifth layers; thus the stacking sequence of the sublaminate and base laminate are [0/90/90/0] and [0/90/90/0]3, respectively. In order to expatiate on the delamination size factor, two different delamination sizes 40 × 40 mm (AD/A = 0.25) and 50.6 × 50.6 mm (AD/A = 0.4) are considered. The thermal bifurcation paths of the composite plates with square embedded delamination are illustrated in Fig. 6 for two ratios of the area of delamination (AD) to the area of the plate (A). The buckled configurations of the delaminated composite plates are illustrated in Fig. 7 at different temperatures. Figs. 6 and 7 clearly show that local mode of postbuckling first happens. Therefore, the buckling initiation of the sublaminate is restrained by the base laminate for an extra out-of-plane deflection. The reason is that the sublaminate buckles firstly, while the base laminate has not buckled due to the higher flexural rigidity of the base laminate compared to the sublaminate. As temperature is increased, the base laminate deflects considerably and reduces the restraint on the sublaminate. It is clearly visible that a delaminated composite plate has a lower ability to resist compressive thermal loading and the reduction in this ability depends on the size and position of the delamination. The variation of the non-dimensional load-end shortening for the same composite plate as that discussed in Fig. 5 is demonstrated in Fig. 8. It is found that area of delamination plays major role in the load carrying capacity of the laminates and leads to significant decline in the load carrying capacity of the composite plate. For instance, the load carrying capacity of delaminated composite plate with non-dimensional delamination area AD/A = 0.4 is about 78% of the critical buckling load of a composite plate without any delamination at ΔT /ΔT Cr = 1. This clearly implies that the plate has experienced more than 22% loss in its load carrying capacity due to the presence of delamination at the aforementioned level of thermal loading. Thus, an accurate thermal stability analysis of delaminated composite structures similar to that developed in the current study is an important step to enhance their reliability. In Fig. 9, distribution of the transverse shear strain γxz through the thickness of delaminated composite plate at delamination crack tip is depicted for two different non-dimensional delamination areas. It is clearly visible from the figure that the transverse shear strain is equal to zero at delamination crack tip. It is interesting to note that maximum value of the transverse shear strain depends on the area of delamination and it occurs at a point on the midplane of the base laminate. For instance at constant thermal loading (ΔT /ΔT Cr = 1.2), increasing the non-dimensional delamination area from 0.25 to 0.4 leads to drastic increase of the maximum value of the transverse shear strain γxz about 77%. Fig. 10 shows the variations of the normal strain (ε¯xx ) at the top surface of the composite plate with square embedded delamination. It can be concluded from this figure that the distribution of normal strain is continuously nonlinear with respect to x direction, expect at delamination crack tips where normal strain is discontinuous and singular. It is interesting to mention that the total normal strain is positive in the middle of the delamination area whereas it is negative in the rest of the delamination zone.

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Fig. 6. Effect of the delamination area on the equilibrium paths of the composite plate with embedded delamination.

Fig. 8. Non-dimensional load-end variations of a delaminated square plate with embedded delamination.

5.3.2. Through-the-width delamination It is assumed that the delamination is centrically located between the fourth and fifth layers; thus the stacking sequence of [0/90/90/0] and the sublaminate and base laminate are [0/90/90/0]3, respectively. In order to elaborate the effect of delamination size, two different delamination sizes LD/L = 0.5 and LD/L = 0.7 are considered. The nonlinear primary-secondary equilibrium paths of the composite plate with through-the-width delamination are depicted in Figs. 11 and 12 for SFSF and CFCF boundary conditions with two different ratios of length of delamination (LD) to length of plate (L ). As expected, due to the higher flexural rigidity of the clamping condition, the critical buckling temperature of the CFCF composite plate without delamination is 2.78 times higher than that of the SFSF composite plate without delamination. As seen in these figures, the thermal stability behavior of the delaminated composite plate is sensitive to delamination length and increasing the delamination length results in hastening the thermal bifurcation point of the composite plate. It can be concluded from these

figures that the buckling is initiated in the sublaminate which is being restrained by the rest of the laminated against further deflection. It is worthwhile noting that due to the out-of-plane thickness ratio of the sublaminate to that of the base laminate, the base laminate essentially acts like a rigid surface constraining the out-of-plane deformations of the plate. It should be noted that with increasing the thermal loading, the base laminate commences to deflect substantially, which is resulted in reduction of its restraint on the sublaminate. The latter global mode of postbuckling leads to a significant loss in the stiffness of the laminate. It is worth noting that delaminated composite plate with CFCF supports buckles in higher temperature as compared to the SFSF conditions. The reason is the higher flexural rigidity of the clamping condition. In Fig. 13, the in-plane displacement fields of the composite plate with through-the-width delamination are depicted at delamination zone (z = 0.5) for two different non-dimensional delamination lengths. As one may conclude, at the delamination zone, the sublaminate has a higher lateral deflection as compared to the

Fig. 7. The buckled configurations of the composite plate with embedded delamination as that investigated in Fig. 6.

S.F. Nikrad, H. Asadi / Thin-Walled Structures 97 (2015) 296–307

Fig. 9. Through the thickness distribution of the shear strain of delaminated composite plate at the delamination crack tip.

305

Fig. 11. Influence of delamination length on the thermal postbuckling responses of composite plate with through-the-width delamination with SFSF boundary condition.

Fig.10. Through the length distribution of normal strain at the top surface of the delaminated composite plate.

base laminate. As a result, the in-plane displacement of the sublaminate shows strong nonlinear behavior in contrast with base laminate which demonstrations linear behavior. Moreover, the nonlinear in-plane response of thinner sublaminate is more profound at higher temperature levels because of increasing the transverse deflection of the sublaminate. Furthermore, it is clearly visible from the figure that the displacement continuity condition is satisfied at the boundary of delaminated region. The in-plane responses of the composite plate with throughthe-width delamination are presented in Fig. 14, at the top and bottom (z¼1 and 1) of the composite plate for two different delamination sizes. As can be seen in Fig. 10, base laminate and

Fig. 12. Effect of delamination size on the thermal bifurcation paths of composite plate with through-the-width delamination with CFCF boundary condition.

sublaminate of the composite plate buckle in the same direction. As a result, at the left half of laminates, the sublaminate experiences the positive in-plane displacement opposed to the base laminate. However, at the right half of the plate, the sublaminate is subjected to negative in-plane displacement in contrast with the base laminate. Furthermore, at the delamination region, the sublaminate has a higher lateral deflection as compared to the base laminate, which is resulted in a strong nonlinear response of inplane displacement of the sublaminate.

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Fig. 13. Effect of delamination length on the variations of in-plane displacement at the delamination surface of the composite plate with through-the-width delamination.

6. Concluding remarks A formulation considering higher order shear deformation theory in conjunction with the von Karman strain–displacement relation for providing an accurate solution for the nonlinear thermal stability of the delaminated composite plates is developed. Two types of delaminations including embedded and through-the-width delaminations are considered. Thermomechanical properties of constituents are assumed to be temperature-dependent. The analytical method is based on the Rayleigh–Ritz technique by the implementation of the simple and complete polynomial series. Newton–Raphson iterative method is employed to solve the resulting nonlinear algebraic equations. The effect of two types of delaminations, delamination size, boundary conditions and thermal loading on the nonlinear thermal stability of delaminated composite plate is brought out through a number of numerical studies. The following major findings of this paper could be useful to enhance reliability of delaminated composite structures

Fig. 14. Influence of delamination size on the variations of in-plane displacement at the top and bottom surface of the composite plate with through-the-width delamination.

number of degree of freedoms, very good results for a rather complicated problem are obtained. Hence, the developed method is significantly efficient from the computational point of view.

Appendix A The assumed out-of-plane displacement functions for CFCF boundary condition are M1

w (1) =

⎛

2 2 L⎞ ⎛ L⎞ ⎟ ⎜ x − ⎟ x m yn 2⎠ ⎝ 2⎠

⎛

2 2 L⎞ ⎛ L⎞ ⎟ ⎜ x − ⎟ x m yn ⎠ ⎝ ⎠ 2 2

m= 0 n= 0 M1

w (2) =

N1

(1) ⎜ x+ ∑ ∑ Wmn ⎝

m= 0 n= 0 M2

+

N2

(2) ∑ ∑ Wmn ( x − L1)2 ( x − L2 )2xmyn m= 0 n= 0

M1

i. It is seen that the critical buckling temperature of delaminated composite plates decreases by increasing delamination size. While increasing delamination size results in a considerable increase in lateral deflection of delaminated plates in the postbuckling regime. ii. It is concluded that the buckling mode of delaminated plates strongly depends on the delamination size. Increasing the delamination size of the plate results in increasing the effect of global mode of postbuckling, which causes a noticeable reduction in the stiffness of the laminates. iii. It is found that presence of delamination plays a major role in load carrying capacity of composite structures. The results reveal the fact that the load carrying capacity of composite plates decreases by increasing delamination size. iv. The analytical method is capable of analyzing mixed mode of local buckling of the delaminated sublaminates with global buckling of the base laminate. Moreover, it is seen that by implementing complete polynomials, and by using fairly small

N1

(1) ⎜ x+ ∑ ∑ Wmn ⎝

w (3) =

N1

⎛

(1) ⎜ x+ ∑ ∑ Wmn ⎝

m= 0 n= 0 M3

+

2 2 L⎞ ⎛ L⎞ ⎟ ⎜ x − ⎟ x m yn 2⎠ ⎝ 2⎠

N3

(3) ∑ ∑ Wmn ( x − L1)2 ( x − L2 )2xmyn m= 0 n= 0

M1

w (4) =

N1

⎛

(1) ⎜ x+ ∑ ∑ Wmn ⎝

m= 0 n= 0

2 2 L⎞ ⎛ L⎞ ⎟ ⎜ x − ⎟ x m yn 2⎠ ⎝ 2⎠

Appendix B The assumed rotational functions for CFCF boundary condition are

ϕx(1) =

J1

K1

⎛

1) ⎜x + ∑ ∑ Φ(Xjk ⎝ j=0 k=0

L ⎞⎛ L⎞ ⎟ ⎜ x − ⎟ x jyk 2 ⎠⎝ 2⎠

S.F. Nikrad, H. Asadi / Thin-Walled Structures 97 (2015) 296–307 J1

J

K1

K

2 2 L ⎞⎛ L⎞ 1) ⎛ ⎜ x + ⎟ ⎜ x − ⎟ x jy k + ∑ ∑ Φ (2) ( x − L1)( x − L2 ) x jy k Φ (Xjk Xjk ⎝ 2 ⎠⎝ 2⎠ j =0 k =0 j =0 k =0

ϕx(2) =

∑ ∑

−

⎞ 4h22 ⎛ L2 − x ⎞ ⎛ ∂w (1) |x = L1⎟ ⎟ ⎜ ϕx(1) |x = L1 + ⎜ ∂x h 2 ⎝ L2 − L1 ⎠ ⎝ ⎠

+

⎞ 4h22 ⎛ L1 − x ⎞ ⎛ ∂w (1) | x = L2 ⎟ ⎜ ⎟ ⎜ ϕx(4) |x = L2 + ∂x h 42 ⎝ L2 − L1 ⎠ ⎝ ⎠

1

J1

J

K1

K

3 3 L ⎞⎛ L⎞ 1) ⎛ ⎜ x + ⎟ ⎜ x − ⎟ x jy k + ∑ ∑ Φ (3) ( x − L1)( x − L2 ) x jy k Φ (Xjk Xjk ⎝ 2 ⎠⎝ 2⎠ j =0 k =0 j =0 k =0

ϕx(3) =

∑ ∑

⎞ L2 − x ⎞ ⎛ (1) ∂w (1) |x = L1⎟ ⎟ ⎜ ϕ |x = L1 + L2 − L1 ⎠ ⎝ x ∂x ⎠

−

4h32 ⎛ ⎜ h12 ⎝

+

⎞ 4h32 ⎛ L1 − x ⎞ ⎛ ∂w (1) | x = L2 ⎟ ⎟ ⎜ ϕx(4) |x = L2 + ⎜ ∂x h 42 ⎝ L2 − L1 ⎠ ⎝ ⎠

ϕx(4) =

J1

K1

⎛

1) ⎜x + ∑ ∑ Φ(Xjk ⎝ j =0 k =0

ϕy(1) =

I1

L ⎞⎛ L⎞ ⎟ ⎜ x − ⎟ x jy k 2 ⎠⎝ 2⎠

T1

(1 ) i t xy ∑ ∑ ΦYit i =0 t =0

ϕy(2) =

ϕy(3) =

I1 T1 I2 T 2 (1 ) i t (2 ) x y + ∑ ∑ ΦYit x − L1 x − L2 x iy t ∑ ∑ ΦYit i =0 t =0 i =0 t =0 ⎞ 4h22 ⎛ L2 − x ⎞ ⎛ ∂w (1) − |x = L1⎟ ⎟ ⎜ ϕy(1) |x = L1 + ⎜ ∂y h12 ⎝ L2 − L1 ⎠ ⎝ ⎠ ⎞ 1 ( ) 4h22 ⎛ L1 − x ⎞ ⎛ ∂w | x = L2 ⎟ + ⎟ ⎜ ϕy(4) |x = L2 + ⎜ ∂y h 42 ⎝ L2 − L1 ⎠ ⎝ ⎠

(

I1

T1

∑ ∑ ΦY(1)it xiyt

I3

+

)(

)

T3

(3 ) ∑ ∑ ΦYit ( x − L1)( x − L2 ) xiyt

i =0 t =0 i =0 t =0 ⎞ 4h32 ⎛ L2 − x ⎞ ⎛ ∂w (1) ( |x = L1⎟ − ⎟ ⎜ ϕy1) |x = L1 + ⎜ ∂y ⎠ h12 ⎝ L2 − L1 ⎠ ⎝ ⎞ (1 ) 4h32 ⎛ L1 − x ⎞ ⎛ 4 w ∂ + | x = L2 ⎟ ⎜ ⎟ ⎜ ϕy( ) |x = L2 + ∂y ⎠ h 42 ⎝ L2 − L1 ⎠ ⎝

ϕy(4) =

I1

T1

(1 ) i t xy ∑ ∑ ΦYit

i =0 t =0

References [1] H. Chai, C.D. Babcock, W.G. Knauss, One dimensional modeling of failure in laminated plates by delamination buckling, Int. J. Solids Struct. 17 (1981) 1069–1083. [2] W.J. Bottega, A. Maewal, Delamination buckling and growth in laminates, J. Appl. Mech. 50 (1983) 184–189. [3] K.N. Shivarkumar, J.D. Withcomb, Buckling of sublaminate in a quasi isotropic composite laminates, J. Compos. Mater. 19 (1985) 2–18. [4] J.S. Anastasiadis, G.J. Simitses, Spring simulated delamination of axially loaded flat laminates, Compos. Struct. 17 (1991) 67–85. [5] B.D. Davidson, Delamination buckling: theory and experiment, J. Compos. Mater. 25 (1991) 1351–1378. [6] C.H. Piao, Shear deformation theory of compressive delamination buckling and growth, AIAA 29 (5) (1991) 813–819. [7] H. Suemasu, Compressive behavior of fiber reinforced composite plates with a center delamination, Adv. Compos. Mater. 1 (1) (1991) 23–37. [8] H. Suemasu, Effects of multiple delaminations on compressive buckling behaviors of composite panels, J. Compos. Mater. 27 (12) (1993) 1172–1192. [9] M. Adan, I. Sheinman, E. Altus, Buckling of multiply delaminated beams, J.

307

Compos. Mater. 28 (1) (1994) 77–90. [10] D. Shu, Buckling of multiple delaminated beams, Int. J. Solids struct. 25 (13) (1998) 1451–1456. [11] J.S. Wang, S.H. Cheng, Local buckling of delaminated beams and plates using continuous analysis, J. Compos. Mater. 29 (10) (1995) 1374–1402. [12] J.S. Wang, J.T. Huang, Strain energy release rate of delaminated composite plates using continuous analysis, J. Compos. Eng. 7 (1994) 731–744. [13] K.W. Shahwan, A.A. Wass, A mechanical model for the buckling of unilaterally constrained rectangular plates, Int. J. Solids Struct. 31 (1) (1994) 75–87. [14] D.W. Sleight, J.T. Wang, Buckling analysis of debonded sandwich panel under compression, 1995 (NASA Technical Memorandum). [15] K.C. Jane, H.W. Liao, W. Hong, Validation of Rayleigh–Ritz method for the postbuckling analysis of rectangular plates with application to delamination, Mech. Res. Commun. 30 (2003) 531–538. [16] N.G. Andrews, R. Massabo, B.N. Cox, Elastic interaction of multiple delaminations in plates subject to cylindrical bending, Int. J. Solids Struct. 43 (2006) 855–886. [17] H.R. Ovesy, H. Hosseini-Toudeshky, M. Kharazi, Buckling analysis of laminates with multiple through-the-width delaminations by using spring simulated model, in: Proceedings of III European Conference on Computational Mechanics Solids, Structures and Coupled Problems in Engineering, Lisbon, Portugal, 2006. [18] M. Kharazi, H.R. Ovesy, M. Taghizadeh, Buckling of the composite laminates containing through-the-width delaminations using different plate theories, Compos. Struct. 92 (2010) 1176–1183. [19] H.R. Ovesy, M. Taghizadeh, M. Kharazi, Post-buckling analysis of composite plates containing embedded delaminations with arbitrary shape by using higher order shear deformation theory, Compos. Struct. 94 (2012) 1243–1249. [20] H.R. Ovesy, M. Kharazi, Compressional stability behavior of composite plates with through-the-width and embedded delaminations by using first order shear deformation theory, Comput. Struct. 89 (2011) 1829–1839. [21] M. Kharazi, H.R. Ovesy, M. Asghari Mooneghi, Buckling analysis of delaminated composite plates using a novel layerwise theory, Thin-Walled Struct. 74 (2014) 246–254. [22] H.R. Ovesy, M. Asghari Mooneghi, M. Kharazi, Post-buckling analysis of delaminated composite laminates with multiple through-the-width delaminations using a novel layerwise theory, Thin-Walled Struct. 94 (2015) 98–106. [23] W.M. Ostachowicz, S. Kaczmarczyk, Vibration of composite plates with SMA fibers in a gas stream with defects of type of delaminations, Compos. Struct. 54 (2001) 305–311. [24] H. Asadi, A.H. Akbarzadeh, Q. Wang, Nonlinear thermo-inertial instability of functionally graded shape memory alloy sandwich plates, Compos. Struct. 120 (2015) 496–508. [25] H.H. Ibrahim, M. Tawfik, H.M. Negm, Thermal buckling and nonlinear flutter behavior of shape memory alloy hybrid composite plates, J. Vib. Control 17 (3) (2011) 321–333. [26] H. Asadi, M. Eynbeygi, Q. Wang, Nonlinear thermal stability of geometrically imperfect shape memory alloy hybrid laminated composite plates, Smart Mater. Struct. 23 (7) (2014) 075012. [27] H.H. Ibrahim, H.H. Yoo, K.S. Lee, Aero-thermo-mechanical characteristics of imperfect shape memory alloy hybrid composite panels, J. Sound Vib. 325 (2009) 583–596. [28] H. Asadi, Y. Kiani, M. Shakeri, M.R. Eslami, Exact solution for nonlinear thermal stability of geometrically imperfect hybrid laminated composite timoshenko beams embedded with SMA fibers, J. Eng. Mech. 141 (4) (2015) 04014144. [29] S.F. Nikrad, H. Asadi, A.H. Akbarzadeh, Z.T. Chen, On thermal instability of delaminated composite plates, Compos. Struct. 132 (2015) 1149–1159. [30] H. Asadi, M.M. Aghdam, Large amplitude vibration and post-buckling analysis of variable cross-section composite beams on nonlinear elastic foundation, Int. J. Mech. Sci. 79 (2014) 47–55. [31] H. Hosseini-Toudeshky, S. Hosseini, B. Mohammadi, Buckling and delamination growth analysis of composite laminates containing embedded delaminations, Appl. Compos. Mater. 17 (2010) 95–109. [32] J.S. Park, J.H. Kim, S.H. Moon, Vibration of thermally post-buckled composite plates embedded with shape memory alloy fibers, Compos. Struct. 63 (2004) 179–188. [33] H. Asadi, Y. Kiani, M. Shakeri, M.R. Eslami, Exact solution for nonlinear thermal stability of hybrid laminated composite Timoshenko beams reinforced with SMA fibers, Compos. Struct. 108 (2014) 811–822. [34] J.N. Reddy, Mechanics of Laminated Composite Plates and Shells, 2nd ed, CRC Press, Boca Raton, 2004. [35] J.N. Reddy, An Introduction to Nonlinear Finite Element Analysis, Oxford University Press, UK, 2004.