Nonlocal buckling analysis of laminated composite plates considering surface stress effects

Composite Structures 226 (2019) 111216

Contents lists available at ScienceDirect

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

Nonlocal buckling analysis of laminated composite plates considering surface stress effects

T

K. Shivaa, P. Raghua, A. Rajagopala, , J.N. Reddyb ⁎

a b

Department of Civil Engineering, IIT Hyderabad, India Department of Mechanical Engineering, Texas A&M University, College Station, TX, USA

ABSTRACT

In this work nonlocal buckling analysis of laminated composite plates considering surface stress effects is presented. For computation of critical uniaxial and biaxial buckling loads, both approximate solutions and finite element solutions are presented. Approximate solutions based on Navier’s method for simply supported boundary conditions and based on Levy’s method for other type of boundary conditions are presented. The analysis of laminate is based on Reddy’s third order shear deformation theory (TSDT) (Reddy, 1984). Eringen’s non-local differential material model (Eringen, 1983) has been used in the present work. The surface stress effects are incorporated using the Gurtin Murdoch theory (Gurtin and Murdoch, 1975). Various types of laminates have been considered for the analysis. A parametric study is carried out, considering geometric and material properties to understand the effect of non-locality and surface stress on buckling loads. It is found that the inclusion of non-locality and surface effect has a significant effect on the critical buckling load and this varies with various aspect ratios and span to thickness ratios, and boundary conditions of the laminated plate. The results obtained are compared with those available in the literature for validation.

1. Introduction Non-local continuum mechanics finds its application in structures where the geometric length scales are extremely small [4] or large [5]. Classical continuum theories do not take into account the micro structural effects and hence are not adequate to describe certain physics. On the other hand, molecular dynamics simulations are computationally very expensive [6]. Non-local continuum mechanics theories account for certain micro structural aspects in a continuum frame work. The concept of non-locality was first introduced in the works of Kröner [7] and Eringen [8]. In Eringen’s non-local stress-gradient model, constitutive relation was modified by introducing a material parameter to include the length scale of the material/structure. Eringen’s stress-gradient model has been widely applied to study many problems, such as lattice dispersion of elastic waves, wave propagation in composites, fracture mechanics, dislocation mechanics, and surface tension in fluids. The model has been applied for analysis of various problems such as stress singularity at the crack tip (see [9,10]), wave propagation in materials (see [11–14]), analysis of beams using Euler-Bernoulli beam theory (see [15–23]), analysis of beams using Timoshenko beam theory (see [24–30]), analysis of plates (see [31–35]) and analysis of functionally graded plates (see [36–38]). Reddy [33] has presented nonlinear finite element formulations for non-local bending analysis of beams and plates, using classical and

⁎

shear deformation theories. It has been observed, the deflections increase with an increase in non-local and surface stress parameters. Nowruzpour and Reddy [39] have recently proposed a theory for the unification of local and non-local models and it has been applied for analysis of defects. In this present work, we use the Reddy’s nonlocal TSDT for analysis of laminated plates. At smaller length scales the amount of material present on the surface is significant in proportion to the proportion in bulk. The surface effects hence become important at smaller length scales. It is noted that the equilibrium inter atomic distance between surface atoms is not the same as that for atoms in the interior or bulk material. The structural coherency of the surface and bulk atoms causes an interaction and results in directional forces on the surface, which are termed surface stresses. A model to incorporate the surface effects was given by Gurtin and Murdoch [3] and was later used in wave propagation studies (see [40,41]). There have been several other works on understanding surface elasticity (see [42–45]). The effect of surface stresses has been included in the analysis of beams and plates (see [46–51]). In this work, we have incorporated the surface effects for the nonlocal analysis of laminated plates. Laminated Composites find several structural applications in view of their very high specific strength ratio (strength/weight), high specific stiffness ratio (stiffness/weight) and high fatigue life. Buckling due to compressive in-plane loads is one of the fundamental failure mechanisms in laminated composite plates. A review of the buckling analysis of

Corresponding author. E-mail address: [email protected] (A. Rajagopal).

https://doi.org/10.1016/j.compstruct.2019.111216 Received 29 April 2019; Received in revised form 30 May 2019; Accepted 9 July 2019 Available online 15 July 2019 0263-8223/ © 2019 Elsevier Ltd. All rights reserved.

Composite Structures 226 (2019) 111216

K. Shiva, et al.

composite plates has been presented by Leissa [52]. Effects of lamination schemes and nonlinear stress-strain relations, complicating effects, and hydrothermal effects are summarized in this study [52]. Reddy and Khdeir [53] studied the exact and finite element solutions for free vibration and buckling of rectangular composite laminates using classical, first-order, and third-order laminate plate theories under various boundary conditions. Reddy and Phan [54] using a higher-order shear deformation theory, studied stability and vibration. Buckling of composite plates subjected to various types of loads and boundary conditions was studied (see [55–65]). The buckling behavior of composite skew plates with different skew angles are also studied (see [66–70]). The non-local elasticity theory applied to buckling of composites, plates and graphene sheets has been studied by several authors to understand the length scale effects on buckling loads (see [71–82]). An extensive review on the field of non-local continuum based theories applied to nanoscopic structures, to study buckling behavior has been given in [83,84]. There have been some recent studies on the application of non-local theories for vibration analysis of functionally graded nanoplates [85]. In a recent study micro structured beam-grid model, which replaces curved lines of deformation by straight lines, has been compared with non-local plate model to find the significance of Eringen’s small length scale coefficient [86]. The ability of three different kinds of non-local plate theories to consider the micro structured scale effects has been reported (see [87,88]). From the literature review, it is clear that small-scale effects and surface effects become critical in micro or nano structured plates subjected to compressive in-plane loads and hence, there is a need to look at buckling problem of laminated composite plates considering these effects. In this work nonlocal buckling analysis of laminated composite plates considering surface stress effects is presented. For computation of critical uniaxial and biaxial buckling loads, both approximate solutions and finite element solutions are presented. Approximate solutions based on Navier’s method for simply supported boundary conditions and based on Levys method for other type of boundary conditions are presented. The analysis of laminate is based on Reddy’s third order shear deformation theory (TSDT) ([1,89]). Eringen’s non-local differential material model [2] has been used in the present work. The surface stress effects are incorporated using the Gurtin Murdoch theory [3]. Various types of laminates have been considered for the analysis. A parametric study is carried out, considering geometric and material properties to understand the effect of non-locality and surface stress on buckling loads. It is found that the inclusion of non-locality and surface effect has a significant effect on the critical buckling load and this varies with various aspect ratios and span to thickness ratios, and boundary conditions of the laminated plate. The results obtained are compared with those available in the literature for validation. The paper is organized as follows: In the following Section 2 we present the description of nonlocal TSDT with surface stress effect. In Section 3 we first present the Navier’s solution for simply supported laminates and then discuss the Levy’s solution of laminates with generic boundary conditions. In Section 4 we present the finite element formulation for nonlocal analysis of laminates with surface stress effects, In Section 5 we present the numerical examples for the parametric study and comparison of the analytical solutions with the FEM solutions and with those solutions taken from litreature. The last section present the conclusions and references.

(

x

+

w0 x

(

y

+

w0 y

u (x , y, z )= u0 x , y + z

x

c1 z 3

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

y

c1 z 3

)

)

(1)

w (x , y , z ) = w0 (x , y ) where u 0, v0, w0 are in-plane displacements of a point on the mid-plane 4 (i.e., z = 0 ), c1 = 2 , and ( x , y ) denote the rotations of a transverse 3h normal line at the mid-plane about the y- and x- axes respectively (i.e., u v x = z and y = z at z = 0 ). The total thickness of the laminate is denoted by h. The Eringen’s non-local elasticity model (see [8,2]), in differential form is used here for obtaining the non-local constitutive relation between the stresses as given below 2l 2

(1

2)

nl

(2)

=

(e 0 a ) 2

where = l2 , e0 being a material constant and a and l are the internal and external characteristic lengths, respectively, and 2 is a three-dimensional Laplace operator. The non-local parameter µ is defined as µ = 2l2 . To incorporate the effect of surface, the stress resultants must also include the contribution from the surface stresses. Following the works of Gurtin and Murdoch [3], the surface stresses are given by s

=

s

s 3

=

su 3,

+ 2(µ s

s)

s

+(

s) u

+

+

,

su

(3a)

,

(3b)

where , , = 1, 2, and are the Lame’s constants and is the surface stress parameter. The superscript ‘s’ is used to denote the quantities corresponding to the surface. Eq. (3) can be expanded to write individual surface stresses. s

s

µs

yy

+ 1+

u x

s

xx

+ 1+

v y

s

s xx

= 2µ s +

s

s

s yy

= 2µ s +

s

s

s xy

= 2µ s

s

s xz

=

s

w x

(7)

s yz

=

s

w y

(8)

xy

+

xx

+

s

+

s

yy

+

s

+

s

s

u y

(4)

(5) (6)

These stresses satisfy the surface equilibrium equation (see [3,40]) given by s i ,

+

i3

su ¨

=

(9)

i

where i = 1, 2, 3 and = 1, 2 . Eq. (9) will not be satisfied since we have taken zz to be zero. So to satisfy the surface equilibrium condition, zz is assumed to vary linearly through the thickness and the expression for the same is given in [49]. s xz

+

x zz

2. Non-local TSDT with surface stress effects The nonlocal TSDT with surface stress effects has recently been formulated and presented by authors in [49]. For the sake of brevity, a brief description is outlined here. The displacement field for TSDT [1] is given by

s yz

s w t2

y

=

+

x

attop

s yz

s w t2

y

atbottom

2 s xz

x

+

s xz

+

+

s yz

y

s w t2

s xz

x

attop

h

+

s yz

y

s w t2

atbottom

z

(10) 2

Composite Structures 226 (2019) 111216

K. Shiva, et al.

The strain-displacement relations, stress-strain relations obtained for TSDT are given in Appendix A. The stress resultants are given below:

Nxx Nyy Nxy

(0) xx (0) yy

A11 A12 A16 A12 A22 A26 A16 A26 A66

=

(0) xy (3) xx (3) yy

E11 E12 E16 + E12 E22 E26 E16 E26 E66

+ 2µ s +

s

(3) xy

Mxx Myy Mxy

B11 B12 B16 B12 B22 B26 B16 B26 B66

(11)

=

+

+

(0) xy (3) xx (3) yy

F11 F12 F16 F12 F22 F26 F16 F26 F66

(1) xx (1) yy

D11 D12 D16 D12 D22 D26 D16 D26 D66

(3) xy

T

=

+

(3) xx (3) yy

(12)

s

(0) xz

s

(0) yz

= s

(0) xz

s

(0) yz

( (

2b + h +

(2) xz

2a + h +

(2) yz

bh2 2 ah2 2

+

h3 12

+

h3 12

O11 + O21 O31

(3) xy

)+ )+

(

bh2 2

Nxy

( ( (

(2) xz

bh4 8

(2) yz

ah4 8

(1) xx (1) yy

Qx x

+ +

h3 12

+

h5 80

+

h5 80

=

(0) xy

(3) xx (3) yy

u0 x v0 y u0 y

+

x

=

y

c1

y

(0) xz

x

y

=

y

+

w0 y

x

+

w0 x

+ ,

+ + y

x

2w

x

x

0

0

(2) yz (2) xz

+

µ

2

xy

w0 x

+ 1 2

y

+2

Myy y

(N

2P xy

x y

+

x

(N

xx

+ Nyy 2P yy y2

w0 x w0 y

+ Nxy

w0 y

)

)

+ 1

µ

2

q=0

(19)

Qx = 0

(20)

Qy = 0

(21)

=M

c1 P

( ,

= 1, 2),

Q =Q

c2 R

( = 4, 5)

3.1. The Navier solution

y2

+2

2P xx x2

(18)

3. Analytical solutions

x2 2w

=0

In Eq. (22), and take the symbols x and y, M and P are normal stress resultants, Q and R are shear stress resultants. Eqs. (17)–(21) are the governing equations of equilibrium with non-local and surface effects for buckling analysis of laminated composites. Nxx , Nxy , Nyy are the constant in-plane edge loads and q is transverse loading. We can see that nonlocal parameter is coupled with constant in-plane edge loads and transverse loading in Eq. (19). In the present study, we present analytical solutions to these set of governing equations.

y

+

+ c1

(17)

(22)

(14)

y y

µ

M

y

=

(16)

where

x

(1) xy

v0 x

x

(3) xy (0) yz

,

y

+ 1

x

x

(1) xx (1) yy

Qy

+

Mxy

where (0) xx (0) yy

y

Mxy Mxx + x y

) ) ) )

Nyy

+

(1) xy

(13)

h3 12

ah2 2

K

Nxy Nxx + =0 x y

F11 F12 F16 F12 F22 F26 F16 F26 F66

(0) xy

H11 H12 H16 H12 H22 H26 H16 H26 H66

+

Qxz Qyz Rxz Ryz

(0) xx (0) yy

E11 E12 E16 E12 E22 E26 E16 E26 E66

U+ V

0

x Pxx Pyy Pxy

(15)

where U , V and K are the virtual strain energy, virtual work done by applied forces and virtual kinetic energy respectively. The equations of equilibrium are obtained as,

(1) xy

L11 L21 L31

+

Qij(k ) (1, z, z 2, z 3, z 4 , z 6 ) dz ,

where Aij are extensional stiffnesses, Dij are bending stiffnesses, Bij are bending extensional coupling stiffnesses, Eij , Fij , Hij contains fourth or higher power of thickness which are defined in terms of laminate stiffnesses as above. With these definitions, the equations of equilibrium are now derived using the principle of virtual displacements as,

0= (0) xx (0) yy

zk

i, j = 1, 2, 6

(1) xy

Z11 Z21 Z31

zk+ 1

k=1

(1) xx (1) yy

B11 B12 B16 B12 B22 B26 B16 B26 B66

+

N

{Aij , Bij , Dij , Eij, Fij , Hij} =

2w

0

The boundary conditions and displacement approximations are shown in Fig. 1 and Table 1 respectively. The ordered pair denotes the buckling mode of the laminate.

x y

=

c2

y

+

w0 y

x

+

w0 x

3.1.1. Stiffness coefficients Substitution of Navier’s approximation of displacement into the equations of equilibrium in terms of displacement, we get laminate

where c1 = 2 and c2 = 3c1. The expressions for Zi1, Li1 and Oi1 3h (i = 1, 2, 3) are given in Appendix A. 4

3

Composite Structures 226 (2019) 111216

K. Shiva, et al.

Fig. 1. SS-1 and SS-2 boundary conditions.

+ N0 (

Table 1 Approximation of displacements for SS-1 and SS-2 boundary conditions, where n m = a and = b . U0, V0, W0, X 0 and Y0 are coefficients that are to be de-

y (x ,

y y y y

S35= A 44

u0 (x , y, t ) = U0 (t )sin u0 (x , y, t ) = V0 (t )cos u0 (x , y, t ) = W0 (t )sin x (x , y , t ) = X0 (t )cos

y, t ) = Y0 (t )sin x cos y

y (x ,

x cos x sin x sin x sin

y y y y

Nyy

S11= A11 2 + A66 S12= (A12 + A66 ) S13=

2

c1 E11

S14= B11

2

+ B33

2

S43= S34 +

2

c1 E22

s )( 2h

2

+ E12 + 2E66 2

+ 2µ s +

s

(

c1 2µ s + c1 h4 32

2

+ bh

s

(

h4 3 32

c1 bh3

2

4

2

+

is

S31= S13 S32= S33= +

2

2

+ (2µ s +

s )( 2h

)

bh3 3 4

2

+ E12 + 2E66

2

+ 2µ s +

bh3 c1 8 3 (4µ s

s

)

(

s

h4 3 32

+

h6 (bc12 32

+ 4

+

4

+

+

(

c1 h4 2 32

+ ah

c1 ah3 2 4

2

c12 h7 3 448

c1 h4 b 3 8

2

c1

+

c12 h6 32

b

3

+

c12 bh6 2 32

a

3

2 2]

bh3 2 8

)

c1 h4 s 40(1 )

h3 12

3

+

2

c1 bh4 4

2

2 c1 h5 2 40

2

,

c12 h7 2 448

+

S51 = S15 s

h2 s 6(1 ) s

)

(ah

2

c1

c1 h4 s 40(1 ) c1 h5 3 80

+

+ D22

2

)

3

ah3 2 8

) 2

+

c12 h7 3 448

c1 h4 a 3 8

+

c12 h6 32

S54= S45

+

c 1 2µ s +

2]

+

)

ah3 3 4

c12 h7 2 448

+

2

+ 2µ s +

s

h3 2 12

c1 h5 2 40

2 2

c1 ah4 4

2

c12 bh6 2 32

where

)

Aij= Aij

c1 Dij , B = Bij

c1 Eij , Dij = Dij

Fij= Fij

c1 Hij , Aij = Aij

c1 Dij = Aij

c1 Fij

(i , j = 1, 2, 6)

2c1 Dij + c12 Fij

(i, j = 1, 2, 6) (24)

s)

ah3 S23 c1 8 3 (4µ s + s ) A55 2 + A 44 2 + c12 [H11 4 4

(

+ 2µ s +

2a)

+

2 ))

+ (F12 + 2F66)

+ D66

S52= S25 + 2µ s +

S55= A 44 + D33

+ B22

c1 h 4 s 40(1 )

s

2

S45= (D12 + D66)

S24= S15 S25= B66

h2 s 6(1 )

+ 2µ s +

2b)

+

(

S44= A55 + D11

S53= S35 +

S15= (B12 + B26) S21= S12 S22= A66 2 + A22 S23=

+ (2µs +

+

(bh

s

+

S42= S24

stiffness matrix for a particular boundary condition, which in turn depends on the lamination scheme. For buckling analysis, we assume that the only applied loads are the in-plane forces (q = 0, Nxy = 0 ): For uniaxial buckling: Nxx = N0, Nyy = 0 (k = 0 ). Nxx

3

c1 [F22

2

+ (F12 + 2F66)

c1 h5 3 80

s

S41= S14 + 2µ s +

y, t ) = Y0 (t )sin x cos y

For biaxial buckling: Nxx = N0, Nyy = N0 (k = 1) where k = the buckling load ratio. The stiffness coefficients Sij for the SS-1 case are:

3

c1 [F11

+ 2µ s +

SS-2

u0 (x , y, t ) = U0 (t )cos x sin v0 (x , y, t ) = V0 (t )sin x cos w0 (x , y, t ) = W0 (t )sin x sin x (x , y , t ) = X0 (t )cos x sin

+ k 2)(1 + µ (

S34= A55

termined.

SS-1

2

+ 2(H12 + 2H66)

+ 2µ s +

s

2 2

h7 (c 2 448 1

4

+ H22

4]

+ c12

4)

Dij= Dij Aij= Aij

c1 Fij = Dij c2 Dij = Aij

2c1 Fij + c12 Hij 2c2 Dij +

c12 Fij

(i, j = 1, 2, 6) (i , j = 4, 5)

The stiffness coefficients Sij for the SS-2 case are:

ac12 4 ) (23)

4

Composite Structures 226 (2019) 111216

K. Shiva, et al.

S11= A11 2 + A66 S12= (A12 + A66 ) S13=

2

c1 3E16

S23=

+ (2µs +

2 2

2,

+ B26 + A22

2

2

c1 E16

s )( 2h

2

+ E26

+ 2µ s +

S14= 2B16 S15= B16 S22= A66

2

c1 2µ s +

(

s

2b)

+

c1 h4 32

2

s

(

h4 3 32

+

bh3 3 4

Fij= Fij

)

)

2

c1 2µ s +

s

(

h4 3 32

+

)

ah3 3 4

S24= S15 + 2µ s +

S25= 2B26

S33= A55 +

2

+

s)

ah3 c1 8 3 (4µ s

+

s)

8

4

h6 (bc12 32

+ N0 (

2

2

+ A 44

c1 h4 s 40(1 )

+

c1 h 4 2 32

3 (4µ s

S31= S13 + c1 S32= S23 +

bh3

(

s

+

+ c12 [H11

4

4

+ 2µ s +

+ ac12

4

2 2

+

S34= A55

c1 [F11

+ 2µ s + S35= A 44

3

c1 [F22

+ 2µ s +

S41= S14 + 2µ s +

+

+

(bh

s

s

h7 (c 2 448 1

4

2

+

+ H22

4]

+ c12

4)

c12 h7 3 448 2

c1

+

c12 h6 32

b

3

S43= S34 +

(

S44= A55 + D11 +

2µ s

2

s

+

+ D66 h3 12

S45= (D12 + D66)

c1 h 4 a 3 8

(

h2 s 6(1 )

+ 2µ s +

s

)

3

+

c12 h6 32

a

3

)

Aij= Aij

c2 Dij = Aij

+

c1 bh4 4

2

(ah

2

c1 h 4 s 40(1 ) c1 h5 3 80

+

+ D22

2

c1

)

3

+

c12 h7 2 448

+

c12 bh6 2 32

)

ah3 2 8

+

c12 h7 3 448

+

c12 h7 2 448

+

2

+ 2µ s +

2

c1 h 4 a 3 8

s

(i, j = 1, 2, 6) (i , j = 4, 5)

S12 S22 S32 S42 S52

S13 S23 S33 S43 S53

S14 S24 S34 S44 S54

S15 S25 S35 S45 S55

U0 0 V0 0 W0 = 0 0 X0 0 Y0

(27)

W X Y

0 = 0 0

(28)

S13 S22 S S S31 + 13 21 S12 S21 S11 S22

S23 S11 S32 + S33 S12 S21

(29)

b=

S24 S12 S11 S22

S14 S22 S S S31 + 14 21 S12 S21 S11 S22

S24 S11 S32 + S34 S12 S21

(30)

c=

S25 S12 S11 S22

S15 S22 S S S31 + 15 21 S12 S21 S11 S22

S25 S11 S32 + S35 S12 S21

(31)

d=

S23 S12 S11 S22

S13 S22 S S S41 + 13 21 S12 S21 S11 S22

S23 S11 S42 + S43 S12 S21

(32)

e=

S24 S12 S11 S22

S14 S22 S S S41 + 14 21 S12 S21 S11 S22

S24 S11 S42 + S44 S12 S21

(33)

f=

S25 S12 S11 S22

S15 S22 S S S41 + 15 21 S12 S21 S11 S22

S25 S11 S42 + S45 S12 S21

(34)

g=

S23 S12 S11 S22

S13 S22 S S S51 + 13 21 S12 S21 S11 S22

S23 S11 S52 + S53 S12 S21

(35)

+

c12 h6 32

a

3

h=

S24 S12 S11 S22

S14 S22 S S S51 + 14 21 S12 S21 S11 S22

S24 S11 S52 + S54 S12 S21

(36)

i=

S25 S12 S11 S22

S54= S45 S55= A 44 + D33

2c2 Dij + c12 Fij

S23 S12 S11 S22

S51 = S15 s

2c1 Fij + c12 Hij

a=

2

2 c1 h5 2 40

2

,

S52= S25 + 2µ s + S53= S35 +

c1 h4 s 40(1 )

(i, j = 1, 2, 6)

where

S42= S24 h2 s 6(1 )

c1 Fij = Dij

a b c d e f g h i

2 2]

bh3 2 8

2c1 Dij +

(i , j = 1, 2, 6) c12 Fij

For a nontrivial solution of Eq. (27), the determinant of coefficient matrix must be zero, which gives an expression for buckling load. The other way of solving for buckling load is “Static condensation”, which is followed here [1]:

2] c1 h4 b 3 8

Dij= Dij

S11 S21 S31 S41 S51

2 ))

c12 h7 3 448

c1 Dij = Aij

c1 Fij

3.1.2. Solution of the Eigenvalue problem The eigenvalue problem that results from the above formulation is

(25)

+ (F12 + 2F66)

c1 h5 3 80

s

2 2

+ 2(H12 + 2H66)

+ (F12 + 2F66)

c1 h5 3 80

s

)

c1 ah3 2 4

4)

+ k 2)(1 + µ ( 3

2

+ ah

c1 Hij , Aij = Aij

c1 Eij , Dij = Dij

Eqs. (23) and (25) gives the stiffness coefficients for SS-1 and SS-2 case respectively, considering nonlocal and surface stress effects. For SS-1 and SS-2 case, all stiffness coefficients except S12, S15, S21, S24, S42, S45, S51 and S54 have stiffness contributions from surface terms. S33 coefficient in both cases consists of nonlocal parameter and is responsible for nonlocal effect on buckling load.

S21 = S12 + (2µ s + s )( 2h + 2 2a)

+ 3E26

c1 Dij , B = Bij

(26)

c1 bh3 2 4

2

+ bh

Aij= Aij

h3 2 12

c1 h5 2 40

c1 ah4 4

2

S15 S22 S S S51 + 15 21 S12 S21 S11 S22

S25 S11 S52 + S55 S12 S21

(37)

Using Eq. (28) we get:

c12 bh6 2 32

X=

where

5

fg ei

di W fh

(38)

Composite Structures 226 (2019) 111216

K. Shiva, et al.

dh ei

Y=

eg W fh

(39)

Now substituting X , Y into aW + bX + cY = 0 and making coefficient of W to zero, we obtain

fg ei

a=

di b fh

dh ei

eg c fh

(40)

where

a=

S23 S12 S11 S22

S33 = A55 +

S13 S22 S S S31 + 13 21 S12 S21 S11 S22

2

2

+ A 44

c1 h4 40(1

s

)

h7 c12 448

4

2

1+µ

+ c12 H11

4

4

2 2

4

+

+

(41)

2 2

+ 2 H12 + 2H66 + 2µ s +

h6 + bc12 32

c12 4

+

S23 S11 S32 + S33 S12 S21

4

+

+ H22

4

s

ac12 4

+ N0

2

+k

Fig. 2. The coordinate system and boundary conditions used on the simply supported edges for Levy solutions.

2

u0 (x , y ) Um (x )sin y v 0 (x , y ) Vm (x )cos y w0 (x , y ) = Wm (x )sin y Xm (x )sin y x (x , y ) Ym (x )cos y y (x , y )

2

+

(42)

Now rewriting Eq. (40) using Eqs. (41) and (42)

(l + j )(ei

N0 =

fh)

b (di fg ) c (eg L fh) mn mn (ei

dh)

Here = m / b . Um, Vm, Wm, Xm and Ym are the functions to be determined. Substitution of Eq. (44) into Eqs. (17)–(21) and doing some simplification, we get a system of ordinary differential equations along x-axis. So here we have converted partial differential equations to ordinary differential equations.

(43)

where the negative sign indicates that N0 is a compressive in-plane load. We have 2 + mn= (1 + µ ( Lmn= ( 2 + k 2)

j=

S23 S12 S11 S22

l= A55 +

2

2 ))

S13 S22 S S12 S21 31

+

2

+

+ A 44

c1 h 4 s 40(1 )

+ 2µ s +

4

s

+

4

S13 S21 S23 S11 S S11 S22 S12 S21 32 2 4 c1 [H11 + 2(H12

+

h7 (c 2 448 1

+ 2H66)

2 2

+ H22

4]

2 2

4

+ c12

4)

+

h6 (bc12 32

4

+ ac12

(44)

4)

There is a unique value of N0 for each pair of m and n, critical buckling load is the smallest of all N0 (m , n) .

Um = T1 Um + T2 Vm + T3 Wm + T4 Wm + T5 Xm + T6 Ym

(45)

Vm = T7 Um + T8 Vm + T9 Wm + T10 Wm + T11 Xm + T12 Ym

(46)

Wm = T13 Um + T14 Vm + T15 Wm + T16 Wm + T17 Xm + T18 Ym

(47)

Xm = T19 Um + T20 Vm + T21 Wm + T22 Wm + T23 Xm + T24 Ym

(48)

Ym = T25 Um + T26 Vm + T27 Wm + T28 Wm + T29 Xm + T30 Ym

(49)

Prime denotes derivative with respect to x. The nonlocal coefficient matrix terms are given in Appendix A. In order to solve Eqs. (45)–(49), we introduce the following variables:

3.2. Antisymmetric cross-ply laminates with two opposite edges simply supported

Z1 = Um, Z2 = Um, Z3 = Vm, Z4 = Vm, Z5 = Wm, Z6 = Wm Z7 = Wm, Z8 = Wm , Z9 = Xm , Z10 = Xm , Z11 = Ym, Z12 = Ym

The application of Naviers’s method is limited to simply supported boundary conditions. Levy’s solution procedure along with nonlocal definition of stress and state space concept is applied for laminates with two opposite edges simply supported and the other two edges with arbitrary boundary conditions. Let us consider a laminate with edges y = 0, b simply supported, while the other edges i.e., x = ± a/2 may have combinations of free(F), simply supported(S) and clamped(C) boundary conditions as shown in Fig. 2 The simply supported (SS-1) boundary conditions at the edges y = 0, b can be satisfied through the following displacement expressions:

(50)

Now using above variables, Eqs. (45)–(49) can be expressed in the matrix form:

{Z } = [A]{Z } where the coefficient matrix [A] is

6

(51)

Composite Structures 226 (2019) 111216

K. Shiva, et al.

0 1 0 0 0 0 0 0 0 0 0 0 T1 0 0 T2 0 T3 0 T4 T5 0 0 T6 0 0 0 1 0 0 0 0 0 0 0 0 0 T7 T8 0 T9 0 T10 0 0 T11 T12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 [A] = 0 0 0 0 0 0 0 1 0 0 0 0 0 T13 T14 0 T15 0 T16 0 0 T17 T18 0 0 0 0 0 0 0 0 1 0 1 0 0 T19 0 0 T20 0 T21 0 T22 T23 0 0 T24 0 0 0 0 0 0 0 1 0 0 0 1 0 T25 T26 0 T27 0 T28 0 0 T29 T30 0

0=

0=

+ Nxy

[Q (x )] =

0 e

11 x

e

12 x

0=

(55)

0=

w

Clamped: u 0 = v0 = w0 = x0 = x = y = 0 . Fixed: Nxx = Nxy = Mxx = Pxx = Mxy c1 Pxy = 0 .

Qx

c2 Rx + c1

(

+

Pxy y

+ Mx

w0 y

nx

w0

w0 } dxdy

n x w0

Pxy x

w0 ny y

x,x

Pxy

ny +

y

+ Pxy

+ Mxy

w0 ny x

+ Pxy

x , y ) dxdy

e

w0 nx y

(Mxx n x

ds

x

+ Mxy n y

(61) y ) ds

e

(Q y

y

+ My

+ Mxy

y, y

y , x ) dxdy

e

(Myy n y

y

+ Mxy n x

x ) ds

w

w

2w

(u, v , w, x , y , x y , x , y ) are the primary variables. We have used Lagrange interpolation functions for approximation of (u, v , x , y ) and Hermite interpolation function for approximation of w. The finite element approximations are as follows:

(56)

For homogeneous equations, nontrivial solution exists if the determinant of [K ] is zero It may be computationally more convenient to substitute {D} = [U ] 1 {Z (0)} (see [90,91]).

K =0 U

x

+ Nxy

2] q

(63)

)=0

|([K ][U ] 1)| = 0

(Q x

µ

n y w0

ny +

+ Pyy

w0 x

w0 + Nxx

[1

Nxx , Nxy , Nyy are the constant in-plane edge loads and q is transverse loading. We note from the boundary terms in Eqs. (59)–(63) that

Now by imposing the x = ± a/2 edge boundary conditions on Eq. (53), we get a set of homogeneous algebraic equations of the form

[K ]{D} = 0

y

w0, x

(62)

The boundary conditions for edges x = ± a/2 are as follows: Simply Supported: v0 = w0 = y = Nxx = Mxx = Pxx = 0

Pxx x

e

Pyy

+

w0 nx x

c1 Pxx

w0 y

+ Nyy

Pxx nx x

+ c1

2x

0

w0 x

+ Nxy

w0 y

+ Nxy

(60)

w0, y

Qx nx + Qy n y

where [U ] is a matrix of eigen vectors of [A], and [Q (x )] is another fundamental matrix which is diagonal.

e

w0 y

+ Nyy

w0 x

c1 (Pxx w0, xx + Pyy w0, yy + 2Pxy w0, xy )

(54)

1x

w0 x

n x Nxy v0 + n y Nyy v0 ds

e

Qx w0, x + Qy w0, y + Nxx

e

(52)

Here D is an unknown vector of constants which we have to find using other edge boundary conditions. When the eigen values of coefficient matrix [A] are distinct, the fundamental matrix [ (x )] is given by:

e

[Nxy v0, x + Nyy v0, y ] dxdy

e

0=

(53)

[ (x )] = [U ][Q (x )]

n x Nxx u 0 + n y Nxy u 0 ds

e

(59)

The general solution to Eq. (51) is given by: [89]

{Z {x }} = [ (x )]{D}

[Nxx u0, x + Nxy u 0, y ] dxdy

e

m

u x , y, t

(57)

Uj (t )

(1) j

x, y

Vj (t )

(1) j

x, y

j=1 m

(58)

v x , y, t

Here finding the critical buckling load is not straight forward. Some of the elements of coefficient matrix [A] contains buckling load N0 . The value of N0 for which the determinant becomes zero or changes sign is critical buckling load.

j=1

n

w x, y, t

¯ j (t )

j

x, y

j=1

4. Finite element formulation

n x

The weak forms of governing equations of motion are developed first. Substitution of finite element approximations into the developed weak forms gives finite element equations. The weak forms for the governing equations of motion are obtained by multiplying Eqs. (17)–(21) with u 0 , v0, w0, x and y respectively, integrating over the element domain and applying integration by parts to weaken the differentiability of u 0 , v0, w0, x and y . Here the virtual displacements ( u 0 , v0, w0, x and y ) takes the role of weight functions in the development of weak forms:

x , y, t

(2) j (t ) j

x, y

(2) j (t ) j

x, y

j=1 n y

x , y, t j=1

(1) j

x , y and

(2) j

(64)

(65)

(66)

(67)

(68)

x , y both represent Lagrangian interpolation func-

tions. In general we can use different Lagrangian interpolation functions to represent inplane displacements and shear [92]. But in this

7

Composite Structures 226 (2019) 111216

K. Shiva, et al.

Table 2 Effect of E1/ E2 ratio on the uniaxial buckling loads, N = No (a/ h = 10 , n = Number of layers). Source

a2

of symmetric cross-ply laminate without non-local and surface stress effects i.e., µ = 0 and

E2 h3

n

=0

N

E1/E2 = 3

10

20

30

40

Elasticity theory [93] Local FEM [94] Local analytical HSDT [94] Non-local analytical (Present) Non-local FEM (Present)

3

5.3044 5.3950 5.3933 5.3898 5.3995

9.7621 9.9427 9.9406 9.8325 9.8785

15.0191 15.3001 15.2980 14.8897 14.9578

19.3040 19.6752 19.6740 18.8778 18.9254

22.8807 23.3398 23.3400 22.1210 22.3012

Elasticity theory [93] Local FEM [94] Local analytical HSDT [94] Non-local analytical (Present) Non-local FEM (Present)

5

5.3255 5.4112 5.4096 5.4033 5.5500

9.9603 10.1524 10.1500 10.0663 10.1150

15.6527 16.0100 16.0080 15.7217 15.7912

20.4663 21.0023 20.99990 20.4630 20.5452

24.5959 25.3086 25.3080 24.5112 24.5925

Table 3 Comparison of dimensionless uniaxial buckling loads, N = No

a2 E2 h3

study we have used the same degree of interpolation for (u, v ) and ( x , y ) . Substitution of approximations Eqs. (64)–(68) into weak forms Eqs. (59)–(63), we obtain the following finite element equations.

of symmetric

cross-ply (0°/90°/90°/0°) laminate with out non-local and surface stress effects i.e, µ = 0 and s = 0(a/ h = 10) . a/h

Local Navier [89]

Non-local Navier (present)

Non-local FEM (present)

5 10 20 50 100

11.997 23.340 31.660 35.347 35.953

11.9970 23.3400 31.6595 35.3467 35.9526

12.1870 23.4821 31.9484 35.8356 36.1315

Table 4 Dimensionless uniaxial buckling loads, N = No

a2 E2 h3

where K is stiffness matrix, F is force vector and vector.

In this section, we present several examples to illustrate the effect of non-locality and surface stress on dimensionless critical buckling loads of cross-ply and angle-ply laminates for uniaxial (k = 0) and biaxial (k = 1) edge loads. Specifically, the following three examples are considered:

of symmetric cross-ply

a/h = 10

a /h = 20

a /h = 50

a/h = 100

0 0 0 0

0 1.7 3.4 6.8

23.3400 23.3397 23.3394 23.3387

31.6596 31.6591 31.6586 31.6577

35.3467 35.3462 35.3457 35.3446

35.9526 35.9521 35.9515 35.9504

1 3 5

0 0 0

19.4924 14.6592 11.7466

30.1707 27.5770 25.3939

35.0698 34.5288 34.0043

35.8818 35.7410 35.6013

1 3 5

1.7 3.4 6.8

19.4924 14.6588 11.7459

30.1703 27.5762 25.3924

35.0693 34.5278 34.0022

35.8813 35.7399 35.5991

is displacement

5. Numerical examples

(1)Symmetric cross-ply (0°/90°/90°/0°) laminated plates. (2)Antisymmetric cross-ply (0°/90°/0°/90°) laminated plates. (3)Antisymmetric angle-ply (30°/ 30°/30°/ 30°) laminated plates.

N

s

(69)

[K][ ] = [F]

(0°/90°/90°/0°) laminate considering non-local and surface effects. µ

s

For all the numerical examples the following material properties are G12 = E2 = 4.375 × 103 MPa, E1 = 175 × 103 MPa, considered: 3 3 G23 =2.1875 × 103 MPa, G13 = 2.625 × 10 MPa, 2.625 × 10 MPa, 12 = 0.25, 13 = 0.25, 21 = (E2/ E1) × 12 . The critical buckling mode for the above mentioned laminates both for uniaxial and biaxial buckling is obtained. The buckling load ratio, which is the ratio of the buckling 0 ) to that obtained using load obtained using non-local elasticity ( µ the local elasticity theory ( µ = 0 ) is computed. A square plate (a/ b = 1) is considered for the analysis. The effect of orthotropy and number of layers on the dimensionless buckling load is investigated. The SS-1

Fig. 3. Dimensionless critical buckling load versus a/ h for different non-local parameters for symmetric cross-ply laminate. (a) uniaxial (b) biaxial. 8

Composite Structures 226 (2019) 111216

K. Shiva, et al.

boundary conditions are considered for analysis of symmetric cross-ply and antisymmetric cross-ply laminate, whereas SS-2 boundary conditions are considered for antisymmetric angle-ply laminate. Antisymmetric cross-ply laminate with two parallel edges simply supported and other edges arbitrary is considered for Levy solution. 5.1. Symmetric cross-ply (0°/90°/90°/0°) laminates Simply supported symmetric cross-ply laminated plate subjected to uniaxial and biaxial compressive load are considered. Tables 2 and 3 show comparison of buckling loads obtained from the present non-local formulation with results from the existing literature for different E1/ E2 and a/ h values. Table 4 shows the variation of dimensionless buckling a2 load N = No 3 with increasing values of non-local parameter µ , surE2 h

face stress parameter s , for various a/ h values. It is observed that an increase in non-local parameter and the surface stress parameter decreases the critical buckling load. This decrease in the value may be attributed to the reduction in stiffness because of the inclusion of nonlocal and surface effects. Fig. 3(a) and (b) shows the variation of dimensionless uniaxial and biaxial buckling load N with increasing values of a/ h , for different nonlocal parameters. In Fig. 4 the variation of buckling load ratio with increasing values of a/ h is plotted. Table 5 shows the effect of E1/ E2 on the buckling load. It is observed that an increase in the modulus in the direction of in-plane loading increases buckling load. Fig. 5(a) and (b) shows the variation of dimensionless uniaxial and biaxial buckling load N with increasing values of a/ h for different E1/ E2 values.

Fig. 4. Variation of uniaxial and biaxial buckling load ratio with a/ h for different non-local parameters for symmetric cross-ply laminate.

Table 5 Effect of E1/ E2 ratio on the uniaxial buckling loads, N = No

a2 E2 h3

of symmetric

cross-ply (0°/90°/90°/0°) laminate considering non-local and surface effects (a/ h = 10) . µ

N

s

E1/E2 = 10

E1/E2 = 20

E1/E2 = 30

E1/E2 = 40

0 0 0 0

0 1.7 3.4 6.8

9.9406 9.9405 9.9403 9.9401

15.2984 15.2982 15.2980 15.2976

19.6748 19.6741 19.6738 19.6733

23.340 23.3397 23.3394 23.3387

1 3 5

0 0 0

8.3018 6.2434 5.0029

12.7764 9.6085 7.6994

16.4310 12.3569 9.9017

19.4924 14.6592 11.7466

1 3 5

1.7 3.4 6.8

8.3018 6.2432 5.0027

12.7763 9.6082 7.6990

16.4308 12.3566 9.9012

19.4921 14.6588 11.7459

5.2. Antisymmetric cross-ply (0°/90°/0°/90°) laminates An antisymmetric cross-ply laminated plate subjected to uniaxial and biaxial compressive loads is considered. It is observed that with an increase in non-local and surface parameter there is a decrease in critical buckling load. Table 6 shows comparison of buckling load values with already existing literature, for different E1/ E2 values and Table 8 shows comparison for different boundary conditions. Tables 7 and 9 a2

shows the variation of dimensionless buckling load N = No 3 with E2 h increasing values of non-local parameter µ , surface stress parameter s , for various a/ h values.

Fig. 5. Dimensionless critical buckling load versus a/ h for different E1/ E2 values for symmetric cross-ply laminate. (a) uniaxial (b) biaxial.

9

Composite Structures 226 (2019) 111216

K. Shiva, et al.

Table 6 Effect of E1/ E2 ratio on the uniaxial buckling loads, N = No stress effects (a/ h = 10 , n is number of layers). Source

a2 E2 h3

of antisymmetric cross-ply (0°/90°…) laminate without non-local and surface

n

N

E1/E2 = 3

10

20

30

40

3D elasticity theory [93] Local FEM [94] Local analytical HSDT [94] Non-local analytical (Present) Non-local FEM (Present)

2

4.6948 4.7769 4.7749 4.7824 4.8713

6.1181 6.2756 6.2721 6.3299 6.4200

7.8196 8.1198 8.1151 8.2544 8.5483

9.3746 9.8751 9.8695 10.0923 10.2890

10.8170 11.5690 11.5630 11.8695 12.0075

3D elasticity theory [93] Local FEM [94] Local analytical HSDT [94] Non-local analytical (Present) Non-local FEM (Present) 3D elasticity theory [93] Local FEM [94] Local analytical HSDT [94] Non-local analytical (Present) Non-local FEM (Present)

4

5.1738 5.2540 5.2523 5.2580 5.4380 5.2670 5.3440 5.3420 5.3450 5.5350

9.0164 9.2344 9.2315 9.2757 9.3650 9.6051 9.7788 9.7762 9.7989 9.9061

13.743 14.2580 14.2540 14.3608 14.5510 15.0010 15.3550 15.3520 15.4067 15.6015

17.783 18.6710 18.6670 18.8375 18.9385 19.6390 20.2038 20.2010 20.2889 20.4665

21.2800 22.5820 22.5790 22.8137 22.9818 23.6690 24.4620 24.4600 24.5807 24.7698

Table 7 Dimensionless uniaxial buckling loads, N = No

6

a2 E2 h3

increasing values of a/ h for different E1/ E2 values. It is observed that an increase in elastic modulus in the direction of loading increases the value of critical buckling load.

of antisymmetric cross-ply

(0°/90°/0°/90°) square laminate considering non-local and surface effects. µ

0 0 0 0 1 3 5 1 3 5

N

s

0 1.7 3.4 6.8 0 0 0 1.7 3.4 6.8

5.3. Antisymmetric angle-ply (30°/ 30°/30°/ 30°) laminates

a/h = 10

a /h = 20

a /h = 50

a/h = 100

22.8137 22.8134 22.8130 22.8124 19.0528 14.3286 11.4817 19.0526 14.3282 11.4810

28.1627 28.1622 28.1617 28.1607 26.8383 24.5310 22.5891 26.8378 24.5302 22.5875

29.8752 29.8746 29.8741 29.8730 29.6411 29.1839 28.7405 29.6406 29.1828 28.7384

28.0361 28.0356 28.0350 28.0339 24.9809 27.8711 27.7621 27.9804 27.8700 27.7600

An antisymmetric angle-ply laminated plate subjected to uniaxial and biaxial compressive edge loads are considered. Table 10 shows the variation of dimensionless buckling load N = No Table 9 Dimensionless uniaxial buckling loads, N = No

ditions (a/ h = 10 , n is number of layers).

a2 E2 h3

with increasing

of antisymmetric cross-ply

(0°/90°) laminate considering non-local effects for various boundary conditions (a/ h = 10 ). N

µ

0 1 3 5

The variation of critical buckling load with a/ h for different nonlocal parameters is shown in Fig. 6(a) and (b). Fig. 7 shows the variation of the buckling load ratio with increasing values of a/ h . Fig. 8(a) and (b) shows the variation of uniaxial and biaxial buckling load N with Table 8 Dimensionless uniaxial buckling loads, N = No

a2 E2 h3

a2 E2 h3

SS

SC

CC

FF

FS

FC

11.237 9.382 7.045 5.468

16.309 13.455 9.932 8.138

19.937 16.448 12.281 10.028

4.699 3.889 2.885 2.345

5.221 4.318 3.216 2.545

6.028 4.973 3.708 3.026

of antisymmetric cross-ply (0°/90°/ …) laminate without nonlocal and surface effects for various boundary con-

Theory

n

SS

SC

CC

FF

FS

FC

TSDT (Analytical-Present) TSDT (FEM-Present) FSDT [89,95] CLPT [89,95]

2

11.237 11.229 11.353 12.957

16.309 16.108 16.437 21.116

19.937 19.916 20.067 31.280

4.699 4.678 4.851 5.425

5.221 5.231 5.351 6.003

6.028 6.016 6.166 6.968

TSDT(Analytical -Present) TSDT(FEM -Present) FSDT [89,95] CLPT [89,95]

10

25.322 25.312 25.450 35.232

32.496 32.487 32.614 59.288

34.627 34.527 34.837 89.770

11.973 11.956 12.092 16.426

12.323 12.325 12.524 17.023

14.235 14.211 14.358 19.389

10

Composite Structures 226 (2019) 111216

K. Shiva, et al.

Fig. 6. Dimensionless critical buckling load versus a/ h for different non-local parameters for antisymmetric cross-ply laminate. (a) uniaxial (b) biaxial. Table 10 Dimensionless uniaxial buckling loads, N = No

a2 E2 h3

antisymmetric angle-ply

(30°/ 30°/30°/ 30°) square laminate considering non-local and surface effects (a/ h = 10) . µ

N

s

a/h = 10

a/h = 20

a/h = 50

a /h = 100

0 0 0 0

0 1.7 3.4 6.8

33.5397 33.5395 33.5392 33.5387

45.7612 45.7608 45.7603 45.7594

53.0971 53.0965 53.0960 53.0949

69.0701 69.0695 69.0690 69.0679

1 3 5

0 0 0

28.0106 21.0653 16.8799

43.6092 39.8602 36.7047

52.6806 51.8685 51.0805

68.9340 68.6635 68.3950

1 3 5

1.7 3.4 6.8

28.0104 21.0650 16.8794

43.6088 39.8594 36.7033

52.6806 51.8674 51.0784

68.9335 68.6624 68.3929

Fig. 7. Variation of uniaxial and biaxial buckling load ratio with a/ h for different non-local parameters for antisymmetric cross-ply laminate.

direction of loading increases the value of critical buckling load. values of non-local parameter µ , surface stress parameter s , for different a/ h values. The variation of critical buckling load with a/ h for different non-local parameters is shown in Fig. 9(a) and (b). Fig. 10 shows the variation of the buckling load ratio with increasing values of a/ h . Fig. 11(a) and (b) shows that an increase in elastic modulus in the

6. Conclusions Equations of equilibrium of Reddy’s third order shear deformation theory for the buckling analysis of laminated composite plates are derived considering Eringen’s non-local differential constitutive model

Fig. 8. Dimensionless critical buckling load versus a/ h for different E1/ E2 values for antisymmetric cross-ply laminate. (a) uniaxial (b) biaxial.

11

Composite Structures 226 (2019) 111216

K. Shiva, et al.

Fig. 9. Dimensionless critical buckling load versus a/ h for different non-local parameters for antisymmetric angle-ply laminate. (a) uniaxial (b) biaxial.

and Gurtin’s surface stress model. Equations of motion are analytically solved to obtain an expression for buckling loads of laminates with all edges simply supported. Antisymmetric cross-ply laminates are solved considering two parallel edges simply supported and other edges as arbitrary. Numerical results are presented to bring out the parametric effects such as the non-local parameter, modular ratio E1/ E2 , side-tothickness ratio a/ h on buckling behavior of laminated composite plates with different lamination schemes. The inclusion of non-local elasticity and surface stress reduces the buckling loads, with surface stress having relatively very less effect. It is observed that as a/ h value of the laminate increases (i.e., as the plate gets thinner) the difference between non-local elasticity solution and classical elasticity solution decreases, which means the non-local effect is more profound in the case of thick plates. Variation of buckling load ratio [ratio of the buckling load obtained using non-local elasticity 0 ) to that obtained using local elasticity theory ( µ = 0 )] is in(µ significant with the change in elastic modulus, thickness, lamination scheme of a selected laminated plate. Increase in material anisotropy (i.e., increase in E1/ E2 ratio) increases the critical buckling load.

Fig. 10. Variation of uniaxial and biaxial buckling load ratio with a/ h for different non-local parameters for antisymmetric angle-ply laminate.

Fig. 11. Dimensionless critical buckling load versus a/ h for different E1/ E2 values for antisymmetric angle-ply laminate. (a) uniaxial (b) biaxial.

12

Composite Structures 226 (2019) 111216

K. Shiva, et al.

Appendix A A.1. Strain displacements relations

u , x

v y

xx

=

xy

=

1 2

u v + , y x

xz

=

yz

=

1 2

v w + , z y

zz

=

xx yy

=

yy

(0) xx (0) yy

xy

yz xz

=

(70)

(1) xx (1) yy

+z

(0) xy

u w + z x w z (3) xx (3) yy

+ z3

(1) xy

(0) yz

=

1 2

(2) yz

+ z2

(0) xz

(3) xy

(2) xz

(71)

where (0) xx (0) yy

u0 x v0 y

=

(0) xy

u0 y

,

x

(3) xx (3) yy

x

=

y

c1

y

(3) xy

x

y

(0) yz

=

(0) xz

where c1 =

+

w0 y

x

+

w0 x

4 3h2

2w

+

y

=

y

x

x

y

y

+

(72)

x

0

x2 2w

+ y

+

y

x

(1) xy

v0 x

+

x

(1) xx (1) yy

0

y2

+2

2w

(2) yz

,

0

(73)

x y

=

(2) xz

c2

y

+

w0 y

x

+

w0 x

(74)

and c2 = 3c1.

A.2. Stress strain relations xx yy xy

Q11 Q12 Q16 = Q12 Q22 Q26 Q16 Q26 Q66

{ }

xx yy

yz

,

xz

xy

Q44 Q45 Q45 Q55

=

yz xz

(75)

where

Q11 = Q11cos4 + 2(Q12 + 2Q66)sin2 cos2 + Q22sin4 Q12 = (Q11 + Q22 4Q66)sin2 cos2 + Q12 (sin4 + cos4 ) Q16 = (Q11 Q12 2Q66)sin cos3 + (Q12 Q22 + 2Q66)sin3 cos Q22 = Q11sin4 + 2(Q12 + 2Q66)sin2 cos2 + Q22cos4 )sin3

Q26 = (Q11 Q12 2Q66 Q66 = (Q11 + Q22 2Q12 Q44 = Q44cos2 + Q55sin2 Q45 = (Q55 Q44)cos sin

Q55 = Q44sin2 Q11 =

E1 1

12 21

cos + (Q12 2Q66)sin2 cos2

(76)

cos3

Q22 + 2Q66)sin + Q66 (sin4 + cos4 ) (77)

+ Q55cos2 ,

Q12 =

12 E1

1

Q16 = Q26 = 0,

12 21

,

Q22 =

Q44 = G23,

E2

,

Q66 = G12,

Q45 = G12,

Q55 = G13

1

12 21

(78)

where is the orientation measured in counterclockwise from the positive x-axis to the fiber direction, E1 and E2 are elastic moduli, poisson’s ratios, and G12, G13 and G23 are the shear moduli. 13

12

and

21

are

Composite Structures 226 (2019) 111216

K. Shiva, et al.

A.3. Stress resultants (0) xx

Z11 Z21 = Z31

L11 L 21 = L31

(0) yy

(1) xx bh

(b + h ) +

(1) yy ah

(a + h ) +

+

(3) h4 xx ( 32

+

(3) h4 yy ( 32

+

bh3 ) 4

+

s (b

+ h)

+

ah3 ) 4

+

s (a

+ h) (79)

0 h6 s 6(1 ) h6 s 6(1 )

O11 O21 = O31

( (

2w

2w

+

x2 2w

y2 2w

+

x2

y2

)+p ) + p¯

(80)

0 h4 s 40(1 ) h4 s 40(1 )

( (

2w

x2 2w

x2

+ +

2w

y2 2w

y2

)+r ) + r¯

(81)

0

p = 2µ s +

s

p¯ = 2µ s +

s

r = 2µ s +

s

r¯ = 2µs +

s

(0) xx bh

+

(1) xx

h3 bh2 + + 12 2

(3) xx

h5 bh4 + 80 8

+

s

(0) yy ah

+

(1) yy

h3 ah2 + + 12 2

(3) yy

h5 ah4 + 80 8

+

s

3 (0) bh xx

(82)

a+h

+

(1) xx

h5 bh4 + + 80 8

(3) xx

h7 bh6 + 448 32

+

s

ah3 + 4

(1) yy

h5 ah4 + + 80 8

(3) yy

h7 ah6 + 448 32

+

s

4

(0) yy

b+h

(83)

b+h

a+h

(84) (85)

A.4. Nonlocal Levy coefficients

T1 = (e7 e30 e3 e34)/ e0, T2 = (e2 e30 e3 e29)/ e0 T3 = (e6 e30 e3 e33)/e0 , T4 = (e5 e30 e3 e32)/ e0 T5 = (e8 e30 e3 e35)/ e0, T6 = (e4 e30 e3 e31)/ e0 T7 = (e9 e39 e12 e36)/ T0, T8 = (e14 e39 e12 e41)/ T0 T9 = (e16 e39 e12 e43)/T0 T10 = (e13 e39 e12 e40)/ T0 T11 = (e11 e39 e12 e38)/ T0, T12 = (e15 e39 e12 e42 )/ T0 T13 = a 0 (T1 e21 + T7 a1 + T25 a2 + T19 e23 + e26) T14 = a0 (T8 a1 + T2 6a2 + e27), T15 = a0 (T9 a1 + T27 a2 + e20 + µ 4 + 2kN0 ) T16 = a0 ((T3 e21 + T21 e23 + T10 a1 + T28 a2 + e18) (N0 + µ 2kN0 + µ 2N0)) T17 = a 0 (T5 e21 + T23 e23 + T11 a1 + T29 a2 + e17) T18 = a 0 (e19 + T12 a1 + T30 a2) T19 = (e1 e34 e7 e28)/ e0, T20 = (e1 e29 e2 e28)/ e0 T21 = (e1 e33 e6 e28)/ e0, T22 = (e1 e32 e5 e28)/ e0 T23 = (e1 e35 e8 e28)/ e0, T24 = (e1 e31 e4 e28)/ e0 T25 = (e10 e36 e9 e37)/ T0, T26 = (e10 e41 e14 e37)/ T0 T27 = (e10 e43 e16 e37)/T0, T28 = (e10 e40 e13 e37 )/ T0 T29 = (e10 e38 e11 e37 )/T0, T30 = (e10 e42 e15 e37)/ T0 e0 = e3 e28 e1 e30, T0 = e12 e37 e10 e39 a 0 = 1/(T4 e21 + T22 e23 + e25 + µN0 ) a1 = T2 e21 + T20 e23 + e22, a2 = T6 e21 + T24 e23 + e24

(86)

where

14

Composite Structures 226 (2019) 111216

K. Shiva, et al.

e1 = A11 , e4 = e6 =

e2 =

(A12 + A66 ),

(B12 + B 66), 2c (E + 2E ), 1 12 66

e8 =

B66,

e12 = B66,

e13 =

2A , 22

e14 = e16 =

e9 =

c1 3e22,

e18 = A55 +

e2 ,

A 44 + c1 3F22

e20 =

2A

e24 e26 e30 e32 e36 e40

e5,

2B

e17 = A55

c12

e11 =

e4

c1 (E12 + 2E66 )

e19 = e21 =

e10 = A66 ,

e15 =

2c12 2 (H12

44

e3 = B11

e5 = c1 E11 2A e7 = 66

22

c1

2 (F 12

+ 2F66) (87)

+ 2H66)

4H 22

e22 = e13,

e23 = c1 F11

= c1 (F12 + 2F66), e25 = c12 H11 = e6, e27 = e16, e28 = e3, e29 = e4 = D11, e31 = (D12 + D66) 2D = e23, e33 = e17, e34 = e8, e35 = A55 66 = e4, e37 = e12, e28 = e31, e39 = D66 2D , = e24 , e41 = e15, e42 = A 44 e43 = e19 22

k=

Nyy Nxx

nonlocal formulation and meshless method. Int J Eng Sci 2011;49:976–84. [25] Kitipornchai S, Ramesh SS, Zhang YY, Wang CM. Buckling analysis of micro- and nano-rod/tubes based on nonlocal Timoshenko beam theory. J Phys D: Appl Phys 2006;39:3904–9. [26] Zhang YQ, Wang JS, Liu GR. Small-scale effects on buckling of multiwalled carbon nanotubes under axial compression. Phys Rev B 2004;70. [27] Reddy JN, El-Borgi Sami. Eringen’s nonlocal theories of beams accounting for moderate rotations. Int J Eng Sci 2014;82:159–77. [28] Xu M. Free transverse vibrations of nano-to-micron scale beams. Proc R Soc A 2006;462:2977–95. [29] Zhang K, Meng-Hua C, Zhao Ge, Deng Zi-Chen, Xu Xiao-Jian. Free vibration of nonlocal Timoshenko beams made of functionally graded materials by symplectic method. Compos Part B 2019;156:174–84. [30] Reddy JN, El-Borgi Sami, Romanoff J. Non-linear analysis of functionally graded microbeams using Eringen’s non-local differential model. Int J Non-Linear Mech 2014;67:308–18. [31] Aghababaei R, Reddy JN. Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates. J Sound Vib 2009;326:277–89. [32] Lu P, Zhang PQ, Lee HP, Wang CM, Reddy JN. Nonlocal elastic plate theories. Proc R Soc A 2007;463:3225–40. [33] Reddy JN. Nonlocal nonlinear formulations for bending of classical and deformation theories of beams and plates. Int J Eng Sci 2010;48:1507–18. [34] Raghu P, Rajagopal A, Reddy JN. Nonlocal nonlinear finite element analysis of composite plates using TSDT. Compos Struct 2018;185:38–50. [35] Babu B, Patel BP. Analytical solution for strain gradient elastic kirchhoff rectangular plates under transverse static loading. Eur J Mech 2019;73:101–11. [36] Daneshmehr A, Rajabpoor A, Pourdavood M. Stability of size dependent functionally graded nanoplate based on non-local elasticity and higher order plate theories and different boundary conditions. Int J Eng Sci 2014;82:84–100. [37] Srividhya S, Raghu P, Rajagopal A, Reddy JN. Nonlocal nonlinear analysis of functionally graded plates using third-order shear deformation theory. Int J Eng Sci 2018;125:1–22. [38] Khorshidi K, Fallah A. Buckling analysis of functionally graded rectangular nanoplate based on non-local exponential shear deformation theory. Int J Mech Sci 2016;113:94–104. [39] Nowruzpour M, Reddy JN. Unification of local and nonlocal models with in a stable integral formulation for analysis of defects. Int J Eng Sci 2018;132:45–59. [40] Gurtin ME, Murdoch IA. Surface stress in solids. Int J Solids Struct 1978;14:431–40. [41] Gurtin ME, Murdoch IA. Effect of surface stress on wave propagation in solids. J Appl Phys 1976;47:4414–21. [42] Steinmann P. On boundary potential energies in deformational and configurational mechanics. J Mech Phys Solids 2008;56:772–800. [43] Papastavrou A, Steinmann P, Kuhl E. On the mechanics of continua with boundary energies and growing surfaces. J Mech Phys Solids 2013;61:1446–63. [44] Hamilton JC, Wolfer WG. Theories of surface elasticity for nanoscale objects. Surf Sci 2009;603:1284–91. [45] Ansari R, Sahmani S. Surface stress effects on the free vibration of nanoplates. Int J Eng Sci 2011;49:1204–15. [46] Ansari R, Sahmani S. Bending behaviour and buckling of nanobeams including surface stress effects corresponding to different beam theories. Int J Eng Sci 2011;49:1244–55. [47] Preethi K, Rajagopal A, Reddy JN. Surface and non-local effects for non-linear analysis of Timoshenko beams. Int J Non-Linear Mech 2015;76:100–11.

References [1] Reddy JN. A simple higher-order theory for laminated composite plates. J Appl Mech 1984;51:745–52. [2] Eringen AC. On differential equations of nonlocal elasticity and solutions of screw dislocations and surface waves. J Appl Phys 1983;54:4703–10. [3] Gurtin ME, Murdoch IA. A continuum theory of elastic material surfaces. Arch Rational Mech Anal 1975;57:291–323. [4] Reddy JN. Microstructure-dependent couple stress theories of functionally graded beams. J Mech Phys Solids 2011;59:2382–99. [5] Romanoff J, Reddy JN. Experimental validation of the modified couple stress Timoshenko beam theory for web-core sandwich panels. Compos Struct 2014;111:130–7. [6] John P, George RB, Richard PM. Application of nonlocal continum models to nano technology. Int J Eng Sci 2003;41:305–12. [7] Kröner E. Elasticity theory of materials with long range cohesive forces. Int J Solids Struct 1967;3:731–42. [8] Eringen AC. Nonlocal polar elastic continua. Int J Eng Sci 1972;10:1–16. [9] Zhen-Gong Z, Jie-Cai H, Shan-Yi D. Investigation of a Griffith crack subject to antiplane shear by using the nonlocal theory. Int J Solid Struct 1999;36:3891–901. [10] Reha A, Tanju Y. Rectangular rigid stamp on a nonlocal elastic half plane. Int J Solid Struct 1996;33:3577–86. [11] Lim CW, Zhanga G, Reddy JN. A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J Mech Phys Solids 2015;78:298–313. [12] Vardhan VK, Wang Q. Vibration of carbon nanotubes studied using nonlocal continuum mechanics. Smart Mater Struct 2006;15:659–66. [13] Xie XY, Liu GR, Zhang YQ. Free transverse vibrations of double walled carbon nanotubes using a theory of nonlocal elasticity. Phys Rev B 2004;71. [14] Wang Q. Wave propagation in carbon nanotubes using nonlocal continuum mechanics. J Appl Phys 2005;98. [15] Saikat S, Reddy JN. Exploring the source of non-locality in the Euler-Bernoulli and Timoshenko beam model. Int J Eng Sci 2016;104:110–5. [16] Reddy JN. Nonlocal theories for bending, buckling and vibration of beams. Int J Eng Sci 2007;45:288–307. [17] Li X, Li Y, Hu Z Ding, Deng W. Bending buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory. Compos Struct 2017;165:250–65. [18] McNitt RP, Buchanan GG, Peddieson J. Application of nonlocal continuum models to nanotechnology. Int J Eng Sci 2003;41:305–12. [19] Romano G, Baretta R. Nonlocal elasticity in nano beams:the stress-driven integral model. Int J Eng Sci 2017;115:14–27. [20] Sudak LJ. Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics. J Appl Phys 2003;94:7281–7. [21] Bahrami A, Teimourian A. Nonlocal scale effects on buckling, vibration and wave reflection in nanobeams via wave propagation approach. Compos Struct 2015;134:1061–75. [22] Tuna M, Kirca M. Bending, buckling and free vibration analysis of Euler-Bernoulli nanobeams using eringen’s nonlocal integral model via finite element method. Compos Struct 2017;179:269–84. [23] Romano G, Baretta R, Diaco M. On nonlocal intergral models for elastic nanobeams. Int J Mech Sci 2017;131–132:490–9. [24] Roque CMC, Ferreira AJM, Reddy JN. Analysis of Timoshenko nanobeams with a

15

Composite Structures 226 (2019) 111216

K. Shiva, et al.

Stab Dyn 2011;3:411–29. [73] Hashemi SH, Samaei T. A Buckling analysis of micro/nanoscale plates via nonlocal elasticity theory. Phys E 2011;43:1400–4. [74] Nejad MZ, Hadi A, Rastgoo A. Buckling analysis of arbitrary two-directional functionally graded euler-bernoulli nano-beams based on nonlocal elasticity theory. Int J Eng Sci 2016;103:1–10. [75] Pradhan SC. Buckling of single layer graphene sheet based on non-local elasticity and higher order shear deformation theory. Phys Lett A 2009;373:4182–8. [76] Pradhan SC. Buckling analysis and small scale effect of biaxially compressed graphene sheets using non-local elasticity theory. Sadhana: Acad Proc Eng Sci 2012;37:461–80. [77] Peng Xiang wu, Guo Xing ming, Liu L, Wu Bing jie. Scale effects on non-local buckling analysis of bilayer composite plates under non-uniform uniaxial loads. Appl Math Mech 2015;36:1–10. [78] Samaei AT, Abbasion S, Mirsayar MM. Buckling analysis of a single-layer graphene sheet embedded in an elastic medium based on non-local mindlin plate theory. Mech Res Commun 2011;38:481–5. [79] Anjomshoa A, Shahidi AR, Hassani B, Jomehzadeh E. Finite element buckling analysis of multi-layered graphene sheets on elastic substrate based on nonlocal elasticity theory. Appl Math Modell 2014;38:5934–55. [80] Golmakani ME, Rezatalb J. Nonuniform buckling of orthotropic nanoplates embedded in an elastic medium based on nonlocal mindlin plate theory. Compos Struct 2015;119:238–50. [81] Chih-Ping Wu, Li Wei-Chen. Asymptotic nonlocal elasticity theory for the buckling analysis of embedded single layered nanoplates/graphene sheets under biaxial compression. Phys E 2017;89:160–9. [82] Golmakani ME, Malikan M, Sadraee Far MN, Majidi HR. Bending and buckling formulation of graphene sheets based on non-local simple first-order shear deformation theory. Mater Res Express 2018. [83] Rafii-Tabar H, Ghavanloo E, Fazelzadeh SA. Nonlocal continuum-based modeling of mechanical characteristics of nanoscopic structures. Phys Rep 2016;638:1–97. [84] Eltaher MA, Khater ME, Emam SA. A review on nonlocal elastic models for bending, buckling, vibration, and wave propagation of nanoscale beams. Appl Math Modell 2016;40:4109–28. [85] Norouzzadeh A, Ansari R. Isogeometric vibrational analysis of functionally graded nanoplates with the consideration of nonlocal and surface effects. Thin-Walled Struct 2018;127:354–72. [86] Zhang Z, Wang CM, Challamel N. Eringen’s length scale coefficient for buckling of nonlocal rectangular plates from microstructured beam-grid model. Int J Solids Struct 2014;51:4307–15. [87] Challamel N, Hache F, Elishakoff I, Wang CM. Buckling and vibrations of microstructured rectangular plates considering phenomenological and lattice-based continuum models. Compos Struct 2016;149:145–56. [88] Hache F, Challamel N, Elishakoff I. Lattice and continualized models for the buckling study of nonlocal rectangular plates including shear effects. Int J Mech Sci 2018;145:221–30. [89] Reddy JN. Mechanics of Laminated Composite Plates and Shells. 2nd edition Boca Raton, FL: CRC Press; 2004. [90] Nosier A, Reddy JN. Vibration and stability analysis of cross-ply laminated circular cylindrical shells. J Sound Vib 1992;157(1):139–59. [91] Khdeir AA. A remark on the state-space concept applied to bending, buckling and free vibration of composite laminates. Comput Struct 1997;149:113–32. [92] Reddy JN. On locking-free shear deformable beam finite elements. Comput Methods Appl Mech Eng 1996;59(5):813–7. [93] Noor AK. Stability of multilayered composite plates. Fibre Sci Technol 1975;8:81–9. [94] Putcha NS, Reddy JN. Stability and natural vibration analysis of laminated plates by using a mixed element based on a refined plate theory. J Sound Vib 1986;104:285–300. [95] Khdeir AA. Free vibration and buckling of unsymmetric cross-ply laminated plates using a refined theory. J Sound Vib 1988;128(3):377–95.

[48] Radebe IS, Adali S. Effect of surface stress on the buckling of non-local nanoplates subject to material uncertainity. Latin Am J Solids Struct 2015;12:1666–76. [49] Raghu P, Preethi K, Rajagopal A, Reddy JN. Nonlocal third-order shear deformation theory for analysis of laminated plates considering surface stress effects. Compos Struct 2016;139:13–29. [50] Mercan K, Civalek O. Buckling analysis of silicon carbide nanotubes (SiCNTs) with surface effect and nonlocal elasticity using the method of HDQ. Compos Part B 2017;114:34–45. [51] Preethi K, Raghu P, Rajagopal A, Reddy JN. Nonlocal nonlinear bending and free vibration analysis of a rotating laminated nano cantilever beam. Mech Adv Mater Struct 2018;25:439–50. [52] Leissa AW. A review of laminated composite plate buckling. Appl Mech Rev 1987;40:575–91. [53] Reddy JN, Khdeir AA. Buckling and vibration of laminated composite plates using various plate theories. AIAA J 1989;27:1808–17. [54] Reddy JN, Phan ND. Stability and vibration of isotropic, orthotropic and laminated plates according to a higher-order shear deformation theory. J Sound Vib 1985;98:157–70. [55] Khdeir AA, Reddy JN. Analytical solutions of refined plate theories of cross-ply composite laminate. J Pressure Vessels Technol 1991;113:570–8. [56] Qatu MS, Leissa AW. Buckling or transverse deflections of unsymmetrically laminated plates subjected to in-plane loads. AIAA J 1993;31:189–94. [57] Timarci T, Aydogdu M. Buckling of symmetric cross-pluy square plates with various boundary conditions. Compos Struct 2005;68:381–9. [58] Owhadi A, Shariat BS. Stability analysis of symmetric laminated composite plates with geometric imperfections under longitudinal temperature gradient. J Mech 2011;25:161–5. [59] Ouinas D, Achouri B. Buckling analysis of laminated composite plates containing an elliptical notch. Composites 2013;55:575–9. [60] Ghorbanpour Arani A, Maghamikia Sh, Mohammadimehr M, Arefmanesh A. Buckling analysis of laminated composite plates reinforced by SWCNTs using analytical and finite element methods. J Mech Sci Technol 2011;25:809–20. [61] Panda SK, Ramachandra LS. Buckling of rectangular plates with various boundary conditions loaded by non-uniform inplane loads. Int J Mech Sci 2010;52:819–28. [62] Park M, Don-Ho C. A two-variable first-order shear deformation theory considering in-plane rotation for bending, buckling and free vibration analysis of isotropic plates. Appl Math Modell 2018;61:49–71. [63] John P, George RB, Richard PM. New analytic buckling solutions of rectangular thin plates with all edges free. Int J Mech Sci 2018;144:67–73. [64] Li Shi-Rong, Batra RC. Relations between buckling loads of functionally graded Timoshenko and homogeneous Euler-Bernoulli beams. Compos Struct 2013;95:5–9. [65] Aydogdu M, Aksencer T. Buckling of cross-ply composite plates with linearly varying inplane loads. Compos Struct 2018;183:221–31. [66] Reddy ARK, Palaninathan R. Buckling of laminated skew plates. Thin-Walled Struct 1995;22:241–59. [67] Hsuan-Teh Hu, Yang Chia-Hao, Lin Fu-Ming. Buckling analyses of composite laminate skew plates with material nonlinearity. Compos Part B 2006;37:26–36. [68] Singh SK, Chakrabarti A. Static, vibration and buckling analysis of skew composite and sandwich plates under thermo mechanical loading. Int J Appl Mech Eng 2013;18:887–98. [69] Kumar A, Panda SK, Kumar R. Buckling behaviour of laminated composite skew plates with various boundary conditions subjected to linearly varying inplane edge loading. Int J Mech Sci 2015;100:136–44. [70] Shahriari B, Shirvani S. Small-scale effects on the buckling of skew nanoplates based on nonlocal elasticity and second-order strain gradient theory. J Mech 2018;34:443–52. [71] Narendar S. Buckling analysis of micro-/nanoscale plates based on two-variable refined plate theory incorporating non-local scale effects. Compos Struct 2011;93:3093–103. [72] Pradhan SC, Phadikar JK. Non-local theory for buckling of nanoplates. Int J Struct

16