COMPOSITE PLATE THEORIES

CHAPTER2 COMPOSITE PLATE THEORIES 2.1 GENERAL Analysis of composite plates is usually done based on one of the following the ries. 1. Equivalent single-layer theories a. Classical laminate theory b. Shear deformation laminate theories 2. Three-dimensional elasticity theories a. Traditional three-dimensional elasticity formulation b. Layer-wise theories 3. Multiple model methods 2.2 EQUIVALENT SINqLE-LA YER THEORIES The equivalent single-layer laminate theories are those m which a heterogeneous laminated plate is treated as a statically equivalent single layer having a complex constitutive behaviour, reducing the three-dimensional problem to a two-dimensional one. The simplest equivalent single-layer laminate theory is the 'classical laminated plate theory'. 2.2.1 Classical Laminated Plate Theory If the transverse deflection, w, of a plate is small in comparison with its thickness, h, a very satisfactory approximate theory of bending of the plate by lateral loads can be developed with the help of the following assumptions. 16

1. Straight lines which are perpendicular to the mid-surface (i.e., transverse normals) before deformation remain straight and perpendicular to the mid-surface even after deformation. In other words, the transverse normals rotate such that they remain perpendicular to the mid-surface after deformation. 2. The transverse normals do not experience elongation. i.e., the normal strains and, hence, stresses in a direction transverse to the plate can be disregarded. In other words, the effect of transverse stresses, O'z, 'txz and 'tyz, is neglected. Using these assumptions, all stress components can be expressed in terms of the deflection of the plate, which is a function of the two co-ordinates in the plane of the plate. These assumptions are due to Kirchoff and the theory is known as 'classical plate theory' [101]. The extension of classical plate theory to laminated composite plates results in 'classical laminated plate theory'. The geometry of an edge of a plate before and after deformation under Kirchoffs assumptions is shown in Fig. 2.1. z -i- --- Wo,,----+---------------x j_ Uo Figure 2.1 Undeformed and deformed geometries of an edge of a plate ( classical plate theory) 17

Classical laminated plate theory neglects both the transverse shear and the transverse normal effects, thereby assuming that the deformation is entirely due to bending and in-plane stretching. Moreover, this theory introduces second derivatives in the strain-displacement relations. Hence, continuity conditions between elements have to be imposed not only on the transverse deflection but also on its derivatives. Classical laminated plate theory is based on the displacement field, u(x, y, z) = u 0 (x, y )-z 8: 0 v{x,y,z) = V 0 (x,y)-z OW o oy w(x,y,z)= w 0 (x, y) (2.1) where llo, v O and w Oare the mid-plane displacements. The strains associated with the displacements are E x au au o a 2 w o ax ax ax 2 av av o a 2 w E y = - = - z Yxy oy oy o y 2 au av au o av o 28 2 w -+- --+- 0 oy ax oy ax axoy o E o X = E o y +z 0 Yxy Kx Ky (2.2) Kxy The strain components considered are extensional strains (membrane strains) and flexural strains (bending strains). The constitutive relation connecting the stress and strain at any point is {cr} = [c) {E}, (2.3) {cr} and { E} being the stress and strain components with respect to the material axes. 18

In Eq. (2.3), 0 ] 0 ' C 33 where (2.4) Since a laminate is made of several orthotropic layers, with their material axes oriented arbitrarily with respect to the laminate co-ordinates, the constitutive equations of each layer must be transformed to the laminate axes (plate axes) by the co-ordinate transformation [c] = [T] [c ][T] T, where [T]= n 2 m2 mn -mn -2mn 2mn m 2 -n 2 in which m = cos a, n = sin a and a is the angle between the plate axis (X - axis) an the principal material axis (I-axis). The plate and material axes of a typical lamina are shown in Fig. 2.2. Then, the stress components with respect to the laminate axes is related to the corresponding strain components as {cr}=[c]{e} (2.5) 19

f Figure 2.2 Plate and material axes of a lamina 2.2.1.1 Laminate constitutive matrix (ABD matrix) Laminate constitutive matrix is developed by establishing a relation between the force and moment resultants and the strains and curvatures at a point (x, y) on the reference surface of the laminate. The stress resultants in a laminate include the three force resultants and the three moment resultants. The normal force resultants in the X-direction (N x ) and in the Y-direction (N y ) and the shear force resultant (Nx y ) are obtained by integrating the respective stresses through the thickness of the laminate, which requires layer-wise integration, and is given by (2.6) 20

The bending moment resultants, Mx and M y about Y-direction and X-direction respectively, and the twisting moment, Mx y, are defined as! M My cry cr X h/2 X NLZk+l X = f zdz= f cry zdz -h/2 k-1 Zk M l \ ) l l x y 'tx y 'tx y k (2.7) Fig. 2.3 illustrates a small element of a laminate surrounding a point (x, y) on the geometric mid-plane, along with the direction of these stress resultants. Figure 2.3 Force and moment resultants on a plate element 21

Using Eqns. (2.5) and (2.2), Eq. (2.6) is written as A 11 A12 A1 3 ] t o N x X N = y [ A21 A22 A23 ty o + B 21 B22 B23 Ky [B 11 B 12 B1 3 Jr) N xy A31 A32 A33 0 B33 Yxy Kxy t' Similarly, Eq. (2.7) is written as, ) - B 11 B 1 2 B t o 1 X D 3 11 D 12 D 13 My - [ B21 B22 B23 ] ty o + [ D21 D22 D23 Ky B31 B32 Jr) Mxy B3 1 B32 B33 0 D31 D32 D33 Kxy Yxy (2.8) (2.9) Thus, the constitutive equations relating the stress resultants and the strains of a laminate are expressed as N x A11 A 12 A 13 B 11 B 12 B 13 t o N y A21 A 22 A 23 B 21 B 22 B 23 go y N x y A 31 A 32 A 33 B 31 B 32 B 33 = Yxy M X B 11 B 12 B 13 D 11 D 12 D 13 Kx (2.10) M y B 21 B 22 B 23 D 21 D 22 D 23 Ky M x y B 31 B 32 -B 33 D 31 D 32 D 33 K xy where NLZk+l ( A ij,b ij, D ij )= I f [c ij t (l,z,z 2 )dz, i,j=l,2,3. k=j Zk The constitutive matrix of the laminate is [Q) = [[Bl f (2.11) 22

2.2.1.2 Remarks on classical laminated plate theory Classical laminated plate theory underpredicts deflections and overpredicts natural frequencies and buckling loads. This is because transverse shear strains are neglected in this theory. For plates made of advanced composites like graphite-epoxy and boronepoxy, whose elastic modulus to shear modulus ratios is very high, the errors in deflections, stresses, natural frequencies and buckling loads are even higher [ 19]. Moreover, this theory leads to considerable errors when thick plates are analysed. Hence, classical laminated plate theory is inadequate for the analysis of composite plates. This has led to the development of an adequate theory, which takes into account the effect of transverse shear strains, for the analysis of composite plates. It has been experienced that the adoption of first-order shear deformation theory based on Mindlin's plate theory, along with proper shear correction factor, overcomes the drawbacks of classical laminated plate theory. 2.2.2 First-Order Shear Deformation Theory The next theory in the hierarchy of equivalent single-layer theories is the 'first-order shear deformation theory'. The assumption of transverse normals being perpendicular to the :mid-surface even after deformation is relaxed in the first-order shear deformation theory. This theory assumes that the straight lines which are normal to the mid-surface before deformation remain straight but not normal to the deformed mid-surface. The geometry of an edge of a plate before and after deformation, based on first-order shear deformation theory, is shown in Fig. 2.4. 23

z Wo _, x j_ 1 Uo -----1 1 Figure 2.4 Undeformed and deformed geometries of an edge of a plate (first-order shear deformation theory) In first-order shear deformation theory, the transverse shear stresses are constant through the laminate thickness because the transverse shear strains are assumed to be constant. But, it is well known that the transverse shear stress varies parabolically through the laminate thickness, with zero values at the top and bottom surfaces of the plate. This discrepancy between the actual stress state and the constant stress state predicted by this theory is corrected by modifying the transverse shear stiffness, using a shear correction coefficient. Usually, a shear correction coefficient of 5/6 is employed [2]. The displacement field of first-order shear deformation theory is of the form u(x, y,z) = u o (x, y)-ze x (x, y) v(x,y,z)= v)x,y)-ze Y (x,y) w(x,y,z)= w 0 (x,y) (2.12) 24

In Eq. (2.12), Sx and Sy are the rotations of _the cross-section about Y-axis and X-axis respectively. In first-order shear deformation theory, the strain components considered are extensional strains, bending strains and shear strains. The strains associated with the displacements are E x ou ox 0V E y = - = oy OUo ox ov o oy -z aex ox ae y oy E o X K x = E o y +z K y (2.13) Yxy ou 0V -+ay ox OUo ox OVo aex ae y --+ay -+ay ox 0 Yxy K xy and {::}= az ox = 0V aw ou aw -+- -+az ay aw o -8 OX aw o -8 ay Stress-strain relationship of an orthotropic layer with refer,ence to the plate axes is written as cr x C 11 C12 C 13 0 0 E x cr y C 21 C 22 C 23 0 0 E y [cr] = t xy = C 31 C32 C33 0 0 Yxy = [c] {E } (2.15) t xz 0 0 0 C44 C4s Yxz t yz 0 0 0 C54 C ss Yyz 25

where Cij are the material constants transformed to the plate axes using [C] = [T] [c ][T] T, in which C11 C1 2 0 0 0 C 21 C 22 0 0 0 [c]= 0 0 C33 0 0 (2.16) 0 0 0 C 44 0 0 0 0 0 Css and m 2 n 2-2mn 0 0 n 2 m 2 2mn 0 0 [T]= mn -mn m 2 -n 2 0 0 0 0 0 m -n 0 0 0 n m In Eq. (2.16), C 44 = G13 and C 55 = G 23. All other elements are the same as in Eq. (2.4). Presence of transverse shear strain terms in this theory results in shear resultants in addition to the force and moment resultants given in Eqs. (2.8) and (2.9). The shear resultants are given by {Q } h/2 { } QY - h/2 'tyz X - f 't xz dz - NL Zk+I I f [cijt { Y xz } dz, k=i Zk Yyz i, j = 4, 5 (2.17) which can be written as (2.18) 26

NL 2k +l where Aij = L f [ci+j, j+j]k k=l Zk dz, 1, J = 1, 2 and Ks is the shear correction factor. [A] [B] [0 [. [o] [o] [A The constitutive matrix of the laminate is [Q]= [B] [D] [o] (2.19) 2.2.2.1 Remarks on first-order shear deformation theory The form of the finite element formulation of the first-order shear deformation theory requires only C 0 continuity of the solution, i.e., only the generalized displacement degrees of freedom (not their derivatives) need be continuous across element interfaces. Though the first-order shear deformation theory with proper shear correction factor predicts the response of thin plates reasonably well, accuracy is less in the case of thick plates. The shear correction factors are difficult to determine arbitrarily for laminated composite plate structures. These factors depend not only on the lamination and geometric parameters, but also on the loading and boundary conditions. Also, the assumption of constant distribution of transverse shear strain, and hence the transverse shear stress, across the thickness of the plate does not satisfy the condition of zero transverse shear stress at the top and bottom surfaces of the plate with parabolic variation across the thickness. This necessitates the inclusion of higher-order terms of thickness co-ordinate in the displacement field. Then the assumption of straightness of transverse normal is no longer necessary. This has led to the development of higher-order shear deformation theories. 27

2.2.3 Higher-Order Shear Deformation Theory Second and higher-order laminated plate theories use higher-order polynomials in the expansion of the displacement components through the thickness of the laminate. Though it is possible to express the displacement field in terms of the thickness up to any desired degree, the algebraic complexity and computational effort restrict the number of higherorder terms. A quadratic variation of transverse shear strains and transverse shear stresses across the thickness of the plate can be achieved by expressing the displacement up to the cubic terms in the thickness co-ordinate and may be referred to as third-order plate theory. In this theory, the assumption on the straightness and normality of a transverse normal after deformation is avoided by expressing the displacements as cubic functions of the thickness co-ordinate. The deformation of a transverse normal according to thirdorder plate theory is shown in Fig. 2.5.,, x _l I- - Uo --+! I Figure 2.5 Undeformed and deformed geometries of an edge of a plate (third-order shear deformation theory) 28

Third-order shear deformation theory [19] is based on the displacement field u{x, y,z) = u 0 (x, y)- z0x (x, y)-z 2 \jl x (x, y)-z 3 cj>x (x, y) v(x, y, z) = v O (x, y)-z0 y (x, y)- z 2 \jl Y (x, y)-z 3 cj> y (x, y) (2.20) w(x,y,z)= w 0 (x,y) Imposing the condition of zero shear strain at top and bottom surfaces of the plate, we get 'l'x = \jl y = 0, <l>x = _i.._ 2 (aw - exj and <!>y = _i.._ 2 (aw -e y ) 3h ax 3h oy Thus, Eq. (2.20) reduces to u = u.-{e, + tn:-0,j] v=v 0 -z [ 0y +tj(:-ey)] (2.21) The strains associated with the displacements in Eq. (2.21) are Ex au au a(aw I ax o - ej aex ax ax ax ax av av o Ey = - = - z ae y 4z 3 3h 2 - a(aw1oy-e y ) oy oy oy oy Yxy au av au o av o aex ae y a(aw1ax-ej a(aw1oy-e y ) -+- -+- -+- + oy ax ax oy ax oy ax

t ) 0 K x y E o K x E o + y Z Ky + ::, K y Yxy K x y and {::} au ow --e ow (: -e x X az ax ax ) 4 z2 av ow ow h -+- --e 2 y az ay (:-ey J -+- = = -- = ay {: }+ : t} The strains are rewritten as o ( 2 * ) Yxy = Yxy +z K xy +C 1z K xy, 0 2 y yz = y yz + C2z y yz 0 2 Yxz = Yxz +C2z Yxz (2.22) 4 where C 1 = - 3h 2 and The higher-order terms in the displacement field lead to higher-order strain terms such as K * = -(8 2 w _ a0yj y ay2 r, = -(:-eyj ay ' (2.23) 30

The stress-strain relation for an orthotropic layer is same as in Eq. (2.15). But the presence of higher-order strain terms leads to higher-order stress resultants as M X cr h/2 x M = y f cr y z 3.dz -h/2 Mxy 'txy and t:}= hs2 Qy }, r -h/2 'tyz 2 Accordingly, the stress resultants are written as N E o x A X 11 A12 A 13 B11 B12 B13 E11 E 12 E13 N y A21 A 22 A23 B21 B22 B23 E21 E 22 E23 N x y A31 A32. A3 3 B31 B32 B3 3 E 31 E32 E 33 0 Yxy M x B11 B12 B13 D11 D12 D13 F11 F12 F13 K x M y = B21 B22 B23 D21 D22 D23 F21 F22 F23 K y (2.24) M x y M X M y M x y B 31 B32 B3 3 D31 D32 D 3 3 F 31 F32 F 33 K x y E11 E12 E13 F11 Fi2 F13 H11 H12 H13 K x E21 E22 E23 F21 F22 F23 H21 H22 H23 K y E31 E 32 E 33 F 31 F 32 F 3 3 H 31 H 32 H3 3 K x y E o y and 31

Qx A 11 A 12 E n E 12 0 Yxz Qy A 21 A 22 E 21 E 22 0 Yyz Q: = E 11 E 12 D 11 D 12 Yxz Q E 21 E 22 D 21 D 22 Yyz Combining Eqs. (2.24) and (2.25), in compact notation, {P} = [Q]{E} (2.26) [A] [B] 0 0 [B] [D] [F] 0 0 where [Q]= [ E ] [F]. [H] 0 0 (2.27) 0 0 0 [A] [E] 0 0 0 [E] [b] [ E ] Coefficients in the above expression are defined as and 2.2.3.1 Remarks on higher-order shear deformation theory Several studies [15, 19, 102] have shown that higher-order theories are often necessary in order: to get a good estimate, not only of the local state of strain and stress but also of global characteristics of the response such as middle plane deflections, eigen frequencies and critical buckling loads. This theory avoids the use of shear correction coefficient. 32

Higher-order theories yield more accurate deflections, vibration frequencies, critical buckling loads and inter-laminar stress distribution in the case of thick plates. In short, when the main aim of the analysis is to determine the overall global response like gross deflection, vibration frequencies, critical buckling loads, etc., equivalent single-layer models can be used with reasonable accuracy. These theories are inherently simple and involve less computational effort. 2.3 THREE-DIMENSIONAL THEORIES In the three-dimensional elasticity theory or in a layer-wise theory, each layer is modelled as a three dimensional solid. Layer-wise theories are developed by assuming that the displacement components are continuous through the laminate thickness, but the derivatives of the displacements with respect to the thickness co-ordinate may be discontinuous at various points across the thickness. This assumption allows the transverse stresses to be continuous at the interfaces of dissimilar materials. Layer-wise theories are of two types: (i). Partial layer-wise theory: This theory [2] uses layer-wise expansions for the in-plane displacement components, but not for the transverse displacement component. The introduction of discrete layer transverse shear effects into the displacement field makes it possible to describe the kinematics of composite laminates more realistically. (ii). Full layer-wise theory: This theory [2] uses layer-wise expansions for all the three displacement components and adds both discrete layer transverse shear effects and transverse normal effects. 33

Layer-wise theories allow the in-plane displacements to vary in a layer-wise fashion through the thickness of the laminate. Since the number of field equations and edge boundary conditions depend upon the number of layers, the layer-wise models are computationally expensive. 2.4 MULTIPLE MODEL METHODS The equivalent single-layer models and the layer-wise models have their own advantages and disadvantages in terms of solution accuracy, solution economy and ease of implementation. However, by combining these model types in a multiple model analysis or global-local analysis, a wide variety of laminate problems can be solved with maximum accuracy and minimum cost. The term 'multiple model analysis' denotes any analysis method that uses different mathematical models and/or distinctly different levels of discretisation. The analysis of composite laminates has provided the incentive for the development of many of the reported multiple model methods (103-113], due mainly to the heterogeneous nature of composite materials and the wide range of scales of interest. 34