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.

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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)

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Classical laminated plate theory neglects both the transverse shear and the transversenormal effects, thereby assuming that the deformation is entirely due to bending andin-plane stretching. Moreover, this theory introduces second derivatives in thestrain-displacement relations. Hence, continuity conditions between elements have to beimposed 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) = u0 (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)

where llo, v O and w Oare the mid-plane displacements.

The strains associated with the displacements are

au auo

a2w __ o

ax ax ax2

Ex

av avo a

2w Ey = - = - z __ o

oy oy oy2

Yxy au av auo avo 282w-+- --+- 0

oy ax oy ax axoy

Eo

X

= Eo

y +z0

Yxy

(2.1)

Kx

Ky (2.2)Kxy

The strain components considered are extensional strains (membrane strains) and flexuralstrains (bending strains). The constitutive relation connecting the stress and strain at anypoint is

{cr} = [c) {E} , (2.3){cr} and {°E} being the stress and strain components with respect to the material axes.

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In Eq. (2.3),

0 ] 0 '

C33

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]= n2 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

(Nx) and in the Y-direction (Ny) 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)

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The bending moment resultants, Mx and My

about Y-direction and X-direction

respectively, and the twisting moment, Mxy, are defined as

!M

l \

cr

) l

cr

l

X h/2 X NLZk+l X

My

= f cry

zdz=� f cry zdzM

-h/2 k-1 Zkxy 'txy 'txy 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

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Using Eqns. (2.5) and (2.2), Eq. (2.6) is written as

Nx [ A11 A12 A13] to

[B11 B12 B13 Jr) X

Ny = A21 A22 A23 t

o + B21 B22 B23 Ky y

Nxy A31 A32 A33 0 B31 B32 B33 KxyYxy

(2.8)

Similarly, Eq. (2.7) is written as, t' )-[B11 B1 2 B13] t

o

[D11 D12 D13Jr) X

My - B21 B22 B23 to + D21 D22 D23 Ky y

Mxy B31 B32 B33 0 D31 D32 D33 KxyYxy

(2.9)

Thus, the constitutive equations relating the stress resultants and the strains of a laminate

are· expressed as

Nx A11 A12 A13 B11 B12 B13 t

o

Ny A21 A22 A23 B21 B22 B23 go

y Nxy A31 A32 A33 B31 B32 B33 = Yxy M

X B11 B12 B13 D11 D12 D13 Kx

My B21 B22 B23 D21 D22 D23 KyMxy B31 B32 -B33 D31 D32 D33 Kxy

where

NLZk+l

(Aij ,Bij ,D

ij )= I f [cij t (l,z,z2)dz, i,j=l,2,3.

k=J Zk

The constitutive matrix of the laminate is [Q) = [[Bl f � �

22

(2.10)

(2.11)

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 boron­

epoxy, 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.

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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) = uo (x, y)-ze x (x, y)

v(x,y,z)= v)x,y)-ze Y(x,y)

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

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(2.12)

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

ou

Exox

0V Ey

= - =

oy

Yxy ou 0V -+-

ay ox

and

ou aw -+-

{::}= az ox

=

0V aw -+-

az ay

OUo

ox

ovo

oy

OUo OVo --+-

ay ox

awo -8OX

awo -8ay

aex

ox

-z

aey

oy

aex aey

-+-

ay ox

Eo

X Kx

= Eo

y +z Ky (2.13)

0

Yxy Kxy

Stress-strain relationship of an orthotropic layer with refer,ence to the plate axes is written

as

crx C11 C12 C13 0 0 Ex

cry C21 C22 C23 0 0 Ey

[cr] = txy = C31 C32 C33 0 0 Yxy = [c]{E} (2.15)

txz 0 0 0 C44 C4s Yxz

tyz 0 0 0 C54 Css Yyz

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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

C21 C22 0 0 0

[c]= 0 0 C33 0 0 (2.16) 0 0 0 C44 0

0 0 0 0 Css

and

m2 n2 -2mn 0 0

n2 m2 2mn 0 0

[T]= mn -mn m2 -n2 0 0

0 0 0 m -n0 0 0 n m

In Eq. (2.16), C44 = G13 and C55 = G23 . 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

X - f 'txz dz -{Q } h/2 { }

QY -h/2

'tyz

which can be written as

I f [cijt dz,

NL Zk+I

{Yxz

} k=I Zk

Yyz

26

i, j = 4, 5 (2.17)

(2.18)

NL 2k+l where Aij = L f [ci+J, j+J]k dz , 1, J = 1, 2 and Ks is the shear correction factor.

k=l Zk

· [[A] [B] [0� The constitutive matrix of the laminate is [Q]= [B] [D] [o] . [o] [o] [A

2.2.2.1 Remarks on first-order shear deformation theory

(2.19)

The form of the finite element formulation of the first-order shear deformation theory

requires only C0 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.

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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 higher-

order 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 third-

order 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)

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Third-order shear deformation theory [19] is based on the displacement fieldu{x, y,z) = u 0(x, y)- z0x (x, y)-z2

\jl x (x, y)-z3cj>x (x, y)v(x, y, z) = vO (x, y)-z0y(x, y)- z2

\jl Y (x, y)-z3cj>y(x, y)w(x,y,z)= w0(x,y)

Imposing the condition of zero shear strain at top and bottom surfaces of the plate,

we get 'l'x = \jly = 0, <l>x = _i.._2 (aw - exJ and <!>y = _i.._2 (aw -ey)3h ax 3h oy

Thus, Eq. (2.20) reduces to

(2.20)

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

au auoaex a(aw I ax - eJ

ax ax ax ax Ex ae

y 4z3 a(aw1oy-ey ) av avo

Ey = - = - z

- 3h2oy oy oy oy Yxy au av auo

avo aex aey

a(aw1ax-eJ a(aw1oy-ey)

-+- -+- -+- +

oy ax � ax oy ax oy ax

Eo •

t ) Kx

Eo + Z Ky + ::, Ky y 0 Kxy

•

Yxy Kxy

and

au ow ow

(:-ex)

{::} -+- --eaz ax ax

X

4z2

= = --

av ow ow h2

(:-ey J -+- --e az ay ay

y

= {:�}+ :� t} The strains are rewritten as

o ( 2 * ) Yxy = Yxy +z Kxy +C1z Kxy , 0 2 •Yxz = Yxz +C2z Yxz

0 2 • y yz =

y yz + C2z y yz

4 where C 1

= - and3h2

(2.22)

The higher-order terms in the displacement field lead to higher-order strain terms such as

K* = -(82w _ a0yJ y ay2 ay '

r�, =-(:-eYJ

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(2.23)

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• crx X

h/2 M• = f cry z3 .dzy

• Mxy

-h/2'txy

and

t:}= hs2 r� },2 �

Qy -h/2 'tyz

Accordingly, the stress resultants are written as

Nx A11 A12 A13 B11 B12 B13 E11 E12 E

o

E13 X

Ny A21 A22 A23 B21 B22 B23 E21 E22 E23

Eo

y

Nxy A31 A32. A33 B31 B32 B33 E31 E32 E33

0

Yxy

Mx B11 B12 B13 D11 D12 D13 F11 F12 F13 Kx

My= B21 B22 B23 D21 D22 D23 F21 F22 F23 Ky (2.24)

MxyB31 B32 B33 D31 D32 D33 F31 F32 F33 Kxy

M• E11 E12 E13 F11 Fi2 F13 H11 H12 H13

•

Kx X

M• E21 E22 E23 F21 F22 F23 H21 H22 H23

Kyy E31 E32 E33 F31 F32 F33 H31 H32 H33

• •

Mxy Kxy

and

31

Qx A11 A12 En E120

Yxz

Qy A21 A22 E21 E220

Yyz

Q: =

E11 E12 D11 D12 Yxz

Q� E21 E22 D 21 D 22 Yyz

Combining Eqs. (2.24) and (2.25), in compact notation,

{P} = [Q]{E} (2.26)

[A] [B] [E] 0 0

[B] [D] [F] 0 0where [Q]= [E] [F] . [H] 0 0 (2.27)

0 0 0 [A] [E]0 0 0 [E] [b]

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.

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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.

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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.

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