Mechanics Analysis Composite Materials Episode 8 ppt

Mechanics of laminates 233 To compare two possible forms of constitutive equations for transverse shear, consider for the sake of brevity an orthotropic layer for which For transverse s

Chapter 5 Mechanies of laminates 23 1

displace-actual ones, i.e.,

However, according to Eqs (5.11), shear strains are linear combinations of shear stresses So we can use the same law to introduce average shear strains as

(5.13)

Average shear strains ’yx and yv can be readily expressed in terms of displacements

if we substitute Eqs (5.9) into Eqs (5.13), i.e.,

These equations, in contrast to Eqs (5.9), do not include derivatives with respect

to z So we can substitute Eqs (5.1) and (5.2) to get the final result

232 Mechanics and analysis of composite materials

Consider Eqs (5.10) and (5.11) Integrating them over the layer thickness and using Eqs (5.12) and (5.13) we get

Because the actual distribution of stresses and strains according to the foregoing reasoning is not significant, we can change them for the corresponding average stresses and strains:

These expressions, in general, do not coincide with Eqs (5.17)

Thus, the constitutive equations for transverse shear are specified by Eqs (5.15), and there exist two, in general different, approximate forms of stiffnesscoefficients -Eqs (5.17) and (5.19) The fact that equations obtained in this way are approximate

is quite natural because the assumed displacement field, Eqs (5.1) and (5.2), is also approximate

Chapter 5 Mechanics of laminates 233

To compare two possible forms of constitutive equations for transverse shear, consider for the sake of brevity an orthotropic layer for which

For transverse shear in the xz-plane Eqs (5.15) yield

If the shear modulus does not depend on z, both equations, Eqs (5.21) and (5.22),

give the same result S55 = G,h

Using the energy method applied in Section 3.3 we can show that the Eqs (5.21) and (5.22) provide the upper and the lower bounds for the actual transverse shear

stiffness Indeed, consider a strip with unit width experiencing transverse shear

induced by force Y , as in Fig 5.7, Assume that Eq (5.20) links the actual force K

with the actual angle yx = A / l through the actual shear stiffness SSSwhich we do not know and which we would like to evaluate To do this, we can use two variational principles described in Section 2.11 According to the principle of minimum total potential energy

Y

Fig 5.7 Transverse shear of a strip with unit width

234

where

Mechanics and anatysis composite materials

Tact = u&,-& t , Tadm = u&, -Aadm

are the total energies of the actual state and some admissible kinematic state

expressed in terms of the strain energy, U, and work A performed by force V, on

displacement A (see Fig 5.7) For both states

and condition (5.23) reduces to

Chapter 5 Mechanics qf laminates 235

For the admissible state we should apply

and use some admissible distribution for L- The simplest approximation is z,

-e

Substitution into condition (5.27) yields

Thus, Eq (5.22) provides the lower bound for S55, and the actual stiffness satisfies the following inequality:

So, constitutive equations for the generalized layer under study are specified with Eqs ( 5 5 ) and (5.15) Stiffness coefficients that are given by Eqs (5.6)-(5.8), and (5.17) or (5.19) can be written in a form more suitable for calculations To do this, introduce new coordinate t = z +e such that 0 d t d h (see Fig 5.8) Transforming the integrals to this new variable we get

where rnn = 1 1, 12, 22, 14, 24, 44 and

(5.28)

Fig 5.8 Coordinates of an arbitrary point A

236 Mechanics and analysis of composite materials

Transverse shear stiffnesses, Eqs (5.17) and (5.19), acquire the form

5.2 Stiffness coefficientsof a homogeneous layer

Consider a layer whose material stiffness coefficients A,, do not depend on

Both Eqs (5.30) and (5.31) give the same result for S,, As follows from the second

of these equations, membrane-bending coupling coefficients C,, become equal to zero if we take e = h / 2 , Le., if the reference plane coincides with the middle plane of

the layer shown in Fig 5.9 In this case, Eqs (5.5) and (5.15) acquire the following

de-coupIed form:

Chapter 5 Mechanics of Inminates 231

X

Y

Fig 5.9 Middle plane of a laminate

As can be seen, we have arrived at three independent groups of constitutive

equations for in-plane stressed state of the layer, bending and twisting, and

transverse shear Stiffness coefficients, Eqs (5.34), become

For an orthotropic layer, there are no in-plane stretching-shear coupling (B14 =

B24 = 0) and transverse shear coupling (s56= 0) Then, Eqs (5.35) reduce to

In terms of engineering elastic constants material stiffness coefficients of an orthotropic layer can be expressed as

(5.38)

238 Mechanics and analysis of composite materials

Finally, for an isotropic layer, we have

(5.40)

where E =E / ( I -v’)

5.3 Stiffness coefficients of a laminate

Consider a general case, Le., a laminate consisting of an arbitrary number of

layers with different thicknesses hi and stiffnessesA!), (i = 1,2,3, ,k).Location of

an arbitrary ith layer of the laminate is specified by coordinate ti, which is the distance from the bottom plane of the laminate to the top plane of the ith layer (see Fig 5.10) Assuming that material stiffness coefficients do not change within the

thickness of the layer and using piece-wise integration we can write parameter I,,,,, in

Chapter 5 Mechanics of laminates 239

(5.41)

where I =0, 1 , 2 and to = 0, tk = h (see Fig 5.10) For thin layers, Eqs (5.41) can

be reduced to the following form, which is more suitable for calculations:

where hi = ti -ti-l is the thickness of the ith layer

Thus, membrane, coupling, and bending stiffness coefficients of the laminate are

specified with Eqs (5.28) and (5.42) Consider transverse shear stiffnesses which have two diflerent forms determined by Eqs (5.30) and (5.31) Because both

equations coincide for a homogeneous layer (see Section 5.2), we can expect that the difference shows itself in laminates consisting of layers with different transverse shear stiffnesses The laminate for which this difference is the most pronounced is a sandwich structure with metal facings (inner and outer layers) and a foam core (middle layer) that has very low shear stiffness For such

a sandwich, experimentally found transverse shear stiffness is S =389 kN/m

(Aleksandrov et al., 1960), while Eqs (5.30) and (5.31) yield, respectively, S =

37200 kN/m and S = 383 kN/m Thus, Eq (5.31) provides much more accurate

result for sandwich structures This conclusion is also valid for composite

laminates (Chen and Tsai, 1996)

A particular case, important for applications, is an orthotropic laminate for

which Eqs (5.5) and (5.15) acquire the form:

(5.43)

where, membrane, coupling, and bending stiffnesses, B,,, C,,,,, and D,,,,, are

specified by Eqs (5.28) and (5.42), while transverse shear stiffnesses are

240 Mechanics and analysis of composite materials

Some typical layers considered in Chapter 4 were actually quasi-homogeneous

laminates (see Sections 4.4, and 4.5),but being composed of a number of identical

plies, they were treated as homogeneous layers The accuracy of this assumption is evaluated below

5.4.I Laminate composed of identical homogeneous Kayers

Consider a laminate composed of layers with different thicknesses but the same

stiffnesses, Le., such that A:; =A,, for all i = 1 , 2 , 3 , .k Then, Eqs (5.29) and (5.32) yield

This result coincides with Eqs (5.33), which means that the laminate consisting of the layers with the same mechanical properties is a homogeneous laminate (layer) studied in Section 5.2

5.4.2 Laminate composed of inhomogeneous orthotropic layers

Let the laminate have the structure [0"/90"],, where p = 1,2,3,. specifies the number of elementary cross-ply couples of 0" and 90" plies In Section 4.4, this laminate was treated as a homogeneous layer with material stiffness coefficients specified by Eqs (4.100) Taking 60 = i g o = 0.5 in these equations we get

In accordance with Eqs (5.36), stiffness coefficients of this layer should be

Chapter 5 Mechanics of laminates 24 1

To calculate the actual stiffnesses of the laminate, we should put hi= 6, ti = is,

unidirectional ply Then, Eqs (5.28) and (5.42) yield

bending stiffnesses are also the same for mn = 12,44 There is no difference between

the models for a = 1 because the laminate reduces in this case to a homogeneous

layer

Summing up the series in Eqs (5.48) and using Eqs (5.47) we arrive at

Taking into account that in accordance with Eqs (5.46) and accepted notations

we can conclude that the only difference between the homogeneous and the

laminated models is associated with coupling coefficients C I ~and C?? which are

242 Mechanics and analysis of composite materials

equal to zero for the homogeneous model and are specified by Eqs (5.49) for the laminated one Because p6 = h / 2 , we can write these coefficients in the form

showing that C,, 4 0 for 6 +0

Consider a laminate with the structure [+$/- +Ip, wherep is the number of layers

each consisting of +4 and -4 unidirectional plies Constitutive equations (5.5) for this laminate are

where

(5.50)

where, h is the laminate thickness, 6 the ply thickness, and A,, are material stiffness

coefficients specified by Eqs (4.72) As can be seen, the laminate is anisotropic

because +4 and -4 plies are located in different planes Homogeneous model of the laminate ignores this fact and yields c 1 = c 2 4 = 0 Calculations show that these coefficients, not being actually equal to zero, practically do not influence the

laminate behavior for h / 6 220

Laminates in which any ply or layer with orientation angle +#Jis accompanied

by the same ply or layer but with angle -4 are referred to as balanced laminates Being composed of only angle-ply layers these laminates have no shear-extension coupling (B14 =B24 = 0 ) , bending-stretching and shear-twisting coupling (CIl=

C12 = C22 = CU = 0) As follows from Eqs (5.50), only stretching-twisting and bending-shear coupling can exist in balanced laminates These laminates can

include also 0" and 90" layers, but membrane-bending coupling can appear in such

laminates

Chapter 5 Mechanics of laminates 243

5.5 Quasi-isotropic laminates

The layers of the laminate can be arranged in such a way that the laminate will behave as an isotropic layer under in-plane loading Actually, the laminate is not isotropic (that is why it is called a quasi-isotropic laminate) because under transverse (normal to the laminate plane) loading and under interlaminar shear its behavior is different from that of an isotropic (e.g., metal) layer

To derive the conditions that should be met by the structure of a quasi-isotropic laminate consider in-plane loading with stresses o ~ ,o,., and z.~, that are shown in Fig 5.1 and induce only in-plane strains E:, E;, and Y:,~ Taking IC, = IC, = K ~ ~ ,= 0 in

Eqs (5.5) and introducing average (through the laminate thickness 6) stresses as

we can write the first three equations of Eqs (5.5) in the following form:

where in accordance with Eqs (5.28) and (5.42)

where, hi is the thickness of the ith layer normalized to the laminate thickness and

constitutive equations analogous to Eqs (5.51) are

0, = E(8: + v&;), I = E(&: + V & ! ) , Z = Gy:-v , (5.53) where

Eqs (4.72) and Section 5.4.3, this means that the laminate should be balanced, Le.,

it should be composed of O", &4i(or d i and IT -4i),and 90"layers only Because the laminate stiffness in the x- and the y-directions must be the same, we require that Bll = B 2 2 Using Eqs (4.72), taking hi= h for all i, and performing some transformation we arrive at the following condition:

244 Mechanics and analysis of composite materials

k

C C O S 2 4 = 0

i= 1

As can be checked by direct substitutions, for k = I this equation is satisfied if

= 45" and for k = 2 if dl = 0 and 42= 90" Naturally, such one- and layered materials cannot be isotropic even in one plane So consider the case k23, for which the solution has the form

and calculating stiffness coefficients in Eqs (5.52) and (4.72) we get

These stiffnesses provide constitutive equations in the form of Eqs (5.53) and satisfy conditions (5.54) which can be written as

if

(5.56)

Possible solutions (5.55) providing quasi-isotropic properties of the laminates with different number of layers are listed in Table 5.1 for k < 6

Chapter 5 Mechanics of laminates 245 All quasi-isotropic laminates having different structures determined by Eqn

Poisson’s ratio specified by Eqs (5.56) For typical advanced composites with

properties listed in Table 3.5, these characteristics are presented in Table 5.2

As follows from Tables 5.2 and I 1,specific stiffness of quasi-isotropic composites with carbon and boron fibers exceeds the corresponding characteristic of traditional isotropic structural materials -steel, aluminum, and titanium

it from the condition C I I= 0 Then,

246 Mechanics and analysis of composite materials

and

(5.58)

Introduce a new coordinate of an arbitrary point A in Fig 5.1 1 as z = t -( h / 2 )

Changing t for z we can present Eq (5.29) in the form

Substituting these integrals into Eqs (5.57) and (5.58) we get

Now decompose All as a function of z into symmetric and antisymmetric

Fig 5.1 1 Coordinate of point A referred to the middle plane

Chapter 5 Mechanics of laminates

Then, Eq (5.61) yields

247

As can be seen from Eq (5.60), Dll reaches its maximum value if J:;) = 0 or = 0

Thus, symmetric laminates provide the maximum bending stiffness for a given number and mechanical properties of layers and, being referred to the middle-plane,

do not have membrane-bending coupling effects This essentially simplifies behavior

of the laminate under loading and constitutive equations which have the form specified by Eqs (5.35) For a symmetric laminate with the layer coordinates shown

in Fig 5.12, stiffness coefficients are calculated as

The transverse shear stiffness coefficients are given by Eq (5.31) in which

To indicate symmetric laminates, contracted stacking-sequence notation is used, e.g., [0"/90"/45"],~instead of [0"/90"/45"/45"/90'/0"]

Fig 5.12 Layer coordinates of a symmetric laminate