Module-6: Laminated Composites-II Learning Unit-1: M6.1 M 6.1 Structural Mechanics of Laminates Classical Lamination Theory: Laminate Stiffness Matrix To this point in the development of classical lamination theory it is clear that if we are given the strains and curvatures at a point (x, y) on the reference surface— , , , ,x y xy x yK Kε ε γo o o o o and

xyK o —then we can compute the strain distributions through the thickness of the laminate. By using the stress-strain relations, we can compute the stress distributions. By using the transformation relations, we can determine the stresses and strains in the principal material system, and by using the definitions of the stress resultants, we can compute the force and moment resultants acting at that point. If we specify that the given strains and curvatures are the same at every point on the reference surface of a laminate of a given length and width, then we can determine the total forces and moments acting on the edges of the laminate. These steps are all a result of the plane stress assumption and the Kirchhoff’s hypothesis. Figure M6.1.1 illustrates the connection between these steps. What remains in the development of classical lamination theory is to be able to specify the force and moment resultants acting at a point (x, y) on the reference surface, and then to be able to compute the strains and curvatures of that point on the reference surface that these resultants cause. We want to fill in the missing link in the upper left portion of Figure M6.1.1, namely, compute the reference surface strains and curvatures, knowing the stress resultants. With these computed reference surface strains and curvatures, we can then, as in the previous examples, compute the strain and stress distributions through the thickness of the laminate. Relating the stress resultants to the reference surface strains and curvatures is an important step. In the application of composite materials to structures, we are often given the forces and moments that act on a laminate, and we want to know the stresses and strains that are caused by these loads. Having a relation between the force and moment resultants and the strains and curvatures at a point (x, y) on the reference surface will allow us to do this. In what follows we shall develop this key relationship.

Figure M6.1.1 Interrelations among concepts developed so far

M6.1.1 Formal Definition of Force and Moment Resultants Though we have introduced and computed them in the examples, we have not formally defined in one location the three force resultants and three moment resultants that are a result of integrating the stresses through the thickness of the laminate. The stress resultants in the x direction, xN in the y direction, and in shear, yN xyN are defined as,

(M6.1.1)

The resultants xN and will be referred to as the normal force resultants, and yN xyN will be referred to as the shear force resultant. Figure M6.1.2 illustrates a small element of laminate surrounding a point (x, y) on the geometric mid-plane, and the figure indicates the directions of these three stress resultants. The units of the stress resultants are force per unit length; the unit of length is a unit of length in the x direction or in the y direction. If the small element of laminate has dimensions by dy , then the force in the x direction due to

dxxN , the force in the x direction due to xyN , the force in the y direction due

to , and the force in the y direction due to yN xyN are, respectively,

(M6.1.2) The moment resultants, ,x yM M and xyM are defined as

(M6.1.3)

The resultants xM and yM will be referred to as bending moment resultants and ‘ ’ will be referred to as the twisting moment resultant. In Figure M6.1.3 the sense of these three moment resultants is illustrated, and the senses shown are important.

xy

Figure M6.1.2 Definition of force resultants xN , and yN xyN

Figure: M6.1.3 Definition of moment resultants ,x yM M , and xyM

Shortly we shall be relating a positive xM , for example, with a positive curvature in the x

direction, xK o . Knowing the correct sense of positive quantities is critical. The sense illustrated in Figure M6.1.3 is consistent with the cross-product definition of moments learned in elementary mechanics courses (i.e. ( = × )M r F . The units of the moment resultants are moment per unit length, again the unit of length being a unit of length in either the x direction or y direction. If a small element of laminate has dimensions ‘ ’ by ‘ ’, then the moments about the positive y and the positive x axis due to

dx dy,x yM M and xyM are, respectively,

(M6.1.4)

We will use these formal definitions of ,........,x xyN M in computing the laminate stiffness matrix. Specifically, we will use the force and moment resultants when studying the equilibrium of a small element of laminated plate. As the definitions of the moment resultants involve integrals with respect to the coordinate ‘z’, and because ‘z’ is measured relative to the reference surface, the moment resultants have to be thought of as producing moments on the reference surface. The force resultants can be considered to act anywhere through the thickness of the laminate. But when force resultants are used in conjunction with moment resultants, it is most consistent to think of the force resultants as producing forces on the reference surface. This will also be consistent when we begin discussing the relationship between the resultants and reference surface response, namely, ,..........,x xyKε o o . From this point forward, then, the force and moment resultants will be considered to act on the reference surface. M6.1.2 Laminate Stiffness: The ABD Matrix Equation (M6.1.1) and Equation (M6.1.3) give the definitions of the force and moment resultants. These definitions involve the stresses ,x yσ σ , and xyτ . From the stress-strain relations, the stresses can be written in terms of the strains by the now-familiar relation,

(M6.1.5)

and the strains can, in turn, be written in terms of the strains and curvatures of the reference surface to produce another familiar relation, namely,

(M6.1.6)

We have emphasized repeatedly that in a laminate the stresses are functions of ‘z’ because the transformed reduced stiffness ijQ ; change from layer to layer, and also because of the linear dependence on ‘z’- The reference surface strains and curvatures, by definition, do not depend on ‘z’- let us examine the expression for the normal force resultant xN by substituting for xσ from Equation (M6.1.6) into the integrand of the first equation of Equation (M6.1.1), specifically:

(M6.1.7)

and expanding the integrand leads to,

(M6.1.8)

The integration can be distributed over the six terms to give,

(M6.1.9)

(M6.1.9)

Because the reference surface deformations are not functions of ‘z’, they can be taken outside of the integrals, resulting in,

(M6.1.10)

The integrals through the thickness now involve just the material properties, specifically, the reduced stiffnesses. We need to look at these integrals in detail. The first integral in Equation (M6.1.10) is

(M6.1.11)

Because the reduced stiffnesses are materials properties, they are constant within a layer. However, from layer to layer the values of the reduced stiffnesses change. Because of this piecewise-constant property of the reduced stiffnesses, the integral through the thickness can be expanded to give,

(M6.1.12)

In the above expansion the locations of the layer interfaces have been used and the subscript on

11Q indicates layer number. The systematic scheme introduced earlier to identify the layer numbers and the ‘z’ location of the layer interfaces will be used to advantage in what follows. Because within each layer 11Q is a constant; in the above series of integrals it can be removed from each integral and the series of integrals written as

(M6.1.13)

But the integrations with respect to ‘z’ are quite simple, Equation (M6.1.13) becoming,

(M6.1.14)

or

(M6.1.15)

The difference in the ‘z’ values is recognized as simply the thickness of the layer, so Equation (M6.1.15) becomes,

thk kh

(M6.1.16)

The integral of 11Q through the thickness is denoted by 11A ; that is:

(M6.1.17)

In a similar manner, we find, referring to the third and fifth terms in Equation (M6.1.10), that

(M6.1.18)

and

(M6.1.19)

Thus the first, third, and fifth terms on the right side of Equation (M6.1.10) become,

(M6.1.20)

The second, fourth, and sixth terms on the right side of Equation (M6.1.10) involve integrals of

11 12,× ×Q z Q z and 16 ×Q z rather than just 11 12,Q Q and 16Q . Specifically, the second term is,

(M6.1.21)

Again using the fact that 11Q is constant within each layer but changes from layer to layer, we find that,

(M6.1.22)

Here the integrals on ‘z’ are also quite simple. Equation (M6.1.22) becomes,

(M6.1.23)

or

(M6.1.24)

This integral is denoted as ; that is:

11B

(M6.1.25)

In like manner, referring to the fourth and sixth terms in Equation (M6.1.10), we find,

(M6.1.26)

and

(M6.1.27)

The expression for xN in Equation (M6.1.10) becomes, after slight rearrangement,

(M6.1.28)

Because they are summations of material and geometric properties,

11 12 16 11 12 16, , , , ,andA A A B B B are easy to compute and are inherent properties of the laminate. Because the anij ijA d B ; are integrals, they represent smeared or integrated properties of the

laminate. Equation (M6.1.28) represents a relation between the normal force resultant xN and the six reference surface deformations. This is part of the missing link in Figure M6.1.1. We will develop five other equations that will lead to a total of six relations between the six stress resultants and the six reference surface deformations. The missing link will then be complete. For the four-layer [0/90] 5 laminate, as each layer is the same thickness ‘ ’, by Equation (M6.1.17),

h

(M6.1.29)

or

(M6.1.30)

By knowing the values of ijQ , we can get value of , and, 11A 12A 16A

1A some numericalvalue1( C )11 = − = (M6.1.31)

Likewise,

2A some numerical value1( C )11A 0 (For cross ply la min ate)16

= − == −

(M6.1.32)

For all cross-ply laminates 16A will be zero because for each layer 16 0Q = . The ijB can be computed from Equations (M6.1.25)-(M6.1.27) as,

(M6.1.33)

or

(M6.1.34)

We find, using the numerical values of the interface locations for the [ laminate, Equation (M6.1.21) leads to,

]0 /90S

(M6.1.35) The quantity is exactly zero because the contributions to from the layers at negative z-locations exactly cancel the contributions to from the layers at the positive z locations. This is a characteristic of a symmetric laminate. The values of all the

11B 11B11B

ijB will be zero for all symmetric laminates because of this cancellation effect. For symmetric laminates there is no need to go through the algebra to compute the values of the ijB . They are all exactly zero; thus,

(M6.1.36)

16B is zero not only because it is a symmetric laminate, but also because the 16Q , are zero for every layer. For the six-layer [ ]30 / 0

S± laminate,

(M6.1.37)

Using layer orientation, we find that,

(M6.1.38)

For 1j = , substituting numerical values for the appropriate ijQ , we find

3A some numericalvalue1( C )11 = − = (M6.1.39) and for 2j = ,

4A some numericalvalue1( C )12 = − = (M6.1.40) and for 6j = ,

A 016 = (M6.1.41) The term 16A is exactly zero because the contributions from the +30° layers exactly cancel the contributions from the —30° layers. This is characteristic of the 16-component of the ijA , for

what we will come to call a balanced laminate. Balanced laminates are characterized by θ± pairs of layers. Because the laminate is symmetric, the components, , and are exactly zero. 11 12,B B 16B

That is, for Symmetric Laminates. 11

12

16

B 0BB 0

===

0

] When turning to the [ laminate, because relative to the six-layer laminate there are

exactly one-half the number of layers in any particular direction, the Ay- for the [ ]30 / 0

T±

30 / 0T

±

laminate are one-half the values of the ijA for the [ ]30 / 0S

± laminate. However, as

the[ , is not a symmetric laminate, the ]30 / 0T

± ijB will not be zero. Specifically, referring to equations (M6.1.39), (M6.1.40) and (M6.1.41), we find that,

A 0 N /16 = m (M6.1.42)

For the ijB ,

(M6.1.43)

For 1j = and using the interface locations, we find,

5B some numericalvalue1( C )11 = − = (M6.1.44) For 2and6j = ,

6

7

B some numericalvalue1( C )12B some numericalvalue1( C )16

= − = −= − = −

(M6.1.45)

To continue with the development of the laminate stiffness matrix, we can substitute the expressions for yσ and xyτ into the definitions in Equation (M6.1.1) for and ‘ ’, respectively, to yield,

yN xy

(M6.1.46)

and

(M6.1.47)

As in going from Equation (M6.1.8) to (6.1.9), Equation (M6.1.9) to (6.1.10), and finally to Equation (M6.1.28), and continuing to define quantities ijA and ijB rearranging, we find that

(M6.1.48)

and

(M6.1.49)

If we combine these definitions with the definitions associated with xN , we can write the results in matrix notation as,

(M6.1.50)

The above matrix equation constitutes three of the six relations between the six stress resultants and the six reference surface deformations. Three equations have yet to be derived, but with the above notation, it is clear that the ijA , and ijB are components of three-by-three matrices. To derive the remaining three equations, we use the moment expressions, Equation (M6.1.3), by substituting for ax from Equation (M6.1.6) into the first equation of Equation (M6.1.3) to yield,

(M6.1.51)

where, as before, the integration can be distributed over the six terms to give,

(M6.1.52)

Removing the reference surface deformations from the integrals, we find that,

(M6.1.53)

The first integral on the right hand side of Equation (M6.1.56) is,

(M6.1.54)

and this is immediately recognizable, from Equation (M6.1.25), as . In fact, the first, third, and fifth terms on the right side of Equation (M6.1.53) become,

11B

(M6.1.55)

The appearance of in both the force resultant and the moment resultant expressions; this is important to remember.

ijB

The second, fourth, and sixth terms on the right hand side of Equation (M6.1.53) are new; the second term is,

(M6.1.56)

which can be expanded to,

(M6.1.57)

Because of the constancy of the reduced stiffnesses within layers,

(M6.1.58)

or

(M6.1.59)

This integral is denoted as ; that is: 11D

(M6.1.60)

Referring to the fourth and sixth terms in Equation (M6.1.53), we find that,

(M6.1.61)

And

(M6.1.62)

With the definitions of the , and the earlierijD ijB , the expression for the bending moment

resultant xM in Equation (M6.1.53) becomes,

(M6.1.63)

The , are similar to the ijD ijB in that they involve powers of the ‘z’ locations of the layer interfaces. Although the ijA also involved ‘z’ location of the layer interfaces, it was really only the thickness of each layer that was involved. The actual location of the layers through the thickness is not important. However, with the locations ijB , and through the thickness of the laminate, as well as the thickness of the layers, are important.

ijD

To continue with the numerical examples, the values of and , for the example laminates will be computed for illustration. For the four-layer

11 12,D D 16D[ ]S0 / 90 laminate,

(M6.1.64)

or

(M6.1.65)

The ensuing algebra leads to,

8D some numericalvalue1( C )11 = − = (M6.1.66) For , 12D

(M6.1.67)

or,

(M6.1.68)

Numerically,

9D some numericalvalue1( C )12 = − = (M6.1.69) Because is zero for each layer, 16Q

D 016 = (M6.1.70) For the [ ]30 / 0

S± laminate,

(M6.1.71)

or

(M6.1.72)

We find, referring to the interface locations for this laminate that the algebra leads to,

8

9

10

D some numericalvalue1( C )11D some numericalvalue1( C )12D some numericalvalue1( C )16

= − == − == − =

(M6.1.73)

Note that the value of is not zero. Unlike the situation with16D 16A , where the contribution of the -30° layers to the 16A , cancels the contribution of the +30° layers, the contributions of the layers with opposite orientation do not cancel in the calculation of . This is because the locations of the layers, in addition to the layer thicknesses, are involved in the computation of the . We

will see later in this module that the existence of a nonzero , term has important consequences. Also, we will see the dependence of the

16DijD

16D16A on the θ± layer arrangement is

identical to the dependence of 16A , and the dependence of is identical to the dependence of .

26D16D

Finally, for the three-layer [ laminate, ]30 / 0

T±

(M6.1.74)

or,

(M6.1.75)

Using numerical values leads to,

8

9

10

D some numericalvalue1( C ) N m11D some numericalvalue1( C ) N m12D some numericalvalue1( C ) N m16

= − = −= − = −= − = −

(M6.1.76)

To finish the development of the laminate stiffness matrix, we can substitute the expressions for

yσ and xyτ into the definitions in Equation (M6.1.3) for andy xyM M . The result is,

(M6.1.77)

and,

(M6.1.78)

As in going from Equation (M6.1.51) to (6.1.52), Equation (M6.1.52) to (6.1.53), and finally to Equation (M6.1.63), and continuing to define additional , we find that, ijD

(M6.1.79)

and,

(M6.1.80)

We can write the results for the three moments , andy y xyM M M in matrix notation as,

(M6.1.81)

The above analysis constitutes the remaining three relations of the six relations between the six stress resultants and the six reference surface deformations. We can combine the six relations just derived to form one matrix relation between the six stress resultants and the six reference surface deformations. The combined matrix relation is,

(M6.1.82)

Of course, in the above,

(M6.1.83)

The six-by-six matrix in Equation (M6.1.82) consisting of the components , is called the laminate stiffness matrix. For obvious reasons it is also known as the ABD matrix. The ABD matrix defines a relationship between the stress resultants (i.e., loads) applied to a laminate, and the reference surface strains and curvatures (i.e., deformations). This form is a direct result of the Kirchhoff hypothesis, the plane-stress assumption, and the definition of the stress resultants. The laminate stiffness matrix involves everything that is used to define the laminate—layer material properties, fiber orientation, thickness, and location.

, 1,2,6; 1,2,6ijD i j= =

The relation of Equation (M6.1.82) can be inverted to give,

(M6.1.84a)

where,

(M6.1.84b)

Equation (M6.1.82) and its inverse Equation (M6.1.84) are very important in the analysis of composite structures, particularly the inverse form. Knowing the loads, we can determine the reference surface strains and curvatures uniquely from Equation (M6.1.84). If we know the reference surface strains and curvatures, as we have seen in the previous examples, then we can determine the strains and stresses in each layer. Figure M6.1.4 illustrates the importance of the ABD matrix, namely the fact that it closes the loop on the analysis at composite laminates (see Figure M6.1.1). It is appropriate at this point to present the numerical values of the remaining components of the ABD matrix for each of the three laminates we have been discussing. So far only limited numerical values have been illustrated. It will be left as an exercise to verify the numerical values presented for all the components of the ABD matrix for the three laminates. It is important to note the units on the various components of the ABD matrix. The units are consistent with the units of the reference surface strains and curvatures and the units of the stress resultants.

Figure M6.1.4 Closing the loop on concepts developed

Writing the numerical values in full form is quite revealing; the coupling, or lack thereof, between the various stress resultants and reference deformations becomes quite obvious. The coupling is important in determining how much interaction there will be between the specific loads and specific reference surface strains and curvatures. Essentially, the more zeros there are in the ABD matrix, or its inverse, the less coupling there is between the stress resultants and reference surface deformations. For example, for the [0 and [0 laminates, because all components of the ‘B’ matrix are zero, the reference surface curvatures are decoupled from the force resultants, and the reference surface strains are decoupled from the moment resultants. The six-by-six relation of Equation (M6.1.82) reduces to two three-by-three relations, namely,

/ 90]S / 30]S

(M6.1.85)

And

(M6.1.86)

Such a reduction is not possible when there are elements of the ‘B’ matrix present (i.e., when the laminate is unsymmetric). We will have more to say regarding the couplings within the ABD matrix in the next section, where the degree of coupling, and its effects on laminate response, are categorized in a systematic way. Moreover, we can do the categorization a priori, before any analysis is started, and we can use the advantages of knowing the level of coupling to simplify the analysis procedure. Learning Unit-2: M6.2 M6.2 Special Classification of Laminates We have seen in the numerical results presented in the previous section that the form of the ABD matrix depends strongly on whether the laminate is what we referred to as symmetric, whether it consists of just 0° and 90° layers, or whether for every layer at orientation θ+ there is a layer at θ− . In this section we would like to formally classify laminates as to their stacking arrangement, and indicate the effects of the various classifications on the ABD matrix. In a subsequent section, physical interpretations will be given to particular components of the ABD matrix. As a sidenote, in connection with stress-strain equations and their inverse relations, it was stated that the presence of the zeros in the compliance and stiffness matrices in the principal material coordinate system was associated with a fiber-reinforced material being orthotropic. Similarly, with the lack of any zero terms in Equation (M6.1.82), the elastic behavior of a general laminate is sometimes termed nonorthotropic, or anisotropic. M6.2.1 Symmetric Laminates A laminate is said to be symmetric if for every layer to one side of the laminate reference surface with a specific thickness, specific material properties, and specific fiber orientation, there is another layer the identical distance on the opposite side of the reference surface with the identical thickness, material properties, and fiber orientation. If a laminate is not symmetric, then it is referred to as an unsymmetric laminate. Note that this pairing on opposite sides of the reference surface must occur for every layer. Although we have restricted our examples to laminates with layers of the same material and same thickness, this does not have to be the case for a laminate to be symmetric. A four-layer laminate with the two outer layers made of glass-reinforced material oriented at +45° and the two inner layers made of graphite-reinforced material oriented at —30° is a symmetric laminate. For a symmetric laminate all the components of the B matrix are identically zero. Consequently, the full six-by-six set of equations in Equation (M6.1.82) decouples into two three-by-three sets of equations, namely,

(M6.2.1a)

And

(M6.2.1b)

With symmetric laminates, the inverse relation, Equation (M6.1.84), also degenerates into two three-by-three relations, specifically,

(M6.2.2a)

(M6.2.2b)

In the above the ‘a’ matrix is the inverse of the ‘A’ matrix and has components,

(M6.2.3a)

and the ‘d’ matrix is the inverse of the ‘D’ matrix and has components,

(M6.2.3b)

where the definitions,

(M6.2.3c)

and,

(M6.2.3d)

have been used. Because neither the ‘A’ nor ‘D’ matrices have any zero terms, a general symmetric laminate is anisotropic in both inplane and bending behavior. M6.2.2 Balanced Laminates A laminate is said to be balanced if for every layer with a specified thickness, specific material properties, and specific fiber orientation, there is another layer with the identical thickness, material properties, but opposite fiber orientation somewhere in the laminate. The layer with opposite fiber orientation does not have to be on the opposite side of the reference surface, nor immediately adjacent to the other layer, nor anywhere in particular. The other layer can be anywhere within the thickness. A laminate does not have to be symmetric to be balanced. The symmetric [ ]30 / 0

S± laminate and the unsymmetric [ ]30 / 0

T± laminate are both balanced

laminates. If a laminate is balanced, then the stiffness matrix components 16A , and 26A are

always zero because the 16Q and 26Q from the layer pairs with opposite orientation are of opposite sign, and the net contribution to 16A , and 26A from these layer pairs is then zero. To classify as balanced, all off-axis layers must occur in pairs. As a special case, all laminates consisting entirely of layers with their fibers oriented at either 0 or 90°, to be discussed shortly, are balanced, as 16Q and 26Q are zero for every layer, resulting in 16A , and 26A being zero. The ABD matrix of a balanced but otherwise general laminate is not that much simpler than the ABD matrix of Classification of Laminates and Their Effect on a general unsymmetric, unbalanced laminate. The full six-by-six form, Equation (M6.1.85), applies but with the 16A , and 26A components set to zero. To obtain the inverse relation, Equation (M6.1.87a), the full six-by-six with the zero entries for 16A , and 26A must be inverted. M6.2.3 Symmetric Balanced Laminates A laminate is said to be a symmetric balanced laminate if it meets both the criterion for being symmetric and the criterion for being balanced. If this is the case, then Equation (M6.1.82) takes the decoupled form,

(M6.2.4a)

with,

and,

(M6.2.4b)

The inverse form involving the force resultants is,

(M6.2.5a)

with,

The inverse form involving the moments is no different than for symmetric but unbalanced laminates, namely,

(M6.2.5b)

For the symmetric balanced case, the third of Equation (M6.1.82) decouples from all other equations. The components of the a matrix in Equation (M6.2.5a) are,

(M6.2.6a)

and the components of the ‘d’ matrix are the same as the symmetric but balanced case, Equation (M6.2.3b), namely,

(M6.2.6b)

with being given by Equation (M6.2.3d). Becausedet[D] 16A , and 26A are zero, a symmetric balanced laminate is orthotropic with respect to inplane behavior. M6.2.4 Cross-Ply Laminates A laminate is said to be a cross-ply laminate if every layer has its fibers oriented at either 0° or 90°. If this is the case, because 16Q and 26Q are zero for every layer, then

16 26 16 26 16 26, , , , andA A B B D D are zero, and there is some decoupling of the six equations. In

particular, the six equations decouple to a set of four equations and a set of two equations, namely,

(M6.2.7a)

(M6.2.7b)

M6.2.5 Symmetric Cross-Ply Laminates A laminate is said to be a symmetric cross-ply laminate if it meets the criterion for being symmetric and the criterion for being cross-ply. This results in the simplest form of the ABD matrix. All the ijB , are zero, and 16 26 16 26, , andA A D D are zero. The six equations of Equation (M6.1.82) decouple significantly. Specifically,

(M6.2.8a)

with,

and,

(M6.2.8b)

with

The inverted form is,

(M6.2.9a)

and

with

(M6.2.9b)

with,

The components of the ‘a’ matrix are the same as the symmetric balanced case,

(M6.2.10a)

and the components of the ‘d’ matrix are,

(M6.2.10b)

Compared to the most general case, the symmetric cross-ply laminate is trivial. Because

16 26 16, ,A A D and are zero, a symmetric cross-ply laminate is orthotropic with respect to both inplane and bending behavior.

26D

M6.2.6 Single Isotropic Layer As a comparison, it is of interest to record the ABD matrix for the case of a single isotropic layer, the subject of studies in classical plate theory. What we have presented in the development of classical lamination theory is a generalization of classical plate theory. The isotropic case is a special case of this generalization, and for a single isotropic layer of thickness ‘H’,

(M6.2.11)

In the above equation, ‘E’ is the extensional modulus, referred to as Young's modulus when dealing with isotropic materials, and ‘ν ’ is Poisson's ratio of the single layer of material. Because of Equation (M6.2.11), the ABD matrix for a single isotropic layer decouples and greatly simplifies, and Equation (M6.1.82) becomes,

(M6.2.12a)

with,

and,

(M6.2.12b)

with,

The inverse relation is,

(M6.2.13a)

with

and

(M6.2.13b)

with,

Finally, as we have implied, there are no simplifications in the ABD matrix if the laminate is unsymmetric, if it is unbalanced, and if it is not a cross-ply. One must work with the full six-by-six matrix of Equation (M6.1.82). In Equation (M6.1.84), one must invert the full six-by-six relation. M6.2.7 Comments Whether we invert the full six-by-six ABD matrix or take advantage of the simplifications just discussed, it is clear we are now in a position to start at any point in the loop of Figure M6.4 and calculate our way to any other point. Despite this fact, it is the loads that are usually known, and these are converted to force and moment resultants by dividing by the appropriate lengths. We use the laminate material properties (materials, fiber orientations, etc.) to compute the ABD matrix. We can then compute the reference surface strains and curvatures and we can determine the stress and strain response, in the x-y-z global coordinate system, or the 1-2-3 principal material coordinate system. An important alternative to this process is determining the laminate material properties that will produce a specific response to a known set of loads. This process is called laminate design. Optimal laminate design goes one step further and seeks the laminate material properties that minimize, for example, laminate stresses in the presence of a given set of loads. Design and optimization are important facets of composite analysis, and in many instances they are the primary activities. However, they both depend on having a thorough understanding of the mechanics of composite materials, and all of the steps that go into closing the loop of Figure M6.1.4. Certain subtleties are involved in some of the steps of Figure M6.1.4, and one subtlety, the coupling effects that occur in laminates, is discussed next. These couplings are reflected in the various components of the ABD matrix and its inverse. Because the ABD matrix is a stiffness matrix, these couplings are properly referred to as elastic couplings. We have discussed two of these important couplings. One was the coupling that occurs in an unsymmetric laminate due to the presence of the B matrix. These couplings do not occur with metallic materials and hence are not fully appreciated by many who do not work with composites. M6.2.8 Effective Engineering Properties of a Laminate It is often convenient to have what can be referred to as effective engineering properties of a laminate. These consist of the effective extensional modulus in the x direction, the effective extensional modulus in the y direction, the effective Poisson's ratios, and the effective shear modulus in the x-y plane. These particular effective properties can be defined for general laminates but they make the most sense when considering the inplane loading of symmetric balanced laminates. Let us restrict our discussion to the inplane loading of symmetric balanced or symmetric cross-ply laminates. In this case the identical equations (6.2.5a) and (6.2.9a) are applicable. To introduce effective engineering properties, define the average laminate stress in the ‘x’ direction, the average laminate stress in the ‘y’ direction, and the average laminate shear stress to be, respectively,

(M6.2.14)

The average stresses, like all average quantities, do not really exist. They are simply a definition. In the definitions of the average stresses, the integrals are actually the definitions of the normal and shear force resultants, Equation (M6.1.1); that is:

(M6.2.15)

or

(M6.2.16)

with ‘H’ being the thickness of the laminate. Knowing the force resultants and the laminate thickness, we can compute the average stresses easily. Substituting the above form into Equation (M6.1.105a), we find

(M6.2.17)

If we compare the form of this equation with the form of the compliance for a single layer in a state of plane stress, from the compliances equations, it is possible to define the above three-by-three matrix as the laminate compliance matrix. Accordingly, by analogy with compliances equations, the laminate's effective extensional modulus in the ‘x’ direction, effective extensional modulus in the ‘y’ direction, effective shear modulus, and two Poisson's ratios are given by,

(M6.2.18)

In terms of the elements of the ‘A’ matrix, from Equation (M6.2.6a),

(M6.2.19)

Of course, the reciprocity relation,

(M6.2.20)

is valid and so andyx xyν ν are not independent. As an example, for the four-layer[0 laminate for which thickness , substituting into Equation (M6.2.19) from Equation (M6.1.88a), we find that the effective engineering properties are,

/ 90]S

12H = Numerical value=C mm

x 13

y 14

xy 15

xy 16

yx 17

E = Numerical value =C GPa

E = Numerical value =C GPa

G = Numerical value =C GPa

= Numerical value=C

= Numerical value=C

ν

ν

(M6.2.21)

For this particular laminate, the x and y directions have identical effective extensional moduli, and the effective shear modulus is the same as the shear modulus for a single layer . This is not surprising, considering that there are two layers with their fibers in the ‘x’ direction and two with their fibers in the ‘y’ direction, and that the two directions respond the same to an applied load. Furthermore, because both the 0° and the 90° layers have their fibers aligned with the global coordinate directions, the shear modulus for the laminate is identical to the shear modulus for a single layer.

12G