A naly! i! of Composite Laminate! 9 Mechanic! of Comp o!ite Laminate.

The th irty six coefficients Cij are not all independent of each other. Thenumber of independent constants depends on the material constitution. First weshow that C kj = Cjk i that is, they are symmetric for materials for which the strainenergy density function Uo is such that

(2.2 - 3)

Some anisotropic materials may possess material symmetries and their consti­tutive behavior can be described with fewer than 21 constants. When the elasticcoefficients at a point have the same values for every pair of coordinate systemswhich are mirror images of each other in a certain plane, that plane is called aplane of elastic .!ymmetry for the material at that point . Materials with one planeof symmetry are called monoclinic materials, and the number of elastic coeffi­cients for such materials reduces to 13. If the plane of symmetry is :1: 3 = 0, theconstitutive relations become:

(2.2 - 4)

C 13 0 0 C 16 El

C23 0 0 C26 E2

C 33 0 0 C36 E3 (2.2 - 8)C44 C45 0 E4

C55 0 E5

C66 E6sym.

=

Note that the out-of-plane shear st resses , namely 0"4 and 0"5, are independent ofnormal strains and the inplane shear strain.

If a material system has three mutually perpendicular planes of elastic sym­metry, then the number of independent elastic coefficients can be reduced to nine .Such materials are referred to as orthotropic. The stress-strain relations for anorthotropic material are given by

(2.2 - 5)

To illustrate this, we consider the strain energy density of the material which maybe expressed as

Substituting equation (2.2-1) into equation (2.2-4) and integrating, we obtain

1u, = -CyYE" Ej2 K j(

Substituting for Uo from Eq . (2.2-5) into Eq . (2.2-3), we arrive at the expression

(2.2 - 6)

By comparing expressions (2.2-6) and (2.2-1), we conclude that Ckj = Cjk'

Because of this symmetry, there are only 21 independent elastic constants foranisotropic materials. In matrix form Eq. (2.2-1) can be expressed as

=

sym.

C 13 0 0 0 El

C 23 0 0 0 E2

C 33 0 0 0 E3 (2.2 - 9)C 44 0 0 E4

055 0 E5

066 E6

sym.

C 13 CH C 15 C 16 ElC 23 C 24 C 25 C 26 E2C 33 C 34 C 35 C 36 E3

(2.2 - 7)C 44 C 45 C 46 E4

C 55 C 56 E5

C 66 E6

Note that there are no interactio~s between extensional and shear components fororthotropic materials when loaded along the material coordinates.

The stiffness coefficients C ij for an orthotropic material may be expressed interms of the engineering constants by (see Reddy [4])

It is understood from Eq. (2.2-7) that, in general, the elastic coefficientsOij relating the Cartesian components of stress and st rain depend on the coor ­dinate system"( :1: 1 , :1:2, :1:3) used. Referred to another Cartesian coordinate system(Xl, X2, X3) , the elastic coefficients are Gij , and in general Gij :.j:. Oij. If Gij = Oij,

then they are independent of the coordinate system and the material is said to beisotropic.

a - 1 - 1123 V32 a _ 1121 + 1131 1123 = 1112 + 1132 1113

11 - !:lE2E3 ' 12 - !:lE2 E3 !:lE1 E2

a - 1131 + 1121 1132 1113 + 1112 1123 a = 1 - 1113 1131

13 - !:lE2E3 !:lE1 E2 22 !:lE1 E3

a _ 1132 + 1112 1131 = 1123 + 1121 1113 a = 1 - 1112 1121

23 - !:lE1E3 !:lE1E2 33 !:lE1E2

jack

tensor rotation rule for sigma.