1992 ochoa & reddy - finite element analysis of composite laminates
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Ochoa & Reddy - Finite Element Analysis of Composite LaminatesTRANSCRIPT
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