Use of Invariant Laminate Stiffnesses in Design

Two simple invariants, U, and U5, were shown in the previous subsubsection to be the basic indicators of average laminate stiffnesses. For isotropic materials, these invariants reduce to U. =Qi. and U5 = Qqq, the extensional stiffness and shear stiffness. Accordingly, Tsai and Pagano suggested the orthotopic invariants U., and U5 be called the isotropic stiffness and isotropic shear rigidity, respectively [7-16 and 7-17]. They observed that these isotropic properties are a realistic measure of the minimum stiffness capability of composite laminates. These isotropic properties can be compared directly to properties of isotropic materials as well as to properties of other orthotropic laminates. Obviously, the comparison criterion is more complex than for isotropic materials because now we have two measures, and U5, instead of the usual isotropic stiffness or E. Comparison of values of U., alone is not fair because of the degrading influence of the usually low values of U5 for composite materials. 

Tsai and Pagano further defined the isotropic stiffness and shear rigidity [7-16] to be 

the usual emphasis on the value of E is badly misplaced. Obviously, the value of E2 enters the representative average properties quite strongly. These approximations are quite accurate as can be verified by simple calculations. 

1 Show that A, + A22 + A,2 is invariant under rotation about the z-axis, that is, that 

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