Invariant Laminate Stiffnesses

Invariant stiffness concepts as developed by Tsai and Pagano [7-16 and 7-17] can be used as an aid to understanding the stiffnesses of laminates of arbitrary orientation and how those stiffnesses can be varied. The concepts and their use are discussed in Sections 7.7.3.1 through 7.7.3.3. 

The topic of invariant transformed reduced stiffnesses of orthotropic and anisotropic laminae was introduced in Section 2.7. There, the rearrangement of stiffness transformation equations by Tsai and Pagano [7-16 and 7-17] was shown to be quite advantageous. In particular, certain invariant components of the lamina stiffnesses become apparent and are heipful in determining how the iamina stiffnesses change with transformation to non-principal material directions that are essential for a laminate. 

The invariant stiffness concepts for a iamina will now be extended to a laminate. All results in this and succeeding subsections on invariant laminate stiffnesses were obtained by Tsai and Pagano [7-16 and 7-17]. The laminate is composed of orthotropic laminae with arbitrary orientations and thicknesses. The stiffnesses of the laminate in the x-y plane can be written in the usual manner as 

When all orthotropic laminae are of the same material, the constants U, U2, and Ug can be brought outside the integrals  

The final result is given in Table 7-6 along with the values for all the stiffnesses. There, the 3 qj are 

---

The analytical tools to accomplish laminate design are at least twofold. First, the invariant laminate stiffness concepts developed by Tsai and Pagano [7-16 and 7-17] used to vary laminate stiffnesses. Second, structural optimization techniques as described by Schmit [7-12] can be used to provide a decision-making process for variation of iami-nate design parameters. This duo of techniques is particularly well suited to composite structures design because the simultaneous possibility and necessity to tailor the material to meet structural requirements exists to a degree not seen in isotropic materials. 

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. 

Quasi-isotroplc laminates do not behave like Isotropic homogeneous materials. Discuss why not, and describe how they do behave. Why is a two-ply laminate with a [0°/90°] sacking sequerx and equat-thickness layers not a quasi-isotropic laminate Determine whether the extensional stiffnesses are the same irrespective of the laminate axes for the two-ply and three-ply cases. Hint use the invariant properties In Equation (2.93). 

Note that both short and continuous fibers are handled in the same manner. These calculations, while tedious, are analytically simple. The plane stress , the Qy terms, are employed because lamination neglects the mechanical properties through the ply thickness. These stiffnesses are sometimes regrouped into new constants called invariants , the Uj terms, for analytical simplicity. To compute the properties of the laminate one then sums the ply (hj ) properties through the thickness of the laminate, weighted by the thickness (h ) of each oriented ply... 

---