Cross-ply laminates

Telega J.J., Lewinski T. (1994) Mathematical aspects of modelling the macroscopic behaviour of cross-ply laminates with intralaminar cracks. Control and Cybernetics 23 (3), 773-792. 

The fundamental analysis of a laminate can be explained, in principle, by use of a simple two-layered cross-ply laminate (a layer with fibers at 0° to the x-direction on top of an equal-thickness layer with fibers at 90° to the x-direction). We will analyze this laminate approximately by considering what conditions the two unbonded layers in Figure 4-3 must satisfy in order for the two layers to be bonded to form a laminate. Imagine that the layers are separate but are subjected to a load in the x-direction. The force is divided between the two layers such that the x-direction deformation of each layer is identical. That is, the laminae in a laminate must deform alike along the interface between the layers or else fracture must existl Accordingly, deformation compatibility of layers is a requirement for a laminate. Because of the equal x-direction deformation of each layer, the top (0°) layer has the most x-direction ress because it is stiffer than the bottom (90°) layer in the x-direction./ Trie x-direction stresses in the top and bottom layers can be shown to have the relation... 

If the cross-ply laminate is symmetric about the middle surface in both material properties and geometry, then the By all vanish as is easily shown. 

The purpose of the remainder of this section is to discuss two important classes of antisymmetric laminates, the antisymmetric cross-ply laminate and the antisymmetric angle-ply laminate. Neither laminate is used much in practice, but both add to our understanding of laminates. 

An antisymmetric cross-ply laminate consists of an even number of orthotropic laminae laid on each other with principal material directions alternating at 0° and 90° to the laminate axes as in the simple example of Figure 4-19. A more complicated example is given in Table 4-4 (where the adjacent layers do not always have the sequence 0°, then 90°, then 0°, etc.). Such laminates do not have A g, Agg, D g, and Dgg, but do have bending-extension coupling. We will show later that the coupling is such that the force and moment resultants are... 

A regular antisymmetric cross-ply laminate is delined to have laminae all of equal thickness and is common because of simplicity of fabrication. As the number of layers increases, the bending-extension coupling stiffness B.,., can be shown to approach zero. 

A cross-ply laminate in this section has N unidirectionally reinforced thotropic) layers of the same material with principal material directions srnatingly oriented at 0° and 90° to the laminate coordinate axes. The sr direction of odd-numbered layers is the x-direction of the laminate, e fiber direction of even-numbered layers is then the y-direction of the linate. Consider the special case of odd-numbered layers with equal kness and even-numbered layers also with equal thickness, but not essarily the same thickness as that of the odd-numbered layers, te that we have imposed very special requirements on how the fiber sntations change from layer to layer and on the thicknesses of the ers to define a special subclass of cross-ply laminates. Thus, these linates are termed special cross-ply laminates and will be explored his subsection. More general cross-ply laminates have no such con-ons on fiber orientation and laminae thicknesses. For example, a neral) cross-ply laminate could be described with the specification t/90° 2t/90° 2t/0° t] wherein the fiber orientations do not alter-e and the thicknesses of the odd- or even-numbered layers are not same however, this laminate is clearly a symmetric cross-ply lami-e. 

For the special cross-ply laminates, two geometrical parameters important N, the total number of layers, and M, the ratio of the total kness of odd-numbered layers to the total thickness of even-nbered layers (called the cross-ply ratio). Thus,... 

Special Cross-Ply Laminates with N Even (Antisymmetric)... 

For both odd- and even-layered special cross-ply laminates, the extensional stiffnesses, Aj., are independent of N, the number of layers (although the N individual lamina thickneWs can be summed to get the total laminate thickness t, so N is implicit im quations (4.78) and (4.82)). However, A., and A22 depend on M, the cross-ply ratio, and on F, the lamina stiffness ratio, as shown in Figures 4-22 and 4-23. For a typical glass-fiber-reinforced lamina, F =. 3, so A, varies from. 65Q.nt to... 

Q t as M changes from 1 to 10. Similarly, varies from to 38Aii over the same range of M. The stiffnesses A g Agg are independent of M and F. The remaining stiffnesses A g and Agg are zero for all cross-ply laminates. 

Two- and three-layered special cross-ply laminates were shown to have extrema of behavior in the preceding section. Thus, comparisons between theoretical and measured stiffnesses for such laminates should... 

The individual laminae used by Tsai [4-6] consist of unidirectional glass fibers in a resin matrix (U.S. Polymeric Co. E-787-NUF) with moduli given in Table 2-3. A series of special cross-ply laminates was constructed with M = 1,2,3,10 for two-layered laminates and M = 1,2,5,10 for three-layered laminates. The laminates were subjected to axial loads and bending moments whereupon surface strains were measured. Accordingly, the stiffness relations as strains and curvatures in terms of forces and moments, that is. 

The measured stiffnesses for two- and three-layered special cross-ply laminates are shown with symbols in Figure 4-28, and the theoretical results are shown with solid lines. In all cases, the load was kept so low that no strain exceeded SOOp. Thus, the behavior was linear and elastic. The agreement between theory and experiment is quite good. Both the qualitative and the quantitative aspects of the theory are verified. Thus, the capability to predict cross-ply laminate stiffnesses exists and is quite accurate. 

Figure 4-28a Theoretical and Measured Special Cross-Ply Laminate Stiffnesses (U. S. Standard Units) (After Tsai [4-6])...

The theoretical and measured stiffnesses are shown in Figure 4-32. As with cross-ply laminates, very good agreement was obtained. Thus, the predictions of laminate stiffnesses are quite accurate. 

Derive the extensional stiffnesses for reguiar symmetric speciai cross-ply laminates, that is, derive Equation (4.78) for the special case in which t = t gn = VN. 

The procedure of laminate strength analysis outlined in Section 4.5.2, with the Tsai-Hill lamina failure criterion will be illustrated for cross-ply laminates that have been cured at a temperature above their service or operating temperature in the manner of Tsai [4-10]. Thus, the thermal effects discussed in Section 4.5.3 must be considered as well. For cross-ply laminates, the transformations of lamina properties are trivial, so the laminate strength-analysis procedure is readily interpreted. 

The particular cross-ply laminate to be examined [4-10] has three layers, so is symmetric about its middle surface. Thus, no coupling exists between bending and extension. Under the condition N = N and all other loads and moments are zero, the stresses in the (symmetric) outer layers are identical. One outer layer is called the 1-layer and has fibers in the x-direction (see Figure 4-39). The inner layer is called the 2-layer and has fibers in the y-direction. The other outer layer is the 3-layer, but because of symmetry there is no need to refer to it. The cross-ply ratio, M, is, 2, so the thickness of the inner layer is ten times that of each of the outer layers (actually, the inner layer" is ten like-oriented lamina Each lamina is. 005 in (.1270 mm) thick, so the total laminate thickness is. 060 in (1.524 mm). 

Thus, all numbers are in hand for calculation of the stresses in the example cross-ply laminate. 

The failure criterion must be applied to determine the maximum values of Nx and AT that can be sustained without failure of any layer. Actually, the failure criterion is applied to each layer separately. For the special orientation of cross-ply laminates, the Tsai-Hill failure criterion for each layer can be expressed as... 

After a layer fails, the behavior of the laminate depends on how the mechanical and thermal interactions between layers uncouple. Actually, failure of a layer might not mean that it can no longer carry load. In the present example of a cross-ply laminate, the inner layer with fibers at 90° to the x-axis has failed, but, because of the orientation of the fibers (perpendicular to the main failure-causing stress), the failure should be only a series of cracks parallel to the fibers. Thus, stress can still be carried by the inner layer in the fiber direction (y-direction). 

Figure 4-42 Unbonded View of a Three-Layered Cross-Ply Laminate with All Layers Degraded (Cracked)...

Figure 4-43 Strength of a Cross-Ply Laminate with M =. 2 (After Tsai [4-10]) Strength and Stiffness for Other Cross-Ply Ratios...

Theoretical and measured strengths and stiffnesses of three-layer cross-ply laminates with cross-ply ratios ranging from. 2 to 4 are shown in Figure 4-44. The scatter in the data is partially due to the difficulty of making good tensile specimens the characteristic dog-bone shape is formed by routing that often damages the 90° layer. 

Angle-ply laminates have more complicated stiffness matrices than cross-ply laminates because nontrivial coordinate transformations are involved. However, the behavior of simple angle-ply laminates (only one angle, i.e., a) will be shown to be simpler than that of cross-ply laminates because no knee results in the load-deformation diagram under uniaxial loading. Other than the preceding two differences, analysis of angle-ply laminates is conceptually the same as that of cross-ply laminates. 

The example considered to illustrate the strength-analysis procedure is a three-layered laminate with a [4-15°/-15°/+15°] stacking sequence [4-10]. The laminae are the same E-glass-epoxy as in the cross-ply laminate example with thickness. 005 in (.1270 mm), so that the total laminate thickness is. 015 in (.381 mm). In laminate coordinates, the transformed reduced stiffnesses are... 

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