Laminate stiffness

These values are determined by experiment. It is, however, by no means a trivial task to measure the lamina compressive and shear strengths (52,53). Also the failure of the first ply of a laminate does not necessarily coincide with the maximum load that the laminate can sustain. In many practical composite laminates first-ply failure may be accompanied by a very small reduction in the laminate stiffness. Local ply-level failures can reduce the stress-raising effects of notches and enhance fatigue performance (54). 

First, the stress-strain behavior of an individual lamina is reviewed in Section 4.2.1, and expressed in equation form for the k " lamina of a laminate. Then, the variations of stress and strain through the thicyiess of the laminate are determined in Section 4.2.2. Finally, the relation of the laminate forces and moments to the strains and curvatures is found in Section 4.2.3 where the laminate stiffnesses are the link from the... 

The stress-strain relations in arbitrary in-plane coordinates, namely Equation (4.5), are useful in the definition of the laminate stiffnesses because of the arbitrary orientation of the constituent laminae. Both Equations (4.4) and (4.5) can be thought of as stress-strain relations for the k layer of a multilayered laminate. Thus, Equation (4.5) can be written as... 

Some engineers have tried to characterize laminates with effective laminate stiffnesses, E, Ey, v y, and G y, and indeed such properties can be determined for a laminate by the usual measurements. However, it is crucial to recognize that with an effective laminate stiffness approach... 

This section is devoted to those special cases of laminates for which the stiffnesses take on certain simplified values as opposed to the general form in Equation (4.24). The general force-moment-strain-curvature relations in Equations (4.22) and (4,23) are far too comprehensive to easily understand. Thus, we build up our understanding of laminate behavior from the simplest cases to more complicated cases. Some of the cases are almost trivial, others are more specialized, some do not occur often in practice, but the point is that all are contributions to the understanding of the concept of laminate stiffnesses. Many of the cases result from the common practice of constructing laminates from laminae that have the same material properties and thickness, but have different orientations of the principal material directions relative to one another and relative to the laminate axes. Other more general cases are examined as well. 

For a single isotropic layer with material properties, E and v, and thickness, t, the laminate stiffnesses of Equation (4.24) reduce to... 

For a single specially orthotropic layer of thickness, t, and lamina stiffnesses, Qjj, given in Equation (2.61), the laminate stiffnesses are... 

The term quasi-isotropic iaminate is used to describe laminates that have isotropic extensionai stiffnesses (the same in all directions in the plane of the laminate). As background to the definition, recall that the term isotropy is a material property whereas laminate stiffnesses are a function of both material properties and geometry. Note also that the prefix quasi means in a sense or manner. Thus, a quasi-isotropic laminate must mean a laminate that, in some sense, appears isotropic, but is not actually isotropic in all senses. In this case, a quasi-isotropic... 

In preceding sections, laminate stiffnesses were predicted on the basis of combination of lamina stiffnesses in accordance with classical lamination theory. However, the actual, practical realization of those laminate stiffnesses remains to be demonstrated. The purpose of this section is to compare predicted laminate stiffnesses with measured laminate stiffnesses to determine the validity of classical lamination theory. Results for two types of laminates, cross-ply and angle-ply laminates, are presented. 

Before the predicted stiffnesses are compared with measured stiffnesses, however, a slight reinterpretation of laminate stiffnesses 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 laminate behavior of these special angle-ply laminates can be described with the number of layers, N, the laminae orientation, a, and the laminae stiffnesses, Q , in addition to the total laminate thickness, t. The laminate stiffnesses,... 

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. 

The laminate stress-analysis elements are affected by the state of the material and, in turn, determine the state of stress. For example, the laminate stiffnesses are usually a function of temperature and can be a function of moisture, too. The laminae hygrothermomechanical properties, thicknesses, and orientations are important in determining the directional characteristics of laminate strength. The stacking sequence... 

The significance of interlaminar stresses relative to laminate stiffness, strength, and life is determined by Classical Lamination Theory, i.e., CLT stresses are accurate over most of the laminate except in a very narrow boundary layer near the free edges. Thus, laminate stiffnesses are affected by global, not local, stresses, so laminate stiffnesses are essentially unaffected by interlaminar stresses. On the other hand, the details of locally high stresses dominate the failure process whereas lower global stresses are unimportant. Thus, laminate strength and life are dominated by interlaminar stresses. 

A specially orthotropic laminate has either a single layer of a specially orthotropic material or multiple specially orthotropic layers that are symmetrically arranged about the laminate middle surface. In both cases, the laminate stiffnesses consist solely of A, A 2> 22> 66> 11> D 2, D22, and Dgg. That is, neither shear-extension or bend-twist coupling nor bending-extension coupling exists. Thus, for plate problems, the transverse deflections are described by only one differential equation of equilibrium ... 

Once the deflections are known, the stresses are straightforwardly obtained by substitution in the stress-strain relations. Equation (4.16), after the strains are found from Equation (4.12). Note that the solution in Equation (5.31) is expressed in terms of only the laminate stiffnesses D., Di2. D22. and Dgg. This solution will not be plotted here, but will be used as a baseline solution in the following subsections and plotted there in comparison with more complicated results. 

A collection of the basic building block, a lamina, was bonded together to form a laminate in Chapter 4. The behavior restrictions were covered in the section on classical lamination theory. Special cases of laminates were discussed to learn about laminate characteristics and behavior. Predicted and measured laminate stiffnesses were favorably compared to give credence to classical lamination theory. Then, the strength of laminates was discussed and found to be reasonably predictable. Finally, interlaminar stresses were analyzed because of their apparent strong influence on laminate strength (and life). 

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... 

Table 7-6 Laminate Stiffnesses as a Function of Lamina Properties After Tsai and Pagano [7-17])...

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