Classical laminate theory

The characteristic features of a cord—mbber composite have produced the netting theory (67—70), the cord—iaextensible theory (71—80), the classical lamination theory, and the three-dimensional theory (67,81—83). From stmctural considerations, the fundamental element of cord—mbber composite is unidirectionaHy reinforced cord—mbber lamina as shown in Figure 5. From the principles of micromechanics and orthotropic elasticity laws, engineering constants of tire T cord composites in terms of constitutive material properties have been expressed (72—79,84). The most commonly used Halpin-Tsai equations (75,76) for cord—mbber single-ply lamina L, are expressed in equation 5 ... 

Ciassicai lamination theory is derived in Section 4.2. Then, special stiffnesses of practical interest are classified and examined in Section 4.3. Next, the theoretical stiffnesses obtained by classical lamination theory are compared with experimental results in Section 4.4. In Section 4.5, the strengths of various laminates are predicted. Finally, the stresses between the laminae of a laminate are examined in Section 4.6 and found to be a proba lei causg of delamination of some laminates. 

Classical lamination theory consists of a coiiection of mechanics-of-materials type of stress and deformation hypotheses that are described in this section. By use of this theory, we can consistentiy proceed directiy from the basic building block, the lamina, to the end result, a structural laminate. The whole process is one of finding effective and reasonably accurate simplifying assumptions that enable us to reduce our attention from a complicated three-dimensional elasticity problem to a SQlvable two-dimensinnal merbanics of deformable bodies problem. 

Actually, because of the stress and deformation hypotheses that are an inseparable part of classical lamination theory, a more correct name would be classical thin lamination theory, or even classical laminated plate theory. We wiiruS ffi bmmon term classical lamination theory, but recognize that it is a convenient oversimplification of the rigorous nomenclature. In the composite materials literature, classical laminationtheoryls en abbreviated as CLT. 

In conclusion, classical lamination theory enables us to calculate forces and moments if we know the strains and curvatures of the middle surface (or vice versa). Then, we can calculate the laminae stresses in laminate coordinates. Next, we can transform the laminae stresses from laminate coordinates to lamina principal material directions. Finally, we would expect to apply a failure criterion to each lamina in its own principal material directions. This process seems straightfonward in principle, but the force-strain-curvature and moment-strain-curvature relations in Equations (4.22) and (4.23) are difficult to completely understand. Thus, we attempt some simplifications in the next section in order to enhance our understanding of classical lamination theory. 

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. 

The predicted strengths in Figure 4-44 are generally somewhat above the measured values. The predicted and observed stiffnesses, both initial (below the knee) and final, are in very good agreement. Thus, the stiffness aspects of classical lamination theory, as well as the present strength-analysis procedure, are verified. 

Moreover, classical lamination theory often implies values of Oy and where they cannot possibly exist, namely at the edge of a laminate. Physical grounds will be used to establish that ... 

The analysis of such a laminate by use of classical lamination theory revolves about the stress-strain relations of an individual orthotropic lamina under a state of plane stress in principal material directions... 

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. 

Demonstrate that use of classical lamination theory leads to... 

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). 

Pagano s exact solution for the stresses and displacements is too complex to present here. The corresponding classical lamination theory result stems from the equilibrium equations, Equations (5.6) to (5.8), which simplify to... 

Even though in classical lamination theory by virtue of the Kirchhoff hypothesis we assume the stresses and are zero, we can still obtain these stresses approximately by integration of the stress equilibrium equations... 

Obviously, the classical lamination theory stresses in Pagano s example converge to the exact solution much more rapidly than do the displacements as the span-to-thickness ratio increases. The stress errors are on the order of 10% or less for S as low as 20. The displacements are severely underestimated for S between 4 and 30, which are common values for laboratory characterization specimens. Thus, a practical means of accounting for transverse shearing deformations is required. That objective is attacked in the next section. 

The preceding subsection was devoted to a comparison of a special exact elasticity solution with classical lamination theory results. The importance of transverse shear effects was clearly demonstrated. However, that demonstration was for a special problem of rather narrow interest. The objective of this subsection is to display approaches and results for the approximate consideration of transverse shear effects for general laminated plates. 

The constants and are found from continuity conditions for u and V at layer interfaces and the symmetry condition that u and v vanish at the laminate middle surface. Obviously, because of the presence of and Oy, u and v are not linear functions of z as in classical lamination theory. 

The boundary conditions for these equilibrium equations are more complicated than for classical lamination theory. However, they are more logical because the Kirchhoff shear force or free-edge condition, in which... 

The results shown in Figure 6-21 for the present shear-deformation approach versus classical lamination theory are quite similar qualitatively to the comparison between the exact cylindrical bending solution and classical lamination theory in Figure 6-17. 

The first observation is that the cured shape of an unsymmetric cross-ply laminate is often cylindrical, whereas we would predict it to be a saddle shape (hyperbolic paraboloid) from classical lamination theory (the curvatures can be shown to be = - Ky or - = Ky). A thick lami-... 

However, these transverse shearing stresses were neglected implicitly when we adopted the Kirchhoff hypothesis of lines that were normal to the undeformed middle surface remaining normal after deformation in Section 4.2.2 on classical lamination theory. That hypothesis is interpreted to mean that transverse shearing strains are zero, and, hence, by the stress-strain relations, the transverse shearing stresses are zero. The Kirchhoff hypothesis was also adopted as part of classical plate theory in Section 5.2.1. 

The analytical quantification of the residual stresses on the macroscopic scale is generally based on a simple ID model [2] or the 2D plane-stress classical lamination theory [3] (CLT). 

Based on these properties, the level of residual stress in the 90° layer for the different cross-ply lay up used in this study as calculated with the classical lamination theory (CLT) range from 0 to 43.5 MPa as shown in Table 2. 

Classical Lamination Theory provides dome contour known as Balanced Ovaloid (solution of 1 order ODE)... 

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