Laminated plate theory

Fig. 13. Schematic of a laminate with the coordinates and ply notation used ia laminated plate theory.

The strength of laminates is usually predicted from a combination of laminated plate theory and a failure criterion for the individual larnina. A general treatment of composite failure criteria is beyond the scope of the present discussion. Broadly, however, composite failure criteria are of two types noninteractive, such as maximum stress or maximum strain, in which the lamina is taken to fail when a critical value of stress or strain is reached parallel or transverse to the fibers in tension, compression, or shear or interactive, such as the Tsai-Hill or Tsai-Wu (1,7) type, in which failure is taken to be when some combination of stresses occurs. Generally, the ply materials do not have the same strengths in tension and compression, so that five-ply strengths must be deterrnined ... 

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. 

Laminated papers, in food packaging, 18 36 Laminated plate theory, 26 780 Laminated-textile bags, 18 10-11 Laminate floorings, melamine resins in, 15 791... 

In the [ 45]j tensile test (ASTM D 3518,1991) shown in Fig 3.22, a uniaxial tension is applied to a ( 45°) laminate symmetric about the mid-plane to measure the strains in the longitudinal and transverse directions, and Ey. This can be accomplished by instrumenting the specimen with longitudinal and transverse element strain gauges. Therefore, the shear stress-strain relationships can be calculated from the tabulated values of and Ey, corresponding to particular values of longitudinal load, (or stress relations derived from laminated plate theory (Petit, 1969 Rosen, 1972) ... 

The longitudinal and transverse stresses in the individual plies are given by the laminated plate theory as [19]... 

Equation 8.24 is valid when the overall dimensions of the laminate are small so that the deflections are small. For large laminates, however, the deflections may become too large to satisfy the assumptions of linear theory. In this case a large-deflection plate theory must be used [15]. For long, narrow strips, the small-deformation laminated plate theory can still be used with the modified condition k2 = 0 in place of M2 = 0. This results in a cylindrical curvature, kc, given by... 

During the processing of composite materials in a hot press or an autoclave, the laminate is usually kept flat until cure is complete. If the platen surfaces are assumed frictionless, the effect of the constraints is to require that the curvatures K] and k2 be zero throughout cure. To develop the elastic solution under these constrained conditions, the laminated plate theory may be used with conditions of N = 0 and jc, = 0. The resulting midplane strains are given by... 

There are several resources available for designing filament-wound cylinders. In general, filament-wound cylinders are classified as cylindrically orthotropic. Adjacent helical plies, loading conditions can be determined by following the principles of laminated plate theory [7]. When applying laminated plate theory, the plate consists of the cylinder wall. In this case, the effect of cylinder curvature is neglected, and the Q and z axes are considered the planar axes of the plate. Failure criteria applied in laminated plate theory, such as maximum stress or strain, or the quadratic Tsai-Wu failure criteria [7] may also be applied. Several specialized loading cases have been studied. 

Laminated plate theory Laminated products Laminated safety glass Laminated-textile bags Laminated windshields Laminate materials Laminates... 

Fiber-reinforced composites are also highly anisotropic when fibers are aligned. In addition, they are nonhomogeneous and usually laminated with individual plies oriented in several different directions. Because of these difficulties, once again the isotropic failure criteria are of little use. However, reasonably successful failure criteria for individual plies have been developed (9-11) and applied to the failure of laminates by laminated plate theory ( , 12). 

CLPT classical laminate plate theory CPET chlorinated polyethylene terephthalate... 

Alternatively, tests can be used to obtain the basic stiffness properties of the material form and their corresponding range measured by some statistical property such as the standard deviation. In two-dimensional cases where there are no significant loads in the out-of-plane direction, the basic orthotropic stiffness properties in Eqn (6.1) can be measured experimentally. Then, the classical laminated plate theory described in previous sections for determination of stiffness can be used effectively to model these sttuctures. Alternatively, the four basic stiffiiesses for 3-D woven composites can be... 

However, especially for three-dimensional stmcture, or, even, in two-dimensional stmcture with significant out-of-plane loads, the stiffness averaging of the classical laminated plate theory is not sufficient. In such cases, a specialized finite element model such as the binary model by Cox, Carter, and Fleck [25] can be used. In this model, Monte Carlo simulation allows the user to randomly orient tows and to randomly assign strength properties to the different components. 

Once the new matrix properties accounting for porosity are known, the ply-level stiffnesses can be determined using micromechanics, and then the classical laminated plate theory outlined in previous sections can be used. 

For the case of an orthotropic laminate under conditions of plane stress, which can be assumed when the thickness of a laminate ply is small compared to its length and width, the stress in each constituent ply of a laminate can be determined with the aid of classical laminate plate theory [5-7]. The elastic stress-strain relationship for an orthotropic unidirectional plate then becomes... 

It was assmned that the complex 0 wound layers could be treated as two separate layers. It was also assmned that the matrix infiltrated the fibre layers completely and could be used as the matrix material for both the 0 wound layers. The composite laminated plate theory was used... 

From a fundamental imderstanding of the mechanics of winding fibers onto a cylinder, it is possible to predict, nsing micromechanics and the classical composite laminate plate theory, the effective elastic properties of the finished filament wormd catheter/tubing. 

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