Laminate displacements

The implications of the Kirchhoff hypothesis on the laminate displacements u, V, and w in the x-, y-, and z-directions are derived by use of the laminate cross section in the x-z plane shown in Figure 4-4. The displacement in the x-direction of point B from the undeformed middle surface to the deformed middle surface is Uo (the symbol nought (°) is used to designate middle-surface values of a variable). Because line ABCD remains straight under deformation of the laminate, the displacement at point C is... 

Novel polyester compositions have also been derived from dicyclopentadiene [77-73-6] (DCPD), which can enter into two distinct reactions with maleic anhydride to modify properties for lower cost. These compositions have effectively displaced o-phthaUc resins in marine and bathtub laminating apphcations. 

Thermoformed acryHc sheet is displacing gel coats in some bathtub appHcations spas have converted almost exclusively to formed acryHc sheet reinforced with glass-reinforced polyester laminations due to higher temperatures and higher stmctural requirements. 

Inert gas flush packing in plastic-laminated pouches, although less effective than vacuum packing, can remove or displace 80—90% of the oxygen in the package. These packages offer satisfactory shelf life and are sold primarily to institutions. 

Then, the displacement, u, at any point z through the laminate thickness is... 

The laminate strains have been reduced to e, Sy, and by virtue of the Kirchhoff hypothesis. That is, ez = Yw -r 0- small strains (linear elasticity), the remaining strains are ned in terms of displacements as... 

If the laminate is subjected to uniform axial extension on the ends X = constant, then all stresses are independent of x. The stress-displacement relations are obtained by substituting the strain-displacement relations, Equation (4.162), in the stress-strain relations. Equation (4.161). Next, the stress-displacement relations can be integrated under the condition that all stresses are functions of y and z only to obtain, after imposing symmetry and antisymmetry conditions, the form of the displacement field for the present problem ... 

Figure 4-56 Axial Displacement Distribution at the Laminate Surface, z = 2ho (After Pipes and Daniel [4-13])...

Obvious and sometimes drastic simplifications occur when the laminate is symmetric about the middle surface (By = 0), specially orthotropic (all the terms with 16 and 26 subscripts vanisn in addition to the By), homogeneous ( = 0 and Djj = Ayr/12), or isotropic. In all those cases. Equations (5.6) and (5.7) are coupled to each other, but uncoupled from Equation (5.8). That is. Equation (5.8) contains derivatives of the transverse displacement w only, and Equations (5.6) and (5.7) contain both u and V but not w. Accordingly, only Equation (5.8) must be solved to determine the transverse deflections of a plate with the aforementioned... 

Boundary conditions used to be thought of as a choice between simply supported, clamped, or free edges if all classes of elastically restrained edges are neglected. The real situation for laminated plates is more complex than for isotropic plates because now there are actually four types of boundary conditions that can be called simply supported edges. These more complicated boundary conditions arise because now we must consider u, v, and w instead of just w alone. Similarly, there are four kinds of clamped edges. These boundary conditions can be concisely described as a displacement or derivative of a displacement or, alternatively, a force or moment is equal to some prescribed value (often zero) denoted by an overbar at the edge ... 

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... 

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. 

Whitney and Pagano [6-32] extended Yang, Norris, and Stavsky s work [6-33] to the treatment of coupling between bending and extension. Whitney uses a higher order stress theory to obtain improved predictions of a, and and displacements at low width-to-thickness ratios [6-34], Meissner used his variational theorem to derive a consistent set of equations for inclusion of transverse shearing deformation effects in symmetrically laminated plates [6-35]. Finally, Ambartsumyan extended his treatment of transverse shearing deformation effects from plates to shells [6-36]. 

Polyurethane foams are lighter than foam rubber and have displaced many of its applications, such as in bedding, cushions, car seats, armrests, and crash pads. Laminates are used widely in clothing as padding. 

Figure 12.7 Sketch over the flow in a sandwich laminate (the displacement of the core is exaggerated). The flow of resin is from left to right...

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