Computation of interlaminar stresses from finite element solutions to plate theories
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Interlaminar stresses are estimated from plate theories by equilibrium. The elasticity equations of equilibrium are integrated with respect to the thickness coordinate z using the linear distribution in z of the in-plane stresses. This procedure, for example, requires fourth order derivatives of the out-of-plane displacement w with respect to the in-plane coordinates x and y to compute the interlaminar normal stress. Since compatible elements for the plate bending problem at most require the displacement and its first derivatives to be continuous across element boundaries, low degree interpolation polynomials are used. Thus, fourth order derivatives of the finite element polynomials are either meaningless, or at least inaccurate.
In order to compute high order derivatives, an approximate polynomial solution of high degree to the governing partial differential equation for w(x,y) is determined using the flnite element solution as a first approximation. A rectangular subdomain that may consist of several elements is selected from the finite element model. The displacement w(x,y) over the subdomain is expanded in a Chebyshev series. Then collocation is used to determine the unknown Chebyshev coefficients such that the Chebyshev series matches displacement wand its normal derivative from the flnite element solution at discrete points on the boundary of the subdomain, and the partial differential equation is enforced at discrete points within the subdomain. Interlaminar shear and normal stresses are computed from the third and fourth derivatives, respectively, of the Chebyshev series at the collocation points.
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