General Representations of Polynomial Elastic Fields

Abstract

For hybrid finite elements based, for example, on complementary energy functional, polynomial trial functions of stress fields are required. Systematic schemes are given for the 2D and 3D elasticity, respectively. For 2D problems, the paper shows that there are maximal four independent polynomials for the nth order homogeneous polynomial Airy stress functions: two for the first order, three for the second order, and four for the nth order (n not less than 3). For 3D problems, there are 3(2n + 1) independent polynomials for the nth order homogeneous polynomial displacement. The corresponding stress field can be given directly. As examples, explicit expressions of the corresponding independent polynomials are listed: up to the tenth order for the 2D case and up to the third order for the 3D case.

Polynomial stress functions of anisotropic plane problems and their applications in hybrid finite elements

“…It is proved that for the study of homogeneous polynomial stress fields, one can equivalently investigate the stress function U (x, y) in a homogeneous polynomial [2]. Assume that U (x, y) = a 0 x n + a 1 x n−1 y + · · · + a n−1 x y n−1 + a n y n ,…”

“…In this presented work, according to Lekhnitskii's theory of anisotropic solid [4], general expressions can be obtained in explicit form for all the possible independent polynomials of arbitrary n-th order homogenous polynomial stress functions for anisotropic plane problems. The results can be degenerated to isotropic solutions in [2]. It will be shown in this work that the n-th order homogeneous polynomials have similar characteristics as for the isotropic case, i.e., there are at most four independent polynomials for arbitrary n-th order homogeneous polynomials: two for n equal to one, three for n equal to two, and four for n greater than or equal to three.…”

“…where 2,3,4), and μ k , (k = 1, 2, 3, 4) are the roots of the characteristic equation [4,5] given by…”

Three dimensional hybrid‐Trefftz stress finite elements for plates and shells

“…The same variational formulation of Freitas and Bussamra is used in the present work to develop 3‐D hybrid‐Trefftz stress elements for plates and shells. Now, the systematic scheme provided by Wang et al is adopted to construct sets of Papkovitch‐Neuber solutions. The harmonic functions are derived from simple polynomials, which may have their order in the plate and shell thickness direction controlled.…”

“…The general representation of polynomials of elastic fields is introduced by Wang et al where 3(2n + 1) independent polynomials of displacement and stress fields are obtained through a harmonic polynomial vector. One can construct a similar field by limiting the maximum degree the polynomial can assume in the coordinate measured along the thickness, the monomials of z(n z max ), on the n th‐order homogeneous harmonic polynomial used to construct the displacement field and stress fields.…”