On the elastostatic significance of four boundary integrals involving biharmonic functions
type: Item
instance of: scholarly article
title: On the elastostatic significance of four boundary integrals involving biharmonic functions
zbMATH Open document ID: 0903.73010
DOI: 10.1007/BF01172797
published in: Acta Mechanica
publication date: 1998-04-20T00:00:00Z
review text: Four boundary integrals are formulated for a biharmonic function \(U\) depending on two space variables. Expressions for boundary integrals contain the next four functions: \(U\), \({\partial U\over \partial \nu}\), \(\Delta U\), \({\partial\over \partial\nu} \Delta U\). Here \(\partial/ \partial\nu\) denotes the differentiation in the direction of outward normal to the boundary, and \(\Delta\) is the Laplace operator. The purpose is to give an elastostatic interpretation of the above-mentioned boundary integrals if \(U\) is an Airy stress function for a plane strain problem. At the first step, the author collects ``all possible'' biharmonic functions in polar coordinates. The second step is to compute for these functions strains, stresses, displacements and resultant forces couples. The third step is to evaluate the boundary integrals by means of Maple V. Then the author is able to explain the mechanical meaning of the results obtained. It turns out that the first two integrals are related to single-valued displacements, and the interpretation of the third integral seems to be a new result. The vanishing of the fourth integral does not give more information as do the first three integrals. The results presented can be used in the solution of plaine strain problems by means of boundary integral equations.
Mathematics Subject Classification ID: 74B10
Mathematics Subject Classification ID: 31A30
zbMATH DE Number: 1143345
zbMATH Keywords: Laplace operator
zbMATH Keywords: Airy stress function
zbMATH Keywords: plane strain problem
zbMATH Keywords: polar coordinates
zbMATH Keywords: Maple V
zbMATH Keywords: single-valued displacements
author: Søren Christiansen
reviewed by: István Ecsedi
describes a project that uses: Maple
MaRDI profile type: Publication
cites work: An Investigation of a Pair of Integral Equations for the Biharmonic Problem
cites work: Q3285143
cites work: Stress-Intensity Factors and Complex Path-Independent Integrals
cites work: Application of the Green and the Rayleigh-Green reciprocal identities to path-independent integrals in two- and three-dimensional elasticity
cites work: Q5822939
cites work: Q5508838
cites work: Q5810374
cites work: A note on the general solution of the two- dimensional linear elasticity problem in polar coordinates.
cites work: Comments on "A Note on the General Solution of the Two-Dimensional Linear Elasticity Problem in Polar Coordinates"
cites work: Q5787011
cites work: Q5844404
cites work: Q3232806
cites work: Plane problem of elasticity theory in a multiconnected domain
cites work: A direct integral equation method for computing the hoop stress at holes in plane isotropic sheets
full work available at URL: https://doi.org/10.1007/bf01172797
OpenAlex ID: W2080891062
description: scientific article; zbMATH DE number 1143345