Abstract
The bi-harmonic Green's functionG(r′,r) for the infinite strip region -1≤y≤1, -∞<x<∞, with the boundary conditionsG=∂G/∂y ony=±1, is obtained in integral form. It is shown thatG has an elegant bi-linear series representation in terms of the (Papkovich-Fadle) eigenfunctions for the strip. This representation is then used to show that any function ϕ bi-harmonic in arectangle, and satisfying the same boundary conditions asG, has a unique representation in the rectangle as an infinite sum of these eigenfunctions. For the case of the semi-infinite strip, we investigate conditions on ϕ sufficient to ensure that ϕ is exponentially small asx→∞. In particular it is proved that this is so, solely under the condition that ϕ be bounded asx→∞.
A corresponding pattern of results is established for the wedge of general angle. The Green's function is obtained in integral form and expressed as a bilinear series of the (Williams) eigenfunctions. These eigenfunctions are proved to be complete for all functions bi-harmonic in anannular sector (and satisfying the same boundary conditions as the Green's function). As an application it is proved that if an elastostatic field exists in a corner region with ‘free-free’ boundaries, and with either (i) the total strain energy bounded, or (ii) the displacement field bounded, then this field has a unique representation as a sum of those Williams eigenfunctions whichindividually posess the properties (i), (ii).
The methods used here extend to all other linear homogeneous boundary conditions for these geometries.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Benthem, J. P., A Laplace transform method for the solution of semi-infinite and finite strip problems in stress analysis.Quart. J. Mech. Appl. Math. 16 (1963), 413–429.
Buchwald, V. T., Eigenfunctions of plane elastostatics I. The strip.Proc. Roy. Soc. A 227 (1964), 385–400.
Buchwald, V. T., Eigenfunctions of plane elastostatics III. The wedge.J. Austral. Math. Soc. 5 (1965), 241–257.
Bueckner, H. F., Field singularities and related integral expressions, Chap. 5 ofMethods of analysis and solutions of crack problems, edited by G. C. Sih (Noordhoff, 1973).
Duffin, R. J., Continuation of biharmonic functions by reflection.Duke Math. J. 22 (1955), 313–324.
Johnson, M. W. and Little, R. W., The semi-infinite strip.Quart. Appl. Math. 22 (1965), 335–344.
Joseph, D. D., The convergence of biorthogonal series for biharmonic and Stokes flow edge problems, Part I.S.I.A.M. J. Appl. Math. 33 (1977), 337–347.
Joseph, D. D. and Sturges, L., The free surface on a liquid filling a trench heated from its side.J. Fluid Mech. 69 (1975), 565–589.
Joseph, D. D. and Sturges, L., The convergence of biorthogonal series for biharmonic and Stokes flow edge problems, Part II.S.I.A.M. J. Appl. Math. 34 (1978), 7–26.
Karp, S. N. and Karal, F. C., The elastic field behaviour in the neighborhood of a crack of arbitrary angle.Comm. Pure Appl. Math. 15 (1962), 413–421.
Knowles, J. K. and Pucik, T. A., Uniqueness for plane crack problems in linear elastostatics.J. of Elasticity 3 (1973), 155–160.
Koiter, W. T. and Alblas, J. B., On the bending of cantilever rectangular plates.Proc. Acad. Sci. Amst. B57 (1954) 549–557.
Koiter, W. T. and Sternberg, E., The wedge under a concentrated couple: A paradox in the two-dimensional theory of elasticity.J. Appl. Mech. 25 (1958), 575–581.
Liu, C. H. and Joseph, D. D., Stokes flow in wedge shaped trenches.J. Fluid Mech. 80 (1977), 443–464.
Liu, C. H. and Joseph, D. D., Stokes flow in conical trenches.S.I.A.M. J. Appl. Math. 34 (1978), 286–296.
Love, A. E. H.,A treatise on the mathematical theory of elasticity (Dover, 1944)
Morley, L. S. D., Simple series solution for the bending of a clamped rectangular plate under uniform normal load.Quart. J. Mech. Appl. Math. 16 (1963), 109–114.
Smith, R. T. C., The bending of a semi-infinite strip.Austral. J. Sci. Res. A5 (1952), 227–237.
Smith-White, W. B. and Buchwald, V. T., A generalisation ofZ!J. Austral. Math. Soc. 4 (1964), 327–341.
Uflyand, Y. S.,Survey of articles on the applications of integral transforms in the theory of elasticity, edited by I.N. Sneddon (North Carolina State University, 1965).
Williams, M. L., Stress singularities resulting from various boundary conditions in angular corners of plates in extension.J. Appl. Mech. 19 (1952), 526–528.
Author information
Authors and Affiliations
Additional information
On leave of absence at the University of British Columbia, Vancouver, B.C. Canada, during 1977–79. This work was supported in part by N.R.C. grants Nos. A9259 and A9117.
Rights and permissions
About this article
Cite this article
Gregory, R.D. Green's functions, bi-linear forms, and completeness of the eigenfunctions for the elastostatic strip and wedge. J Elasticity 9, 283–309 (1979). https://doi.org/10.1007/BF00041100
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00041100