Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 

 

Plane harmonic functions in the presence of a surface layer of arbitrary stiffness


Author: George Ireneus Zahalak
Journal: Quart. Appl. Math. 37 (1980), 337-353
MSC: Primary 73C40; Secondary 31A05
DOI: https://doi.org/10.1090/qam/564728
MathSciNet review: 564728
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Complex variable techniques are employed to characterize two-dimensional solutions $ u\left( {x, y} \right)$ of Laplace's equation which satisfy the boundary condition $ {\left[ {\beta \left( {{\partial ^2}u/ \\ \partial {y^2}} \right) + \left( {\partial u/\partial x} \right)} \right]_{x = 0}} = 0$, where $ \beta $ is referred to as the surface-stiffness parameter. Simple closed-form singular solutions are derived which satisfy this boundary condition and represent source and dislocation singularities. The former is used to synthesize the field generated by a small inclusion of arbitrary shape on which $ u = 1$, in the presence of a boundary at $ y = 0$ on which $ u = 0$. At points not near the inclusion the field has the form of a function of position and surface stiffness multiplied by a strength factor which depends on the size and shape of the inclusion and the surface stiffness. Detailed calculations are presented for two extreme shapes of inclusions--a shallow, wide inclusion on the surface and a deep, narrow inclusion penetrating below the surface--which exhibit the relation between the field near the inclusion and the distant field, and show explicitly the dependence of the strength factor on surface stiffness and inclusion size and shape. The nature and strength of the singularities at the tips of the inclusions are also examined and it is found that a tip singularity at the surface changes character as the surface stiffness varies.


References [Enhancements On Off] (What's this?)

  • [1] F. C. Goodrich, The hydrodynamical theory of surface shear viscosity, Prog. Surf. Memb. Sci. 7, 151-181 (1973)
  • [2] D. A. Simons, Scattering of a Love wave by the edge of a thin surface layer, J. Appl. Mech. (4) 42, 842-846 (1975)
  • [3] R. J. Mannheimer and R. A. Burton, A theoretical estimation of the viscous-interaction effects with a torsional (knife-edge) surface viscometer, J. Coll. Int. Sci. (1) 32, 73-80 (1970)
  • [4] F. C. Goodrich and L. H. Allen, The theory of absolute surface shear viscosity, part V. The effect of finite ring thickness, J. Coll. Int. Sci. (3) 40, 329-336 (1972).
  • [5] M. Abramowitz and J. Stegun, eds., Handbook of mathematical functions, National Bureau of Standards, Washington, 1965, p. 228
  • [6] J. R. Rice, Mathematical analysis in the mechanics of fracture in Fracture, vol. 2 (H. Liebowitz, ed.), Academic Press, New York, 1968, p. 191
  • [7] Milton Van Dyke, Perturbation methods in fluid mechanics, Annotated edition, The Parabolic Press, Stanford, Calif., 1975. MR 0416240
  • [8] George F. Carrier, Max Krook, and Carl E. Pearson, Functions of a complex variable: Theory and technique, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0222256

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 73C40, 31A05

Retrieve articles in all journals with MSC: 73C40, 31A05


Additional Information

DOI: https://doi.org/10.1090/qam/564728
Article copyright: © Copyright 1980 American Mathematical Society


Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2017 Brown University
Comments: qam-query@ams.org
AMS Website