An application of the dynamic Betti-Rayleigh reciprocal theorem to moving-point loads in elastic media

Author:
R. G. Payton

Journal:
Quart. Appl. Math. **21** (1964), 299-313

DOI:
https://doi.org/10.1090/qam/155477

MathSciNet review:
155477

Full-text PDF

Abstract | References | Additional Information

Abstract: Two problems connected with the transient motion of an elastic body acted upon by a moving-point force are solved by an application of the dynamic Betti-Rayleigh reciprocal theorem. This theorem, which is the analogue of Green's theorem for the scalar wave equation, permits the solution to be written as a single expression, irrespective of the value of the (constant) moving-force velocity . In particular, the displacement field in an infinite elastic body, due to a transient-point body force moving, in a straight line, is given in a simple form. Next the surface motion of an elastic half-space acted upon by a transient pressure spot moving in a straight line is analyzed for a material for which Poisson's ratio is one-fourth. The normal displacement is expressed in a simple manner, but the tangential displacement is quite complicated and is not fully expressible in terms of elementary functions. Singularities of the displacement fields are identified and discussed.

**[1]**I. S. Sokolnikoff,*Mathematical theory of elasticity*, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1956. 2d ed. MR**0075755****[2]**G. Eason, J. Fulton, and I. N. Sneddon,*The generation of waves in an infinite elastic solid by variable body forces*, Philos. Trans. Roy. Soc. London. Ser. A.**248**(1956), 575–607. MR**0078181**, https://doi.org/10.1098/rsta.1956.0010**[3]**Jean Mandel and André Avramesco,*Déplacements produits par une charge mobile à la surface d’un semi-espace élastique*, C. R. Acad. Sci. Paris**252**(1961), 3730–3732 (French). MR**0128170****[4]**Dang Dinh Ang,*Transient motion of a line load on the surface of an elastic half-space*, Quart. Appl. Math.**18**(1960/1961), 251–256. MR**0114399**, https://doi.org/10.1090/S0033-569X-1960-0114399-0**[5]**F. L. DiMaggio and H. H. Bleich,*An application of a dynamic reciprocal theorem*, J. Appl. Mech.**26**(1959), 678–679. MR**0109454****[6]**Leon Knopoff,*Scattering of shear waves by spherical obstacles*, Geophysics**24**(1959), 209–219. MR**0108090****[7]**P. M. Morse and H. Feshback,*Methods of Theoretical Physics*, McGraw-Hill Book Company, Inc., New York (1953), p. 1790**[8]**I. N. Sneddon and D. S. Berry,*Handbuch der Physik*, vol. 6, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1959, p. 88**[9]**C. L. Pekeris,*The seismic buried pulse*, Proc. Nat. Acad. Sci. U.S.A.**41**(1955), 629–639. MR**0072679****[10]**C. L. Pekeris,*The seismic surface pulse*, Proc. Nat. Acad. Sci. U.S.A.**41**(1955), 469–480. MR**0072002**

Additional Information

DOI:
https://doi.org/10.1090/qam/155477

Article copyright:
© Copyright 1964
American Mathematical Society