Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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

   
 
 

 

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
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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 $ v$. 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.


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DOI: https://doi.org/10.1090/qam/155477
Article copyright: © Copyright 1964 American Mathematical Society

American Mathematical Society