Solving the Navier-Lamé Equation in Cylindrical Coordinates Using the Buchwald Representation: Some Parametric Solutions with Applications (Advances in Applied Mathematics and Mechanics, 2018)
Using a separable Buchwald representation in cylindrical coordinates, weshow how under certain conditions the coupled equations of motion governing theBuchwald potentials can be decoupled and then solved using well-known techniquesfrom the theory of PDEs. Under these conditions, we then construct three parametrizedfamilies of particular solutions to the Navier-Lamé equation in cylindrical coordinates.In this paper, we specifically construct solutions having 2π-periodic angular parts.These particular solutions can be directly applied to a fundamental set of linear elasticboundary value problems in cylindrical coordinates and are especially suited to problemsinvolving one or more physical parameters. As an illustrative example, we considerthe problem of determining the response of a solid elastic cylinder subjected to atime-harmonic surface pressure that varies sinusoidally along its axis and we demonstratehow the obtained parametric solutions can be used to efficiently construct anexact solution to this problem. We also briefly consider applications to some relatedforced-relaxation type problems.