Volume 10, Issue 4
Solving the Navier-Lamé Equation in Cylindrical Coordinates Using the Buchwald Representation: Some Parametric Solutions with Applications

Jamal Sakhr & Blaine A. Chronik

Adv. Appl. Math. Mech., 10 (2018), pp. 1025-1056.

Published online: 2018-07

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  • Abstract

Using a separable Buchwald representation in cylindrical coordinates, we show how under certain conditions the coupled equations of motion governing the Buchwald potentials can be decoupled and then solved using well-known techniques from the theory of PDEs. Under these conditions, we then construct three parametrized families 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 elastic boundary value problems in cylindrical coordinates and are especially suited to problems involving one or more physical parameters. As an illustrative example, we consider the problem of determining the response of a solid elastic cylinder subjected to a time-harmonic surface pressure that varies sinusoidally along its axis and we demonstrate how the obtained parametric solutions can be used to efficiently construct an exact solution to this problem. We also briefly consider applications to some related forced-relaxation type problems.

  • Keywords

Navier-Lamé equation, cylindrical coordinates, Buchwald representation, Buchwald potentials, exact solutions.

  • AMS Subject Headings

35Q74, 35G35, 74H05, 31B35

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-10-1025, author = {}, title = {Solving the Navier-Lamé Equation in Cylindrical Coordinates Using the Buchwald Representation: Some Parametric Solutions with Applications}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {10}, number = {4}, pages = {1025--1056}, abstract = {

Using a separable Buchwald representation in cylindrical coordinates, we show how under certain conditions the coupled equations of motion governing the Buchwald potentials can be decoupled and then solved using well-known techniques from the theory of PDEs. Under these conditions, we then construct three parametrized families 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 elastic boundary value problems in cylindrical coordinates and are especially suited to problems involving one or more physical parameters. As an illustrative example, we consider the problem of determining the response of a solid elastic cylinder subjected to a time-harmonic surface pressure that varies sinusoidally along its axis and we demonstrate how the obtained parametric solutions can be used to efficiently construct an exact solution to this problem. We also briefly consider applications to some related forced-relaxation type problems.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2017-0203}, url = {http://global-sci.org/intro/article_detail/aamm/12507.html} }
TY - JOUR T1 - Solving the Navier-Lamé Equation in Cylindrical Coordinates Using the Buchwald Representation: Some Parametric Solutions with Applications JO - Advances in Applied Mathematics and Mechanics VL - 4 SP - 1025 EP - 1056 PY - 2018 DA - 2018/07 SN - 10 DO - http://dor.org/10.4208/aamm.OA-2017-0203 UR - https://global-sci.org/intro/aamm/12507.html KW - Navier-Lamé equation, cylindrical coordinates, Buchwald representation, Buchwald potentials, exact solutions. AB -

Using a separable Buchwald representation in cylindrical coordinates, we show how under certain conditions the coupled equations of motion governing the Buchwald potentials can be decoupled and then solved using well-known techniques from the theory of PDEs. Under these conditions, we then construct three parametrized families 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 elastic boundary value problems in cylindrical coordinates and are especially suited to problems involving one or more physical parameters. As an illustrative example, we consider the problem of determining the response of a solid elastic cylinder subjected to a time-harmonic surface pressure that varies sinusoidally along its axis and we demonstrate how the obtained parametric solutions can be used to efficiently construct an exact solution to this problem. We also briefly consider applications to some related forced-relaxation type problems.

Jamal Sakhr & Blaine A. Chronik. (2020). Solving the Navier-Lamé Equation in Cylindrical Coordinates Using the Buchwald Representation: Some Parametric Solutions with Applications. Advances in Applied Mathematics and Mechanics. 10 (4). 1025-1056. doi:10.4208/aamm.OA-2017-0203
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