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On Spencer’s displacement function approach for problems in second-order elasticity theory

The paper describes the displacement function approach first proposed by AJM Spencer for the formulation and solution of problems in second-order elasticity theory. The displacement function approach for the second-order problem results in a single inhomogeneous partial differential equation of the...

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Detalles Bibliográficos
Autor principal: Selvadurai, APS
Formato: Online Artículo Texto
Lenguaje:English
Publicado: SAGE Publications 2022
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC9810831/
https://www.ncbi.nlm.nih.gov/pubmed/36619893
http://dx.doi.org/10.1177/10812865221096771
Descripción
Sumario:The paper describes the displacement function approach first proposed by AJM Spencer for the formulation and solution of problems in second-order elasticity theory. The displacement function approach for the second-order problem results in a single inhomogeneous partial differential equation of the form [Formula: see text] , where [Formula: see text] is Stokes’ operator and [Formula: see text] depends only on the first-order or the classical elasticity solution. The second-order isotropic stress [Formula: see text] is governed by an inhomogeneous partial differential equation of the form [Formula: see text] , where [Formula: see text] is Laplace’s operator and [Formula: see text] depends only on the first-order or classical elasticity solution. The introduction of the displacement function enables the evaluation of the second-order displacement field purely through its derivatives and avoids the introduction of arbitrary rigid body terms normally associated with formulations where the strains need to be integrated. In principle, the displacement function approach can be systematically applied to examine higher-order effects, but such formulations entail considerable algebraic manipulations, which can be facilitated through the use of computer-aided symbolic mathematical operations. The paper describes the advances that have been made in the application of Spencer’s fundamental contribution and applies it to the solution of Kelvin’s concentrated force, Love’s doublet, and Boussinesq’s problems in second-order elasticity theory.