Abstract
Lord Kelvin (William Thompson, 1824–1907) solved the problem that was later named after him in 1848. The problem consists in finding the equilibrium state of a linearly elastic, isotropic material body occupying the whole space and being subject to a point load.
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Notes
- 1.
- 2.
Consistent with definition (1.19), here \(\left[ \!\left[ \varPsi \right] \!\right] :=\varPsi ^+-\varPsi ^-\) denotes the jump of the field \(\varPsi \) at a suture plane, in terms of the limits \(\varPsi ^\pm \) of \(\varPsi \) when the point of interest is attained from one or the other part of that plane.
- 3.
We also see from (2.47) that, under the same circumstances, for the stored energy to stay finite the volume changes must become smaller and smaller as \(\nu \) approaches \(1/2\).
- 4.
Needless to say, the same developments follow by an application of definition (2.2)\(_{2}\) to the field (6.1). Recall that each symmetric tensor \({\varvec{A}}\) can be additively split into uniquely defined deviatoric and spheric parts:
$$\begin{aligned} {\varvec{A}}=\mathrm{dev }\,{\varvec{A}}+\mathrm{sph }\,{\varvec{A}},\quad \mathrm{sph }\,{\varvec{A}}:=\frac{1}{3}\mathrm{tr }\,{\varvec{A}},\;\;\mathrm{dev }\,{\varvec{A}}:={\varvec{A}}-\mathrm{sph }\,{\varvec{A}}. \end{aligned}$$ - 5.
More about internal constraint in linear elasticity is found in [3], Chapter III, Sections 17 and 18.
- 6.
Alternative terminological choices are ‘power’ (or ‘power expenditure’) for ‘working’ and ‘virtual’ for ‘test’; an alternative version of the italicized sentence above would read: the stress power equals the load power for whatever virtual velocity field.
References
Favata A (2012) On the Kelvin problem. J Elast 109:189–204
Love AEH (1927) A treatise on the mathematical theory of elasticity. Cambridge University Press, Cambridge
Podio-Guidugli P (2000) A primer in elasticity. Kluwer, Dordrecht
Podio-Guidugli P (2004) Examples of concentrated contact interactions in simple bodies. J Elast 75:167–186
Sokolnikoff IS (1956) Mathematical theory of elasticity. McGraw-Hill, New York
Thompson W, (Lord Kelvin), (1848) Note on the integration of the equations of equilibrium of an elastic solid. Cambr Dubl Math J 3:87–89
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Podio-Guidugli, P., Favata, A. (2014). The Kelvin Problem. In: Elasticity for Geotechnicians. Solid Mechanics and Its Applications, vol 204. Springer, Cham. https://doi.org/10.1007/978-3-319-01258-2_6
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