Initial stress and elastic instability

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Abstract
By a well-known theorem of Kirchhoff the strain energy of an elastic solid is less in the equilibrium position than in any other position satisfying the same boundary conditions, and under the same body forces. The theorem contradicts the fact that elastic instability can occur, since two or more positions of equilibrium can then exist, and both cannot have the smaller strain energy. Numerous writers (Bryan (1), Southwell (2), Dean (3)) have explained the apparent discrepancy as due to the neglect of second-order terms in the elastic equations. This is correct so far as it goes, but it does not explain why the usual discussions of elastic instability give the right answers. The elastic constants vary somewhat with stress and in any case will be different according as a stressed or an unstressed state is taken as the standard. If any second-order terms should be included, we might expect that this variation would make a contribution comparable with those hitherto considered. Further, several theories of elasticity now exist that differ in the higher terms (Dean (3), Seth (8), Murnaghan (6)), and it may be asked whether they should lead to the same estimates of the critical loads.

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