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Cauchy’s stress theorem and tensor fields with divergences in \(L^ p\). (English) Zbl 0776.73003

The proofs of Cauchy’s stress theorem were based on the assumptions: (1) the force can be associated with every oriented surface, (2) the force acting on a surface is bounded by the area of that surface, and (3) the integral form of the balance equation holds for every part of the body. The author asserts that these assumptions can be weakened to: (1’) the force can be associated with “almost every” surface, (2’) the force acting on a surface is “weakly” bounded, and (3’) the integral form of the balance equation holds for “almost every” part of the body. It is shown that under these weakened assumptions, the existence of the stress field follows, and the stress field automatically has a weak divergence in the Lebesgue class \(L^ p\). To give the appropriate notions of “almost every” and “weakly bounded” is one of the aims of this paper.
Reviewer: S.Minagawa (Tokyo)

MSC:

74A20 Theory of constitutive functions in solid mechanics
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