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On the divergence theorem for submanifolds of Euclidean vector spaces within the theory of second-gradient continua. (English) Zbl 1491.53101

Summary: In the theory of second-gradient continua, the internal virtual work functional can be considered as a second-order distribution in which the virtual displacements take the role of test functions. In its easiest representation, the internal virtual work functional is represented as a volume integral over a subset of the three-dimensional Euclidean vector space and involves first and second derivatives of the virtual displacements. In this paper, we show by an iterative integration by parts procedure how an alternative representation of such a functional can be obtained when the integration domain is a subset that contains also edges and wedges. Since this procedure strongly relies on the divergence theorem for submanifolds of a Euclidean vector space, it is a main goal to derive this divergence theorem for submanifolds starting from Stokes’ theorem for manifolds. To that end, results from Riemannian geometry are gathered and applied to the submanifold case.

MSC:

53Z05 Applications of differential geometry to physics
53Z30 Applications of differential geometry to engineering
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
74A99 Generalities, axiomatics, foundations of continuum mechanics of solids
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