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Force Representations

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Geometric Continuum Mechanics and Induced Beam Theories

Part of the book series: Lecture Notes in Applied and Computational Mechanics ((LNACM,volume 75))

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Abstract

This chapter introduces the concept of force, states the principle of virtual work of a continuous body, discusses admissible force representations and concludes with the application to classical nonlinear continuum mechanics. In Sect. 3.1, forces are defined as linear functionals on the space of virtual displacements and the principle of virtual work for the continuous body is formulated. Subsequently, the force representation of Segev [1] by smooth tensor measures is introduced. In Sect. 3.2 the applied forces are restricted to a subclass of possible forces and the equations of motion of a continuous body mapped to the Euclidean vector space are derived.

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Notes

  1. 1.

    We refer to [9], Proposition 9.20, for a similar representation theorem for functions of the Sobolev space \(W^{1,p}_0(\varOmega )\) on a subset \(\varOmega \subset \mathbb {R}^n\).

  2. 2.

    Notice, symmetry condition does not mean that the two first components of \({\varvec{\pi }}\) are symmetric. Such a statement is meaningless, since both components belong to different vector spaces. Hence, the symmetry condition will include metric information as well as the tangent map \(T\kappa \).

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Authors and Affiliations

  1. Institute for Nonlinear Mechanics, University of Stuttgart, Stuttgart, Germany

    Simon R. Eugster

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  1. Simon R. Eugster
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Correspondence to Simon R. Eugster .

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Eugster, S.R. (2015). Force Representations. In: Geometric Continuum Mechanics and Induced Beam Theories. Lecture Notes in Applied and Computational Mechanics, vol 75. Springer, Cham. https://doi.org/10.1007/978-3-319-16495-3_3

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  • DOI: https://doi.org/10.1007/978-3-319-16495-3_3

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-16494-6

  • Online ISBN: 978-3-319-16495-3

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