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Newtonian and Eshelbian Mechanics

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Book cover Error Estimates for Advanced Galerkin Methods

Part of the book series: Lecture Notes in Applied and Computational Mechanics ((LNACM,volume 88))

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Abstract

The objective of this chapter is to present an introduction to the theory of continuum mechanics of elastic structures. Classical continuum mechanics deals with finding the spatial configuration of an elastic body that is subjected to external forces. This forward problem is attributed to Sir Isaac Newton and therefore termed Newtonian mechanics. In the associated inverse problem, which is attributed to John Douglas Eshelby and therefore termed Eshelbian mechanics, we are concerned with the forces applied to the spatial configuration. In Newtonian mechanics, the applied forces are of a physical nature, and as a result, the associated stress that arises in the spatial configuration of the elastic body is well known as the (physical) Cauchy stress. In Eshelbian mechanics, on the other hand, the deformed elastic body is subjected to so-called material forces, and the resulting stress in the initial configuration (of the forward problem) is termed the (material) Eshelby stress. In this chapter, the stress tensors that naturally appear in both Newtonian and Eshelbian mechanics are systematically derived for both compressible and (nearly) incompressible materials.

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Notes

  1. 1.

    Note that it is conventional practice to denote the divergence operator equivalently as \(\nabla _{x} \cdot (\cdot )\). However, this is not always possible, as we shall see in Sect. 2.2.4. Moreover, this notation is not consistent with the (spatial) gradient operator \((\cdot ) \otimes \nabla _{x}\).

  2. 2.

    Note that it is customary in the literature to express the material tractions \({\varvec{T}}_{\text {mat}}\) by \({\varvec{\Sigma }}^{\sharp } \cdot {\varvec{N}}\), according to the stress theorem (2.24).

  3. 3.

    To be consistent with the tensor gradient operators \((\cdot ) \otimes \nabla _{x}\) and \((\cdot ) \otimes \nabla _{X}\), we use the notations \((\cdot ) \nabla _{x}\) and \((\cdot ) \nabla _{X}\) in the vector case.

  4. 4.

    Note that the material part \(\mathbb {A}_{\text {mat}}\) is not related to material forces in Eshelbian mechanics.

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  1. Department of Civil and Environmental Engineering, University of California Los Angeles, Los Angeles, CA, USA

    Marcus Olavi Rüter

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Rüter, M.O. (2019). Newtonian and Eshelbian Mechanics. In: Error Estimates for Advanced Galerkin Methods. Lecture Notes in Applied and Computational Mechanics, vol 88. Springer, Cham. https://doi.org/10.1007/978-3-030-06173-9_2

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  • DOI: https://doi.org/10.1007/978-3-030-06173-9_2

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