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2014 | OriginalPaper | Chapter

4. Stress

Author : Alan D. Freed

Published in: Soft Solids

Publisher: Springer International Publishing

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Abstract

The concept of stress traces back nearly two centuries to the published works of Cauchy (1827). Cauchy generalized Euler’s concept of pressure and the hydrodynamic laws that Euler derived some 70 years earlier. Cauchy made the notion of stress precise. He surmised that a body responds to externally applied loads by transmitting forces internally throughout the body via a matrix valued field that now bears his name: Cauchy stress. Not only did Cauchy develop the concept of stress, but he also derived the physical conservation laws that apply to stress. In doing so, he generalized Euler’s theory for an inviscid fluid.

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previous chapter Strain
next chapter Explicit Elasticity
Footnotes
1
Cauchy stress is expressed in an uppercase font, as if it were a Lagrangian field, but it is not; Cauchy stress is an Eulerian field. The notation https://static-content.springer.com/image/chp%3A10.1007%2F978-3-319-03551-2_4/MediaObjects/315668_1_En_4_Figq_HTML.gif is adopted for historical reasons. https://static-content.springer.com/image/chp%3A10.1007%2F978-3-319-03551-2_4/MediaObjects/315668_1_En_4_Figr_HTML.gif is the commonly accepted notation for Cauchy stress when written in a roman font [cf. Truesdell and Noll (2004)]. The engineering stress that associates with the infinitesimal strain https://static-content.springer.com/image/chp%3A10.1007%2F978-3-319-03551-2_4/MediaObjects/315668_1_En_4_Figs_HTML.gif of Eq. (3.​22) is typically denoted as https://static-content.springer.com/image/chp%3A10.1007%2F978-3-319-03551-2_4/MediaObjects/315668_1_En_4_Figt_HTML.gif , which appears in the linear theory of elasticity.
 
2
Seventy years before Cauchy’s work, Euler derived a field equation that describes Newton’s laws of motion for an inviscid fluid, i.e., https://static-content.springer.com/image/chp%3A10.1007%2F978-3-319-03551-2_4/MediaObjects/315668_1_En_4_Figahsda_HTML.gif
where https://static-content.springer.com/image/chp%3A10.1007%2F978-3-319-03551-2_4/MediaObjects/315668_1_En_4_Figai_HTML.gif is the internal hydrodynamic pressure, which Cauchy described in terms of stress as https://static-content.springer.com/image/chp%3A10.1007%2F978-3-319-03551-2_4/MediaObjects/315668_1_En_4_Figaj_HTML.gif . Seventy additional years passed before Cauchy’s genius was able to generalize Euler’s governing equation for hydrodynamics to that of a general material. Of this, Truesdell (1961) wrote:
“Nothing is harder to surmount than a corpus of true but too specific knowledge; to reforge the tradition of his forebears is the greatest originality a man can have.”
 
3
The force of reaction is understood to exist and to be present, but it is not drawn so as to keep the schematic simple and uncluttered. This is true of most schematics drawn for this chapter. This force is carried through a clamped boundary condition, which is drawn as a hatched surface.
 
4
To capture https://static-content.springer.com/image/chp%3A10.1007%2F978-3-319-03551-2_4/MediaObjects/315668_1_En_4_Figjm_HTML.gif in fluid experiments, shear flows are set up along planes with curvature. Analysis of these BVPs exceeds the scope of this book. The interested reader is referred to the texts by Bird et al. (1987a), Ferry (1980), and Lodge (1974).
 
5
In the presence of curvature, e.g., for a balloon, a pressure https://static-content.springer.com/image/chp%3A10.1007%2F978-3-319-03551-2_4/MediaObjects/315668_1_En_4_Figjv_HTML.gif acting normal to the surface of a membrane can exist. In such cases, this normal pressure is balanced by the four in-plane stress components https://static-content.springer.com/image/chp%3A10.1007%2F978-3-319-03551-2_4/MediaObjects/315668_1_En_4_Figjw_HTML.gif , i, j = 1, 2 that are carried along the curved membrane surface. A study of curvature lies beyond the intended scope of this introductory text.
 
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Metadata
Title
Stress
Author
Alan D. Freed
Copyright Year
2014
Book
Soft Solids

Print ISBN: 978-3-319-03550-5

Electronic ISBN: 978-3-319-03551-2

Copyright Year: 2014

DOI
https://doi.org/10.1007/978-3-319-03551-2_4

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