Abstract
Up to this time we have restricted the discussion to so-called small deformation. This permitted the employment of the strain tensor of the form:
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Notes
- 1.
See Novozhilov, V. V.: “Foundations of the Non-Linear Theory of Elasticity,” Graylock Press, Rochester, 1953.
- 2.
This interface then has resulted from the deformation of an element originally in the x 1 x 2 plane.
- 3.
We have thus related the traction vectors for the rectangular parallelopiped in the deformed geometry, namely \(\mathop {\bf{T}}\limits^{*} {^{({\xi _j})}}\), with a traction vector \(\mathop {\bf{T}}\limits^{*} {^{({n^\prime_3})}}\) on an interface which in the undeformed geometry may be considered a face of a rectangular parallelopiped, namely one with normal \({\bf{\hat n}}_3\).
- 4.
Note that \({{\bf{\hat n^\prime}}_1},\,\,{{\bf{\hat n^\prime}}_2}\), and \({{\bf{\hat n^\prime}}_3}\) in general will not be orthogonal to each other.
- 5.
Note again that n is being used to give orientations in the undeformed geometry and n′ is being used to give orientations in the deformed geometry with the subscripts of the latter tying back to normal directions of the element when it was in the undeformed geometry.
- 6.
Note in this regard that u k,α = e kα + ω kα .
- 7.
Actually, when using Eq. (8.86) we get
$${1 \over 2}{\partial \over {\partial {x_i}}}\left[ {{\tau _{ij}}\left( {{\delta _{kj}} + {\omega _{kj}}} \right)} \right] + {1 \over 2}{\partial \over {\partial {x_\iota }}}\left[ {{\tau _{kj}}\left( {{\delta _{\iota j}} - {\omega _{\iota j}}} \right)} \right] + {B_k} = 0$$ - 8.
If we neglect ω ki note that we get back to the familiar Cauchy formula.
- 9.
In the Russian literature these terms are often referred to as “reduced loads” of the in-plane forces or the “reduced forces.”
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Dym, C.L., Shames, I.H. (2013). Nonlinear Elasticity. In: Solid Mechanics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6034-3_8
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DOI: https://doi.org/10.1007/978-1-4614-6034-3_8
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