Abstract
The subject of Elasticity is concerned with the determination of the stresses and displacements in a body as a result of applied mechanical or thermal loads, for those cases in which the body reverts to its original state on the removal of the loads. In this book, we shall further restrict attention to the case of linear infinitesimal elasticity, in which the stresses and displacements are linearly proportional to the applied loads, and the displacement gradients are small compared with unity. These restrictions ensure that linear superposition can be used and enable us to employ a wide range of series and transform techniques which are not available for non-linear problems.
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Notes
- 1.
In fact, the only practical way to examine the effect of these approximations is to relax them, by considering the same problem, or maybe a simpler problem with similar features, in the context of the theory of Elasticity.
- 2.
For the elasticity solution of this problem, see Chapter 18.
- 3.
In two dimensional problems, the same rules can be used to define \(\epsilon _{ij}\), with \(i,j=1,2\).
- 4.
A similar argument can be used to show that if a laminated fibre-reinforced composite is laid up with equal numbers of identical, but not necessarily symmetrical, laminæ in each of 3 or more equispaced orientations, it must be elastically isotropic within the plane. This proof depends on the properties of the fourth order Cartesian tensor \(c_{ijkl}\) describing the stress-strain relation of the laminæ (see equations (1.40 1.43) below).
- 5.
Notice that the matrix of direction cosines is not symmetric.
- 6.
J. R. Barber, Intermediate Mechanics of Materials, Springer, Dordrecht, 2nd. edn. (2011), §2.2.
- 7.
Students of Mechanics of Materials will have already used a similar result when they express the slope of a beam as du/dx, where x is the distance along the axis of the beam and u is the transverse displacement.
- 8.
In some books, \(\omega _x, \omega _y, \omega _z\) are denoted by \(\omega _{yz}, \omega _{zx}, \omega _{xy}\) respectively. This notation is not used here because it gives the erroneous impression that \(\boldsymbol{\omega }\) is a second order tensor rather than a vector.
- 9.
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Barber, J.R. (2022). Introduction. In: Elasticity. Solid Mechanics and Its Applications, vol 172. Springer, Cham. https://doi.org/10.1007/978-3-031-15214-6_1
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