Abstract
In this chapter we cover the rudiments of solid mechanics as will be required in later development.
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Notes
- 1.
For more information about conditions that \({\mathbf f}\) is subjected to we refer the interested reader to Gurtin (1981, Chap. III).
- 2.
The subscript ‘F’ is included to distinguish this rotation tensor from \({\mathsf {R}}\) defined by (2.1) that relates frames \({\mathscr {O}}\) and \({\mathscr {I}}\).
- 3.
Cf. (2.54).
- 4.
Indeed, such an \({\mathsf {E}}\) is expressible as \({\mathsf {E}} = \sum _{i=1}^3 \lambda ^2_i\hat{\mathbf E}_i\otimes \hat{\mathbf E}_i,\) where \(\lambda ^2_i\) are the eigenvalues and \(\hat{\mathbf E}_i\) are the corresponding eigenvectors. Then \(1= {\mathbf X}\cdot {\mathsf {E}}\cdot {\mathbf X} = \lambda _i^2X_i^2\) represents an ellipsoid with semi-axes of length \(\lambda ^{-1}_i\) lying along \(\hat{\mathbf E}_i\).
- 5.
While symmetry is obvious, the following computation shows that \({\mathsf {e}}\) is positive definite. For arbitrary \({\mathbf u},\) we have \({\mathbf u}\cdot {\mathsf {e}}\cdot {\mathbf u} = {\mathbf u}\cdot {\mathsf {H}}^{-T} \cdot {\mathsf {E}} \cdot {\mathsf {H}}^{-1}\cdot {\mathbf u} = \left( {\mathsf {H}}^{-1}\cdot {\mathbf u}\right) \cdot {\mathsf {E}} \cdot \left( {\mathsf {H}}^{-1}\cdot {\mathbf u}\right) = {\mathbf y}\cdot {\mathsf {E}}\cdot {\mathbf y} > 0,\) as \({\mathsf {E}}\) is positive definite and \({\mathsf {H}}\) is invertible.
- 6.
Other stress tensor definitions depending on the choice of the body’s configuration are possible; see Spencer (1980, Chap. 5).
- 7.
In contrast, the work function in elasticity relates stress to strain, not its increment.
- 8.
We label any substantial Solar System object, which is not a major planet as a minor planet.
- 9.
Ratio of the free volume in an aggregate to the volume occupied by grains.
- 10.
Ratio of the deviation of the void ratio from its maximum achievable value to the total possible change in the void ratio at a given confining pressure.
- 11.
Ratio of volume occupied by grains to total volume.
- 12.
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Sharma, I. (2017). Continuum Mechanics. In: Shapes and Dynamics of Granular Minor Planets. Advances in Geophysical and Environmental Mechanics and Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-40490-5_2
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