Abstract
In this introductory chapter, we work in a one-dimensional (1-D) setting. Firstly, we exemplify the nonstandard integration method we are going to use systematically in Part II. Secondly, we exemplify the Green-kernel integration method to be exploited, in particular, for the problems collected in Part III. Finally, we use these two integration methods to solve the 1-D versions of Kelvin’s and Mindlin’s problems.
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Notes
- 1.
The dyadic product of two vectors \({{\varvec{a}}},{{\varvec{b}}}\) is defined as follows:
$$\begin{aligned} ({{\varvec{a}}}\otimes {{\varvec{b}}}){\varvec{v}}:=({{\varvec{b}}}\cdot {\varvec{v}}){{\varvec{a}}},\quad \mathrm{for\; all\; vectors }\;\,{\varvec{v}}. \end{aligned}$$ - 2.
To obtain (1.18), it is sufficient to note that
$$\begin{aligned} \int _\mathcal{R }Nv^\prime dz=N(0-)\int _\mathcal{R ^-}v^\prime dz+N(0+)\int _\mathcal{R ^+}v^\prime dz=-v(0)\big (N(0+)-N(0-)\big ), \end{aligned}$$
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© 2014 Springer International Publishing Switzerland
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Podio-Guidugli, P., Favata, A. (2014). One-Dimensional Paradigms. In: Elasticity for Geotechnicians. Solid Mechanics and Its Applications, vol 204. Springer, Cham. https://doi.org/10.1007/978-3-319-01258-2_1
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DOI: https://doi.org/10.1007/978-3-319-01258-2_1
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