Abstract
We start with the formulation of the different boundary value problems of linear elasticity: mixed problem, stress problem, and displacement problem. Theorems of existence and unicity of solutions of these problems are proved in suitable functional spaces. Then, the solution of Boussinesq–Papkovich–Neuber of linear elasticity is given. The Saint–Venant conjecture is discussed together with all the fundamental solutions of linear elasticity it supplies. The remaining part of the chapter contains a wide analysis of waves in linear elasticity in which important phenomena as reflection waves, refraction of waves, and Rayleigh surface waves are analyzed.
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Notes
- 1.
If the portion \(\Sigma _{2}\) is nonempty, then conditions (10.8) are replaced by
$$\displaystyle{\left \{\begin{array}{l} \int _{\Sigma _{1}}\mathbf{t}\,d\sigma +\int _{C_{{\ast}}}\rho _{{\ast}}\mathbf{b}\,dc +\int _{\Sigma _{2}}\boldsymbol{\phi }\,d\sigma = \mathbf{0}, \\ \int _{\Sigma _{1}}\mathbf{r} \times \mathbf{t}\,d\sigma +\int _{C_{{\ast}}}\rho _{{\ast}}\mathbf{r} \times \mathbf{b}\,dc +\int _{\Sigma _{1}}\mathbf{r} \times \boldsymbol{\phi }\,d\sigma = \mathbf{0},\end{array} \right.}$$where \(\boldsymbol{\phi }\) is the reaction provided by the constraints fixing the portion \(\Sigma _{2}\) of the boundary. These conditions of global equilibrium are certainly satisfied due to the presence of the above reactions, provided that they are assured by the constraints.
- 2.
- 3.
This is related to the assumption of plane waves, so that all the physical quantities are independent of z.
- 4.
The same argument applies to (10.75).
- 5.
It can be proved, starting from (10.87), that the expression
$$\displaystyle{C\sin \beta + D\cos \beta }$$can be written as
$$\displaystyle{C^{{\prime}}e^{i\beta } + D^{{\prime}}e^{-i\beta },}$$with
$$\displaystyle{C^{{\prime}} + D^{{\prime}} = C,\,i(C^{{\prime}}- D^{{\prime}}) = D.}$$ - 6.
If the media are bounded together, the transverse slip can also occur at lubricated interfaces, a case not considered here but of interest in ultrasonics, when transducers on lubricated plates are used to generate and receive waves from solids.
References
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R. Marcolongo, Teoria Matematica dell’Equilibrio dei Corpi Elastici (Manuali Hoepli, Serie Scientifica, Milano, 1904)
R. Toupin, Saint-Venant’s Principle. Arch. Rat. Mech. Anal. 18, (1965)
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Romano, A., Marasco, A. (2014). Linear Elasticity. In: Continuum Mechanics using Mathematica®. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-1604-7_10
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DOI: https://doi.org/10.1007/978-1-4939-1604-7_10
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