Abstract
In this chapter we will investigate some problems of linear elasticity in cylindrical and spherical coordinates. First we consider several circular geometries, namely a disc, cylinders with and without holes, and a sphere, subjected to centrifugal and thermal loads as well as internal and external pressure. We will also discuss the appropriate mathematical tools and procedures that allow us to obtain closed-form solutions, depending on prescribed boundary conditions. Last, but not least, we analyze a more complicated geometry, namely an elliptic hole in a plate subjected to biaxial tensile stress. In the limit of a vanishing half axis the hole degenerates into a pointed slit, the so-called Griffith crack.
The United States Constitution has proven itself the most marvelously
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Franklin Delano Roosevelt
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Notes
- 1.
Strictly speaking, during the tetragonal to monoclinic phase transition the volume of Zirconia increases and we observe a shear as well: The right-angled, tetragonal unit cell transforms into a slightly inclined monoclinic configuration. In other words the angle β of Fig. 9.5 is not quite 90°. Consequently shear strains will arise and a more complex state of stress will result. However, in our calculations we will ignore this effect.
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Müller, W.H. (2014). Problems of Linear Elasticity. In: An Expedition to Continuum Theory. Solid Mechanics and Its Applications, vol 210. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7799-6_9
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DOI: https://doi.org/10.1007/978-94-007-7799-6_9
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