Deformations of spherical and cylindrical pressure vessels induced by pressure applied to inner and outer surfaces are axisymmetric in the sense that they are independent of the angular position of a material particle. The flow of a fluid in a circular pipe studied in Sec. 7.3 is also an axisymmetric problem. Here we study deformations of a pressure vessel when the material is either Hookean or neo-Hookean. These problems are more easily formulated in spherical and cylindrical coordinates. Here we use rectangular Cartesian coordinates.
Consider a spherical pressure vessel made of a homogeneous and isotropic linear elastic material deformed by applying pressures p inn and p out to its inner and outer surfaces whose radii in the undeformed configurations are R 1 and R 2, respectively. Thus equilibrium Eq. (6.1) is to be solved under the following boundary conditions.
Due to the symmetry of the geometry and the loading, we assume that material points move only radially. The displacement u of a material point with respect to rectangular Cartesian coordinate axes with their origin at the center of the sphere is given by
where
is the radial coordinate of a point or its distance from the origin. Note that (x 1 /r, x 2 /r, x 3 /r) is a unit vector in the radial direction. Differentiation of Eq. (10.2) with respect to x j yields
where u ? = du/dr,...
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