Using the example of a hollow sphere consisting of idealized Hookean material that allows unlimited compression and expansion, this paper formulates the exact strain and stress differential equations that remain valid for arbitrarily large strains and stresses. These equations are derived in terms of the displaced radius r’ rather than un-displaced r. The tangential strain-radial strain equation is nonlinear while the tangential stress-radial stress equation is linear. The solution to these equations is a generalization of Lamé’s solution for a hollow sphere. The results are similar to the effects of nonlinear elasticity, except here they arise from geometric nonlinearity. For large stains the theory predicts unphysical results if the Poisson ratio ν is nonzero. For an elastic material to be indestructible, ν would need to change for large strains and approach zero. As an example of the theory, the pressure dependence of the breathing mode frequency of an elastic sphere with an incompressible core is derived. Insights from elasticity theory that avoids small strain approximations may be applicable to improved understanding of deep-sea marine creature survival, improved underwater vessel design for large depths, and safer containers of fluids at high pressures.

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