The End Problem of Cylinders

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Abstract
When self-equilibrating rotationally symmetric normal and shear tractions sk(r), tk(r) act on an end face of a cylinder, stresses ensue which decay with distance from the loaded end as e−αk2, where αk is the real part of an eigenvalue parameter γk. The analysis is carried out in a variational approximation; the method is based on Sadowsky-Sternberg stress functions Φk(z, r) which are so modified as to identically satisfy the equilibrium equations. The factors Fk(r) of the functions Φk = Gk(z) Fk(r) or Φk = Hk(z)Fk(r) (k = 2, 4, 6, …) are polynomials of degree k + 2, which are orthogonal in a “generalized sense”; their appropriate derivatives constitute sk(r) and tk(r). The polynomials sk(r), tk(r), of degrees k and k + 1, respectively, form two complete sets of tractions in terms of which arbitrary rotationally symmetric self-equilibrating end tractions may be expanded. The factors Gk(z), Hk(z) of the stress functions Φk are exponentially decaying sinusoidals; Gk(z) is appropriate for the normal load problem, Hk(z) for the shear-load problem. The function Φ2 leads, for ν = 0.3, to the fundamental variational eigenvalue γ2 = 2.69 + i 1.34, as contrasted with the rigorous value γ2 = 2.72 + i 1.35.

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