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Existence and Uniqueness of Azimuthal Shear Solutions in Compressible Isotropic Nonlinear Elasticity

The Behrend College, Division of Science Penn State Erie, Erie, PA 16563

Debra Polignone Warne

Mathematics Department, University of Tennessee, Knoxville, TN 37996-1300

Paul G. Warne

Division of Mathematics and Computer Science, Maryville College, Maryville, TN 37804

The authors consider the two-point boundary-value problem resulting from the equations of nonlinear elastostatics for azimuthal shear of a Blatz-Ko tube. Previous work on this problem by Simmonds and Warne includes a numerical study of these equations and indicates that smooth radial deformation solutions (no kinks) should exist regardless of the aspect ratio of the tube, provided that the dimensionless applied torque r is small enough (r <-0.72). The numerics of Simmonds and Warne also indicated that the existence of smooth solutions for r >-0.72 depends on the geometry of the tube, and that for r = A, no smooth solution exists. Motivated by this numerical work, the authors prove via a topological shooting argument the existence and uniqueness of smooth solutions to this problem for r < tr, = 3/44 S 0.69, and the nonexistence of smooth solutions for r = a.

Mathematics and Mechanics of Solids, Vol. 3, No. 1, 53-69 (1998)
DOI: 10.1177/108128659800300104


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