Existence, uniqueness, and decay estimates for solutions in the nonlinear theory of elastic, edge-loaded, circular tubes

Solutions to the two, coupled, ordinary nonlinear differential equations for a semi-infinite circular elastic tube subjected to edge loads and undergoing small axisymmetric strains, but arbitrarily large axisymmetric rotations—the simplified Reissner equations—are analyzed. First, with the aid of a Green’s function, the differential equations and boundary conditions are transformed to a complex-valued integral equation. From this equation existence, uniqueness, boundedness, and rate of decay are extracted for a dimensionless stress function, $f$, and an angle of rotation, $\beta$ , for sufficiently small edge data. It is shown that these unique solutions must decay at least as fast as the linear solution. Second, it is shown that any solution that decays to zero must, at a sufficiently large distance from the edge, decay at the linear rate. Third, rates of decay are established for any solution for which the ${L_2}$ or the sup norm of $\beta$ has certain bounds. Finally, an energy (or Lyapunov) function $E$, defined on solutions of the differential equations, is constructed and under certain a priori restrictions on the angle of rotation, three different upper bounds on $E$ are obtained. These also provide exponential decay estimates for solutions. The energy approach is examined with a view to more general shells where a Green’s function may not be readily available.