The effect of elastic surface coating on the finite deformation and bifurcation of a pressurized circular annulus
The problem considered is that of the plane strain deformation, under internal and external pressure, of a thick walled cylindrical tube which has an elastic coating of uniform thickness on one or both of its cylindrical boundaries. Both the nominal stress (the first Piola-Kirchhoff stress tensor) \(S\) in the annular core and the tension \(F\) and bending moment \(M\) in the coating are derivable from separate strain energies \(W(A)\) and \(U(\lambda,\kappa)\), \(A\) the strain tensor in the core material, \(\lambda\) and \(\lambda^{-1}\kappa\) the stretch and curvature in the deformed coating. The coating is also subject to a shear force \(G\) in the outward normal direction which contributes nothing to the energy \(U\), just as the pressure in the incompressible annular core contributes nothing to the energy \(W\). The standard equilibrium equation for the solid core is consolidated by two others for the coating, one for the force \(F\) with components \(F\) and \(G\), and the other for the moment \(M\). The first of these additional equations acts as a traction compatibility condition at the common boundary of the core and coating. Bifurcation of the basic radially symmetric deformation is examined by seeking solutions periodic in the polar angle \(\theta\) of the linearized equations governing superimposed incremental deformations. From the authors' summary: The effect of the coating on the symmetrical response of the annular cross-section of the tube is determined first. It is noted, in particular, that while the pressure may exhibit a maximum followed by a minimum during inflation for an uncoated tube, it may be a monotonic increasing function of the radius for a coated tube with coating elastic modulus sufficiently large. Next, the possibility of bifurcation from a symmetric configuration is examined, and again the influence of the coating is analyzed. The effect of a coating on the outer boundary is compared with that (of a coating) on the inner boundary. Specifically, during compression, coating on the outer boundary delays bifurcation compared with the uncoated case. On the other hand, when the coating is on the inner boundary, bifurcation is either delayed or advanced relative to be uncoated situation depending on the values of the bending stiffness and tube thickness parameters. Generally, bifurcation is delayed by an increase in the magnitude of the bending stiffness of the coating at fixed values of the other parameters.
scientific article; zbMATH DE number 1089101