Localized bulging in an inflated cylindrical tube of arbitrary thickness – the effect of bending stiffness

Abstract

We study localized bulging of a cylindrical hyperelastic tube of arbitrary thickness when it is subjected to the combined action of inflation and axial extension. It is shown that with the internal pressure P and resultant axial force F viewed as functions of the azimuthal stretch on the inner surface and the axial stretch, the bifurcation condition for the initiation of a localized bulge is that the Jacobian of the vector function (P, F ) should vanish. This is established using the dynamical systems theory by first computing the eigenvalues of a certain eigenvalue problem governing incremental deformations, and then deriving the bifurcation condition explicitly. The bifurcation condition is valid for all loading conditions, and in the special case of fixed resultant axial force it gives the expected result that the initiation pressure for localized bulging is precisely the maximum pressure in uniform inflation. It is shown that even if localized bulging cannot take place when the axial force is fixed, it is still possible if the axial stretch is fixed instead. The explicit bifurcation condition also provides a means to quantify precisely the effect of bending stiffness on the initiation pressure. It is shown that the (approximate) membrane theory gives good predictions for the initiation pressure, with a relative error less than 5%, for thickness/radius ratios up to 0.67. A two-term asymptotic bifurcation condition for localized bulging that incorporates the effect of bending stiffness is proposed, and is shown to be capable of giving extremely accurate predictions for the initiation pressure for thickness/radius ratios up to as large as 1.2.

Localized bulging of rotating elastic cylinders and tubes

“…We emphasize that this correspondence is lost when, for instance, it is the λ z that is held fixed in rotating the tube. Drawing upon the results of [Fu et al 2016], we may then further conjecture that when the inner surface is traction-free or subjected to a hydrostatic pressure P the bifurcation condition for localized bulging is simply J ( , F) = 0, whether it is the F or λ z that is fixed in rotating the tube. We shall verify in the next section that this is indeed the case.…”

“…However, it can be shown that this Jacobian is a nonzero multiple of J ( , F) when the connection (2-10) is used. The case previously studied by [Fu et al 2016] can now be viewed as a special case, corresponding to ≡ 0, of the current more general formulation. The observations made in that paper about J ( P, F) can be extended to the case when is nonzero but is held fixed.…”

“…As explained in [Fu et al 2016], the bifurcation condition can be derived by first looking for a solution of the form…”

“…We follow the same strategy as in [Fu et al 2016]. After formulating the problem and summarizing the expressions for the angular speed ω, the internal pressure P, and the resultant axial force F associated with the primary shape-preserving deformation in the next section, we first derive the bifurcation condition in Section 3, and then conjecture and verify that the bifurcation condition in each case can be written simply in terms of the derivatives of ω, F, and/or P with respect to the stretches as stated in the abstract.…”

Methods of Investigation of Equations that Describe Waves in Tubes with Elastic Walls and Application of the Theory of Reversible and Weak Dissipative Shocks

“…If the geometric nonlinearity is not taken into account then P b = P bs = −Dr and the bending pressure is included only in the equation for r. Only the term −br is added to the left part in this A non-differential term related with the latitudinal (circular) bending resistance [8] may be included in pressure formula. But if the wall of the tube is weakly stretched (this is a typical case when inclusion of meridional bending resistance is required) or if h/R is close to zero this term vanishes.…”