Localized Bulging In An Inflated Cylindrical Tube Of Arbitrary Thickness—The Effect Of Bending Stiffness

Rubber tubes not only play an irreplaceable role in daily life,but also have non-substitution function in industrial production and national high-tech zone.And at the same time,human tissues are hyperelastic material.In response to human's vascular disease,researchers will simulate vein as a hyperelastic tube to find the cause of human's vascular disease.So it has important theoretical significance and broad application prospect to study the bifurcation problem of the hyperelastic tube.And membrane theory is mainly used to simplify the problem in related fields of study.But according to the application of the hyperelastic tube,the thickness is not always trancendentally small.Based on above purposes,corresponding research is finished in this paper.The main results are summarized as follow.1.Basic theory of this paper is given,including the basic theory and basic equations of continuum mechanics,the expression forms of incremental equations in cylindrical coordinates,the basic processing method of the perturbation method,and the determinant method,which is used to solve the eigenvalue problem.2.Based on the research of Von K a?rm a?n and Tsien(1941),Hutchinson(1965),Hunt(1991),we have Von K a?rm a?n-Donnell equation.Three resonance modes of the problem are obtained by using the perturbation method.3.The problem of localized bulging in an inflated cylindrical hyperelastic tube of arbitrary thickness has been considered in this paper.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 J(P,F)should vanish.And the incremental equation,the strain energy function and the equilibrium equation are used to model the problem.Then incremental equilibrium equation and its boundary conditions are obtained.4.Using the dynamical systems theory and the matrix method to compute the eigenvalues of a certain eigenvalue problem.And get an function about the azimuthal stretch on the inner surface and the axial stretch.The function 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.And then using the perturbation method derive the bifurcation condition explicitly.5.Considering the effect of bending stiffness.Using different materials in the same situation of fixed axial force,we compare the different paths with the contour plot of bifurcation condition.Then the critical pressure of the membrane theory,the exact equation and the approximate solution are compared.The main results are summarized as follows.It is shown that even if localized bulging cannot take place when the axial force is fixed,it is still possible when 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.At the same time,by comparing the approximate solution with the exact solution,the error mainly comes from the truncation of the power series expansion of the pressure.The analytical results in present paper may be helpful as a theoretical reference for the further study of the bifurcation problem.And an approximate algorithm is provided for applications.