Bifurcation And Post-bifurcation Analysis Of Localized Bulging Of An Inflated Hyperelastic Tube Of Arbitrary Wall Thickness

The problem of bulging in an infinitely tube of arbitrary thickness is a very common problem in our daily life and production practice and is even closely related to human life and health.The problem of bulging has appeared widely from the rupture of hemangioma/aneurysm to the irregular deformation of the balloon and the power generation process of the"electric eel"underwater,etc.These kinds of problems can be explained from the perspective of mathematics and mechanics.In our daily life,there are often large deformation problems similar to this type,and if this type of problem is not well protected,it will have disastrous consequences for our lives and production practices.Therefore,it is very important to study the instability/bulge problem of round tubes.In this paper,we conduct a buckling analysis to study the effects of materials,modulus,thickness and other conditions on bulging and propagation;perform a post-buckling analysis to study the morphological changes after instability,and give theoretical and numerical explanations.We can apply this part of the research to the reality of life,which has high theoretical and practical value.This thesis contains two parts of work:for the buckling stage:we use mathematical tools such as continuum mechanics,numerical analysis methods,bifurcation theory,etc to study the effect of parameters to the bifurcation;for the post-buckling stage:we use the membrane theory and incremental theory in the continuum mechanics framework,singular perturbation method,bifurcation theory,numerical software etc to derive the post-bifurcation amplitude equation.We have the following conclusions,Buckling stage:1,Basic solution and the bifurcation conditions of the double-layer circular tube are derived and their effectiveness are verified by numerical software Abaqus;2,Using the hyperelastic constitutive model(Gent material),it is found that whether the single-layer round tube will bulge depends on the obtained parameters J_m^c and ?_z^c and if J_m^c or ?_z<?_z^c it will not bulge.Critical geometrical parameters marking the transition between bulging and no bulging are determined.Further,by designing a reasonable shear modulus,the stability of an inflated bilayer tube of arbitrary thickness can be enhanced,which offers a possible way to avoid bulging formation in a cylindrical tube while retaining moderate extensibility.3,We study a bilayer tube subject to inflation and axial stretching to reveal the effects of,which is the ratio of the shear modulus of the outer layer to that of the inner layer,the interfacial radius,and different constitutive models on the bulge initiation.By use of the internal volume ratioas the bifurcation parameter,a parametric study is carried out.It is found that a largerproduces a more stable bilayer tube.If the thickness of the bilayer tube is specified,the composite tube is more stable when the stiffer part occupies the outer layer.Moreover,the critical volume ratio ?_cr as a function ofhas a maximum if>1 but a minimum if<1.In addition,there is a similar pattern in the propagation stage of the round tube.This provides a possible method for adjusting stability to the maximum by adjusting parameters.Post-buckling stage:1,A weakly nonlinear analysis is conducted for localized bulging of an inflated hyperelastic cylindrical tube of arbitrary wall thickness.Analytical expressions are obtained for the coefficients in the amplitude equation despite the fact that the primary deformation is inhomogeneous and the incremental governing equations have variable coefficients;2,It is shown that even for thin-walled tubes the membrane approximation becomes poorer and poorer as the tube is subjected to increasingly larger and larger axial stretch or force prior to inflation;3,It is shown that for each value of wall thickness a localized bulging solution does indeed bifurcate subcritically from the primary solution for almost all values of fixed axial force or fixed axial stretch for which the bifurcation condition is satisfied,as reported in all previous experimental studies,but there also exist extreme cases of fixed axial stretch for which localized bulging gives way to localized necking.Validation is carried out by comparing with results obtained under the membrane assumption and with fully numerical simulations based on Abaqus.