An Asymptotic Shell Model for Incompressible Hyperelastic Materials: Theory and Applications

不可壓縮超彈性材料的漸近殼模型: 理論與應用

Student thesis: Doctoral Thesis

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Award date19 Jul 2021

Abstract

This thesis is devoted to the derivation of an asymptotic shell model for incompressible hyperelastic materials and its applications to two problems in human arteries.

In the first part, we derive a refined dynamic finite-strain shell theory for incompressible hyperelastic materials. Based on the previous work for the static problem, we first derive one form of consistent dynamic finite-strain shell equations that involve three shell constitutive relations. In order to single out the bending effect as well as to reduce the number of shell constitutive relations, a further refinement is performed, leading to a refined dynamic finite-strain shell theory with only two shell constitutive relations (deducible from the given three-dimensional strain energy function) and some new insights are also given. By choosing the appropriate virtual displacement in the edge term in the Lagrange functional, we establish physically meaningful boundary conditions. The two-dimensional shell virtual work principle is also deduced from the weak forms of the shell equations. As a benchmark problem, we consider the extension and inflation of an artery. The good agreement between the asymptotic solution based on the shell equations and that from the 3d exact one gives the validation of the former.

The second part of this thesis investigates the propagation of waves in a pressurized fiber-reinforced tube based on the refined shell model. We first derive the linearized incremental theory associated with the refined shell theory and then use it to investigate wave propagation in a fiber-reinforced hyperelastic tube that is subjected to an axial pre-stretch and internal pressure. We obtain the dispersion relations for both axisymmetric and non-axisymmetric waves and discuss their accuracy by comparing them with the exact dispersion relations. The bending effect is also examined by comparing the dispersion curves based on the present theory and membrane theory, respectively. It is shown that the present theory is more accurate than the membrane theory in studying wave propagation and the bending effect plays an important role in some wave modes for relatively large wavenumbers. The effects of the pressure, axial pre-stretch and fiber angle on the dispersion relations are displayed. These results provide a theoretical foundation for wave propagation in arteries, which can be used to determine arterial properties.

Finally, we study the localized bulging in an inflated hyperelastic tube using the refined shell theory. We consider the problem in the static and asymmetric setting and give the refined shell equations in this case. The comparison between the refined shell equations and their membrane counterparts shows that the membrane theory can provide results with errors of O2), where α is the ratio of the thickness against the radius of the middle surface. By analyzing types of equilibrium points of the shell equations, we show that the bifurcation condition is simply that the Jacobian of the vector function (P,N) should vanish, where P is the internal pressure and N is the resultant axial force, and both are viewed as functions of the azimuthal stretch on the middle surface and the axial stretch. To understand the post-bifurcation behavior, we conduct a weakly nonlinear analysis for localized bulging based on the refined shell equations and obtain an asymptotic solution of solitary type. The asymptotic solution is validated by comparing it with the exact weakly bulging solution. In the last part, we study the bifurcation condition derived from exact theory of nonlinear elasticity. We prove that the exact bifurcation condition is equivalent to the vanishing of the Jacobian of (P,N) analytically. These results give useful insights into aneurysm formation in human arteries.