Post-bifurcation Analysis for Deformations of Thin Tubes Composed of Nonlinearly Elastic Materials under Compression and End Twist

When instability occurs, it can induce damage to the material or structure. Sometimes, one can also use instabilities for advantages. An example is the design of a car driver's wheel such that it will buckle under certain threshold impact so the driver can be protected. Theoretically, it is also important to study instabilities in nonlinear elasticity. Mathematically, instability is associated with a solution bifurcation, and nonlinearity is essential for the post-bifurcation solution behavior. So, studying instability is an important aspect of "understanding and exploring nonlinearity". Although it has been a research topic for a long time, analytical solutions for post-bifurcation states are rarely obtained within the framework of two/three-dimensional nonlinear elasticity. The difficulty is that the governing equations are coupled nonlinear partial differential equations and there lack of mathematical tools to solve nonlinear PDEs analytically. Analytical solutions can shed light on the physical roles played by material and geometric nonlinearity and provide clear pictures how the geometric parameters influence the post-bifurcation states. In this project, we shall study instabilities in thin tubes composed of nonlinearly elastic materials under compression and end twist. Tubes are common in numerous devices and structures, while instabilities are often observed in tubes. We intend to construct analytical (asymptotic) solutions for the post-bifurcation states within the framework of three-dimensional field equations for nonlinearly elastic materials. A methodology of coupled series-asymptotic expansions developed earlier by us will be used to derive asymptotic model equations. These equations with lubricated end conditions will be first studied analytically and numerically. We shall analyze the equilibrium point and do a linear bifurcation analysis to obtain the bifurcation condition. The multiple scale method will be used to construct the analytical post-bifurcation solutions, which will lead to analytical formulas for the amplitudes. Then, we can have clear pictures on the post-bifurcation states; in particular, it is interesting to determine when the tube bulges inward or outward. A variety of cases can arise, probably including bifurcation phenomena not reported before for nonlinear boundary-value problems. Numerical computations will also be carried out, and the numerical solutions will be compared with the analytical ones for the verification purpose. A few related nontrivial problems will also be treated in this project. It is hoped that this project could help to further understand the effects of nonlinearity on instability phenomena in solids and the analytical formulas for the amplitudes could be useful for engineering designs.