Analytical studies on the corner instability and finite deformations of a nonlinear elastic cylinder
Student thesis: Master's Thesis
Related Research Unit(s)
In this thesis, we study several three-dimensional axisymmetric boundary-value problems of a slender cylinder composed of a nonlinear elastic material subject to axial forces. This thesis is divided into two parts. For the ¯rst part our purpose is to study the mechanism of the corner instability and describe the characters of di®erent corners. It seems that no one has done any analytical study on it. For the second part our purpose is to obtain the analytical solution of a higher dimen- sional boundary-value problem. In both parts, starting from the ¯eld equations for di®erent constitutive relationships, after a transformation and proper scalings, we identify a small variable and two small parameters. Then, by a novel approach in- volving compound series-asymptotic expansions, a nonlinear second-order ODE is derived, which governs the axial strain (the ¯rst-term in the series expansion). In the ¯rst part, one more small term represents the interaction of the material nonlin- earity and geometrical size is kept in this ODE compared with the second part. The Euler-Lagrange equation can also lead to the same ODE which justi¯es our method in deriving this ODE. This ODE can be rewritten as a ¯rst-order system. It is a singular dynamical system and singular lines appear in phase planes. We point out that singular lines in phase planes are the mechanisms of the corner formations. By phase plane analysis, analytical solutions under di®erent values of the external force are found by imposing several di®erent end conditions. In each phase plane there are maybe more than one solutions. We calculate the total potential energy value for each solution to identify which is the preferred solution. In the second part, by imposing the zero radial displacement conditions at two ends, we manage to get the analytical solution of the axial strain, from which all other physical quantities can be deduced and thus the three-dimensional displacement ¯eld can be determined. Graphical results are presented, which show that there are two boundary layers near the two ends while the middle part is in a state of almost uniform extension. Asymp- totic structures of the analytical solution are derived, which o®er clear explanations to the structure of the deformed con¯guration and show that the thickness of both boundary layers is of the order of the radius. We also point out the relevance of the present results to the St. Venant's problem. In particular, we obtain the ex- plicit uniformly-valid exponentially small error term, when the obtained deformed con¯guration is compared to the con¯guration of a uniform extension.
- Fracture mechanics, Cylinders, Elastic plates and shells