Bifurcation in the solution set of the von Kármán equations of an elastic disk lying on an elastic foundation
Volume 77 / 2001
Abstract
We investigate bifurcation in the solution set of the von Kármán equations on a disk ${\mit \Omega }\subset {\mathbb R}^{2}$ with two positive parameters $\alpha $ and $\beta $. The equations describe the behaviour of an elastic thin round plate lying on an elastic base under the action of a compressing force. The method of analysis is based on reducing the problem to an operator equation in real Banach spaces with a nonlinear Fredholm map $F$ of index zero (to be defined later)\ that depends on the parameters $\alpha $ and $\beta $. Applying the implicit function theorem we obtain the following necessary condition for bifurcation: if $(0,p) $ is a bifurcation point then $\mathop {\rm dim}\nolimits \mathop {\rm Ker}F_{x}^{\prime }(0,p) >0$. Next, we give a full description of the kernel of the Fréchet derivative of $F$. We study in detail the situation when the dimension of the kernel is one. We prove that $(0,p) $ is a bifurcation point by the use of the Lyapunov–Schmidt finite-dimensional reduction and the Crandall–Rabinowitz theorem. For a one-dimensional bifurcation point, analysing the Lyapunov–Schmidt branching equation we determine the number of families of solutions, their directions and asymptotic behaviour (shapes).
