Bifurcation in the solution set of the von Kármán equations of an elastic disk lying on an elastic foundation

Abstract

Abstract. We investigate bifurcation in the solution set of the von Kármán equations on a disk Ω ⊂ R 2 with two positive parameters α and β. 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 α and β. Applying the implicit function theorem we obtain the following necessary condition for bifurcation: if (0, p) is a bifurcation point then dim Ker F x (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).

Subcritical bifurcation of free elastic shell of biological cluster

“…Other examples of applications of the Crandall-Rabinowitz theorem to elasticity theory, biology and fluid mechanics can be found resp. in [12][13][14][15][16][17][18].…”

Symmetry-Breaking Bifurcation for Free Elastic Shell of Biological Cluster, Part 2.

“…Moreover, we want to adapt Friedman and collaborator's approach to symmetry breaking bifurcations in free boundary problems (see [4,8,9,10]). The scheme of application of the CrandallRabinowitz theorem is similar to that in [2,3,11,12,13,14,16].…”

Solvability and stability of von Karman model without rotational inertia with nonlinear forces

“…The coefficient α ≥ 0 corresponds to rotational forces. Some problems concerning bifurcation of solutions for stationary von Karman equations depending on parameter we find in [9][10][11]. F 0 represents in-plane forces within the plate, F x is nonlinear transverse force and Lx is a first order differential operator modeling a nonconservative force.…”

“…without rotational forces. In order to study (1) we construct the variational method quite different than that in [9,10] not only because we consider evolution equation, but mainly because we consider stronger nonlinearity as well we take into account the internal force F 0 and a first order differential operator Lx modeling a nonconservative force which is of not variational type. the results often depend on the structure of nonlinearity.…”