Bifurcation to a corner-like formation in a slender nonlinearly elastic cylinder: asymptotic solution and mechanism

In this paper, we study the corner-like formation in a slender nonlinearly elastic cylinder due to compression. Mathematically, this is a very challenging problem: one needs to study the bifurcation of the nonlinear field equations (complicated nonlinear partial differential equations) and show that there is a bifurcation leading to a solution with a corner-like profile. We also aim to obtain the asymptotic expression for this post-bifurcation solution. As far as we know, there is no available analytical method to obtain the post-bifurcation solution of nonlinear partial differential equation(s). Here we use a novel approach to tackle the present problem. Through a method of coupled series–asymptotic expansions, we manage to derive the normal form equation of the original nonlinear field equations, which can be written as a singular ordinary differential equation (ODE) system (the vector field has a singularity at one point). With welding end conditions, the problem is reduced to study the boundary-value problem of a singular ODE system. It seems that such a boundary-value problem of a singular ODE system was not formulated or studied before in the context of the present problem. With the help of phase planes, we manage to solve such a boundary-value problem. It turns out that there is a vertical singular line in phase planes, which causes the bifurcation to a corner-like solution. The expression for this post-bifurcation solution is also obtained. From the analytical results obtained, we reveal that the coupling effect of the material nonlinearity and geometrical size is the physical mechanism that causes the formation of a corner-like profile.

The Two-sided FD-method of Solving the First Boundary Problem for Singular ODE of the Second Order in the Half-axis

To solve the boundary problems for singular ordinary differential equations of the second order the functional-discrete (FD-) method is proposed. The method gives the possibility of obtaining an approximate solution in the numerical-analytical form in the whole half-axis with a guaranteed accuracy. Sufficient conditions of convergence of the method at a rate of a geometrical progression are found. The two-sided approximations are established and an explicit evaluation of the two-sided estimation width is given. In conclusion the results of numerical experiments obtained by the system of computer algebra are presented.