Sequential Buckling: A Variational Analysis

Abstract

Abstract. We examine a variational problem from elastic stability theory: a thin elastic strut on an elastic foundation. The strut has infinite length, and its lateral deflection is represented by tL : lR -+ R Deformation takes place under conditions of prescribed total shortening, leading to the variational problem (0.1) Solutions of this minimization problem solve the Euler-Lagrange equationThe foundation has a nonlinear stress-strain relationship F', combining a destiffening character for small deformation with subsequent st·iffening for large deformation. We prove that for every value of the shortening >. > 0 the minimization problem has at least one solution. Iu the limit >. -+ oo these solutions converge on bounded intervals to a periodic profile that is characterized by a related variational problem.We also examine the relationship with a bifurcation branch of solutions of (0.2), and show numerically that all minimizers of (0.1) lie on this branch This information provides an interesting insight into the structure of the solution set of (0.1).

Chevron folding patterns and heteroclinic orbits

“…More complex nonlinear elastic foundation models can be used, and have been extensively investigated in the engineering and mathematical literature ( [19], [23]) and give rise to a plethora of complex behaviour. Here however, we investigate the influence of the geometric nonlinearity in the folding process and the role of this in fold selection through a heteroclinic connection.…”

“…9 (right), solutions form a heteroclinic connection in u x leading to a chevron type solution in (x, u), characterised by the expected straight limbs of slope ±v * connected by a region of large curvature over a small length-scale ≈ 1/ √ P. The phase-plane clearly shows the evolution of the small amplitude periodic solution to the near heteroclinic orbit. 23 We also include a series of calculations made with the Matlab routine bvp4c for the particular case of P = 600 and k 2 h 2 = 10000 for which 1/μ = 0.06 and it follows from the earlier analysis and formulas (80) and (82) that the saddle point occurs when z = √ 1 + u 2 = 1.0744, so that the predicted maximum value of |u | = 0.3929. In these calculations we find that the calculated maximum value of |u | = 0.3928 very close to the predicted value.…”

“…In this paper we will discuss the nature of these patterns from the perspective of dynamical systems theory, and will derive a model in which we demonstrate that sinusoidal folding patterns are consistent with (low amplitude) periodic orbits arising at bifurcation points whereas non-smooth, chevron-type behaviour, in which we see zig-zag patterns with straight limbs, can be described in terms of near heteroclinic orbits, and that there is a continuous transition between these two types of behaviour. The focus of the existing modelling work has considered the buckling or folding of a single layered imbedded in a softer matrix [3,27,28,25] and has more recently received much interested from an engineering and mathematical perceptive on localised and sequential folding [19,23,6].…”

Generalized monotonicity from global minimization in fourth-order ordinary differential equations

“…The function F : JR. ~ JR. is similar to the potential energy of an elastic spring; we will make assumptions on F below. We refer the reader to [14] for a detailed derivation of this model.…”

“…Remark 3. In [14] it was proved that as A-+ oo, solutions of (3) exhibit a form of convergence. Ifwe choose a solution U>.. for every A > 0 (note that solutions of (3) are not necessarily unique and that the set { u >..} >..>O need not be a continuum), then for any sequence An -+ oo there exists a subsequence An' such that, after an appropriate translation, uniformly on compact sets.…”