Structural localization phenomena and the dynamical phase-space analogy

Abstract

The localization of buckle patterns in elastic structures is reviewed from three complementary viewpoints: ( a ) from a modal perspective, ( b ) from a formulation which allows the amplitude to modulate in an asymptotically defined ‘slow’ space and ( c ) from a dynamical analogy in phase space suggested by the form of the underlying differential equation. A simple strut on an (asymmetric) nonlinear foundation provides a typical illustrative example. The three approaches emphasize different features of the localization phenomenon. The modal view illustrates the distinctive effects of boundary conditions, the modulated approach generates a convenient second-order differential equation in the amplitude function and the dynamical phase-space analogy suggests a useful interpretation of localization as a homoclinic connection. Comparisons are also made with nonlinear numerical solutions. As the strut length approaches infinity it is shown that the fully localized solution represents the unstable post-buckled state with the lowest energy, allowing evaluation of the minimum energy barrier relevant to dynamical impact studies. Attention is drawn to the possibility of spatial chaos, becoming manifest as a randomly spaced sequence of localizations caused by a regular sinusoidal spatial imperfection.

Effects of imperfections on localized bulging in inflated membrane tubes

“…For the Ogden material, we have u cr = −3.2329, g cr = −1.3369 and so the localized solution (2.18) represents a bulge (since y 1 > 0). We observe that for subcritical bifurcations into sinusoidal patterns, a weakly nonlinear near-critical analysis that takes into account amplitude modulation would lead to an equation similar to (2.17), but with the quadratic term replaced by a cubic term (see [43] or [44]). As a result, the sech 2 on the r.h.s.…”

Biomechanical Modelling of Colorectal Crypt Budding and Fission

“…The linearisation of this simple problem is considered in, for example, (Brush and Almroth, 1975) for a simply supported beam. Variations on the model include considering that the beam is under load rather than compression (Timoshenko and Lessells, 1928), or that the springs are nonlinear in response (Hunt et al, 1989). We will then extend this model to include the extensibility of the tissue.…”

Single-molecule magnetic tweezer tests on DNA: bounds on topoisomerase relaxation

“…This form was, however, never observed experimentally in our mechanical tests on rubber and metal specimens, and large-deflection nonlinear theory shows that the preferred deformation will be a localized one as shown. This type of spatial localization has been greatly illuminated by work employing the static-dynamic analogy (Hunt et al 1989;Champneys et al 1997;Hunt 2006), where the localized form corresponds to a homoclinic orbit of an equivalent dynamical system. The equivalent dynamical system for a rod is a spinning top (with arc length of the rod corresponding to the time for the top), as observed by Kirchhoff.…”

“…This is akin to the Maxwell equal-energy criterion, which was originally conceived for stochastically driven thermodynamic processes under Brownian agitation. It has re-emerged in theoretical mechanics (Hunt et al 1989;Hunt & Neto 1993;van der Heijden et al 2002;Hunt 2006), where it is deeply associated with the development of localization in the static-dynamic analogy. In the present X a Figure 9.…”