Beyond the fold: experimentally traversing limit points in nonlinear structures

Abstract

<jats:p>Recent years have seen a paradigm shift regarding the role of nonlinearities and elastic instabilities in engineering science and applied physics. Traditionally viewed as unwanted aberrations, when controlled to be reversible and well behaved, nonlinearity can enable novel functionalities, such as shape adaptation and energy harvesting. The analysis and design of novel structures that exploit nonlinearities and instabilities have, in part, been facilitated by advances in numerical continuation techniques. An experimental analogue of numerical continuation, on the other hand, has remained elusive. Traditional quasi-static experimental methods control the displacement or force at one or more load-introduction points over the test specimen. This approach fails at limit points in the control parameter, as the immediate equilibrium beyond limit points is statically unstable, causing the structure to snap to a different equilibrium. Here, we propose a quasi-static experimental path-following method that can continue along stable and unstable equilibria, and traverse limit points. In addition to controlling the displacement at the main load-introduction point, the technique relies on overall shape control of the structure using additional actuators and sensors. The proposed experimental method enables extended testing of the emerging class of structures that exploit nonlinearities and instabilities for novel functionality.</jats:p>

Experimental path-following of equilibria using Newton’s method. Part I: Theory, modelling, experiments

“…However, as previously established through virtual testing, accurate results can nonetheless be achieved with greater values of the convergence tolerance with the benefit of shorter experimentation time. To conclude, we note that the experimental results correlate better with the numerical benchmark equilibrium curve than the previous experimental results of the 'step-scan' method [22].…”

“…9f. Compared with the wedged ends used in previous work [22], the effects of slip and friction are smaller for the pinned configuration used here.…”

“…In addition to loading the structure via the main actuation points, additional actuators and sensors are attached at so-called 'probe points'. The purpose of the probe points is to control the overall shape of the structure and stabilise otherwise unstable equilibria [22]. Furthermore, the probes are used to compute an experimental tangent stiffness matrix.…”

“…The geometric and material properties of the arch are as presented in Section 3. Test pieces were laser cut from sheets of arcylic; the mechanical properties were found from coupon test results [22] The probes were attached to the arch using 2 mm diameter pins through loops in the arch, as shown in Fig. 9d.…”

“…The present authors recently presented an experimental path-following technique [22] based on the concept of 'shape control' [23]. A shallow arch was loaded via rigid loading (displacement control) at the mid-span (main actuation point), while the overall shape of the structure was controlled via two coupled probe points at the quarter-and three-quarter span.…”