Exact and approximate mechanisms for pure bending of sheets

Abstract

Direct measurement of the moment-curvature response of sheets or wires up to high curvatures can aid in modeling creasing, pleating, and other forming operations. We consider theoretical issues related to the geometry of pure bending. We present a linkage design that, for homogeneous deformation of the sample, results in an exact pure bending state up to arbitrarily high curvature, by combining a cochleoidal trajectory with an angle-doubling mechanism. The full mechanism is single degree of freedom and position controlled, a desirable feature for measurement of soft materials. We also present an optimal approximation to the cochleoid using a more easily implemented circular trajectory, and compare this with an existing commercial system for fabric testing. We quantify the error of the approximate test by calculating the deformation of an Euler elastica, a structure with linear moment-curvature response. While the circular mechanism can approximate the exact boundary conditions quite well up to moderately large curvatures, the resulting curvature of the test sample still deviates significantly from homogeneous response. We also briefly discuss expectations of localization behavior in pure bending tests.

Avoiding localization instabilities in rotary pleating

“…Although we have treated the crease strength and stiffness as independent parameters, they are linked through the scoring process that weakens a pristine sheet to form the creases. To obtain parameters related to the pleat facets, one can measure the moment-curvature response, up to the localization curvature, of a sheet using a bend tester [15][16][17]. Measuring the response of the crease is, however, not a standardized process.…”

“…The inextensible rod model of Equations ( 1) can take as input any elastic moment-curvature relation M (κ) for a sheet, as measured using a suitable bend tester [15][16][17]. For this theoretical study, we use an adjustable function that mimics the observed elastic-plastic behavior of soft sheets during forward loading.…”