A computational approach to obtain nonlinearly elastic constitutive relations of strips modeled as a special Cosserat rod

Abstract

This study presents a computational approach to obtain nonlinearly elastic constitutive relations of strip/ribbon-like structures modeled as a special Cosserat rod. Starting with the description of strips as a general Cosserat plate, the strip is first subjected to a strain field which is uniform along its length. The Helical Cauchy–Born rule is used to impose this uniform strain field which deforms the strip into a six-parameter family of helical configurations — the six parameters here correspond to the six strain measures of rod theory. Two vector variables are introduced to model the position of the deformed centerline of the strip’s cross-section and to model orientation of thickness lines along the strip’s width. The minimization of the strip’s plate energy under the constraint of the aforementioned uniformity in strain field reduces the partial differential equations of plate theory from the entire mid-plane of the strip to just a system of nonlinear ordinary differential equations along the strip’s width line for the above mentioned two vector variables. A nonlinear finite element formulation is further presented to solve this set of ordinary differential equations. This, in turn, yields the strip’s stored energy per unit length as well as the induced internal force, moment and stiffnesses of the strip for every prescribed set of six strain measures of rod theory. The proposed scheme is used to study uniform bending, twisting and shearing of a strip. For the case of uniform twisting and shearing, the strip is also seen to buckle along its width into a more complex configuration which are accurately captured by the presented scheme. We demonstrate that the presented scheme is more general and accurate than the existing schemes available in the literature.