Constrained Buckling of Variable Length Elastica: Solution by Geometrical Segmentation

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https://doi.org/10.13020/D62Q21
https://hdl.handle.net/11299/190821

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Constrained buckling

elastica

closed-form solution

asymptotics

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Constrained Buckling of Variable Length Elastica: Solution by Geometrical Segmentation

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2017-11-06

Authors

Liakou, Anna

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Emmanuel Detournay

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Liakou, Anna
liako005@umn.edu

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Abstract

The associated paper to this dataset proposes a method to analyze the post-buckling response of a planar elastica subjected to unilateral constraints. The method rests on assuming a deformed shape of the elastica that is consistent with an assumed buckling mode and given unilateral constraints, and on uniquely segmenting the elastica so that each segment is a particular realization of the same canonical problem. An asymptotic solution of the canonical problem, which is characterized by clamped-pinned boundary conditions and monotonically varying curvature, is derived using a perturbation technique. The complete solution of the constrained elastica is constructed by assembling the solution for each segment. It entails solving a nonlinear system of algebraic equations that embodies the continuity conditions between the segments and the contact constraints. The method is then applied to analyze the post-buckling response of a planar weightless elastica compressed inside two rigid frictionless horizontal walls. The length of the elastica could be either constant, or variable, but the focus of the analysis is on the response of a variable length elastica, which is gradually inserted inside the conduit. In the insertion problem, a configurational force is generated at the insertion point, which is not present in the classical problem of a constant length elastica (Bigoni et al, in Mechanics of Materials, 2015)[10]. The proposed approach is shown to lead to a simple and accurate numerical technique to simulate the constrained buckling of an elastica. The optimal sequence of equilibrium configurations of the elastica associated with a monotonic force- or displacement-control loading is deduced in accordance with the principle of minimum energy.

Description

The MATLAB codes demonstrate the example of a free-standing fold described in Section 4.3 of the related paper (under review). In particular, the MATLAB code "Example_Secondary" provides the solution for a pinned-pinned elastica of constant length, characterized by two symmetric discrete contacts between the elastica and the upper wall and a free standing fold. The MATLAB code, "fbnd_secondary," solves a system of equations for this example, using the nonlinear least-squares MatLab solver "lsqnonlin." With this solver, convergence of the solution is achieved even with a poor initial guess. The supplementary paper provides an analytical derivation of the buckling response of a constrained elastica clamped at both ends, as opposed to the response of pinned-pinned elastica analyzed in the paper.

Referenced by

Liakou, A. & Detourney, E. (2017). Constrained buckling of variable length elastica: Solution by geometrical segmentation. International Journal of Non-Linear Mechanics, 99, 204-217.
https://doi.org/10.1016/j.ijnonlinmec.2017.12.001

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Liakou, Anna. (2017). Constrained Buckling of Variable Length Elastica: Solution by Geometrical Segmentation. Retrieved from the Data Repository for the University of Minnesota (DRUM), https://doi.org/10.13020/D62Q21.

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