Browsing by Author "Liakou, Anna"
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Item Constrained Buckling of Variable Length Elastica(2017-10) Liakou, AnnaThe physical understanding of the response of slender elastic bodies restrained inside constraints under various loading and boundary conditions is of a great importance in engineering and medical applications. The research work presented in this thesis is especially concerned with the buckling response of an elastic rod (the elastica) subjected to unilateral constraints under axial compression. It seeks to address two main issues: (i) the conditions that lead to the onset of instability, and (ii) the factors that define the bifurcation diagram. Two distinct classes of problems are analyzed; (i) the classical buckling problem of a constant length elastica and (ii) the insertion buckling problem of a variable length elastica. Their main difference is the generation of a configurational force at the insertion point of the sliding sleeve in the insertion problem, which is not present in the classical problem. The thesis describes two distinct methodologies that can solve these constrained buckling problems; (1) a geometry-based method, and (2) an optimal control method. The geometry-based method is used to analyze the post-buckling response of a weightless planar elastica subjected to unilateral constraints. The method rests on assuming a deformed shape of the elastica and on uniquely segmenting the elastica consistent with a single canonical segment (clamped-pinned). An asymptotic solution of the canonical problem is then derived and the complete solution of the constrained elastica is constructed by assembling the solution for each segment. Nevertheless, the application of the optimal control method is more generic. It can be used to solve any constrained buckling problem under general boundary and loading conditions. Based on Hamiltonian mechanics, the optimality conditions, which constitute the Pontryagin’s minimum principle, involve the minimization of the Hamiltonian with respect to the control variables, the canonical equations and the transversality conditions. The main advantage of the optimal control method is the assumption of strong rather than weak variation of the involved variables, which leads to the additional Weierstrass necessary condition (“optimal” equilibrium state). Based on it, several factors such as the effect of the self-weight of the elastica and the clearance of the walls are investigated.Item Constrained Buckling of Variable Length Elastica: Solution by Geometrical Segmentation(2017-11-06) Liakou, Anna; liako005@umn.edu; Liakou, Anna; Emmanuel DetournayThe 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.Item MATLAB codes for "Fast In-Plane Dynamics of a Beam with Unilateral Constraints"(2016-06-02) Liakou, Anna; Detournay, Emmanuel; Denoël, Vincent; aliakou81@gmail.com; Liakou, AnnaA computationally efficient technique to simulate the dynamic response of a beam colliding with rigid obstacles is described in this paper. The proposed method merges three key concepts. First, a low order discretization scheme that maximizes the number of nodes of the discrete model, where impacts are detected, at the expense of the degree of continuity of the constructed displacement field. Second, the constrained problem is transformed into an unconstrained one by formulating the impact by means of a complementarity Signorini's law involving the impulse generated by the collision and the pre- and post- impact velocity linked via a coefficient of restitution. Third, Moreau's midpoint time-stepping scheme developed within the context of colliding rigid bodies, is used to advance the solution.