Equilibria of a clamped Euler beam (Elastica) with distributed load: Large deformations
28 Apr 2017-Mathematical Models and Methods in Applied Sciences (World Scientific Publishing Company)-Vol. 27, Iss: 08, pp 1391-1421
TL;DR: In this paper, the authors characterized the properties of the minimizers of total energy, determine the corresponding Euler-Lagrange conditions and prove, by means of direct methods of calculus of variations, the existence of curled local minimizers.
Abstract: We present some novel equilibrium shapes of a clamped Euler beam (Elastica from now on) under uniformly distributed dead load orthogonal to the straight reference configuration. We characterize the properties of the minimizers of total energy, determine the corresponding Euler–Lagrange conditions and prove, by means of direct methods of calculus of variations, the existence of curled local minimizers. Moreover, we prove some sufficient conditions for stability and instability of solutions of the Euler–Lagrange, that can be applied to numerically found curled shapes.
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TL;DR: In this paper, the global mean Green operator for the entire embedded fiber network is obtained from one of infinite planar alignments of infinite fibers, which the network can be seen as an interpenetrated set of, with the fiber interactions being fully accounted for in the alignments.
Abstract: Composites comprising included phases in a continuous matrix constitute a huge class of meta-materials, whose effective properties, whether they be mechanical, physical or coupled, can be selectively optimized by using appropriate phase arrangements and architectures. An important subclass is represented by “network-reinforced matrices,” say those materials in which one or more of the embedded phases are co-continuous with the matrix in one or more directions. In this article, we present a method to study effective properties of simple such structures from which more complex ones can be accessible. Effective properties are shown, in the framework of linear elasticity, estimable by using the global mean Green operator for the entire embedded fiber network which is by definition through sample spanning. This network operator is obtained from one of infinite planar alignments of infinite fibers, which the network can be seen as an interpenetrated set of, with the fiber interactions being fully accounted for in the alignments. The mean operator of such alignments is given in exact closed form for isotropic elastic-like or dielectric-like matrices. We first exemplify how these operators relevantly provide, from classic homogenization frameworks, effective properties in the case of 1D fiber bundles embedded in an isotropic elastic-like medium. It is also shown that using infinite patterns with fully interacting elements over their whole influence range at any element concentration suppresses the dilute approximation limit of these frameworks. We finally present a construction method for a global operator of fiber networks described as interpenetrated such bundles.
60 citations
TL;DR: It is proved that, also when shear deformation effects are of relevance, the enriched, yet simple, model and numerical computation scheme herein proposed can be profitably used for efficient structural analyses of non-linear mechanical systems in rather nonstandard situations.
Abstract: Among the most studied models in mathematical physics, Timoshenko beam is outstanding for its importance in technological applications. Therefore it has been extensively studied and many discretizations have been proposed to allow its use in the most disparate contexts. However, it seems to us that available discretization schemes present some drawbacks when considering large deformation regimes. We believe these drawbacks to be mainly related to the fact that they are formulated without keeping in mind the mechanical phenomena for describing which Timoshenko continuum model has been proposed. Therefore, aiming to analyze the deformation of complex plane frames and arches in elastic large displacements and deformation regimes, a novel intrinsically discrete Lagrangian model is here introduced whose phenomenological application range is similar to that for which Timoshenko beam has been conceived. While being largely inspired by the ideas outlined by Hencky in his renowned doctoral dissertation, the presented approach overcomes some specific limitations concerning the stretch and shear deformation effects. The proposed model is applied to get the solutions for some relevant benchmark tests, both in the case of arch and frame structures. It is proved that, also when shear deformation effects are of relevance, the enriched, yet simple, model and numerical computation scheme herein proposed can be profitably used for efficient structural analyses of non-linear mechanical systems in rather nonstandard situations.
59 citations
TL;DR: A novel directly discrete three-dimensional beam model is presented and discussed, in the framework of geometrically nonlinear analysis, which presents a convenient balance between accuracy and computational cost.
Abstract: Complex problems such as those concerning the mechanics of materials can be confronted only by considering numerical simulations. Analytical methods are useful to build guidelines or reference solutions but, for general cases of technical interest, they have to be solved numerically, especially in the case of large displacements and deformations. Probably continuous models arose for producing inspiring examples and stemmed from homogenization techniques. These techniques allowed for the solution of some paradigmatic examples but, in general, always require a discretization method for solving problems dictated by the applications. Therefore, and also by taking into account that computing powers are nowadays more largely available and cheap, the question arises: why not using directly a discrete model for 3D beams? In other words, it could be interesting to formulate a discrete model without using an intermediate continuum one, as this last, at the end, has to be discretized in any case. These simple considerations immediately evoke some very basic models developed many years ago when the computing powers were practically inexistent but the problem of finding simple solutions to beam deformation problem was already an emerging one. Actually, in recent years, the keynotes of Hencky and Piola attracted a renewed attention [see, one for all, the work (Turco et al. in Zeitschrift fur Angewandte Mathematik und Physik 67(4):1–28, 2016)]: generalizing their results, in the present paper, a novel directly discrete three-dimensional beam model is presented and discussed, in the framework of geometrically nonlinear analysis. Using a stepwise algorithm based essentially on Newton’s method to compute the extrapolations and on the Riks’ arc-length method to perform the corrections, we could obtain some numerical simulations showing the computational effectiveness of presented model: Indeed, it presents a convenient balance between accuracy and computational cost.
55 citations
TL;DR: In this article, a nonlinear model for the dynamics of a Kirchhoff rod in the 3D space is developed in the framework of a discrete elastic theory, which avoids the use of Euler angles for t
Abstract: A nonlinear model for the dynamics of a Kirchhoff rod in the three-dimensional space is developed in the framework of a discrete elastic theory. The formulation avoids the use of Euler angles for t...
54 citations
Cites background from "Equilibria of a clamped Euler beam ..."
...Precisely, the initial conditions are the main equilibrium configuration (see [30] for details about other possible equilibrium...
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TL;DR: This work discretizes the links according to the Hencky bar-chain model, which is an application of the lumped parameters techniques, and studies the cases of a linear and a parabolic trajectory with a polynomial time law chosen to minimize the onset of possible vibrations.
33 citations
References
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02 Mar 2010
1,709 citations
TL;DR: In this paper, a study of the linear theory of elasticity in which the potential energy-density depends on the gradient of the strain in addition to the strain was performed, and the relations connecting the stresses in the three forms and the boundary conditions in three forms were derived.
1,323 citations
Additional excerpts
...those described by second gradient energy models [25, 19]....
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TL;DR: The advantages of a systematic application of the method of virtual power are exemplified by the study of continuous media with microstructure, and the equations of motion for the general micromorphic medium are established for the first time.
Abstract: The advantages of a systematic application of the method of virtual power are exemplified by the study of continuous media with microstructure. Results on micromorphic media of order one are easily found and the equations of motion for the general micromorphic medium are established for the first time. Various interesting special cases, which have been previously considered in the literature, may be derived upon imposing some convenient constraints. The paper ends with some comparisons with other related work.
763 citations
Additional excerpts
...those described by second gradient energy models [25, 19]....
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TL;DR: In this paper, the authors derived the solution for large deflection of a cantilever beam based on the fundamental Bernoulli-Euler theorem, which states that the curvature is proportional to the bending moment.
Abstract: The solution for large deflection of a cantilever beam cannot be obtained from elementary beam theory since basic assumptions are no longer valid. Specifically the elementary theory neglects the square of the first derivative in the curvature formula and provides no correction for the shortening of the moment arm as the loaded end of the beam deflects. For large finite loads, it gives deflections greater than the length of the beam! The square of the first derivative and correction factors for the shortening of the moment arm become the major contribution to the solution of large deflection problems. The following theory, which utilizes these corrections, is in agreement with experimental observations. The derivation is based on the fundamental Bernoulli-Euler theorem, which states that the curvature is proportional to the bending moment. It is assumed also that bending does not alter the length of the beam. Considering a long, thin cantilever leaf spring, let L denote the length of beam, ∆ the horizontal component of the displacement of the loaded end of the beam, δ the Corresponding vertical displacement, P the concentrated vertical load at the free end, B the flexural rigidity, that is B =EI, when cross-sectional dimensions are of the same order of magnitude, and B =EI / (I ν2) for 'wide' beams, where ν is the Poisson ratio. The exact expression for the curvature of the elastic line may be stated conveniently in terms of arc length and slope angle denoted by s and φ, respectively, so that if x is the horizontal coordinate measured from the fixed end of the beam, the product of B and the curvature of the beam equals the bending moment M:
566 citations
"Equilibria of a clamped Euler beam ..." refers background in this paper
...In the recent past, however, the awareness of the importance of large deformations problems in structural mechanics came back in both theoretical [12, 16, 17, 1] and computational [22] directions, and this importance will probably increase whether further substantial progresses will be achieved, in particular since large deformations play a relevant role in topical research lines as the design of metamaterials [8, 24, 9, 27]....
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TL;DR: In this article, a third gradient theory has been proposed to describe the homogenized energy associated with a microscopic structure using pantographic-type structures, where the deformation energies involve combinations of nodal displacements in the form of second-order or third-order finite differences.
Abstract: Until now, no third gradient theory has been proposed to describe the homogenized energy associated with a microscopic structure. In this paper, we prove that this is possible using pantographic-type structures. Their deformation energies involve combinations of nodal displacements havin the form of second-order or third-order finite differences. We establish the Gamma convergence of these energies to second and third gradient functionals. Some mechanical examples are provided so as to illustrate the special features of these homogenized models.
441 citations
"Equilibria of a clamped Euler beam ..." refers background in this paper
...Exactly as in many other conceivable metamaterials, pantographic ones base their exotic behavior on the particular geometrical and mechanical micro-structure of beam lattices [2] and on the deformation energy localization allowed by the onset of large deformations in portion of beams located in some specific areas of the lattice structure....
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