A targeted review on large deformations of planar elastic beams: extensibility, distributed loads, buckling and post-buckling

In this paper, we give a targeted review of the state of the art in the study of planar elastic beams in large deformations, also in the presence of geometric nonlinearities. The main scope of this work is to present the different methods of analysis available for describing the possible equilibrium forms and the motions of elastic beams. For the sake of completeness, we start by giving an overview of the nonlinear theories introduced for approaching this argument and then we account for the variational principles and deformation energies introduced for modelling beams undergoing large deformations and displacements. We then consider different kinds of loads treated in the literature and the corresponding induced beam deformations. We conclude by accounting for the available analysis for stability and some considerations about problems where live loads are applied, as well as by describing some relevant numerical methods of use in the applications we have in mind. The selection criterion for the reviewed papers is dictated by the need to study large deformations and the dynamics of pantographic sheets. (Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium. Proc R Soc A 2016; 472(2185): 20150790), dell’Isola et al. (Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence. Z Angew Math Phys 2015; 66(6): 3473–3498), Turco et al. (Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Z Angew Math Phys 2016; 67(4): 1–28)].

Download Full-text

Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models

Zeitschrift für angewandte Mathematik und Physik ◽

10.1007/s00033-016-0681-8 ◽

2016 ◽

Vol 67 (4) ◽

Cited By ~ 133

Author(s):

Emilio Turco ◽

Francesco dell’Isola ◽

Antonio Cazzani ◽

Nicola Luigi Rizzi

Keyword(s):

Discrete Model ◽

Continuum Models ◽

Numerical Comparison ◽

Pantographic Structures ◽

Second Gradient ◽

Second Gradient Continuum

Download Full-text

Hydromechanical Modelling of an Initial Boundary Value Problem: Studies of Non-uniqueness with a Second Gradient Continuum

Springer Series in Geomechanics and Geoengineering - Bifurcation and Degradation of Geomaterials in the New Millennium ◽

10.1007/978-3-319-13506-9_8 ◽

2014 ◽

pp. 47-52

Author(s):

F. Marinelli ◽

Y. Sieffert ◽

R. Chambon

Keyword(s):

Boundary Value Problem ◽

Boundary Value ◽

Initial Boundary ◽

Initial Boundary Value Problem ◽

Second Gradient ◽

Initial Boundary Value ◽

Second Gradient Continuum

Download Full-text

Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2015.0790 ◽

2016 ◽

Vol 472 (2185) ◽

pp. 20150790 ◽

Cited By ~ 177

Author(s):

F. dell’Isola ◽

I. Giorgio ◽

M. Pawlikowski ◽

N. L. Rizzi

Keyword(s):

Large Deformations ◽

Numerical Solutions ◽

Three Dimensional ◽

Beam Theory ◽

Beam Model ◽

Computationally Efficient ◽

Nonlinear Beam ◽

Pantographic Structures ◽

Diffusion Of Technology ◽

Second Gradient

The aim of this paper is to find a computationally efficient and predictive model for the class of systems that we call ‘pantographic structures’. The interest in these materials was increased by the possibilities opened by the diffusion of technology of three-dimensional printing. They can be regarded, once choosing a suitable length scale, as families of beams (also called fibres) interconnected to each other by pivots and undergoing large displacements and large deformations. There are, however, relatively few ‘ready-to-use’ results in the literature of nonlinear beam theory. In this paper, we consider a discrete spring model for extensible beams and propose a heuristic homogenization technique of the kind first used by Piola to formulate a continuum fully nonlinear beam model. The homogenized energy which we obtain has some peculiar and interesting features which we start to describe by solving numerically some exemplary deformation problems. Furthermore, we consider pantographic structures, find the corresponding homogenized second gradient deformation energies and study some planar problems. Numerical solutions for these two-dimensional problems are obtained via minimization of energy and are compared with some experimental measurements, in which elongation phenomena cannot be neglected.

Download Full-text

Cellular buckling from mode interaction in I-beams under uniform bending

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2011.0400 ◽

2011 ◽

Vol 468 (2137) ◽

pp. 245-268 ◽

Cited By ~ 36

Author(s):

M. Ahmer Wadee ◽

Leroy Gardner

Keyword(s):

Variational Principles ◽

Structural Strength ◽

Local Buckling ◽

Numerical Continuation ◽

Mode Interaction ◽

Lateral Torsional Buckling ◽

Post Buckling ◽

Strength And Stiffness ◽

First Time ◽

Differential And Integral Equations

Beams made from thin-walled elements, while very efficient in terms of the structural strength and stiffness to weight ratios, can be susceptible to highly complex instability phenomena. A nonlinear analytical formulation based on variational principles for the ubiquitous I-beam with thin flanges under uniform bending is presented. The resulting system of differential and integral equations are solved using numerical continuation techniques such that the response far into the post-buckling range can be portrayed. The interaction between global lateral-torsional buckling of the beam and local buckling of the flange plate is found to oblige the buckling deformation to localize initially at the beam midspan with subsequent cellular buckling (snaking) being predicted theoretically for the first time. Solutions from the model compare very favourably with a series of classic experiments and some newly conducted tests which also exhibit the predicted sequence of localized followed by cellular buckling.

Download Full-text

Bias extension test on an unbalanced woven composite reinforcement: Experiments and modeling via a second-gradient continuum approach

Journal of Composite Materials ◽

10.1177/0021998316643577 ◽

2016 ◽

Vol 51 (2) ◽

pp. 153-170 ◽

Cited By ~ 15

Author(s):

Gabriele Barbagallo ◽

Angela Madeo ◽

Ismael Azehaf ◽

Ivan Giorgio ◽

Fabrice Morestin ◽

...

Keyword(s):

Constitutive Expression ◽

Woven Fabrics ◽

Woven Composite ◽

Plane Shear ◽

Second Gradient ◽

Proposed Model ◽

Bias Extension Test ◽

Bias Extension ◽

Deformation Patterns ◽

Second Gradient Continuum

The classical continuum models used for the woven fabrics do not fully describe the whole set of phenomena that occur during the testing of those materials. This incompleteness is partially due to the absence of energy terms related to some microstructural properties of the fabric and, in particular, to the bending stiffness of the yarns. To account for the most fundamental microstructure-related deformation mechanisms occurring in unbalanced interlocks, a second-gradient, hyperelastic, initially orthotropic continuum model is proposed. A constitutive expression for the strain energy density is introduced to account for (a) in-plane shear deformations, (b) highly different bending stiffnesses in the warp and weft directions, and (c) fictive elongations in the warp and weft directions which eventually describe the relative sliding of the yarns. Numerical simulations which are able to reproduce the experimental behavior of unbalanced carbon interlocks subjected to a bias extension test are presented. In particular, the proposed model captures the macroscopic asymmetric S-shaped deformation of the specimen, as well as the main features of the associated deformation patterns of the yarns at the mesoscopic scale.

Download Full-text

Hydromechanical modeling of an initial boundary value problem: Studies of non-uniqueness with a second gradient continuum

International Journal of Solids and Structures ◽

10.1016/j.ijsolstr.2014.10.012 ◽

2015 ◽

Vol 54 ◽

pp. 238-257 ◽

Cited By ~ 4

Author(s):

F. Marinelli ◽

Y. Sieffert ◽

R. Chambon

Keyword(s):

Boundary Value Problem ◽

Boundary Value ◽

Initial Boundary ◽

Initial Boundary Value Problem ◽

Second Gradient ◽

Initial Boundary Value ◽

Second Gradient Continuum

Download Full-text

Pantographing Self-Adaptive Gap Elements

Tire Science and Technology ◽

10.2346/1.2150993 ◽

1985 ◽

Vol 13 (3) ◽

pp. 154-182

Author(s):

J. Padovan ◽

R. Moscarello ◽

J. Stafford ◽

F. Tabaddor

Keyword(s):

Large Deformations ◽

Benchmark Problems ◽

Inelastic Behavior ◽

Time Stepping ◽

Type Contact ◽

Deformation Kinematics ◽

Post Buckling ◽

Element Type ◽

Stiffness Characteristics ◽

Self Adaptive

Abstract A pantographing self-adaptive gap element type contact strategy is presented. Due to the manner of its formulation, it can handle large deformations in the contact zone, contact initiation in a structure that has either positive or indefinite stiffness characteristics, kinematic and material nonlinearities, as well as self-adaptive load (time) stepping. Contact in pre- and post-buckling structures can be treated in this context. Several illustrative benchmark problems are given. These include coming into contact with a fiat surface, and involve large deformation kinematics and inelastic behavior as well as pre- and post-buckling stiffness characteristics.

Download Full-text