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

2016 ◽  
Vol 67 (4) ◽  
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

Construction of second gradient continuum models for random fibrous networks and analysis of size effects

2017 ◽  
Vol 181 ◽  
pp. 347-357 ◽  
Author(s):  
K. Berkache ◽  
S. Deogekar ◽  
I. Goda ◽  
R.C. Picu ◽  
J.-F. Ganghoffer
Keyword(s):  
Size Effects ◽  
Continuum Models ◽  
Second Gradient ◽  
Second Gradient Continuum

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

2018 ◽  
Vol 24 (1) ◽  
pp. 258-280 ◽  
Author(s):  
Mario Spagnuolo ◽  
Ugo Andreaus
Keyword(s):  
Large Deformations ◽  
Variational Principles ◽  
Selection Criterion ◽  
Numerical Comparison ◽  
Elastic Beams ◽  
Second Gradient ◽  
Post Buckling ◽  
Live Loads ◽  
Second Gradient Continuum ◽  
Math Phys

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)].

Are higher-gradient models also capable of predicting mechanical behavior in the case of wide-knit pantographic structures?

2020 ◽  
Vol 26 (1) ◽  
pp. 18-29 ◽  
Author(s):  
Mario Spagnuolo ◽  
M Erden Yildizdag ◽  
Ugo Andreaus ◽  
Antonio M Cazzani
Keyword(s):  
Continuum Model ◽  
Central Theme ◽  
Fiber Density ◽  
Pantographic Structures ◽  
Numerical Predictions ◽  
Second Gradient ◽  
Bias Extension ◽  
Second Gradient Continuum ◽  
Gradient Models ◽  
Good Agreement

The central theme of this study is to investigate a remarkable capability of a second-gradient continuum model developed for pantographic structures. The model is applied to a particular type of this metamaterial, namely the wide-knit pantograph. As this type of structure has low fiber density, the applicability of such a continuum model may be questionable. To address this uncertainty, numerical simulations are conducted to analyze the behavior of a wide-knit pantographic structure, and the predicted results are compared with those measured experimentally under bias extension testing. The results presented in this study show that the numerical predictions and experimental measurements are in good agreement; therefore, in some useful circumstances, this model is applicable for the analysis of wide-knit pantographic structures.

scholarly journals Murray’s law for discrete and continuum models of biological networks

2019 ◽  
Vol 29 (12) ◽  
pp. 2359-2376
Author(s):  
Jan Haskovec ◽  
Peter Markowich ◽  
Giulia Pilli
Keyword(s):  
Biological Networks ◽  
Discrete Model ◽  
Continuum Model ◽  
Transportation Network ◽  
Continuum Models ◽  
Scaling Relation ◽  
Murray's Law ◽  
Murray’S Law ◽  
Rectangular Mesh ◽  
The Continuum

We demonstrate the validity of Murray’s law, which represents a scaling relation for branch conductivities in a transportation network, for discrete and continuum models of biological networks. We first consider discrete networks with general metabolic coefficient and multiple branching nodes and derive a generalization of the classical 3/4-law. Next we prove an analogue of the discrete Murray’s law for the continuum system obtained in the continuum limit of the discrete model on a rectangular mesh. Finally, we consider a continuum model derived from phenomenological considerations and show the validity of the Murray’s law for its linearly stable steady states.

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

2016 ◽  
Vol 472 (2185) ◽  
pp. 20150790 ◽  
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.

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

2016 ◽  
Vol 51 (2) ◽  
pp. 153-170 ◽  
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.

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