Large deformations induced in planar pantographic sheets by loads applied on fibers: Experimental validation of a discrete Lagrangian model

Turco, Emilio; Gołaszewski, Maciej; Cazzani, Antonio; Rizzi, Nicola Luigi

doi:10.1016/j.mechrescom.2016.07.001

Cited by 102 publications

(79 citation statements)

References 44 publications

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“…In a series of papers (see previous studies), it has been proven that, rather unexpectedly, even in presence of relatively few degrees of freedom, discrete systems show a very similar behaviour to their continuous homogenised counterparts. However, we wanted to include a rather numerous set of grains (about 900) to make the continuous identification of the whole assembly more credible.…”

Section: Numerical Simulations and Their Interpretationmentioning

confidence: 94%

A nonlinear Lagrangian particle model for grains assemblies including grain relative rotations

Turco

Dell’Isola²,

Misra

2019

Num Anal Meth Geomechanics

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Summary We formulate a discrete Lagrangian model for a set of interacting grains, which is purely elastic. The considered degrees of freedom for each grain include placement of barycenter and rotation. Further, we limit the study to the case of planar systems. A representative grain radius is introduced to express the deformation energy to be associated to relative displacements and rotations of interacting grains. We distinguish inter‐grains elongation/compression energy from inter‐grains shear and rotations energies, and we consider an exact finite kinematics in which grain rotations are independent of grain displacements. The equilibrium configurations of the grain assembly are calculated by minimization of deformation energy for selected imposed displacements and rotations at the boundaries. Behaviours of grain assemblies arranged in regular patterns, without and with defects, and similar mechanical properties are simulated. The values of shear, rotation, and compression elastic moduli are varied to investigate the shapes and thicknesses of the layers where deformation energy, relative displacement, and rotations are concentrated. It is found that these concentration bands are close to the boundaries and in correspondence of grain voids. The obtained results question the possibility of introducing a first gradient continuum models for granular media and justify the development of both numerical and theoretical methods for including frictional, plasticity, and damage phenomena in the proposed model.

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Section: Numerical Simulations and Their Interpretationmentioning

confidence: 94%

A nonlinear Lagrangian particle model for grains assemblies including grain relative rotations

Turco

Dell’Isola²,

Misra

2019

Num Anal Meth Geomechanics

Self Cite

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“…As discussed in full detail in [3] Lagrangian discrete models can be a viable option in the description of pantographic sheets: the possibility to predict their shapes in large deformation equilibrium configurations using said discrete models has been the object of the investigations presented in [4]. Referring the reader to just cited papers for a detailed discussion of the model and the algorithmic procedure we use to determine equilibrium configurations, we delineate here the main ideas on which we base our solution strategy when dealing with both elastic and rupture deformation phenomena.…”

Section: Lagrangian Discrete Model For Pantographic Structuresmentioning

confidence: 99%

Fiber rupture in sheared planar pantographic sheets: Numerical and experimental evidence

Turco

Dell’Isola

Rizzi

et al. 2016

Mechanics Research Communications

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“…More precisely: in order to find a class of materials whose deformation energy depends on both first and second gradient of placement field in [22] a microstructured (pantographic) fabric is introduced and its homogenized continuum model (which we call pantographic sheet) is determined. Various aspects of modelling of pantographic lattices are considered in [8,52,53,56,[62][63][64][65][66] where discrete and homogenized models are considered. Let us note that one of the sources of generalized continua and models of metamaterials is the homogenization of heterogeneous materials, see, e.g., [18,54,61] and reference therein.…”

Section: Introductionmentioning

confidence: 99%

Linear Pantographic Sheets: Existence and Uniqueness of Weak Solutions

Eremeyev

Dell’Isola

Boutin

et al. 2017

J Elast

129

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The well-posedness of the boundary value problems for second gradient elasticity has been studied under the assumption of strong ellipticity of the dependence on the second placement gradients (see, e.g., Chambon and Moullet in Comput. Methods Appl. Mech. Eng. 193:2771-2796 and Mareno and Healey in SIAM J. Math. Anal. 38:103-115, 2006.The study of the equilibrium of planar pantographic lattices has been approached in two different ways: in dell 'Isola et al. (Proc. R. Soc. Lond. Ser. A 472:20150, 2016) a discrete model was introduced involving extensional and rotational springs which is also valid in large deformations regimes while in Boutin et al. (Math. Mech. Complex Syst. 5:127-162, 2017) the lattice has been modelled as a set of beam elements interconnected by internal pivots, but the analysis was restricted to the linear case. In both papers a homogenized second gradient deformation energy, quadratic in the neighbourhood of non deformed configuration, is obtained via perturbative methods and the predictions obtained with the obtained continuum model are successfully compared with experiments. This energy is not strongly elliptic in its dependence on second gradients. We consider in this paper also the important particular case of pantographic lattices whose first gradient energy does not depend on shear deformation: this could be considered either a pathological case or an important exceptional case (see Stillwell et al. in Am. Math. Mon. 105:850-858, 1998 and Turro in Angew. Chem., Int. Ed. Engl. 39:2255Engl. 39: -2259Engl. 39: , 2000. In both cases we believe that such a particular case deserves some attention because of what we can understand by studying it (see Dyson in Science 200:677-678, 1978). This circumstance motivates the present paper, where we address the well-posedness of the planar linearized equilibrium problem for homogenized pantographic lattices. To do so: (i) we introduce a class of subsets of anisotropic Sobolev's space as the most suitable energy space E relative to assigned boundary conditions; (ii) we prove that the considered strain energy density is coercive and positive definite in E; (iii) we prove that the set of placements for which the strain energy is vanishing (the so-called floppy modes) must strictly include rigid motions; (iv) we determine the restrictions on displacement boundary conditions which assure existence and uniqueness of linear static problems. The presented results represent one of the first mechanical applications of the concept of Anisotropic Sobolev space, initially introduced only on the basis of purely abstract mathematical considerations.

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Large deformations induced in planar pantographic sheets by loads applied on fibers: Experimental validation of a discrete Lagrangian model

Cited by 102 publications

References 44 publications

A nonlinear Lagrangian particle model for grains assemblies including grain relative rotations

A nonlinear Lagrangian particle model for grains assemblies including grain relative rotations

Fiber rupture in sheared planar pantographic sheets: Numerical and experimental evidence

Linear Pantographic Sheets: Existence and Uniqueness of Weak Solutions

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