Homogenization of periodic graph-based elastic structures

Abdoul-Anziz, Houssam; Seppecher, Pierre

doi:10.5802/jep.70

Cited by 24 publications

(43 citation statements)

References 40 publications

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“…For instance, we provide in Fig. 6 the value of C 1,1 111111 as function of the number of periods. It is observed that the component C 1,1 111111 decreases with the number of periods while the quantity N 2 C 1,1 111111 is independent.…”

Section: Basic Results With Periodic Boundary Conditionsmentioning

confidence: 99%

“…More recently, it has been demonstrated that the effective elastic energy of lattice materials having soft modes of deformation can be of generalized type [3]. Some general results have been obtained in [2,1] combining asymptotic analysis and Γ-convergence method. However even if a method to determine the higher-order parameters was proposed in this last reference it fails when the first order elasticity is not degenerated.…”

Section: Introductionmentioning

confidence: 99%

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Strain-gradient homogenization: A bridge between the asymptotic expansion and quadratic boundary condition methods

Monchiet

Auffray

Yvonnet

2020

Mechanics of Materials

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In this paper we deal with the determination of the strain gradient elasticity coefficients of composite material in the framework of the homogenization methods. Particularly we aim to eliminate the persistence of the strain gradient effects when the method based on quadratic boundary conditions is considered. Such type of boundary conditions is often used to determine the macroscopic strain gradient elastic coefficients but leads to contradictory results, particularly when a RVE is made up of a homogeneous material. The resulting macroscopic equivalent material exhibits strain gradient effects while it should be expected of Cauchy type. The present contribution is to provides new relationship to correct the approach based on the quadratic boundary condition. To this purpose, we start from the asymptotic homogenization approach, we establish a connection with the method based on quadratic boundary conditions and we highlight the correction required to eliminate the persistence of the strain gradient effects. An application to a composite with fibers is provided to illustrate the method.

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Section: Basic Results With Periodic Boundary Conditionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Strain-gradient homogenization: A bridge between the asymptotic expansion and quadratic boundary condition methods

Monchiet

Auffray

Yvonnet

2020

Mechanics of Materials

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“…assuming that, inside each cell, only pairs of nodes (1, 2), (1,3), (1,4), (1,5), (2,5), (3,4), (4,6), (5,6) are interacting, assuming that nodes 4 and 5 of each cell are respectively interacting with nodes 3 and 2 of the next cell following εt 1 (with t 1 = (1, 0, 0)) and assuming moreover that node 6 of each cell is interacting with both nodes 3 and 2 of the next cell following εt 1 , we get a pantographic beam as shown in Figure 6. Assuming moreover that node 1 of each cell is interacting with node 1 of the next cells following εt 2 and εt 3 (with t 2 = (0, √ 3, −1) and t 3 = (0, √ 3, 1)) we get a 3D structure as shown in Figure 3.…”

Section: Pantographic Structuresmentioning

confidence: 99%

Homogenization of frame lattices leading to second gradient models coupling classical strain and strain-gradient terms

Abdoul-Anziz¹,

Seppecher

Bellis

2019

Mathematics and Mechanics of Solids

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We determine in the framework of static linear elasticity the homogenized behavior of three-dimensional periodic structures made of welded elastic bars. It has been shown that such structures can be modeled as discrete systems of nodes linked by extensional, flexural/torsional interactions corresponding to frame lattices and that the corresponding homogenized models can be strain-gradient models, i.e., models whose effective elastic energy involves components of the first and the second gradients of the displacement field. However, in the existing models, there is no coupling between the classical strain and the strain-gradient terms in the expression of the effective energy. In the present article, under some assumptions on the positions of the nodes of the unit cell, we show that classical strain and strain-gradient strain terms can be coupled. In order to illustrate this coupling we compute the homogenized energy of a particular structure which we call asymmetrical pantographic structure.Strain-gradient models describe materials whose energy depends not only on the strain but also on its gradient. Equivalently, elastic energy density is a function of the strain e(u), that is the symmetric part of the gradient of the displacement field u, and of the strain-gradient ∇e(u). Clearly the components of the strain-gradient can be Prepared using sagej.cls

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“…Actually generalized continua can be proven to be Γ-limits of systems described at micro-level as first gradient continua. [2,3,91,92] On the other hand, it is possible to regard a beam as an energetic boundary curve on a two-dimensional manifold in a three-dimensional space and employ the frameworks developed in [73,77,78] and even account for higher gradients elaborated in [75]. The advantage of this approach is that the fictitious bulk material can regularize the behavior of the beam and this allows to analyze buckling in a computational framework.…”

Section: F I G U R E 1 Some Illustrative Examples Of Pantographic Strmentioning

confidence: 99%

“…To make this discussion complete, and make happy those who consider continuum models more important, we can recall here that recently some Γ‐convergence results have been proven that Hencky's model is a fully reliable approximation of continuous inextensible and extensible Euler–Bernoulli beams. Actually generalized continua can be proven to be Γ‐limits of systems described at micro‐level as first gradient continua . On the other hand, it is possible to regard a beam as an energetic boundary curve on a two‐dimensional manifold in a three‐dimensional space and employ the frameworks developed in [] and even account for higher gradients elaborated in [].…”

Section: Introductionmentioning

confidence: 99%

Nonlinear dynamics of uniformly loaded Elastica: Experimental and numerical evidence of motion around curled stable equilibrium configurations

et al. 2019

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It has been numerically observed and mathematically proven that for a clamped Euler's Elastica, which is uniformly loaded, there exist, in large deformations, some ‘undocumented’ equilibrium configurations which resemble a curled pending wire. Even if Elastica is one of the most studied model in mathematical physics, we could not find in the literature any description of an equilibrium like the one whose existence was forecast theoretically in [36]. In this paper, we prove that this kind of equilibrium configurations can be actually observed experimentally when using ‘soft’ beams. We mean with soft beams: Elasticae whose ratio between the applied load intensity and the bending stiffness is large enough. Moreover, we prove experimentally that such equilibrium configurations are actually stable, by observing their oscillations around the considered nonstandard equilibrium configuration. To describe theoretically such oscillations we consider, instead of a ‘soft’ Elastica model, directly a Hencky‐type discrete model, i.e. a ‘masses‐springs’ finite dimensional Lagrangian model. In this way we formulate, avoiding the use of an intermediate continuum model, a model for which numerical simulations can be performed without the introduction of any further discretization. In this way, we can also predict quantitatively the motions of soft beams, in the regime of large displacements and deformations. Postponing to future investigations more careful quantitative measurements, we report here that it was possible to get a rather promising qualitative agreement between observed motions and predictive numerical simulations.

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Homogenization of periodic graph-based elastic structures

Cited by 24 publications

References 40 publications

Strain-gradient homogenization: A bridge between the asymptotic expansion and quadratic boundary condition methods

Strain-gradient homogenization: A bridge between the asymptotic expansion and quadratic boundary condition methods

Homogenization of frame lattices leading to second gradient models coupling classical strain and strain-gradient terms

Nonlinear dynamics of uniformly loaded Elastica: Experimental and numerical evidence of motion around curled stable equilibrium configurations

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