A Note on Reduced Strain Gradient Elasticity

Eremeyev, Victor A.; Dell’Isola, Francesco

doi:10.1007/978-3-319-72440-9_15

Cited by 20 publications

(12 citation statements)

References 47 publications

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“…Unlike to the general framework of the strain-gradient elasticity given by Toupin-Mindlin, the model of pantographic beam lattices relates to a strain energy density which depends on functions having different differential properties in different spatial directions, see [22][23][24]28]. We say that such a model belongs to reduced strain-gradient elasticity [29] and we claim that the theory of anisotropic Sobolev spaces [30] finds an interesting application in the study of pantographic structures.…”

Section: Introductionmentioning

confidence: 93%

On existence and uniqueness of weak solutions for linear pantographic beam lattices models

Eremeyev

Alzahrani

Cazzani

et al. 2019

Continuum Mech. Thermodyn.

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In this paper, we discuss well-posedness of the boundary-value problems arising in some “gradient-incomplete” strain-gradient elasticity models, which appear in the study of homogenized models for a large class of metamaterials whose microstructures can be regarded as beam lattices constrained with internal pivots. We use the attribute “gradient-incomplete” strain-gradient elasticity for a model in which the considered strain energy density depends on displacements and only on some specific partial derivatives among those constituting displacements first and second gradients. So, unlike to the models of strain-gradient elasticity considered up-to-now, the strain energy density which we consider here is in a sense degenerated, since it does not contain the full set of second derivatives of the displacement field. Such mathematical problem was motivated by a recently introduced new class of metamaterials (whose microstructure is constituted by the so-called pantographic beam lattices) and by woven fabrics. Indeed, as from the physical point of view such materials are strongly anisotropic, it is not surprising that the mathematical models to be introduced must reflect such property also by considering an expression for deformation energy involving only some among the higher partial derivatives of displacement fields. As a consequence, the differential operators considered here, in the framework of introduced models, are neither elliptic nor strong elliptic as, in general, they belong to the class so-called hypoelliptic operators. Following (Eremeyev et al. in J Elast 132:175–196, 2018. 10.1007/s10659-017-9660-3) we present well-posedness results in the case of the boundary-value problems for small (linearized) spatial deformations of pantographic sheets, i.e., 2D continua, when deforming in 3D space. In order to prove the existence and uniqueness of weak solutions, we introduce a class of subsets of anisotropic Sobolev’s space defined as the energy space E relative to specifically assigned boundary conditions. As introduced by Sergey M. Nikolskii, an anisotropic Sobolev space consists of functions having different differential properties in different coordinate directions.

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Section: Introductionmentioning

confidence: 93%

On existence and uniqueness of weak solutions for linear pantographic beam lattices models

Eremeyev

Alzahrani

Cazzani

et al. 2019

Continuum Mech. Thermodyn.

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“…1a and an example of its practical realization in Fig. 1b and c, has been first introduced in the context of homogenized generalized media in [45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60]. In order to achieve the associated homogenized macro-model, it is required to examine a structure consisting of n pantographic elements and to analyze the behavior of the limit material when n tends to infinity.…”

Section: The Pantographic Microstructurementioning

confidence: 99%

“…Longly, it has been assumed that the second gradient energetic terms were always irrelevant [16]. This assumption was demonstrated to be inadequate at the very beginning of the modern continuum mechanics by Gabrio Piola [12,13], albeit some scholars have longly ignored the opinion of Gabrio Piola in this context while praising some of his results [51].…”

Section: The Pantographic Microstructurementioning

confidence: 99%

Contact interactions in complex fibrous metamaterials

Spagnuolo

Cazzani

2021

Continuum Mech. Thermodyn.

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In this work, an extension of the strain energy for fibrous metamaterials composed of two families of parallel fibers lying on parallel planes and joined by connective elements is proposed. The suggested extension concerns the possibility that the constituent fibers come into contact and eventually scroll one with respect to the other with consequent dissipation due to friction. The fibers interact with each other in at least three different ways: indirectly, through microstructural connections that could allow a relative sliding between the two families of fibers; directly, as the fibers of a family can touch each other and can scroll introducing dissipation. From a mathematical point of view, these effects are modeled first by introducing two placement fields for the two fiber families and adding a coupling term to the strain energy and secondly by adding two other terms that take into account the interdistance between the parallel fibers and the Rayleigh dissipation potential (to account for friction).

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“…( 3) and ( 4) give 1D and 3D scalar examples of (2). Other physical models with nonsymmetric appearance of second-order derivatives are discussed in [27]. We call the model based on (2) gradient-incomplete as W does not contain all second-order derivatives.…”

Section: Strain Energy Densitymentioning

confidence: 99%

Weak Solutions within the Gradient-Incomplete Strain-Gradient Elasticity

Eremeyev

Dell’Isola

2020

Lobachevskii J Math

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In this paper we consider existence and uniqueness of the three-dimensional static boundary-value problems in the framework of so-called gradient-incomplete strain-gradient elasticity. We call the straingradient elasticity model gradient-incomplete such model where the considered strain energy density depends on displacements and only on some specific partial derivatives of displacements of first-and second-order. Such models appear as a result of homogenization of pantographic beam lattices and in some physical models. Using anisotropic Sobolev spaces we analyze the mathematical properties of weak solutions. Null-energy solutions are discussed.

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A Note on Reduced Strain Gradient Elasticity

Cited by 20 publications

References 47 publications

On existence and uniqueness of weak solutions for linear pantographic beam lattices models

On existence and uniqueness of weak solutions for linear pantographic beam lattices models

Contact interactions in complex fibrous metamaterials

Weak Solutions within the Gradient-Incomplete Strain-Gradient Elasticity

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