Micropolar Shells as Two-dimensional Generalized Continua Models

Altenbach, Holm; Eremeyev, Victor A.; Lébedev, Leonid P.

doi:10.1007/978-3-642-19219-7_2

Cited by 23 publications

(21 citation statements)

References 44 publications

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“…give the orientation of the microstructure in the deformed configuration as an e↵ect of the rotations 1 R and 2 R. We remark that the proposed continuum di↵ers from a typical micropolar (or Cosserat-type) continuum as, e.g., proposed in [39,40], because of the presence of two distinct rotations.…”

Section: A 2d Continuum With (Latent) Internal Microstructurementioning

confidence: 70%

Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis

Giorgio¹,

2017

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To cite this version:I. Giorgio, N. Rizzi, E. Turco. Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2017The, , 473 (2207The, ), 10.1098The, /rspa.2017 Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis Giorgio, I. and Rizzi, N. L. and Turco, E.Abstract A nonlinear 2D continuum with a latent internal structure is introduced as a coarse model of a plane network of beams which, in turn, is assumed as a model of a pantographic structure made up by two families of equispaced beams, superimposed and connected by pivots. The deformation measures of the beams of the network and that of the 2D body are introduced and the former are expressed in terms of the latter by making some kinematical assumptions. The expressions for the strain and kinetic energy densities of the network are then introduced and given in terms of the kinematic quantities of the 2D continuum. In order to account for the modelling abilities of the 2D continuum in the linear range, the eigenmode and eigenfrequencies of a given specimen are determined. The buckling and postbuckling behaviour of the same specimen, subjected to two di↵erent loading conditions are analysed as tests in the nonlinear range. The problems have been solved numerically by means of COMSOL Multiphysics FE software.

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Section: A 2d Continuum With (Latent) Internal Microstructurementioning

confidence: 70%

Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis

Giorgio¹,

2017

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“…This kind of structure is inspired by several natural examples; indeed, some biological tissues present fibers that can be modeled in a similar way (see, e.g., [Melnik and Goriely 2013;Federico and Grillo 2012;Grillo et al 2015] for some possible applications). Their interest was initially related to the possibility of proving the existence of purely second gradient continua [Alibert et al 2003], but subsequently their practical applicability has been proven for woven fabrics and some "ad hoc" designed metamaterials (see, e.g., [Del Vescovo and Giorgio 2014;Bîrsan et al 2012;Altenbach et al 2011;dell'Isola et al 2015b;Eremeyev 2016]). On the other hand, some interesting theoretical results were presented in [Chesnais et al 2015;Boutin et al 2010;Boutin and Soubestre 2011].…”

Section: Introductionmentioning

confidence: 99%

Linear pantographic sheets: Asymptotic micro-macro models identification

Boutin

Dell’Isola

Giorgio

et al. 2017

Math. Mech. Compl. Sys.

166

120

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In this paper we consider linear pantographic sheets, which in their natural configuration are constituted by two orthogonal arrays of straight fibers interconnected by internal pivots. We introduce a continuous model by means of a micro-macro identification procedure based on the asymptotic homogenization method of discrete media. The rescaling of the mechanical properties and of the deformation measures is calibrated in order to comply with the specific kinematics imposed by the quasi-inextensibility of the fibers together with the large pantographic deformability. The obtained high-order continuum model shows interesting and exotic features related to its extreme anisotropy and also to the subcoercivity of its deformation energy. Some initial numerical simulations are presented, showing that the model can account for experimental uncommon phenomena occurring in pantographic sheets. The paper focuses on the precise analysis and the understanding of the effective behavior based on a well-calibration of the extension and bending phenomena arising at the local scale. In an upcoming work, the analysis will be extended to oblique arrays, some analytical solutions to proposed equations and some further applications.

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“…In order to improve the results of the classical theory of elasticity various theories of generalized continua have been developed [3][4][5]. The mechanics of generalized continua has long history of development.…”

Section: Introductionmentioning

confidence: 99%

Micropolar curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models

Zozulya

2017

Curved and Layered Structures

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New models for micropolar plane curved rods have been developed. 2-D theory is developed from general 2-D equations of linear micropolar elasticity using a special curvilinear system of coordinates related to the middle line of the rod and special hypothesis based on assumptions that take into account the fact that the rod is thin. High order theory is based on the expansion of the equations of the theory of elasticity into Fourier series in terms of Legendre polynomials. First stress and strain tensors, vectors of displacements and rotation and body forces have been expanded into Fourier series in terms of Legendre polynomials with respect to a thickness coordinate. Thereby all equations of elasticity including Hooke's law have been transformed to the corresponding equations for Fourier coefficients. Then in the same way as in the theory of elasticity, system of differential equations in term of displacements and boundary conditions for Fourier coefficients have been obtained. The Timoshenko's and EulerBernoulli theories are based on the classical hypothesis and 2-D equations of linear micropolar elasticity in a special curvilinear system. The obtained equations can be used to calculate stress-strain and to model thin walled structures in macro, micro and nano scale when taking into account micropolar couple stress and rotation effects.

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Micropolar Shells as Two-dimensional Generalized Continua Models

Abstract: To cite this version:Holm Altenbach, Victor Eremeyev, Leonid Lebedev. Micropolar Shells as Two-dimensional Generalized Continua Models.

Cited by 23 publications

References 44 publications

Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis

Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis

Linear pantographic sheets: Asymptotic micro-macro models identification

Micropolar curved rods. 2-D, high order, Timoshenko’s and Euler-Bernoulli models

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