Waves in Linear Elastic Media with Microrotations, Part 2: Isotropic Reduced Cosserat Model

Grekova, Elena F.; Kulesh, M.; Herman, Gérard C.

doi:10.1785/0120080154

Cited by 53 publications

(36 citation statements)

References 16 publications

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“…We recall some results from [12]. The compression wave for the linear isotropic elastic reduced Cosserat continuum with zero initial stress is the same as in the classical continuum (non-dispersive).…”

Section: Dispersion Relation For the Shear-rotational Perturbationsmentioning

confidence: 99%

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Small perturbations of the spherical prestressed state in a nonlinear isotropic elastic reduced Cosserat medium: Waves and instabilities

Grekova

2011

Proceedings of the International Conference Days on Diffraction 2011

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We consider small deviations from a nonlinear equilibrium in a nonlinear elastic reduced Cosserat medium (a Cosserat continuum which does not resist to the gradient of rotation). We obtain the equations for small deviations for an arbitrary nonlinear equilibrium and any type of elastic energy, which can be explored into the Taylor series near this equilibrium. Then we consider an isotropic material in the spherically symmetric stress state and show that the equations for small deviations coincide with the equations of motion for the reduced linear elastic Cosserat continuum. For a wide class of materials, strong compression leads to the instability of the medium caused by shear perturbations, and strong tension -to the instability with respect to rotational perturbations. In the stable domain, shear-rotational wave has a forbidden band of frequencies, demonstrates strong dispersion near this zone, and there is a resonant frequency corresponding to the independent rotational oscillations. In the forbidden zone localisation phenomena near heterogeneities are present.

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Section: Dispersion Relation For the Shear-rotational Perturbationsmentioning

confidence: 99%

“…Define ϕ = A s θ, consider I = IE. The equations for perturbations appear to be the same as those for the reduced linear Cosserat medium with zero initial stress [12]:…”

Section: Spherically Symmetrical Stress State In An Isotropic Materialsmentioning

confidence: 99%

Small perturbations of the spherical prestressed state in a nonlinear isotropic elastic reduced Cosserat medium: Waves and instabilities

Grekova

2011

Proceedings of the International Conference Days on Diffraction 2011

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“…materials with special and unique properties, which are not present in natural materials. From this perspective, among general Cosserat solids there is a special type of models where the second derivative of rotational degree of freedom in corresponding equation for rotations is absent, so-called reduced Cosserat models [6,7]. Understanding of mathematical and physical properties of such models, as well as general approach to describe the behavior of such media is quite nontrivial and interesting task.…”

Section: Introductionmentioning

confidence: 99%

Modeling of static deformations and dynamics of localized rotation in a chain of finite size particles

Vasiliev¹,

Miroshnichenko²

2017

LoM

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In beam-like structures and granular media the rotations are usually quite small. Linear generalized continuum models were developed and successfully applied for the modelling of such deformations. But, they do not describe full continuous rotations. The development of generalized continuum models describing not only small oscillations, but full continuous rotations as well is quite challenging and an open problem. This manuscript extends the results by Vasiliev A. A, Dmitriev S. V. Discrete and Multifield Models of a Cosserat Chain: One-Period and Two-Period Solutions. Russian Physics Journal. 59, 961 (2016). We derive and analyse three types of discrete models: i) nonlinear springs model, ii) linearized model with respect to displacement of particles and full sine-dependence of the angle of rotations; and iii) linearized one with respect to both, displacements and rotations. The latter is the discrete analogue of the classical Cosserat model. As a test example, we consider a chain of finite size particles with fixed boundaries in viscous medium with an applied momentum load to the central particle. By employing continuous Cosserat model, we find an approximated stationary solution for the full and reduced discrete Cosserat models. Numerically calculated static deformations of the chain under small momentum loads by using all three models are very similar. Under strong load localized rotation is observed. It is not described by the linear model (iii). Our results indicate the possibility of modelling of localized rotations in the nonlinear model (i) and the model containing nonlinear sine-dependence of the angle of rotation (ii).

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“…So far, four effective methods have been developed. The first is to study seismic rotational component based on focal mechanism and seismic wave propagation theory (Knopoff and Chen, 2009;Kulesh, 2009;Grekova et al, 2009). The second, which is based on elastic wave theory, is to calculate seismic rotational component by three seismic translation component records from a single seismographic station.…”

Section: Introductionmentioning

confidence: 99%

Torsional phenomena in 2008 great Wenchuan earthquake

2010

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It is well known that there are some torsional damages in earthquakes. In Taibai park, Jiangyou city, Sichuan province, most of the stone statues, which were placed upon the banisters of one zigzag bridge, exhibited different torsional phenomena in 2008 Wenchuan earthquake. This paper introduces the torsional phenomena of all the statues on the zigzag bridge firstly. Then one eccentricity model is established and the equivalent rotational accelerations are calculated in order to analyze the causes of the torsional damage. In addition, the torsional components are synthesized by using translation accelerations recorded at Jiangyou station in the Wenchuan earthquake. The results show that the equivalent rotational acceleration is larger than the synthesized rotational components, which suggests that the torsional phenomena of the statues on the zigzag bridge might mainly come from its eccentricity. The comparison between the estimated torsional component at Jiangyou and that presented by Trifunac shows that they are in the same order. The research implies that the torsional phenomena in earthquakes are very complicated, and not only caused by torsional motions.

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Waves in Linear Elastic Media with Microrotations, Part 2: Isotropic Reduced Cosserat Model

Cited by 53 publications

References 16 publications

Small perturbations of the spherical prestressed state in a nonlinear isotropic elastic reduced Cosserat medium: Waves and instabilities

Small perturbations of the spherical prestressed state in a nonlinear isotropic elastic reduced Cosserat medium: Waves and instabilities

Modeling of static deformations and dynamics of localized rotation in a chain of finite size particles

Torsional phenomena in 2008 great Wenchuan earthquake

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