Micropolar modelling of rotational waves in seismology

Abreu, Rafael; Kamm, Jochen; Reiß, Anne-Sophie

doi:10.1093/gji/ggx211

Cited by 22 publications

(32 citation statements)

References 36 publications

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“…As an extension of [Mindlin 1964], one may treat a more general Cosserat thermo-viscoelasticity by addition to the list of variables in (2) and (5) the Cosserat rotation vector ϑ = [ϑ i ] and the moment stress σ (2) = [σ (2) i j ], conjugate to ∇ϑ, and by replacing the stress σ by a non-symmetric tensor with antisymmetric part defining a vector conjugate to ϑ. Under the rubric of micropolar elasticity, [Eringen 1984] has already given a comprehensive analysis for the isothermal case that leads to a cubic in k 2 as dispersion relation, and [Abreu et al 2017] provide a similar analysis with a view to the emerging field of rotational seismology.…”

Section: Extension To Solids and Cosserat Mediamentioning

confidence: 99%

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2018

Math. Mech. Compl. Sys.

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We study a system of N layers with a Kac horizontal interaction of parameter γ > 0 and a Kac vertical interaction of parameter γ 1/2 . We shall prove that the limit free energy functional is the rate function of the large deviations of the Gibbs measure (of a canonical constrained magnetization). The limit free energy functional is achieved as a -limit for γ → 0 for magnetizations with fixed average. Among all such magnetizations there exists a quasiconstant magnetization that minimizes the energy.Communicated by Raffaele Esposito. MSC2010: 82B24. 267 268 MICHELE ALEANDRI AND VENANZIO DI GIULIO between nearest neighbor horizontal lines (say (x, i) and (y, i + 1)) interact via the Kac potential λJ γ 1/2 (x, y), λ > 0; that is, the vertical interaction is much more local than the horizontal one.We study the mesoscopic limit γ → 0. The mesoscopic state of the system is a collection m ≡ {m(r, i) : r ∈ [0, ], i = 1, . . . , N } of measurable functions with values in [−1, 1]. Its statistical properties are then described by a free energy functional F(m). According to the Gibbs theory such a functional is the limit as γ → 0 of −1/β times the log of a constrained partition function where the spin configurations are required to be "close" to the mesoscopic state m (this involves a coarse grain procedure which is specified in Section 2). This is not as in the classical Lebowitz-Penrose [1966;Penrose and Lebowitz 1971] procedure because there are two scales, γ −1 for the horizontal interaction and γ −1/2 for the vertical one. Thus, there could be oscillations on the scale γ −1/2 which do not appear in m because the latter is defined by averages over ≈ γ −1 but which could affect the free energy of m. These oscillations actually do not occur if λ is small; indeed by Theorem 4.1 the optimal profile is quasiconstant on the scale γ −α with α ∈ (0, 1). However, if λ is large enough we can provide an example where such a phenomena occurs.The paper is organized as follows. In Section 2 we introduce the microscopic and mesoscopic models and enunciate the main results. In Section 3 we introduce the coarse graining procedure used to prove the Lebowitz-Penrose limit. In Section 4 we prove a key result, that is, Theorem 4.1, in which we provide a technique to minimize the free energy. This theorem is needed to prove the main results in Section 2. In Section 5 we prove the Lebowitz-Penrose limit for our model. In Section 6 we prove the -limit result. The proofs of Theorem 2.4 and Proposition 3.1 are deferred to Appendix A. In Appendix B, finally, we illustrate the case in which Theorem 2.3 fails for the parameter λ large enough.Similar model have been studied in [Cassandro et al. 2016;Fontes et al. 2014;. A numerical investigation of the mesoscopic limit for lattice gas model was also recently tackled in [Colangeli et al. 2016;.This work is the first step of a research program pointed towards the characterization of the surface tension associated to free energy in the thermodynamic limit. Model and main resultsWe consider an Ising spin ...

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Section: Extension To Solids and Cosserat Mediamentioning

confidence: 99%

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2018

Math. Mech. Compl. Sys.

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“…Specialists in the rotational seismology, a science which has developed during the past decades, agree that rotational degrees of freedom of blocks and heterogeneities in the geomedium must be taken into account [36][37][38]. However, there is a strong discussion on the most appropriate model.…”

Section: Introductionmentioning

confidence: 99%

Reduced Linear Constrained Elastic and Viscoelastic Homogeneous Cosserat Media as Acoustic Metamaterials

2020

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We consider the reduced constrained linear Cosserat continuum, a particular type of a Cosserat medium, for three different material behaviors or symmetries: the isotropic elastic case, a special type of elastic transversely isotropic case, and the isotropic viscoelastic case. Such continua, in which stresses do not work on rates of microrotation gradients, behave as acoustic metamaterials for the (pure) shear waves and also for one branch of the mixed wave in the considered anisotropic material case. In elastic media, those waves do not propagate for frequencies exceeding a certain threshold, whence these media exhibit a single negative acoustic metamaterial behavior in this range. In the isotropic viscoelastic case, dissipation destroys the bandgap and favors wave propagation. This curious effect is, probably, due to the fact that the bandgap is associated not with the dissipation, but with the wave localization which can be destroyed by the viscosity. The dispersion curve is now decreasing in some part of the former bandgap, above a certain frequency, whence the medium is a double negative acoustic metamaterial. We prove the existence of a boundary wavenumber in the viscoelastic case and estimate its value. Below the characteristic frequency corresponding to the boundary of the elastic bandgap, the wave attenuation (logarithmic decrement) is a growing function of the viscous dissipation parameter. Above this frequency, the attenuation decreases as the viscosity increases.

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“…Rotational waves are another mode of mechanical waves which have recently caught a lot of attention. They exist due to the media having rotational degrees of freedom [1,170]. Cosserat continuum theory takes into account some of the rotational degrees of freedom for the internal heterogeneities of disordered media [39] and has been used for various analyses ranging from localized failure problems [131], dynamics of ballast on railway tracks [189], shear band formation and dissolution [149] due to wave propagation [182,190,191].…”

Section: Signature Of Mechanical Wave Propagation In Disordered Mediamentioning

confidence: 99%

“…porous solids, cracked solids, soil, rocks, composites, polycrystals, biological materials, etc. [1]. Effective Medium Theory (E.M.T) gives the approximate modeling of such media in the large wavelength, low resolution, homogeneous limit, which allows predicting some macroscopic/material properties [15,48,51].…”

Section: Disordered Media: Granular Mediamentioning

confidence: 99%

“…where M is a diagonal mass matrix with entries b (1) , b (2) , b (3) , ..., b (N ) and zero otherwise; K is a matrix with diagonal entries −( of the matrix A give the eigendomain of the granular chain that are independent of the external driving. The square roots of the eigenvalues, ω j , are the natural frequencies of the chain.…”

Section: Effect Of Mass Disorder On Bulk Sound Wave Speed: a Multiscamentioning

confidence: 99%

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Towards stochastic and deterministic modeling of mechanical waves in disordered media

Shrivastava¹

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Samenvatting 159 Acknowledgments 161 Curriculum Vitae 163 • Non-destructive testing of materials for geotechnical investigations (artificial mechanical waves are sent into the material and the received response is studied using piezoelectric devices such as electronic transducers [215] or bender elements [142, 176]). A mechanical wave propagating in a disordered medium has a characteristic signature, whether the media is Earth [4] or Moon [43, 133]. The schematic illustration in Fig. 1.2 depicts the signature in time. The longitudinal mode (Pwave) arrives first, then comes the transverse mode (S-wave) and then arrive the Surface waves. Similar signatures of mechanical waves were also observed in laboratory experiments [76, 77, 93, 148, 214, 215]. However, on zooming over a particular mode of a mechanical wave in a time series profile, the me-1.5 Mesoscopic Wave Phenomena: Diffusion, Coherent Backscattering Effect & Weak Localization The diffusive transport of energy in disordered systems is a well known fact. This fact has been readily exploited to develop a diffusion model for propagation of multiply scattered mechanical waves [4, 67, 68, 147, 148, 204, 212]. However, unusual wave phenomena have been observed at intermediate or higher frequencies such as large variations in acoustic wave speed and suppression of wave propagation due to wave localization (Anderson localization [9], similar to the localization observed in ultracold atoms [23, 32, 157], microwave and light [3, 30, 31, 44, 171, 188, 213]). Amongst many definitions of Anderson localizations, two of them rely on vanishing diffusion and decay of spatially lo-1.6.2 Experimental Investigations Earthquakes are naturally occurring mechanical waves; they have been measured and recorded since a very long time; they have been quintessential in determining the internal structure of Earth, hence, their study is one of the early experimental investigations of mechanical waves in soil [55, 174]. Ultrasound experiments have become instrumental in recent years to study the unique vibrational response of granular materials [102, 103, 104, 147]. For kHz-MHz frequency signals (mechanical waves) that are sent through materials, the received signals are analyzed for identifying the internal structure of Effect of Mass Disorder on Bulk Sound Wave Speed: A Multiscale Spectral Analysis relations for disordered chains; these dispersion relations confirm the decrease in cutoff frequency and thus wave speed with increasing disorder. 1

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Micropolar modelling of rotational waves in seismology

Cited by 22 publications

References 36 publications

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Reduced Linear Constrained Elastic and Viscoelastic Homogeneous Cosserat Media as Acoustic Metamaterials

Towards stochastic and deterministic modeling of mechanical waves in disordered media

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