1,142 research outputs found

    A Semi-classical calculus of correlations

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    The method of passive imaging in seismology has been developped recently in order to image the earth crust from recordings of the seismic noise. This method is founded on the computation of correlations of the seismic noise. In this paper, we give an explicit formula for this correlation in the "semi-classical" regime. In order to do that, we define the power spectrum of a random field as the ensemble average of its Wigner measure, this allows phase-space computations: the pseudo-differential calculus and the ray theory. This way, we get a formula for the correlation of the seismic noise in the semi-classcial regime with a source noise which can be localized and non homogeneous. After that, we show how the use of surface guided waves allows to image the earth crust.Comment: To appear in a special issue "Imaging and Monitoring with Seismic Noise" of the series "Comptes Rendus G\'eosciences", from the French "Acad\'emie des sciences

    A homogenization result in finite plasticity and its application to high-contrast media

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    We carry out a variational study for integral functionals that model the stored energy of a heterogeneous material governed by finite-strain elastoplasticity with hardening. Assuming that the composite has a periodic microscopic structure, we firstly establish the {\Gamma}-convergence of the energies in the limiting of vanishing periodicity. Then, in the second part of the paper, we use the result to derive a macroscopic description for an elastoplastic medium with high-contrast microstructure. Specifically, we consider a composite obtained by filling the voids of a periodically perforated stiff matrix by soft inclusions. Again, we study the {\Gamma}-convergence of the related energy functionals as the periodicity tends to zero. The main challenge is posed by the lack of coercivity brought about by the degeneracy of the material properties in the soft part. We prove that the {\Gamma}-limit, which we compute with respect to a suitable notion of convergence, is the sum of the contributions resulting from each of the two components separately

    Energy partition of seismic coda waves in layered media: theory and application to Pinyon Flats Observatory

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    We have studied the partition of shear, compressional and kinetic energies in the coda of ten earthquakes recorded on a dense array, located at Pinyon Flats Observatory (PFO), California. We observe a clear stabilization of the shear to compressional (Ws/WpW^s/W^p) energy ratio in the coda, with an average value of about 2.8. The ratio between the vertical and horizontal kinetic energies (V2/H2V^2/H^2) can be measured from 5 to 25Hz and shows an abrupt transitionfrom 0.1 in the 5-10Hz band, to about 0.8 in the 15-25Hz band. These measured values are in sharp contrast with the theoretical prediction for equipartitioned elastic waves in a homogeneous half-space. To explain these observations, we have developed a theory of equipartition in a layered elastic half-space. Using a rigorous spectral decomposition of the elastic wave equation, we define equipartition as a white noise distributed over the complete set of eigenfunctions. The theory predicts that close to the resonance frequency of a low-velocity layer, the ratio between shear and compressional energies strongly decreases. Using a detailed model of the subsurface at PFO, this conterintuitive result is found to be in good qualitative and quantitative agreement with the observations

    On periodic homogenization of highly contrasted elastic structures

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    While homogenization of periodic linear elastic structures is now a well-known procedure when the stiffness of the material varies inside fixed bounds, no homogenization formula is known which enables us to compute the effective properties of highly contrasted structures. Examples have been given in which the effective energy involves the strain-gradient but no general formula provides this strain-gradient dependence. Some formulas have been proposed which involve such terms and provide a small correction to the classical effective energy still when the stiffness of the material varies inside fixed bounds. The goal of this paper is to check the applicability of these formulas for highly contrasted structures. To that aim we focus on structures whose limit energy is already known and we compare the energies given by (i) the convergence results, (ii) the corrective formulas and (iii) by a direct numerical simulation of the complete structure

    Damage-driven fracture with low-order potentials: asymptotic behavior, existence and applications

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    We study the Γ\Gamma-convergence of damage to fracture energy functionals in the presence of low-order nonlinear potentials that allows us to model physical phenomena such as fluid-driven fracturing, plastic slip, and the satisfaction of kinematical constraints such as crack non-interpenetration. Existence results are also addressedComment: 41 pages, 4 Figure

    Seismic Wave Propagation in Stratified Media

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    Seismic Wave Propagation in Stratified Media presents a systematic treatment of the interaction of seismic waves with Earth structure. The theoretical development is physically based and is closely tied to the nature of the seismograms observed across a wide range of distance scales – from a few kilometres as in shallow reflection work for geophysical prospecting, to many thousands of kilometres for major earthquakes. A unified framework is presented for all classes of seismic phenomena, for both body waves and surface waves. Since its first publication in 1983 this book has been an important resource for understanding the way in which seismic waves can be understood in terms of reflection and transmission properties of Earth models, and how complete theoretical seismograms can be calculated. The methods allow the development of specific approximations that allow concentration on different seismic arrivals and hence provide a direct tie to seismic observations
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