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with variable coefficients has drawn much attention (see, e.g., [ 27 , 28 , 7 ]). For perturbed linear elastic wave equations, the second author [ 40 ] proved a KSS-type estimate where the weight is a negative power of 〈 x 〉 {\langle x\rangle} . A full history of KSS-type estimates can be found in [ 40 ]. In this paper, we first revisit [ 40 ] to present a refinement in that the KSS estimate thereby obtained in [ 40 ] remains to hold even for the weight of the form of a negative power of 〈 x 〉 2 ⁢ δ ⁢ | x | 1 - 2 ⁢ δ {\langle x\rangle^{2\delta}\lvert x\rvert^{1

fractional parameters by the use of the simplest mechanical systems based on these equations. Mech. Time-Depend. Mat. 5, No 2 (2001), 131–175. http://dx.doi.org/10.1023/A:1011476323274 [83] Y. A. Rossikhin and M. V. Shitikova, Application of fractional calculus for dynamic problems of solid mechanics: Novel trends and recent results. Appl. Mech. Rev. 63 (2010), 010801–1-25. http://dx.doi.org/10.1115/1.4000563 [84] D. Royer and E. Dieulesaint, Elastic Waves in Solids, Vol. I. Springer, Berlin, 2000. http

5 Elastic waves In this chapter, we give an overview of the theory of elastic waves in solids, including piezoelectricity, plus an original part on finite element modeling of elementary elas- tic wave problems. This presentation aims at providing some basics required for the presentation of phononic crystals that follows in the next chapters. 5.1 Elastodynamic equations Let us first introduce the basic concepts and relations for linear elastic waves. 5.1.1 Strain tensor Let us consider some solid material. The solid is certainly composed of a very large number of

5 Elastic waves In this chapter, we give an overview of the theory of elastic waves in solids, including piezoelectricity, plus an original part on finite element modeling of elementary elas- tic wave problems. This presentation aims at providing some basics required for the presentation of phononic crystals that follows in the next chapters. 5.1 Elastodynamic equations Let us first introduce the basic concepts and relations for linear elastic waves. Strain tensor. Let us consider some solid material. The solid is certainly composed of a very large number of

Elastic Waves in Projectiles N. DAVIDS, B. P. GUPTA, and H. R. MINNICH INTRODUCTION Propagation of Elastic Waves In the past, rigid dynamics was applied to projectiles, where it is assumed that every point is set in motion instantaneously when forces are applied to the body. This was sufficiently accurate for calculating target pressures or penetration velocities. However, for a more refined study of projectile failure the rapidity of the loading and distribution of the stresses are of critical importance, especially, as in this analysis, when repeated

Chapter Ten Elastic Waves The field equations of physics (yielding the dynamic and thermodynamic variables) for a given continuous medium arise from three conservation equations: conservation of mass, momentum and energy. For an elastic medium these equations of motion are called the Navier equations. From them, a rich variety of stress waves is obtained (unlike waves in fluids, which are rather limited). . . . When we deal with an elastic solid we are concerned with the dynamic variables: stress and strain. The energy equation for an elastic solid yields a

The Elastic-Wave Pattern We have seen how the energy released by detonation of an explosive first crushes rock in the immediate vicinity, then moves through the next few feet as a shock front or strain pulse. This quickly becomes an oscillatory wave in which the particles along the path of travel move in orbits that repeat cyclically. The simplest two-dimensional illustration of such an oscillation is a wave shape with a crest and a trough. From there on, the energy produces movements in rock or other materials that are within their elastic limit. In

⁢ ( 𝐮 ) {\mathscr{B}\mathbf{u}=t(\mathbf{u})} on ∂ ⁡ D {\partial D} . The same results can be extended in a similar way to the Dirichlet boundary condition case. The direct problem in this paper is the unique solvability and uniform boundedness of solutions to the fluid-solid interaction problem ( 1.1 ). The inverse problem we considered is that for the coefficients λ , μ {\lambda,\mu} satisfying some physical and mathematical assumptions (but not necessarily known), determine the location and shape of the boundary between acoustic and elastic waves from the

References Beard M.D. and Lowe M.J.S. (2003): Non-destructive testing of rock bolts using ultrasonic guided waves. - International Journal of Rock Mechanics and Mining Sciences, vol.40, pp.527-536. Chróścielewski, Rucka M., Wilde K. and Witkowski W. (2012): Diagnostics of concrete beams during bending process using elastic wave propagation (in Polish). - Scientific Letters of Rzeszow University of Technology, No.283, pp.349-356. Gołaski L., Goszczyńska B., Świt G. and Trąmpczyński W. (2012): System for the global monitoring and evaluation of damage processes

Elastic wave velocity anomalies of anorthite in a subducting plate: In situ experiments KyoKo N. MatsuKage1,*, yu Nishihara2, FuMiya NoritaKe3, KatsuyuKi KawaMura3, Noriyoshi tsujiNo4, Moe saKurai5, yuji higo6, juNichi NaKajiMa7, aKira hasegawa7 aNd eiichi taKahashi5 1Earth and Planetary Sciences, Kobe University, Rokkoudai, Nada-ku, Kobe 657-8501, Japan 2Geodynamics Research Center, Ehime University, Bunkyocho, Matsuyama, Ehime 790-8577, Japan 3Environmental Science and Technology, Okayama University, Tsushimanaka, Kita, Okayama 700-8530, Japan 4Institute for