Apart from these recent experiments carried out in the physics community, it is worth recalling that many interest- ing measurements of the elastic or acoustical characteristics of granular materials are to be found in the soil mechanics and geotechnical engineering literature 共关 18 兴 is a recent re- view stressing the need for sophisticated rheological mea- surements of soils, including elastic properties 兲 . The elastic properties of granular soils have been investigated from qua- sistatic stress-strain dependencies, as measured with a tri- axial 关19兴 or a hollow cylinder 关20兴 apparatus, by “resonant column” devices 关21,22兴 共which measure the frequency of the long wavelength eigenmodes of cylindrically shaped samples兲, and from sound propagation velocities 关18,20,23–25兴. One remarkable result is the consistency of moduli values obtained with various techniques, provided the applied strain increments are small enough, i.e., typically lower than 10 −5 兲. Thus, the agreement between sound propa-
similar equipartition phenomenon was studied numerically in an earlier work  for weakly coupled chains. This phenomenon is of interest in the creation of new acoustic wave guides, delay lines and stress mitigating materials. Energy transfer and equipartition phenomena in weakly coupled one-dimensional granular chains were studied [125, 126], and in  have been studied through a macroscopic realization of the Landau-Zener tunneling quantum effect. The energy equipartition principle is well known for elastic waves. Seismic waves for example have well known regimes where the P and S wave energy density equilibrates in a unique way that is inde- pendent of the details of the scattering. Interaction of solitons in coupled nonlinear lattices (scalar models) have been considered for various classical configurations such as coupled Toda lattices , coupled nonlinear Schr¨ odinger equations [128, 129] or coupled Ablowitz-Ladik chains  for example. In the case of coupled Toda lat- tices, it was shown numerically that solitonic excitations supplied to each one of the coupled chains may result in the two distinct dynamical regimes (attractors) . A non-uniform initial excitation (solitons with different amplitudes and/or phase mis- matches) may lead to the formation of the identical solitons on each one of the chains, propagating with the same speed and zero phase mismatch (first attractor) as well as the formation of two unequal solitons (i.e. with different amplitudes and phases also propagating with the same speed, second attractor). Here, we report an extension of energy equipartition phenomena in 2D granularmedia perturbed by lines of intruders.
Dunin et al. studied  the effect of gas bubbles on the P1- and P2-waves. They found that the dispersion curve of the P2-wave consists of two branches: a low-frequency branch and a high-frequency branch. In this work, a simple stress-strain relation was used, Ee (in standard notations). Nikolaevskiy used a much more complicated stress-strain relation that involves higher-order time derivatives of the stress and strain e. This relation is the result of the rheological scheme shown in Fig.1. Eventually, it leads to the higher-order partial differential Eq.(1.1). However, the original rheological scheme  does not include gas bubbles. Nikolaevskiy and Strunin  pointed out the place in this scheme that the bubbles should take, see Fig.1. In the present work, we aim to include the bubble into the rheological scheme and, based on this, derive the Nikolaevskiy-type Eq.(1.1), where the coefficients A p will depend on the bubble-related parameters.
We investigated the sensitivity of elastic thickness to water using recent laboratory data, and found that water weakening effect results in a significantly thinner elastic layer. Gravity and topography data indicate that the elas- tic thickness of the Martian lithosphere was extremely thin during its early history (prior to ~ 3.7 Ga), which can be explained by the presence of water-rich rheo- logical layers. In contrast, a dry rheology can account for the relatively thick elastic layer during the later evolution of Mars. The transition from a water-rich to a dry rheo- logical stratification roughly coincides with the dis- appearance of oceans that covered the northern plains of Mars, indicating that water was sequestered into a dee- per part of the planets. However, we should note the limitation regarding to the thermal evolution that highly influence the estimate of elastic thickness. The proposed model of a water-rich early Martian lithosphere needs to be further tested during the ongoing lander mission (InSight) through systematic heat-flux measurements of Mars.
Figure 1 shows the positive influence of magnetic field H 0 on non-dimensional temperature distribution, displacement distribution, normal force distribution, force stress distribution and carrier density, respectively with respect to axis x. the different values of magnetic field are shown by solid ( H 0 0 . 0 ), dashed ( H 0 10 ) and dash-dot ( H 0 20 )lines respectively when 3 1 . 8 and hydrostatic initial stress P 10 .The values of temperature for a non-magnetic thermoelastic medium under the influence of initial stress decrease sharply in the beginning and smooth increases to arrive the maximum value and then oscillate uniformly. The variations of temperature are similar in nature for both differences in magnitude of magnetic field. The heat conduction and the dynamical heat take the same behavior. Under the influence of magnetic field, the values of normal force stress first increases in the first range and then oscillate uniformly. Also the values of shear force stress for a thermoelastic medium without magnetic field lie in a very short range and have a very similar to the variation of normal force stress with difference in different values of magnetic field. The Carrier density start from its minimum value, increase with an increasing of x tend to zero as x tends to infinity but u starts from maximum, decreases with an increasing of axis x tends to zero as x tends to infinity, this indicate to the vanishing of all components for the large values of x.
The range of validity of the SFS approach is not yet clearly mapped out, and additional studies to answer the following key questions are called for. i) How does the stress tensor evolve when the ﬂow rate is increased? The stress tensor in the Pouliquen approach for fast ﬂows is similar to ours, but with the restrictions that P = P and that µ eﬀ (I) is a function of the local strain rate only . Here µ eﬀ apparently depends on geometry, and the crossover from geometry (θ) to inertial number (I) dependence needs to be explored. ii) We have seen here that P and P are systematically diﬀerent, as was also seen in simulations of chute ﬂows , and moreover, that P /P is not a constant. Though we do not understand the cause, nor the precise relevance, of this, it cannot be a priori ignored given the crucial role played by such variation of µ eﬀ in the formation of the wide shear zones in the linear geometry. iii) What distinguishes the zone where the stresses are in the SFS form from the region where they are not? Underlying the SFS picture is the assumption that the ﬂuctuations are suﬃciently strong and fast, and one imagines that far away from the shear zones this no longer holds true, thus leading to a breakdown of co-linearity. Preliminary data suggest, however, that the ﬂuctuation ﬂuidized region, most of which is established after a short transient, very slowly expands as a function of time . Possibly, after suﬃciently long time, all the material has experienced ﬂow and the stress tensor takes the SFS form everywhere, but this may be hard to verify numerically. Similar questions on the validity of the SFS framework can also be asked when the driving rate is made excessively slow. Ultimately, these questions are related to the puzzling nature of the transition between the static and ﬂowing state of granularmedia [17,19]. iv) Is the variation of the eﬀective friction the cause or eﬀect of the smoothness of our shear proﬁles? We suggest that the spreading of contact force fluctuations, from the rapidly shearing center to the tails of the shear zones, may elucidate the microscopic mechanism by which the width of the shear zones are selected. In this picture, the spread of ﬂuctuations would also drive the subtle variations of
The coordination number plays an important role in the description of granularmedia. It can be related to the microstructural evolution of a powder during sintering (cf. Fig. 1b) and to several physical or mechanical properties of sintered materials (Artz, 1982; Jernot, 1991; Tancret et al., 1997). It has been estimated in the past by using several methods (German, 1989; Guyon and Troadec, 1994). We will focus here on its direct measurement by image analysis without any assumption about the particle shape.
tine minerals are hydrous phyllosilicates composed of a 1:1 layer that is formed by linking one silicon-occupied tetra- hedral sheet (Fig. 1(a)) with one magnesium-occupied oc- tahedral sheet (Fig. 1(b)). The lateral dimensions of an ideal magnesium-occupied octahedral sheet (b ∼ 9.4 × 10 −10 m) are larger than those of an ideal silicon-occupied tetrahe- dral sheet (b ∼ 9.1 × 10 −10 m). This misﬁt between sheets leads to the three serpentine structures, each with a differ- ent solution to the misﬁt problem (Wicks and O’Hanley, 1988). In lizardite, the misﬁt is accommodated within the normal, planar 1:1 layer structure (Fig. 1(c)); in chrysotile, the misﬁt is partly overcome by the cylindrical or spiral cur- vature of the layers (Fig. 1(d)); in antigorite, the misﬁt is overcome by the curvature of the alternating wave modu- lation (Fig. 1(e)). The 1:1 layers are bonded by hydrogen bonds in lizardite, while by Si-O bonds in antigorite. Con- sequently, a higher elastic stiffness is expected in antigorite than lizardite. Chrysotile is considered to be more compli- ant that lizardite due to its lack of interlayer bonding.
We firstly consider the shape of pores in granite rock samples on the basis of the confining pressure depend- ence of velocities. For simplicity, a pore is supposed to have a spheroidal shape. The closure pressure given by Eq. (1) depends on its aspect ratio and elastic properties of the solid matrix. Since the influence of pores on elas- tic properties is sufficiently suppressed at high pressures, elastic properties of the solid phase are estimated from velocities in a dry rock sample (AJG02) at 177 MPa (Table 2). The pores closed below 125 MPa must thus have aspect ratios less than 2 × 10 − 3 . Such oblate spher- oidal pores can be treated as circular cracks, and their influence on velocities are formulated in terms of the crack density parameter defined by
This is the goal of the present paper. We will show that the velocity averaging method is based on an erroneous fun- damental assumption. To do so, we perform a demonstration for the cases of cluster and girdle ice textures with VTI. It is found that a wave propagating along the vertical axis is associated with two different shear velocities, which is un- physical since the polarizations of the shear waves are in the plane of isotropy. In Bennett (1968) a unique expression of the shear velocity was obtained, starting with modified ex- pressions of the two shear wavevelocities in ice single crys- tals. The modification consisted in attributing symmetrical weights to both velocities in order to get the same value after the velocity averaging. It resulted in a velocity value close to the harmonic mean of the two unphysical shear velocities derived from the direct use of the velocity averaging method. Considering the low elastic anisotropy of the ice single crystal, the accuracy of the inversion procedure from the measured velocities to ice texture is essential. In particular, a model relying on a rigorous mathematical formalism is re- quired, in order to describe the case of complex textures, be- yond the simple case of VTI clusters. With regard to this re- quirement, we will show that the model based on the effective medium theory appears more reliable.
Ganular media are collections of macroscopic, athermal grains which interact through dissipative, frictional contact forces. In the presence of gravity and in the absence of addi- tional external forces, they jam in metastable configurations; however, external forcing can easily lead to yielding and flow [1–10]. The best known scenario that leads to granular flow is by exerting shear stresses that exceed the yield stress, as when tilting a quiescent layer of sand sufficiently far away from the horizontal [1,2]. To understand such dense granular flows, it is becoming increasingly clear that both stress and mechanical agitations play a crucial role. Indeed, a given stress can give rise to a wide range of flow rates depending on the magnitude of these agitations [5,11,13,14]. Moreover, agitations make granularmedia lose their rigidity, although in the absence of shear stresses this does not need to cause flow [5,11–13]. We note here that the idea that both the stress and the amount of agitations determine the flow rate lies at the basis of numerous models for slowly flowing disordered materials [15–18].
The two-station method was first introduced by Sato 1955. It has been used to measure the dispersion curves of surface waves (Knopoff 1972). Meier et al. (2004) imple- mented the cross-correlation approach, which allows for the measurements of dispersion curves in a broad period range (in our case, between 5 and 250 s). We follow the conventional two-station approach in constructing the Rayleigh-wave phase-velocity maps in northern Vietnam. In the first step, we measure the phase-velocity using broadband regional waveform records (Figure 4) to deter- mine the individual interstation dispersion curves for a specific path connecting two stations. In the second step, we invert the obtained collection of dispersion curves for regional Rayleigh-wave maps at selected periods.
Early studies (Toba & Koga, 1986; Zhao & Toba, 2001; Zhao et al., 2003) which relate breaking conditions, whitecapping, and gas transfer velocities to Reynolds numbers focus on wind-sea statistics, ignoring the importance of swell. It is typically assumed that wave breaking is governed by the wind-sea component of the wave spectrum and that properties of the wind-sea partition are most relevant for air-sea processes. This study, however, suggests that consideration of swell is important for gas transfer. Indeed, considering only the wind-sea partition of the wave spectra did not reduce the scatter in k, yielding poor ﬁ ts that differ substantially between data sets. Arguably, this could be due to the paucity of data collected under wind- sea conditions especially during SO GasEx or the dif ﬁ culty of separating wind-sea and swell. More data are needed to verify this. However, nonbreaking wave-induced mixing has been shown to signi ﬁ cantly contri- bute to upper ocean turbulence through Langmuir circulation (Fan & Grif ﬁ es, 2014; Li et al., 2016). This sug- gests that mixing that arises from all components of the wave spectra should be considered. Furthermore, if large waves indeed inhibit interfacial gas transfer, the swell component needs to be considered. Reynolds numbers computed from the full spectrum can, however, not account for this effect.
On the other hand, similar to conventional elastic optical networks, the coarse granular routing elastic optical network also can support single or multiple modulation formats flexibly and dynamically. Each lightpath can be assigned to a pre-determined modulation format (single modulation format scenario) or an appropriate modulation format according to its distance (called distance-adaptive scenario). In distance-adaptive scheme, for a given traffic capacity, modulating optical signal with a higher-order format offers higher capacity per spectrum slot and consequently, requires less number of spectrum slots. It means that applying higher-order modulation format obtains higher spectrum efficiency but its optical transparent reach is shortened and consequently, more frequent regeneration and/or more regeneration resources are required. Inversely, utilizing lower-order modulation formats might lower the spectrum slot capacity and hence, may cause an increment in the required spectrum slot number. Hence,
amplitude at sensor location C with respect to location A (Figure 4.1), capturing the main features of the wave fronts. An amplitude ratio greater than one, as observed with the delrin spheres and steel intruders, indicates a wave front consistent with the impact energy being diverted away from the central line of impact. A ratio near one, as observed for the steel spheres and steel intruders, indicates a uniform, circular wave front. Similarly, as the ratio decreases below one we observed more amplitude being centrally directed within the system. While there is notable variation between experiments, the agreement between the mean response in experiments and numerical simulations is quite good. To visualize the variation in wave front shapes between successive experiments, the experiments most closely representing the standard devia- tions are also plotted for a packing of each test configuration (Figure 4.7 Bottom).We plotted the wave front amplitude ratio against both the sphere-cylinder material stiff- ness ratio and mass ratio (Figure 4.7 Top). A decreasing trend was observed for both stiffness and mass ratios as the wavefront began to follow a more pointed triangular form. However, only the stiffness ratio had a monotonically decreasing relationship with the shape of the particle velocity wave front. This suggests that the material stiffness had a more significant effect on the wave front shape than the mass ratios for the material combinations tested.
The time dG method considered herein is a displacement and velocity two-ﬁelds method, which is a speciﬁc case of a general time discontinuous space–time ﬁnite ele- ment method for the second order hyperbolic elastodynamic equations developed by Hughes and Hulbert [1–3]. In the weak formulation of the general method, additional stabilizing terms of least-squares type are introduced so it can be viewed as a Petrov– Galerkin method. However, these least-squares terms are not necessary for the stability of the method and can be omitted. Without these terms, we get a time discontinuous Galerkin method. In this case, the only appropriate deﬁnition of upwind ﬂuxes (due to the causality), together with the use of the elastic energy inner product to enforce the displacement–velocity compatibility, is suﬃcient to make the time dG method stable (see “Displacement–velocity two ﬁelds time dG method” section). By choosing speciﬁc space– time elements, the time dG method can be ﬁnally written as a time-stepping scheme and results in an implicit solver. As one important advantage, the method provides an appro- priate framework to develop adaptive computing as it remains unconditionally stable even if the space discretization changes over time [4–8]. From one time step to another, the energies that are dissipated in the jumps in time of both displacement and velocity ﬁelds guarantee the unconditional stability.
The present article concentrates on the propagation of generalized surface acoustic waves in a composite struc- ture consisting of piezoelectric and non-piezoelectric semiconductor media. The mathematical model of the problem is depicted by a set of partial differential equations of motion, Gauss equation in piezoelectric and elec- tron diffusion equation in semiconductor along with boundary conditions to be satisfied at the interface. The secular equation that governs the propagation of surface waves has been derived in compact form after obtaining the formal solution. The analytic expressions for displacements, stresses, piezoelectric potential and electron concentration during the surface wave propagation at the interface have also been obtained. The numerical solu- tion of the secular equation is carried out for the cadmium selenide and silicon composite by employing fixed point functional iteration numerical method along with irreducible Cardano method. The computer simulated results with the help of MATLAB software in respect of dispersion curves, attenuation coefficient, displace- ments, stresses, carrier concentration and piezoelectric potential are presented graphically. This work may be useful in surface acoustic wave (SAW) devices and electronic industry.
In this paper, in order to investigate the precompression force range of solitary wave impulses existence, we will study the acoustic wave propagation in a chain of sphere with a harmonic input, using an algorithm of molecular dynamics simulation to solve the discrete dynamic equations of the particles directly. For a certain length chain, with increase of the precompression force, we will sort out the solitary wave impulses with different periods and analyze their characteristics.