Wave Transmission and Reflection Involving Auxetic Solids

Stress wave micro–macro attenuation in ceramic plates made of tiles during ballistic impact

This paper establishes the effect of auxeticity on stress wave transmission and reflection. Specifically, investigation was made on wave transmission across two perfectly bonded isotropic solids in which the Poisson’s ratios ranged between −1 and 0.5. The results show that the combined use of auxetic and conventional solids at extreme Poisson’s ratio is helpful to multiply or even to eliminate stress waves, under the prescribed density and modulus relations. These results suggest the usefulness of auxetic solids as smart materials and in smart structures for effective control of stress wave transmission.

Transmission and reflection of acoustic waves from a layer of a bubble medium into a liquid is theoretically studied, with subsequent reflection of the waves that arise from the rigid wall. The amplitudes of the emerging waves are determined through the amplitude of the initial wave, analytical expressions are obtained for the coefficients of reflection and transmission of waves across the interfaces.

Reflected and transmitted waves in a channel with side porous mattresses

This study presents the first experimental investigation of stress wave propagation in 1D granular chains of closed-cell PVC foam disks. Average impact velocities for H130 and H250 foams ranged from 17.6 to 38.1 m/s. The analysis focuses solely on the incident stress wave, excluding the reflected wave. The mid-planes of the disks were chosen for analysis due to their uniaxial force components along the chain's length. The results show that the stress wave speed is faster in the H250 foam chain due to its higher stiffness. Wave speed increases with impact velocity but decreases as it travels along the chain, with a more pronounced reduction in the H130 foam compared to the H250 foam. The peak normal forces in the H250 foam chain disks are approximately three times greater than those observed in the H130 foam chain disks at comparable impact velocities. The peak normal forces in both foam chains decrease rapidly with increasing impact velocity, especially over the first few disks. As the wave propagates further from the impact source, the attenuation rate slows, with a more gradual force reduction in the H250 foam due to its higher density and stiffness. Energy loss is governed by viscoelastic and plastic dissipation at disk contacts, which becomes more significant at higher impact velocities. This study provides new insight into dissipative wave phenomena in granular systems of deformable elements and offers experimental data for future modeling of strongly nonlinear, dissipative granular media.

The instability of double-cavern structure subjected to dynamic disturbances is a key issue for deep rock engineering. To investigate the dynamic responses of deep double U-shaped caverns, comprehensive analyses are conducted by Particle Flow Code (PFC2D), and the influences of incident directions of stress wave, cavern clearances, and cavern height ratios are discussed. The results indicate that the decreasing cavern clearance aggravates the static stress concentration on the intermediate rock pillar. When the stress wave is horizontally incident, the presence of the incident side cavern reduces peak tangential stress and kinetic energy on the non-incident side cavern; the higher the incident side cavern, the less damage on the non-incident side cavern. A vertically incident stress wave causes more severe damage in the intermediate rock pillar compared to a horizontally incident stress wave; the smaller the cavern clearance, the more violent the rockburst in the intermediate rock pillar. Comparatively, the cavern with a lower height exhibits more severe failure at the adjacent sidewall compared to the cavern with a higher height. This work can provide guidelines for disaster prevention of deep double-cavern structures.

This work describes the results of a test campaign aimed to measure the propagation of longitudinal, torsional, and flexural stress waves on a drill bit during percussive rock drilling. Although the stress wave propagation during percussive drilling has been extensively modeled and studied in the literature, its experimental characterization is poorly documented and generally limited to the detection of the longitudinal stress waves. The activity was performed under continuous drilling while varying three parameters, the type of concrete, the operator feeding force, and the drilling hammer rotational speed. It was found that axial stress wave frequencies and spectral amplitudes depend on the investigated parameters. Moreover, a relevant coupling between axial and torsional vibrations was evidenced, while negligible contribution was found from the bending modes. A finite element model of the drill bit and percussive element was developed to simulate the impact and the coupling between axial and torsional vibrations. A strong correlation was found between computed and measured axial stress spectra, but additional studies are required to achieve a satisfactory agreement between the measured and the simulated torque vibrations.

In this paper, the reflection and transmission of plane waves are examined from a complex potential. Such potentials have the property of absorbing wave packets incident on them and are used widely in time-dependent quantum scattering theory. The purpose of the study is to determine the optimal form of potential to be used for absorbing wave packets near the edges of finite grids in coordinate space. The best potentials for such purposes lead to the minimum possible transmission and reflection of the incident wave packet. The Jeffreys–Wentzel–Kramers–Brillouin (JWKB) theory is used to address this problem and a new form for the optimal complex potential is proposed. A scaled dimensionless form of the Schrödinger equation is also derived, so that the parameters of any optimized potential obtained for a particular collision energy and mass combination may be readily converted to apply to a new set of masses and energies.

SUMMARY The reflection and transmission of elastic waves are of great significance for predicting reservoir physical properties, interpreting seismic data and detecting crustal structures. Most studies only consider the initial vertical stress state when studying the reflection and transmission of elastic waves at imperfectly bonded interfaces, but few studies consider the influence of initial stress on boundary conditions. Moreover, the effect of initial stress on the energy distribution of elastic waves at imperfectly bonded interfaces has rarely been investigated. We propose a unified method to calculate the energy reflection and transmission coefficients for different incident waves at welded or imperfectly bonded interfaces in stressed media. The effects of initial stress on the equation of motion, the elastic properties of the medium and the boundary conditions at the interface are considered. The elastic properties of rocks under initial stress are described by acoustoelasticity theory. In addition, we define a new stress tensor to modify the linear-slip model for describing boundary conditions at the imperfectly bonded interface in the presence of initial stress. Numerical results show that the energy reflection and transmission coefficients at the non-welded interface in stressed media depend on the elastic properties of the incident and transmitted media, the initial stress, the type and magnitude of the interfacial compliance and the frequency and propagation direction of the incident wave. The initial vertical and horizontal stresses dominate the reflection and transmission coefficients at small and large angles, respectively. The discontinuity in displacement across the imperfectly bonded boundary results in the frequency dependence of the reflection and transmission coefficients. Imperfect bonding enhances P-wave and SV-wave energy reflection and weakens P-wave energy transmission. However, imperfect bonding can enhance the energy transmission of the SV wave for the imperfectly bonded interface with high contrast between tangential and normal compliance and a resonance peak appears at a specific frequency. We also notice that imperfectly bonded interfaces with interfacial compliance less than $1.0 \times {10}^{ - 11}$ m Pa−1 can be regarded as welded interfaces in the seismic frequency band (lower than 100 Hz). In addition, the initial stress greatly influences the reflection coefficients at high frequencies and the transmission coefficients at low frequencies. The initial vertical stress can reduce the energy transmission of SV waves at imperfectly bonded interfaces. In contrast, the initial horizontal stress can significantly increase the energy transmission of low-frequency SV waves and may lead to the disappearance of the resonant peak in the transmission coefficients.

Upon stress wave interaction with joints, incident energy is reflected and refracted. To gain insight into the mechanisms associated with dynamic processes such as blasting, percussion drilling and seismicity, knowledge of the amplitudes of the reflected and refracted waves is of importance. This paper compiles stress, energy, displacement and velocity amplitudes of stress waves, which are reflected and refracted by a variety of interface types. In addition to covering the well-known wave interactions with standard discontinuities, such as a free boundary and welded interface, some new interface conditions are analysed, e.g. interfaces with friction and shear strength, and interfaces with normal and shear stiffness related to measurable rock properties. The aim of this tutorial paper is to act as an educational guide and reference for the blasting engineer, seismologist, geophysicist and rock engineer, and to assist in further understanding stress wave interactions with rock mass discontinuities.

When stress waves interact with joints and fractures in a rock mass, the incident energy is reflected from and refracted (transmitted) across the interface. Knowledge of the amplitudes of reflected and transmitted waves is of importance to understand stress wave propagation through discontinuous rock, and to assess the stability and ultimately the safety of structures in rock loaded transiently. In this paper the dynamic behaviour of joints containing fluids or thin layers of comparatively soft material are analysed. A generalised Kelvin model is used to simulate the interactions of incident P- and SV-waves. It is found that the amplitudes of the reflected and refracted waves are dependent on the joint stiffness and viscosity, as well as the frequency of the incident wave. The energy dissipation of the viscous interface is analysed and parameters influencing the maximum dissipation are quantified and discussed.

In this paper, we study wave interaction with a porous structure. A nonlinear diffusive wave equation is used to describe gravity surface wave propagation in a porous media. We solve the equations using asymptotic expansion method and numerically using a staggered finite volume method. We then derive the dispersion relation that holds for gravity waves inside a porous structure. This dispersion relation explains the diffusive mechanism of wave amplitude inside the porous structure. Analysis of the dispersion relation shows that amplitude reduction depends on porous medium parameters such as porosity, friction coefficient, length of the structure, and wave frequency. To validate our numerical scheme, we compare the wave reduction amplitude from the numerical result with the asymptotic solution. A good agreement of the comparison is observed. Furthermore, the numerical model is employed to investigate the effectiveness of porous media in dissipating wave energy of an incoming wave. The results from this paper can be used to determine the optimum dimension of the porous medium so that the incoming wave can be reduced as much as possible to protect shoreline. References H.J.S. Fernando, S.P. Samarawickrama, S. Balasubramanian, S.S.L. Hettiarachchi, and S. Voropayev. Effects of porous barriers such as coral reefs on coastal wave propagation. Journal of Hydro-environment Research, 1(3):187 – 194, 2008. Pengzhi Lin and S. A. Karunarathna. Numerical study of solitary wave interaction with porous breakwaters. Journal of Waterway, Port, Coastal, and Ocean Engineering, 133(5):352–363, 2007. Philip L.-F. Liu, Pengzhi Lin, Kuang-An Chang, and Tsutomu Sakakiyama. Numerical modeling of wave interaction with porous structures. Journal of Waterway, Port, Coastal, and Ocean Engineering, 125(6):322–330, 1999. Patrick J. Lynett, Philip L.-F. Liu, Inigo J. Losada, and Cesar Vidal. Solitary wave interaction with porous breakwaters. Journal of Waterway, Port, Coastal, and Ocean Engineering, 126(6):314–322, 2000. O.S. Madsen. Wave transmission through porous structures. Journal of Waterways, Harbors, Coastal Engineering, 100:169–188, 1974. I. Magdalena, S. R. Pudjaprasetya, and L. H. Wiryanto. Wave interaction with an emerged porous media. Advances in Applied Mathematics and Mechanics, 6(5):680–692, 06 2015. S.R. Pudjaprasetya and I. Magdalena. Momentum conservative scheme for dam break and wave run up simulations. East Asia Journal on Applied Mathematics, 4(2):152–165, 2014. Panagiotis D. Scarlatos and Vijay P. Singh. Long-wave transmission through porous breakwaters. Coastal Engineering, 11(2):141 – 157, 1987. C.K. Sollitt and R.H. Cross. Long-wave transmission through porous breakwaters. Proc. 13th Coastal Eng. Conf. 3, pages 1827–1846, 1972. W. Sulisz. Wave reflection and transmission at permeable breakwaters of arbitrary cross-section. Coastal Engineering, 9:371–386, 1985. M. R. A. Van Gent. Wave interaction with permeable coastal structures. International Journal of Rock Mechanics and Mining Sciences and Geomechanics Abstracts, 33(6):277A, 1996.

In an infinite elastic medium dilatational waves exhibit much more nonlinear distortion than do shear waves, but such behavior cannot be assumed when a boundary, such as a free surface, is present. Because of the process of mode conversion, a dilatational (P) wave that has already undergone significant distortion prior to its incidence at the boundary must impart its distortion to both the reflected P and vertically polarized shear (SV) wave. The problem of oblique incidence at a free surface was previously considered by us [1,2]. The present paper generalizes that analysis by permitting the boundary conditions at the planar interface to be arbitrary, and also by considering critical and supercritical angles of incidence.

Wave transmission in layered composite rock mass comprising parallel joints and different rock materials

The problem of the reflection and transmission of Rayleigh waves in an elastic wedge discussed in an earlier paper for the case of incidence from infinity is now studied in more detail and for the more general case of an incidence from a finite distance from the corner. A detailed application is made of the effect of the critical regions of Rayleigh waves in Lamb's half-space problem when a series of interactions with the wedge faces are considered. These interactions are two-fold, viz. those due to the inwardly progressing waves and those due to the outwardly receding waves. Both lead to contributions given by certain integral equations. While in the latter case the integral equations behave like the Fredholm equations, in the case of the former the behaviour is like Volterra equations of second kind at lower range of wedge angles and like the Fredholm equations at higher range and there is a mixed character in the intermediate values. These approximations lead to dividing the range of the wedge angle, which we take to be from 0° to 180°, into five parts at points depending on the critical angles of Lamb's problem. The solutions in these parts are piecewise continuous. A brief outline of the corner wave effects is also included. The numerical results show that the present theory can explain well some of the important experimental features of the problem that were only partially achieved by previous theories.