Isotropic Elastic Medium Research Articles

Propagation characteristics of plane waves in nonlocal isotropic diffusive materials

The size-dependent theoretical modeling of coupled phenomena between deformation and mass diffusion is of current interest in the fields of nanomechanics and nanotechnology. The theory of nonlocal elasticity is applied to study the propagation of plane waves in a linear and isotropic diffusive elastic material. The governing equations of motion are specialized in a plane and are solved for time-harmonic plane wave solutions. The velocity equations are derived to show the possible propagation of three homogeneous plane waves in an isotropic diffusive elastic medium. A reflection phenomenon of plane waves is also considered to obtain the reflection and energy coefficients of reflected waves. A particular numerical example is taken to illustrate graphically the effects of elastic and diffusive nonlocal parameters on the speeds of plane waves, reflection coefficients and energy ratios of reflected waves.

Seismic analysis of a lined tunnel in a multi-layered TI saturated half-space due to qP1-and qSV-waves

A special indirect boundary element method (IBEM) is presented to study the in-plane scattering by a lined tunnel embedded in a multi-layered TI saturated half-space. By integrating the full-space and half-space Green’s functions for the inclined distributed load and pore pressure, these singular-free kernels enables the method to treat an arbitrarily shaped tunnel with a superior flexibility and high efficiency in comparison to the traditional IBEM. For the total system, the substructure process is implemented to decompose the model into a layered TI saturated half-space region and a lining region filled with isotropic elastic medium. The dynamic stiffness matrix method developed by the authors is adopted to solve the free field of the exterior half-space, and the scattered fields existed in the exterior and interior lining regions are simulated by the half-space and full-space Green’s functions in a single-layer potential representation, respectively. Then the total wave-fields in the half-space region can be expressed as the superposition of free-field and scattered field, and the tunnel region merely contains the scattered field. The compatibility conditions on inner and outer interfaces enclosing the lining are satisfied to couple these sub-regions to the original one. The present method is calibrated by comparison with available limited cases. Displacement and dynamic hoop stress concentration at the inner surface of the lining are calculated in the frequency domain to portray the effects of anisotropy degree, drainage condition, incident angle, frequency and layering on the seismic response. Results show that the decrease of anisotropy degree generally leads to larger amplitudes of surface displacement and the stress. Prominent amplification of displacement and stress can be observed for the two-phase medium model in comparison to the one-phase model although the vertical displacement is hardly affected by the drainage condition at the lining-rock interface. When taking the layering into account, the vertical displacement maximum is increased by about 32%, while the DHSC amplitudes at different cases tend to become larger on the whole compared to those for the homogeneous half-space.

Finite-difference modelling of 2-D wave propagation in the vicinity of dipping interfaces: a comparison of anti-aliasing and equivalent medium approaches

SUMMARY Finite-difference (FD) modelling of seismic waves in the vicinity of dipping interfaces gives rise to artefacts. Examples are phase and amplitude errors, as well as staircase diffractions. Such errors can be reduced in two general ways. In the first approach, the interface can be anti-aliased (i.e. with an anti-aliased step-function, or a lowpass filter). Alternatively, the interface may be replaced with an equivalent medium (i.e. using Schoenberg & Muir (SM) calculus or orthorhombic averaging). We test these strategies in acoustic, elastic isotropic, and elastic anisotropic settings. Computed FD solutions are compared to analytical solutions. We find that in acoustic media, anti-aliasing methods lead to the smallest errors. Conversely, in elastic media, the SM calculus provides the best accuracy. The downside of the SM calculus is that it requires an anisotropic FD solver even to model an interface between two isotropic materials. As a result, the computational cost increases compared to when using isotropic FD solvers. However, since coarser grid spacings can be used to represent the dipping interfaces, the two effects (an expensive FD solver on a coarser FD grid) equal out. Hence, the SM calculus can provide an efficient means to reduce errors, also in elastic isotropic media.

Elastic isotropic and anisotropic full-waveform inversions using automatic differentiation for gradient calculations in a framework of recurrent neural networks

We have implemented multiparameter full-waveform inversions (FWIs) in the framework of recurrent neural networks in elastic isotropic and transversely isotropic media. A staggered-grid velocity-stress scheme is used to solve the first-order elastodynamic equations for forward modeling. The gradients of loss with respect to model parameters are obtained by automatic differentiation. Multiple elastic model parameters are simultaneously inverted with a minibatch optimizer. We prove the equivalency of full-batch automatic differentiation and the conventional adjoint-state method for inversions in elastic isotropic media. Synthetic tests on elastic isotropic models show that the minibatch configuration has a better convergence rate and higher inversion accuracy than full-batch elastic FWIs. Inversions with data that contain incoherent and coherent noise are tested, respectively. With automatic differentiation, we determine the ease of extension to anisotropic media with two parameterizations, and the potential to implement it for more general media.

Finite-Difference Modeling and Characteristics Analysis of Love Waves in Anisotropic-Viscoelastic Media

ABSTRACTIn this study, the characteristics of Love waves in viscoelastic vertical transversely isotropic layered media are investigated by finite-difference numerical modeling. The accuracy of the modeling scheme is tested against the theoretical seismograms of isotropic-elastic and isotropic-viscoelastic media. The correctness of the modeling results is verified by the theoretical phase-velocity dispersion curves of Love waves in isotropic or anisotropic elastic or viscoelastic media. In two-layer half-space models, the effects of velocity anisotropy, viscoelasticity, and attenuation anisotropy of media on Love waves are studied in detail by comparing the modeling results obtained for anisotropic-elastic, isotropic-viscoelastic, and anisotropic-viscoelastic media with those obtained for isotropic-elastic media. Then, Love waves in three typical four-layer half-space models are simulated to further analyze the characteristics of Love waves in anisotropic-viscoelastic layered media. The results show that Love waves propagating in anisotropic-viscoelastic media are affected by both the anisotropy and viscoelasticity of media. The velocity anisotropy of media causes substantial changes in the values and distribution range of phase velocities of Love waves. The viscoelasticity of media leads to the amplitude attenuation and phase velocity dispersion of Love waves, and these effects increase with decreasing quality factors. The attenuation anisotropy of media indicates that the viscoelasticity degree of media is direction dependent. Comparisons of phase velocity ratios suggest that the change degree of Love-wave phase velocities due to viscoelasticity is much less than that caused by velocity anisotropy.

ON THE INFLUENCE OF THE SIZE OF FLAKE GRAPHITE INCLUSIONS AND ITS VOLUME CONTENT ON THE ACOUSTIC PARAMETERS OF CAST IRON AND THE CHARACTERISTICS OF ULTRASONIC LONGITUDINAL WAVE SIGNALS

Algorithms have been presented for calculating the velocity (in the approximation of a fine-layered model) and the attenuation coefficient of a longitudinal ultrasonic wave in cast iron, depending on the average size of graphite elements and its volume content, the calculation results for which are qualitatively confirmed experimentally. The calculation was performed using a fine-layered model of the structure, the graphite inclusions were described in the form of plane-parallel layers placed in an isotropic elastic medium (metal base). Computer simulation of acoustic paths for a mediumcast iron with flake graphite for standard direct converters is carried out in order to study the influence of such a medium on the characteristics of transmitted and received signals during ultrasonic testing. In the course of the research, a previously developed model was used to calculate the attenuation coefficient of longitudinal waves in cast iron with flake graphite due to their Rayleigh and phase scattering on graphite inclusions. Computer simulation of the acoustic characteristics of the signals of a direct linear probe with a phased array in cast iron with flake graphite was carried out, during which the shape of the acoustic pulses of the longitudinal wave was calculated, depending on the distance traveled by the wave and the value of the attenuation coefficient for various models of cast iron. The main modeled characteristics of the transducer include the directivity characteristic and the change in the signal amplitude along the acoustic axis. It is shown that for cast iron with flake graphite, there are cases when the directivity of the probe with a phased array transmitted into the cast iron is practically absent.

Propagation of waves in an incompressible rotating transversely isotropic nonlocal elastic solid

In this paper, the nonlocal elasticity theory is applied to study the propagation of plane wave and Rayleigh-type surface wave in an incompressible, rotating and transversely isotropic material. The governing equations of motion for an incompressible, rotating, transversely isotropic and nonlocal elastic medium are specialized for a plane. The medium is assumed rotating about an axis perpendicular to the plane. The transverse isotropy axis is taken perpendicular to the surface. The specialized governing equations are first applied to derive a velocity equation for homogeneous plane wave. The specialized governing equations along with traction free boundary conditions are also applied to derive the secular equation governing the wave speed of Rayleigh wave. The speeds of plane wave and Rayleigh wave are computed and illustrated graphically to observe the effects of nonlocality, rotation, anisotropy, frequency and propagation direction. It is noticed from the theory and numerical results that the speeds of both plane wave and Rayleigh wave decrease sharply with an increase in nonlocal parameter or rotation parameter. The speeds of plane wave and Rayleigh wave increase logarithmically with anisotropy material parameter. The feasible ranges of nonlocality, rotation or anisotropy parameters for the existence of plane wave or Rayleigh surface wave are determined for a given wave speed when the values of other parameters are fixed.

A highly accurate boundary integral method for the elastic obstacle scattering problem

Consider the scattering of a time-harmonic plane wave by a rigid obstacle embedded in a homogeneous and isotropic elastic medium in two dimensions. In this paper, a novel boundary integral formulation is proposed and its highly accurate numerical method is developed for the elastic obstacle scattering problem. More specifically, based on the Helmholtz decomposition, the model problem is reduced to a coupled boundary integral equation with singular kernels. A regularized system is constructed in order to handle the degenerated integral operators. The semi-discrete and full-discrete schemes are studied for the boundary integral system by using the trigonometric collocation method. Convergence is established for the numerical schemes in some appropriate Sobolev spaces. Numerical experiments are presented for both smooth and nonsmooth obstacles to demonstrate the superior performance of the proposed method.

Roadbed Sedimentation on Folded Bases with Weak Soils

Purpose. The main purpose of the work is to construct accurate analytical dependences of the sedimentation of the roadbed base on its coordinates and the trapezoidal external load applied to the upper limit of the base. This will allow the foundation settlement profile to be calculated within and outside the embankment using the layer-by-layer stacking method. Methodology. Theoretical studies of geomechanical processes using analytical and numerical mathematical methods were applied to achieve the purpose, as well as analysis and generalization of the results of theoretical research. Findings. Analytical dependences of vertical and horizontal normal deformations on the half-plane coordinates to the upper limit of which the trapezoidal load is applied were obtained within the framework of the base model in the form of a linear elastic isotropic medium and the calculated scheme of planar deformation. On this basis, an algorithm for constructing a profile of a roadbed sedimentation made of soil materials has been developed. Originality. Analytical dependences of vertical normal deformations in the roadbed base with a trapezoidal profile on its coordinates are obtained. Practical value. The research materials presented in this work make it possible to construct the design profile of the roadbed of soil materials, taking into account the strong compressibility of the soil. In addition, the results obtained can be used to determine the lower limit of the compressible strata of the bases, to the upper limit of which a distributed load of either a triangular or a trapezoidal form is applied.

An axisymmetric problem for a penny-shaped crack under the influence of the Steigmann–Ogden surface energy

A problem for a nanosized penny-shaped fracture in an infinite homogeneous isotropic elastic medium is considered. The fracture is opened by applying an axisymmetric normal traction to its surface. The surface energy in the Steigmann–Ogden form is acting on the boundary of the fracture. The problem is solved by using the Boussinesq potentials represented by the Hankel transforms of certain unknown functions. With the help of these functions, the problem can be reduced to a system of two singular integro-differential equations. The numerical solution to this system can be obtained by expanding the unknown functions into the Fourier–Bessel series. Then the approximations of the unknown functions can be obtained by solving a system of linear algebraic equations. Accuracy of the numerical procedure is studied. Various numerical examples for different values of the surface energy parameters are considered. Parametric studies of the dependence of the solutions on the mechanical and the geometric parameters of the system are undertaken. It is shown that the surface parameters have a significant influence on the behaviour of the material system. In particular, the presence of surface energy leads to the size-dependency of the solutions and smoother behaviour of the solutions near the tip of the crack.

Topography and structural heterogeneities in surface ground deformation: a simulation test for Somma-Vesuvius volcano

Abstract. We simulate the deformation of Somma-Vesuvius volcano due to some overpressure sources by means of a finite element 3D code. The main goal of these simulations is to investigate the influence of topography and structural heterogeneity on ground deformation. In our model the sources of deformation are embedded in an elastic linear isotropic medium and located at various depths. Geometry (shape and lateral extension) of the sources is mainly constrained by the results coming from recent seismic tomography studies. The structural heterogeneity has been modelled in terms of dynamic elastic parameters (Young's modulus) retrieved from previous seismic tomography and gravity studies. A high-resolution digital terrain model is used for the topography of the volcano subaerial edifice. Evidences from our results suggest that real topography and structural heterogeneities are key factors governing the ground deformation, which often turns being one of the most relevant problems in volcano monitoring. A large deviation from the axially symmetrical model of the displacement field is the main result of our modelling. Such an asymmetry is routinely unaccounted for when Mogi's simplistic modelling in a homogeneous medium with simplified topography is used. Our study clearly demonstrate that a better knowledge of deformation patterns can significantly help in the location of monitoring sensors as well as in the design of an efficient geodetic network.

Dynamics of plane strains in heteromodular isotropic elastic media

Dynamics of plane strains in heteromodular isotropic elastic media

A hybrid dislocation

ABSTRACT The shear of a crystalline interface by a matrix dislocation is an elementary phenomenon of plasticity that generates a three-dimensional elastic field difficult to express. However, in the case of a flat interface and two elastic isotropic media, the present work shows that this field can be described starting from an explicit form of the Green tensor associated with a point force exerted in one of the crystals. Indeed, the knowledge of this tensor makes it possible to express the displacement field of a dislocation in the final form of a surface integral. Because the shear is limited by a matrix branch and an interfacial branch, the dislocation can be called ‘hybrid dislocation’ or HD. If the two branches are straight, the field is that of an angular HD. Moreover, if the crystals are identical, the field becomes that of an angular matrix dislocation whose field is well known since Yoffe (1960). By applying the superimposition theorem, it becomes possible to determine the field of more complex configurations, such as those sometimes observed during the transmission of the plastic slip through the interfaces.

Dynamics of Plane Strains in Heteromodular Isotropic Elastic Media

Dynamics of Plane Strains in Heteromodular Isotropic Elastic Media

Recovery of discontinuous Lamé parameters from exterior Cauchy data

Consider an isotropic elastic medium whose Lamé parameters are piecewise smooth. In the elastic wave initial value inverse problem, we are given the solution operator for the elastic wave equation, but only outside Ω and only for initial data supported outside Ω. Using the recently introduced scattering control series in the acoustic case, we prove that piecewise smooth Lamé parameters are uniquely determined by this map under certain geometric conditions.

Analytical solution to Scholte’s secular equation for isotropic elastic media

In terms of a method based on Cauchy integrals, we have obtained a robust analytic expression to predict a unique physical solution for the Scholte slowness in all range of possible elastic and isotropic media. Proper analysis of the discontinuities of the secular Scholte equation allows the identification of the velocity of the evanescent wave in one of three possible regimes. When the liquid phase tends to vanish, it was observed: a) the Rayleigh wave solution or the free surface limit, and b) the rarefied fluid medium limit, where there exists a gradual extinction of the Scholte wave as both the density and velocity of the fluid decrease. In general terms, the results show that the propagation speed of a Scholte wave is less than or equal to that of a Rayleigh wave.

Dual-phase-lag model on thermo-microstretch elastic solid Under the effect of initial stress and temperature-dependent

The present paper attempts to investigate the propagation of plane waves in an isotropic elastic medium under the effect of initial stress and temperature-dependent properties. The modulus of elasticity is taken as a linear function of the reference temperature. The formulation is applied under the thermoelasticity theory with dual-phase-lag; the normal mode analysis is used to obtain the expressions for the displacement components, the temperature, the stress, and the strain components. Numerical results for the field quantities are given in the physical domain and illustrated graphically. Comparisons are made with the results predicted by different theories (Lord–Shulman theory, the classical coupled theory of thermoelasticity and the dual-phase-lag model) in the absence and presence of the initial stress as well as the case where the modulus of elasticity is independent of temperature.

Granular impacts: how far the Boussinesq model can go?

A key problem on granular impacts deals with the determination of the mechanical response of the grains due to the impact of the intruder. This topic has been poorly addressed in the literature so far, a gap to which this study aims to contribute by measuring the pressure distribution at the bottom of a loose and dry sandy bed, impacted by a heavy sphere of fixed diameter. Exploring different bed thicknesses and intruder’s dropping height, we have found that the structure of this distribution is very similar to the Boussinesq model, initially proposed for a static point-force acting over an isotropic-elastic medium. This surprising result opens up many challenging questions that could help validate or refute this model in other scenarios.

Analysis of surface effects using the theory of second-order strain gradient elasticity

In this study, we investigate the effect of surface energy on the nano-scale object using the theory of second strain gradient elasticity. This theory adds the first- and second-order gradients of strain to constitutive equations of the classical elasticity. The characteristic length scales, which represents the size of the microscopic structure of the material, and the surface energy are introduced into the constitutive equations. We solve the variational problem by using the isogeometric analysis; Galerkin method with non-uniform B-spline basis function. Strain energy density is simplified by using the irreducible decomposition of elasticity tensors under the general linear group GL(3). We conducted a numerical analysis for a hollow torus-shaped isotropic elastic medium. We observed that the surface displacement tends to dominate as the size of the material approaches the nanoscale.

Analysis and Assessment of Seismic Hazard for the Azov-Black Sea Recreation Area

One of the geological structures accountable for the implementation of seismic potential of the region is the largest vertical faults in the Earth’s interior, where earthquake foci are usually located. This article is aimed at developing a better method for calculation of stresses and strains that occur in such seismogenic areas. According to the results of the analysis of data collected during the expeditionary work, the geophysical medium is modeled by a block structure in the form of a half-space with a cut rectangular parallelepiped, which is divided into five block elements. The state of material in the geological medium is described in each block by motion equations for a homogeneous, isotropic elastic medium in the Lamé form. Following the block element method, the algorithm of the differential factorization method is implemented in each block. Based on the numerical analysis results, the main trends in contact stresses and dynamics of displacement amplitudes were determined depending on the mechanical property values of the block material and the geometric parameters of the structure.