Lamb's Problem Research Articles

Lamb's problem on random mass density fields with fractal and Hurst effects.

This paper reports on a generalization of Lamb's problem to a linear elastic, infinite half-space with random fields (RFs) of mass density, subject to a normal line load. Both, uncorrelated and correlated (with fractal and Hurst characteristics) RFs without any weak noise restrictions, are proposed. Cellular automata (CA) is used to simulate the wave propagation. CA is a local computational method which, for rectangular discretization of spatial domain, is equivalent to applying the finite difference method to the governing equations of classical elasticity. We first evaluate the response of CA to an uncorrelated mass density field, more commonly known as white-noise, of varying coarseness as compared to CA's node density. We then evaluate the response of CA to multiscale mass density RFs of Cauchy and Dagum type; these fields are unique in that they are able to model and decouple the field's fractal dimension and Hurst parameter. We focus on stochastic imperfection sensitivity; we determine to what extent the fractal or the Hurst parameter is a significant factor in altering the solution to the planar stochastic Lamb's problem by evaluating the coefficient of variation of the response when compared with the coefficient of variation of the RF.

On attenuation of free and forced waves in an infinitely long visco-elastic layer of a constant thickness

The conventional concepts of a loss factor and a complex-valued elastic module are used to study wave attenuation in a visco-elastic layer. The hierarchy of reduced order models is employed to assess attenuation levels of free and forced waves in various situations. First, the free waves are considered. In the low frequency limit, the attenuation of these waves is found to be in the excellent agreement with the existing knowledge. At high frequencies, predictions of the reduced order models fully agree with the solutions of exact Rayleigh–Lamb problem. Alternative excitation cases are considered for the forcing problem and a measure of the attenuation level is proposed and validated. The differences between two regimes, the low frequency one, when a waveguide supports only one propagating wave, and the high frequency one, when several waves are supported, are demonstrated and explained.

Seismic waves in a three-dimensional block medium

We study numerically the propagation of seismic waves in a three-dimensional block medium. The medium is modelled by a spatial lattice of masses connected by elastic springs and viscous dampers. We study Lamb's problem under a surface point vertical load. The cases of both step and pulse load are considered. The displacements and velocities are calculated for surface masses. The influence of the viscosity of the dampers on the attenuation of perturbations is studied. We compare our numerical results for the block medium with known analytical solutions for the elastic medium.

Exterior 3D lamb problem: Harmonic load distributed over a surface

The solutions of the exterior Lamb problem with a distributed harmonic surface load acting on the boundary of an elastic half-space are studied. A load normal to the surface and distributed over the surface as the Poisson kernel is considered. The solution is constructed with the use of integral transforms and the finite-element method.

Elastic orthotropic or transversely-isotropic half-plane under the action of normal unsteady surface perturbations

Abstract The unsteady plane problem of the action of different normal perturbations on an elastic orthotropic or transversely-isotropic uniform half-plane is considered. Its solution is represented in the form of convolutions of the perturbations with surface influence functions. Explicit expressions for these functions are found using a Laplace transform with respect to time and a Fourier transform with respect to a spatial coordinate. The corresponding preimages are determined using their analytical representations. Without major changes, this method enables us to obtain explicit formulae for the remaining influence functions (such as, for example, those corresponding to tangential perturbations). The reliability of the method was evaluated by passing to the limit of an isotropic medium, as a result of which the well-known formulae for plane Lamb problem were obtained. The special features of the behaviour of influence functions are explained. The influence functions found allow us to investigate the stress-strain state of a half-plane for any normal perturbation using quadratures.

Interface waves in almost incompressible elastic materials

We study the problem of two elastic half-planes in contact and the Stoneley interface wave that may exist at the interface between two different elastic materials, emphasis being put on the case when the half-planes are almost incompressible. We show that numerical simulations involving interface waves require an unexpectedly high number of grid points per wavelength as the materials become more incompressible. Let λ, μ, ρ and λ′, μ′, ρ′ be the Lamé parameters and densities of the first and second half-plane, respectively. A theoretical study shows that if K is a real constant, λ′=Kλ, μ′=Kμ, ρ′=Kρ and μ→0, then for an accurate solution the required number of grid points per wavelength scales as (μ/λ)−1/p, where p is the order of accuracy of the numerical method. This requirement becomes very restrictive close to the incompressible limit μ≪λ, especially for lower order methods i.e., a small p. The theoretical findings are supported by numerical experiments that illustrate the demanding resolution requirement as well as the superiority of higher order methods. The scaling is also seen to hold for a more general choice of Lamé parameters. Numerical experiments when one of the half-planes is a vacuum are also presented, where the higher resolution requirement is illustrated in a numerical solution of Lamb's problem.

Time-Harmonic Lamb Problem for a System Consisting of a Prestressed Piezoelectric Covering Layer and a Prestressed Half-Plane

Within the framework of a piecewise homogeneous body model, with the use of exact equations of the linearized theory of electroelasticity for initially stressed bodies, the time-harmonic Lamb problem for a system consisting of a prestressed piezoelectric covering layer and a half-plane is studied. The boundary value problem considered is solved by employing the Fourier exponential transformation technique with respect to the coordinates along the interface. An algorithm is proposed and employed to obtain numerical results for the distribution of the normal and shear stresses acting on the interface plane.

Wave fields and domination regions for the interior Lamb problem

The domination regions of wave fields in the epicentral region are analyzed for the interior Lamb problem on the action of a lumped force applied inside an elastic half-plane. The solutions obtained by integral representations and finite-element approximations are compared. The domination regions are distinguished for the first time for all types of acoustic waves observed near the epicenter.

Reducing spurious oscillations in discontinuous wave propagation simulation using high-order finite elements

In this paper, a modified explicit time integration scheme is proposed for simulating the propagation of discontinuous waves. The spatial domain is discretized by the finite element method. To obtain accurate results, the standard finite element method requires a very fine mesh, which is why the computational effort can be very time consuming. The use of high-order finite element methods–such as the spectral element method based on Lagrange polynomials through Gauss–Lobatto–Legendre points or the iso-geometric analysis using non-uniform rational B-splines–can reduce the enormous computational costs significantly, compared to the standard finite element method. However, explicit time integration schemes such as the central difference method cannot eliminate the spurious oscillation near the front wave. The procedure proposed here is basically similar to the explicit method of Noh and Bathe, but the semi-discrete equation of motion is modified by introducing a damping parameter to suit the high-order FEM analysis of the discontinuous wave propagation. The performance due to this modification is tested for one-dimensional wave propagation problems using both lumped and consistent mass matrices. Then, the idea of a combination of the consistent and row sum lumped mass matrices is evaluated. The proposed method is studied also in two-dimensions by considering the Lamb problem of wave propagation. The results are promising enough to provide a better simulation of the discontinuous wave propagation problem.

Influence of polled direction on the stress distribution in piezoelectric materials

In this paper, the influence of the polled direction of piezoelectric materials on the stress distribution is studied under time-harmonic dynamical load (time-harmonic Lamb's problem). The system considered in this study consists of piezoelectric covering layer and piezoelectric half-plane, and the harmonic dynamical load acts on the free face of the covering layer. The investigations are carried out by utilizing the exact equations of motion and relations of the linear theory of electro-elasticity. The plane-strain state is considered. It is assumed that the perfect contact conditions between the covering layer and half-plane are satisfied. The boundary value problems under consideration are solved by employing Fourier exponential transformation techniques with respect to coordinates directed along the interface line. Numerical results on the influence of the polled direction of the piezoelectric materials such as PZT-5A, PZT-5H, PZT-4 and PZT-7A on the normal stresses, shear stresses and electric potential acting on the interface plane are presented and discussed. As a result of the analyses, it is established that the polled directions of the piezoelectric materials play an important role on the values of the studied stresses and electric potential.

Planar internal Lamb problem: Waves in the epicentral zone of a vertical power source

We analyze the field in the epicentral zone for the internal Lamb problem of the action of a concentrated force within an elastic half-plane. We compare the solutions obtained using integral representations, geometric-optical methods, and finite-element approximations.

Second-order seismic wave simulation in the presence of a free-surface by pseudo-spectral method

In this paper second-order seismic wave equations are adopted to simulate the wave propagation in the presence of a free-surface. In the simulations, horizontal and vertical spatial derivatives are solved by Fourier and Chebyshev pseudo-spectral algorithms respectively. In addition, Fourier–Chebyshev pseudo-spectral method is also used to solve the boundary conditions of the free-surface, which are Neumann boundaries. That makes results at the free-surface keep same high precision with inner domain. The new scheme is tested against Lamb's problem in a uniform elastic half space. Agreement between numerical and analytical results is every good. Compared with the former simulation scheme based first-order velocity–stress equations, the new scheme based on second-order equations is more efficient because it has fewer controlling equations.

The plane lamb problem for a 2D discrete medium

The propagation of unsteady waves under the effect of a vertical stepped concentrated load at the boundary of a discrete medium is investigated. The problem is solved by two methods. Asymptotic solutions for a prolonged time from the beginning of impact are found. The solution over the entire temporal range is found numerically. These solutions are compared with each other and the applicability limits of the analytical solution are determined.

Probabilistic and Possibilistic Estimates of Generating-Set Vibrations at HPP Due to Water Pulsation in Toe Basin

The problem of computational assessment of the vibrations of a generating set at an HPP due to water pulsation in the toe basin with consideration of incomplete (doubtful) information on the properties of the bed material is examined. Asolution is derived on the basis of the Lamb problem of the propagation of elastic waves in a half plane due to the dynamic effect applied to a portion of its boundary. Point (probabilistic) and interval (with use of possibility theory) estimates are obtained for the level of vibration and compared with results of measurements on, for example, a generating set at the Sayano-Shushenskaya HPP.

Time-Harmonic Lamb Problem for a System Consisting of an Imperfectly Bonded Orthotropic Covering Layer and an Orthotropic Half-Plane

The time-harmonic Lamb problem for a system consisting of an imperfectly bonded covering layer and an orthotropic half-plane is studied. It is assumed that the materials of constituents of the system considered are orthotropic, and the imperfectness of contact conditions is modeled by using a shear-spring model. A plane strain state is considered, and an algorithm for calculating the dynamical stress field is developed and employed. Numerical results on the influence of the shear-spring imperfectness parameter on the distribution of normal contact stresses are presented and discussed. In particular, it is established that the numerical results obtained at limit values of the parameter coincide with the results obtained in the previous work of the authors.

A hierarchy of high-order theories for symmetric modes in an elastic layer

Abstract A hierarchy of high-order theories for symmetric modes in an infinitely long elastic layer of the constant thickness is derived by means of the inertia-corrected polynomial approximations. For each member of the hierarchy, boundary conditions for layers of the finite length are formulated. The forcing problems at several approximation levels are solved with the use of the bi-orthogonality conditions. Accuracy of these approximations is assessed by comparison of results with the exact solution of the Rayleigh–Lamb problem. Special attention is paid to the power flow analysis in alternative modal excitation cases and to the applicability of the Saint-Venant׳s principle in stationary elasto-dynamics.

The discrete Lamb problem: Elastic lattice waves in a block medium

We study the propagation of transient waves under the action of a vertical step point load on the surface of a half-space filled by a block medium. The block medium is modeled by a square lattice of masses connected by springs in the directions of the axes x,y, and in the diagonal directions. The problem is solved by two methods. Analytically, we obtain asymptotic solutions in the vicinity of the Rayleigh wave at large time intervals. Numerically, we obtain a solution for any finite time interval. We compare these solutions with each other and with the solution to the Lamb problem for an elastic medium.

Time-harmonic Lamb's problem for a system comprising a piezoelectric layer and piezoelectric half-plane

The time-harmonic Lamb problem is considered for the system consisting of a piezoelectric covering layer and piezoelectric half-plane. The investigations are carried out within the framework of the piecewise homogeneous body model by utilizing the exact equations of motion and relations of the linear theory of electro-elasticity. The plane-strain state is considered and it is assumed that between the covering layer and half-plane perfect contact conditions are satisfied. The boundary value problems under consideration are solved by employing Fourier exponential transformation techniques with respect to coordinates directed along the interface line. An algorithm is proposed and employed to obtain numerical results on the distribution of the normal and shear stresses acting on the interface plane. Moreover, numerical results on the distribution of the electric potential in the covering layer are also presented and discussed. The piezoelectric materials PZT-5A, PZT-5H, PZT-4 and PZT-7A are examined. In particular, it is established that the piezoelectricity of the covering layer material (half-plane material) causes to decrease (increase) the absolute values of the aforementioned stresses acting on the interface plane.

On the effect of damping on dispersion curves in plates

This paper presents a study on quantitative prediction and understanding of time-harmonic wave characteristics in damped plates. Material dissipation is modelled by using complex-valued velocities of free dilatation and shear waves in an unbounded volume. As a numerical example, solution of the classical Rayleigh–Lamb problem for a viscoelastic plate is presented to illustrate and discuss the role of dissipation in the cut-off phenomenon and in the phenomenon of veering for dispersion curves. These phenomena are explained in more detail considering a simple model, which allows accurate asymptotic analysis of the perturbation of dispersion curves in the regions of cut-off and veering.

Lamb's problem at its simplest

This article revisits the classical problem of horizontal and vertical point loads suddenly applied onto the surface of a homogeneous, elastic half-space, and provides a complete set of exact, explicit formulae which are cast in the most compact format and with the simplest possible structure. The formulae given are valid for the full range of Poisson's ratios from 0 to 0.5, and they treat real and complex poles alike, as a result of which a single set of formulae suffices and also exact formulae for dipoles can be given.