scholarly journals Lamb's problem at its simplest

2013 ◽  
Vol 469 (2149) ◽  
pp. 20120462 ◽  
Author(s):  
Eduardo Kausel
Keyword(s):  
Half Space ◽  
Full Range ◽  
Classical Problem ◽  
Elastic Half Space ◽  
Lamb’S Problem ◽  
Complete Set ◽  
Explicit Formulae ◽  
Poisson's Ratios ◽  
Poisson’S Ratios ◽  
Single Set

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.

Anisotropic and Inhomogeneous Tensile Behavior of the Human Anulus Fibrosus: Experimental Measurement and Material Model Predictions

2000 ◽  
Vol 123 (3) ◽  
pp. 256-263 ◽  
Author(s):  
Dawn M. Elliott ◽  
Lori A. Setton
Keyword(s):  
Tensile Modulus ◽  
Material Model ◽  
Material Behavior ◽  
Anulus Fibrosus ◽  
Linear Region ◽  
Material Behaviors ◽  
Model Predictions ◽  
Complete Set ◽  
Poisson's Ratios ◽  
Poisson’S Ratios

The anulus fibrosus (AF) of the intervertebral disc exhibits spatial variations in structure and composition that give rise to both anisotropy and inhomogeneity in its material behaviors in tension. In this study, the tensile moduli and Poisson’s ratios were measured in samples of human AF along circumferential, axial, and radial directions at inner and outer sites. There was evidence of significant inhomogeneity in the linear-region circumferential tensile modulus (17.4±14.3 MPa versus 5.6±4.7 MPa, outer versus inner sites) and the Poisson’s ratio ν21 (0.67±0.22 versus 1.6±0.7, outer versus inner), but not in the axial modulus (0.8±0.9 MPa) or the Poisson’s ratios ν12 (1.8±1.4) or ν13 (0.6±0.7). These properties were implemented in a linear anisotropic material model of the AF to determine a complete set of model properties and to predict material behaviors for the AF under idealized kinematic states. These predictions demonstrate that interactions between fiber populations in the multilamellae AF significantly contribute to the material behavior, suggesting that a model for the AF as concentric and physically isolated lamellae may not be appropriate.

On Evaluation of Lamb's Integrals for Waves in a Two-Dimension Elastic Half-Space

2000 ◽  
Vol 16 (2) ◽  
pp. 109-124 ◽  
Author(s):  
Chau-Shioung Yeh ◽  
Tsung-Jen Teng ◽  
Wen-I Liao
Keyword(s):  
Half Space ◽  
Phase Function ◽  
Wave Scattering ◽  
Elastic Half Space ◽  
Steepest Descent ◽  
Inverse Mapping ◽  
Lamb’S Problem ◽  
Admissible Path ◽  
Formal Solutions ◽  
Steepest Descent Path

ABSTRACTIn this paper, a modified version of the method of steepest descent is proposed for the evaluation of Lamb's integrals which can be considered as basis functions dealing with the development of the transition matrix method which can be used to study the wave scattering in a two-dimensional elastic half-space. The formal solutions of the generalized Lamb's problem are studied and evaluated on the basis of the proposed method. After defining a phase function which presents in wavenumber integral, an exact mapping and an inverse mapping can be obtained according to the phase function. Thus, the original integration path can be deformed into an equivalent admissible path, namely, steepest descent path which passed through the saddle point, and then mapped onto a real axis of mapping plane, finally, resulted in an integral of Hermite type. This integral can be efficiently evaluated numerically in spite of either near- to far-field or low to high frequency. At the same time, the asymptotic value can easily be obtained by applying the proposed method. The numerical results for generalized Lamb's solutions are calculated and compared with analytic, asymptotic or other existing data, the excellent agreements are found. The properties of generalized Lamb's solutions are studied and discussed in details. Their possible applications for wave scattering in elastic half-space are also pointed out.

scholarly journals Surface instability of elastic half-spaces by using the energy method

2018 ◽  
Vol 474 (2213) ◽  
pp. 20170854 ◽  
Author(s):  
Yi-chao Chen ◽  
Shengyou Yang ◽  
Lewis Wheeler
Keyword(s):  
Eigenvalue Problem ◽  
Half Space ◽  
Elastic Half Space ◽  
Surface Instability ◽  
Second Variation ◽  
Equilibrium Equations ◽  
Stability And Instability ◽  
Complete Set ◽  
Instability Regions ◽  
The Second Variation

Finding the complete set of stability conditions of an elastic half-space has been an open problem ever since Biot (Biot 1963 Appl. Sci. Res. 12 , 168–182 ( doi:10.1007/BF03184638 )) first studied the surface instability of half-spaces by seeking solutions of the incremental equilibrium equations. Towards solving this problem, a method based on the energy stability criterion is developed in the present work. A variational problem of minimizing the elastic energy associated with a half-space is formulated. The second variation condition is derived and is converted to an eigenvalue problem. For a half-space of neo-Hookean materials, the eigenvalue problem is solved, which leads to complete descriptions of stability and instability regions in the deformation space.

Solutions for Axisymmetric Problems of Layered Generalized Gibson Subgrade

2014 ◽  
Vol 574 ◽  
pp. 53-57
Author(s):  
Jian Ming Jin
Keyword(s):  
Half Space ◽  
Hankel Transform ◽  
Surface Load ◽  
Shear Moduli ◽  
Space Solution ◽  
Axisymmetric Surface ◽  
Axisymmetric Problems ◽  
The Relationship ◽  
Poisson's Ratios ◽  
Poisson’S Ratios

This article presents the solutions for displacements and stress of layered generalized Gibson subgrade subjected to an axisymmetric surface load. We assume that the material has constant Poisson’s ratios (μ=0.5), and its shear moduli varies linearly with depth. During the solutions, the Hankel transform in a cylindrical co-ordinate system is employed. The relationship between the solution and half-space solution is also examined and discussed. At last, the analysis flow is presented .

Exact closed-form solutions for Lamb’s problem—III: the case for buried source and receiver

2020 ◽  
Vol 224 (1) ◽  
pp. 517-532
Author(s):  
Xi Feng ◽  
Haiming Zhang
Keyword(s):  
Closed Form ◽  
Half Space ◽  
Closed Form Solution ◽  
Time History ◽  
Natural Generalization ◽  
Elastic Half Space ◽  
Form Solution ◽  
Elementary Functions ◽  
Heaviside Step Function ◽  
Lamb’S Problem

SUMMARY In this paper, we derive the exact closed-form solution for the displacement in the interior of an elastic half-space due to a buried point force with Heaviside step function time history. It is referred to as the tensor Green’s function for the elastic wave equation in a uniform half-space, also a natural generalization of the classical 3-D Lamb’s problem, for which previous solutions have been restricted to the cases of either the source or the receiver or both are located on the free surface. Starting from the complex integral solutions of Johnson, we follow the similar procedures presented by Feng and Zhang to obtain the closed-form expressions in terms of elementary functions as well as elliptic integrals. Numerical results obtained from our closed-form expressions agree perfectly with those of Johnson, which validates our explicit formulae conclusively.

scholarly journals Weighted Residual Method for Diffraction of Plane P-Waves in a 2D Elastic Half-Space Revisited: On an Almost Circular Arbitrary-Shaped Canyon

2015 ◽  
Vol 2015 ◽  
pp. 1-21 ◽  
Author(s):  
Vincent W. Lee ◽  
Heather P. Brandow
Keyword(s):  
Half Space ◽  
Plane Waves ◽  
Classical Problem ◽  
Strong Motion ◽  
Elastic Half Space ◽  
Shear Stresses ◽  
Weighted Residual Method ◽  
P Waves ◽  
Closed Form Solutions ◽  
P And Sv Waves

Scattering and diffraction of elastic in-plane P- and SV-waves by a surface topography such as an elastic canyon at the surface of a half-space is a classical problem which has been studied by earthquake engineers and strong-motion seismologists for over forty years. The case of out-of-plane SH-waves on the same elastic canyon that is semicircular in shape on the half-space surface is the first such problem that was solved by analytic closed-form solutions over forty years ago by Trifunac. The corresponding case of in-plane P- and SV-waves on the same circular canyon is a much more complicated problem because the in-plane P- and SV-scattered-waves have different wave speeds and together they must have zero normal and shear stresses at the half-space surface. It is not until recently in 2014 that analytic solution for such problem is found by the author in the work of Lee and Liu. This paper uses the technique of Lee and Liu of defining these stress-free scattered waves to solve the problem of the scattering and diffraction of these in-plane waves on an on an almost-circular surface canyon that is arbitrary in shape.

The efficiency of application of virtual cross-sections method and hypotheses MELS in problems of wave signal propagation in elastic waveguides with rough surfaces

2016 ◽  
pp. 3564-3575 ◽  
Author(s):  
Ara Sergey Avetisyan
Keyword(s):  
Half Space ◽  
Cross Sections ◽  
Classical Method ◽  
Signal Propagation ◽  
Elastic Half Space ◽  
Layered Systems ◽  
Surface Layers ◽  
Inhomogeneous Surface ◽  
Wave Signal ◽  
The Impact

The efficiency of virtual cross sections method and MELS (Magneto Elastic Layered Systems) hypotheses application is shown on model problem about distribution of wave field in thin surface layers of waveguide when plane wave signal is propagating in it. The impact of surface non-smoothness on characteristics of propagation of high-frequency horizontally polarized wave signal in isotropic elastic half-space is studied. It is shown that the non-smoothness leads to strong distortion of the wave signal over the waveguide thickness and along wave signal propagation direction as well.  Numerical comparative analysis of change in amplitude and phase characteristics of obtained wave fields against roughness of weakly inhomogeneous surface of homogeneous elastic half-space surface is done by classical method and by proposed approach for different kind of non-smoothness.

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