Elementary solutions to Lamb's problem for a point source and their relevance to three-dimensional studies of spontaneous crack propagation

1979 ◽  
Vol 69 (4) ◽  
pp. 947-956
Author(s):  
Paul G. Richards
Keyword(s):  
Free Surface ◽  
Arbitrary Shape ◽  
Three Dimensional ◽  
Elastic Half Space ◽  
P Wave ◽  
Displacement Discontinuity ◽  
Shear Cracks ◽  
Lamb’S Problem ◽  
Square Roots ◽  
Tension Cracks

abstract Certain exact solutions to Lamb's problem (the transient response of an elastic half-space to a force applied at a point) involve the computation merely of three square roots, and about ten arithmetic operations (+, −, ×, /). They arise when both source and receiver lie on the free surface. It is just these solutions which are needed in a method due to Hamano for obtaining the slip function (displacement discontinuity), as a function of space and time, for planar tension cracks and shear cracks which grow spontaneously with arbitrary shape. The solutions are described here in detail, for an elastic medium with general Poisson's ratio. They include perhaps the simplest-possible example of the P-wave.

Transient interior analytical solutions of Lamb’s problem

2019 ◽  
Vol 24 (11) ◽  
pp. 3485-3513 ◽  
Author(s):  
Mohamad Emami ◽  
Morteza Eskandari-Ghadi
Keyword(s):  
Analytical Solutions ◽  
Three Dimensional ◽  
Time History ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Integral Transforms ◽  
Elastic Half Space ◽  
Point Load ◽  
Lamb’S Problem ◽  
Interior Points

The classical three-dimensional Lamb’s problem is considered for an inclined surface point load of Heaviside time dependence. Attention is focused upon the acquisition of the transient elastodynamic analytical solutions for interior points through a unified method of analysis that is valid for arbitrary Lamé constants. The method of elastodynamic potentials is employed jointly with integral transforms to treat the corresponding initial boundary value problem. To derive the time-domain solutions, some integral equations are encountered, the solutions of which are found via a modified version of the Cagniard–Pekeris method. The final solutions are obtained as finite integrals that are amenable to numerical calculations. They are also expressed in the form of Green’s functions. The limit case of infinite time is investigated analytically to derive the closed-form expressions for the limits of the solutions as the temporal variable tends to infinity. As expected, the results are found to be equivalent to Boussinesq–Cerruti solutions in elastostatics. The elastodynamic solutions are also evaluated numerically to plot several time-history diagrams, depicting the transient motions of the interior points, especially of the points close to the boundary so as to illustrate the formation of forced Rayleigh waves at shallow depths within the elastic half-space.

Self-similar circular shear cracks lacking cohesion

1974 ◽  
Vol 64 (6) ◽  
pp. 1789-1808 ◽  
Author(s):  
Robert Burridge ◽  
Cesar Levy
Keyword(s):  
Wave Speed ◽  
Three Dimensional ◽  
Analytic Solutions ◽  
Two Dimensions ◽  
P Wave ◽  
Shear Cracks ◽  
Antiplane Strain ◽  
S Wave ◽  
Complete Evaluation ◽  
Crack Problems

abstract It has recently been shown (Burridge, 1973) that in two dimensions plane-strain shear cracks lacking cohesion may run at speeds ranging from the Rayleigh-wave to the S-wave speed or possibly at the P-wave speed. On the other hand, it has long been known that in antiplane strain, cracks lacking cohesion must run at least at the S-wave speed. Since locally at the edge of a three-dimensional crack there is a combination of plane and antiplane strain, we have sought and found solutions for circular shear cracks expanding at the S-wave speed and at the P-wave speed. These have finite shear tractions ahead of the crack and so may correspond to frictional sliding in the absence of cohesion. The analysis combines the method of Kostrov (1964b) with the results of Burridge (1973). We carry out a complete evaluation for the displacements and tractions on the fault plane, and far-field radiation for the S-wave-speed crack. The corresponding evaluations for the P-wave speed are not elementary and are not attempted here. As far as the authors are aware, these are the first analytic solutions of three-dimensional crack problems which satisfy a physically plausible fracture criterion for failure under shear.

scholarly journals Scattering of Surface Waves by a Three-Dimensional Cavity of Arbitrary Shape: Analytical and Experimental Studies

2019 ◽  
Vol 9 (24) ◽  
pp. 5459 ◽  
Author(s):  
Jaesun Lee ◽  
VanTrung Ngo ◽  
Haidang Phan ◽  
TruongGiang Nguyen ◽  
Duy Kien Dao ◽  
...  
Keyword(s):  
Surface Waves ◽  
Cylindrical Cavity ◽  
Arbitrary Shape ◽  
Experimental Studies ◽  
Three Dimensional ◽  
Scattered Wave ◽  
Ultrasonic Pulse ◽  
Elastic Half Space ◽  
Cavity Surface ◽  
Vertical Displacements

The scattering of surface waves by a three-dimensional shallow cavity of arbitrary shape at the surface of a homogenous, isotropic, linearly elastic half-space is theoretically investigated. A novel analytical approach based on a reciprocity consideration is introduced in this article to determine the particle displacements of the scattered wave field generated by the interaction between the surface waves and the cavity. In the usual manner, the scattered field was shown to be equivalent to the radiation from the distribution of tractions, calculated from the incident wave, on the surface of the cavity. The radiation of surface waves subjected to the computed tractions applied at a single location was found using reciprocity theorems. The field scattered by the cavity was subsequently obtained from the superposition of displacements due to all the forces applied on the cavity surface. Solutions for the scattering of surface waves by a spherical, a circular cylindrical (coin-shaped) and a square cylindrical cavity are presented in detail. We here derive the closed-form expressions of the displacement amplitudes, which represent the far-field scattered waves produced by each of the cavities. An experimental setup using the ultrasonic pulse-echo technique was then carried out to record the scattered echoes of surface waves from these cavities in order to provide practical validation of the analytical findings. The vertical displacements measured at a significant distance of about twenty-five wavelengths from the cavities of the same width and different depth were compared with the corresponding theoretical predictions. The comparisons show excellent agreement for the case of a spherical cavity and good agreement in the cases of a circular and a cylindrical cavity in terms of trends and magnitudes. It is followed by a discussion on the results of the comparison and the limitations of the proposed approach regarding the degree of smoothness and the size of cavity.

scholarly journals An Analytical Solution for Acoustic Emission Source Location for Known P Wave Velocity System

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Longjun Dong ◽  
Xibing Li ◽  
Gongnan Xie
Keyword(s):  
Acoustic Emission ◽  
Analytical Solution ◽  
Linear Equations ◽  
Three Dimensional ◽  
Source Location ◽  
Emission Source ◽  
P Wave ◽  
Square Roots ◽  
Existence And Multiplicity ◽  
Acoustic Emission Source

This paper presents a three-dimensional analytical solution for acoustic emission source location using time difference of arrival (TDOA) measurements from N receivers, N⩾5. The nonlinear location equations for TDOA are simplified to linear equations, and the direct analytical solution is obtained by solving the linear equations. There are not calculations of square roots in solution equations. The method solved the problems of the existence and multiplicity of solutions induced by the calculations of square roots in existed close-form methods. Simulations are included to study the algorithms' performance and compare with the existing technique.

Reflected Amplitude of Elastic Wave at Free Surface

2011 ◽  
Vol 287-290 ◽  
pp. 1888-1891
Author(s):  
Xue Ying Yang ◽  
Bo Zhang ◽  
Dun Fu Zhang
Keyword(s):  
Free Surface ◽  
Elastic Wave ◽  
Analytic Solution ◽  
Harmonic Wave ◽  
Wave Theory ◽  
Elastic Half Space ◽  
Wave Functions ◽  
P Wave ◽  
Sh Wave ◽  
Propagation Media

The analytic solution of incident P wave, SV wave and SH wave amplitudes reflected at free surface in elastic medium was presented in this paper. Taking the assumption that the wave propagation media are in elastic half space and the seismic wave is plane simple harmonic wave, the incident wave and reflected wave functions of P wave, SV wave and SH wave were deduced with elastic wave theory, and the wave amplitudes reflected at free surface in elastic media was presented

Prediction of Ship Motions Via a Three-Dimensional Time-Domain Method Following a Quad-Tree Adaptive Mesh Technique

2020 ◽  
Vol 27 (1) ◽  
pp. 29-38
Author(s):  
Teng Zhang ◽  
Junsheng Ren ◽  
Lu Liu
Keyword(s):  
Free Surface ◽  
Incident Wave ◽  
Time Domain ◽  
Three Dimensional ◽  
Adaptive Mesh ◽  
Wave Profile ◽  
Time Domain Method ◽  
Ship Motions ◽  
Quad Tree ◽  
Mesh Technique

AbstractA three-dimensional (3D) time-domain method is developed to predict ship motions in waves. To evaluate the Froude-Krylov (F-K) forces and hydrostatic forces under the instantaneous incident wave profile, an adaptive mesh technique based on a quad-tree subdivision is adopted to generate instantaneous wet meshes for ship. For quadrilateral panels under both mean free surface and instantaneous incident wave profiles, Froude-Krylov forces and hydrostatic forces are computed by analytical exact pressure integration expressions, allowing for considerably coarse meshes without loss of accuracy. And for quadrilateral panels interacting with the wave profile, F-K and hydrostatic forces are evaluated following a quad-tree subdivision. The transient free surface Green function (TFSGF) is essential to evaluate radiation and diffraction forces based on linear theory. To reduce the numerical error due to unclear partition, a precise integration method is applied to solve the TFSGF in the partition computation time domain. Computations are carried out for a Wigley hull form and S175 container ship, and the results show good agreement with both experimental results and published results.

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