Elastic Full Space Research Articles

Bayesian Inversion of MDJ Seismograms for Source Parameters and Yields of DPRK Underground Nuclear Explosions 2 to 6

AbstractThe Democratic People’s Republic of Korea (DPRK) conducted six underground nuclear explosions at the Punggye-ri nuclear test site at Mount Mantap, a granite peak. Test 1 was separate from tests 2 to 6, which were within about 1 km of each other. Using seismograms recorded at Mudanjiang (MDJ) seismic station in China, I propose a new approach to obtain source parameters, source time functions and yields of events 2 to 6, assuming they share the same Green’s function from Punggye-ri to MDJ. Each source is modelled as a spherical cavity in a homogeneous isotropic elastic full space, with four independent parameters constrained by published data on the properties of granite and analysis of the recorded MDJ seismograms. The effect of the ground surface is included as a planar reflection that modifies the pressure at the cavity boundary. The Green’s function for each event is estimated by deconvolving the seismogram for the estimated source time function. Very fast simulated annealing (VFSA) is used to search the parameter space to minimise the root-mean square difference among the estimated Green’s functions and their mean. The estimated Green’s functions are similar and differ in amplitude by less than a factor of 2. Green’s functions from Punggye-ri to other seismic stations may be obtained by deconvolving the seismograms for the corresponding source time functions. Mapping the nonlinear zones surrounding each explosion indicates that this part of the site was destroyed by the underground nuclear tests before its official destruction on 24 May 2018.

The relations between the corner frequency, seismic moment and source dynamic parameters derived from the spontaneous rupture of a circular fault

SUMMARYThe stress drop is an important dynamic source parameter for understanding the physics of source processes. The estimation of stress drops for moderate and small earthquakes is based on measurements of the corner frequency ${f_c}$, the seismic moment ${M_0}$ and a specific theoretical model of rupture behaviour. To date, several theoretical rupture models have been used. However, different models cause considerable differences in the estimated stress drop, even in an idealized scenario of circular earthquake rupture. Moreover, most of these models are either kinematic or quasi-dynamic models. Compared with previous models, we use the boundary integral equation method to simulate spontaneous dynamic rupture in a homogeneous elastic full space and then investigate the relations between the corner frequency, seismic moment and source dynamic parameters. Spontaneous ruptures include two states: runaway ruptures, in which the rupture does not stop without a barrier, and self-arresting ruptures, in which the rupture can stop itself after nucleation. The scaling relationships between ${f_c}$, ${M_0}$ and the dynamic parameters for runaway ruptures are different from those for self-arresting ruptures. There are obvious boundaries in those scaling relations that distinguish runaway ruptures from self-arresting ruptures. Because the stress drop varies during the rupture and the rupture shape is not circular, Eshelby's analytical solution may be inaccurate for spontaneous dynamic ruptures. For runaway ruptures, the relations between the corner frequency and dynamic parameters coincide with those in the previous kinematic or quasi-dynamic models. For self-arresting ruptures, the scaling relationships are opposite to those for runaway ruptures. Moreover, the relation between ${f_c}$ and ${M_0}$ for a spontaneous dynamic rupture depends on three factors: the dynamic rupture state, the background stress and the nucleation zone size. The scaling between ${f_c}$ and ${M_0}$ is ${f_c} \propto {M_0^{ - n}}$, where n is larger than 0. Earthquakes with the same dimensionless dynamic parameters but different nucleation zone sizes are self-similar and follow a ${f_c} \propto {M_0^{ - 1/3}}$ scaling law. However, if the nucleation zone size does not change, the relation between ${f_c}$ and ${M_0}$ shows a clear departure from self-similarity due to the rupture state or background stress.

Periodic crack problems for an elastic layer

The problems of periodic mode I 3 D cracks for an elastic layer are reduced to an integro-differential equation (IDE). The principal part of the IDE is the same as in the classical Griffith problem for one crack in an elastic full space. The periodic system of elliptic cracks, featuring equal spaces between neighboring cracks, lies in the middle plane of the layer. Each crack is parallel to the layer’s faces. Four types of boundary conditions are considered on the layer’s faces (sliding or rigid support, stress-free state, sliding support in between elastic layers). The regular asymptotic method has been applied to solving the problems as it proves effective for some fairly big spaces between cracks as well as for relatively thick layers. The stress intensity factor (SIF) has been calculated for a variety of geometric and mechanical parameters. It was compared to that for the well-known case of a single crack in a layer. It has been shown that the SIF for the crack system can be smaller or greater than that for a single crack depending on the boundary conditions on the layer’s faces.

Fundamental solutions in 3D elastodynamics for the BEM: A review

In this work, we attempt to collect all available fundamental solutions in 3D elastodynamics that would be suitable for use in Boundary Element Method (BEM) formulations. Given that research on transient and steady state cases involving the elastic, homogeneous and isotropic continuum commenced in the 1950′s, it is practically impossible to reference all work done since then, so we apologize a priori for any omissions on our part. When we use the term fundamental solution, what comes to mind is the elastic full space under a point load in space and time and with the radiation condition arising because the medium has no boundaries, i.e. it extends to infinity. However, this is far from being the end: If the full space becomes a half-space, then boundary conditions enter the picture and now we are talking about Green's functions. These are more desirable for BEM formulations because it now becomes unnecessary to discretize the free surface. In general, the more particular features of the problem at hand are included in the Green's function, the less discretization is necessary in the BEM formulation. What we aim for in this review is to present and briefly discuss the basic fundamental solutions and more specialized Green's functions in a 3D elastic continuum, in either the frequency or the time domain, for the following type of materials: (1) Isotropic and homogeneous; (2) isotropic and inhomogeneous; (3) anisotropic and homogeneous; (4) anisotropic and inhomogeneous. We also look at poroelastic materials with all above possible combinations. We note at this point that the continuously inhomogeneous (e.g., functionally graded) material is understood as one having its material parameters as functions of position. There is, of course, the category of layered media, considered here as discretely inhomogeneous materials. One final note of caution has to do with the numerical implementation of these solutions: Some are very difficult to program, despite the fact that they come in closed form, because they may involve integrals, or there may be turning points in the solution where the form available depends on the frequency, or because of round-off errors in computing special functions. In closing, there is a trade-off between easy to implement fundamental solutions that require substantial spatial discretization effort and advanced ones based on Green's functions that are difficult to implement but require minimal discretization effort. This points out the relevance of speeding up BEM computations, a subject that is treated in the Appendix.

Periodic crack system in a layered elastic wedge

ABSTRACTThree-dimensional periodic mode I crack problems for an elastic layered wedge are reduced to an integro-differential equation principal part of which is the same as in the classical Griffith problem for one crack in an elastic full space. The infinite system of elliptic cracks, with equal spaces between neighbouring cracks, is situated in the middle of the wedge parallel to its edge. The faces of the wedge with cracks are layered, with sliding support, by identical elastic incompressible wedge-shaped layers so that the system of three wedges have the same edge (for example, rubber–metal–rubber system). Three types of boundary conditions on the outer layered wedge faces are considered (sliding or rigid support, stress-free state). The regular asymptotic method is used to solve the problems effective for fairly big spaces between the cracks as well as for big distances between the cracks and the edge of the wedge. The stress intensity factor (SIF) is calculated for different geometric parameters. It is shown that the solution, including the SIF, for close cracks does not practically depend on the type of the outer boundary conditions.

Combined Effect of Pressure and Shear Stress on Penny-Shaped Fluid-Driven Cracks

Penny-shaped fluid-driven cracks are often detected in many fluid–solid interaction problems. We study the combined effect of pressure and shear stress on the crack propagation in an impermeable elastic full space. Boundary integral equations are presented, by using the integral transform method, for a penny-shaped crack under normal and shear stresses. The crack propagation criterion of stress intensity factor is examined with the strain energy release rate. Dominant regimes are obtained by using a scaling analysis. Asymptotic solution of the toughness-dominant regime is derived to show the effect of shear stress on the crack opening, crack length, and pressure distribution. The results indicate that a singular shear stress can dominate the asymptotic property of the stress field near the crack tip, and the stress intensity factor cannot be calculated even though the energy release rate is finite. Shear stress leads to a smaller crack opening, a longer crack, and a slightly larger wellbore pressure. A novel dominant-regime transition between shear stress and pressure is found. Unstable crack propagation occurs in the shear stress-dominant regime. This study may help in understanding crack problems under symmetrical loads and modeling fluid–solid interactions at the crack surfaces.

THE FMM-IBEM SOLUTION OF THREE-DIMENSIONAL SCATTERING OF ELASTIC WAVES

Boundary element method has unique advantages to solve elastic wave scattering problems, while the asymmetric and dense matrix equation greatly restrict the application of traditional boundary element in practical engineering. Therefore, based on the single-layer potential theory, combined with the fast multipole expansion technique, a new rapid indirect boundary element method is developed by using the Taylor series expansion of compressible and shear wave potential function for solving large scale elastic wave scattering problems efficiently and accurately. It is verified that this method greatly improves the computing speed and reduces the calculation storage. Then the elastic wave scattering problem for millions of DOFs can be solved effectively and accurately on an ordinary personal workstation. Finally, the scattering problem of P wave by ellipsoidal cavities group in elastic full-space is solved, and some important characteristics for the displacement and stress fields in steady state are discussed based on the numerical results. This method is especially suitable for solving the multiple scattering problems at low dimensionless frequency (ka<5.0).

Dynamic response of a frame-foundation-soil system: a coupled BEM–FEM procedure and a GPU implementation

Graphics processing units have experienced an increasing demand for general-purpose computer applications such as engineering, science, finance, among other areas. This study uses such devices to analyze the performance of a code designed to evaluate the dynamic response of frame-foundation-soil system. In the present article, a direct version of the boundary element method (BEM) is applied to synthesize the 3D dynamic compliance matrix of a rigid and massless foundation interacting with unbounded soil profiles. The foundation compliance matrix is coupled to a frame, modeled by the finite element method (FEM) leading to the dynamic response of a coupled frame-foundation-soil system. The direct version of the 3D boundary element method is built based on the stationary fundamental solution of an elastic full-space. Viscoelastic effects are incorporated by means of the elastic–viscoelastic correspondence principle. The article describes the methodology applied to couple rigid bodies with the BEM mesh. The soil-foundation interaction result is given in terms of a dynamic compliance matrix for a rigid and massless foundation. The frame structural analysis is based on FEM where bar and beam elements are coupled. This strategy allows performing frequency domain analysis of such structural systems. The performance analysis is done by comparing two codes: one was exclusively developed in C language and the other one in CUDA C (Compute Unified Device Architecture). The level of speedup was achieved by implementing some functions in the frame model, i.e., on the FEM based code.

Line-integral representations of the displacement and stress fields due to an arbitrary Volterra dislocation loop in a transversely isotropic elastic full space

Transversely isotropic materials or hexagonal crystals are commonly utilized in various engineering fields; however, dislocation solutions for such special materials have not been fully developed. In this paper, we present a comprehensive study on this important topic, where only Volterra dislocations of the translational type are considered. Based on the potential theory of linear elasticity, we extend the well-known Burgers displacement equation for an arbitrarily shaped dislocation loop in an isotropic elastic full space to the transversely isotropic case. Both the induced displacements and stresses are expressed uniformly in terms of simple and explicit line integrals along the dislocation loop. We introduce three quasi solid angles to describe the displacement discontinuities over the dislocation surface and extract a simple step function out of these angles to characterize the dependence of the displacements on the configuration of the dislocation surface. We also give a new explicit formula for calculating accurately and efficiently the traditional solid angle of an arbitrary polygonal dislocation loop. From the present line-integral representations, exact closed-form solutions in terms of elementary functions are further obtained in a unified way for the displacement and stress fields due to a straight dislocation segment of arbitrary orientation. The non-uniqueness of the elastic field solution due to an open dislocation segment is rigorously discussed and demonstrated. For a circular dislocation loop parallel to the plane of isotropy, a new explicit expression of the induced elastic field is presented in terms of complete elliptic integrals. Several numerical examples are also provided as illustration and verification of the derived dislocation solutions, which further show the importance of material anisotropy on the dislocation-induced elastic field, and reveal the non-uniqueness feature of the elastic field due to a straight dislocation segment.

A boundary-volume integral equation method for the analysis of wave scattering

A method for the analysis of wave scattering in 3-D elastic full space is developed by means of the coupled boundary-volume integral equation, which takes into account the effects of both the boundary of inclusions and the uctuation of the wave field. The wavenumber domain formulation is used to construct the Krylov subspace by means of FFT. In order to achieve the wavenumber domain formulation, the boundary-volume integral equation is transformed into the volume integral equation. The formulation is also focused on this transform and its numerical implementation. Several numerical results clarify the accuracy and effectiveness of the present method for scattering analysis.

The effect of a twin tunnel on the propagation of ground-borne vibration from an underground railway

Accurate predictions of ground-borne vibration levels in the vicinity of an underground railway are greatly sought after in modern urban centres. Yet the complexity involved in simulating the underground environment means that it is necessary to make simplifying assumptions about this system. One such commonly made assumption is to ignore the effects of neighbouring tunnels, despite the fact that many underground railway lines consist of twin-bored tunnels, one for the outbound direction and one for the inbound direction. This paper presents a unique model for two tunnels embedded in a homogeneous, elastic fullspace. Each of these tunnels is subject to both known, dynamic train forces and dynamic cavity forces. The net forces acting on the tunnels are written as the sum of those tractions acting on the invert of a single tunnel, and those tractions that represent the motion induced by the neighbouring tunnel. By apportioning the tractions in this way, the vibration response of a two-tunnel system is written as a linear combination of displacement fields produced by a single-tunnel system. Using Fourier decomposition, forces are partitioned into symmetric and antisymmetric modenumber components to minimise computation times. The significance of the interactions between two tunnels is quantified by calculating the insertion gains, in both the vertical and horizontal directions, that result from the existence of a second tunnel. The insertion-gain results are shown to be localised and highly dependent on frequency, tunnel orientation and tunnel thickness. At some locations, the magnitude of these insertion gains is greater than 20 dB. This demonstrates that a high degree of inaccuracy exists in any surface vibration prediction model that includes only one of the two tunnels. This novel two-tunnel solution represents a significant contribution to the existing body of research into vibration from underground railways, as it shows that the second tunnel has a significant influence on the accuracy of vibration predictions for underground railways.

Stiffness of a Flexible Circular Footing Embedded in an Elastic Half-Space

This paper addresses the problem of a circular footing of finite but nonzero stiffness embedded in an isotropic nonhomogeneous elastic half-space. The problem is solved by coupling the scaled boundary finite-element method with axisymmetric shell finite elements. The coupling of the two methods is validated by comparing computed solutions with analytical solutions for a flexible footing subjected to a uniformly distributed vertical load embedded in an elastic full-space. Based on the finding that the bending stiffness of the footing dictates the response for vertical and moment load cases, whereas the normal stiffness of the footing dominates the horizontal response, a convenient method for estimating dimensionless elastic stiffness coefficients is presented graphically. New results are presented for homogeneous and Gibson soil profiles and Poisson’s ratios of 0.2 and 0.499, to represent both sand and clay. An example demonstrating a practical application of these results is also provided.

Dynamic response of lined circular tunnel to plane harmonic waves

Two dimensional harmonic response of lined circular tunnels in elastic full space medium against plane P–SV waves is investigated. The solution uses hybrid boundary, and finite element methods for modelling of media and lining, respectively. In the proposed ring element used in modelling of lining, the radial and tangential deformations are defined by Fourier series expansion. Therefore, the direct finite element unknowns of the problem are introduced as coefficients of these series. The non-dimensional shear and hoop stresses in the lining, and the same parameters in its interface with surrounding media are presented.

2-D ELASTODYNAMIC FUNDAMENTAL SOLUTION FOR DISTRIBUTED LOADS AND BEM TRANSIENT RESPONSE ANALYSES OF HALFPLANE PROBLEMS

A closed form solution for the 2-dimensional problem to evaluate the displacement and stress of an elastic fullspace subject to sudden distributed forces is developed. The force is expressed in a form of multinormal function of space and time variables over strips and time increments. The solution procedure is to utilize the Fourier-Laplace domain transform with the Cagniard-de Hoop method for the inversion. The application to the boundary-initial value problems is demonstrated for the Lamb's problem and for the seismic wave scattering propagation problem. The former makes the validation of the present solution and the latter provides the useful information on ground motions at irregular sites.

Shear Stress Intensity Factors for a Planar Crack With Slightly Curved Front

Recent work (Rice, 1985a) has presented the calculations of the first order variation in an elastic displacement field associated with arbitrary incremental planar advance of the location of the front of a half-plane crack in a loaded elastic full space. That work also indicated the relation of such calculations to a three-dimensional weight function theory for crack analysis and derived an expression for the distribution of the tensile mode stress intensity factor along a slightly curved crack front, to first order accuracy in the deviation of the crack front location from a reference straight line. Here we extend the results on stress intensity factors to the shear modes, solving to similar first order accuracy for the in-plane (Mode 2) and antiplane (Mode 3) shear stress intensity factors along a slightly curved crack front. Implications of results for the configurational stability of a straight crack front are discussed. It is also shown that the concept of line tension, while qualitatively useful in characterizing the crack extension force (energy release rate) distribution exerted on a tough heterogeneity along a fracture path as the crack front begins to curve around it, does not agree with the exact first order effect that is derived here.