APPLICATION OF A NEW FAST MULTIPOLE BEM FOR SIMULATION OF 2D ELASTIC SOLID WITH LARGE NUMBER OF INCLUSIONS

A fast multipole method(FMM)is applied for BEM to reduce both the operation and memory requirement in dealing with very large scale problems.In this paper,a new version of fast multipole BEM for 2D elastostatics is presented and used for simulation of 2D elastic solid with a large number of randomly distributed inclusions combined with a similar subregion approach.Generalized minimum residual method(GMRES)is used as an iterative solver to solve the equation system formed by BEM iteratively.The numerical results show that the scheme presented is applicable to certain large scale problems.

Theoretical analysis of the spatio-temporal evolution of the bulk-strain field based on a rheologic inclusion model

Based on the theoretical expression of the three-dimension rheologic inclusion model, we analyze in detail the spatio-temporal changes on the ground of the bulk-strain produced by a spherical rheologic inclusion in a semi-infinite rheologic medium. The results show that the spatio-temporal change of bulk-strain produced by the hard inclusion has three stages of different characteristics, which are similar to most of those geodetic deformation curves, but those by a soft inclusion do not. The α-stage is a long stage in which the precursors in both the near source region and the far field develop from the focal region to the periphery. The β-stage indicates a very rapid propagation of the precursors, so that they almost appear everywhere. During the γ-stage, the precursors in the far-field converge from the periphery, and the precursors in the near source region develop outwards. The theoretical results have been used to explain tentatively the stage characteristics of the spatio-temporal change of earthquake precursors.

THE INTERACTION PROBLEM BETWEEN THE ELASTIC LINE INCLUSIONS

Using the engineering model of elastic line inclusion and the basic solutions of a single inclusion, the interaction problem between line inclusions in an elastic solid it-as investigated. A set of standard Cauchy-type singular equations of the problem was presented. The stress intensity factors at points of inclusions and the interface stresses of two sides of the inclusion were calculated. Several numerical examples were given. The results could be regarded as a reference to engineering.

BOUNDARY ELEMENT ANALYSIS OF INTERACTION BETWEEN AN ELASTIC RECTANGULAR INCLUSION AND A CRACK

The interaction between an elastic rectangular inclusion and a kinked crack in an infinite elastic body was considered by using boundary element method.The new complex boundary integral equations were derived.By introducing a complex unknown function H(t) related to the interface displacement density and traction and applying integration by parts,the traction continuous condition was satisfied automatically.Only one complex boundary integral equation was obtained on interface and involves only singularity of order l/r.To verify the validity and effectiveness of the present boundary element method,some typical examples were calculated.The obtained results show that the crack stress intensity factors decrease as the shear modulus of inclusion increases.Thus,the crack propagation is easier near a softer inclusion and the harder inclusion is helpful for crack arrest.

INTERACTION BETWEEN A RIGID LINE INCLUSION AND AN ELASTIC CIRCULAR INCLUSION

In this paper, the interaction problem of a rigid line inclusion and an elastic circular inclusion has been reduced to solve a normal Cauchy-type singular integral equation. The stress intensity, factors at the ends of the rigid line inclusion and the interface stresses of the inclusions are obtained.

Three-dimensional elastostatic solutions for transversely isotropic functionally graded material plates containing elastic inclusion

Based on the generalized England-Spencer plate theory, the equilibrium of a transversely isotropic functionally graded plate containing an elastic inclusion is studied. The general solutions of the governing equations are expressed by four analytic functions α（ζ）, β（ζ）, φ（ζ）, and ψ（ζ） when no transverse forces are acting on the surfaces of the plate. Axisymmetric problems of a functionally graded circular plate and an infinite func- tionally graded plate containing a circular hole subject to loads applied on the cylindrical boundaries of the plate are firstly investigated. On this basis, the three-dimensional （3D） elasticity solutions are then obtained for a functionally graded infinite plate containing an elastic circular inclusion. When the material is degenerated into the homogeneous one, the present elasticity solutions are exactly the same as the ones obtained based on the plane stress elasticity, thus validating the present analysis in a certain sense.

3-D rheologic model of earthquake preparation (I)—Displacement field

Based on the three-dimensional ela stic inclusion model proposed by Dobrovolskii, we developed a three-dimensional rheologic inclusion model and theory to study the earthquake preparation process. By using correspondence principle in the theory of rheologic mechanics, we derived the analytic expression of the viscoelas-tic displacement at an arbitrary point (x, y, z) in three directions of x, y and z-axes (i. e., U(r, t), V*(r, t) and W(r, t)) produced by a three-dimension inclusion in the semi-infinite rheologic medium defined by the standard linear rheologic model.

3-D rheologic model of earthquake preparation (Ⅱ): Strain field and its applications

On the basis of the three-dimensional elastic inclusion model, the analytic expression of viscoelastic strain field is derived, i.e., the analytic expression of viscoelastic strain at an arbitrary point (x, y, z) in x-axis, y-axis and z-axis produced by three-dimension inclusion in the semi-infinite rheologic medium defined by the standard linear rheologic model, namely the normal strains exx(r, t), eyy(r, t) and ezz(r, t), the shear strains exy(r, t) and eyx(r, t), eyz(r, t) and ezy(r, t), exz(r, t) and ezx(r, t), and the bulk-strain q (r, t). By computing the spatial-temporal variation of bulk strain on the ground produced by a spherical rheologic inclusion in a semi-infinite rheologic medium, we obtained some significant results that the bulk-strain variation with time produced by a hard inclusion has three stages (a, b, g) with different characteristics, which are similar to those of most geodetic deformation curves, but not the case for those by a soft inclusion. It is meaningful that these theoretical results have been applied to explain preliminarily the characteristics of stage variation of spatial-temporal evolution, the pattern and quadrant distribution of earthquake precursors, the changeability, spontaneity and complexity of short-term and imminent-term precursors. It offers a theoretical base to found the physical model of earthquake precursors and a reference to predict physically the earthquakes.

3-D rheologic model of earthquake prepa-ration (III): Precursor field

On the basis of the theory of viscoelastic displacement and strain field for the three-dimensional rheologic model of earthquake preparation, this paper mainly studies the theoretical solution of precursor field for the three-dimensional rheologic model of earthquake preparation. We derive the viscoelastic analytical expressions of the ground tilt, underground water level, earth resistivity at an arbitrary point （x, y, z） in the rheologic medium, and analyzed the earth resistivity preliminarily, providing a certain theoretical basis for the precursor analysis of seismogenic process.