Abstract This paper reports new developments on the complex variables boundary element approach for solving three-dimensional problems of cracks in elastic media. These developments include implementation of higher order polynomial approximations for the boundary displacement discontinuities and more efficient analytical techniques for evaluation of integrals. The approach employs planar triangular boundary elements and is based on the integral representations written in a local coordinate system of an element. In-plane components of the fields involved in the representations are separated and arranged in certain complex combinations. The Cauchy–Pompeiu formula is used to reduce the integrals over the element to those over its contour and evaluate the latter integrals analytically. The system of linear algebraic equations to find the unknown boundary displacement discontinuities is set up via collocation. Several illustrative numerical examples involving a single (penny-shaped) crack and multiple (semi-cylindrical) cracks are presented.

This paper examines major techniques for modeling elastostatic crack problems. The foundations of these techniques and fundamental papers that introduced, developed, and applied them are reviewed. The goal is to provide a “translation” between the different academic languages that describe the same problem.

Summary The article presents a new complex variables-based approach for analytical evaluation of threedimensional integrals involved in boundary element method (BEM) formulations. The boundary element is assumed to be planar and its boundary may contain an arbitrary number of straight lines and/or circular arcs. The idea is to use BEM integral representations written in a local coordinate system of an element, separate in-plane components of the fields involved, arrange them in certain complex c...

Last. Joseph F. Labuz(UMN: University of Minnesota)H-Index: 32

view all 3 authors...

Abstract This paper presents a new boundary element-based approach for solving three-dimensional problems of an elastic medium containing multiple cracks of arbitrary shapes. The medium could be loaded by far-field stress (for infinite domains), surface tractions (including those at the cracks surfaces), or point loads. Constant body forces are also allowed. The elastic fields outside of the cracks are represented by integral identities. Triangular elements are employed to discretize the boundar...

We provide a brief historical background of the development of hydraulic fracturing models for use in the petroleum and other industries. We discuss scaling laws and the propagation regimes that control the growth of hydraulic fractures from the laboratory to the field scale. We introduce the mathematical equations and boundary conditions that govern the hydraulic fracturing process, and discuss numerical implementation issues including: tracking of the fracture footprint, the control of the gro...

Preface. Preface Z. Olesiak. Introduction. Part I: Method of Potentials. 1. Real Potentials of Elasticity Theory. 2. Singular Solutions and Potentials in Complex Form. 3. Complex Integral Equations of the Indirect Approach. 4. Complex Integral Equations of the Direct Approach. Part II: Methods Based on the Theory by Kolosov-Muskhelishvili. 5. Functions of Kolosov-Muskhelishvili and Holomorphicity Theorems. 6. Complex Variable Integral Equations. 7. Periodic Problems. 8. Doubly Periodic Problems....

The effective numerical algorithm to solve a wide range of plane elasticity problems is presented. The method is based on the use of the complex hypersingular boundary integral equation (CHBIE) for blocky1 systems and bodies with cracks and holes. The BEM technique is employed to solve this equation. The unknown functions (displacement discontinuities (DD) or tractions) are approximated by Lagrange polynomials of the arbitrary degree. For the tip elements the asymptotics for the DD are taken int...

In this paper the representation of three-dimensional displacement fields in linear elasticity in terms of six complex valued functions is considered. The representation includes the complex Muskhelishvili formulation for plane strain as a special case. The completeness of the complex representation for regular solutions is shown and a relationship to the Neuber/Papkovich solutions is given.

For the treatment of plane elasticity problems the use of complex functions has turned out to be an elegant and effective method. The complex formulation of stresses and displacements resulted from the introduction of a real stress function which has to satisfy the 2-dimensional biharmonic equation. It can be expressed therefore with the aid of complex functions. In this paper the fundamental idea of characterizing the elasticity problem in the case of zero body forces by a biharmonic stress fun...

Abstract Closed-form solutions are obtained for a penny-shaped crack in a transversely isotropic elastic body, with the crack faces subjected to an arbitrary normal and shear loading. These results are important for investigating crack interactions. A simple and direct relationship is established between the limiting values of the tangential displacements and the stress intensity factors. A new governing integral equation is derived which is valid for an arbitrary crack under shear loading. This...

Abstract The paper describes a flexible computational method that can be used to represent the complex geometric evolution of a fluid-driven fracture front using an unstructured triangular element mesh. A specific motivation for the method is to allow subsequent treatment of non-planar fracture growth problems. Elastic interactions between the crack opening displacements and the surrounding medium are calculated using displacement discontinuity boundary element influence functions. A novel featu...

Abstract This paper develops a closed-form analytical solution for the problem of an infinite transversely isotropic thermoporoelastic rock weakened by an elliptic crack lying in the transverse plane. The crack is symmetrically subjected to three pairs of uniform generalized loads, i.e., mechanical pressure, pore pressure and temperature increment, on the upper and lower crack surfaces. Based on the general solution, the three-dimensional (3D) steady-state thermoporoelastic field in the rock are...

Among the numerical approaches to fracture mechanics analysis of cracked anisotropic solids, the boundary element method is notable for high accuracy and performance due to its semi-analytical nature and the use of only boundary mesh. Various boundary element techniques were proposed for 3D fracture mechanics analysis. However, the main problem of these approaches is the treatment of singular and hypersingular integrals, which can demand analytic evaluation of coefficient at kernel singularity a...

Abstract The quasi-static propagation of fracture in brittle materials was studied in several recent publications. A variational formulation (Salvadori, 2008; Salvadori and Carini, 2011; Salvadori and Fantoni, 2013) led to three-dimensional crack tracking strategies (Salvadori and Fantoni, 2014 [2,3]; Salvadori and Fantoni, 2016). One of the complexities of this new class of algorithms is the evaluation of high-order terms of the expansion of the crack opening and sliding. In this paper, new typ...

#2Xi Zhang(CSIRO: Commonwealth Scientific and Industrial Research Organisation)H-Index: 26

Last. Zuorong Chen(CSIRO: Commonwealth Scientific and Industrial Research Organisation)H-Index: 18

view all 3 authors...

Abstract Volcanic dikes and sills, sometimes exposed in outcrops, are examples of natural hydraulic fractures that interact with faults and natural fractures. The industrial use of hydraulic fracturing for stimulation of naturally fractured reservoirs and to modify rock strength for mining has motivated study of hydraulic fracture growth in naturally fractured rock in the petroleum, geothermal, and mining industries. Predicting the path and overall geometry of a hydraulic fracture growing throug...

Abstract The paper presents a numerical study of the three-dimensional problem of cracks interacting with a cylindrical uniformly pressurized borehole. The theoretical developments describe general case in which the axis of the borehole can be inclined to the vertical direction, the cracks are either located outside of the borehole or emanate from it, and the in-situ stresses are uniform with major principal stress acting in vertical direction. The tractions are prescribed at the cracks surfaces...

Abstract In this paper, it is shown that the equations of the displacement discontinuity method (DDM) for solving 3-D crack problems are exactly the same as the equations of the boundary element method (BEM) based on the boundary integral equation (BIE). Therefore, many of the results in the BEM research can be applied directly to the DDM, such as the results of analytical integration and fast solution methods, due to this equivalence or connection. A couple of examples are presented to show the...

#2Yan Jin(China University of Petroleum)H-Index: 49

Abstract Since only the boundary of the domain requires discretization, the boundary element method (BEM) is very efficient for the semi-infinite or infinite rock-related engineering problems, e.g., hydraulic fracturing in reservoir stimulation and rock cutting during excavation. A real fracture in the solid is usually of an arbitrary geometry in three dimensions, which usually requires a three-dimensional displacement discontinuity method (3D DDM) to determine the deformation and stress field i...