Anizotrop cisimlerde yer değiştirme süreksizliği yöntemi

Abstract

Bir çok mühendislik probleminin çözümünde, sayısal hesap yöntemlerinden olan sınır eleman yöntemi (BEM) kullanılmaktadır. Sınır eleman yönteminde sınırdaki ayrıklaştırma direkt veya indirekt olarak iki ayrı yaklaşımla yapılmaktadır. Direkt sınır eleman yönteminde sınırdaki bilinmeyenler doğrudan elde edilir. İndirekt yöntemde ise önce sınırda fiktif değerler elde edilir daha sonra bu fiktif değerler yardımıyla diğer bilinmeyenler hesaplanır. Şuurdaki bu fiktif değerler ise seçilen temel bilinmeyenler bakımından iki kısımda incelenebilir. Buna göre eğer şuurdaki bilinmeyenler fiktif gerilmeler ise fiktif değerler olarak fiktif gerilmeler alınır, eğer bilinmeyenler yer değiştirme süreksizlikleri ise fiktif değerler yerine, yer değiştirme süreksizlikleri alınır, Bu çalışmada, izotrop ve anizotrop ortamlar için, indirekt şuur eleman yöntemlerinden olan, fiktif gerilme yöntemi (FSM) ve yerdeğştirme süreksizliği yöntemi (DDM) kullanılmıştır. Her iki yöntemde de Kelvin ve Melan temel çözümleri kullanılarak, şuur integralleri sabit ve doğrusal olan elemanlar üzerinde kapalı olarak elde edilmiştir. Bu çalışmanın bir özelliğide, Kelvin temel çözümleri kullanılarak, sonsuz ve yan- sonsuz düzlemlerde izotrop-anizotrop ortamlar için tekil yüklerden dipoller oluşturulmuş ve bu dipoller yardımıylada DDMdeki temel çözümler elde edilmiştir. Ayrıca izotrop ortamdan farklı olarak, anizotrop ortamlarda, elastik sabitler doğrultusundaki eksenlerden her hangi bir açı yapacak şekilde bir eleman gözönüne alınmış olup, türevler ve integraller bu doğrultuda alınarak elemanlar daki bilinmeyen yer değiştirme süreksizlikleri için lineer denklem sistemi elde edilmiştir. Sayısal uygulama olarak sonsuz ve yan-sonsuz düzlemlerde, çeşitli boşluk ve çatlak problemleri incelenmiştir. Bunlardan, değişik şuur koşullan altında sonsuz ve yan- sonsuz düzlemlerde, dairesel kesitli bir boşluk için seçilen noktalarda meydana gelen gerilme ve yerdeğiştirme bileşenleri hesaplanmıştır, ikinci örnek olarak, iki malzemeli dairesel kesitli boşluk problemi DDM ile incelenmiştir. DDM için özel uygulama olması açısından, yan sonsuz düzlemde, yüzeyden itibaren derinlikle doğru orantılı değişen yükleme altında çatlak ve fay problemi ve ayrıca Mohr-Coulomb hipotezi kullanılarak, sonsuz düzlemde dairesel boşluk ve fay etkileşim problemi çözülmüştür. Son olarak anizotrop malzeme için, eksenel yük etkisi altında sonsuz düzlemde dairesel ve kare kesitli boşluk ile hidrostatik iç basınç ve eksenel yük etkisi altında yan sonsuz düzlemde dairesel boşluk problemleri çözülmüştür. Geliştirilen yöntemle çözülmüş olan tüm örnek problemlerde ankotropinin sonuçlan önemli ölçüde etkilediği ve gözönünde tutulması gerekli bir malzeme özelliği olduğu sonucuna varılmışta.

Realistic problems of engineering mechanics, usually formulated in terms of a set of governing differential equations together with appropriate boundary and initial conditions, are characterized by great complexities in geometry and material constitution. As a result of that, analytical solutions of these problem are practically impossible to obtain and resort has to be made to numerical methods for the determination of an approximate solution. Numerical solutions of very complicated problems are feasible today due to the availability of powerful digital computers. A numerical method is of the differential or the integral type depending on whether the numerical analysis precedes or follows the integration of the governing equations. The general problem two-dimensional elastostatics is that of determining the stresses, ay (i,j = 1,2), and displacements, u;, in a body of known shape subject to prescribed boundary conditions (traction, displacements). In a additions to these may be body forces which act throughout the interior of the body. The two-dimensional elastostatics problems considered here are those of plane strain and plane stress. The conditions to be satisfied by any solution for a particular problem geometry and loading condition are the differential equations of equilibrium, the constitutive equations, the equations of compatibility and the boundary conditions for the problem. The treatment of problems two-dimensional elastostatics simplifies somewhat when the body forces do not appear in differential equations. Since the equations for continuos, homogeneous form by finding one of the infinitely many particular integrals. The boundary element method has been used widely in geotechnical analysis and has performed well for elastic analysis of excavations in semi-infinite or infinite bodies. In this method, the problem is solved in terms of the conditions imposed at the surfaces of openings the problem domain. However, there are situations in geotechnical engineering for which it is difficult to use the boundary element method, especially those practical problems involving sequential excavation or construction, inhomogenius materials, material non-linearities, and the presence of joints. Natural rock masses are usually composed of blocks of intact material separated by joints or discontinuities. The behavior of these rock masses is complex as it is governed not only by the properties of the discontinuities. In general, the overall behavior of a jointed rock mass will be anisotropic. XU Most boundary element analyses carried out in geotechnical engineering assume that the rock mass can be modeled as an isotropic elastic material. There have been a few reported attempts to account for the structure of the rock mass in analyses, and examples of boundary element treatments of rock with distinct joints have been presented by Hocking and Brady et al however, when the spacing between the joint is small in comparison with the length scale of interest (such as tunnel width or foundation size), a simulation incorporating the jointing explicitly is very difficult and costly to implement. The problem of dealing with the structure of the rock mass explicitly can be avoided in cases where the jointing is closely spaced and regular, because it is convenient in such problems to idealize the rock mass and anisotropic elastic medium ; that is, the effects of the discontinuities is implicit in the choice of the stress-strain model adopted for the equivalent rock mass continuum. Some of the boundary element treatments of anisotropic materials relevant to jointed rock masses are reviewed briefly in this work. Increasing structural use is being made of materials with anisotropic elastic material properties. In recent years, various kinds of composite materials have been developed and used for structural components. The major reasons for their success are to be found in the fact that they can fulfill all the requirements for a given application. Orginally, structural composites were developed for the aerospace industry as they offered attractive properties of stiffness and strengthcompared to their weight. Further advantages such as high corrosion resistance and design flexibilty made composite materials the ideal replacement to the aluminum alloys previously used. Rizzo and Shippy and Crouch and Starfield have assumed that in some cases rock masses may be represented as transversely anisotropic materials, and they have implemented into boundary element formulations the fundamental solutions due to Green for a point force in an infinite sheet of transversely anisotropic elastic material. Brebbia has also suggested an iterative perturbation boundary element analysis for general anisotropic materials, and Dumir and Mehte have analyzed orthotropic half- plane problems using boundary element techniques using appropriate Airy stress functions. A boundary element formulation for general anisotropic materials has also been presented by Carter and Alehossein, using the fundamental solutions derived by Lekhnitskii. The most widely known and used numerical methods of solution are the Finite Difference Method (FDM) and the Finite Element Method (FEM), both of the differential type. The FDM replaces the differential equations by algebraic ones, valid at a set of nodes within the domain, through the approximation of derivatives by finite differences, while the FEM replaces the domain itself by a set of finite sub domains or elements connected through their nodes and approximately reproducing the behavior of the sub domain that represent. The FEM presents some very distinct advantages over the FDM such as better conformity to the domain geometry, much easier handling of the boundary conditions and easier construction of variable-size meshes. These advantages have made the FEM the most popular numerical method among scientists and engineers. Even the FEM, however, presents some disadvantages such as rather high costs for the preparation and input of data and time-consuming solutions of three-dimensional problems, especially those with distant boundaries. Xlll The Boundary Element Method (BEM), which is based on an integral formulation of the problem, has emerged during the last 15 years as a new powerful computational tool. This method usually requires only a surface discretization and not a discretization of both the interior and the surfaces of the domain of interest as it is the case with "domain-type" techniques, such as the FDM and the FEM. This fact makes the BEM more efficient than the FEM for quite a number of classes of problems. For some other problems the BEM might be equally good or inferior to the FEM, while in certain cases a combination of the two methods usually creates the optimum numerical scheme. The BEM has been successfully used to a great variety of problems in engineering science, such as potential theory, elastostatics, elastodynamics, elastoplasticity, viscoelasticity, viscoplasticity, fracture analysis, fluid mechanics, acoustics, heat conduction, electromagnetism, soil-structure interaction and fluid- structure interaction. There are basically two kinds of BEMs, the indirect and the direct ones. In the indirect approach the discretized integral equations are first solved for the density of the singular solutions over the boundary surface and then the remaining boundary quantities are computed in terms of these densities which have no physical significance. In the direct approach the discretized integral equations are formulated with the help of certain fundamental integral theorems and connect directly the unknown with the known boundary quantities. Even thought it has been shown by Brebbia and Butterfield in 1978 that the indirect and direct BEMs are formally equivalent, more emphasis is usually given to direct BEMs because they are more appealing to scientists and engineers. In addition, direct BEMs appear to be, at least so far, more easily amenable to improvements of the FEM type for the development of advanced BEMs than indirect ones. The term semi-direct BEM has also been used for some time in the past to indicate the treatment of some potential and elastostatic problems by a direct formulation with unknowns functions analogous to stream or stress function from which the physical quantities of the problem can be obtained by differentiation after the solution of the integral equations. This BEM category, however, is very limited and of a special character to be an independent one and for this it is usually considered as part of either the direct or, more frequently, the indirect BEM. In the direct boundary element formulation based on the point-force fundamental solution, considerable difficulties arise since the displacement discontinuities across the fracture surfaces are not explicitly accounted for. One approach to tackle this problem is partition the medium (which must be finite) into sub-regions However, this approach becomes inefficient when there are two or more cracks, or even for a single crack which propagates out-of-plane in mixed mode loading. In the indirect boundary element (or fictitious) formulation, difficulties similar to that of the direct boundary element method arise because continuous displacements are implicitly assumed. XIV The indirect boundary element method (IBEM) makes use of "fictitious forces" distributed along the boundary of the region of interest. By means of the principle of superposition, an integral equation for these fictitious forces is derived. By representing the boundary by a set of elements and assuming some variation of fictitious forces over each element, a system of simultaneous equations approximating to the integral equation is obtained. Once these fictitious forces have been obtained by solving the equations, the stresses and displacements anywhere with in the region and on the boundary can be calculated by integration. For the sake of simplicity, represent a two-dimensional continuous, homogeneous, isotropic, linear and elastic region V bounded by the surface S\ acted upon by a surface traction t;'(x). We now want to find the stresses and displacements in the region V and on the boundary S\ However, it often is easier to find the solution to the relevant partial differential equations in an infinite region, because we do have the Kelvin solution which gives the displacements at the point x (referred to as the field point), U;(x), due to a concentrated unit force, Fj(£), acting at the point £ (referred to as the load point) in an elastic body of infinite extent. The displacement discontinuity method (DDM), as a means of solving boundary value problems in elasticity, has become a popular numerical method in the field of geomechanics. The most significant characteristic is its ability of handling rock discontinuities and fractures. The successful application of the DDM, however, relies on the derivation of the fundamental solution for a displacement discontinuity singularity, or mathematically a point dislocation in the infinite space. The displacement discontinuity method is particularly well suited to model fractures which have relative displacements across their surface (hence a displacement discontinuity results). In addition, it can also be used to model an ordinary boundary of a 2 or 3 dimensional body. However, as will be shown later strong stress singularities at the ands of the displacement elements make it undesirable to model boundaries with finite and smoothly distributed applied loads. The formulation of the displacement discontinuity method is identical to that of the fictitious stress method except that displacement discontinuities (or "fictitious cracks") are used instead of fictitious stresses. (Naturally, the fundamental solution in also be different.) Hybridizing these two methods so that the advantages of both can be utilized seems to be very promising. In addition, the boundary element methods have further advantages over the finite element method when some special types of boundaries a present. For example, for a semi-infinite medium with a circular cavity. In this study investigated the indirect boundary element solution for isotrpic and anisotropic problems. The complete fundamental solutions due to unit loads within the infinite and semi-infinite plane are given. Expressions for stresses and displacements at internal points are also given. This formulation İs applied and compared to some classical problems. This solution procedure is highly accurate and computational more efficient than the boundary element formulation using the Kelvin fundamental solution for orthotopic problems. Throughout this work the so- called Cartesian tensor notation is used. This notation is not only a time-saver in XV writing long expressions, but is also extremely useful in derivation and in the proof of theorems. Such notation makes use of subscript indices (1,2,3) to represent (x,y,z). The general problems two dimensional elastostatics are explained in chapter 2. This chapter is partly devoted to introducing some basic concepts of the theory of elasticity needed for developing boundary element models. The chapter starts by reviewing the small starin theory of elasticity. The complete Kelvin fundamental solutions are given in this chapter. The indirect boundary element method and numerical procedure are given in chapter 3. The fundamental solutions for a two-dimensional displacement discontinuity are also given in this chapter. The DDM is used to model joints and faults fillet with weak material in chapter 4. In this case the dispalcement discontinuity element surfaces are connected by springs which have shear stiffness K« and normal stiffness Kn. The other special application is to use DDM for inhomogeneous bodies in this chapter. In the last chapter anisotropic elasticity problems are investigated in the indirect boundary element method. Kelvin fundamental solutions for any direction are used and singular solutions found in the DDM.