A boundary element method for two dimensional linear elastic fracture analysis

A boundary element method is developed for the analysis of fractures in two-dimensional solids. The solids are assumed to be linearly elastic and isotropic, and both bounded and unbounded domains are treated. The development of the boundary integral equations exploits (as usual) Somigliana's identity, but a special manipulation is carried out to regularize certain integrals associated with the crack line. The resulting integral equations consist of the conventional ordinary boundary terms and two additional terms which can be identified as a distribution of concentrated forces and a distribution of dislocations along each crack line. The strategy for establishing the integral equations is first outlined in terms of real variables, after which complex variable techniques are adopted for the detailed development. In the numerical implementation of the formulation, the ordinary boundary integrals are treated with standard boundary element techniques, while a novel numerical procedure is developed to treat the crack line integrals. The resulting numerical procedure is used to solve several sample problems for both embedded and surface-breaking cracks, and it is shown that the technique is both accurate and efficient. The utility of the method for simulating curvilinear crack propagation is also demonstrated.     

A finite element formulation for highly anisotropic incompressible elastic solids

A numerical approach is developed for the solution of problems of materials with extremely strong directions. Small deformations of a transversely isotropic linear elastic solid, reinforced by a single family of inextensible fibres, are considered. The kinematic constraint equations of incompressibility and inextensibility in the fibre direction lead to the appearance of an arbitrary hydrostatic pressure and an arbitrary tension stress in the constitutive equations. A Galerkin approach is used to discretize the virtual work and weak form of the constraint equations. Independent interpolation of the displacement, pressure and tension fields leads to a mixed system of equations, with characteristic zero-diagonal terms. The assumption of plane stress conditions in the plane of the fibres results in a simplified displacement-tension formulation, analogous to the primitive-variable formulation of Stokes flow. A mixed penalty approximation is then employed to solve for displacement and tension stress fields. Computations are carried out using a biquadratic displacement element with discontinuous bilinear tension stress interpolation. The formulation is used to solve a number of simple beam problems and the results compared to closed-form solutions.     

An anisotropic elastic formulation for configurational forces in stress space

A new variational principle for an anisotropic elastic formulation in stress space is constructed, the Euler–Lagrange equations of which are the equations of compatibility (in terms of stress), the equilibrium equations and the traction boundary condition. Such a principle can be used to extend recently obtained configurational balance laws in stress space to the case of anisotropy.     

An improved boundary element formulation for wave propagation problems

In this paper the dual reciprocity formulation for scalar wave equations and elastodynamic problems developed by Nardini & Brebbia is extended to the problem of waves propagating in an infinite domain by applying the Sommerfeld's radiation condition on a suitable artificial boundary. The free surface condition of first order can also be taken into consideration. To validate the present scheme, some examples have been worked out and compared with analytical solutions.     

An infinite boundary element formulation for three-dimensional potential problems

An infinite boundary element (IBE) is presented for the analysis of three-dimensional potential problems in an unbounded medium. The IBE formulations are done to allow their coupling with the finite element (FE) matrices for finite domains and to obtain the overall matrices without destroying the banded structure of the FE matrices. The infinite body is divided into a number of zones whose contributions are expressed in terms of the nodal quantities at FE nodes by employing suitable decay functions and performing mainly analytical integrations of the boundary element kernels. The continuity and compatibility conditions for the potential and the flux at the FE-IBE interface are developed. The relationships for the contributions of the IBE flux vectors to the FE load vectors are given. The final equations for the IBE are obtained in the usual FE stiffness-load vector form and are easily assembled with the FE matrices for the finite object. A series of numerical examples in heat transfer and electromagnetics were solved and compared with alternative solutions to demonstrate the validity of the present formulations.     

An accelerated symmetric time-domain boundary element formulation for elasticity

Wave propagation phenomena occur often in semi-infinite regions. It is well known that such problems can be handled well with the boundary element method (BEM). However, it is also known that the BEM, with its dense matrices, becomes prohibitive with respect to storage and computing time. Focusing on wave propagation problems, where a formulation in time domain is preferable, the mentioned limit of the method becomes evident. Several approaches, amongst them the adaptive cross approximation (ACA), have been developed in order to overcome these drawbacks mainly for elliptic problems.The present work focuses on time dependent elastic problems, which are indeed not elliptic. The application of the presented fast boundary element formulation on such problems is enabled by introducing the well known Convolution Quadrature Method (CQM) as time stepping scheme. Thus, the solution of the time dependent problem ends up in the solution of a system of decoupled Laplace domain problems. This detour is worth since the resulting problems are again elliptic and, therefore, the ACA can be used in its standard fashion.The main advantage of this approach of accelerating a time dependent BEM is that it can be easily applied to other fundamental solutions as, e.g., visco- or poroelasticity.     

Greens functions for boundary element analysis of anisotropic bimaterials

We present several Greens functions for anisotropic bimaterials for two-dimensional elasticity and steady-state heat transfer problems. The details of the various Greens functions for perfect, slipping, and cracked interfaces are given for mechanical loading conditions. Previously reported formulations for cubic materials are extended to materials with general anisotropy in which plane strain deformations can exist. We also give the steady-state Greens function for thermal loading of a bimaterial with a perfectly bonded interface. The Greens functions are incorporated in boundary integral formulations and method of fundamental solutions formulations for analysis of finite solids under general boundary conditions.     

A new boundary element formulation for wave load analysis

A new boundary element (BEM) formulation is proposed for wave load analysis of submerged or floating bodies. The presented formulation, through establishing an impedance relation, permits the evaluation of the hydrodynamic coefficients (added mass and damping coefficients) and the coefficients of wave exciting forces systematically in terms of system matrices of BEM without solving any special problem, such as, unit velocity or unit excitation problem. It also eliminates the need for scattering analysis in the evaluation of wave exciting forces. The imaginary and real parts of impedance matrix give, respectively, added mass and damping matrices whose elements describe the fluid resistance against the motion of the body. The formulation is explained through the use of a simple fluid-solid system under wave excitations, which involves a uniform fluid layer containing a solid cylindrical body. In the formulation, the solid body is taken first as deformable, then, it is specialized when it is rigid. The validity of the proposed method is verified by comparing its result with those available in literature for rigid submerged or floating bodies.     

Dual boundary element formulation for half-space cathodic protection analysis

In this paper, a boundary element formulation is developed and used for the analysis of cathodic protection systems of buried slender structures. The slenderness of the structure brings numerical difficulties into the classical boundary element method. To avoid this problem, the dual boundary element method is implemented: combination of standard and hypersingular integral equations to form a system of equations free from the singularity behavior of the standard approach when the thickness of the body tends to zero. Regularity conditions in infinite domains are analyzed for both standard and hypersingular equations. Besides numerical tests to validate the formulation, a simple experiment is carried out where a galvanized metallic sheet is buried alongside two copper electrodes, in parallel, with the objective of simulating a two-dimensional problem. The soil resistivity properties are measured along the depth and the relation between the current density and the electrochemical potential at the metallic sheet is investigated. The proposed dual approach is applied to model the experiment and results are compared with potential measurements at the ground surface.     

An asymptotic general model for linear elastic homogeneous anisotropic rods

This paper covers the mathematical justification and generalization of classical anisotropic rod theories using asymptotic analysis as the area of the cross-section tends to zero in the three-dimensional elasticity model after a rescaling in the unknowns and data, together with convergence results.     

A scaled boundary finite element formulation for dynamic elastoplastic analysis

This study presents the development of the scaled boundary finite element method (SBFEM) to simulate elastoplastic stress wave propagation problems subjected to transient dynamic loadings. Material nonlinearity is considered by first reformulating the SBFEM to obtain an explicit form of shape functions for polygons with an arbitrary number of sides. The material constitutive matrix and the residual stress fields are then determined as analytical polynomial functions in the scaled boundary coordinates through a local least squares fit to evaluate the elastoplastic stiffness matrix and the residual load vector semianalytically. The treatment of the inertial force within the solution of the nonlinear system of equations is also presented within the SBFEM framework. The nonlinear equation system is solved using the unconditionally stable Newmark time integration algorithm. The proposed formulation is validated using several benchmark numerical examples.     

A multi-domain boundary element formulation for acoustic frequency sensitivity analysis

The determination of acoustic sensitivity characteristics for vibrating structures with respect to the design parameters is a necessary and an important step of the acoustic design and optimization process. Acoustic frequency sensitivity analysis, as one kind of the sensitivity analysis, is extremely useful for large models in which only a few discrete frequencies can be analyzed due to high computational cost. Through frequency sensitivity analysis, the acoustic performance can be obtained near the initial frequency for less cost than a second full analysis. However, based on single-domain boundary element method (BEM), acoustic sensitivity analysis also has low computation efficiency. When many forces, with the same frequency, act on one vibrating structure, it is hard to decide which one should be improved. On the contrary, by using multi-domain BEM, the sensitivity figure can clearly show which exciting frequency has the greatest influence on acoustic characteristics with lower computational cost. Adopting multi-domain BEM, the expressions of the change of acoustical characteristic with respect to the change of the frequency are presented for exterior problems. The sensitivities of coefficient matrix are computed by analytical differentiation of the discrete Helmholtz integral equation. Finally, several examples are given to demonstrate the validation and availability of the presented method.     

A new boundary element formulation for shear deformable shells analysis

A new domain‐boundary element formulation to solve bending problems of shear deformable shallow shells having quadratic mid‐surface is presented. By regrouping all the terms containing shells curvature and external loads together in equilibrium equation, the formulation can be formed by coupling boundary element formulation of shear deformable plate and two‐dimensional plane stress elasticity. The boundary is discretized into quadratic isoparametric element and the domain is discretized using constant cells. Several examples are presented, and the results shows a good agreement with the finite element method. Copyright © 1999 John Wiley & Sons, Ltd.     

A new boundary element method formulation for three dimensional problems in linear elasticity

Summary A new boundary element method (BEM) formulation for planar problems of linear elasticity has been proposed recently [6]. This formulation uses a kernel which has a weaker singularity relative to the corresponding kernel in the standard formulation. The most important advantage of the new formulation, relative to the standard one, is that it delivers stresses accurately at internal points that are extremely close to the boundary of a body. A corresponding BEM formulation for three dimensional problems of linear elasticity is presented in this paper. This formulation is derived through the use of Stokes' theorem and has kernels which are only 1/r singular (wherer is the distance between a source and a field point) for the displacement equation. The standard BEM formulation for three-dimensional elasticity problems has a kernel which is 1/r ² singular.With 2 Figures     

An extended boundary element method formulation for the direct calculation of the stress intensity factors in fully anisotropic materials

We propose a formulation for linear elastic fracture mechanics in which the stress intensity factors are found directly from the solution vector of an extended boundary element method formulation. The enrichment is embedded in the boundary element method formulation, rather than adding new degrees of freedom for each enriched node. Therefore, a very limited number of new degrees of freedom is added to the problem, which contributes to preserving the conditioning of the linear system of equations. The Stroh formalism is used to provide boundary element method fundamental solutions for any degree of anisotropy, and these are used for both conventional and enriched degrees of freedom. Several numerical examples are shown with benchmark solutions to validate the proposed method. Copyright © 2016 John Wiley & Sons, Ltd.     

Analyzing static three-dimensional elastic domains with a new infinite boundary element formulation

A new two-dimensionally mapped infinite boundary element (IBE) is presented. The formulation is based on a triangular boundary element (BE) with linear shape functions instead of the quadrilateral IBEs usually found in the literature. The infinite solids analyzed are assumed to be three-dimensional, linear-elastic and isotropic, and Kelvin fundamental solutions are employed. One advantage of the proposed formulation over quadratic or higher order elements is that no additional degrees of freedom are added to the original BE mesh by the presence of the IBEs. Thus, the IBEs allow the mesh to be reduced without compromising the accuracy of the result. Two examples are presented, in which the numerical results show good agreement with authors using quadrilateral IBEs and analytical solutions.     

A formulation and solution for boundary element analysis of inhomogeneous-nonlinear problem

A new formulation is presented in this paper for the boundary element analysis of a nonlinear potential-type problem wherein the linear term is governed by the Laplace operator, and the nonlinear term is a function of the spatial coordinates as well as the unknown solution function. The formulation aims to transform the domain integral relevant to the inhomogeneous-nonlinear term to a corresponding boundary integral. The proposed approach is different from the more popular schemes for the purpose, such as the Dual Reciprocity and Multiple Reciprocity Methods. The inhomogeneous-nonlinear term is first approximated by a polynomial in terms of the space coordinates with unknown coefficients. Integral equations on the selected points (referred to “computing points”) on the boundary as well as inside domain are employed to determine the above-mentioned unknown coefficients using the least square method. The number of computing points affects the accuracy of the result, which is discussed through some numerical examples in two-dimensional space.     

Non-singular symmetric boundary element formulation for elastodynamics

In this paper, a non-singular boundary element formulation for 3D-elastostatics and 3D-elastodynamics is presented. The proposed method is based on a generalized variational principle. A weighted superposition of static fundamental solutions is used for the field approximation in the domain, whereas the displacement and stress field on the boundary are interpolated by well-known polynomial shape functions. By separating time- and space-dependence a symmetric equation of motion is derived with time-independent mass and stiffness matrix. The domain integral over inertia terms, leading to the mass matrix, is analytically transformed to the boundary. Thus, a boundary only formulation is derived. Comparing numerical results with analytical solutions clearly shows that the obtained system of equations is well-suited for dynamic problems.     

Fracture mechanics analysis of cracked 2-D anisotropic media with a new formulation of the boundary element method

A new formulation of the boundary element method (BEM) is proposed in this paper to calculate stress intensity factors for cracked 2-D anisotropic materials. The most outstanding feature of this new approach is that the displacement and traction integral equations are collocated on the outside boundary of the problem (no-crack boundary) only and on one side of the crack surfaces only, respectively. Since the new BEM formulation uses displacements or tractions as unknowns on the outside boundary and displacement differences as unknowns on the crack surfaces, the formulation combines the best attributes of the traditional displacement BEM as well as the displacement discontinuity method (DDM). Compared with the recently proposed dual BEM, the present approach doesn't require dua elements and nodes on the crack surfaces, and further, it can be used for anisotropic media with cracks of any geometric shapes. Numerical examples of calculation of stress intensity factors were conducted, and excellent agreement with previously published results was obtained. The authors believe that the new BEM formulation presented in this paper will provide an alternative and yet efficient numerical technique for the study of cracked 2-D anisotropic media, and for the simulation of quasi-static crack propagation.     

Thermodynamic formulation of finite linear viscoelasticity for anisotropic media

Summary The theory of irreversible thermodynamics of continuous media with fading memory is used to formulate general constitutive equations of finite linear viscoelasticity. Specific forms for transversely isotropic and orthotropic media based on both the mechanical and thermodynamic theories are found.

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Zusammenfassung Die Aufstellung allgemeiner Materialgleichungen der endlichen linearen Viskoelastizität wird mit Hilfe der irreversiblen Thermodynamik kontinuierlicher Medien mit Erinnerungsschwund durchgeführt. Spezielle Fassungen für transversal isotrope und orthotrope Medien, basierend auf der mechanischen und der thermodynamischen Theorie, werden angegeben.