Green's functions for dislocations in bonded strips and related crack problems

Green's functions are derived for the plane elastostatics problem of a dislocation in a bimaterial strip. Using these fundamental solutions as kernels, various problems involving cracks in a bimaterial strip are analyzed using singular integral equations. For each problem considered, stress intensity factors are calculated for several combinations of the parameters which describe loading, geometry and material mismatch.     

Numerical solution and utilization of Green's functions

A numerical method for obtaining the Green's functions for Laplace's, Poisson's, and the transient heat diffusion equations is presented. The Green's functions thus obtained are then employed to rapidly obtain numerical solutions of the above equations by matrix multiplication, with subsequent considerable savings in machine time.     

Green's functions for the isotropic or transversely isotropic space containing a circular crack

Summary Green's functions for an infinite three-dimensional elastic solid containing a circular crack are derived in terms of integrals of elementary functions. The solid is assumed to be either isotropic or transversely isotropic (with the crack being parallel to the plane isotropy).     

Green's functions in the solutions of heat-conduction problems for a hollow cylinder

We present solutions of heat-conduction problems for a hollow cylinder for mixed boundary conditions of the second and third kind, the solutions containing rapidly converging series. For the fundamental types of boundary conditions we obtain expressions for the Green's functions which enable us to improve the convergence of the series.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol.21, No. 6, pp. 1096–1100, December, 1971.     

Westergaard stress functions for displacement-prescribed crack problems—I

The single stress function approach of Westergaard has been effective for a certain class of stress-prescribed crack problems. In the present study, the Westergaard approach was successfully extended to displacement-prescribed crack problems. The method presented, which requires no more than the evaluation of integrals, significantly simplifies the analysis. The method is easily extended to crack problems involving displacement-prescribed and stress-prescribed conditions. This initial study laid the ground work for the subsequent extension to the mixed problem.     

Green's functions and three-dimensional steady-state heat-conduction problems in a two-layered composite

A boundary-value problem for steady-state heat conduction in a three-dimensional, two-layered composite is studied. The method of Green's function is used in the study. Green's functions are constructed as double sums in terms of eigenfunctions in two of the three directions. The eigenfunctions in the direction orthogonal to the layers are unconventional and must be defined appropriately. The use of different forms of the Green's functions leads to different representations of the solutions as double sums with different convergence characteristics and it is shown that the method of Green's functions is superior to the classical method of separation of variables.     

Green's functions for boundary-value problems for the heat equation over regions with uniformly moving boundaries

Green's functions are obtained for a semi-infinite straight line with a uniformly moving boundary (10), (11), (12) and for a segment with boundaries moving uniformly and in parallel (16), (17), (18). For the solution a moving coordinate system is introduced and the method of Laplace transforms is applied.     

Approximate Green's functions for singular and higher order terms of an edge crack in a finite plate

An edge crack in a finite plate (FSECP) subjected to wedge forces is solved by the superposition of the analytical solution of a semi-infinite crack, and the numerical solution of a FSECP with free crack faces, which is solved by the Williams expansion. The unknown coefficients in the expansion are determined by a continuous least squares method after comparing it with the direct boundary collocation and the point or discrete least squares methods. The results are then used to validate the stress intensity factor (SIF) formula provided by Tada et al. that interpolates the numerical results of Kaya and Erdogan, and an approximate crack face opening displacement formula obtained in this paper by Castigliano's theorem and the SIF formula of Tada et al. These approximate formulae are accurate except for point forces very close to the outer edge, and can be used as Green's functions in the crack-closure based crack growth analysis, as well as in interpreting the size effect of quasi-brittle materials. Green's functions for coefficients relevant to the second to the fifth terms in the crack tip asymptotic field are also provided. Finally, a FSECP with a uniform pressure over a part of the crack faces is solved to illustrate the application of the obtained Green's functions and to further assess their accuracy by comparing with a finite element analysis.     

General Green's functions for SAW device analysis

The traditional Green's function has been widely used to analyze surface acoustic wave (SAW) devices for many years because of its physical basis and its ability to model many propagating modes. Simplifications are, however, introduced in the derivation of the traditional Green's function that cause it to ignore some important effects, such as mass loading caused by surface electrodes. Recently, a generalized Green's function was introduced that is able to include mechanical loading effects. Based on previous work by E.L. Adler (1994) and R.C. Peach (1995), we use concise matrix notation to deduce the generalized Green's functions that describe the effects of surface stresses and electrical displacement on the three mechanical displacement components and on the electrical potential. Effective permittivities of 15 materials/cuts and the 16-element Green's functions for two materials are presented.     

Dyadic Green's functions for multi-layer SAW substrates

Recent formulations of the dyadic (or generalized) Green's function describe the relationship between sources (both mechanical stresses and electrical charge) and waves (both mechanical displacements and acoustic potential) on the surface of a substrate. The 16 elements of the function intrinsically describe all propagation modes, whether Rayleigh or leaky, and are, therefore, extremely useful in the design of surface acoustic wave devices. In addition to requiring little computational effort, the dyadic Green's function provides much more information than the traditional effective permittivity function. In this paper, we extend the calculation of the dyadic Green's function to multi-layer substrates. We show that its computation involves a simple cascaded matrix multiplication. The resulting function fully contains the substrate characteristics and, once obtained, can be used to describe the surface behavior with no further regard to the substrate's composition     

Reciprocal theorems for self-similar wedge and crack problems and some generalisations

Reciprocal theorems are presented here for a class of time-dependent problems of antiplane strain elasticity. The application of these theorems is illustrated by considering the problem of shear waves in an elastic wedge and the problems of crack forking or kinking. A generalized “Chaplygin” transformation is given for media with a power law radially varying density and this is applied to the problem of rapid tearing of a dynamically loaded wedge of such a material.     

A numerical Green's function and dual reciprocity BEM method to solve elastodynamic crack problems

A computational model based on the numerical Green's function (NGF) and the dual reciprocity boundary element method (DR-BEM) is presented for the study of elastodynamic fracture mechanics problems. The numerical Green's function, corresponding to an embedded crack within the infinite medium, is introduced into a boundary element formulation, as the fundamental solution, to calculate the unknown external boundary displacements and tractions and in post-processing determine the crack opening displacements (COD). The domain inertial integral present in the elastodynamic equation is transformed into a boundary integral one by the use of the dual reciprocity technique. The dynamic stress intensity factors (SIF), computed through crack opening displacement values, are obtained for several numerical examples, indicating a good agreement with existing solutions.     

Green's functions for transversely isotropic piezoelectric multilayered half-spaces

This paper presents Green's functions for transversely isotropic piezoelectric and layered half-spaces. The surface of the half-space can be under general boundary conditions and a point source (point-force/point-charge) can be applied to the layered structure at any location. The Green's functions are obtained in terms of two systems of vector functions, combined with the propagator-matrix method. The most noticeable feature is that the homogeneous solution and propagator matrix are independent of the choice of the system of vector functions, and can therefore be treated in a unified manner. Since the physical-domain Green's functions involve improper integrals of Bessel functions, an adaptive Gauss-quadrature approach is applied to accelerate the convergence of the numerical integral. Typical numerical examples are presented for four different half-space models, and for both the spring-like and general traction-free boundary conditions. While the four half-space models are used to illustrate the effect of material stacking sequence and anisotropy, the spring-like boundary condition is chosen to show the effect of the spring constant on the Green's function solutions. In particular, it is observed that, when the spring constant is relatively large, the response curve can be completely different to that when it is small or when it is equal to zero, with the latter corresponding to the traction-free boundary condition.     

Green's functions for Kirchhoff plates of irregular shape

A method is proposed for the construction of Green's functions for the Sophie Germain equation in regions of irregular shape with mixed boundary conditions imposed. The method is based on the boundary integral equation approach where a kernel vector function B satisfies the biharmonic equation inside the region. This leads to a regular boundary integral equation where the compensating loads and moments are applied to the boundary. Green's function is consequently expressed in terms of the kernel vector function B, the fundamental solution function of the biharmonic equation, and kernel functions of the inverse regular integral operators. To compute moments and forces, the kernel functions are differentiated under the integral sign. The proposed method appears highly effective in computing both displacements and stress components.     

A numerical Green's function BEM formulation for crack growth simulation

This paper presents a crack growth prediction analysis based on the numerical Green's function (NGF) procedure and on the minimum strain energy density criterion for crack extension, also known as S-criterion. In the NGF procedure, the hypersingular boundary integral equation is used to numerically obtain the Green's function which automatically includes the crack into the fundamental infinite medium. When solving a linear elastic fracture mechanisms (LEFM) problem, once the NGF is obtained, the classical boundary element method can be used to determine the external boundary unknowns and, consequently, the stress intensity factors needed to predict the direction and increment of crack growth. With the change in crack geometry, another numerical analysis is carried out without need to rebuilding the entire element discretization, since only the crack built in the NGF needs update. Numerical examples, contemplating crack extensions for two-dimensional LEFM problems, are presented to illustrate the procedure.     

Dislocation and point-force-based approach to the special Green's Function BEM for elliptic hole and crack problems in two dimensions

In this paper we give the theoretical foundation for a dislocation and point-force-based approach to the special Green's function boundary element method and formulate, as an example, the special Green's function boundary element method for elliptic hole and crack problems. The crack is treated as a particular case of the elliptic hole. We adopt a physical interpretation of Somigliana's identity and formulate the boundary element method in terms of distributions of point forces and dislocation dipoles in the infinite domain with an elliptic hole. There is no need to model the hole by the boundary elements since the traction free boundary condition there for the point force and the dislocation dipole is automatically satisfied. The Green's functions are derived following the Muskhelishvili complex variable formalism and the boundary element method is formulated using complex variables. All the boundary integrals, including the formula for the stress intensity factor for the crack, are evaluated analytically to give a simple yet accurate special Green's function boundary element method. The numerical results obtained for the stress concentration and intensity factors are extremely accurate. © 1997 John Wiley & Sons, Ltd.     

Numerical analysis of growing crack problems using particle discretization scheme

This paper presents the particle discretization scheme (PDS) to analyze brittle failure of solids. The scheme uses characteristic functions of Voronoi and Delaunay tessellations to discretize a function and its derivatives, respectively. A discretized function has numerous discontinuities so that these discontinuities are utilized as a candidate of crack path segment in modeling propagating cracks, without making any extra computation to accommodate new displacement discontinuities. When the scheme is implemented to a finite element method (FEM), the resulting stiffness matrix coincides with the one that is obtained by using linear elements. The accuracy of computing a stress intensity factor at a crack tip is examined. It is shown that the accuracy is better than that of a FEM with linear elements when the rotational degree of freedom is included in discretizing displacement functions. Three three‐dimensional growing crack problems are solved by means of the PDS and the results are presented. Copyright © 2009 John Wiley & Sons, Ltd.     

Formulation of the plastic source method for plane inelastic problems Part 1: Green's functions for plane inelastic deformation

Summary Green's function representation of the residual stress caused by any plane inelastic strain is given for the infinite region with a simple defect such as a crack or an elliptic hole. The plane inelastic strain developed within an infinitesimal region is represented by a double couple whose Green's functions (or complex potentials) are derived using the analytic function theory. The inelastic strain over a finite region is, then, represented by a continuous distribution of such double couples and its complex potentials by area integrals. Closed form expression for the stress field arising from the uniform inelastic strain distribution is given for the general polygonal region withn sides.With 5 FiguresThis paper is dedicated to the memory of Aris Phillips, founding Co-Editor of Acta Mechanica, and was presented at the Aris Phillips Memorial-Symposium, Gainesville, Fla., 1987.     

Finite element computation of Green's functions

Green's functions are important mathematical tools in mechanics and in other parts of physics. For instance, the boundary element method needs to know the Green's function of the problem to compute its numerical solution. However, Green's functions are only known in a limited number of cases, often under the form of complex analytical expressions. In this article, a new method is proposed to calculate Green's functions for any linear homogeneous medium from a simple finite element model. The method relies on the theory of wave propagation in periodic media and requires the knowledge of the finite element dynamic stiffness matrix of only one period. Several examples are given to check the accuracy and the efficiency of the proposed numerical Green's function.