Numerical evaluation of harmonic Green's functions for triclinic half‐space with embedded sources—Part I: a 2D model

The Green's functions for a triclinic half‐space for embedded harmonic line load are considered. Corresponding displacement and stress fields are expressed in terms of double Fourier integrals. The first integral was evaluated using contour integration while the second one was computed through the Gauss–Legendre quadrature. The resulting Green's functions algorithm avoids repeated calculations of the same quantities and utilizes the vector computational features within MATLAB environment. Extensive testing of the results has been performed for both displacement and stress fields. The tests demonstrate the accuracy of the proposed procedure for evaluating the Green's functions without any restrictions upon material properties, frequency, and location of the source and observation points. Copyright © 2006 John Wiley & Sons, Ltd.     

Three‐dimensional time‐harmonic Green's functions for a triclinic full‐space using a symbolic computation system

Time‐harmonic Green's functions for a triclinic anisotropic full‐space are evaluated through the use of a symbolic computation system.This procedure allows evaluation of the Green's functions for the most general anisotropic materials. The proposed computational algorithms are programmed in a MATLAB environment by incorporating symbolic calculations performed using Maple Computer Algebra System. Extensive testing of the numerical results has been performed for both displacement and stress fields. The tests demonstrate the accuracy of the proposed algorithm in evaluating the Green's functions. Copyright © 2001 John Wiley & Sons, Ltd.     

The 3‐D BEM implementation of a numerical Green's function for fracture mechanics applications

The use of Green's functions has been considered a powerful technique in the solution of fracture mechanics problems by the boundary element method (BEM). Closed‐form expressions for Green's function components, however, have only been available for few simple 2‐D crack geometry applications and require complex variable theory. The present authors have recently introduced an alternative numerical procedure to compute the Green's function components that produced BEM results for 2‐D general geometry multiple crack problems, including static and dynamic applications. This technique is not restricted to 2‐D problems and the computational aspects of the 3‐D implementation of the numerical Green's function approach are now discussed, including examples. Copyright © 2000 John Wiley & Sons, Ltd.     

Stress analyses around holes in composite laminates using boundary element method

A three-dimensional (3D) boundary element method (BEM) is developed for the analysis of composite laminates with holes. Instead of using Kelvin-type Green's functions of anisotropic infinite space, 3D layered Green's functions with the materials of each layer being generally anisotropic, derived recently in the Fourier transform domain, are implemented into a 3D BEM formulation. A novel numerical algorithm is designed to calculate layered Green's functions efficiently. It should be noted that since layered Green's functions satisfy exactly the continuity conditions along the interfaces and top and bottom free surfaces a priori, the model becomes truly 2D and discretization is only needed along the hole surface and prescribed traction and/or displacement boundaries. To test the validity and accuracy of the proposed method, the present layered BEM formulation is applied to the problem of an infinite anisotropic plate with a circular hole where the analytical solution is available. It is found that even with a very coarse mesh, the present BEM can predict the hoop stress very accurately along the hole surface. The BEM formulation is then applied to analyze two composite laminates (90/0)_s and (−45/45)_s, under a remote in-plane strain, that have been studied previously with different approaches. For the (90/0)_s case, the hoop stresses along the hole surface predicted by the present layered BEM formulation are in very close agreement with the previous results. For the (−45/45)_s case, however, it is found that a nearly converged solution (less than 5% convergence by doubling the mesh) by the present method is at significant variance with the previous ones that are lack-of-convergence checks. It can be expected that for designing the bolted joints of composites with many layers, a computational tool developed based on the present techniques would be robust and offer a much better solution with regard to accuracy, versatility and design cycle time.     

A time domain FEM approach based on implicit Green's functions for non‐linear dynamic analysis

The present paper describes an unconditionally stable algorithm to integrate the equations of motion in time. The standard FEM displacement model is employed to perform space discretization, and the time‐marching process is carried out through an algorithm based on the Green's function of the mechanical system in nodal co‐ordinates. In the present ‘implicit Green's function approach’ (ImGA), mechanical system Green's functions are not explicitly computed; rather they are implicitly considered through an iterative pseudo‐forces process. Under certain simplifying hypothesis, iterations are not necessary and the ImGA becomes cheaper than standard Newmark/Newton–Raphson algorithm. At the end of the paper numerical examples are presented in order to illustrate the accuracy of the present approach. Copyright © 2004 John Wiley & Sons, Ltd.     

Extended displacement discontinuity Green's functions for three-dimensional transversely isotropic magneto-electro-elastic media and applications

A versatile method is presented to derive the extended displacement discontinuity Green's functions or fundamental solutions by using the integral equation method and the Green's functions of the extended point forces. In particular, the three-dimensional (3D) transversely isotropic magneto-electro-elastic problem is used to demonstrate the method. On this condition, the extended displacement discontinuities include the elastic displacement discontinuities, the electric potential discontinuity and the magnetic potential discontinuity, while the extended forces include the point forces, the point electric charge and the point electric current. Based on the obtained Green's functions, the extended Crouch fundamental solutions are derived and an extended displacement discontinuity method is developed for analysis of cracks in 3D magneto-electro-elastic media. The extended intensity factors of two coplanar and parallel rectangular cracks are calculated under impermeable boundary condition to illustrate the application, accuracy and efficiency of the proposed method.     

Efficient evaluation of three-dimensional Green's functions in anisotropic elastostatic multilayered composites

The three-dimensional Green's functions in anisotropic elastostatic multilayered composites (MLCs) obtained within the framework of generalized Stroh formalism are expressed as two-dimensional integrals of Fourier inverse transform over an infinite plane. Their numerical evaluations involve tremendous computational efforts in particular in the presence of various singularities and near-singularities due to the presence of material mismatches across interfaces. The present paper derives the complete set of the Green's functions including displacement, stress and their derivatives with respect to source coordinates using a novel and computationally efficient approach. It is proposed for the first time that the Green's functions in the MLCs are expressed as a sum of a special solution and a general-part solution, with the former consisting of the first few terms of the trimaterial expansion solution around a source load. Since the zero-order term contains the singularity corresponding to the homogeneous full-space solution and can be evaluated analytically, and the other higher-order terms contain most of the near-singular behaviors and can be reduced to a line integral over a finite interval, the general-part solution becomes regular and the Green's functions overall can be evaluated efficiently. As an example, the Green's functions in a five-layered orthortropic plate are evaluated to demonstrate the efficiency of the proposed approach. Also, the detailed characteristics of these Green's functions are examined in both the transform- and physical-domains. These Green's functions are essential in developing the boundary-integral-equation formulation and numerical boundary element method for composite laminate problems involving regular and cracked geometries.     

An indirect elastodynamic boundary element method with analytic bases

An indirect time‐domain boundary element method (BEM) is presented here for the treatment of 2D elastodynamic problems. The approximated solution in this method is formulated as a linear combination of a set of particular solutions, which are called bases. The displacement and stress fields of a basis are analytically derived by means of solving Lame's displacement potentials. A semi‐collocation method is proposed to be the time‐stepping algorithm. This method is equivalent to a displacement discontinuity method with piecewise linear discontinuities in both space and time. The resulting time‐stepping scheme is explicit. The BEM is implemented to solve three numerical examples, Lamb's problem, half‐plane with a buried crack and Selberg's problem. Though Lamb's problem is considered a difficult problem for numerical methods, the current numerical results for the surface displacements show accurately the characteristics of the Rayleigh wave. This method is efficient and accurate. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

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.     

An efficient multi‐layer planar 3D fracture growth algorithm using a fixed mesh approach

We present a planar three‐dimensional (3D) fracture growth simulator, based on a displacement discontinuity (DD) method for multi‐layer elasticity problems. The method uses a fixed mesh approach, with rectangular panel elements to represent the planar fracture surface. Special fracture tip logic is included that allows a tip element to be partially fractured in the tip region. The fracture perimeter is modelled in a piece‐wise linear manner. The algorithm can model any number of interacting fractures that are restricted to lie on a single planar surface, located orthogonal to any number of parallel layers. The multiple layers are treated using a Fourier transform (FT) approach that provides a numerical Green's function for the DD scheme. The layers are assumed to be fully bonded together. Any fracture growth rule can be postulated for the algorithm. We demonstrate this approach on a number of test problems to verify its accuracy and efficiency, before showing some more general results. Copyright © 2001 John Wiley & Sons, Ltd.     

Green's functions in orthotropic thermoelastic diffusion media

The aim of the present paper is to study the Green's function in orthotropic thermoelastic diffusion media. With this objective, firstly the two-dimensional general solution in orthotropic thermoelastic diffusion media is derived. On the basis of general solution, the Green's function for a steady point heat source in the interior of semi-infinite orthotropic thermoelastic diffusion material is constructed by four newly introduced harmonic functions. The components of displacement, stress, temperature distribution and mass concentration are expressed in terms of elementary functions. From the present investigation, a special case of interest is also deduced, to depict the effect of diffusion on components of stress and temperature distribution.     

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.     

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.     

STRESS INTENSITY FACTORS FOR CRACKS IN NOTCHED FINITE PLATES SUBJECTED TO BIAXIAL LOADING

Stress intensity factors were calculated, based on Bueckner's principle for cracks in both infinite and finite plates with notches subjected to biaxial loading. Approximate Green's functions have been obtained by modifying two existing Green's functions, originally for unnotched plates. Values of stress intensity factors calculated using Bueckner's principle with the approximate Green's functions are in good agreement with published stress intensity factors for cracks in both infinite and finite plates containing a circular notch or an elliptical notch, previously found by the method of boundary collocation.     

Three-dimensional extended displacement discontinuity method for vertical cracks in transversely isotropic piezoelectric media

The conventional displacement discontinuity method is extended to study a vertical crack under electrically impermeable condition, running parallel to the poling direction and normal to the plane of isotropy in three-dimensional transversely isotropic piezoelectric media. The extended Green's functions specifically for extended point displacement discontinuities are derived based on the Green's functions of extended point forces and the Somigliana identity. The hyper-singular displacement discontinuity boundary integral equations are also derived. The asymptotical behavior near the crack tips along the crack front is studied and the ordinary 1/2 singularity is obtained at the tips. The extended field intensity factors are expressed in terms of the extended displacement discontinuity on crack faces. Numerical results on the extended field intensity factors for a vertical square crack are presented using the proposed extended displacement discontinuity method.     

Numerical Green's functions for some electroelastic crack problems

A plane electroelastic problem involving planar cracks in a piezoelectric body is considered. The deformation of the body is assumed to be independent of time and one of the Cartesian coordinates. The cracks are traction free and are electrically either permeable or impermeable. Numerical Green's functions which satisfy the boundary conditions on the cracks are derived using the hypersingular integral approach and applied to obtain a boundary integral solution for the electroelastic crack problem considered here. As the conditions on the cracks are built into the Green's functions, the boundary integral solution does not contain integrals over the cracks. It is used to derive a boundary element procedure for computing the crack tip stress and electrical displacement intensity factors.     

Green's function expansion for exponentially graded elasticity

New computational forms are derived for Green's function of an exponentially graded elastic material in three dimensions. By suitably expanding a term in the defining inverse Fourier integral, the displacement tensor can be written as a relatively simple analytic term, plus a single double integral that must be evaluated numerically. The integration is over a fixed finite domain, the integrand involves only elementary functions, and only low‐order Gauss quadrature is required for an accurate answer. Moreover, it is expected that this approach will allow a far simpler procedure for obtaining the first and second‐order derivatives needed in a boundary integral analysis. The new Green's function expressions have been tested by comparing with results from an earlier algorithm. Copyright © 2009 John Wiley & Sons, Ltd.     

2D SH modelling of ultrasonic testing for cracks near a non-planar surface

A model of 2D SH ultrasonic nondestructive testing for interior strip-like cracks near a non-planar back surface in a thick-walled elastic solid is presented. The model employs a Green's function to reformulate the 2D antiplane wave scattering problem as two coupled boundary integral equations (BIE): a displacement BIE for the back surface displacement and a hypersingular traction BIE for the crack opening displacement (COD). The integral equations are solved by performing a boundary element discretization of the back surface and expanding the COD in a series of Chebyshev functions which incorporate the correct behaviour at the crack edges. The transmitting ultrasonic probe is modelled by prescribing the traction underneath it, enabling the consequent calculation of the incident field. An electromechanical reciprocity relation is used to model the action of the receiving probe. A few numerical examples which illustrate the influence of the non-planar back surface are given.     

Three‐dimensional BEM for scattering of elastic waves in general anisotropic media

Based on the full‐space Green's functions, a three‐dimensional time‐harmonic boundary element method is presented for the scattering of elastic waves in a triclinic full space. The boundary integral equations for incident, scattered and total wave fields are given. An efficient numerical method is proposed to calculate the free terms for any geometry. The discretization of the boundary integral equation is achieved by using a linear triangular element. Applications are discussed for scattering of elastic waves by a spherical cavity in a 3D triclinic medium. The method has been tested by comparing the numerical results with the existing analytical solutions for an isotropic problem. The results show that, in addition to the frequency of the incident waves, the scattered waves strongly depend on the anisotropy of the media. Copyright © 2003 John Wiley & Sons, Ltd.