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.     

Analysis of an arbitrarily oriented crack in a finite piezoelectric plane via the hybrid extended displacement discontinuity-fundamental solution method

In this paper, we analyze an arbitrarily oriented crack in a finite two-dimensional piezoelectric medium with the polarization saturation model near the crack tip. We first derive the extended Green’s functions corresponding to the extended point-displacement discontinuities of an arbitrarily oriented crack based on the Green’s functions of the extended point forces and the Somigliana identity. Then, the extended field intensity factors and the local J-integral near the crack tip are expressed in terms of the extended displacement discontinuity on crack faces. Finally, the nonlinear hybrid extended displacement discontinuity-fundamental solution method is proposed to analyze an electrically nonlinear crack in a finite piezoelectric medium. Numerical examples are carried out for both linear and nonlinear fracture models of the crack under electrically impermeable boundary conditions. The influence of the crack orientation and geometric size on the fracture behaviors of the crack is investigated.     

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 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.     

Numerical method for nonlinear models of penny-shaped cracks in three-dimensional magnetoelectroelastic media

Green functions corresponding to uniformly distributed extended displacement discontinuities on an annular crack element in the isotropic plane of a three-dimensional transversely isotropic magnetoelectroelastic medium are derived. Using the obtained Green functions, an extended displacement discontinuity method is presented to analyze a penny-shaped crack under axisymmetric loadings. Using the electric and magnetic polarization saturation model and the electric and magnetic breakdown model, the electric and magnetic yielding zones, the extended displacement discontinuities, the extended stress intensity factors and the J-integral are numerically calculated. The accuracy and efficiency of the proposed method are demonstrated by comparing the numerical results with those obtained from analytical solutions.     

Displacement Discontinuity Method for Fracture Mechanics Analysis of Reissner Plates: Static and Dynamic

This paper is concerned with the displacement discontinuity method applied to the shear deformable plates (Reissner’s and Mindlin’s theories) with cracks subjected to static and dynamic loads. Fundamental solutions of dislocation are derived using the Fourier transform method and the Laplace transformation technique. Boundary integral equations are presented in terms of rotations/displacement on the crack surfaces. The Chebyshev polynomials of the second kind are used to evaluate the integral equations with hypersingular kernels on the crack boundaries and determine the stress intensity factors at the crack tips. Comparisons are made with other numerical solutions to demonstrate the proposed method is accurate both for static and dynamic problems.     

Displacement discontinuity formulation for modeling cracks in orthotropic shear deformable plates

A displacement discontinuity formulation is presented for modeling cracks in orthotropic Reisnner plates. Fundamental solutions for displacement discontinuity are derived for the first time using a Fourier transform method. Boundary integral equations are presented in terms of discontinuity rotations on the crack surfaces for opening mode problems. As the fundamental solutions have singularity of O (1/r ²), Chebyshev polynomials of the second kind are used to evaluate the integral equations. By solving for coefficients of the Chebyshev polynomials, the stress intensity factors at the crack tips are obtained directly. Comparisons are made with solutions using the finite element method to demonstrate that the displacement discontinuity method is an efficient and accurate method for solving crack problems in orthotropic Reissner plates.     

Numerical method of crack analysis in 2D finite magnetoelectroelastic media

The present paper extends the hybrid extended displacement discontinuity fundamental solution method (HEDD-FSM) (Eng Anal Bound Elem 33:592–600, 2009) to analysis of cracks in 2D finite magnetoelectroelastic media. The solution of the crack is expressed approximately by a linear combination of fundamental solutions of the governing equations, which includes the extended point force fundamental solutions with sources placed at chosen points outside the domain of the problem under consideration, and the extended Crouch fundamental solutions with extended displacement discontinuities placed on the crack. The coefficients of the fundamental solutions are determined by letting the approximated solution satisfy the prescribed boundary conditions on the boundary of the domain and on the crack face. The Crouch fundamental solution for a parabolic element at the crack tip is derived to model the square root variations of near tip fields. The extended stress intensity factors are calculated under different electric and magnetic boundary conditions.     

Stress intensity factors for cracks of arbitrary shape near an interfacial boundary

Modes I, II and III stress intensity factors for a crack of arbitrary planar shape near a bimaterial interface are calculated. The solution utilizes the body-force method and requires Green's functions for perfectly bonded elastic half-spaces. The formulation leads to a system of two-dimensional singular integral equations whose solutions represent the three modes of crack opening displacement. Numerical examples of a semicircular or semielliptical crack terminating at the interface and circular or elliptical cracks contained in one material are given for both internal pressure and farfield tension.     

Weight functions and strip yield solution for two equal-length collinear cracks in an infinite sheet

The problem of two equal-length collinear cracks in an infinite sheet is treated using the weight function method. Exact weight functions for the inner and outer crack tips are derived based on the crack opening displacement solution for a reference load case. These weight functions are used to calculate stress intensity factors for different load cases, plastic zone sizes and crack tip opening displacements of the strip yield model. The approach is validated by the perfect agreement between the present strip yield model solutions and Collins and Cartwright’s analytical results based on the direct complex stress function formulation.     

Collinear Zener-Stroh crack problem in plane elasticity

This paper investigates the collinear Zener-Stroh crack problem in plane elasticity. Two cracks in series are chosen as an example in analysis. Different to the Griffith crack problem, an initial displacement discontinuity exists in the Zener-Stroh crack problem, which in turn is the increment of displacement when a moving point goes around the crack in a closed loop. From the traction free condition along the cracks, the dislocation distribution function as well as the complex potential with undetermined coefficients is suggested. The involved undetermined coefficients can be evaluated from the condition of the assumed initial displacement discontinuity. Finally, a closed form solution for the problem is obtained and the calculated stress intensity factors at crack tips are presented. A problem for two collinear Zener-Stroh cracks with different lengths is also studied.     

Thermal stresses near tips of an insulated line crack in a semi-infinite medium under uniform heat flow

On the basis of the two-dimensional theory of thermoelasticity, the thermal stress field near the tips of a thermally insulated line crack in a semi-infinite medium under steady-state uniform heat flow is discussed. The crack is replaced by continuous distributions of sources of temperature discontinuity and edge dislocations. Then we obtain a set of simultaneous singular integral equations for density functions. The solution is given in the form of the product of the series of Tchebycheff polynomials of the first kind and their weight function. By means of this method, the thermal stress singularities at the crack tips are estimated exactly and the stress intensity factors can be readily evaluated. The effects of the distance from the crack tip to the bounding plane surface of the semi-infinite medium and the angle of inclination of the line crack on the stress intensity factors are shown graphically.     

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.     

Hybrid extended displacement discontinuity-charge simulation method for analysis of cracks in 2D piezoelectric media

The extended displacement discontinuity method (EDDM) and the charge simulation method (CSM) are combined to develop an efficient approach for analysis of cracks in two-dimensional piezoelectric media. In the proposed hybrid EDD–CSM, the solution for an electrically impermeable crack is approximately expressed by a linear combination of fundamental solutions of the governing equations, which includes the extended point force fundamental solutions with the sources placed at chosen points outside the domain of the problem under consideration and the extended Crouch fundamental solutions with the extended displacement discontinuities placed on the crack. The coefficients of the fundamental solutions are determined by letting the approximated solution satisfy the conditions on the boundary of the domain and on the crack face. Furthermore, the hybrid EDD–CSM is applied to solve the problems of cracks under electrically permeable condition, as well as under semi-permeable conditions by using an iterative approach. Two important crack problems in fracture mechanics, the center cracks and the edge cracks in piezoelectric strips, are analyzed by the proposed method. The stress intensity factor and the electric displacement intensity factor are calculated. Meanwhile the effects of strip size and the electric boundary conditions on these intensity factors are studied.     

A BEM analysis of fracture mechanics in 2D anisotropic piezoelectric solids

This paper presents a single-domain boundary element method (BEM) analysis of fracture mechanics in 2D anisotropic piezoelectric solids. In this analysis, the extended displacement (elastic displacement and electrical potential) and extended traction (elastic traction and electrical displacement) integral equations are collocated on the outside boundary (no-crack boundary) of the problem and on one side of the crack surface, respectively. The Green's functions for the anisotropic piezoelectric solids in an infinite plane, a half plane, and two joined dissimilar half-planes are also derived using the complex variable function method. The extrapolation of the extended relative crack displacement is employed to calculate the extended `stress intensity factors' (SIFs), i.e., K_I, K_II, K_III and K_IV. For a finite crack in an infinite anisotropic piezoelectric solid, the extended SIFs obtained with the current numerical formulation were found to be very close to the exact solutions. For a central and inclined crack in a finite and anisotropic piezoelectric solid, we found that both the coupled and uncoupled (i.e., the piezoelectric coefficient e_ijk=0) cases predict very similar stress intensity factors K_I and K_II when a uniform tension σ_yy is applied, and very similar electric displacement intensity factor K_IV when a uniform electrical displacement D_y is applied. However, the relative crack displacement and electrical potential along the crack surface are quite different for the coupled and uncoupled cases. Furthermore, for a inclined crack within a finite domain, we found that while a uniform σ_yy (=1 N m⁻²) induces only a very small electrical displacement intensity factor (in the unit of Cm^−3/2), a uniform D_y (=1 C m⁻²) can produce very large stress intensity factors (in the unit of Nm^−3/2).     

An effective method of stress intensity factor calculation for cracks emanating from a triangular or square hole under biaxial loads

This paper is concerned with stress intensity factors for cracks emanating from a triangular or square hole under biaxial loads by means of a new boundary element method. The boundary element method consists of the constant displacement discontinuity element presented by Crouch and Starfied and the crack‐tip displacement discontinuity elements proposed by the author. In the boundary element implementation, the left or the right crack‐tip displacement discontinuity element is placed locally at the corresponding left or right crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. The method is called a Hybrid Displacement Discontinuity Method (HDDM). Numerical examples are included to show that the method is very efficient and accurate for calculating stress intensity factors for plane elastic crack problems. In addition, the present numerical results can reveal the effect of the biaxial loads on stress intensity factors.     

T-stress in the Zener-Stroh arc crack problem in plane elasticity

A Zener-Stroh curved crack is defined such that the crack undergoes an initial displacement discontinuity. A singular integral equation is suggested to solve the Zener-Stroh curved crack problem. General formulation for evaluating the stress intensity factors and the T-stresses at the crack tips of a Zener-Stroh curved crack is carried out. For the Zener-Stroh arc crack, T-stresses at the crack tips can be evaluated in a closed form.     

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.     

2D Green's functions of defective magnetoelectroelastic solids under thermal loading

Thermomagnetoelectroelastic problems for various defects embedded in an infinite matrix are considered in this paper. Using Stroh's formalism, conformal mapping, and perturbation technique, Green's functions are obtained in closed form for a defect in an infinite magnetoelectroelastic solid induced by the thermal analog of a line temperature discontinuity and a line heat source. The defect may be of an elliptic hole or a Griffith crack, a half-plane boundary, a bimaterial interface, or a rigid inclusion. These Green's functions satisfy the relevant boundary or interface conditions. The proposed Green's functions can be used to establish boundary element formulation and to analyzing fracture behaviour due to the defects mentioned above.