Displacement Discontinuity Boundary Integral Equation Research Articles

Numerical Comparison Research on the Solution of Stress Intensity Factors of Multiple Crack Problems

A newly developed approach without crack surface discretization for modeling 2D solids with large number of cracks in linear elastic fracture mechanics is proposed with the eigen crack opening displacement (COD) boundary integral equations in this paper. The eigen COD is defined as a crack in an infinite domain under fictitious traction acting on the crack surface. Respect to the computational accuracies and efficiencies, the multiple crack problems in finite and infinite plates are solved and compared numerically using three different kinds of boundary integral equations (BIEs): 1) the dual BIEs require crack surface discretization; 2) the BIEs with numerical Green’s functions (NGF) without crack surface discretization, but have to solve a complementary matrix; 3) the eigen crack opening displacement (COD) BIEs in the present paper. With the concept of eigen COD, the multiple crack problems can be solved by using a conventional displacement discontinuity boundary integral equation in an iterative fashion with a small size of system matrix as that in the NGF approach, but without troubles to determine the complementary matrix. Solution of the stress intensity factors of multiple crack problems is solved and compared in some numerical examples using the above three computational algorithms. Numerical results clearly demonstrate the numerical models of eigen COD BIEs have much higher efficiency, providing a newly numerical technique for multiple crack problems. Not only the accuracy and efficiency of computation can be guaranteed, but also the overall properties and local details can be obtained. In conclusion, the numerical models of eigen COD BIEs realize the simulations for multiple crack problems with large quantity of cracks.

Effects of thermal and electric boundary conditions on fracture of three-dimensional thermopiezoelectric media

An arbitrarily shaped planar crack under different thermal and electric boundary conditions on the crack surfaces is studied in three-dimensional transversely isotropic thermopiezoelectric media subjected to thermal–mechanical–electric coupling fields. Using Hankel transformations, Green functions are derived for unit point extended displacement discontinuities in three-dimensional transversely isotropic thermopiezoelectric media, where the extended displacement discontinuities include the conventional displacement discontinuities, electric potential discontinuity, as well as the temperature discontinuity. On the basis of these Green functions, the extended displacement discontinuity boundary integral equations for arbitrarily shaped planar cracks in the isotropic plane of three-dimensional transversely isotropic thermopiezoelectric media are established under different thermal and electric boundary conditions on the crack surfaces, namely, the thermally and electrically impermeable, permeable, and semi-permeable boundary conditions. The singularities of near-crack border fields are analyzed and the extended stress intensity factors are expressed in terms of the extended displacement discontinuities. The effect of different thermal and electric boundary conditions on the extended stress intensity factors is studied via the extended displacement discontinuity boundary element method. Subsequent numerical results of elliptical cracks subjected to combined thermal–mechanical–electric loadings are obtained.

Analysis of arbitrarily shaped, planar cracks in a three-dimensional transversely isotropic thermoporoelastic medium

The displacement discontinuity boundary integral equation method is extended to analyze planar cracks of arbitrary shape embedded in a three-dimensional, transversely isotropic, thermoporoelastic medium. Based on the general solutions and Hankel transform technique, the fundamental solutions for unit-point, extended displacement discontinuities (including the displacement discontinuities, pore pressure discontinuity, and the temperature discontinuity) are derived. The extended displacement discontinuity boundary integral equations are established for an arbitrarily shaped, planar crack in the isotropic plane of the thermoporoelastic medium in terms of the extended displacement discontinuities. Using the boundary integral equations method, the singularities of near-crack front fields are analyzed, and the stress, fluid flux and heat flux intensity factors are derived in terms of the extended displacement discontinuities. To validate the analytical solution, the EDD boundary element method is proposed. The numerical simulation of a penny-shaped crack under combined uniform mechanical-pore pressure–thermal loadings is compared with the analytical solution to validate the correctness of the proposed method. As an application, two coplanar elliptical cracks are numerically simulated. The influences of the applied, combined mechanical-pore-pressure-thermal loadings, the crack distance, the ellipticity ratio as well as the size of cracks are all studied.

Analysis of a three-dimensional arbitrarily shaped interface crack in a one-dimensional hexagonal thermo-electro-elastic quasicrystal bi-material. Part 2: Numerical method

The extended displacement discontinuity boundary integral equation and boundary element method are extended to analyze a three-dimensional (3D) arbitrarily shaped interface crack in a one-dimensional hexagonal, quasicrystal bi-material with both electric and thermal effects under combined phonon-phason-electric-thermal loadings. Based on the analogy with the analysis method for three-dimensional, transversely isotropic magnetoelectrothermoelastic bi-materials (Dang et al., in press) the numerical method is proposed for one-dimensional quasicrystal bi-materials. Firstly, the fundamental solutions for uniformly distributed, extended displacement discontinuities applied over a constant triangular element are obtained by integrating the fundamental solutions for unit-point extended displacement discontinuities given in Part 1 over the triangular area (Zhao et al., 2017). Secondly, in order to eliminate the oscillatory singularity near the crack front, the Delta function in the integral–differential equations is approximated by the Gaussian distribution function, and the Heaviside step function is replaced by the Error function accordingly. Thirdly, the extended stress intensity factors without oscillatory singularities and the energy release rate are all obtained in terms of the extended displacement discontinuities. At last, the extended displacement discontinuity boundary element method is proposed to validate the analytical solution. In the numerical simulation, the multi-physical behavior of an elliptical, interface crack is numerically simulated. The correctness of the proposed numerical method, the influence of the applied, combined phonon-phason-electric-thermal loadings, the material-mismatch, and the ellipticity ratio are all studied.

Extended displacement discontinuity boundary integral equation and boundary element method for cracks in thermo-magneto-electro-elastic media

The extended displacement discontinuity boundary integral equation (EDDBIE) and boundary element method is developed for the analysis of planar cracks of arbitrary shape in the isotropic plane of three-dimensional (3D) transversely isotropic thermo-magneto-electro-elastic (TMEE) media. The extended displacement discontinuities (EDDs) include conventional displacement discontinuity, electric potential discontinuity, magnetic potential discontinuity, as well as temperature discontinuity across crack faces; correspondingly, the extended stresses represent conventional stress, electric displacement, magnetic induction and heat flux. Employing a Hankel transformation, the fundamental solutions for unit point EDDs in 3D transversely isotropic TMEE media are derived. The EDDBIEs for a planar crack of arbitrary shape in the isotropic plane of a 3D transversely isotropic TMEE medium are then established. Using the boundary integral equation method, the singularities of near-crack border fields are obtained and the extended stress field intensity factors are expressed in terms of the EDDs on crack faces. According to the analogy between the EDDBIEs for an isotropic thermoelastic material and TMEE medium, an analogical solution method for crack problems of a TMEE medium is proposed for coupled multi-field loadings. Employing constant triangular elements, the EDDBIEs are discretized and numerically solved. As an application, the problems of an elliptical crack subjected to combined mechanical-electric-magnetic-thermal loadings are investigated.

Fundamental solutions and analysis of three-dimensional cracks in one-dimensional hexagonal piezoelectric quasicrystals

In this study, three-dimensional cracks in one-dimensional hexagonal piezoelectric quasicrystals are investigated. Based on the general solutions and Hankel transform, the fundamental solutions for unit point extended displacement discontinuities (including the phonon and phason displacement discontinuities and the electric potential discontinuity) are derived. The extended displacement discontinuity boundary integral equations for a planar crack of arbitrary shape in the periodic plane of one-dimensional hexagonal piezoelectric quasicrystals are established in terms of the extended displacement discontinuities. Using the boundary integral equation method, the singularities of near-crack border fields are obtained and the stress and electric displacement intensity factors are derived.

Singularity analysis of planar cracks in three-dimensional piezoelectric semiconductors via extended displacement discontinuity boundary integral equation method

The displacement discontinuity boundary integral equation method is extended to analyze the singularity of near-border fields of the planar crack of arbitrary shape in the isotropic plane of a three-dimensional transversely isotropic piezoelectric semiconductor. The hyper-singular boundary integral equations are derived in terms of the displacement, electric potential and carrier density discontinuities across the crack faces, in which body integrals for the carrier density are introduced. Based on the finite-part integrals, singularity exponents and asymptotic expressions of the crack border fields are obtained. The stress, electric displacement and electric current intensity factors are given in terms of the displacement, electric potential and carrier density discontinuities. Finite element results for penny-shaped and line cracks based on the piezoelectric-conductor iterative method are used to verify the derivations of the intensity factors.

Nonlinear solution of the PS model for a semi-permeable crack in a 3D piezoelectric medium

By considering the electric field in the crack cavity, the polarization saturation (PS) model of a penny-shaped crack under electrically semi-permeable boundary condition in a three-dimensional piezoelectric medium is studied via both the extended displacement discontinuity boundary integral equation method and the boundary element method. An approximate analytical solution is derived, and the electric displacement in the crack cavity, the electric yielding zone and the local J-integral are obtained. The extended displacement discontinuity boundary element method with double iterative approaches is adapted to numerically simulate the electrically semi-permeable crack and to validate the analytical solution. The effects of different boundary conditions on the electric yielding zone and the local J-integral are also investigated.

Three-dimensional vertical cracks in magnetoelectroelastic media via the extended displacement discontinuity boundary integral equation method

In this article, the extended displacement discontinuity method is extended to study the fracture problem in a three-dimensional transversely isotropic magnetoelectroelastic medium with a planar cr...

Efficient solution of multiple cracks in great number using eigen COD boundary integral equations with iteration procedure

A newly developed computational approach is proposed in the paper for the analysis of multiple crack problems based on the eigen crack opening displacement (COD) boundary integral equations. The eigen COD particularly refers to a crack in an infinite domain under fictitious traction acting on the crack surface. With the concept of eigen COD, the multiple cracks in great number can be solved by using the conventional displacement discontinuity boundary integral equations in an iterative fashion with a small size of system matrix to determine all the unknown CODs step by step. To deal with the interactions among cracks for multiple crack problems, all cracks in the problem are divided into two groups, namely the adjacent group and the far-field group, according to the distance to the current crack in consideration. The adjacent group contains cracks with relatively small distances but strong effects to the current crack, while the others, the cracks of far-field group are composed of those with relatively large distances. Correspondingly, the eigen COD of the current crack is computed in two parts. The first part is computed by using the fictitious tractions of adjacent cracks via the local Eshelby matrix derived from the traction boundary integral equations in discretized form, while the second part is computed by using those of far-field cracks so that the high computational efficiency can be achieved in the proposed approach. The numerical results of the proposed approach are compared not only with those using the dual boundary integral equations (D-BIE) and the BIE with numerical Green's functions (NGF) but also with those of the analytical solutions in literature. The effectiveness and the efficiency of the proposed approach is verified. Numerical examples are provided for the stress intensity factors of cracks, up to several thousands in number, in both the finite and infinite plates.

Numerical solution of stress intensity factors of multiple cracks in great numbers with eigen crack opening displacement boundary integral equations

Based on the eigen crack opening displacement (COD) boundary integral equations (BIEs), a newly developed computational approach is proposed for the analysis of multiple crack problems. The eigen COD particularly refers to a crack in an infinite domain under fictitious traction acting on the crack surface. With the concept of eigen COD, the multiple cracks in great number can be solved by using the conventional displacement discontinuity BIEs in an iterative fashion with a small size of system matrix. The interactions among cracks are dealt with by two parts according to the distances of cracks to the current crack. The strong effects of cracks in adjacent group are treated with the aid of the local Eshelby matrix derived from the traction BIEs in discrete form. While the relatively week effects of cracks in far-field group are treated in the iteration procedures. Numerical examples are provided for the stress intensity factors of multiple cracks, up to several thousands in number, with the proposed approach. By comparing with the analytical solutions in the literature as well as solutions of the dual BIEs, the effectiveness and the efficiencies of the proposed approach are verified.

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

Abstract 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.

Solution of stress intensity factors of multiple cracks in plane elasticity with eigen COD formulation of boundary integral equation

The concept of eigen crack opening displacement (COD) can be defined as the COD of a crack in infinite plate under the tractions acting on the crack surface. By introducing this concept, the eigen COD formulation of boundary integral equation is proposed in this paper, together with the solution procedures for multiple crack problems in plane elasticity. With the proposed approach, the multiple crack problems can be solved with the conventional displacement discontinuity boundary integral equations in an iterative fashion with a small size of system matrix as that in the numerical Green’s function (NGF) approach but without the trouble to determine the complementary solutions since the standard boundary element discretization on the crack surface is no longer required with the proposed approach. Some numerical examples computing the stress intensity factors are presented and compared with those in literature to show the accuracy and the effectiveness of the proposed approach.

Dugdale model solutions for a penny-shaped crack in three-dimensional transversely isotropic piezoelectric media by boundary-integral equation method

Making use of the Displacement Discontinuity Boundary Integral Equation Method (DDBIEM), the dimension of the plastic zone at the tip of a penny-shaped crack in a three-dimensional elastic medium is determined by the application of the Dugdale model; Furthermore, the solutions for a penny-shaped crack in three-dimensional piezoelectric media are obtained by the use of the Dugdale-like model proposed by Gao et al.[Gao H, Zhang T, Tong P. Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic. J. Mech. Phys. Solids 1997;45:491–510], in which the electrical polarization is assumed to reach a saturation limit in a thin annular region in front of a crack while the mechanical stresses have the ordinary singularity.

The method of analysis of crack problem in three-dimensional non-local elasticity

In this paper, the displacement discontinuity fundamental solution (DDFS) corresponding to the unit concentrated displacement discontinuity for three dimensional (3D) non-local elasticity under symmetrical condition is obtained. Based on the displacement discontinuity boundary integralequation (DDBIE) and boundary-element method (DDBEM) of local (classical) elasticity, a method of analysis of crack in 3D non-local elasticity with wide application is proposed with the DDFS. Through the method, several important problems of fracture mechanics are analysed.

The analysis of crack problems with non-local elasticity

In this paper, the displacement discontinuity fundamental solutions (DDFS) corresponding to the unit concentrated displacement discontinuity for plane problems of non-local elasticity are obtained. Based on the displacement discontinuity boundary integral equation (DDBIE) and boundary element method (BEM), a method of analysis of crack problems in non-local elasticity with generalized purpose is proposed. By using this method, several important problems in fracture mechanics such as edge crack are studied. The study of edge crack shows that the stress concentration factor (SCF) near the crack tip is not a constant but varies with the crack length. With this result the effect of crack length on the fracture toughness KI c is studied. The results obtained in this paper are in accordance with the published ones.

Moment stress intensity factors for collinear and parallel cracks in Reissner's plate: Boundary integral equation approach

Based on Betti's reciprocal theorem, a displacement discontinuity boundary integral equation is derived for a Reissner plate with line cracks. The boundary element method is used to obtain the moment stress intensity factors for two collinear and parallel cracks. Displayed graphically are the numerical results for the moment stress intensity factors.