Analysis of elliptical cracks perpendicular to the interface of two joined transversely isotropic solids

This paper presents a boundary element analysis of elliptical cracks in two joined transversely isotropic solids. The boundary element method is developed by incorporating the fundamental singular solution for a concentrated point load in a transversely isotropic bi-material solid of infinite space into the conventional displacement boundary integral equations. The multi-region method is used to analyze the crack problems. The traction-singular elements are employed to capture the singularity around the crack front. The values of stress intensity factors (SIFs) are obtained by using crack opening displacements. The results of the proposed method compare well with the existing exact solutions for an elliptical crack parallel to the isotropic plane of a transversely isotropic solid of infinite extent. Elliptical cracks perpendicular to the interface of transversely isotropic bi-material solids of either infinite extent or occupying a cubic region are further examined in detail. The crack surfaces are subject to the uniform normal tractions. The stress intensity factor values of the elliptical cracks of the two types are analyzed and compared. Numerical results have shown that the stress intensity factors are strongly affected by the anisotropy and the combination of the two joined solids.     

Stress intensity factors of square crack inclined to interface of transversely isotropic bi-material

This paper analyzes a square crack in a transversely isotropic bi-material solid by using dual boundary element method. The square crack is inclined to the interface of the bi-material. The fundamental solution for the bi-material solid occupying an infinite region is incorporated into the dual boundary integral equations. The square crack can have an arbitrary angle with respect to the plane of isotropy of the bi-material occupying either finite or infinite regions. The stress intensity factor (SIF) values of the modes I, II, and III associated with the square crack are calculated from the crack opening displacements. Numerical results show that the properties of the anisotropic bi-material have evident influences on the values of the three SIFs. The values of the three SIFs are further examined by taking into account the effect of the external boundary of the internally cracked bi-material.     

Three-dimensional fracture analysis in transversely isotropic solids

In this paper a boundary element formulation for three-dimensional crack problems in transversely isotropic bodies is presented. Quarter-point and singular quarter-point elements are implemented in a quadratic isoparametric element context. The point load fundamental solution for transversely isotropic media is implemented. Numerical solutions to several three-dimensional crack problems are obtained. The accuracy and robustness of the present approach for the analysis of fracture mechanics problems in transversely isotropic bodies are shown by comparison of some of the results obtained with existing analytical solutions. The approach is shown to be a simple and useful tool for the evaluation of stress intensity factors in transversely isotropic media.     

Application of finite-part integrals to three-dimensional fracture problems for piezoelectric media Part I: Hypersingular integral equation and theoretical analysis

This paper deals with some basic linear elastic fracture problems for an arbitrary-shaped planar crack in a three-dimensional infinite transversely isotropic piezoelectric media. The finite-part integral concept is used to derive hypersingular integral equations for the crack from the point force and charge solutions with distinct eigenvalues s _i(i=1,2,3) of an infinite transversely isotropic piezoelectric media. Investigations on the singularities and the singular stress fields and electric displacement fields in the vicinity of the crack are made by the dominant-part analysis of the two-dimensional integrals. Thereafter the stress and electric displacement intensity factor K-fields and the energy release rate G are exactly obtained by using the definitions of stress and electric displacement intensity factors and the principle of virtual work, respectively. The hypersingular integral equations under axially symmetric mechanical and electric loadings are solved analytically for the case of a penny-shaped crack.     

Crack transient torsional wave interaction in an elastic bi-material

An infinitely long cylindrical elastic solid of finite radius (fibre or inner medium) is imbedded in another infinite elastic cylindrical solid (matrix or outer medium) to form an elastic bi-material composite perfectly bonded at their bi-material interface. The early-time response of the composite to a sudden twist applied axisymmetrically over a stationary penny-shaped crack which appears over a cross-section of the fibre and terminates orthogonally at the bi-material interface, is investigated. The magnitude of the applied twist is assumed to be less than that for which debonding or crack propagation can take place. For a bi-material composite the square of whose matrix shear-wave speed is less than twice the square of whose fibre shear-wave speed, uniform asymptotic analytic results, whose accuracy can be improved by taking higher order terms in the solution, are obtained for the time-dependent surface displacement gradient normal to the plane of the crack and outside the crack region, the dynamic stress intensity factor, and also the stress singularity strength eigenvalue-all as functions of the fibre-matrix material disparities.     

Axisymmetric crack terminating at the interface of transversely isotropic dissimilar media

In this paper, the axisymmetric elasticity problem of an infinitely long transversely isotropic solid cylinder imbedded in a transversely isotropic medium is considered. The cylinder contains an annular or a penny shaped crack subjected to uniform pressure on its surfaces. It is assumed that the cylinder is perfectly bonded to the medium. A singular integral equation of the first kind (whose unknown is the derivative of crack surface displacement) is derived by using Fourier and Hankel transforms. By performing an asymptotic analysis of the Fredholm kernel, the generalized Cauchy kernel associated with the case of `crack terminating at the interface' is derived. The stress singularity associated with this case is obtained. The singular integral equation is solved numerically for sample cases. Stress intensity factors are given for various crack geometries (internal annular and penny-shaped cracks, annular cracks and penny-shaped cracks terminating at the interface) for sample material pairs.     

Finite element study of a penny-shaped crack along the interface in a bi-material cylinder

A contemporary approach to the analysis of interface cracks in bi-material cylinders using finite elements is presented. From results obtained with a commercial finite element code using regular and singular isoparametric elements, three fracture mechanics techniques are considered to study the interface crack problem and are presented in a fundamental manner. These are the stress intensity factor evaluation by the crack opening displacement method, the strain energy release rate evaluation using the modified crack closure integral method, and the J-integral evaluation using the virtual crack extension technique. Only the finite element results in the vicinity of the crack are then needed. The accuracy of the proposed approach is assessed by solving standard test problems with known solutions. In particular, the mode I problem of a penny-shaped crack in a homogeneous isotropic cylinder under remote tension loading is used as a standard test case. Finally, the mixed-mode (I and II) problem of a penny-shaped crack along the interface in a bi-material cylinder under three loading conditions is studied in detail. Numerical results are presented to quantify the combined effects of geometry and material discontinuities on the strain energy release rate.     

Boundary integral equation method for conductive cracks in two and three-dimensional transversely isotropic piezoelectric media

Using the fundamental solutions and the Somigliana identity of piezoelectric medium, the boundary integral equations are obtained for a conductive planar crack of arbitrary shape in three-dimensional transversely isotropic piezoelectric medium. The singular behaviors near the crack edge are studied by boundary integral equation approach, and the intensity factors are derived in terms of the displacement discontinuity and the electric displacement boundary value sum near the crack edge on crack faces. The boundary integral equations for two dimensional crack problems are deduced as a special case of infinite strip planar crack. Based on the analogy of the obtained boundary integral equations and those for cracks in conventional isotropic elastic material and for contact problem of half-space under the action of a rigid punch, an analysis method is proposed. As an example, the solution to conductive Griffith crack is derived.     

Three-dimensional steady-state general solution for transversely isotropic hygrothermopiezoelectric media and its application in fracture

The potential theory method is utilized to derive the steady-state, general solution for three-dimensional (3D) transversely isotropic, hygrothermopiezoelectric media in the present paper. Two displacement functions are introduced to simplify the governing equations. Employing the differential operator theory and superposition principle, all physical quantities can be expressed in terms of two functions, one satisfies a quasi-harmonic equation and the other satisfies a tenth-order partial differential equation. The obtained general solutions are in a very simple form and convenient to use in boundary value problems. As one example, the 3D fundamental solutions are presented for a steady point moisture source combined with a steady point heat source in the interior of an infinite, transversely isotropic, hygrothermopiezoelectric body. As another example, a flat crack embedded in an infinite, hygrothermopiezoelectric medium is investigated subjected to symmetric mechanical, electric, moisture and temperature loads on the crack faces. Specifically, for a penny-shaped crack under uniform combined loads, complete and exact solutions are given in terms of elementary functions, which serve as a benchmark for different kinds of numerical codes and approximate solutions.     

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.     

Axisymmetric problems for an externally cracked elastic solid. I. Effect of a penny-shaped crack

The present paper examines the problem of a penny-shaped flaw which is located in the plane of an external crack in an isotropic elastic solid. The penny-shaped flaw is subjected to uniform internal pressure. The paper develops power series representations for the stress intensity factors at the boundary of the penny-shaped flaw and at the perimeter of the externally cracked region. These series representations are in terms of a non-dimensional parameter which is the ratio of the radius of the penny-shaped flaw to the radius of the externally cracked region.     

The interaction between a penny-shaped crack and a spherical inhomogeneity in an infinite solid under uniaxial tension

Summary This paper presents an approximate three-dimensional analytical solution to the elastic stress field of a penny-shaped crack and a spherical inhomogeneity embedded in an infinite and isotropic matrix. The body is subjected to an uniaxial tension applied at infinity. The inhomogeneity is also isotropic but has different elastic moduli from the matrix. The interaction between the crack and the inhomogeneity is treated by the superposition principle of elasticity theory and Eshelby's equivalent inclusion method. The stress intensity factor at the boundary of the penny-shaped crack and the stress field inside the inhomogeneity are evaluated in the form of a series which involves the ratio of the radii of the spherical inhomogeneity and the crack to the distance between the centers of inhomogeneity and crack. Numerical calculations are carried out and show the variation of the stress intensity factor with the configuration and the elastic properties of the matrix and the inhomogeneity.     

Punch and crack problems in transversely isotropic bodies

An exact solution is proposed for the mixed boundary-value problem in a transversely isotropic half-space. Here, certain arbitrary shear tractions are prescribed inside a circular region, outside of which certain arbitrary tangential displacements are given. The normal stresses are supposed to be known all over the boundary. A particular case is considered, in detail, where normal stresses vanish all over the boundary with the shear tractions vanishing inside the circular region. A closed form expression is obtained for the tangential displacements inside the circular region directly through the displacements outside. As an example, a penny-shaped crack in an infinite transversely isotropic body is considered with arbitrary shear tractions acting on both sides of the crack. The formulae for the tangential displacements inside the circle and the shear stresses outside are obtained. Special cases where uniform shear and a concentrated tangential force arise are also discussed.     

A frequency-domain BEM for 3D non-synchronous crack interaction analysis in elastic solids

A frequency-domain boundary element method (BEM) is presented for non-synchronous crack interaction analysis in three-dimensional (3D), infinite, isotropic and linear elastic solids with multiple coplanar cracks. The cracks are subjected to non-synchronous time-harmonic crack-surface loading with contrast frequencies. Hypersingular frequency-domain traction boundary integral equations (BIEs) are applied to solve the boundary value problem. A collocation method is adopted for solving the BIEs numerically. The local square-root behavior of the crack-opening-displacements at the crack-front is taken into account in the present method. For two coplanar penny-shaped cracks of equal radius subjected to non-synchronous time-harmonic crack-surface loading, numerical results for the dynamic stress intensity factors are presented and discussed.     

Sensitivities of Two-Dimensional Fracture Problems to the Near-tip Mesh Parameters

Analytical results for a penny-shaped crack with a plastic zone at the crack front are given. The crack is embedded in an infinite transversely isotropic elastic medium and is assumed to be subjected to two identical axisymmetric loads on the upper and lower crack faces. The size of the plastic zone at the crack front is determined by applying Dugdale hypothesis to the elasticity results for a penny-shaped crack. The size of the plastic zone is derived in terms of hyper-geometric functions. Expression of the normal stress outside the plastic zone is also given.     

Stress intensity factors for penny-shaped cracks perpendicular to graded interfacial zone of bonded bi-materials

This paper examines the stress intensity factors that are associated with a penny-shaped crack perpendicular to the interface of a bi-material bonded with a graded interfacial zone. Elastic modulus of the graded interfacial zone is assumed to be an exponential function of the depth. The stress intensity factors are calculated numerically using a so-called generalized Kelvin solution based boundary element method. Three cases of normal or shear tractions acting on the crack surfaces are examined. Values of the stress intensity factors are examined by taking into account the effects of the following four parameters: (a) the crack front position; (b) the non-homogeneity parameter of the graded interfacial zone; (c) the crack distance to the graded interfacial zone; and (d) the graded interfacial zone thickness. The numerical results are compared well with existing solutions under some degenerated conditions. These results are useful to furthering our knowledge on fracture behavior of bi-material systems with or without a graded interfacial zone.     

Dynamic stress intensity factor (Mode I) of a penny-shaped crack in an infinite poroelastic solid

This paper considers the transient stress intensity factor (Mode I) of a penny-shaped crack in an infinite poroelastic solid. The crack surfaces are impermeable. By virtue of the integral transform methods, the poroelastodynamic mixed boundary value problems is formulated as a set of dual integral equations, which, in turn, are reduced to a Fredholm integral equation of the second kind in the Laplace transform domain. Time domain solutions are obtained by inverting Laplace domain solutions using a numerical scheme. A parametric study is presented to illustrate the influence of poroelastic material parameters on the transient stress intensity. The results obtained reveal that the dynamic stress intensity factor of poroelastic medium is smaller than that of elastic medium and the poroelastic medium with a small value of the potential of diffusivity shows higher value of the dynamic stress intensity factor.     

A magneto-electro-elastic material with a penny-shaped crack subjected to temperature loading

Summary The analysis of intensity factors for a penny-shaped crack under thermal, mechanical, electrical and magnetic boundary conditions becomes a very important topic in fracture mechanics. An exact solution is derived for the problem of a penny-shaped crack in a magneto-electro-thermo-elastic material in a temperature field. The problem is analyzed within the framework of the theory of linear magneto-electro-thermo-elasticity. The coupling features of transversely isotropic magneto-electro-thermo-elastic solids are governed by a system of partial differential equations with respect to the elastic displacements, the electric potential, the magnetic potential and the temperature field. The heat conduction equation and equilibrium equations for an infinite magneto-electro-thermo-elastic media are solved by means of the Hankel integral transform. The mathematical formulations for the crack conditions are derived as a set of dual integral equations, which, in turn, are reduced to Abel's integral equation. Solution of Abel's integral equation is applied to derive the elastic, electric and magnetic fields as well as field intensity factors. The intensity factors of thermal stress, electric displacement and magnetic induction are derived explicitly for approximate (impermeable or permeable) and exact (a notch of finite thickness crack) conditions. Due to its explicitness, the solution is remarkable and should be of great interest in the magneto-electro-thermo-elastic material analysis and design.     

Diffraction of high frequency torsion waves by a penny-shaped crack

The diffraction of high frequency torsion waves by a penny-shaped crack situated in an infinite isotropic elastic solid is considered. Asymptotic expressions for the dynamic stress intensity factors are derived for a variety of incident excitations, and the results predict an oscillatory behaviour of these factors at high frequencies.     

Time-domain boundary element analysis of elastic wave interaction with a crack

The mathematical formulation of the problem of transient wave interaction with a crack in a homogeneous, isotropic, linearly elastic solid has been reduced to the solution of an integral equation over the insonified crack face. The integral equation relates the unknown crack-opening displacement, which depends on time and position, to the incident wave field. The integral equation has been solved numerically by a time-stepping method in conjunction with a boundary element discretization of the crack surface. For normal incidence of a longitudinal step-stress wave on a penny-shaped crack, results as functions of time have been obtained for the crack-opening displacement, the elastodynamic Mode-I stress intensity factor and the scattered far-field.