The stress field and intensity factor due to a craze formed at the equator of a spherical inhomogeneity

The problem of a hoop-like craze formed at the equator of a spherical inhomogeneity has been investigated. The inhomogeneity is embedded in an infinitely extended elastic body which is under uniaxial tension. Both the inhomogeneity and the matrix are isotropic but have different elastic moduli. The craze is treated as a crack with parallel fibrils connecting the top and bottom surfaces. The analysis is based on the superposition principle of the elasticity theory, Hankel transform and Eshelby's equivalent inclusion method. The stress field inside the inhomogeneity and the stress intensity factor on the boundary of the craze are evaluated in the form of integral equations which are solved numerically. The result obtained is in good agreement with experimental results given in the literature. By setting the elastic moduli of the inhomogeneity the same as those of the matrix, the stress intensity factor for a thin hoop-like crack embedded in an isotropic matrix can be obtained as a deduction.     

A penny-shaped crack above the pole of a spherical inhomogeneity

Stress investigation for the problem of a penny-shaped crack located above the pole of a spherical particle (inhomogeneity) in 3D elastic solid under tension has been carried out. Both the inhomogeneity and the solid are isotropic but have different elastic moduli. The analysis is based on Eshelby's equivalent inclusion method and superposition theory of elasticity. An approximation according to the Saint-Venant principle is made in order to decouple the interaction between the crack and the inhomogeneity. An analytical solution for the stress intensity factors on the boundary of the crack is thus evaluated. It is found that both Mode I and Mode II intensity factors exist, even the loading applied at infinity is uniform tension. Results obtained show that shielding and anti-shielding (amplifying) effects of the inhomogeneity to the crack are solely determined by the modulus ratios of the inhomogeneity to the matrix. Numerical examples also indicate the interaction between the crack and the inhomogeneity is strongly influenced by the distance between the centers.     

Interaction between a spherical inhomogeneity and two symmetrically placed penny-shaped cracks

Stress analysis is carried out for a three-dimensional elastic solid containing an elastic spherical inhomogeneity and two coplanar penny-shaped cracks. Each of the two cracks is located on either side of the elastic spherical inhomogeneity and the geometry is subjected to uniform tensile stress at infinity. The interaction between the inhomogeneity and the cracks is tackled by the superposition principle of elasticity theory and Eshelby's equivalent inclusion method. Analytical solutions for the stress intensity factors on the boundaries of the cracks and the stress field inside the inhomogeneity are evaluated in series form. Numerical calculations are reported for several special cases, and show the variations of the stress intensity factors and stress field inside the inhomogeneity with the configuration and elastic properties of the solid and the inhomogeneity.     

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.     

On integral equation approaches to the mechanics of fibre-crack interaction

The paper examines the problem of a penny-shaped crack which is formed by the development of a crack in both the fibre and the matrix of a composite consisting of an isolated elastic fibre located in an elastic matrix of infinite extent. The composite region is subjected to a uniform strain field in the direction of the fibre. The paper presents two integral-equation based approaches for the analysis of the problem. The first approach considers the formulation of the complete integral equations governing the associated elasticity problem for a two material region. The second approach considers the boundary integral equation formulation of the problem. Both methods entail the numerical solution of the governing integral equations. The solutions to these integral equations are used to evaluate the stress intensity factor at the boundary of the penny-shaped crack.     

Slow torsional oscillations: The reissner-sagoci and crack problems at low frequencies

The axisymmetric elastic field produced by slowly forced torsional oscillations of a finite circular fiaw in a certain inhomogeneous medium is sought. The problem is reduced to a system of integral equations, which system is shown to cover intrinsically, the two separate cases of the flaw in the form of a penny-shaped crack and the flaw in the form of a rigid disc. The solutions are given in series of the frequency factor. Estimates of the radius of convergence are given as functions of the inhomogeneity parameter. For the flaw in the form of a rigid disc, the solution of the integral equations gives the normal displacement gradient just above the disc, from which simple integration gives the moment of the applied forces necessary to oscillate the disc. In the case of the flaw in the form of a penny-shaped crack, the solution gives the normal displacement over the crack region. This is then used to obtain the surface shear stress just outside the crack rim, from which is obtained the stress intensity factor. These physical results are all given as functions of the frequency factor and inhomogeneity parameter.     

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.     

On Boussinesq’s problem for a cracked halfspace

This paper examines the axisymmetric elastostatic problem that deals with the action of a concentrated normal force on the surface of an isotropic elastic halfspace containing a penny-shaped crack. The mathematical formulation of the elasticity problem should take into consideration the sense of action of the concentrated force. The paper presents the development of Fredholm integral equations of the second-kind that are associated with this category of problem and indicates the numerical technique that is adopted for their solution. The numerical results are presented for the stress intensity factors generated at the penny-shaped crack experiencing either opening or closure.     

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.     

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.     

Inverse crack problem in elasticity

Summary Exact solution is given to the problem of a penny-shaped crack embedded in a transversely isotropic elastic half-space when arbitrary normal displacements are prescribed at its faces. A new integral representation of the kernel of the governing integral equation allowed to obtain closed form expressions for all the quantities of interest like, stresses inside and outside the crack, stress intensity factor, work done to open the crack, directly through the given displacements. Several illustrative examples are considered.     

Elastic compliance of a partially debonded circular inhomogeneity

In the present work, we predict contribution of a partially debonded circular inhomogeneity into the material overall elastic compliance. Debonding at the matrix/inclusion boundary is modeled as interfacial arc cracks. The change in the elastic compliance caused by interface cracking is estimated through the accompanying energy change that is related to the mode I and mode II stress intensity factors at the crack tips. The sum of the crack compliance and the inhomogeneity compliance (with perfect bonding) gives the total compliance of the debonded inhomogeneity. The latter is obtained in terms of the material properties and crack length. Additional analysis shows that the replacement of an interface crack with a crack in a homogenized medium is an inadequate approach when seeking approximate solutions. The paper also provides guidelines how to determine properties of a fictitious perfectly bonded orthotropic inhomogeneity that has the same contribution into the material compliance as the debonded isotropic one. This problem is of practical importance when modeling damage accumulation in composite materials by means of homogenization schemes.     

On the dynamic interaction between a penny-shaped crack and an expanding spherical inclusion in 3-D solid

Three-dimensional stress investigation on the interaction between a penny-shaped crack and an expanding spherical inclusion in an infinite 3-D medium is studied in this paper. The spherical transformation area (the inclusion) expands in a self-similar way. By using the superposition principle, the original physical problem is decomposed into two sub-problems. The transient elastic filed of the medium with an expanding spherical inclusion is derived with the dynamic Green's function. A time domain boundary integral equation method (BIEM) is then adopted to solve the current problem. The numerical scheme applied here uses a constant shape function for elements away from the crack front, and a square root crack-tip shape function for elements near the crack tip to describe the proper behavior of the unknown quantities near the crack front. A collocation method as well as a time stepping scheme is applied to solve the BIEs. Numerical examples for the Mode I stress intensity factor are presented to assess the dynamic effect of the expanding inclusion.     

Diffraction of torsional waves by a penny-shaped crack in an infinitely long cylinder bonded to an infinite medium

This paper contains an analysis of the interaction of torsional waves with penny-shaped crack located in an infinitely long cylinder which is bonded to an infinite medium. Both the cylinder and infinite medium are of homogeneous and elastic but dissimilar materials. The solution of the problem is reduced to a Fredholm integral equation of the second kind which is solved numerically. The numerical solution is used to calculate the stress intensity factor at the rim of the penny-shaped crack.     

Axisymmetric crack with a slightly non-flat surface

A three-dimensional axisymmetric crack with a slightly non-flat surface in an isotropic linear solid under axisymmetric loading is analyzed. The problem is formulated by using the Hankel transform and a perturbation solution is obtained, which is accurate to the first order of the parameter representing the non-flatness. The stress intensity factor for the problem is evaluated. In particular, the stress intensity factor at the onset of axisymmetric kinking from a penny-shaped crack is obtained. It is also shown that the two-dimensional Cotterell-Rice theory for the effect of tensile stress acting parallel to the crack surface on the stability of crack path is valid for the axisymmetric crack.     

Dynamic stress field for torsional impact of a penny-shaped crack in a transversely isotropic functional graded strip

The torsional impact response of a penny-shaped crack in a transversely isotropic strip is considered. The shear moduli are assumed to be functionally graded such that the mathematics is tractable. Laplace and Hankel transforms are used to reduce the problem to solving a Fredholm integral equation. The crack tip stress field is obtained by considering the asymptotic behavior of Bessel function. Investigated are the effects of material nonhomogeneity and orthotropy and strip’s highness on the dynamic stress intensity factor. The peak of the dynamic stress intensity factor can be suppressed by increasing the shear moduli’s gradient and/or increasing the shear modulus in a direction perpendicular to the crack surface. The dynamic behavior varies little with the increasing of the strip’s highness.     

The interaction between a screw dislocation and a nano-inhomogeneity with a semi-infinite wedge crack penetrating the interface

The interaction between a screw dislocation and a circular nano-inhomogeneity with a semi-infinite wedge crack penetrating the interface is investigated. By using Riemann-Schwartz’s symmetry principle integrated with the analysis of singularity of complex functions and the conformal mapping technique, the analytical expressions of the stress field in both the circular nano-inhomogeneity and the infinite matrix, the image force acting on the screw dislocation and the stress intensity factor at the crack tip are obtained. The influence of elastic mismatch of materials, inhomogeneity size, interface stress, wedge crack opening angle and the relative location of dislocation on the image force and on the equilibrium position of the screw dislocation and the shielding effect of the screw dislocation are discussed in detail. The results show that interface stress has a significant impact on the movement of dislocations near the interface, and the effect of interface stress enhances when the inhomogeneity radius decreases. With the decrease in the wedge crack opening angle, the influence of interface stress on the movement of the screw dislocation and on the SIF enhances. With the increment of the relative shear modulus, the influence of interface stress weakens with the screw dislocation locating in the inhomogeneity and strengthens with the screw dislocation locating in the matrix. When the screw dislocation is located in the inhomogeneity, the positive (negative) interface stress increases (decreases) the shielding effect, while this phenomenon is opposite when the screw dislocation locates in the matrix.     

Fracture analysis of a piezoelectric layer with a penny-shaped and energetically consistent crack

The problem of an eccentric penny-shaped crack embedded in a piezoelectric layer is addressed by using the energetically consistent boundary conditions. The Hankel transform technique is applied to solve the boundary-value problem. Then two coupling Fredholm integral equations are derived and solved by using the composite Simpson’s rule. The intensity factors of stress, electric displacement, crack opening displacement and electric potential together with the energy release rate are further given. The effects of the thickness of a piezoelectric layer and the discharge field inside the penny-shaped crack on the fracture parameters of concern are discussed through numerical computations. The observations reveal that an increase of the discharge field decreases the stress intensity factor and the energy release rate. An eccentric penny-shaped crack is easier to propagate than a mid-plane one in a piezoelectric layer, and the geometry of the crack along with the layer thickness have significant influences on the electrostatic traction acting on the crack faces. The solutions for a penny-shaped dielectric crack in an infinite or a semi-infinite piezoelectric material can be obtained easily.     

Boundary element analysis of three-dimensional crack problems in two joined transversely isotropic solids

The present paper presents a boundary element analysis of penny-shaped crack problems in two joined transversely isotropic solids. The boundary element analysis is carried out 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 conventional 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 the stress intensity factors are obtained by using crack opening displacements. The numerical scheme results are verified with the closed-form solutions available in the literature for a penny-shaped crack parallel to the plane of the isotropy of a homogeneous and transversely isotropic solid of infinite extent. The new problem of a penny-shaped crack perpendicular to the interface of a transversely isotropic bi-material solid is then examined in detail. The crack surfaces are subject to the three normal tractions and the uniform shear traction. The associated stress intensity factor values are obtained and analyzed. The present results can be used for the prediction of the stability of composite structures and the hydraulic fracturing in deep rock strata and reservoir engineering.     

Elastodynamic Problem for Two Penny-Shaped Cracks: The Effect of Cracks’ Closure

The paper is devoted to the solution of the three-dimensional elastodynamic problem for linearly elastic, homogeneous and isotropic solid with two penny-shaped in-plane cracks under normally incident harmonic tension-compression wave with allowance for the contact interaction of crack faces. The effect of the distance between cracks on the stress intensity factor (opening mode) is studied for different wave numbers. The results are compared with those obtained neglecting the cracks’ closure.