Sudden twisting of an external circular crack in an infinite medium with a cylindrical inclusion

In this paper the torsional impact response of an external circular crack in an infinite medium bonded to a cylindrical inclusion has been investigated. The infinite medium and cylindrical inclusion are assumed to be of different homogeneous isotropie elastic materials. Laplace and Hankel transforms are used to reduce the problem to the solution of a pair of dual integral equations. These equations are solved by using an integral transform technique and the results are expressed in terms of a Fredhol integral equation of the second kind. By solving Fredholm integral equation of the second kind the numerical results for the dynamic stress-intensity factor are obtained which measure the load transmission on the crack.     

A problem of Reissner-Sagoci type for an elastic cylinder embedded in an elastic half-space

The problem considered is that of the torsion of an elastic cylinder which is embedded in an elastic half-space of different rigidity modulus. It is assumed that there is perfect bonding at the common cylindrical surface and also that the torque is applied to the cylinder through a rigid disk bonded to its flat surface. The problem is reduced, by means of the use of integral transforms and the theory of dual integral equations to that of solving a Fredholm integral equation of the second kind. The results obtained by solving this equation are exhibited graphically in Fig. 2.     

Penny-shaped crack in an infinite elastic cylinder bonded to an infinite elastic material surrounding it

This paper concerns with the state of stress in a long elastic cylinder, with a concentric penny-shaped crack, bonded to an infinite elastic medium. The crack is assumed to be opened by an internal pressure and that the plane of the crack is perpendicular to the axis of the cylinder. The elastic constants of the cylinder and the semi-infinite medium are assumed to be different. The problem is reduced to the solution of a Fredholm integral equation of the second kind. Closed form expressions are obtained for the stress-intensity factor and the crack energy. The integral equation is solved numerically and results are used to obtain the numerical values of the stress-intensity factor and the crack energy which are graphed.     

Thermal stresses near a penny-shaped crack in an elastic sphere embedded in an infinite elastic space

This paper gives an analysis of the distribution of thermal stresses in a sphere which is bonded to an infinite elastic medium. The thermal and the elastic properties of the sphere and the elastic infinite medium are assumed to be different. The penny-shaped crack lies on the diametral plane of the sphere and the centre of the crack is the centre of the sphere. By making a suitable representation of the temperature function, the heat conduction problem is reduced to the solution of a Fredholm integral equation of the second kind. Using suitable solution of the thermoelastic displacement differential equation, the problem is then reduced to the solution of a Fredholm integral equation, in which the solution of the earlier integral equation arising from heat conduction problem occurs as a known function. Numerical solutions of these two Fredholm integral equations are obtained. These solutions are used to evaluate numerical values for the stress intensity factors. These values are displayed graphically.     

A non-axisymmetric cylindrical contact problem

The contact problem under investigation is one whereby a solid circular elastic cylinder of infinite length is rigidly indented by a two piece collar of finite length, each piece being diametrically opposed and extending only partially around one half of the circumference. This case is practically significant in relation to the axisymmetric cylindrical contact problem since in many cases attachment of a component to a cylindrical shaft is achieved by means of a two piece clamp.Shear stresses on the contact interface are taken zero and a radial displacement influence coefficient technique is used to model the integral equation governing this contact problem. Adopting the Papkovich-Neuber solution for the non-axisymmetric cylindrical coordinate case and substituting the appropriate boundary conditions leads to a combined Fourier series, Fourier integral representation for the desired displacements. Convergence of this series—integral is studied and results of interference contact pressure are presented for an illustrative range of the various parameters involved.     

Shear stress intensity factor near semi-infinite cylindrical crack edges under their shock loading

The discontinuous solution of the torsional vibration equation for an elastic medium with a flaw in the form of a semi-infinite cylindrical crack is constructed. The method of solving the integro-differential equation describing the distribution of shear stresses along the edges of a cylindrical crack is presented. The evaluation procedure for a stress intensity factor and its numerical calculation for the case of short times under the shock loading of cylindrical crack edges are given. It is established that the magnitude of a dynamic stress intensity factor can be used to determine the condition of shock wave interactions with structural heterogeneities at the high-rate deformation of treated surfaces containing flaws in the form of cylindrical cracks. Translated from Problemy Prochnosti, No. 3, pp 63–72, May–June, 1999.     

Contact problem of a functionally graded layer resting on a Winkler foundation

The contact problem for a functionally graded layer supported by a Winkler foundation is considered using linear elasticity theory in this study. The layer is loaded by means of a rigid cylindrical punch that applies a concentrated force in the normal direction. Poisson’s ratio is taken as constant, and the elasticity modulus is assumed to vary exponentially through the thickness of the layer. The problem is reduced to a Cauchy-type singular integral equation with the use of Fourier integral transform technique and the boundary conditions of the problem. The numerical solution of the integral equation is performed by using Gauss–Chebyshev integration formulas. The effect of the material inhomogeneity, stiffness of the Winkler foundation and punch radius on the contact stress, the contact area and the normal stresses are given.     

Singular stresses in an infinite solid with a flat annular crack around a spherical cavity under tension

The problem of singular stresses in an infinite elastic solid containing a spherical cavity and a flat annular crack subjected to axial tension is considered. By application of an integral transform method and the theory of triple integral equations the problem is reduced to that of solving a singular integral equation of the first kind. The singular integral equation is solved numerically, and the influence of the spherical cavity upon the stress intensity factor and the influence of the annular crack upon the maximum stress at the surface of the spherical cavity are shown graphically in detail.     

Dislocations and internal loading in a semi-infinite elastic medium with surface stresses

This paper presents analytical solutions for shear and opening dislocations in an elastic half-plane with surface stresses by using the Gurtin–Murdoch continuum theory of elastic material surfaces. The fundamental solutions corresponding to buried vertical and horizontal loads are also presented. Fourier integral transforms are used in the analysis. It is found that a characteristic length parameter that depends on the surface and bulk elastic moduli exists for this class of problems, and it represents the influence of surface stresses on the bulk elastic field. Selected numerical results are presented to demonstrate the influence of surface stresses on the bulk stress field. The fundamental solutions presented in this study can be used to develop boundary integral equation and other methods to analyze complicated fracture and boundary-value problems associated with nano-scale structures and soft elastic solids.     

Fracture criterion for the axisymmetric disking process

Disking is a process designed to cut brittle plates and rods. In axisymmetric disking, a pre-cracked cylindrical rod is placed in an elastic, annular sheath and the composite is subjected to biaxial fluid pressure. At a critical pressure the crack runs across the circular section of the rod producing a clean cut. A linear elastic fracture mechanics analysis is used to develop a fracture criterion for the process. A concentric circular cylinders model is assumed with perfect bonding at the interface. The flaw that initiates fracture in the rod is modeled as an annular crack perpendicular to the interface with one crack tip at the interface. The problem is formulated as a singular integral equation of the first kind with a Cauchy type kernel. The stress intensity factors are determined as a function of crack size and shear moduli.     

Stress intensity factor for the cylindrical interface crack between nonhomogeneous coaxial finite elastic cylinders

The problem of determining the stress intensity factor for a cylindrical interface crack between two dissimilar nonhomogeneous coaxial finite elastic cylinders under axially symmetric longitudinal shear stress is considered. The mixed boundary conditions lead to a pair of dual series equations which are reduced to a Fredholm integral equation of the second kind and then finally to a system of algebraic equations. Numerical values of the stress intensity factor are presented graphically.     

Periodically distributed parallel cracks in a functionally graded piezoelectric (FGP) strip bonded to a FGP substrate under static electromechanical load

In this paper, the Fourier integral transform–singular integral equation method is presented for the problem of a periodic array of cracks in a functionally graded piezoelectric strip bonded to a different functionally graded piezoelectric material. The properties of two materials, such as elastic modulus, piezoelectric constant and dielectric constant, are assumed in exponential forms and vary along the crack direction. The crack surface condition is assumed to be electrically impermeable or permeable. The mixed boundary value problem is reduced to a singular integral equation over crack by applying the Fourier transform and the singular integral equation is solved numerically by using the Lobatto–Chebyshev integration technique. The analytic expressions of the stress intensity factors and the electric displacement intensity factors are derived. The effects of the loading parameter λ, material constants and the geometry parameters on the stress intensity factor, the energy release ratio and the energy density factor are studied.     

Edge crack in an elastic layer resting on Winkler foundation

The paper deals with the stress analysis near a crack tip in an elastic layer resting on Winkler foundation. The edge crack is assumed to be normal to the lower boundary plane. The upper surface of the layer is loaded by given forces normal to the boundary. The considered problem is solved by using the method of Fourier transforms and dual integral equations, which are reduced to a Fredholm integral equation of the second kind. The stress intensity factor is given in the term of solution of the Fredholm integral equation and some numerical results are presented.     

On crack problem in an elastic ponderable layer

The paper deals with the plane problem of stress distribution in an elastic ponderable layer with a stationary edge crack normal to the boundary plane. The layer is situated and fixed on a rigid foundation. The stresses are caused by action of body forces. By using the method of Fourier transforms the problem is reduced to a system of dual integral equations and next, to a Fredholm integral equation of the second kind. The numerical analysis of the Fredholm equation permitted to determine the stress intensity factor and the crack opening displacement. This revised version was published online in July 2006 with corrections to the Cover Date.     

A problem of Reissner-Sagoci type for a semi-infinite composite elastic cylinder

Summary The problem considered is that of the torsion of a semi-infinite elastic cylinder which is embedded in a semi-infinite elastic cylindrical shell of different material. By the use of integral transforms and the theory of dual integral equations, the problem is reduced to the solution of a Fredholm integral equation of the second kind. Numerical solution of the integral equation is obtained and the numerical values of the torque are displayed graphically.     

Elastic analysis of a half-plane with multiple inclusions using volume integral equation method

A volume integral equation method (VIEM) is used to calculate the elastostatic field in an isotropic elastic half-plane containing circular inclusions subject to remote loading parallel to the traction-free boundary. The material of the inclusions may be either isotropic or anisotropic and they are assumed to be distributed in square or hexagonal array. A detailed analysis of the stress field at the interface between the matrix and one of the inclusions is carried out for different distances between the inclusion and the surface of the half-plane. The results of the calculations are compared with available results. The VIEM is shown to be very accurate and effective for investigating the local stresses in the presence of multiple inclusions. The method can be applied to multiple inclusions of arbitrary geometry and elastic properties embedded in extended isotropic elastic media.     

On the problem of two coplanar cracks inside an infinite isotropic elastic solid

A method is proposed for the approximate evaluation of normal displacements and normal stresses on the plane of two coplanar cracks located inside an infinite isotropic elastic solid and subjected to normal internal pressure. The formulation results in a single integral equation for the unknown normal stresses on the plane of the cracks. Numerical results are given for the stress intensity factor K_I of two coplanar circular cracks and two coplanar elliptical cracks opened up under a uniform internal pressure.     

Annular punch on an elastic layer overlying an elastic foundation

The problem solved here is the axisymmetric mixed boundary value problem of the isotropic homogeneous theory of elasticity, in which the normal displacement is specified inside an annular area $\text{a ≤ r ≤ b}$, the normal stress is zero in $\text{r < a, r \# b}$ and the shearing stress is zero on the whole face $\text{z = ?h}$, the upper face of the elastic layer; the continuity of the normal and radial displacements and the normal and shearing stresses is assumed at the interface $\text{z = 0}$ between the elastic layer and the elastic foundation having different elastic constants. The problem is reduced to the solution of a Fredholm integral equation of the first kind. The Fredholm integral equation is further put in terms of four simultaneous Fredholm integral equations of the second kind in four unknown functions. The iterative solution of these integral equations has been obtained for $\text{epsi = b/h ? 1}$, and $\text{λ = a/b ? 1}$ for the case of an annular cylindrical punch. The expressions for the normal stress $\text{σ}{}_{\mathrm{zz}}\text{(r, ?h)}$ for $\text{a ≤ r ≤ b}$ and the total load $\text{P}$ on the punch have been obtained.     

Theoretical stress analysis for a non-isotropic body of cylindrical configuration containing a row of cracks

A problem of stress analysis for a long circular cylinder has been dealt with in this paper by analytical methods. The cylinder is assumed to be made of an elastic material which is not isotropic but the elastic properties are considered to be similar in directions perpendicular to the axis of the cylinder. The body under consideration is supposed to contain an infinite row of penny-shaped cracks which are parallel to each other and located periodically along the cylinder-axis. All the cracks are assumed to be opened by the same distribution of internal pressure on their surfaces. By choosing appropriate potential functions the problem is treated mathematically through the use of integral equation approach. Numerical results for the stress-intensity factor, the strain energy and the critical pressure, obtained on the basis of the analysis are also given.     

An integral equation analysis of inelastic shells

This paper presents an integral equation approach for the analysis of deformation and stresses in inelastic shells of arbitrary shape subjected to arbitrary loading. The proposed mathematical model is completely consistent and is derived by transforming the three-dimensional equations from the Cartesian to the appropriate curvilinear coordinates of the shell. Appropriate kinematic assumptions for the dependence of the displacements on the thickness coordinate of the shell and assumptions regarding the loads at the ends are introduced consistently in the model to take advantage of the thinness of the shell. Numerical implementation and numerical results are presented for elastic and inelastic deformation of axisymmetric shells subjected to axisymmetric loading. These results are compared against exact elasticity, Love-Kirchoff model analysis of inelastic cylindrical shells and finite element solutions.