Axisymmetric finite cylinder with one end clamped and the other under uniform tension containing a penny-shaped crack

This study considers the axisymmetric analysis of a finite cylinder containing a penny-shaped transverse crack. Material of the cylinder is assumed to be linearly elastic and isotropic. One end of the cylinder is bonded to a fixed support while the other end is subjected to uniform axial tension. Solution is obtained by superposing the solutions for an infinite cylinder loaded at infinity and an infinite cylinder containing four cracks and a rigid inclusion loaded along the cracks and the inclusion. When the radius of the inclusion approaches the radius of the cylinder, its mid-plane becomes fixed and when the radius of the distant cracks approach the radius of the cylinder, the ends become cut and subject to uniform tensile loads. General expressions for the perturbation problem are obtained by solving Navier equations with Fourier and Hankel transforms. Formulation of the problem is reduced to a system of five singular integral equations. By using Gauss-Lobatto and Gauss-Jacobi integration formulas, these five singular integral equations are converted to a system of linear algebraic equations which is solved numerically. Stress distributions along the rigid support, stress intensity factors at the edges of the rigid support and the crack are calculated.     

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.     

Cracked semi-infinite cylinder and finite cylinder problems

This work considers the analysis of a cracked semi-infinite cylinder and a finite cylinder. Material of the cylinder is assumed to be linearly elastic and isotropic. One end of the cylinder is bonded to a fixed support while the other end is subjected to axial tension. Solution of this problem can be obtained by superposition of solutions for an infinite cylinder subjected to uniformly distributed tensile load at infinity (I) and an infinite cylinder having a penny-shaped rigid inclusion at z = 0 and two penny-shaped cracks at z = ±L (II). General expressions for the perturbation problem (II) are obtained by solving Navier equations with Fourier and Hankel transforms. When the radius of the inclusion approaches the radius of the cylinder, the end at z = 0 becomes fixed and when the radius of the cracks approach the radius of the cylinder, the ends at z = ±L become cut and subject to uniform tensile load. Formulation of the problem is reduced to a system of three singular integral equations. By using Gauss–Lobatto and Gauss–Jacobi integration formulas, these three singular integral equations are converted to a system of linear algebraic equations which is solved numerically.     

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.     

Two bonded strips containing an infinite row of interface cracks

The plane strain problem of two bonded dissimilar isotropic elastic strips is considered. It is assumed that the composite strip contains an infinite row of interface cracks located symmetrically on the centerline. The case in which each of the cracks is opened out by the same constant pressure is discussed in detail and numerical results are reported for quantities of practical interest.     

Stress intensity factors and crack extension in a cracked pressurised cylinder

This paper considers the plane elastic problem corresponding to single or multiple radial cracks emanating from the internal boundary of a circular ring, under uniform external tension and internal pressure. The stress intensity factors are calculated by using the dual boundary element method with the J-integral technique. Accurate data are found for varying crack depths over a representative range of wall ratios for fracture mechanics applications to pressurised circular cylinders. The interaction of multiple cracks and crack extension are investigated in the case of an internal pressure loading condition. The analysis shows that, for a multi-cracked pressurised cylinder, it is sufficient to calculate the stress intensity of the main crack in isolation for the purposes of safety assessment.     

Incremental elastic compliance and electric resistance of a cylinder with partial loss in the cross-sectional area

Motivated by material science applications, the paper focuses on quantitative characterization and comparison of two microstructural elements typical for lamellar materials - crack and contacting area - in the context of their effect on macroscopic elastic and conductive (thermal or electrical) properties of a body of finite size. The problem is solved in axisymmetric formulation - axial load or axial heat flux is applied to a circular cylinder containing a centered crack, either internal or external. The latter case corresponds to the welding of two halves of the cylinder at the center. The changes in the elastic and conductive properties of the cylinder due to these types of cracks are obtained in explicit analytical form. It is shown that the contributions of internal and external cracks into elastic and conductive properties are similar if the relative loss in the cross-sectional between two parts of the cylinder is up to 70% for elasticity problem and up to 85% for conductivity problem. We also show that the changes in elastic compliance and conductive properties generated by both microstructural elements are interrelated by cross-property connection identical to one obtained for an unbounded material.     

Two moving cracks in a layered composite

This paper deals with the problem of the uniform motion of two cracks located at the central plane of an elastic layer embedded in bonded contact with two elastic half-space regions. The material properties in the layer are assumed to be different from the material properties of the half-space regions. The uniform motion of the cracks is induced by antiplane and in-plane extension modes. The analysis of the problem employs successive application of Galilean and Fourier transformations. The results of primary interest to fracture mechanics, namely the dynamic stress-intensity factors, are illustrated in graphical form.     

Edge cracks in a transversely isotropic hollow cylinder

The analytical solution for the linear elastic, axisymmetric problem of inner and outer edge cracks in a transversely isotropic infinitely long hollow cylinder is considered. The z = 0 plane on which the crack lies is a plane of symmetry. The loading is uniform crack surface pressure. The mixed boundary value problem is reduced to a singular integral equation where the unknown is the derivative of the crack surface displacement. An asymptotic analysis is done to derive the generalized Cauchy kernel associated with edge cracks. It is shown that the stress intensity factor is a function of three material parameters. The singular integral equation is solved numerically. Stress intensity factors are presented for various values of material and geometric parameters.     

Stress intensity factors of two diametrically opposed edge cracks in a thick-walled functionally graded material cylinder

A method is developed to evaluate stress intensity factors for two diametrically-opposed edge cracks emanating from the inner surface of a thick-walled functionally graded material (FGM) cylinder. The crack and the cylinder inner surfaces are subjected to an internal pressure. The thermal eigenstrain induced in the cylinder material due to nonuniform coefficient of thermal expansion after cooling from the sintering temperature is taken into account. First, the FGM cylinder is homogenized by simulating its nonhomogeneous material properties by an equivalent eigenstrain, whereby the problem is reduced to the solution of a cracked homogenized cylinder with an induced thermal and an equivalent eigenstrains and under an internal pressure. Then, representing the cracks by a continuous distribution of edge dislocations and using their complex potential functions, generalized formulations are developed to calculate stress intensity factors for the cracks in the homogenized cylinder. The stress intensity factors calculated for the cracks in homogenized cylinder represents the stress intensity factors for the same cracks in the FGM cylinder. The application of the formulations are demonstrated for a thick-walled TiC/Al₂O₃ FGM cylinder and some numerical results of stress intensity factors are presented for different profiles of material distribution in the FGM cylinder.     

Stress intensity factors in a reinforced thick-walled cylinder

In this paper an elastic thick-walled cylinder containing a radial crack is considered. It is assumed that the cylinder is reinforced by an elastic membrane on its inner surface. The model is intended to simulate pressure vessels with cladding. The formulation of the problem is reduced to a singular integral equation. Various special cases including that of a crack terminating at the cylinder-reinforcement interface are investigated and numerical examples are given. Among the interesting results found one may mention the following: In the case of the crack touching the interface the crack surface displacement derivative is finite and consequently the stress state around the corresponding crack tip is bounded, and generally, for realistic values of the stiffness parameter, the effect of the reinforcement is not very significant.     

Quasistatic indentation of a rubber-covered roll by a rigid roll

The nonlinear elastic problem involving the indentation of a slightly compressible rubber-like layer bonded to a rigid cylinder and indented by another rigid cylinder is analysed by the finite element method. Both the geometric and material nonlinearities are accounted for. The finite element formulation of the problem is based upon a variational principle recently proposed by Cescotto and Fonder, and is valid for both slightly compressible and incompressible materials. The results computed and presented graphically include the shape of the indented surface, the pressure distribution over the contact surface, and the stress distribution at the bond surface. For the same contact width, the results for the compressible material are found to differ significantly from those for the case when the rubber-like layer is assumed to be incompressible.     

Torsion of a non-homogeneous infinite elastic cylinder slackened by a circular cut

We consider the torsional deformation of a non-homogeneous infinite elastic cylinder slackened by an external circular cut. The shear modulus of the material of the cylinder is assumed to vary with the radial coordinate by a power law. It is assumed that the lateral surface of the cylinder as well as the surface of the cut are free of stress. The main object of this study is to establish the effect of the non-homogeneity on the stress intensity factor at the tip of the cut. The problem leads to a pair of dual series relations, the solution of which is governed by a Fredholm integral equation of the second kind with a symmetric kernel. This equation is solved numerically by reducing it to an algebraic system. It is concluded that for any degree of non-homogeneity and for D, the relative depth of the cut, greater than 0.6, the cylinder may be replaced by a half-space. However, as the non-homogeneity increases, D decreases.     

The Influence of the Imperfectness of Contact Conditions on the Critical Velocity of the Moving Load Acting in the Interior of the Cylinder Surrounded with Elastic Medium

The dynamics of the moving-with-constant-velocity internal pressure acting on the inner surface of the hollow circular cylinder surrounded by an infinite elastic medium is studied within the scope of the piecewise homogeneous body model by employing the exact field equations of the linear theory of elastodynamics. It is assumed that the internal pressure is point-located with respect to the cylinder axis and is axisymmetric in the circumferential direction. Moreover, it is assumed that shear-spring type imperfect contact conditions on the interface between the cylinder and surrounding elastic medium are satisfied. The focus is on the influence of the mentioned imperfectness on the critical velocity of the moving load and this is the main contribution and difference of the present paper the related other ones. The other difference of the present work from the related other ones is the study of the response of the interface stresses to the load moving velocity, distribution of these stresses with respect to the axial coordinates and to the time. At the same time, the present work contains detail analyses of the influence of problem parameters such as the ratio of modulus of elasticity, the ratio of the cylinder thickness to the cylinder radius, and the shear-spring type parameter which characterizes the degree of the contact imperfection on the values of the critical velocity and stress distribution. Corresponding numerical results are presented and discussed. In particular, it is established that the values of the critical velocity of the moving pressure decrease with the external radius of the cylinder under constant thickness of that.     

Bifurcation of a solid circular elastic cylinder under finite extension and torsion

Summary The problem of bifurcation of a solid circular cylinder subjected to finite extension and torsion is investigated using the theory of small deformations superposed on large elastic deformations. The material of the cylinder is assumed to be isotropic, elastic, homogeneous and incompressible. A numerical scheme is adopted to solve the system of partial differential equations and the associated boundary conditions governing the problem for a class of strain-energy functions. The numerical results obtained determined the critical twist corresponding to a given extension of the cylinder.     

The stress intensity factors of a star-shaped array of cracks in an infinite elastic solid

The problem of determining the stress intensity factors and crack formation energy of a radial system of line cracks in an infinite elastic solid is reduced to the solution of a singular integral equation. The equation is solved numerically for the special case in which the cracks are opened by a constant pressure.     

Deformation of loaded micropolar elastic cylinders

In this paper we solve the general problem of deformation of a composite elastic cylinder. We consider the case when the cross-section of the cylinder is occupied by different inhomogeneous and anisotropic micropolar elastic materials. The elastic coefficients are assumed to be independent of the axial coordinate and are discontinuous, in general, over the surface of separation of the materials. The cylinder is subjected to body loads and to tractions on the lateral surface, and to appropriate stress and couple-stress resultants over its ends.     

Interaction of torsional waves with an annular crack in an infinitely long cylinder

The Hankel transform is used to obtain a complete solution for the dynamic stresses and displacements around a flat annular surface of a crack embedded in an infinite elastic cylinder, which is excited by normal torsional waves. The curved surface of the cylinder is assumed to be stress free. Solution of the problem is reduced to three simultaneous Fredholm integral equations. By finding the numerical solution of the simultaneous Fredholm integral equations the variations of the dynamic stress-intensity factors are obtained which are displayed graphically.     

Saint-Venant torsion of a circular bar with radial cracks incorporating arbitrarily varied surface elasticity

We analytically investigate the contribution of arbitrarily varied surface elasticity to the Saint-Venant torsion problem of a circular cylinder containing a radial crack. The varied surface elasticity is incorporated by using a modified version of the continuum-based surface/interface model of Gurtin and Murdoch. In our discussion, the surface shear modulus is assumed to be arbitrarily varied along the crack surfaces. Both internal and edge cracks are studied. By using Green's function method, the boundary value problem is reduced to the Cauchy singular integro-differential equation of first order, which can be numerically solved by using the Gauss–Chebyshev integration formula, the Chebyshev polynomials, and the collocation method. The torsion problem of a cylinder containing two symmetric collinear radial cracks of equal length with symmetrically varied surface elasticity is also solved by using a similar method. Our numerical results indicate that the variation of the surface elasticity exerts a significant influence on the strengths of the logarithmic stress singularity at the crack tips, the torsional rigidity, and the jump in warping function.