Nonaxisymmetric compressive failure of a circular crack parallel to a surface of halfspace

Considered in this work is the fracture stability of a circular crack parallel to the surface of a halfspace subjected to around applied compression. A state of subcritical initial strain is assumed. The failure criterion is based on the loss of local stability and quantified in terms of a critical eigen value. Analysis involves reducing the problem to a system of Fredholm equations of the second kind where the solutions are identified with harmonic potential functions. Critical loads are determined for nonaxisymmetric forms of loss in stability in the region local to the crack; their values would depend on the material properties and geometric parameters. Numerical results are displayed graphically for a crack in an elastic solid and a composite made of aluminum/boron/silicate glass with epoxy-maleimic resin. They depend on the number of angularly dependent harmonies that introduce the nonaxisymmetry in the problem.     

Buckling and cracking characteristics of two-layered plate in tension

This study is concerned with the loss of local stability of two-layered metal plates (steel–aluminium alloy) with a crack in tension. The critical stresses corresponding to the initiation of buckling are determined. The post-critical deformation and form of stability loss are investigated. The influence of buckling on the fracture characteristics is also estimated.     

Symmetric failure of the halfspace with penny-shaped cracks in compression

The three-dimensional axisymmetric problem is investigated for a halfspace with penny-shaped crack parallel to the free surface. A uniform compression is applied parallel to the crack plane. The Griffith-Irwin theory is not applicable to this configuration of load and crack geometry since all the stress intensity factors are zero. Instead, a stability criterion will be invoked within the framework of the three-dimensional linearized stability theory. Reference can be made to previous works [5,6] involving compressible and incompressible elastic bodies. Use was made of an arbitrary form of the elastic potential for high subcritical deformations and for two variants of the theory of small subcritical deformations that involved equal and unequal roots of the characteristics equation [6]. It was found that consideration of the mutual influence of the subsurface crack and the free surface results in a considerable reduction of the theoretical strength limit. This was previously derived for infinite material with a crack. Examples of potentials with equal roots are discussed: the Bartenev-Khazanovich (incompressible bodies) and harmonic potential (compressible bodies).     

Axial compression of circular cylindrical bar with coaxial subsurface cylindrical crack of finite length

A fracture stability of a circular cylindrical bar with a coaxial surface cylindrical crack subjected to an axial compression is considered. A state of subcritical initial strain is assumed. A non-classical fracture criterion is based on a local stability loss near the defect. The theory of integral Fourier transforms and series expansions are used to reduce these problems to a system of paired integral equations and then to a system of linear algebraic equations with respect to the contraction parameter.     

Integral formulation for a circular cylindrical cavity in infinite solid and a finite length coaxial cylindrical crack compressed axially

A method for solving problems of fracture of an infinite solid with a circular cylindrical cavity and a coaxial cylindrical crack near the surface under an uniform axial compression is proposed using a non-classical criterial approach associated with a mechanism of a local stability loss near the defect. The theory of integral Fourier transforms and series expansions are used to reduce these problems to a system of paired integral equations and then to a system of linear algebraic equations with respect to the contraction parameter.     

A rigid triangular inclusion partially bonded in an elastic matrix

The two-dimensional problem of a rigid rounded-off angle triangular inclusion partially bonded in an infinite elastic plate is studied. The unbonded part of the inclusion boundary forms an interfacial crack. Based on the complex variable method for curvilinear boundaries, the problem is reduced to a non-homogeneous Hilbert problem and the stress and displacement fields in the plate are obtained in closed form. Special attention is paid in the investigation of the stress field in the vicinity of the crack tip. It is found that the stresses present an oscillatory singularity and the general equations for the local stresses are derived. The singular stress field is coupled with the maximum circumferential stress and the minimum strain energy density criteria to study the fracture characteristics of the composite plate. Results are given for the complex stress intensity factors, the local stresses, the crack extension angles and the critical applied loads for unstable crack growth from its more vulnerable tip or two types of interfacial cracks along the inclusion boundary.     

Complex variable approach in studying modified polarization saturation model in two-dimensional semipermeable piezoelectric media

A modified polarization saturation model is proposed and addressed mathematically using a complex variable approach in two-dimensional(2 D) semipermeable piezoelectric media. In this model, an existing polarization saturation(PS) model in 2D piezoelectric media is modified by considering a linearly varying saturated normal electric displacement load in place of a constant normal electric displacement load, applied on a saturated electric zone. A centre cracked infinite 2D piezoelectric domain subject to an arbitrary poling direction and in-plane electromechanical loadings is considered for the analytical and numerical studies. Here, the problem is mathematically modeled as a non-homogeneous Riemann-Hilbert problem in terms of unknown complex potential functions representing electric displacement and stress components. Having solved the Hilbert problem, the solutions to the saturated zone length, the crack opening displacement(COD), the crack opening potential(COP), and the local stress intensity factors(SIFs) are obtained in explicit forms. A numerical study is also presented for the proposed modified model, showing the effects of the saturation condition on the applied electrical loading, the saturation zone length, and the COP. The results of fracture parameters obtained from the proposed model are compared with the existing PS model subject to electrical loading, crack face conditions, and polarization angles.     

Retardation of a crack with connections between the faces using an induced thermoelastic stress field

This paper considers local temperature variations near the tip of a crack in the presence of regions in which the crack faces interact. It is assumed that these regions are adjacent to the crack tip and are comparable in size to the crack size. The problem of local temperature variations consists of delay or retardation of crack growth. For a crack with connections between the crack faces subjected to external tensile loads, an induced thermoelastic stress field, and the stresses at the connections preventing crack opening, the boundary-value problem of the equilibrium of the crack reduces to a system of nonlinear singular integrodifferential equations with a Cauchy kernel. The normal and tangential stresses at the connections are found by solving this system of equations. The stress intensity factors are calculated. The energy characteristics of cracks with tip regions are considered. The limiting equilibrium condition for cracks with tip regions is formulated using the criterion of limiting stretching of the connections.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 133–143, January–February, 2005     

A theoretical investigation on the vibrational characteristics and torsional dynamic response of circumferentially cracked turbo-generator shafts

Turbo-generator shafts are often subjected to complex dynamic torsional loadings, resulting in generation and propagation of circumferential cracks. Mode III fatigue crack growth generally results in a fracture surface consisting of peaks and valleys, resembling a factory roof. The fracture surface roughness depends on the material microstructure, the material yield strength, and the applied cyclic torque amplitude. This crack pattern can severely affect the vibration characteristics of the shafts. The accurate evaluation of the torsional dynamic response of the turbo-generator shafts entails considering the local sources of energy loss in the crack vicinity. The two most common sources of the energy loss are the local energy loss due to the plasticity at the crack tip and frictional energy loss due to interaction of mutual crack surfaces. A theoretical procedure for evaluating the values of the system loss factors corresponding to these sources of energy loss is presented. Furthermore, the local flexibility is obtained by evaluating the resistance of the cracked section of the shaft to the rotational displacement. The shaft material is assumed to be elastic perfectly plastic. The effects of the applied Mode III stress intensity factor and the crack surface pattern parameters on the energy loss due to the friction and the energy loss due to the plasticity at the crack tip are investigated. The results show that depending on the amplitude of the applied Mode III stress intensity factor, one of these energy losses may dominate the total energy loss in the circumferentially cracked shaft. The results further indicate that the torsional dynamic response of the turbo-generator shaft is significantly affected by considering these two sources of the local energy loss.     

Failure instability of a debonded interface in the presence of a main crack

The problem of fracture initiating from an edge crack in a nonhomogeneous beam made of two dissimilar linear elastic materials that are partially bonded along a common interface is studied by the strain energy density theory. The beam is subjected to three-point bending and the unbonded part of the interface is symmetrically located with regard to the applied loading. The applied load acts on the stiffer material, while the edge crack lies in the softer material. Fracture initiation from the tip of the edge crack and global instability of the composite beam are studied by considering both the local and global stationary values of the strain energy density function, dW/dV. A length parameter l defined by the relative distance between the maximum of the local and global minima of dW/dV is determined for evaluating the stability of failure initiation by fracture. Predictions on critical loads for fracture initiation from the tip of the edge crack, crack trajectories and fracture instability are made. In the analysis the load, the length of the edge crack and the length and position of the interfacial crack remained unchanged. The influence of the ratio of the moduli of elasticity of the two materials, the position of the edge crack and the width of the stiffer material on the local and global instability of the beam was examined. A general trend is that the critical load for crack initiation and fracture instability is enhanced as the width and the modulus of elasticity of the stiffer material increase. Thus, the stiffer material acts as a barrier in load transfer.     

Driving forces and kinking of an oblique crack in bonded nonhomogeneous materials

Summary  Plane elasticity solutions are presented for the problem of an oblique crack in two bonded media. The material model under consideration consists of a homogeneous half-plane with an arbitrarily oriented crack and a nonhomogeneous half-plane. The Fourier integral transform method is employed in conjunction with the coordinate transformations of field variables in the basic elasticity equations. Formulation of the crack problem results in having to solve a system of singular integral equations for arbitrary crack surface tractions. A crack perpendicular to or along the bonded interface between the homogeneous and nonhomogeneous constituents arises as a limiting case. In the numerical results, the values of mixed-mode stress intensity factors are provided for various combinations of relevant geometric and material parameters of the bonded media. Subsequently, the infinitesimal kinks from the tips of a main crack are presumed, with the corresponding local driving forces being evaluated in terms of the stress intensities of the main crack. The criterion of maximum energy release rate is applied with the aim of making some conjectures concerning the likelihood of kinking and the probable kink direction based on the approximation of local homogeneity and brittleness of the crack-tip behavior. Received 25 September 2001; accepted for publication 13 February 2002     

Localisation issues in local and nonlocal continuum approaches to fracture

Continuum approaches to fracture regard crack initiation and growth as the ultimate consequences of a gradual, local loss of material integrity. The material models which are traditionally used to describe the degradation process, however, may predict premature crack initiation and instantaneous, perfectly brittle crack growth. This nonphysical response is caused by localisation instabilities due to loss of ellipticity of the governing equations and—more importantly—singularity of the damage rate at the crack tip. It is argued that this singularity results in instantaneous failure in a vanishing volume, even if ellipticity is not first lost. Adding strong nonlocality to the modelling is shown to preclude localisation instabilities and remove damage rate singularities. As a result, premature crack initiation is avoided and crack growth rates remain finite. Weak nonlocality, as provided by explicit gradient models, does not suffice for this purpose. In implementing the enhanced modelling, the crack must be excluded from the equilibrium problem and the nonlocal interactions in order to avoid unrealistic damage growth.     

Theoretical analysis of crack front instability in mode I+III

This paper focusses on the theoretical prediction of the widely observed crack front instability in mode I+III, that causes both the crack surface and crack front to deviate from planar and straight shapes, respectively. This problem is addressed within the classical framework of fracture mechanics, where the crack front evolution is governed by conditions of constant energy-release-rate (Griffith criterion) and vanishing stress intensity factor of mode II (principle of local symmetry) along the front. The formulation of the linear stability problem for the evolution of small perturbations of the crack front exploits previous results of Movchan et al. (1998) (suitably extended) and Gao and Rice (1986), which are used to derive expressions for the variations of the stress intensity factors along the front resulting from both in-plane and out-of-plane perturbations. We find exact eigenmode solutions to this problem, which correspond to perturbations of the crack front that are shaped as elliptic helices with their axis coinciding with the unperturbed straight front and an amplitude exponentially growing or decaying along the propagation direction. Exponential growth corresponding to unstable propagation occurs when the ratio of the unperturbed mode III to mode I stress intensity factors exceeds some “threshold” depending on Poisson's ratio. Moreover, the growth rate of helical perturbations is inversely proportional to their wavelength along the front. This growth rate therefore diverges when this wavelength goes to zero, which emphasizes the need for some “regularization” of crack propagation laws at very short scales. This divergence also reveals an interesting similarity between crack front instability in mode I+III and well-known growth front instabilities of interfaces governed by a Laplacian or diffusion field.     

Finite plane and anti-plane elastostatic fields with discontinuous deformation gradients near the tip of a crack

In this paper the fully nonlinear theory of finite deformations of an elastic solid is used to study the elastostatic field near the tip of a crack. The special elastic materials considered are such that the differential equations governing the equilibrium fields may lose ellipticity in the presence of sufficiently severe strains.The first problem considered involves finite anti-plane shear (Mode III) deformations of a cracked incompressible solid. The analysis is based on a direct asymptotic method, in contrast to earlier approaches which have depended on hodograph procedures.The second problem treated is that of plane strain of a compressible solid containing a crack under tensile (Mode I) loading conditions. The materials is characterized by the so-called Blatz-Ko elastic potential. Again, the analysis involves only direct local considerations.for both the Mode III and Mode I problems, the loss of equilibrium ellipticity results in the appearance of curves (elastostatic shocks) issuing from the crack-tip across which displacement gradients and stresses are discontinuous.The results communicated in this paper were obtained in the course of an investigation supported by Contract N00014-75-C-0196 with the Office of Naval Research.     

Initiation and growth characterization of corner cracks near circular hole

The finite element analysis of crack problems often incorporates the asymptotic character of the local solution into the formulation. Embedment of stress or strain singularities can impose serious restrictions on the outcome and inconsistencies in predicting crack and/or growth. These restrictions are discussed in connection with the problem of two diametrically opposite corner cracks near a circular hole subjected to remote uniform tension. Enforced in the numerical treatment is the 1/r character of the strain energy density function local to the corner crack border where r is the radial distance measured from the crack front. The tendency for the corner crack to become a through crack is predicted by assuming that each point of the crack border extends by an amount proportional to the strain energy density factor. The path would correspond to the loci of minimum strain energy density function. Numerical results are displayed graphically and discussed in connection with crack initiation and non-self-similar crack growth.     

The weight function for cracks in piezoelectrics

The weight function in fracture mechanics is the stress intensity factor at the tip of a crack in an elastic material due to a point load at an arbitrary location in the body containing the crack. For a piezoelectric material, this definition is extended to include the effect of point charges and the presence of an electric displacement intensity factor at the tip of the crack. Thus, the weight function permits the calculation of the crack tip intensity factors for an arbitrary distribution of applied loads and imposed electric charges. In this paper, the weight function for calculating the stress and electric displacement intensity factors for cracks in piezoelectric materials is formulated from Maxwell relationships among the energy release rate, the physical displacements and the electric potential as dependent variables and the applied loads and electric charges as independent variables. These Maxwell relationships arise as a result of an electric enthalpy for the body that can be formulated in terms of the applied loads and imposed electric charges. An electric enthalpy for a body containing an electrically impermeable crack can then be stated that accounts for the presence of loads and charges for a problem that has been solved previously plus the loads and charges associated with an unsolved problem for which the stress and electric displacement intensity factors are to be found. Differentiation of the electric enthalpy twice with respect to the applied loads (or imposed charges) and with respect to the crack length gives rise to Maxwell relationships for the derivative of the crack tip energy release rate with respect to the applied loads (or imposed charges) of the unsolved problem equal to the derivative of the physical displacements (or the electric potential) of the solved problem with respect to the crack length. The Irwin relationship for the crack tip energy release rate in terms of the crack tip intensity factors then allows the intensity factors for the unsolved problem to be formulated, thereby giving the desired weight function. The results are used to derive the weight function for an electrically impermeable Griffith crack in an infinite piezoelectric body, thereby giving the stress intensity factors and the electric displacement intensity factor due to a point load and a point charge anywhere in an infinite piezoelectric body. The use of the weight function to compute the electric displacement factor for an electrically permeable crack is then presented. Explicit results based on a previous analysis are given for a Griffith crack in an infinite body of PZT-5H poled orthogonally to the crack surfaces.     

ON PROBLEMS OF FRACTURE OF MATERIALS IN COMPRESSION ALONG TWO INTERNAL PARALLEL CRACKS

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The Stress State of a Transversally Isotropic Piezoceramic Body with a Parabolic Crack in a Uniform Heat Flow

The explicit solution is constructed for a static thermoelastic problem for an infinite transversally isotropic piezoceramic body containing a heat-insulated parabolic crack in the isotropy plane. The crack surface is assumed free of forces. The body is under a uniform heat flow, which is perpendicular to the crack surface and is far from the crack itself. The problem is solved for two cases of electric conditions on the crack surface. In the first case, an electric potential is absent on the crack surface and, in the second case, the normal component of the electric-displacement vector is equal to zero. The intensity factors, which depend on the heat flow, crack geometry, and the thermoelectroelastic properties of the piezoceramic body, are determined for the force field and electric displacement near the crack tip     

Stability of thin plates with cracks under biaxial tension

Conclusions A method has been developed for analyzing the stability of thin plates with cracks under biaxial tension which is based on measuring the local buckling of the plate in the crack region with pneumatic instruments and describing the results in the system of plate coordinates.It has been demonstrated that the critical stress corresponding to local loss of stability depends strongly on the magnitude of the force acting along the crack. Values of the bending factor have been determined experimentally for plates of the grade AMg6M aluminum alloy. Unlike in the case of no load along the crack, in the case of biaxial loading the value of this bending factor does depend on the geometrical parameters of the plate.An empirical expression has been derived for the bending factor which, for this particular material, yields values close to experimental ones.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 18, No. 10, pp. 80–85, October, 1982.     

Dependence of the Critical Load on the Geometric Characteristics of a Hinged Plate with a Crack

The plane problem of three-dimensional stability of a hinged plate with a central crack under uniaxial loading along the crack is considered. The net approach is used to solve the problem. The variational difference and gradient methods are used, respectively, to construct a difference scheme and to solve difference problems. The dependence of the critical load on two parameters — the crack length and the thickness ratio — is derived. Formulas for calculation of the critical load are given