Condition Of Finite Stress Research Articles

Subcritical and critical states of a crack with failure zones

• A new semi-analytical technique is developed to solve the problems of crack mechanics within the cohesive zone model . • Proposed technique implicitly takes into account the finite stress condition. • New dependencies for critical crack lengths are obtained for the non-uniform traction-separation laws. The problem on the stress–strain state near a mode I crack in an infinite plate is solved in the frame of a cohesive zone model. The complex variable method of Muskhelishvili is used to obtain the crack opening displacements caused by the cohesive traction , which models the failure zone at the crack tip, as well as by the external load. The finite stress condition and logarithmic singularity of the derivative of the separation with respect to the coordinate at the tip of a physical crack are taken into account. The cohesive traction distribution is sought in a piecewise linear form, nodal values of which are being numerically chosen to satisfy the traction-separation law. According to this law, the cohesive traction is coupled with the corresponding separation and fracture toughness . The tips of the physical crack and cohesive zone (geometric variables) along with the discrete cohesive traction are used as the problem parameters determining the stress-strain state. If the crack length is included in the set, then the critical crack size can be found for the given loading intensity. The obtained determining system of equations is solved numerically. To find the initial point for a standard numerical algorithm, the asymptotic determining system is derived. In this system, the geometric variables can be easily eliminated, which make it possible to linearize the system. In the numerical examples, the one-parameter traction-separation laws are used. Influence of the shape parameters of the law on the critical crack size and the corresponding cohesive length is studied. The possibility of using asymptotic solutions for determining the critical parameters is analysed. It is established that the critical crack length slightly depends on the shape parameter, while the cohesive length shows a strong dependence on the shape of cohesive laws .

Edge crack due to circular rigid punch in incomplete contact

A circular rigid punch with friction is assumed to be in contact with a half-plane and the punch has one end sliding on the half-plane and another end with a sharp corner. The contact length is determined by satisfying the finite stress condition at the sliding end of the punch. An oblique edge crack is assumed to be initiated near the end with the sharp corner where the infinite stresses exist. Coulomb's frictional force is assumed to act on the contact region. The cracked half-plane is mapped into a unit circle by using a rational mapping function and the problem is transformed into a standard Riemann–Hilbert problem, which is solved by introducing a Plemelj function. The contact length, the stress intensity factors of the crack and the resultant moment about the origin of the coordinates are calculated with different frictional coefficients and oblique angles of the crack. The stress distributions on the contact region are also shown.

Circular Rigid Punch With One Smooth and Another Sharp Ends on a Half-Plane With Edge Crack

A circular rigid punch with friction is assumed to contact with a half-plane with one end sliding on the half-plane and another end with a sharp corner. The contact length is determined by satisfying the finite stress condition at the sliding end of the punch. The crack is initiated near the end with a sharp corner where infinite stresses exist. Coulomb’s frictional force is supposed to act on the contact region. The cracked half-plane is mapped into a unit circle by using a rational mapping function, and the problem is transformed into a standard Riemann-Hilbert problem, which is solved by introducing a Plemelj function. The contact length, the stress intensity factors of the crack, and the resultant moment about the origin of the coordinates on the contact region are calculated for different frictional coefficients, Poisson’s ratios of the half-plane, crack lengths, and distances from the crack to the punch, respectively. The stress distributions on the contact region are also shown.

Nonlinear crack mechanics

The characteristic features of the deformation and failure of actual materials in the vicinity of a crack tip are due to their physical nonlinearity in the stress-concentration zone, which is a result of plasticity, microfailure, or a nonlinear dependence of the interatomic forces on the distance. Therefore, adequate models of the failure mechanics must be nonlinear, in principle, although linear failure mechanics is applicable if the zone of nonlinear deformation is small in comparison with the crack length. Models of crack mechanics are based on analytical solutions of the problem of the stress-strain state in the vicinity of the crack. On account of the complexity of the problem, nonlinear models are bason on approximate schematic solutions. In the Leonov-Panasyuk-Dugdale nonlinear model, one of the best known, the actual two-dimensional plastic zone (the nonlinearity zone) is replaced by a narrow one-dimensional zone, which is then modeled by extending the crack with a specified normal load equal to the yield point. The condition of finite stress is applied here, and hence the length of the plastic zone is determined. As a result of this approximation, the displacement in the plastic zone at the abscissa is nonzero.

Inclined pileup of screw dislocations at the crack tip with a dislocation-free zone

A single pileup of screw dislocations extending from the crack tip along an inclined direction has been observed in experiments. It is often associated with dislocation emission mechanisms at the crack tip. This linear pileup is a microplastic slipline emanating from the crack tip. A region near the crack tip is often free from dislocations because of a finite resistance value for the crack tip to emit dislocations. The mathematical problem is solved in this paper by applying the extended Wiener-Hopf method. The condition of finite stress at the end of the plastic zone, the crack opening displacement, and the stress distribution along the slipline are obtained in analytical expressions. Numerical values are calculated and the results can be used to discuss brittle versus ductile fracture for metals as treated in previous studies. A method to approximately calculate the corresponding results for edge dislocations is suggested.

Inclined pileup of screw dislocations at the crack tip without a dislocation-free zone

A linear array of screw dislocations in front of a mode 3 crack lying on a plane inclined to the crack is studied. The problem is formulated in terms of an integral equation describing the equilibrium condition between the dislocations and the crack under an externally applied shear stress. The distribution function for dislocations is derived from a solution of the integral equation by applying the Wiener–Hopf technique. The condition of finite stress at the end of the pileup, or the Bilby, Cottrell, and Swindeman (BCS) condition and the crack opening displacement (COD) are obtained in simple analytical forms as a function of the angle of inclination. Numerical results show that, for a given applied stress, the distribution function, the length of the plastic zone, and the COD decrease only slightly as the angle of inclination increases from 0 to π/2. The quantity of σf COD , where σf is the friction stress, is shown to be equal to the radial component of the complex J integral and, hence, is the total force on the dislocations along the direction of the pileup. A method to estimate results under mode 1 condition involving an inclined pileup of edge dislocations is considered.

Distribution function of dislocations and condition of finite stress for the dislocation-free zone model of fracture

Distribution function of dislocations and condition of finite stress for the dislocation-free zone model of fracture

Dynamic effects in the propagation of a ductile penny-shaped crack

The propagation of an interior penny-shaped crack in a ductile material subject to uniform tension at infinity is investigated using the methods of Hankel and Laplace transforms. The crack is assumed to be the Dugdale crack which has a uniform tensile yield stress around its crack tip. Exact expressions for the finite stress condition at the crack tip, the crack shape, the crack opening displacement and the energy release rate are obtained and written as the product of explicit dimensional quantities and a non-dimensional dynamic correction function. All the expressions obtained reduce to the associated static results when the crack speed tends to zero. Using an electronic computer, the dynamic correction functions were calculated for various values of the non-dimensional parameters involved.The size of the plastically deformed zone around the crack tip is determined from the finite stress condition. A general, convenient method is described to determine the plastic zone size using the non-dimensional dynamic correction curves. It is shown that the plastic zone width shrinks significantly with increasing crack speed. Due to the inertial effect of the material particles, the crack opening displacement and the energy release rate are also found to decrease with increasing crack speed.

Propagation of a Vertical Hydraulic Fracture

Abstract This paper treats the propagation of hydraulic fractures of limited vertical extent and elliptic cross-section with the effect of fluid loss included. Numerical and asymptotic approximate solutions in dimensionless form give the fracture length and width at any value of time or any set of physical parameters. The insight provided by The dimensionless parameters. The insight provided by The dimensionless results and approximate solutions should be useful in the design of fracture treatments. Introduction The theory and practice of hydraulic fracturing has been reviewed by Howard and Fast. Therefore, we confine our discussion of previous investigations to those pertinent to the present study of the propagation of vertical fractures. propagation of vertical fractures. An important theoretical result is Carter's formula for the area of a fracture of constant width formed by injection at constant rate with fluid lost to the formation. For a vertical fracture of constant height, Carter's formula gives fracture length as a function of time. In general, Carter's assumption of constant width is not realistic. However, at large values of time the effect of this assumption becomes insignificant since the effect of fluid loss dominates. The width of a vertical fracture was first investigated by Khristianovic and Zheltov under the assumption that the width does not vary in the vertical direction. Thus, a state of plane strain prevails in horizontal planes and the width can be prevails in horizontal planes and the width can be determined as the solution of a plane elasticity problem. An approximate solution is found in Ref. 3 problem. An approximate solution is found in Ref. 3 upon neglect of fluid loss, fracture volume change, and pressure variation along the fracture. The fracture length is determined by the condition of finite stress at the fracture tip. Baron et al. and Geertsma and de Klerk have included the effect of fluid loss in the approach of Ref. 3. Geertsma and de Klerk give simple approximate formulas for fracture length and width. A different approach to the determination of fracture width was taken by Perkins and Kern. They considered a vertically limited fracture under the assumption of plane strain in vertical planes perpendicular to the fracture plane. The perpendicular to the fracture plane. The cross-section of the fracture is found to be elliptical, and the maximum width decreases along the fracture according to a simple formula that contains the fracture length. In the derivation of this formula, fluid loss and fracture volume change are neglected in the continuity equation and the fracture length is not determined. In a subsequent application, a "reasonable" fracture length was assumed. Carter's formula for length and the width formula of Perkins and Kern are both cited by Howard and Fast, and combined use of the two formulas is believed to be common practice. The present theoretical investigation is concerned with vertically limited fractures of the type studied by Perkins and Kern. However, we include the effects of fluid loss and fracture volume change in the continuity equation. Consequently, fracture length is determined as part of the solution. General results for the variation of fracture width and length with time are obtained in dimensionless form by a numerical method. In addition, asymptotic solutions are derived for large and small values of time. The small-time solution is also the exact solution for the case of no fluid loss to the formation. For large values of time our asymptotic formula for fracture length is identical with Carter's formula at large time. Our large-time formula for fracture width differs from the formula of Perkins and Kern by a numerical factor that varies along the fracture. In comparison with our formula, this formulas overestimates the width by 12 percent at the well and 24 percent at the midlength of the fracture. At early times Perkins and Kern's formulas for width in terms of length is again a fair approximation to our result. However, our formula for length differs from Carter's formula, which is not applicable since the neglected width variation is important at early times. The results for the width of a vertically limited fracture as obtained here and in Ref. 6 differ from the results for vertically constant fractures.

Stress distribution, crack shape and energy for a penny-shaped crack in a plate of finite thickness

The shape of a penny-shaped crack located at the center of an elastic plate of finite thickness is related to the arbitrary axisymmetrical internal pressures applied to the crack surfaces in the form of a Fredholm integral equation, without using the methods of dual-integral equations. General expressions for the stresses in the plane containing the crack are written as the sums of the associated infinite solid stresses and the integrals accounting for the effect of plate thickness. The crack shape due to uniform crack pressures and the fracture criterion for brittle plates subjected to uniform stresses are obtained for various plate thicknesses. Using the finite stress condition, general methods for determining the radii of the plastic zone are described for ideally elastic-plastic materials. The normal stress is shown to be continuous and finite at the outer circle of the plastic zone. The crack shape and the rate of the plastic energy dissipation are calculated for various values of plate thickness and crack pressure.