Stress-intensity factor and crack opening for a linearly viscoelastic strip with a slowly propagating central crack

The stress-intensity factor and the size of the crack opening have been calculated for a linearly viscoelastic strip with a slowly propagating central crack. The edges of the infinitely long strip are displaced normal to the crack and both clamped and shear-free strip edges have been investigated. The results are based on the solution to the problem of a suddenly loaded strip with a stationary crack. The resulting integral equation has been solved numerically for arbitrary crack length and analytical solutions in form of asymptotic series are given for crack length up to about half the strip width. The response to a propagating crack is found by superposition.This work represents part of a Ph.D. Thesis submitted to the California Institute of Technology. The author gratefully acknowledges the support of this work by the National Aeronautics and Space Administration under Research Grant NGL-05-002-005, GALCIT 120.     

The central crack problem for a functionally graded piezoelectric strip

In this paper the dynamic anti-plane problem for a functionally graded piezoelectric strip containing a central crack vertical to the boundary is considered. The crack is assumed to be electrically impermeable or permeable. Integral transforms and dislocation density functions are employed to reduce the problem to Cauchy singular integral equations. Numerical results show the effects of loading combination parameter, material gradient parameter and crack configuration on the dynamic response. With the permeable assumption, the electric impact has no contribution to the crack tip field singularity. With the impermeable assumption, the direction of applied electric impact loading plays a great role in the behavior of dynamic stress intensity factor, and the existence of electric load always enhances the crack propagation. However, the crack is easier to propagate under the negative electric load than that under the positive electric load.     

Ductile fracture before localized necking in a strip under tension

Unexpected cracking with a 22° opening angle grew out of a rough-sheared edge before appreciable necking in an 0.79 by 37 mm mild steel strip, which normally fractures after diffuse and then localized oblique necking. From the crack tip, two shear bands formed at 55° to the load direction, consistent with isotropic plane stress characteristics (53° was predicted from anisotropy, but necking in thin strips occurred at 67°). Photomicrographs showed that the 22° crack growth occurred by first tunnelling at mid-thickness, and then spreading along through-thickness shear planes. Springback on unloading caused a 0.038 mm crack closure and local buckling. This form of cracking illustrates a size effect in fracture under macroscopically plane stress. It also gives an example of a local mechanism triggering a fracture mode that can require more total work than an alternative.Analysis of isotropic localized necking shows the equivalent strain at fracture in thin strips to be uniquely related to the Reduction in Squared Thickness (RST). With smooth edges, width and thickness strains before and during necking differed by factors of 1.4 and 1.7; such measures of anisotropy should be routinely found and reported for strips.     

A circumferential crack in a cylindrical shell under tension

A closed cylindrical shell under uniform internal pressure has a slit around a portion of its circumference. Linear shallow shell theory predicts inverse square-root-type singularities in certain of the stresses at the crack tips. This paper reports the computed strength of these singularities for different values of a dimensionless parameter based on crack length, shell radius and shell thickness.     

Transient analysis of a piezoelectric strip with a permeable crack under anti-plane impact loads

The problem of a through permeable crack situated in the mid-plane of a piezoelectric strip is considered under anti-plane impact loads for two cases. The first is that the strip boundaries are free of stresses and of electric displacements, and the second is that the strip boundaries are clamped rigid electrodes. The method adopted is to reduce the mixed initial-boundary value problem, by using integral transform techniques, to dual integral equations, which are further transformed into a Fredholm integral equation of the second kind by introducing an auxiliary function. The dynamic stress intensity factor and energy release rate in the Laplace transform domain are obtained in explicit form in terms of the auxiliary function. Some numerical results for the dynamic stress intensity factor are presented graphically in the physical space by using numerical techniques for solving the resulting Fredholm integral equation and inverting Laplace transform.     

Stress concentration of a strip with double edge notches under tension or in-plane bending

The stress concentration analysis of 60° V-shaped or partially-circular double edge notches in an infinite strip under tension or in-plane bending is discussed. The stress field induced by a point force in an infinite plate is used to solve these problems. The present results for semicircular notch are in close agreement with other reports. The results calculated on the 60° V-shaped notches show that the Neuber formula gives an underestimated stress concentration factor of about 11% for tension case and in about 9% for bending case. These errors exist for a wide range of notch depth. However, in the case of blunt notches, the Neuber solution of deep hyperbolic notches still gives a sufficient accuracy in engineering use. In addition, the stress concentration factors of 60° V-shaped notches are also represented by diagrams for wide use.     

Transient response of a functionally graded piezoelectric strip with a penny-shaped crack under electric time-dependent loading

Summary The dynamic fracture problem for a functionally graded piezoelectric material (FGPM) strip containing a penny-shaped crack parallel to the free boundaries is considered in this study. It is assumed that the electroelastic properties of the strip vary continuously along the thickness direction of the strip, and that the strip is under time-dependent electric load. Integral transform techniques and dislocation density functions are employed to reduce the problem to the solutions of a system of singular integral equations. The stress and electric displacement intensity factors versus time are presented for various values of dimensionless parameters representing the crack size, the crack location and the material nonhomogeneity.     

Propagation of a through crack in a sheet material under biaxial tension

We present the results of investigation of the effect of the form of the stressed state and low temperatures on the growth of plastic strains at the crack tip in sheet materials subjected to repeated static biaxial loading. Information of this sort enables one to obtain adequate estimates of the load-carrying capacity of plates with cracklike concentrators by determining the plastic component of strains at the crack mouth and to introduce the required corrections in the existing models of fracture mechanics. Institute for Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 4, pp. 52–59, July–August, 1998.     

Impact response of a finite crack in a finite strip under anti-plane shear

The response of a through-the thickness crack with finite dimensions to impact in a finite elastic strip is investigated in this study. The elastic strip is assumed to be subjected to anti-plane shear deformation. Laplace and Fourier transform were used to formulate the mixed boundary value problem. The dynamic stress intensity factor and crack opening displacement are obtained as a function of time and the strip width to crack length ratio, h/a. The results indicate that the intensity of the crack-tip stress field reaches a peak very quickly and then decreases in magnitude oscillating about the static value. In general, the dynamic stress intensity factor is higher for small $\text{h/a}$. Similar behavior has also been found for the crack surface displacement.     

Peripheral edge crack around a spherical cavity under uniaxial tension field

The singular stress problem of a peripheral edge crack around a spherical cavity under a uniaxial tension field is investigated. The problem is analyzed by means of integral transforms and reduced to that of solving a singular integral equation of the first kind. The stress-intensity factors and crack opening displacements are obtained by means of the numerical technique due to Erdogan, Gupta and Cook [6].     

Penny shaped crack in a three-dimensional piezoelectric strip under in-plane normal loadings

Summary The singular mechanical and electric fields in a three-dimensional piezoelectric ceramic strip containing a penny shaped crack under in-plane normal mechanical and electrical loadings based on the continuous electric boundary conditions on the crack surface are considered here. The potential theory and Hankel transforms are used to obtain a system of dual integral equations, which is then expressed as a Fredholm integral equation. All sorts of field intensity factors of Mode I are given, and numerical values for PZT-6B piezoelectric ceramic are graphically shown.     

A penny-shaped crack in a functionally graded piezoelectric strip under thermal loading

The mixed-mode thermoelectromechanical fracture problem for a functionally graded piezoelectric material (FGPM) strip with a penny-shaped crack is considered. It is assumed that the thermoelectroelastic properties of the strip vary continuously along the thickness of the strip, and that the strip is under thermal loading. The crack faces are supposed to be insulated thermally and electrically. The thermal and electromechanical problems are reduced to singular integral equations and solved numerically. The stress and electric displacement intensity factors are presented for different crack size, crack position and material nonhomogeneity.     

Impact response of a strip composite with a finite crack

The dynamic response of a central crack in a strip composite under normal impact is analyzed. The crack is oriented normally to the interfaces. Laplace and Fourier transform techniques are used to reduce the elastodynamic problem to a pair of dual integral equations. The integral equations are solved by using an integral transform technique and the result is expressed in terms of a Fredholm integral equation of the second kind. A numerical Laplace inversion routine is used to recover the time dependence of the solution. The dynamic stress intensity factor is determined and its dependence on time, the material properties and the geometrical parameters is discussed.     

Stress analysis of a strip with a step and a crack

Stress analysis for a strip of which widths are different at the right and left handsides is carried out for before and after occurrence of a crack at a corner of a step. The strip is analyzed as a thin plate bending problem subject to transverse bending and torsion, and as a plane eleastic problem subject to bending in the plane and uniaxial tension. The rational mapping function of a sum of fractional expressions and complex variable method are used. Stress distribution and stress intensity factors are obtained.     

Singular stress and electric fields of a piezoelectric ceramic strip with a finite crack under longitudinal shear

Summary Following the theory of linear piezoelectricity, we consider the problem of determining the singular stress and electric fields in an orthotropic piezoelectric ceramic strip containing a Griffith crack under longitudinal shear. The crack is situated symmetrically and oriented in a direction parallel to the edges of the strip. Fourier transforms are used to reduce the problem to the solution of a pair of dual integral equations. The solution of the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind. Numerical values on the stress intensity factor and the energy release rate for piezoelectric ceramics are obtained, and the results are graphed to display the influence of the electric field.     

Influence of a central straight crack on the buckling behaviour of thin plates under tension,compression or shear loading

Thin-walled structural components, such as plates and shells, are used in several aerospace, naval, nuclear power plant, pressure vessels, mechanical and civil structures. Due to their high slenderness, the safety assessment of such structural components requires to carefully assess the buckling collapse which can strongly limit their bearing capacity. For very thin plate, buckling collapse can occur under shear, compression or even under tension. In the latter case, fracture or plastic failure can also take place instead of elastic instability. In the present paper, the effects of a central straight crack on the buckling collapse of rectangular elastic thin-plates—characterized by different boundary conditions, crack length and orientation—under compression, tension or shear loading are analysed. Accurate FE numerical parametric analyses have been performed to get the critical load multipliers in such loading cases. Moreover the effect of crack faces contact is examined and discussed. Some useful conclusions related to the sensitivity to cracks of the buckling loads for thin plates, especially in the case of shear stresses, are drawn. Cracked plates under tension are finally considered in order to determine the most probable collapse mechanism among fracture, plastic flow or buckling and some failure-type maps are determined.     

Buckling behavior of a central cracked thin plate under tension

The buckling characteristics of cracked plates subject to uniaxial tensile loads are analysed by the aid of the finite element method. Owing to the fact that crack buckling behavior is affected by the in-plane stress distribution around a crack, to get more accurate results, pre-buckling in-plane stress fields are analysed by the finite element method. For the critical loads calculation, the finite element approach adopted is based on Von Karman's linearize theory for buckling of plates subjected to pre-buckling state of plane stress. Several singular elements based on the Willian series are used in this plate bending approach. In this study, the effect of crack length, the effect of boundary condition, the effect of Poison's ratio and the effect of biaxial force on critical loads are analysed and discussed. Furthermore, the effect of initial imperfection is also discussed. There is a good agreement between other researcher's work and present results.     

The solution of a rectangular orthotropic sheet with a central crack under anti-plane shear

Making use of the basic theorem of the Fourier transform and series, the solution of the stress intensity factor of a rectangular orthotropic plate containing a central crack under anti-plane shear, is obtained in this study. The result to the mixed boundary value problem is expressed in terms of a Fredholm integral equation of the second kind. It is easily proved that the problem of a strip with a central crack of mode III, are the special cases of the solution in this article.