Stress intensification due to an edge crack in an anisotropic elastic solid

Integral transform techniques are used to determine the stress intensity factors of a crack at the edge of an anisotropic elastic half space under generalized plane strain conditions. Numerical results are given for a carbon fibre reinforced epoxy in uniaxial tension.     

Dependence of a crack growth path on the elastic moduli of an anisotropic solid

One of the ways to increase the resistance of a structure to catastrophic fracture is to force a main line crack to deviate from its path. For this reason the influence of the elastic moduli of an anisotropic material on crack rotation are studied. In particular a linear elastic problem for a straight Mode I crack, located on a symmetry axis of an orthotropic plane is considered. The strength properties of the material are assumed to be isotropic. Several crack models are considered for studying the direction of a crack growth path. It is shown that a crack modeled as a thin, elongated, elliptical hole leads to more plausible results concerning crack rotation conditions than an ideal cut model. The maximal tensile stresses are taken as a crack growth criterion. It is shown that for a class of orthotropic materials a crack deviates from the straight path just after it starts to grow, even in the conditions of uniaxial normal tension. The problem of the stability of a straight crack path under Mode I loadings is also considered. This problem is reduced to the problem of the fracture direction determination for thin, elongated, elliptical cavities slightly inclined to the initial direction. The conditions of instability are obtained within the framework of the proposed approach. It is shown that for a class of orthotropic materials a straight crack path is unstable in the conditions of uniaxial normal tension. This class of materials is larger than the one for which a crack deviates from the straight crack path just after its start.     

Edge crack in a strip of an elastic solid

The problem of determining the distribution of stress and the deformation of a long strip of an elastic material, damaged by a crack normal to an edge of the strip, is investigated. The strip is deformed by pressure applied to the faces of the crack. The stress intensity factor is calculated and its variation with the depth of the crack, relative to the width of the strip, in the special case of uniform pressure, is illustrated.     

Dynamic effects on the near crack-line fields for crack growth in an elastic perfectly-plastic solid

The dynamic effects on the near crack-line fields for steady-state tensile crack growth in an elastic perfectly-plastic solid are investigated under plane stress condition in this paper. In the plastic loading zone, the stresses and particle velocities near the crack-line are expanded in powers of the distance y to the crack line, with coefficients which depend on the distance of the moving crack tip. Substituting the expansions into the equations of motion, the Huber-Mises yield criterion and the Prandtl-Reuss flow rule yield a system of non-linear ordinary differential equations for the coefficients. This equation system is solved by using the approximate approach proposed by J.D. Achenbach and Z.L. Li. Finally, the crack growth criterion of critical strain is employed to determine the value of the remote elastic stress intensity factor K ₁that would be required for a crack growing steadily at a given Mach number. It is also shown in this paper that the steady-state dynamic solution yields the quasi-static solution as the speed of crack growth tends to zero.

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Résumé On étude les effets dynamiques qu'exercent les champs de contrainte au voisinage de la ligne de fissuration sous des conditions de tension stable dans un solide parfaitement élastique-plastique et pour un état plan de tension.Dans la zone de sollicitation plastique, les contraintes et vitesses élémentaires au voisinage de la ligne de fissuration se distribuent selon une puissance de la distance y à partir de cette ligne, avec des coefficients qui sont eux-mêmes dépendant de la distance à l'extrémité de la fissure en mouvement. En substituant les dilatations dans les équations de mouvement, le critère de plastification de Huber-Von Mises et la loi de'ecoulement de Prandtl-Reuss conduisent à un systéme d'équations différentielles ordinaires non linéaires pour déterminer ces coefficients.On résoud ce système d'équations grâce à une approach proposée par J.D. Achenbach et Z.L. Li.Enfin, le critère de déformation critique entraînant une croissance de la fissure est utilisé pour déterminer la valeur du facteur d'intensité des contraintes élastiques lointaines K ₁qui serait requis pour qu'une fissure croisse régulièrement à un nombre de Mach donné. On montre également dans l'étude qu'une solution dynamique sous conditions stables conduit à une solution quasi-statique lorsque la vitesse de propagation de la fissure tend vers zéro.

     

Characterization of isochromatic fringe patterns for a dynamically propagating interface crack

The isochromatic fringes surrounding a crack propagating along a bimaterial interface have been developed and characterized. A parametric investigation has also been conducted to study the influence of various fracture parameters on this isochromatic fringe pattern. The relevant fracture parameters of interest were the crack-tip velocity, the mode mixity of loading and the non-singular stress field component. In all the cases the fringe pattern was compared with the more familiar patterns that are generated for the case of crack propagation in homogeneous media. It was found that both the crack tip velocity and the mode mixity of loading have a significant effect on the size and shape of the isochromatic fringe pattern surrounding a crack tip propagating along a bimaterial interface. However, the non-singular stress field component was found not to have a substantial effect on the fringe pattern. This is in contrast with the case of crack propagation in homogeneous media, where the non-singular stress field component determines the tilt of the fringe contours. The paper also presents an appropriate scheme to analyze experimental fringe contours to extract the various fracture parameters of interest. Finally, this scheme is employed to analyze actual experimental data from a typical bimaterial interface fracture experiment.     

Numerical simulations of a dynamically propagating crack with a nonlinear cohesive zone

Numerical solutions of a dynamic crack propagation problem are presented. Specifically, a mode III semi-infinite crack is assumed to be moving in an unbounded homogeneous linear elastic continuum while the crack tip consists of a nonlinear cohesive (or failure) zone. The numerical results are obtained via a novel semi-analytical technique based on complex variables and integral transforms. The relation between the properties of the failure zone and the resulting crack growth regime are investigated for several rate independent as well as rate dependent cohesive zone models. Based on obtained results, an hypothesis is formulated to explain the origin of the crack tip velocity periodic fluctuations that have been detected in recent dynamic crack propagation experiments. This revised version was published online in August 2006 with corrections to the Cover Date.     

An arbitrarily-oriented plane crack in an anisotropic elastic slab

The problem of an arbitrarily-oriented plane crack in an anisotropic elastic slab is considered. Through the use of a Fourier transform technique, the problem is reduced to a system of simultaneous Fredholm integral equations of the second kind. Once these integral equations are solved, relevant quantities such as the crack energy can be readily computed. Numerical results pertaining to the stability of a plane crack in a particular elastic slab are given.     

The effects of hyperbolic heat conduction around a dynamically propagating crack tip

Using infrared detectors, Zehnder and Rosakis (1991, J. Mech. Phys. Solids 39(3), 385), Zehnder and Kallivayalil (1991, SPIE ISS4A, 48) and Mason and Rosakis (1992, SN Report 92-2), have recorded the temperature field around a dynamically propagating crack tip travelling at constant velocity in several metals. At the same time, Tzou (1990a, J. Heat Transfer 112, 21, 1990b, Inst. Heat Mass Transfer 33(5), 877) has suggested that the temperature field around a propagating crack tip might exhibit some of the characteristics of hyperbolic heat conduction. In this paper a corrected solution of the hyperbolic heat conduction equation for a travelling point source is derived. Then an experimental estimate of the active plastic zone (heat generating zone) at a crack tip is used for various experimental conditions to examine the possible effects of hyperbolic heat conduction around a propagating crack tip. Finally, using the actual experimental conditions of Zehnder and Rosakis (1991) Zehnder and Kallivayalil (1991) and Mason and Rosakis (1992) it is shown that no effects of hyperbolic heat conduction are observed around a propagating crack tip. It is seen that, due to adiabatic conditions at the crack tip during these experiments, the solution of the hyperbolic heat equation is indistinguishable from the solution of the parabolic heat conduction equation for crack propagation in steel.     

Transient dislocation emission from a crack tip in an anisotropic elastic material

The emission of a dislocation with a general Burgers vector from the tip of a stationary semi-infinite crack in an anisotropic elastic material is examined. The dislocation is assumed to leave the crack tip along the crack extension plane at constant speed. Explicit expressions for the transient shielding stress intensity factors at the crack tip and the drag forces on the dislocations are derived. Numerical results for a class of cubic materials and two hexagonal crystals, zinc and cobalt, are given. Dislocation emission under plane stress wave loading is discussed.     

The dynamic near crack-line fields for mode II crack growth in an elastic perfectly-plastic solid

The dynamic near crack-line fields for mode II crack growth in an elastic perfectly-plastic solid are investigated under plane strain and plane stress conditions. In each case, by expanding the plastic fields and the governing equations in the coordinate y, the problem is reduced to solving a system of nonlinear ordinary differential equations which is similar to that of mode III derived by Achenbach and Z.L.Li. An approximate solution for small values of x is obtained and matched with the elastic field of a blunt crack at the elastic-plastic boundary. The crack growth criterion of critical strain is employed to determine the value of K _II of the far-field that would be required for a steadily growing crack.     

Elastodynamic near-tip fields for a rapidly propagating interface crack

The elastodynamic stress field near a crack tip rapidly propagating along the interface between two dissimilar isotropic elastic solids is investigated. Both anti-plane and in-plane motions are considered. The anti-plane displacements and the in-plane displacement potentials are sought in the separated forms $\text{r}{}^{\mathrm{q}}\text{F(θ)}$, $\text{r}$ and θ being polar coordinates centered at the moving tip. The mathematical statement of the problem reduces to a second-order linear ordinary differential equation in θ, which can be solved analytically. Formulation of the boundary and interface conditions leads to an eigenvalue problem for the singularity exponent $\text{q}$. For the in-plane problem, root $\text{q}$ is found to be complex. Thus, the stresses exhibit violent oscillations within a small region around the crack tip, and the solutions have physical significance only outside this region. The angular stress distributions are plotted for various crack speeds, and it is found that at a high enough speeds the direction θ of maximum stress moves out of the interface. This result indicates that a running interface crack may move into one of the adjoining materials.     

Thermoelastic wave propagation in an elastic solid containing a finite crack

Investigated in this paper is the scattering of plane harmonic thermoelastic waves around the tip of a finite crack. Integral transform techniques are used to formulate the problem and reduce it to Fredholm integral equations of the second kind. The equations are solved numerically and the singular stress field near the crack tip is determined. In particular, the variation of the stress intensity factor with the frequency of the incoming wave is exhibited graphically. The peak in the magnitude of the stress intensity factor is of paramount interest in the application of fracture mechanics to thermal stress problems.     

On a generalised plane strain crack problem for inhomogeneous anisotropic elastic materials

A generalised plane strain crack problem is considered for a class of inhomogeneous anisotropic elastic materials. The problem is reduced to a boundary integral equation involving hypersingular integrals. The boundary integral equation may be solved numerically using standard procedures. Some crack problems for a particular inhomogeneous material are considered in detail and the stress intensity factors are obtained in order to assess the effect of the anisotropy and inhomogeneity on the stress field near the crack tips.     

Dynamic fields near a crack tip growing in an elastic-perfectly-plastic solid

A complete asymptotic solution is given for the fields in the neighborhood of the tip of a steadily advancing crack in an incompressible elastic-perfectly-plastic solid.For Mode I crack growth in the plane strain condition, the following noteworthy results are revealed: (1) The entire crack tip in steady crack growth is surrounded by a plastic region, and no elastic unloading is predicted by the complete dynamic asymptotic solution. Thus, the elastic unloading region predicted by the result of neglecting the important influence of the inertia terms in the equations of motion. (2) Unlike the quasi-static solution, the dynamic solution yields strain fields with a logarithmic singularity everywhere near the crack tip. (3) The stress field varies throughout the entire crack tip neighborhood, but does display behavior which can be approximated by a constant field followed by an essentially centered-fan field and then by another constant field, especially for small crack growth speeds. Indeed, the stress field reduces to that for the stationary crack, as the crack tip velocity - measured by the Mach number, M - reduces to zero; the strain field, however, does not reduce to that for the static solution, as M vanishes. (4) There are two shock fronts emanating from the crack tip across which certain stress and strain components undergo jump discontinuities. The location of the shock fronts and the magnitude of the jumps depend on the crack growth speed. The stress jump vanishes while the strain jump becomes unbounded, as the crack tip speed goes to zero.Finally, the Mode III steady-state crack growth is reviewed and, on the basis of Mode I and Mode III results, it is concluded that ductile fracture criteria for nonstationary cracks must be based on solutions which include the inertia effects, and that for this purpose, quasi-static solutions may be inadequate. Then, a possible ductile fracture criterion is suggested and discussed.One interesting feature of the complete dynamic asymptotic solution is that, unlike the quasi-static solution, it yields the same strain singularity for all three fracture modes.     

A complex variable approach for asymptotic Mode-III crack-tip fields in an anisotropic functionally graded material

This paper addresses asymptotic full crack-tip fields for an anti-plane (Mode-III) stationary crack in an anisotropic functionally graded material. A monoclinic material that has a material symmetry plane is considered. The complex variable approach and the asymptotic analysis are used to solve a perturbed Laplace equation resulting from material anisotropy and gradation. The out-of-plane displacement and stress solutions are provided for a crack in exponentially and linearly graded anisotropic materials by considering material gradation either parallel or perpendicular to the crack. The characteristics of the asymptotic solutions in an anisotropic functionally graded material are compared with those for anisotropic homogeneous and isotropic graded materials. Finally, engineering significance of the present work is discussed.     

Asymptotic crack-tip fields in anisotropic power law material under antiplane shear

A hodograph transformation in conjunction with an appropriate affine transformation are both used to investigate the strain and stress fields near the crack tip in an anisotropic power law material under antiplane shear. Stress and strain exponents as well as angular distributions for the asymptotic stress and strain fields are obtained analytically. All the stress strain exponents are independent of material anisotropy, and the effect of material anisotropy on the asymptotic stress and strain field is discussed including higher order terms.     

Near-tip behavior of a crack in a plane anisotropic elastic body

The asymptotic form of the stress and displacement components near the tip of a straight crack in a generally rectilinear anisotropic plane elastic body are resolved. As in the isotropic analysis, the solutions for the stresses display a $\text{r}{}^{\mathrm{<mtext>?</mtext><mtext>1</mtext><mtext>2</mtext>}}$ dependence, where r is the distance from the tip, while the angular dependence depends upon the anisotropy in a complicated way. The effect of some special anisotropies upon these solutions is fully explored. Finally, these solutions are used to solve the problem of a finite length straight crack in an anisotropic elastic plane when uniform stresses are applied far from the crack. This solution includes obtaining the stress intensity factors, and the nature and magnitude of the crack face displacements.