Plastic zone near the crack tip in a material with strain anisotropy caused by loading along spatial rectilinear trajectories

The equation of the boundary of the plastic zone near the tip of a mode I crack is deduced for the case of a plate made of a material with strain anisotropy. It is assumed that the anisotropy is caused by hardening in the process of plastic deformation performed prior to the appearance of the crack under loading along arbitrary rectilinear trajectories in the space of the stress tensor. An analysis of this equation demonstrates that the main factors affecting the shape and size of the plastic zone are the level of plastic strains accumulated in the process of preloading, their sign, and the orientation of the crack relative to the axes of anisotropy. Translated from Problemy Prochnosti, No. 6, pp. 21–27, November–December, 1997.     

Influence of deformation anisotropy on the size of the plastic zone at the tip of an opening mode crack

The form and dimensions of the plastic zone at the tip of an opening mode crack in a plate made of a material with deformation anisotropy were investigated within the limits of the elastic solution. The anisotropy was caused by strengthening during plastic deformation until formation of cracks by loading in a straight trajectory located in the plane of the plate. It was shown that in the case of anisotropy caused by loading in a trajectory which is oriented on a normal to the crack edges the size of the plastic zone decreases and its boundaries are rotated in the direction opposite to the crack growth. Loading in a trajectory in the direction of crack growth leads to broadening of the plastic zone in the transverse direction.Translated from Problemy Prochnosti, No. 1, pp. 73–76, January, 1990.     

Plastic zones in an orthotropic plate of finite width containing a Griffith crack

The problem of an orthotropic infinite plate of finite width containing a centrally located stressed Griffith crack is considered. The crack is located perpendicular to the edges of the orthotropic plate. The crack tips are fully yielded and in the inelastic zones the material carries only constant normal stresses equal to the yield stress. Dugdale's model is employed to find the effects of the material anisotropy on the size of the plastic zones around the crack tips. Graphical results showing the effects of anisotropy on the length of the plastic zone are also presented.     

Effects of notch radius and anisotropy on the crack tip plastic zone

Factors which influence the shape and size of the plastic zone in the immediate vicinity of a crack tip in isotropic materials at small loads are investigated. The plastic zone dimensions for the opening mode (Mode I) have been calculated over a range of values for the crack tip radius. An increase in tip radius results in an increase in the plastic zone dimension. In anisotropic materials, the orientation of crack slit and the anisotropic yield constants are other factors that affect the plastic zone size and shape. In this paper, typical curves for the shape and size of plastic zone are given to illustrate the influence of normal or shear anisotropic yield constants. For sheet metals the effects of anisotropy on the plastic zone dimensions can be evaluated in terms of R values. Suggested values of constant b for isotropic materials are given if the “radius” approximation is employed for small applied stresses.     

Correlation of a plastic zone length near the tip of a mode I crack in an orthotropic material with anisotropy parameters

We derived equations that relate the length of a plastic zone near a mode I crack tip in a plate made of an orthotropic material with yield strength levels in the direction of the anisotropy axes. The case of crack orientation along one of the anisotropy axes is examined, with the latter being determined by the strain hardening of a material at the stage preceding the crack nucleation. The growth of yield strength along the axes lying in the plane of the plate is shown to result in smaller sizes of the plastic zone. An increase in yield strength in the direction of the normal to the above plane leads to an increase in its length. Ukrainian Regional Research and Design Institute of Civil Engineering, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 4, pp. 32–37, July–August, 1999.     

Effect of plastic anisotropy on crack growth resistance under mode 1 loading

Crack growth in a solid with plastic anisotropy is modeled by representing the fracture process in terms of a traction-separation law specified on the crack plane, and crack growth resistance curves are calculated numerically. A phenomenological elastic-viscoplastic material model is applied, using one of two different anisotropic yield criteria to account for the plastic anisotropy. The analyses are carried out for conditions of small scale yielding, with mode I loading conditions far from the crack-tip. Different initial orientations of the principal axes relative to the crack plane are considered and it is found that the steady-state fracture toughness is quite sensitive to the type of anisotropy and to the angle of inclination of the principal axes relative to the crack plane.     

A low-cycle fatigue approach to fatigue crack propagation

The damage accumulation hypothesis is used to derive a fatigue crack growth rate equation. The fatigue life of a volume element inside the plastic zone is evaluated by using low-cycle fatigue concepts. Crack growth rate is expressed as a function of cyclic material parameters and plastic zone characteristics. For a given material, crack growth increment, is predicted to be a fraction of the plastic zone size which can be expressed in terms of fracture mechanics parameters,K andJ. Hence, the proposed growth rate equation has a predictive capacity and is not limited to linear elastic conditions.     

Plastic deformation in a thick transversely isotropic layer containing a penny-shaped crack

Summary The width of a thin plastic annular zone formed during the deformation of a pennyshaped crack in a transversely isotropic layer of an ideal elasto-plastic material is determined. Considered are the cases where the penny-shaped crack is extended by normal stresses and by torsional stresses. The faces of the layer are shear free and deformation of the plastic zone around the penny-shaped crack occurs according to the Dugdale hypothesis. For each case, the solution of the problem is reduced to a Fredholm integral equation of the second kind. Iterative solutions are obtained for small values of the parameters and numerical results for the width of the plastic zone are determined. Graphical results showing the effect of transverse isotropy upon the width of the plastic zone are also presented.With 6 Figures     

Plastic zones in a transversely isotropic solid cylinder containing a ring–shaped crack

In this study, a problem of a ring shaped-crack contained in a infinitely long solid cylinder of elastic perfectly-plastic material is considered.The problem is formulated for a tranversely isotropic matrial by using integral transform technique under uniform load. Due to the geometry of the configuration, Hankel and Fourier integral transform techniques are chosen and the problem is reduced to a singular integral equation. This integral equation is solved numerically by using Gaussian Quadrature Formulae and the values are evaluated for various for discrete points. The plastic zone widths are obtained by using the plastic strip model. They are plotted for various ring-shaped crack sizes and transversely isotropic matrials. It is found that the width of the plastic zone at the inner tip of the crack is greater than the outer one.     

Fatigue crack propagation in a piezoelectric ceramic strip subjected to mode III loading

Summary Following the theory of linear piezoelectricity, a forth-power stress intensity factor crack growth equation in an orthotropic piezoelectric ceramic strip is developed under mode III loading. The crack is situated symmetrically and oriented in a direction parallel to the edges of the strip. Dugdale's assumption regarding the plastic zone in metals is applied to estimate the effects of yield around the crack tips. Fourier transforms are used to reduce the electroelastic problem to one involving the numerical solution of a Fredholm integral equation of the second kind. A direct approach based on the accumulated plastic displacement criterion for crack propagation is used to develop the equation to predict the fatigue crack growth. Graphical results showing the effect of electroelastic interactions on the fatigue crack growth rate are presented.     

Analysis of temperature distribution near the crack tip under constant amplitude loading

An analytical/numerical method has been developed to find the temperature rise near the crack tip under fatigue loading. The cyclic plastic zone ahead of the crack tip is assumed to be the shape of the source of heat generation and some fraction of plastic work done in cyclic plastic zone as heat generation. Plastic work during fatigue load was found by obtaining stress and strain distribution within the plastic zone by Hutchinson, Rice and Rosengren (HRR) crack tip singularity fields applied to small scale yielding on the cyclic stress strain curve. A two‐dimensional conduction heat transfer equation, in moving co‐ordinates, was used to obtain temperature distribution around the crack tip. Temperature rise was found to be a function of frequency of loading, applied stress intensity factor and thermal properties of the material. A power–law relation was found between the rise in temperature at a fixed point near the crack tip and range of stress intensity factor.     

Ductile penny-shaped crack in a transversely isotropic cylinder

The problem of a penny-shaped crack contained in a transversely isotropic cylinder of elastic perfectly-plastic material is considered for the case when the crack is extended by an axial load. The problem is reduced to solving numerically a Fredholm integral equation of the second kind for the width of the plastic zone. Graphical results are presented showing the effect of transverse isotropy upon the width of the plastic zone and these are compared with the results for isotropic materials.     

Crystallographic analysis on small fatigue crack propagation behaviour of a nickel-based single crystal superalloy

In situ observations by scanning electron microscopy show that small fatigue cracks in a nickel-based single crystal superalloy are inclined to the loading direction and propagate in dominant crystallographic manners. In order to evaluate the driving forces for inclined crack propagation, three-dimensional anisotropic linear elastic finite-element analysis is conducted. The plastic zone size on the dominant slip plane has been calculated and proposed to correlate the fatigue crack growth. It is shown that this parameter takes into account both material anisotropy and octahedral fracture modes, and it can effectively characterize small crack propagation behaviour.     

Fatigue crack growth retardation under constant amplitude and variable mean stress

Fatigue crack growth retardation under stress spectra with constant amplitude and variable mean stress, respectively, was studied. Flat specimens with a central through crack were tested under tension-tension load. The specimens were made with low alloy steel 4 mm thick, with yield strength of 625 Mpa and ultimate tensile strength of 784 MPa. Overload affected crack length increments Δa¹ were studied. The best correlation was obtained between monotonic plane stress plastic zone size 2r_y and Δa¹. The cyclic plastic zone size 2r_pc correlated with crack length increment of minimum crack growth rate after overload. Forman's equation and Willenborg's model of fatigue crack growth retardation were used for theoretical prediction of fatigue crack propagation life. The best agreement between theoretical and experimental results was also obtained using monotonic plastic zone size instead of monotonic plastic zone radius or cyclic plastic zone size. The agreement is reasonably good, even though in the case of one spectrum, cracks were arrested for several thousand cycles.     

Effect of thickness on plastic zone size in BCS theory of fracture

In the fracture theory of Bilby, Cottrell and Swinden, the plastic zone in an infinitely thick specimen was represented as a distribution of dislocations. For a specimen of finite thickness, Eshelby and Stroh have shown that the interaction force between two screw dislocations can be expressed by a modified Bessel function which decays much faster than the inverse first power law as the distance between the dislocations is increased. By assuming a continuous distribution of dislocations, an integral equation based on the Eshelby and Stroh interaction is formulated to examine the effect of thickness on the plastic zone size as well as the relative displacement along the crack surfaces. The distribution function of dislocations is obtained numerically from this integral equation. Numerical results indicate that for a given crack opening displacement the length of the plastic zone decreases with decreasing specimen thickness. Energy release rate is discussed from the dislocation point of view. A previous result that the energy release rate is equal to the total force on the dislocations in the plastic zone is generalized to the case of a plate with finite thickness.     

Plastic zone formation and fatigue crack extension during high cyclic bending of steels

In repeated high cyclic bending, with constant load amplitude, the size and the shape of the plastic zone preceding the propagating crack is controlled by local structural conditions near the tip rather than by stress intensity. No significant correlations were found between the experimentally determined sizes of plastic zone and the theoretically predicted values of Liu and Rice. The plastic zone sizes ahead of the propagating crack cannot be simply expressed as proportional to the rate of fatigue crack propagation, though a simple relationship exists between the rate and the stress intensity factor. The relationship given by Paris, dl/dN = QΔKⁿ, describes the rate of crack propagation only in a limited range of relative crack length, x < 0.5. The extent of this range depends on the structure and on the level of applied cyclic stress. Beyond this range, the Paris equation could not be applied and the crack propagation cannot be related to the stress intensity factor.     

Effects of plastic anisotropy on crack-tip behaviour

For a crack in a homogeneous material the effect of plastic anisotropy on crack-tip blunting and on the near-tip stress and strain fields is analyzed numerically. The full finite strain analyses are carried out for plane strain under small scale yielding conditions, with purely symmetric mode I loading remote from the crack-tip. In cases where the principal axes of the anisotropy are inclined to the plane of the crack it is found that the plastic zones as well as the stress and strain fields just around the blunted tip of the crack become non-symmetric. In these cases the peak strain on the blunted tip occurs off the center line of the crack, thus indicating that the crack may want to grow in a different direction. When the anisotropic axes are parallel to the crack symmetry is retained, but the plastic zones and the near-tip fields still differ from those predicted by standard isotropic plasticity.     

Determination of the energy for crack creation using micro-hardness measures

The growth of crack is related to the existence of a plastic zone at the crack tip; whose formation and growth is accompanied by energy dissipation. The estimation of this energy is generally done by the so called global methods (hysterisis loops) or the micro-gages. In the present study, the micro-hardness measures in the plastic zone are used to evaluate the energy dissipated in the fracture process zone by plastic deformation. The obtained results on the aluminium alloy 7075 T7 and E460 steel are compared to those obtained by other methods.     

Sensitivities of Two-Dimensional Fracture Problems to the Near-tip Mesh Parameters

Analytical results for a penny-shaped crack with a plastic zone at the crack front are given. The crack is embedded in an infinite transversely isotropic elastic medium and is assumed to be subjected to two identical axisymmetric loads on the upper and lower crack faces. The size of the plastic zone at the crack front is determined by applying Dugdale hypothesis to the elasticity results for a penny-shaped crack. The size of the plastic zone is derived in terms of hyper-geometric functions. Expression of the normal stress outside the plastic zone is also given.