An assessment of factors influencing data obtained by the photoelastic stress freezing technique for stress fields near crack tips

Factors which influence crack tip stress field data are identified as: (1) Non-linear zone near crack tip due to crack blunting. (2) Normal stress parallel to crack surface. (3) Location of region for data retrieval.

The Kolosoff-Inglis solution is used in order to assess effects of crack tip blunting. A set of stress freezing photoelastic experiments are conducted on plates containing through cracks and results are compared with the Westergaard solution in order to assess the effect of Item 2 using appropriate Item 3 locations. A data conditioning computer program is employed to yield accurate values of the stress intensity factor from photoelastic data.     

A quantitative evaluation of fatigue crack shielding forces using photoelasticity

The mechanisms controlling the phenomenon of plasticity-induced shielding during fatigue are investigated and quantified by fitting a recently developed model to photoelastic data. The model derives from the Muskhelishvili approach and includes additional terms to describe the effect of plasticity on the elastic stress field around the crack tip. The photoelastic technique used a polycarbonate CT specimen containing a naturally propagating fatigue crack from which full-field data was obtained digitally using the phase-stepping method. The model was fitted to approximately 1000 data values for the isochromatic fringe order around the crack tip and generated values for the stress intensity factor and T-stress plus an interfacial shear stress intensity factor and a retardation intensity factor which together characterize the influence of plasticity on crack growth.     

Application of finite element analysis techniques for predicting crack propagation in lugs

The plane elastic problem corresponding to a through radial crack emanating from the internal boundary of a symmetrical lug is considered. A pin bearing pressure distribution was developed by utilizing photoelastic test data and differs considerably from the usually assumed uniform or cosine pressure distributions. The stress intensity factors at the crack tip were obtained by using recently derived quadratic isoparametric finite elements which embody the inverse square root singularity. Fatique crack growth tests of 17 aluminium, titanium and steel lugs were utilized to verify stress intensity factor solutions.     

Photoelastic determination of stress intensity factors in patched cracked plates

In this paper, the influence of patch parameters on stress intensity factors in edge cracked plates is studied by employing transmission photoelasticity. Edge cracked plates made of photo-elastic material are patched on one side only by $\text{E}$ glass-epoxy and carbon-epoxy unidirectional composites. The patch is located on the crack in such a way that the crack tip is not covered. Magnified isochromatic fringes are obtained by using a projection microscope of magnification 50, converted into a polariscope. Irwin's method is used to compute stress intensity factors from photoelastic data. The reduction in stress intensity factors is presented in graphical form as a function of patch parameters, namely stiffness, location and length. An empirical equation connecting reduction in stress intensity factor and these patch parameters is presented.     

A photoelastic determination of Ki stress intensity factors

The quantitative relation between the exact solution of the stress field at the vicinity of a crack tip derived fron Westergaard's formulation and the well-known Irwin singular solution was established and results obtained were correlated with photoelastic data for the study of the reigon near the crack tip. The maximum shear stress distribution expressed by the isochromatic pattern for the exact and the singular solution were calculated respectively for uniaxial and biaxial tension. The region where accurate measurements in the isochromatic pattern are possible to evaluate the stress intensity factor to any desired decree of accuracy was established and the extrapolation law for the analysis of the region near the crack tip from data obtained fron the far-field of isochromatics was demonstrated. Experimental evidence corroborated this technique. The method was compared with other already existing experimental methods for the determination of K_I.     

Analysis of the optical method of caustics for dynamic crack propagation

In the interpretation of experimental data on dynamic crack propagation in solids obtained by means of the optical method of caustics, it has been customary to neglect the effect of material inertia on the stress distribution in the vicinity of the crack tip. In this paper, the elastodynamic crack tip stress field is used to establish the exact equations of the caustic envelope formed by the reflection of light rays from the surface of a planar solid near the tip of a propagating crack. These equations involve the instantaneous crack tip speed, the material parameters and the instantaneous dynamic stress intensity factor, and they can be used to determine the stress intensity factor for given material parameters and crack tip speed. The influence of inertial effects on stress intensity factor measurements for system parameters typical of experiments with PMMA specimens is considered. It is found that the stress intensity factor values inferred through a dynamic analysis may differ by as much as 30–40% from values based on a quasi-static analysis.     

A simple method for the photoelastic determination of mode I stress intensity factors

A new simple method for the photoelastic determination of Mode I stress intensity factors from isochromatics is proposed. This method takes into account the fact that a considerable part of the error committed in the photoelastic determination of Mode I stress intensity factors K_I at crack tips, based on experimentally obtained isochromatic fringe patterns, is due to ignoring the non-singular part of the stress field near the crack tips for the evaluation of these factors. This error can, in most cases, be minimized by an appropriate selection of the polar direction from the crack tip on which the experimental measurements for the subsequent evaluation of the stress intensity factors K_I are made. The suitable polar direction for determining K_I depends in general on the distance of the point where measurements on the isochromatics are made from the crack tip. The method was applied to the problem of a simple crack inside an infinite medium under uniaxial and biaxial loading. A comparison of the present method whith the employed analogous methods shows the superiority of the proposed method.     

A finite deformation analysis of the near field surrounding the tip of crack-like elliptical perforations

A finite deformation analysis of the region surrounding the tip of crack-like elliptical perforations in an infinite plate under all around tension is presented as a model of the near field behavior of stress freezing photoelastic materials above critical temperature. The study is carried out in the deformed geometry and includes the effects of finite strains and rotations. A stress function is first introduced into the complete compatibility equations through linear constitutive relations; the resulting governing equation is solved through finite differences. The range of root radii investigated varies from one to nine times that of a deformed crack. Normal stresses and stress intensity factors are presented. The results of the analysis are compared to the linear analysis of Inglis. The effects of finite strains and rotations are shown to be large but are concentrated within a few root radii of the tip. Results suggest that the stress distribution near the notch root depends on the size of the root radius: small root radii produce maximum stresses away from the tip, whereas larger root radii produce maximum stresses at the tip itself. These effects are judged too local to influence photoelastic stress intensity determination.     

On the photoelastic determination of complex stress intensity factors

An improvement of the one-parameter extrapolation method of photoelastic determination of complex (mixed-mode) stress intensity factors at straight or curvilinear crack tips in a plane isotropic elastic medium due to Smith et al. [12, 13] can be achieved by measuring the absolute value of such a factor on the isochromatic fringes along properly selected polar directions and not at the maxima of the isochromatic fringes. In this way, the unknown value of the constant term of the stress field near the crack tip is taken into account. It is seen that it is always possible to find at least one appropriate polar direction to measure the absolute value of the stress intensity factor. In the case of Mode I stress intensity factors, these polar angles are = ± 120° and not = ± 90° as generally considered previously. Some numerical results are also presented in this special case and show the efficiency of the present method.     

Dynamic stress intensity factor due to concentrated loads on a propagating semi-infinite crack in orthotropic materials

The elastodynamic response of an infinite orthotropic material with a semi-infinite crack propagating at constant speed under the action of concentrated loads on the crack faces is examined. Solution for the stress intensity factor history around the crack tip is found for the loading modes I and II. Laplace and Fourier transforms along with the Wiener-Hopf technique are employed to solve the equations of motion. The asymptotic expression for the stress near the crack tip is analyzed which lead to a closed-form solution of the dynamic stress intensity factor. It is found that the stress intensity factor for the propagating crack is proportional to the stress intensity factor for a stationary crack by a factor similar to the universal function k(v) from the isotropic case. Results are presented for orthotropic materials as well as for the isotropic case.     

A Griffith crack shielded by a dislocation pile-up

The dislocation free zone at the tip of a mode III shear crack is analyzed. A pile-up of screw dislocations parallel to the crack front, in anti-plane shear, in the stress field of a crack has been solved using a continuous distribution of dislocations. The crack tip remains sharp and is assumed to satisfy Griffith's fracture criteria using the local crack tip stress intensity factor. The dislocation pile-up shield the sharp crack tip from the applied stress intensity factor by simple addition of each dislocation's negative contribution to the applied stress intensity value. The analysis differs substantially from the well known BCS theory in that the local crack tip fracture criteria enters into the dislocation distributions found.     

The condition for the cyclic plastic deformation of the crack tip: the influence of dislocation obstacles

The stress intensity range below which no cyclic plastic deformation at the crack tip and, hence, no fatigue crack propagation occurs is investigated. The emission of dislocations from the crack tip is assumed as mechanism for the dislocation generation. For a mode III crack, a computer simulation is carried out to study the influence of dislocation obstacles. Both the distance between the crack tip and the obstacle and the strength of the obstacle are varied and the characteristic dislocation arrangements are shown.The stress intensity range necessary to return one dislocation to the crack tip is mainly controlled by the critical stress intensity factor sufficient to emit a dislocation. The influence of the obstacles is not very significant.     

Investigations on the stress intensity factor field for unstable dynamic crack growth

In this study, the unstable dynamic crack propagation due to static loading in an elastic material is analyzed for both antiplane and inplane conditions. Of particular concern is the investigation of limitations on the assumption that the stress intensity factor field is fully established over a region of given size near the tip of a growing crack. The transient analysis of the stress for a material particle at a small fixed distance from the moving crack tip is examined in detail. Some estimations are made of the time required for the stress at a point near the moving crack tip to reach the value it would have if the stress field were actually given by the near tip stress intensity factor field. In addition, a simple formulation obtained from the equivalent static problem is proposed which can be used as a good approximation to the associated complicated dynamic transient problem.     

Determination of stress intensity factor solutions for cracks in finite-width functionally graded materials

A generalized method to determine the stress intensity factor equations for cracks in finite-width specimens of functionally graded materials (FGMs), based on force balance in regions ahead of the crack tip is provided. The method uses the Westergaard's stress distribution ahead of the crack in an infinite plate and is based on the requirement of isostrain deformation of layers of varying moduli ahead of the crack tip. It is shown that the modified Westergaard equation describes the normal stress distribution and the singular stress state ahead of the crack tip in a reasonably accurate manner. Based on this, closed-form analytical equations for the stress intensity factors of cracks in finite-width center cracked specimens were derived. Comparisons of the K values from the analytical equations with that obtained from FEM simulations indicate that the derived stress intensity factor equations for FGMs are reasonably accurate. For the finite-width center-cracked-tension (CCT) specimen, the errors are less than 10% for most of the crack lengths for materials with the outer layer modulus ratios varying from 0.2 to 5. The stress intensity factors were found to be sensitive to the absolute values of moduli of the layers, the modulus ratio of the outer layers as well as the nature of gradation including the increasing and the decreasing functional forms. The stress intensity factor equations are convenient for engineering estimates of stress intensity factors as well as in the experimental determinations of fracture toughness of FGMs.     

KI of a circumferential crack emanating from an ellipsoidal cavity obtained by the crack tip stress method in FEM

In high hardness steel, fatigue crack appears from a microcavity in most cases. Therefore, it is important to know the stress intensity factor of a crack emanating from the cavity. Recently, a method for calculating the highly accurate values of two-dimensional stress intensity factors was proposed by Nisitani, based on the usefulness of the stress values at a crack tip calculated by FEM. This method is called the crack tip stress method. In this paper, the crack tip stress method is extended to the problems of an infinite solid having an ellipsoidal cavity with a circumferential crack emanating from the cavity subjected to tension.     

Thermal effect of plastic dissipation at the crack tip on the stress intensity factor under cyclic loading

Plastic dissipation at the crack tip under cyclic loading is responsible for the creation of an heterogeneous temperature field around the crack tip. A thermomechanical model is proposed in this paper for the theoretical problem of an infinite plate with a semi-infinite through crack under mode I cyclic loading both in plane stress or in plane strain condition. It is assumed that the heat source is located in the reverse cyclic plastic zone. The proposed analytical solution of the thermo-mechanical problem shows that the crack tip is under compression due to thermal stresses coming from the heterogeneous stress field around the crack tip. The effect of this stress field on the stress intensity factor (its maximum and its range) is calculated analytically for the infinite plate and by finite element analysis. The heat flux within the reverse cyclic plastic zone is the key parameter to quantify the effect of dissipation at the crack tip on the stress intensity factor.     

Estimating the importance of cyclic thermal loads in thermo-mechanical fatigue

A method for evaluating the effect of cyclic thermal loading on crack tip stress fields is developed. In its development, advantage is taken of the periodic nature of fatigue loading and only harmonic loadings are evaluated. Formulating the problem in this way permits the extraction of time as an explicit variable and replaces its role with a dependence on the frequency of the thermal loading. The means for evaluating the effect of periodic loadings on crack tip stress fields is the stress intensity factor which is calculated from numerically defined stress and displacement fields using a path independent integral. Results obtained indicate that stress intensity factors of cracked components exposed to thermal fatigue conditions have a significant dependence on the frequency of the thermal cycle and the crack geometry. Numerical estimates for mode I thermal stress intensity have been obtained using thermal fatigue test data for a titanium alloy and can be as high as 25 percent of the critical mode I mechanical stress intensity.     

Towards a device independent digital image processing software for photoelastic analysis

The application of digital image processing (DIP) techniques to photoelasticity has greatly simplified data reduction and analysis. The softwares reported so far are hardware dependent and hence not portable. A simple PC based device independent software for collection of photoelastic data from digitised fringe patterns is presented. The software is validated by evaluating the near crack tip stress field parameters for a crack subjected to mode I and mode II loadings.     

Modeling of the Stress–Strain State near Cracks in Anisotropic Linearly Viscoelastic Plates

The photoelastic behavior of a linearly viscoelastic orthotropic plate containing a crack is described by using the well-known Netrebko–Vasyl'chenko equations of photoelasticity for linearly viscoelastic bodies. Relations for the stress intensity factors near the crack are deduced according to the data of polarization optical measurements. It is shown that the behavior of the stress intensity factors as functions of time depends on the angle between the direction of tension and the crack.     

Crack tip stress intensity factors for a crack emanating from a sharp notch

Accurate calibrations are provided for the crack tip stress intensity factor for a crack of finite length emanating from the symmetric tip of a sharp notch, of arbitrary angle, in terms of the generalised stress intensity quantifying remote loading of the notch. The solution is applied to example problems and shown to be accurate for cases where the crack is much shorter then the notch depth.