Estimations of the T-stress for small cracks at notches

The T-stress is increasingly being recognized as an important additional stress field characterizing parameter in the analyses of cracked bodies. Using T-stress as the constraint parameter, the framework of failure assessments including the constraint effect has been established; and the effect of T-stress on fatigue crack propagation rate has been investigated by several researchers. In this paper, a simple method for determining the T-stress for small notch-emanating cracks is presented. First, the background on the T-stress calculation using the superposition principle and the similarities between the elastic notch-tip stress fields described by two parameters: the stress concentration factor (K_t) and the notch-tip radius (ρ), are summarized. Then, the method of estimating T-stress for both short and long cracks at the notches is presented. The method is used to predict T-stress solutions for cracks emanating from an internal hole in a wide plate, and cracks emanating from an U-shaped edge notch in a finite thickness plate. The results are compared to the T-stress results in the literature, and the T-stresses solutions obtained from finite element analysis. Excellent agreements have been achieved for small cracks. The method presented here can be used for a variety of notch crack geometries and loading conditions.     

Universal features of elastic notch-tip stress fields

The results of linear elastic analyses of stress distributions near a wide variety of notches are presented. Notches under tension, tension and bending, and pure bending have been considered. It is demonstrated that notch-tip stress fields are similar to each other regardless of the notch geometry and the loading system. Universal functions describing the stress field in the notch-tip neighbourhood have been derived and tested againts available analytical, numerical and experimental data. The universal expressions can be used to calculate stresses in the region x⩽ 3ρ from the notch tip. These expressions can be particularly useful when using the weight function method to calculate stress intensity factors for cracks emanating from notches.     

Calculation of inelastic notch-tip strain-stress histories under cyclic loading

An application of the equivalent strain energy density method for calculation of elastic-plastic notch-tip strains under cyclic loading is presented. It is shown that the theoretical notch-tip strain calculations can be improved if the stress redistribution due to the plastic yielding around the notch-tip is taken into account. The energy density method, corrected for plastic yielding gave good results almost up to the general plastic yielding, i.e. $\text{S = σ}{}_{\mathrm{ys}}$. It was also found that a universal function for the elastic stress distribution ahead of a notch tip can be derived for both tension and bending loads. Several different notches and materials were analyzed. The equivalent strain energy density concept can easily be used for a simulation of the notch-tip cyclic stress-strain history if the stress concentration factor$\text{K}{}_{\mathrm{t}}$, the cyclic stress-strain curve σ−ϵ and nominal stress/load history are known. Good notch-tip strain predictions were achieved for both the monotonie and cyclic load.     

Crack propagation in ceramics under cyclic loads

Stable crack growth is observed in notched plates of polycrystalline alumina subject to fully compressive far-field cyclic loads at room temperature in a moist air environment andin vacuo. The fatigue cracks propagate at a progressively decreasing velocity along the plane of the notch and in a direction macroscopically normal to the compression axis. The principal failure events leading to this effect are analysed in terms of notch-tip damage under the far-field compressive stress, microcracking, frictional sliding and opening of microcracks, and crack closure. An important contribution to such Mode I crack growth arises from the residualtensile stresses induced locally at the notch-tip when the deformation within the notch-tip process zone leaves permanent strains upon unloading from the maximum nominal compressive stress. It is shown that the phenomenon of crack growth under cyclic compressive stresses exhibits a macroscopically similar behaviour in a wide range of materials spanning the very ductile metals to extremely brittle solids, although the micromechanics of this effect are very different among the various classes of materials. The mechanisms of fatigue in ceramics are compared and contrasted with the more familiar examples of crack propagation under far-field cyclic compression in metallic systems and the implications for fracture in ceramic-metal composites and transformation toughened ceramic composites are highlighted. Strategies for some important applications of this phenomenon are recommended for the study of fracture mechanisms and for the measurement of fracture toughness in brittle solids.     

SMALL FATIGUE CRACK GROWTH IN A LOW CARBON STEEL UNDER TENSION–COMPRESSION AND PULSATING-TENSION LOADING

The growth behaviour of small fatigue cracks has been investigated in a low carbon steel under axial loading at the stress ratios R of –1 (tension-compression) and 0 (pulsating-tension). Crack closure was measured to evaluate the effects of stress ratio and stress level on small crack growth. Except for the accelerated growth at stress levels close to the yield stress of the material, at R=–1 small cracks grow faster than large cracks below a certain crack length, but at R= 0 the crack growth rates for small cracks are coincident with those for large cracks in the whole region of crack length investigated. The critical crack length, 2c_c, above which the growth behaviour of small cracks is similar to that of large cracks depends on stress ratio, being 1–2 mm at R=–1 and smaller than 0.7 mm at R=0. The 2c_c value at R=–1 agrees with that obtained under rotating bending (R=–1). The small crack data are closely correlated with large crack growth rates in terms of the effective stress intensity range, ΔK_eff; thus ΔK_eff is found to be a characterizing parameter for small crack growth including the growth at the higher stress levels.     

Effect of material and geometric parameters on deformations near the notch-tip of a dynamically loaded prenotched plate

We analyze plane strain thermomechanical deformations of a prenotched rectangular plate impacted on one side by a prismatic body of rectangular cross-section and moving parallel to the axis of the notch. Both the plate and the projectile are made of the same material. Strain hardening, strain-rate hardening and thermal softening characteristics of the material are modeled by the Johnson–Cook relation. The effect of different material parameters, notch-tip radius, impact speed and the length of the projectile on the maximum tensile principal stress and the initiation and propagation of adiabatic shear bands at the notch-tip is analyzed. It is found that for high impact speeds or enhanced thermal softening, two shear bands, one at −10° to the notch ligament and the other at −128° to it, propagate from the notch tip. Otherwise, only one shear band nearly parallel to the notch-ligament initiates at the notch-tip. The notch-tip distortion for high strength materials is quite different from that for low strength materials. The maximum tensile principal stress occurs at a point on the upper surface of the notch-tip and for every set of values of material parameters and impact speeds studied equals about 2.3 times the yield stress of the material in a quasistatic simple tension or compression test. We assume that the brittle failure occurs when the maximum tensile principal stress equals twice the yield stress of the material in a quasistatic simple tension test and a shear band initiates when the effective plastic strain at a point equals 0.5. The effect of material and geometric parameters on the time of initiation of each failure mode is computed. It is found that for low impact speeds (<30 m/s) a material will fail due to the maximum tensile principal stress exceeding its limiting value, and at high impact speeds due to the initiation of a shear band at the notch-tip. Results are also computed for a C-300 steel with material parameters given by Zhou et al. For an impact speed of 50 m/s, the shear band speed and the maximum effective plastic strain-rate before a material point melts are found to be 350 m/s and 5×10⁷/s respectively. Key words: Failure-mode transition, shear bands, thermoviscoplasticity, numerical simulations. This revised version was published online in July 2006 with corrections to the Cover Date.     

Stiffness reduction and stress analysis in cracked [0m 0/90n 0]s laminates

Summary A new method of modelling effective properties and stress distribution in cracked laminates [12]–[17] is applied to assess the loss of the effective Young, Kirchhoff and Poisson moduli of the [0 _m ⁰/90 _n ⁰] _s laminates with transverse aligned cracks in the 90° layer and to find the stress distribution between interacting cracks and in the vicinity of isolated cracks. The decaying curve describing the loss of the effective Young's modulus in the tensile direction lies slightly over the curve found by Hashin [7] with the help of similar stress assumptions. The reduction of Kirchhoff's modulus is predicted in the same way as in Hashin [7] and Tsai and Daniel [28]. All components of the state of stress are given by closed-form relations. The stresses within periodicity cells are expressed in terms of macrodeformations. The explicit formulae for stresses derived in the present paper make it possible to interrelate the crack spacing with a magnitude of the applied stress. To this end a local stress-type criterion of the onset of cracks nucleation is used and reasonable good fit with experimental data of Highsmith and Reifsnider [9] is observed.     

Contour integral method for stress intensity factors of interface crack

A general Betti's reciprocal work theorem with interface cracks of a bimaterial is established in this paper, and a path independent contour integral method for the stress intensity factor (SIF) of the interface crack was obtained. When the stress and displacement fields in a specimen are calculated by the finite element method, the SIF K _I and K _II of interface cracks can be obtained immediately by a contour integral. Some solutions of interesting examples, such as two collinear interface cracks, are also given.Presented at the Far East Fracture Group (FEFG) International Symposium on Fracture and Strength of Solids, 4–7 July 1994 in Xi'an China.     

Crack growth behavior and fatigue-life prediction based on worst-case near-threshold data of a large crack for 9310 steel

ABSTRACT Both experimental and analytical investigations were conducted to study crack initiation and growth of small cracks, near‐threshold growth behavior of large cracks at constant R‐ratio/decreasing ΔK and constant K_max/decreasing ΔK, respectively, for 9310 steel. The results showed that a pronounced small‐crack effect was not observed even at R = ?1, small cracks initiated by a slip mechanism at strong slip sites. Worst‐case near‐threshold testing results for large cracks under several K_max values showed that an effect of K_max on the near‐threshold behavior does not exist in the present investigation. A worst‐case near‐threshold test for a large crack, i.e. constant K_max/decreasing ΔK test, can give a conservative prediction of growth behavior of naturally initiated small cracks. Using the worst‐case near‐threshold data for a large crack and crack‐tip constraint factor equations defined in the paper, Newman's total fatigue‐life prediction method was improved. The fatigue lives predicted by the improved method were in reasonable agreement with the experiments. A three‐dimensional (3D) weight function method was used to calculate stress‐intensity factors for a surface crack at a notch of the present SENT specimen (with r/w = 1/8) by using a finite‐element reference solution. The results were verified by limited finite‐element solutions, and agreed well with those calculated by Newman's stress‐intensity factor equations when the stress concentration factor of the present specimen was used in the equations.     

Spannungsfaktoren eines Risses in der Umgebung eines Kreisloches und ihr Einfluß auf das Bruchverhalten

Stress Intensity Factors in the Neighbourhood of a Circular Hole and their Influence on Crack Behavior . A photoelastic method was developed to determine the stress intensity factors K_I and K_II for cracks subjected to mixed-mode loading. The constants in the near field expansion about the crack tip were computed using a non-linear optimization program to give a best fit to the observed isochromatics. Copying the latter onto an equal density film increased their sharpness and, thus, the accuracy of the determination. The method was applied to cracks lying perpendicular to the external stress near a circular hole in a plate under uniaxial tension and the results used to describe the paths of cracks in the neighbourhood of a hole.     

Mixed-mode fracture angle and fracture locus of materials subjected to compressive loading

The incipient fracture angle and fracture loci of prenotched brittle-like material subjected to compressive loading are investigated analytically and experimentally.The analysis of the problem includes parameters whose effects on fracture were pronounced via laboratory tests, namely: notch-tip curvature, subcritical microcracks emanating from the notch and crack closure process. Such considerations, jointly with the well-established fracture criteria in tensile loading (like the critical energy release rate, the critical energy density, J-integral and critical maximum stress used in this work) yielded an associated fracture locus for each criterion. Due to the mixed mode nature of the situation (K₁ and K₂) preevaluation of the fracture angle was instrumental.Data on critical (far-field) compressive load along with measured fracture angles performed on PMMA and Tungsten Carbide specimens are used to depict the most suitable fracture locus and thus to distinguish between the various fracture criteria when extended to fracture under compressive loading. An exact expression for the threshold load for complete closure of 2D elliptical cracks is used to delimit the fracture locus.     

B.E. Analysis of a cracked cylindrical rod twisted by rotation of a flat annular stamp

Conclusions The use of suitable Green functions, in the BEM for axisymmetric bodies with cracks, yields accurate evaluation of the stress solution near the crack boundary. The singular character of the contact shearing stress between the stamp and the elastic medium influences the Mode III stress intensity factor for internal cracks more than for the external cracks. The validity of the numerical method has been verified through some characteristic examples. The accuracy of the results for K _IIIis displayed for internal and external cracks. Their interaction effects with the singularity due to the stamp are reflected as the geometric parameters are varied.Dedicated to G. Rieder     

On the problem of two coplanar cracks inside an infinite isotropic elastic solid

A method is proposed for the approximate evaluation of normal displacements and normal stresses on the plane of two coplanar cracks located inside an infinite isotropic elastic solid and subjected to normal internal pressure. The formulation results in a single integral equation for the unknown normal stresses on the plane of the cracks. Numerical results are given for the stress intensity factor K_I of two coplanar circular cracks and two coplanar elliptical cracks opened up under a uniform internal pressure.     

Sub-region mixed finite element analysis of V-notched plates

In this paper the eigenproblem of elastic plates with V-notches is studied in terms of the complex potentials of elasticity. The variation of the eigenvalues as functions of notch angle is discussed. The phenomenon of bifurcation in the curves of higher-order eigenvalues is discovered and the concept of critical angle is proposed. Furthermore, a singular stress element, according to the stress field around notch-tips, is developed to account for notch-tip singularity. Moreover, conventional regular displacement elements are used outside the singular stress element, and then the basic finite element equations can be established based on the sub-region mixed energy principle. In two numerical examples, the stress intensity factors K _I and K _II of the notched specimens with various opening angles are evaluated, satisfactory accuracy can be obtained with very coarse meshes.     

Long fatigue life critical crack lengths*

A model based on surface strain redistribution and crack closure is presented for prediction of the endurance or fatigue limit stress by determining the threshold stress and critical length of short cracks that develop under microstructural control. The threshold stress first decreases with crack size to a local minimum then increases to a local maximum corresponding to the fatigue limit stress. This occurs at the critical crack length corresponding to about four grain diameters. The model is capable of determining the threshold stress range and depth of propagating and non‐propagating surface cracks as a function of stress ratio, material and grain size. The microstructure is shown to be particularly significant in the very long life regime (N_f ≈ 10⁹ cycles). When the surface cracks become non‐propagating, internally initiated cracks continue growing slowly, eventually reaching the critical crack length with failure occurring after a very high number of cycles (10⁷ < N_f < 10⁹ cycles).     

Shear mode threshold for a small fatigue crack in a bearing steel

The fatigue behaviour of small, semi‐elliptical surface cracks in a bearing steel was investigated under cyclic shear‐mode loading in ambient air. Fully reversed torsion was combined with a static axial compressive stress to obtain a stable shear‐mode crack growth in the longitudinal direction of cylindrical specimens. Non‐propagating cracks less than 1 mm in size were obtained (i) by decreasing the stress amplitude in tests using notched specimens and (ii) by using smooth specimens in constant stress amplitude tests. The threshold stress intensity factor ranges, ΔK_IIth and ΔK_IIIth, were estimated from the shape and dimensions of non‐propagating cracks. Wear on the crack faces was inferred by debris and also by changes in microstructure in the wake of crack tip. These effects resulted in a significant increase in the threshold value. The threshold value decreased with a decrease in crack size. No significant difference was observed between the values of ΔK_IIth and ΔK_IIIth.     

Investigation of fracture mechanism of 6063 aluminum alloy under different stress states

Fracture mechanisms in a 6063 aluminum alloy were investigated and analyzed carefully by in-situ tensile tests in SEM with a vacuum chamber. Specimens used were designed to produce different stress states. Studies indicated that with stress triaxiality (σ _m/σ _e) decreasing, the fracture modes changed from normal fracture to shear fracture and the fracture surfaces changed from the dimples and intragranular dominated fracture mode to the shear dominated fracture mode. The grain boundaries of the 6063 aluminum alloy were the weakest positions. In the case of high stress triaxiality, the grain boundary cracks were produced by normal stress or by the incompatibility of deformation between neighboring grains, and the normal stress dominated the crack propagation. In the case of low stress triaxiality, the boundary cracks were produced by the relative slipping of grains against neighboring grains, and the shear stress dominated the crack propagation. The final fracture of the specimens occurred by connections of cracks through transgranular cracking of the ligaments among these cracks.     

Estimation of Ct for weld cracks including HAZ softening region

C_t is an important parameter in predicting creep crack growth life from the small-scale creep stage to the extensive steady state creep stage. In this paper, weld interface and non-interface creep cracks for compact tension (CT) specimens with heat-affected zone (HAZ) were studied by finite element method (FEM) to systematically research the estimation method of C_t for weld creep cracks. Simulation results showed that C_t of weld interface cracks could be estimated by Yoon’s method. For weld non-interface cracks, C_t under the small-scale and transition creep stages could be exactly estimated by a proposed method. The accuracy of the proposed method was not affected by HAZ, but Yoon’s method was slightly affected by HAZ.     

Stress intensity factors for a semiinfinite crack with branches

The stressed state of an elastic plane weakened by a semiinfinite branching crack whose branches are either shear cracks or cracks of a mixed type is determined by the method of singular integral equations for the case where the stressed state at the tip of the semiinfinite crack without branches is characterized by stress intensity factorsK ₁ ⁰ andK _II ⁰ . The values of these factors are obtained for the cases of one, two, and three branches and different values of geometric parameters. Karpenko Physicomechanical Institute, Ukrainian Academy of Sciences, L'viv. Translated from Fizyko-Khimichna Mekhanika Materialiv, Vol. 33, No. 1, pp. 45–50, January–February, 1997.