The effect of T‐stress on crack‐tip plastic zones under mixed‐mode loading conditions

In this paper, the influence of T‐stress on crack‐tip plastic zones under mixed‐mode I and II loading conditions is examined. The crack‐tip stress field is defined in terms of the mixed‐mode stress intensity factors and the T‐stress using William's series expansion. The crack‐tip stress field is incorporated into the Von Mises yield criteria to develop an expression that determines the crack‐tip plastic zone. Using the resultant expression, the plastic zone is plotted for various combinations of mode II to mode I stress intensity factor ratios and levels of T‐stress. The properties of the plastic zone affected by T‐stress and mixed‐mode phase angle are discussed. The observations obtained on plastic zones variations are important for further fatigue and fracture analyses for defects in engineering structures under mixed‐mode loading conditions.     

A Three-dimensional Numerical Study of Mixed Mode (I and II) Crack Tip Fields in Elastic–plastic Solids

The stress fields near a crack front in a ductile solid are essentially three-dimensional (3D) in nature. The objective of this paper is to investigate the structure of these fields and to establish the validity of two-dimensional (2D) plane stress and plane strain approximations near the crack front under mixed mode (combined modes I and II) loading. To this end, detailed 3D and 2D small strain, elastic–plastic finite element simulations are carried out using a boundary layer (small scale yielding) formulation. The plastic zones and radial, angular and thickness variations of the stresses are studied corresponding to different levels of remote elastic mode mixity and applied load, as measured by the plastic zone size with respect to the plate thickness. The 3D results are compared with those obtained from 2D simulations and asymptotic solutions. It is found that, in general, plane stress conditions prevail at a distance from the crack front exceeding half the plate thickness, although it could be slightly smaller for mode II predominant loading. The implications of the 3D stress distribution on micro-void growth near the crack front are briefly discussed.     

In‐plane and out‐of‐plane constraint parameters along a three‐dimensional crack‐front stress field under creep loading

Full‐field three‐dimensional (3D) numerical analyses was performed to determine in‐plane and out‐of‐plane constraint effect on crack‐front stress fields under creep conditions of finite thickness boundary layer models and different specimen geometries. Several parameters are used to characterize constraint effects including the non‐singular T‐stresses, the local triaxiality parameter, the T_z ‐factor of the stress‐state in a 3D cracked body and the second‐order‐term amplitude factor. The constraint parameters are determined for centre‐cracked plate, three‐point bend specimen and compact tension specimen. Discrepancies in constraint parameter distribution on the line of crack extension and along crack front depending on the thickness of the specimens have been observed under different loading conditions of creeping power law hardening material for various configurations of specimens.     

State‐of‐the‐art review on extended stress intensity factor concepts

The stress intensity factor concept for describing the stress field at pointed crack or slit tips is well known from fracture mechanics. It has been substantially extended since Williams' basic contribution (1952) on stress fields at angular corners. One extension refers to pointed V‐notches with stress intensities depending on the notch opening angle. The loading‐mode‐related simple notch stress intensity factors K₁, K₂ and K₃ are introduced. Another extension refers to rounded notches with crack shape or V‐notch shape in two variants: parabolic, elliptic or hyperbolic notches (‘blunt notches’) on the one hand and root hole notches (‘keyholes’ when considering crack shapes) on the other hand. Here, the loading‐mode‐related generalised notch stress intensity factors K_1ρ, K_2ρ and K_3ρ are defined. The concepts of elastic stress intensity factor, notch stress intensity factor and generalised notch stress intensity factor are extended into the range of elastic–plastic (work‐hardening) or perfectly plastic notch tip or notch root behaviour. Here, the plastic notch stress intensity factors K_1p, K_2p and K_3p are of relevance. The elastic notch stress intensity factors are used to describe the fatigue strength of fillet‐welded attachment joints. The fracture toughness of brittle materials may also be evaluated on this basis. The plastic notch stress intensity factors characterise the stress and strain field at pointed V‐notch tips. A new version of the Neuber rule accounting for the influence of the notch opening angle is presented.     

Effects of non-singular stresses on crack-tip fields for pressure-sensitive materials,Part 1: Plane strain case

Mode I near-tip stress fields for elastic perfectly plastic pressure-sensitive materials under plane strain and small-scale yielding conditions are presented. A Coulomb-type yield criterion described by a linear combination of the effective stress and the hydrostatic stress is adopted in the analysis. The finite element computational results sampled at the distance of a few crack opening displacements from the tip show that, as the pressure sensitivity increases, the magnitudes of the normalized radial and hoop stress ahead of the tip decrease, the total angular span of the singular plastic sectors decreases, and the angular span of the elastic sectors bordering the crack surfaces increases. When non-singular T stresses are considered along the boundary layer of the small-scale yielding model, the near-tip stresses decrease as the T stress decreases. The plastic zone shifts toward the crack surfaces as the T stress increases. When the discontinuities of the radial stress and the out-of-plane normal stress along the border between the plastic sector and the elastic sector are allowed, the angular variations of the asymptotic crack-tip fields agree well with those of the finite element computations. Variation of the Q stresses for pressure-sensitive materials can be found from the asymptotic solutions when the plastic zone size ahead of the tip is relatively larger than the crack opening displacement. In addition the T stress is shown to have strong effects on the plastic zone sizes and shapes which could affect the toughening of pressure-sensitive materials.     

Three‐dimensional analyses of in‐plane and out‐of‐plane crack‐tip constraint characterization for fracture specimens

Three‐dimensional elastic–plastic finite element analyses have been conducted for 21 experimental specimens with different in‐plane and out‐of‐plane constraints in the literature. The distributions of five constraint parameters (namely T‐stress, Q, h, T_z and A_p) along crack fronts (specimen thickness) for the specimens were calculated. The capability and applicability of the parameters for characterizing in‐plane and out‐of‐plane crack‐tip constraints and establishing unified correlation with fracture toughness of a steel were investigated. The results show that the four constraint parameters (T‐stress, Q, h and T_z) based on crack‐tip stress fields are only sensitive to in‐plane or out‐of‐plane constraints. Therefore, the monotonic unified correlation curves with fracture toughness (toughness loci) cannot obtained by using them. The parameter A_p based on crack‐tip equivalent plastic strain is sensitive to both in‐plane and out‐of‐plane constraints, and may effectively characterize both of them. The monotonic unified correlation curves with fracture toughness can be obtained by using A_p. In structural integrity assessments, the correlation curves may be used in the failure assessment diagram (FAD) methodology for incorporating both in‐plane and out‐of‐plane constraint effects in structures for improving accuracy.     

An investigation of T‐stresses in a plate with an inclined crack under biaxial loadings

For plates with an inclined crack of wide‐range aspect ratios under biaxial loadings, T‐stress values are calculated with three‐dimensional finite element method. The results show that the normalized T‐stress is crack length and orientation dependent. A linear equation for the relationship between normalized T‐stresses and biaxility factors is proposed to describe the normalized T‐stresses for different crack lengths and crack angles under different biaxial loadings, which is more convenient and involves wider biaxility ratios compared with the existing solutions. The plate thickness effect and the trend of normalized T‐stresses along the crack front thickness are also studied for mode I and I–II mixed‐mode cracks. Based on the analyses and comparisons, it is necessary to take the thickness effect into consideration when the crack length is long enough (a/W = 7/10). When the component of mode II is significant (β > 45°), and the biaxility ratios are negative, T‐stresses near the free surface are lower than those at other positions, which are the opposite of mode I crack and most of I–II mixed‐mode crack.     

T-stress effects on steady crack growth in a thin, ductile plate under small-scale yielding conditions: Three-dimensional modeling

The non-singular T-stress provides a first-order estimate of geometry and loading mode, e.g. tension vs. bending, effects on elastic–plastic, crack-front fields under mode I conditions. The T-stress has a pronounced effect on measured crack growth resistance curves for ductile metals – trends most computational models confirm using a two-dimensional setting. This work examines T-stress effects on three-dimensional (3D), elastic–plastic fields surrounding a steadily advancing crack for a moderately hardening material in the framework of a 3D, small-scale yielding boundary-layer model. A flat, straight crack front advances at a constant quasi-static rate under near invariant local and global mode I loading. The boundary-layer model has thickness B that defines the only geometric length-scale. The material flow properties and (local) toughness combine to limit the in-plane plastic-zone size during steady growth to at most a few multiples of the thickness (conditions obtainable, for example, in large, thin aluminum components). The computational model requires no crack growth criterion; rather, the crack front extends steadily at constant values of the plane-stress displacements imposed on the remote boundary for the specified far-field stress intensity factor and T-stress. The specific numerical results presented demonstrate similarity scaling of the 3D near-front stresses in terms of two non-dimensional loading parameters. The analyses reveal a strong effect of T-stress on key stress and strain quantities for low loading levels and less effect for higher loading levels, where much of the plastic zone experiences plane-stress conditions. To understand the combined effects of T-stress on stresses and plastic strain levels, normalized values from a simple void-growth model, computed over the crack plane for low loading, clearly reveal the tendency for crack-front tunneling, shear-lip formation near the outside surfaces, and a minimum steady-state fracture toughness for T = 0 loading.     

Effects of an overload event on crack closure in 3-D small-scale yielding: finite element studies

This paper describes the effects of a single overload event, within otherwise constant amplitude cycles, on the plasticity‐induced closure process for mode I fatigue crack growth in the small‐scale yielding (SSY) regime. The 3‐D finite element (FE) analyses extend the initially straight, through‐thickness crack front by a fixed amount in each load cycle, using a node release procedure. Crack closure during reversed loading occurs when nodes behind the growing crack impinge on a frictionless, rigid plane. A bilinear, purely kinematic hardening model describes the constitutive response of the elastic–plastic material. Extensive crack growth in the analyses, both before and after the overload, allows the crack to grow out of the initial and the post‐overload transient phases, respectively. The work presented here shows that the large plastic deformation in the overload cycle reduces the crack driving force through enhanced closure. Further, the residual plastic deformations due to the overload cause a disconnected pattern of closure in the wake long after the crack front passes through the overload plastic zone. The computational studies demonstrate that the 3‐D scaling relationship for crack opening loads established in our earlier work for constant amplitude cycling (with and without a T‐stress) also holds before, during and after the overload event. For a specified ratio of overload‐to‐constant amplitude loading (R_OL=K^OL_max/K_max) , the normalized opening load (K_op/K_max) at each location along the crack front remains unchanged when the constant amplitude peak load (K_max) , thickness (B) and material flow stress (σ₀) all vary to maintain a fixed value of . The paper concludes with a comparison of the post‐overload response predicted by the 3‐D analyses and by the conventional Wheeler model.     

Variation of elastic T-stresses along slightly wavy 3D crack fronts

The non-singular terms in the series expansion of the elastic crack-tip stress field, commonly referred to as the elastic T-stresses, play an important role in fracture mechanics in areas such as the stability of a crack path and the two-parameter characterization of elastic-plastic crack-tip deformation. In this paper, a first order perturbation analysis is performed to study some basic properties of the T-stress variation along a slightly wavy 3D crack front. The analysis employs important properties of Bueckner-Rice 3D weight function fields. Using the Boussinesq-Papkovitch potential representation for the mode I weight function field, it is shown that, for coplanar cracks in an infinite isotropic and homogeneous linear elastic body, the mean T-stress along an arbitrary crack front is independent of the shape and size of the crack. Further, a universal relation is discovered between the mean T-stress and the stress field at the same crack front location under the same loading but in the absence of a crack. First-order-accurate solutions are given for the T-stress variation along a slightly wavy crack front with nearly circular or straight confifurations. Specifically, cosine wave functions are adopted to describe smooth polygonal and slightly undulating planar crack shapes. The results indicate that T ₁₁, the 2D T-stress component acting normal to the crack front, increases with the curvature of the crack front as it bows out but T ₃₃, acting parallel to the crack front, decreases with the crack front curvature.     

<Emphasis Type="Italic">T</Emphasis><Subscript><Emphasis Type="Italic">z</Emphasis></Subscript> constraints of semi-elliptical surface cracks in elastic plates subjected to uniform tension loading

The out-of-plane constraints T_z around the semi-elliptical surface cracks in an elastic plate subjected to uniform tension loading have been investigated through detailed three-dimensional (3D) finite element (FE) analyses. The distributions of T_z are obtained in the vicinity of the crack border with aspect ratios of 0.2, 0.4, 0.5, 0.6, 0.8 and 1.0. T_z drops from Poissons ratio at the crack tip to approximate zero beyond certain radial distance in the normal plane of the crack front line, and increases gradually from the free surface to the mid-plane at the same radial distance. By fitting the numerical results, empirical formulae are obtained to describe the 3D distributions of T_z for semi-elliptical surface cracks with a sufficient accuracy in the wide aspect ratio range of 0.2a/c 1.0 except very near the free surface, where T_z is extremely low. T_z, combining with the corresponding K and T or J and Q, can be applied to establish the three-parameter dominated stress field, which can characterize the 3D crack front field completely as an attempt.     

On improved crack tip plastic zone estimates based on T‐stress and on complete stress fields

Cracked ductile structures yield locally to form a plastic zone (pz) around their crack tips, which size and shape controls their structural behaviour. Classical pz estimates are based solely on stress intensity factors (SIF), but their precision is limited to very low σ_n/S_Y nominal stress to yield strength ratios. T‐stresses are frequently used to correct SIF‐based pz estimates, but both SIF and SIF plus T‐stress pz estimates are based on truncated linear elastic (LE) stress fields that do not satisfy boundary conditions. Using Griffith's plate complete LE stress field to avoid such truncated pz estimates, the influence of its Williams’ series terms on pz estimation is evaluated, showing that T‐stress improvements are limited to medium σ_n/S_Y values. Then, corrections are proposed to introduce equilibrium requirements into LE pz estimates. Finally, these improved estimates are compared with pz calculated numerically by an elastic–plastic finite element analysis.     

Low‐cycle fatigue/high‐cycle fatigue interactions in notched Ti‐6Al‐4V*

Combined low‐cycle fatigue/high‐cycle fatigue (LCF/HCF) loadings were investigated for smooth and circumferentially V‐notched cylindrical Ti–6Al–4V fatigue specimens. Smooth specimens were first cycled under LCF loading conditions for a fraction of the previously established fatigue life. The HCF 10⁷ cycle fatigue limit stress after LCF cycling was established using a step loading technique. Specimens with two notch sizes, both having elastic stress concentration factors of K_t = 2.7, were cycled under LCF loading conditions at a nominal stress ratio of R = 0.1. The subsequent 10⁶ cycle HCF fatigue limit stress at both R = 0.1 and 0.8 was determined. The combined loading LCF/HCF fatigue limit stresses for all specimens were compared to the baseline HCF fatigue limit stresses. After LCF cycling and prior to HCF cycling, the notched specimens were heat tinted, and final fracture surfaces examined for cracks formed during the initial LCF loading. Fatigue test results indicate that the LCF loading, applied for 75% of total LCF life for the smooth specimens and 25% for the notched specimens, resulted in only small reductions in the subsequent HCF fatigue limit stress. Under certain loading conditions, plasticity‐induced stress redistribution at the notch root during LCF cycling appears responsible for an observed increase in HCF fatigue limit stress, in terms of net section stress.     

J-Dominance in Mixed Mode Ductile Fracture Specimens

The asymptotic mixed mode crack tip fields in elastic-plastic solids are scaled by the J-integral and parameterized by a near-tip mixity parameter, M _{_p} . In this paper, the validity and range of dominance of these fields are investigated. To this end, small strain elastic-plastic finite element analyses of mixed mode fracture are first performed using a modified boundary layer formulation. Here, a two term expansion of the elastic crack tip field involving the stress intensity factor |K| the elastic mixity parameter M _{_e} as well as the T-stress is prescribed as remote boundary conditions. The analyses are conducted for different values of M _{_e} and the T-stress. Next, several commonly used mixed mode fracture specimens such as Compact Tension Shear (CTS), Four Point Bend (4PB), and modified Compact Tension specimen are considered. Here, the complete range of loading from contained yielding to large scale yielding is analyzed. Further, different crack to width ratios and strain hardening exponents are considered. The results obtained establish that the mixed mode asymptotic fields dominate over physically relevant length scales in the above geometries, except for predominantly mode I loading and under large scale yielding conditions. This revised version was published online in August 2006 with corrections to the Cover Date.     

The effect of T-stress on plane strain dynamic crack growth in elastic–plastic materials

In this paper, the effects of T‐stress on steady, dynamic crack growth in an elastic–plastic material are examined using a modified boundary layer formulation. The analyses are carried out under mode I, plane strain conditions by employing a special finite element procedure based on moving crack tip coordinates. The material is assumed to obey the J₂ flow theory of plasticity with isotropic power law hardening. The results show that the crack opening profile as well as the opening stress at a finite distance from the tip are strongly affected by the magnitude and sign of the T‐stress at any given crack speed. Further, it is found that the fracture toughness predicted by the analyses enhances significantly with negative T‐stress for both ductile and cleavage mode of crack growth.     

Elastic T stress estimates for circumferential surface‐cracked cylinders*

On the basis of detailed three‐dimensional (3D) elastic finite element (FE) analyses, this paper provides tractable approximations for elastic T stress solutions for circumferential inner‐surface cracks in cylinders. Internal pressure and global bending moment were considered. The FE model and analysis procedure employed in the analysis were verified using existing solutions for both elastic stress intensity factor and T stress. To cover a practical range, three different values of the ratio of the mean radius of cylinder to the thickness, R_m/t, were selected; furthermore, four different values of the ratio of the crack depth to the thickness, a/t, ranging from 0.1 to 0.75 and three different values of θ/π ranging from 0.1 to 0.4 were selected. On the basis of FE analyses results, polynomial approximations were proposed at three different locations: surface point, middle point and deepest point. On the basis of the detailed 3D elastic FE analysis, the solutions presented are believed to be the most accurate, and thus provide valuable information for structural integrity assessment considering a crack‐tip constraint.     

On the use of an anti‐symmetric four‐point bend specimen for mode II fracture experiments

The edge‐cracked beam specimen subjected to anti‐symmetric four‐point bend (ASFPB) loading has been conventionally used in the past for investigating the pure mode II fracture experiments in many engineering materials. However, it is shown through finite element analysis that the ASFPB specimen sometimes fails to produce pure mode II conditions. For anti‐symmetric loads applied close to the crack line, there are considerable effects from K_I and T‐stress in the ASFPB specimen. Pure mode II is provided only when the applied loads are sufficiently far from the crack plane.     

Characterization of crack tip stress fields in test specimens using mode mixity parameters

The aim of this study is to represent the combined effect of mode mixity, specimen geometry and relative crack length on the $T$ -stress, elastic–plastic stress fields, integration constant $I_{n}$ , angle of initial crack extension, and the plastic stress intensity factor. The analytical and numerical results are obtained for the complete range of mixed modes of loading between mode I and mode II. For comparison purposes, the reference fields for plane mixed-mode problems governing the asymptotic behavior of the stresses and strains at the crack tip are developed in a power law elastic–plastic material. For the common experimental fracture mechanics specimen geometries considered, the numerical constant of the plastic stress field $I_{n}$ and the $T$ -stress distributions are obtained as a function of the dimensionless crack length and mode mixity. A method is also suggested for calculating the plastic stress intensity factor for any mixed-mode I/II loading based on the $T$ -stress and power law solutions. It is further demonstrated that in both plane stress and the plane strain, the plastic stress intensity factor can be used to characterize the crack tip stress fields for a variety of specimen geometries and different mixed-mode loading. The applicability of the plastic stress intensity factor to analysis of the in-plane and out-of-plane constraint effect is also discussed.     

Mixed mode crack tip parameters for different wheel positions relative to a vertical crack at the rail foot

Finite element method is used to analyze a rail with a vertical bottom up crack at its foot, under the axle load and surface traction of a wheel. The possibility of crack formation at the foot of the rail in the neighborhood of a welding connection is discussed. A brief review on the importance of T‐stress in brittle fracture is presented. Seven cases with different locations of the crack relative to rail's sleeper contact region are considered. Numerous positions of the wheel are considered, and in each case, 3 crack parameters K_I, K_II, and T‐stress are calculated. Then, the biaxiality ratio and the mixity parameter for each loading and crack condition are calculated. It is shown that the location of crack and wheel can create mixed mode loading in the cracked rail and that the magnitude of crack tip parameters are strongly dependent on these geometric variables. In particular, the magnitudes of T‐stress and biaxiality ratio are significant in some cases. The effect of friction between the crack faces in the presence of compressive mode I loading on the mode II stress intensity factor is studied. Under mixed mode loading, due to the axle load and surface traction, the most critical condition is the formation of vertical cracks near the sleeper contact region.     

Transferability of the two‐parameter fracture criterion for 2219 aluminium alloy cracked configurations

Two‐dimensional elastic–plastic finite‐element fracture simulations with the critical crack‐tip‐opening‐angle fracture criterion were used to evaluate the two‐parameter fracture criterion (TPFC). Three different crack configurations under tension and bending loads made of thin‐sheet 2219‐T87 aluminium alloy were analysed. A very wide range of widths (w = 76 to 2440 mm) and initial crack‐length‐to‐width ratios (c_i/w = 0.05 to 0.95) were considered. A relation from the original TPFC was shown to fit the simulated fracture behaviour fairly well for the three different specimen types for net‐section stresses less than the yield stress (σ_y) of the material. Comparisons were also made on measured and simulated fracture tests on middle‐crack‐tension specimens. A relation between the elastic stress‐intensity factor, K_Ie, and net‐section stress, S_n, at failure was found to be linear for S_n < σ_y. The results demonstrated the transferability of the TPFC for different crack configurations for S_n < σ_y, but further study is needed for S_n > σ_y.