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.     

Two-parameter characterization of elastic-plastic crack front fields: Surface cracked plates under tensile loading

In this paper the J-Q two-parameter characterization of elastic-plastic crack front fields is examined for surface cracked plates under uniaxial and biaxial tensile loadings. Extensive three-dimensional elastic-plastic finite element analyses were performed for semi-elliptical surface cracks in a finite thickness plate, under remote uniaxial and biaxial tension loading conditions. Surface cracks with aspect ratios a/c = 0.2, 1.0 and relative depths a/t = 0.2, 0.6 were investigated. The loading levels cover from small-scale to large-scale yielding. In topological planes perpendicular to the crack fronts, the crack stress fields were obtained. In order to facilitate the determination of Q-factors, modified boundary layer analyses were also conducted. The J-Q two-parameter approach was then used in characterizing the elastic-plastic crack front stress fields along these 3D crack fronts. Complete distributions of the J-integral and Q-factors for a wide range of loading conditions were obtained. It is found that the J-Q characterization provides good estimate for the constraint loss for crack front stress fields. It is also shown that for medium load levels, reasonable agreements are achieved between the T-stress based Q-factors and the Q-factors obtained from finite element analysis. These results are suitable for elastic-plastic fracture mechanics analysis of surface cracked plates.     

Experimental T33-stress formulation of test specimen thickness effect on fracture toughness in the transition temperature region

This paper describes a study of the test specimen thickness effect on fracture toughness of a material, in the transition temperature region, for CT specimens. In addition we studied the specimen thickness effect on the T₃₃-stress (the out-of-plane non-singular term in the series of elastic crack-tip stress fields), expecting that T₃₃-stress affected the crack-tip triaxiality and thus constraint in the out-of-plane direction. Finally, an experimental expression for the thickness effect on the fracture toughness using T₃₃-stress is proposed for 0.55% carbon steel S55C. In addition to the fact that T₃₃ (which was negative) seemed to show an upper bound for large B/W, these results indicate the possibility of improving the existing methods for correlating fracture toughness obtained by test specimen with the toughness of actual cracks found in the structure, using T₃₃-stress.     

The T-stress solutions for through-wall circumferential cracks in cylinders subjected to general loading conditions

The elastic T-stress is a parameter used to define the level of constraint at a crack tip. It is important to provide T-stress solutions for practical geometries to apply the constraint-based fracture mechanics methodology. In the present work, T-stress solutions are provided for circumferential through-wall cracks in thin-walled cylinders. First, cylinders with a circumferential through-wall crack were analyzed using the finite element method. Three cylinder geometries were considered; defined by the mean radius of the cylinder (R) to wall thickness (t) ratios: R/t = 5, 10, and 20. The T-stress was obtained at eight crack lengths (θ/π = 0.0625, 0.1250, 0.1875, 0.2500, 0.3125, 0.3750, 0.4375, and 0.5000, θ is the crack half angle). Both crack face loading and remote loading conditions were considered including constant, linear, parabolic and cubic crack face pressures and remote tension and bending. The results for constant and linear crack face pressure were used to derive weight functions for T-stress for the corresponding cracked geometries. The weight functions were validated against several linear and non-linear stress distributions. The derived weight functions are suitable for T-stress calculations for circumferential cracks in cylinders under complex stress fields.     

Elastic T-stress solutions for flat elliptical cracks under tension and bending

Exact solutions for elastic T-stress of a flat elliptical crack in an infinite body under tension and bending are obtained in this paper. Many papers have been devoted to the problems for elliptical cracks in an elastic medium, but all their attention has been concentrated on the determination of stress intensity factors. In the current paper, elastic T-stress solutions are derived by means of the potential method and a specific collection of harmonic functions. The formulas of the elastic T-stress for a penny-shaped crack [Wang X. Elastic T-stress solutions for penny-shaped cracks under tension and bending. Engng Fract Mech 2004;71:2283-98] follow from the present results as a special case. It is obtained that under tension loading, the elastic T-stress is always compressive along the elliptical crack front. In both tension and bending cases, T-stress essentially depends on the Poisson’s ratio of the material, a parametric angle and semi-axes of the ellipse.     

The effective T-stress estimation and crack paths emanating from U-notches

The concept of the T-stress as a local constraint factor has been extended to U-notch tip stress distribution as the effective T-stress. The effective T-stress has been estimated as the average value of the T-stress in the region corresponding to the effective (characteristic) distance ahead of the notch tip. The T-stress is evaluated by finite element method using the experimental load for crack initiation and computing the difference between principal stresses along ligament. A large range of critical effective T-stress values is investigated for different specimen configurations and notch aspect ratios. Crack stabilisation and crack bifurcation for fracture emanating from notches according to the critical effective T-stress is discussed. A model involving the influence of the critical effective T-stress on void growth for ductile failure in the vicinity of the notch tip has been proposed.     

Elastic T-stress solutions for semi-elliptical surface cracks in finite thickness plates

Three-dimensional finite element analyses have been conducted to calculate the elastic T-stress for semi-elliptical surface cracks in finite thickness plates. Far-field tension and bending loads were considered. The analysis procedures and results were verified using both exact solutions and approximate solutions. The T-stress solutions are presented along the crack front for cracks with a/t values of 0.2, 0.4, 0.6 or 0.8 and a/c values of 0.2, 0.4, 0.6 or 1.0. Based on the present finite element calculations for T-stress, empirical equations for the T-stress at three locations: the deepest, the surface and the middle points of the crack front under tension or bending are presented. The numerical results are approximated by empirical formulae fitted with an accuracy of 1% or better. They are valid for 0.2?a/c?1 and 0?a/t?0.8. These T-stress results together with the corresponding K or J values for surface cracks are suitable for the analysis of constraint effects for surface cracked components.     

Elastic T-stress for cracks in test specimens subjected to non-uniform stress distributions

Finite element analyses have been conducted to calculate elastic T-stress solutions for cracked test specimens. The T-stress solutions are presented for single edge cracked plates, double edge crack plates and centre cracked plates. Uniform, linear, parabolic or cubic stress distributions were applied to the crack face. The results for uniform and linear stress distributions were used to derive weight functions for T-stress for the corresponding specimens. The weight functions for T-stress are then verified against several linear and non-linear stress distributions. The derived weight functions are suitable for the T-stress calculation for cracked specimens under any given stress field.     

T-stress of cracks loaded by near-tip tractions

T-stress solutions were derived for tractions acting on the crack-faces near a crack tip. Such solutions are of interest for the determination of the leading term of a weight function representation of T-stresses and the computation of an “intrinsic” T-stress for cracks growing in a material with a rising crack growth resistance. First, the type of the Green’s function for T-stresses is theoretically established. Then, results of finite element computations are reported for edge-cracked bars, DCB and CT specimens, which are suited for the determination of the first series term. As an application of the Green’s functions, the T-stresses caused by bridging interactions very close to the crack tip are computed.     

T-Stress solutions for a semi-elliptical axial surface crack in a cylinder subjected to mode-I non-uniform stress distributions

This paper presents the T-stress solutions (T₁₁ and T₃₃) for semi-elliptical axial surface cracks in a cylinder subjected to mode-I non-uniform stress on the crack surface. Two cylindrical geometries with inner radius (R_i) to wall thickness (t) ratios R_i/t = 5 and 10 were considered. The T-stresses were applied along the crack front for normalized crack depth values a/t of 0.2, 0.4 and 0.5 and aspect ratios a/c of 0.2, 0.4, 0.6 and 1.0. Three stress distribution; uniform, linear and parabolic were applied to the crack face. In addition to these solutions, concrete formulation of the superposition principle is given for the T₃₃-stress, which is known as an elastic parameter that describes the out-of-plane crack tip constraint effect. Then, the validity of the formulation was shown through application of our T-stress solutions to the problem of an axial semi-elliptical surface crack in a cylinder subjected to internal pressure, and checking that the principle of superposition holds for the problem.     

An experimental investigation of the effect of biaxial loading on the master curve transition temperature in RPV steels

During the 1990s considerable work was conducted to characterize the effect of biaxial loading on the ductile to brittle transition temperature. The work centered on a series of tests using large cruciform bend specimens from an experimental A533B test plate denoted as HSST Plate 14 (Heavy Section Steel Technology Plate 14). Recently a series of similar biaxial cruciform tests has been conducted on the steel used for an extensive European Round Robin that investigated the ductile-to-brittle transition master curve and associated T₀ reference temperature. The results of these tests have been used to promote the concept of a “Biaxial Effect” which corresponds to a shift in the shallow crack transition master curve of +20 °C or more when biaxial stresses are present, in comparison with the master curve for uniaxially loaded shallow crack specimens. A comprehensive analysis of the all of the available HSST Plate 14 data and data from two other structural steels was performed to investigate the extent of a biaxial effect on the reference temperature, T₀. The analysis included many additional biaxial cruciform test results on three different materials. The results of all three materials discussed in this paper fail to clearly demonstrate that biaxial loading, imposed through the use of a cruciform specimen geometry, has an effect on the fracture toughness, characterized using a master curve approach and reference temperature T₀. The analysis utilized in this paper assumes that the toughness distribution and temperature dependence of shallow cracked specimens can be modeled by using the master curve approach. This assumption has not been rigorously validated and would benefit from further study. Additional detailed stress analysis of the constraint evolution in the cruciform specimens may better define the precise conditions under which a biaxial effect on the fracture toughness could be realized.     

Crack-tip constraint in mode II deformation

Studies of cracked specimens loaded in mode I have shown that the stresses near the crack tip depend significantly on the level of constraint. The stresses can be determined near the crack tip using the HRR solution, but only for high constraint specimens. For other levels of constraint, O'Dowd and Shih's Q parameter may be used to adjust the stresses derived from the HRR solution. Only limited research has been carried out to study the effect of constraint in mode II. In this paper a mode II boundary layer formulation is used to study the effect of far field elastic stresses on the size and shape of the plastic zone around the crack tip and on the stresses inside the plastic zone. It is shown that in mode II, both positive and negative values of remote T-stress influence the tangential stress along the direction of maximum tangential stress. In the spirit of O'Dowd and Shih, a dimensionless parameter Q _II is introduced to quantify the constraint for mode II specimens failing by brittle fracture. The relation between Q _II and T/₀ is determined for different values of the strain hardening coefficient n. To investigate the range of validity of the Q–T diagram for real specimens, the constraint parameter Q _II is calculated directly from finite element analysis for three mode II specimens and compared with the evaluation using the Q–T diagram.     

Two-parameter fracture assessment of surface cracked cylindrical shells during collapse

The present study addresses the use of CTOD and T-stress in fracture assessments of surface cracked shell structures. A new software is developed for this purpose, denoted LINK_pipe. It is based on a combination of a quadrilateral assumed natural deviatoric strain thin shell finite element and an improved linespring finite element. Plasticity is accounted for using stress resultants. A power law hardening model is used for shell and linespring materials. A co-rotational formulation is employed to represent nonlinear geometry effects. With this, one can carry out nonlinear fracture mechanics assessments in structures that show instabilities due buckling (local/global), ovalisation and large rigid body motion. Many constraint-measuring parameters have been proposed, with the Q-parameter or the T-stress being the most popular ones. Solid finite element meshing for complex structures such as pipes containing semi-elliptical surface cracks in order to compute Q is at present not a feasible approach. However, shell structures are most conveniently meshed with shell finite elements, and the linespring finite element is a natural way of accounting for surface cracks. The T-stress is readily obtained from the linespring membrane force and bending moment along the surface crack. In this study we present a new approach to analyse cracked shell structures subjected to large geometric changes. By numerical examples it is shown how geometric instabilities and fracture compete as governing failure mode.     

Quasicontinuum simulation of crack propagation in bcc-Fe

We have investigated fracture in bcc-Fe through multiscale simulations. The quasicontinuum (QC) method with an embedded atom method (EAM) interatomic potential is applied. The analyses have been carried out assuming different crystallographic orientations and different T-stress under Mode I loading. Both anisotropic and isotropic formulations of the modified boundary layer (MBL) approach has here been investigated and compared. The results show that the mechanisms at the crack tip and the critical stress intensity factor K_Ic are sensitive to both the crystallographic orientation and whether or not the formulation of the boundary conditions are isotropic or anisotropic. Mechanisms such as cleavage crack propagation, twinning, and dislocation emission are observed in the analyses. A short literature review on atomistic and multiscale simulations of fracture in bcc-Fe has been performed and evaluated, and also compared with the current results.     

Quantifying crack tip displacement fields with DIC

Crack paths under both fatigue and fracture conditions are governed by the crack tip displacement field and the material deformation characteristics, including those influenced by metallurgical anisotropy. Experimental techniques such as thermoelasticity and photoelasticity have been successfully used to characterise the elastic stress fields around cracks but they do not take into account either plasticity or anisotropy. Considerable work has been carried out to characterise crack tip stress fields from displacement measurements. The current method of choice for obtaining displacement field data is digital image correlation (DIC) which has undergone significant advances in the recent years. The ease of use and capabilities of the technique for full field displacements has led to improved methods for characterising crack tip displacement fields based on data obtained from DIC. This paper gives an overview of some of the applications of DIC for crack tip characterisation such as K, T-stress and crack tip opening angle (CTOA) measurements as well as data obtained from 3D measurements of a propagating crack.     

On the full set of elastic T-stress terms of internal elliptical cracks under mixed-mode loading condition

Analytical expressions for the elastic constant stress terms of the asymptotic field, the so called T-stresses, for internal mixed-mode elliptical cracks in infinite homogeneous and isotropic elastic solids are addressed. To solve the problem the mixed-mode crack problem is divided into sub-problems using the superposition method, and each sub-problem is then solved for the asymptotic stress field. Considering the expansion of the local stress field at the crack front, the elastic T-stress terms are derived for each sub-problem. The results are superimposed to give the analytical expressions of the so far missing elastic T-stresses for mixed-mode elliptical cracks.The effect of the T-stresses on the size and shape of the plastic zone at the crack tip is discussed, and analytical results are compared to the ones from finite element analyses, both for the T-stress components and the size of the plastic zone. For an accurate prediction of the plastic zone all singular and constant terms (T-stresses) in the stress expansion formulae should be considered. It is observed that negative T-stresses increase the size of the plastic zone, while positive ones reduce it.     

The influence of the T-stress on the directional stability of cracks

The sign of the T-stress has been widely used for deciding whether directional stability prevails for straight cracks subjected to mode I loading under small scale yielding. However, there is little evidence for the reliability of such a criterion. On the contrary, it is shown that a local criterion is not applicable and that directional stability generally depends on body and load geometry as well as on material parameters, whereas the sign of the T-stress is totally irrelevant in most cases.     

Crack paths and the problem of global directional stability

Crack paths of original Griffith crack and edge crack under biaxial remote mode-I loading after different local disturbances are calculated by using integro-differential equations of first-order perturbation and numerical simulation with FEM respectively. Considering the asymptotic behaviour for large crack lengths the problem of global directional stability is reinvestigated extending the work by Melin. For the Griffith crack, correct power functions for the asymptotic path with one or both crack tips growing have been determined. The well- known critical stress biaxiality ratio R _c = 1 for the global directional stability has been obtained independently whether the crack is disturbed by local imperfections in geometry or in loading. For an edge crack the calculated critical stress biaxiality ratio for the global directional stability R _c  = 0.616, also irrespective of the local disturbances, corresponds to a positive T-stress and is considerably smaller than the value R  >  0.95 estimated by Melin (2002). In general, cracks need not propagate asymptotically in the direction perpendicular to the largest principal stress (without crack). This is found to be due to the effect of the boundaries. Considering the initial crack growth exclusively, it is shown that the solution for crack path prediction in series expansion form as derived by Cotterell and Rice (1980) for traction-free crack faces (after correction of a misprint) is exact in the two first terms in all cases. Thus, for small crack growth the Cotterell and Rice solution is universal with respect to all loading and geometrical situations.