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.     

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.     

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.     

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.     

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.     

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.     

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.     

Two-parameter fracture criterion (Kρ,c-Tef,c) based on notch fracture mechanics

A two-parameter fracture criterion has been proposed to predict fracture conditions of notched components. This criterion includes the critical notch stress intensity factor K _ρ,c , which represents fracture toughness of a material with a notch of radius ρ, and the effective T-stress. The effective T-stress T _ef has been estimated as the average value of the T-stress distribution in the region ahead of the notch tip at the effective distance X _ef . These parameters were derived from the volumetric method of notch fracture mechanics. The results of numerical T _ef,c -stress estimation are compared to the T _ef,c -stress results obtained from experimental analysis. The material failure curve or master curve K _ρ,c = f(T _ef,c ) has been established as a result of the notched specimen tests. A large T _ef,c range was covered from −0.80 σ _Y to +0.19 σ _Y using SENT, CT, RT (roman tile) and DCB specimens. It was shown that the notch fracture toughness is a linear decreasing function of the T _ef,c -stress. The use of the material failure curve to predict fracture conditions was demonstrated on gas pipes with the surface notch.     

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

Analytical expressions for all non-singular stress terms of the asymptotic crack tip field, the so-called T-stresses of internal mixed mode circular (penny shaped) 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 of internal mixed mode penny shaped 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 the singular terms and the 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.     

Influence of crack tip constraint on void growth in ductile FCC single crystals

Effect of constraint (stress triaxiality) on void growth near a notch tip in a FCC single crystal is investigated. Finite element simulations within the modified boundary layer framework are conducted using crystal plasticity constitutive equations and neglecting elastic anisotropy. Displacement boundary conditions based on mode I, elastic, two term K-T field are applied on the outer boundary of a large circular domain. A pre-nucleated void is considered ahead of a stationary notch tip. The interaction between the notch tip and the void is studied under different constraints (T-stress levels) and crystal orientations. It is found that negative T-stress retards the mechanisms of ductile fracture. However, the extent of retardation depends on the crystal orientation. Further, it is found that there exists a particular orientation which delays the ductile fracture processes and hence can potentially improve ductility. This optimal orientation depends on the constraint level.     

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.     

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.     

Dynamic fracture mechanics analysis of failure mode transitions along weakened interfaces in elastic solids

Dynamic fracture mechanics theory was employed to analyze the crack deflection behavior of dynamic mode-I cracks propagating towards inclined weak planes/interfaces in otherwise homogenous elastic solids. When the incident mode-I crack reached the weak interface, it kinked out of its original plane and continued to propagate along the weak interface. The dynamic stress intensity factors and the non-singular T-stresses of the incident cracks were fitted, and then dynamic fracture mechanics concepts were used to obtain the stress intensity factors of the kinked cracks as functions of kinking angles and crack tip speeds. The T-stress of the incident crack has a small positive value but the crack path was quite stable. In order to validate fracture mechanics predictions, the theoretical photoelasticity fringe patterns of the kinked cracks were compared with the recorded experimental fringes. Moreover, the mode mixity of the kinked crack was found to depend on the kinking angle and the crack tip speed. A weak interface will lead to a high mode-II component and a fast crack tip speed of the kinked mixed-mode crack.     

Perturbation method for the solution of a Zener–Stroh crack with a slightly curved configuration

This paper investigates the Zener–Stroh crack with curved configuration in plane elasticity. A singular integral equation is suggested to solve the problem. Formulae for evaluating the SIFs and T-stress at the crack tip are suggested. If the curve configuration is a product of a small parameter and a quadratic function, a perturbation method based on the singular integral equation is suggested. In the method, the singular integral equation can be expanded into a series with respect to the small parameter. Therefore, many singular integral equations can be separated from the same power order for the small parameter. These singular integral equations can be solved successively. The solution of the successive singular integral equations will provide results for stress intensity factors and T-stress at the crack tip. It is found that the behaviors for the solution of SIFs and T-stress in the Zener–Stroh crack and the Griffith crack are quite different. This can be seen from the presented comparison results.     

T-Stress in Elastic-plastic Crack-tip Fields

A ligament field ahead of a crack tip for Model-I plane strain problem in a power-law hardening material is developed. Based on this solution, a formula for the critical value of J-integral in terms of T-stress for an elastic-plastic material is derived. T-stress effects on fracture toughness are discussed.     

Maximum Average Tangential Stress Criterion for Prediction of the Crack Path

The concept of the average stress has been employed to propose the maximum average tangential stress (MATS) criterion for predicting the direction of fracture angle. This criterion states that a crack grows when the maximum average tangential stress in the fracture process zone ahead of the crack tip reaches its critical value and the crack growth direction coincides with the direction of the maximum average tangential stress along a constant radius around the crack tip. The tangential stress is described by the singular and non-singular (T-stress) terms in the Williams series solution. The predicted directions of fracture angle are consistent with the experimental data for the mixed mode I/II crack growth behavior of Guiting limestone.     

On crack path evolution rules

The models of crack growth in mixed mode conditions are reviewed for the plane and three-dimensional (3D) states of stress. Both critical load value and crack path or surface growth are predicted by different criteria in terms of elastic singular stress states and T-stress component. Monotonic and cyclic loading induced crack growth is considered. The energy and critical plane criteria based on local or non-local measure of stress and strain are most useful in developing predictive crack growth simulation. The finite critical distance from the crack tip should be specified to provide averaged or local stress and strain states. The application of MK-criterion of crack growth expressed in terms of volumetric and deviatoric strain energies is presented for several specific cases of monotonic and cyclic loading. The concepts of smooth and rough crack surface are discussed with application to 3D crack surface growth.     

Mode I fatigue crack growth under biaxial loading

This paper is centred on the role of the T-stress during mode I fatigue crack growth. The effect of a T-stress is studied through its effect on plastic blunting at crack tip. As a matter of fact, fatigue crack growth is characterized by the presence of striations on the fracture surface, which implies that the crack grows by a mechanism of plastic blunting and re-sharpening (Laird C. The influence of metallurgical structure on the mechanisms of fatigue crack propagation. In: Fatigue crack propagation, STP 415. Philadelphia: ASTM; 1967. p. 131–68 [8]). In the present study, plastic blunting at crack tip is a global variable ρ, which is calculated using the finite element method. ρ is defined as the average value of the permanent displacement of the crack faces over the whole K-dominance area. The presence of a T-stress modifies significantly the evolution of plastic deformation within the crack tip plastic zone as a consequence of plastic blunting at crack tip. A yield stress intensity factor K_Y is defined for the cracked structure, as the stress intensity factor for which plastic blunting at crack tip exceeds a given value. The variation of the yield stress intensity factor was studied as a function of the T-stress. It is found that the T-stress modifies significantly the yield point of the cracked structure and that the yield surface in a (T, K_I) plane is independent of the crack length. Finally, a yield criterion is proposed for the cracked structure. This criterion is an extent of the Von-Mises yield criterion to the problem of the cracked structure. The proposed criterion matches almost perfectly the results obtained from the FEM. The evolution of the yield surface of the cracked structure in a (T, K_I) plane was also studied for a few loading schemes. These results should develop a plasticity model for the cracked structure taking into account the effect of the T-stress.     

Influence of crack tip constraint on void growth in pressure sensitive plastic solids - I: 2D analysis

In this work, the effect of constraint on void growth near a notch tip under mode-I loading is investigated in materials exhibiting pressure sensitive yielding and plastic dilatancy. To this end, large deformation elastic-plastic finite element analyses are performed using a two-dimensional (2D) plane strain, modified boundary layer formulation by prescribing the elastic K − T field as remote boundary conditions. The analyses are conducted for different values of K (or J) and T-stress. The material is assumed to obey a finite strain, Extended Drucker-Prager yield condition. The roles of pressure sensitivity, plastic dilatancy and yield locus shape on the interaction between the notch and a nearby void are studied by examining the distribution of hydrostatic stress and plastic strain in the ligament connecting them as well as the growth of notch and the void. The results show that void growth with respect to J is enhanced due to pressure sensitivity, and more so when the plastic flow is non-dilatational. By contrast, irrespective of pressure sensitivity, loss of crack tip constraint can significantly retard void growth.     

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.