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.     

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.     

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.     

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.     

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.     

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.     

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.     

Calculation of K-factor and T-stress for cracks in anisotropic bimaterials

The problem of a crack in a thin layer terminating perpendicular to a layer/substrate interface is analyzed for a general case of elastic anisotropy. The crack is modelled by means of continuous distribution of dislocations, which is assumed to be singular at the crack tip. A system of simultaneous functional equations is obtained that enables to find the singularity exponent λ. The reciprocal theorem (ψ-integral) is used to compute the generalized stress intensity factor (GSIF) through the remote stress and displacement field for a particular specimen geometry and boundary conditions using FEM. The results obtained are compared with the evaluation of GSIF based upon the dislocation arrays technique. Existing semi-analytical solution for singularities in anisotropic trimaterials is applied and its validity for the specimen investigated is checked by FEM. The evaluation of T-stress using the dislocation arrays technique is performed.     

Evaluation of power-logarithmic singularities, T-stresses and higher order terms of in-plane singular stress fields at cracks and multi-material corners

The scaled boundary finite-element method is extended to analyze the in-plane singular stress fields at cracks and multi-material corners. A complete singular stress field is represented semi-analytically as a series of matrix power functions of the radial coordinate originating from the singular point. This method is capable of directly evaluating orders of singularity, stress intensity factors, T-stresses, higher order terms, angular distributions of stresses for each term and, if present, power-logarithmic singularities. In this method, the singular functions are represented analytically and are not evaluated close to the singular point. Numerical examples are calculated to demonstrate the simplicity and to evaluate the accuracy of the scaled boundary finite-element method.     

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.     

Evaluation of dynamic stress intensity factors and T-stress using the scaled boundary finite-element method

A technique to evaluate the dynamic stress intensity factors and T-stress is developed by extending the scaled boundary finite-element method. Only the boundary of the problem domain is discretized. The inertial effect at high frequencies is modeled by a continued fraction solution of the dynamic stiffness matrix without introducing an internal mesh. Standard time-stepping scheme is applied to perform response history analyses directly in the time domain. The internal displacement field is obtained by numerical integration after removing the stress singularity. The dynamic stress intensity factors and the T-stress are evaluated directly based on their definitions. No asymptotic solution around the singular point is required. Numerical examples of cracks in homogeneous and bi-material plates demonstrate the simplicity and accuracy of this technique.     

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.     

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.     

A non-singular boundary integral formula for determining the T-stress for cracks of arbitrary geometry

A simple, yet accurate 2-D boundary integral equation (BIE) for determining the T-stress for cracks of arbitrarily geometry is introduced in this paper. The formulation is based upon the asymptotic expansion for the stress field in the vicinity of a crack tip. It can be conveniently implemented in the post-processing stage of a boundary element fracture analysis. As demonstrated in this work, the proposed BIE is non-singular, and thus it can directly be collocated at the crack tip under consideration. The technique requires a similar computational effort as that used in calculating the stress components at an interior point of a domain. Consequently, this new approach is very computationally effective and accurate for evaluating the elastic T-stress. Five test examples, involving straight, kink and curved cracks, are studied to validate the proposed technique and to assess its accuracy.     

Stress intensity factor K and the elastic T-stress for corner cracks

The stress intensity factor K and the elastic T-stress for corner cracks have been determined using domain integral and interaction integral techniques. Both quarter-circular and tunnelled corner cracks have been considered. The results show that the stress intensity factor K maintains a minimum value at the mid-plane where the T-stress reaches its maximum, though negative, value in all cases. For quarter-circular corner cracks, the K solution agrees very well with Pickard's (1986) solution. Rapid loss of crack-front constraint near the free surfaces seems to be more evident as the crack grows deeper, although variation of the T-stress at the mid-plane remains small. Both K and T solutions are very sensitive to the crack front shape and crack tunnelling can substantially modify the K and T solutions. Values of the stress intensity factor K are raised along the crack front due to crack tunnelling, particularly for deep cracks. On the other hand, the difference in the T-stress near the free surfaces and at the mid-plane increases significantly with the increase of crack tunnelling. These results seem to be able to explain the well-observed experimental phenomena, such as the discrepancies of fatigue crack growth rate between CN (corner notch) and CT (compact tension) test pieces, and crack tunnelling in CN specimens under predominantly sustained load.     

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.     

Elastic J-integral solution for an elbow with axial through-wall crack at crown under in-plane bending

Piping elbows under bending moment are vulnerable to cracking at crown. The structural integrity assessment requires knowledge of the J-integral. The J-integral values for axially through-wall cracked thick elbows are not available in the open literature over a certain range. This paper presents a closed form expression for elastic J-integral for 90°, long radius elbows subjected to bending moment. The expression is derived, based on the results of a large number of finite element analyses covering a wide range of standard geometries. The analyses were performed using WARP3D software. The present study enables the estimation of the elastic J-integral over a range of R_m/t from 5 to 25 (R_m/t = 5, 6, 7.5, 9, 12, 15, 20 and 25) and thus extends the range of earlier solution towards the thicker elbows used in nuclear industry. The crack angles considered were 9°, 18°, 27° and 36°.     

J and CTOD estimation procedure for circumferential surface cracks in pipes under bending

This work provides an estimation procedure to determine the J-integral and CTOD for pipes with circumferential surface cracks subjected to bending load for a wide range of crack geometries and material (hardening) based upon fully-plastic solutions. A summary of the methodology upon which J and CTOD are derived sets the necessary framework to determine nondimensional functions h₁ and h₂ applicable to a wide range of crack geometries and material properties characteristic of structural, pressure vessel and pipeline steels. The extensive nonlinear, 3-D numerical analyses provide a definite full set of solutions for J and CTOD which enters directly into fitness-for-service (FFS) analyses and defect assessment procedures of cracked pipes and cylinders subjected to bending load.