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.     

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 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.     

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.     

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°.     

Approximate J estimates for tension-loaded plates with semi-elliptical surface cracks

This paper provides a simplified engineering J estimation method for semi-elliptical surface cracked plates in tension, based on the reference stress approach. Note that the essential element of the reference stress approach is the plastic limit load in the definition of the reference stress. However, for surface cracks, the definition of the limit load is ambiguous (“local” or “global” limit load), and thus the most relevant limit load (and thus reference stress) for the J estimation should be determined. In the present work, such limit load solution is found by comparing reference stress based J results with those from extensive 3-D finite element (FE) analyses. Based on the present FE results, the global limit load solution proposed by Goodall for surface cracked plates in combined bending and tension was modified, in the case of tension loading only, to account for a weak dependence on w/c and was defined as the reference normalizing load. Validation of the proposed equation against FE J results based on actual experimental tensile data of a 304 stainless steel shows excellent agreements not only for the J values at the deepest point but also for those at an arbitrary point along the crack front, including at the surface point. Thus the present results provide a good engineering tool for elastic-plastic fracture analyses of surface cracked plates in tension.     

Finite element modeling of plasticity-induced crack closure with emphasis on geometry and mesh refinement effects

Two-dimensional, elastic-perfectly plastic finite element analyses of middle-crack tension (MT) and compact tension (CT) geometries were conducted to study fatigue crack closure and to calculate the crack-opening values under plane-strain and plane-stress conditions. The behaviors of the CT and MT geometries were compared. The loading was selected to give the same maximum stress intensity factor in both geometries, and thus approximately similar initial forward plastic zone sizes. Mesh refinement studies were performed on both geometries with various element types. For the CT geometry, negligible crack-opening loads under plane-strain conditions were observed. In contrast, for the MT specimen, the plane-strain crack-opening stresses were found to be significantly larger. This difference was shown to be a consequence of in-plane constraint. Under plane-stress conditions, it was found that the in-plane constraint has negligible effect, such that the opening values are approximately the same for both the CT and MT specimens.     

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.     

Three-dimensional stress state around quarter-elliptical corner cracks in elastic plates subjected to uniform tension loading

Detailed full-field three-dimensional (3D) finite element analyses have been conducted to study the out-of-plane stress constraint factor T_z around a quarter-elliptical corner crack embedded in an isotropic elastic plate subjected to uniform tension loading. The distributions of T_z are studied in the forward section (0° ? θ ? 90°) of the corner cracks with aspect ratios a/c of 0.2, 0.4, 0.5, 0.6, 0.8 and 1.0. In the normal plane of the crack front line, T_z drops radially from Poisson’s ratio at the crack tip to zero beyond certain radial distances. Strong 3D zones (T_z > 0) exist within a radial distance r/a of about 4.6-0.7 for a/c = 0.2-1.0 along the crack front, despite the stress-free boundary conditions far away. At the same radial distance along the crack front in the 3D zones, T_z increases from zero on one free surface to a peak value in the interior, and then decreases to zero on another free surface. The distributions of T_z near the corner points are also discussed. Empirical formulae describing the 3D distributions of T_z are obtained by fitting the numerical results, which prevail with a sufficient accuracy in the valid range of 0.2 ? a/c ? 1.0 and 0° ? θ ? 90° except very near the free surfaces where T_z is extremely low. Combined with the K-T solution, the transition of approximate plane-stress state near the surfaces to plane-strain state in the interior can be characterized more accurately.     

A finite element based interior collocation method for the computation of stress intensity factors and T-stresses

A new, simple and efficient method for simultaneous estimation of the mixed-mode stress intensity factors (SIFs) and T-stresses using finite element computations is proposed in this paper. The current work is based on the formation of overdetermined system of equations using the displacement components near the crack tip. The proposed method can be easily implemented in the existing finite element codes. The results obtained from the present investigation for plane stress problems are validated by comparing with the published results and found to be in very good agreement with them.     

Finite element analysis of plasticity-induced fatigue crack closure: an overview

Finite element analysis is perhaps the most commonly used numerical method to model plasticity-induced fatigue crack closure. The state-of-the-art is reviewed and a comprehensive overview is presented, summarizing issues which must be considered and emphasizing potential difficulties. These include mesh refinement level, crack advancement schemes, crack shape evolution, geometry effects, and crack opening value assessment techniques.     

Fully plastic J-integral solutions for surface cracked plates under biaxial loading

In this paper, surface cracked plates under biaxial tension are studied. Three-dimensional elastic-plastic finite element analyses have been carried out to calculate the J-integral for surface cracked plate for a wide range of geometry, biaxiality and material properties. Fully plastic J-integral solutions along the front of the surface cracks are presented for Ramberg-Osgood power law hardening material of n = 3, 5, 10 and 15. Geometries considered are a/c = 0.2, 1.0 and a/t = 0.2, 0.4, 0.6 and 0.8 and the biaxial ratios of 0, 0.5 and 1. Based on these results, the J-integral along the crack front for general elastic-plastic loading conditions can be estimated using the EPRI scheme. These solutions are suitable for fracture analyses for surface cracked plates under biaxial loading.     

J-T characterized stress fields of surface-cracked metallic liners bonded to a structural backing - I. Uniaxial tension

Surface crack-tip stress fields in a tensile loaded metallic liner bonded to a structural backing are developed using a two-parameter J-T characterization and elastic-plastic modified boundary layer (MBL) finite element solutions. The Ramberg-Osgood power law hardening material model with deformation plasticity theory is implemented for the metallic liner. In addition to an elastic plate backed surface crack liner model, elastic-plastic homogeneous surface crack models of various thicknesses were tested. The constraint effects that arise from the elastic backing on the thin metallic liner and the extent to which J-T two parameter solutions characterize the crack-tip fields are explored in detail. The increased elastic constraint imposed by the backing on the liner results in an enhanced range of validity of J-T characterization. The higher accuracy of MBL solutions in predicting the surface crack-tip fields in the bonded model is partially attributed to an increase in crack-tip triaxiality and a consequent increase in the effective liner thickness from a fracture standpoint. After isolating the effects of thickness, the constraint imposed by the continued elastic linearity of the backing significantly enhanced stress field characterization. In fact, J and T along with MBL solutions predicted stresses with remarkable accuracy for loads beyond full yielding. The effects of backing stiffness variation were also investigated and results indicate that the backing to liner modulus ratio does not significantly influence the crack tip constraint. Indeed, the most significant effect of the backing is its ability to impose an elastic constraint on the liner. Results from this study will facilitate the implementation of geometric limits in testing standards for surface cracked tension specimens bonded to a structural backing.     

Theoretical and numerical modelling of creep crack growth in a carbon-manganese steel

This paper presents an analytical and numerical study of time dependent crack growth at elevated temperatures. A triaxiality dependent damage model is used to represent the multiaxial creep ductility of the material and an analytical model to predict steady state crack growth in terms of the fracture parameter C^∗, designated the NSW-MOD model, is presented. This model is an enhancement of the earlier NSW model for creep crack growth as it accounts for the dependence of stress and strain on angular position around the crack tip. Elastic-creep and elastic-plastic-creep finite element analyses are performed for a cracked compact tension specimen and the crack propagation rate in the specimen is predicted. It is found that in general the NSW-MOD model gives an accurate estimate of the crack growth rate when compared to the finite element predictions and experimental data for a carbon-manganese steel. However, crack growth rates predicted from the finite element analysis at low values of C^∗ may be higher than those predicted by either the NSW or NSW-MOD model. This enhanced level of crack growth may be associated with the non-steady state conditions experienced at the crack tip.     

Mismatch limit loads and approximate <Emphasis Type="Italic">J</Emphasis> estimates for tensile plates with constant-depth surface cracks in the center of welds

The present work provides mismatch limit loads and approximate J estimates for tensile plates with constant-depth, part-through surface cracks in the center of the weld metal. Based on systematic three-dimensional FE limit analyses, effects of strength mismatch related variables on limit loads are firstly quantified by the strength mismatch ratio and one geometry-related parameter. Mismatch limit loads for part-through surface cracks are then correlated to those for two-dimensional, through-wall crack problems. Based on the proposed limit load solutions, the applicability of the reference stress based J estimates is also investigated. When the reference stress is defined by the mismatch limit load, predicted J values agree overall well with FE results.