Material constraint parameters for the assessment of shallow defects in structural components. Part II: Constraint-based assessment of shallow defects

This Part II paper describes a series of constraint-based R6 assessments of shallow-cracked specimens under equibiaxial loading using material constraint parameters obtained from lookup tables presented in the companion Part I paper. Parameters are derived for an A533B-1 steel forging from a knowledge of the yield and flow behaviour of the material and a calibration of the Beremin model parameter m. Resulting assessments, which use both T- and Q-stress to quantify constraint, are found to be conservative with respect to the available experimental data. The results are also used to predict the influence of T-stress on the Master Curve reference temperature. Available data are broadly in agreement with the predicted trend curve. Finally, the results demonstrate that out-of-plane biaxial loading increases constraint to the extent that the inherent conservatism of the elastic T-stress at high L_r is eroded. Out-of-plane constraint effects are only properly accounted for by the hydrostatic Q-stress and for this reason, the use of T to assess biaxial loading situations should be undertaken with a degree of caution.     

Method of constraint loss correction of CTOD fracture toughness for fracture assessment of steel components

This paper presents a procedure for transferring the CTOD fracture toughness obtained from laboratory specimens to an equivalent CTOD for structural components, taking constraint loss into account. The Weibull stress criterion is applied to correct the CTOD for constraint loss, which leads to an equivalent CTOD ratio, β, defined as β = δ/δ_WP, where δ and δ_WP are CTODs of the standard fracture toughness specimen and the structural component, respectively, at the same level of the Weibull stress. The CTOD ratio β is intended to apply to the fracture assessment of ferritic steel components to stress levels beyond small-scale yielding. Nomographs are given to determine the β-value as a function of the crack type and size in the component, the yield-to-tensile ratio of the material and the Weibull shape parameter m. Examples of the fracture assessment using β are shown within the context of a failure assessment diagram (FAD). An excessive conservatism observed in the conventional procedure is reduced reasonably by applying the equivalent CTOD ratio, β.     

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.     

Correlation of fracture behavior in high pressure pipelines with axial flaws using constraint designed test specimens. Part II: 3-D effects on constraint

This study describes an extensive set of 3-D analyses conducted on conventional fracture specimens, including pin-loaded and clamped SE(T) specimens, and axially cracked pipes with varying crack configurations. The primary objective is to examine 3-D effects on the correlation of fracture behavior for the analyzed crack configurations using the J-Q methodology. An average measure of constraint over the crack front, as given by an average hydrostatic parameter, denoted Q_avg, is employed to replace the plane-strain measure of constraint, Q. Alternatively, a local measure of constraint evaluated at the mid-thickness region of the specimen, denoted Q_Z0, is also utilized. The analysis matrix considers 3-D numerical solutions for models of SE(T) fracture specimens with varying geometries (i.e., different crack depth to specimen width ratio, a/W, as well as different loading point distance, H) and test conditions (pin-loaded ends vs. clamped ends). The 3-D numerical models for the cracked pipes cover different crack depth to pipe wall thickness ratio, a/t, and a fixed crack depth to crack length ratio, a/c. The extensive 3-D numerical analyses presented here provide a representative set of solutions which provide further support for using constraint-designed SE(T) specimens in fracture assessments of pressurized pipes and cylindrical vessels.     

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.     

Influence of statistical and constraint loss size effects on cleavage fracture toughness in the transition - A model based analysis

A database derived from tests on specimens with a large range of ligament (b) and thickness (B) dimensions was systematically analyzed to evaluate constraint loss and statistical size effects on cleavage fracture toughness. The objectives were to: (1) decouple size effects related to constraint loss, mediated by b and B, from those arising from statistical effects, primarily associated with B; and, (2) develop procedures to transfer toughness data to different conditions of constraint and B. The toughness database for a Shoreham pressure vessel steel plate, tested at a common set of conditions, was described in a companion paper. Quantification of constraint loss was based on an independently calibrated 3D finite-element critical stress-area, σ^∗-[K_Jm/K_Jc], model. The measured toughness data, K_Jm, were first adjusted using computed [K_Jm/K_Jc] constraint loss factors to the corresponding values for small scale yielding conditions, K_Jc=K_Jm/[K_Jm/K_Jc]. The K_Jc were then statistically adjusted to a K_Jr for a reference B_r = 25.4 mm. The B adjustment was based on a critically stressed volume criterion, modified to account for a minimum toughness, K_min, consistent with modest modifications of the ASTM E 1921 Standard procedure. The combined σ^∗-[K_Jm/K_Jc]-K_min adjustment procedure was applied to the Shoreham b − B database, producing a homogeneous population of K_Jr data, generally within the expected scatter. The analysis suggests that: (1) there may be a maximum B beyond which statistical size effects diminish, and (2) constraint loss in the three-point bend specimens begins at a relatively low deformation level. A corresponding analysis, based on a Weibull stress, σ_w-[K_Jm/K_Jc]-K_min, adjustment procedure, yielded similar, but somewhat less satisfactory, results. The optimized adjustment procedure was also applied to other K_Jm data for the Shoreham plate from this study, as well as a large database taken from the literature. The population of 489K_Jr data points, covering an enormous range of specimen sizes, geometries and test temperatures, was found to be consistent with the same master curve T₀ = −84 °C derived from the b − B database. Thus, calibrated micromechanical models can be used to treat size and geometry effects on K_Jm, facilitating using small specimens and data transfer to predict the fracture limits of structures.     

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.     

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.     

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.     

Accounting for constraint effects in fracture mechanics analysis of floating production, storage and off-loading vessels and ships

Research has been performed to study the effect of constraint dependent fracture toughness of parent metal and HAZ for steels applied for fabrication of oil and gas floating production, storage and off-loading (FPSO) vessels and ships. A test method was employed to study the HAZ crack tip opening displacement (CTOD) fracture toughness at various levels of constraint avoiding excessive scatter usually associated with conventional HAZ CTOD testing, which may obscure effects of constraint.The objective of the research was to determine the material dependent constraint parameters in Ainsworth and O'Dowd's constraint modified fracture assessment approach and to develop a method based on information from the literature pertaining to the structural constraint in plates with semi-elliptical surface cracks. Results due to Wang and Parks and Nakamura and Parks computed using line-spring and FE analyses were used to establish polynomial expressions for the structural constraint in tension and bending applicable to fracture assessment of shallow fatigue cracks initiated at weld toes in floating offshore structures such as FPSOs.The results of the analyses clearly show that fracture assessment of semi-elliptical surface cracks in tension would be overly conservative if constraint effects are not accounted for in the Option 1 or Option 2 fracture assessment curves or in fracture toughness. It was also found from the tests conducted on specially prepared wide-plate test specimens, that the constraint modified Option 1 curve was conservative for fracture assessment of semi-elliptical surface cracks located in the HAZ of a 500 MPa minimum specified yield strength quenched and tempered steel for offshore application.     

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.     

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.     

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.     

Loading rate effects on parameters of the Weibull stress model for ferritic steels

This study investigates the effects of loading rate on parameters of the Weibull stress model for prediction of cleavage fracture in a low strength, strongly rate-sensitive A515-70 pressure vessel steel. Based on measured, dynamic fracture toughness data from deep- and shallow-cracked SE(B) specimens, the calibrated Weibull modulus (m) at shows little difference from the value calibrated previously using static toughness data. This newly obtained result supports the hypothesis in an earlier study [Gao X, Dodds RH, Tregoning RL, Joyce JA. Weibull stress model for cleavage fracture under high-rate loading. Fatigue Fract Engng Mater Struct 2001;24:551-64] that the Weibull modulus likely remains rate independent for this material over the range of low-to-moderate loading rates. Additional experimental and computational results for higher rates show that a constant m-value remains applicable up to the maximum loading rate imposed in the testing program . Rate dependencies of the scale parameter (σ_u) and the threshold parameter (σ_w-min) are computed using the calibrated m, and the results indicate that σ_u decreases and σ_w-min increases with higher loading rates. The predicted cumulative probability for cleavage fracture exhibits a strong sensitivity to small changes in σ_u. Consequently, σ_u must be calibrated using dynamic fracture toughness data at each loading rate of interest in an application or selected to make the Weibull stress model predict a dynamic master curve of macroscopic toughness for the material.     

A probabilistic model for the pearlite-induced cleavage of a plain carbon structural steel

The cleavage fracture of a carbon structural steel is shown to deviate from what is typical of a ferritic matrix. This occurs at temperatures ranging from −150 to 20 °C and affects the scatter of fracture toughness, and fractographic features. To explain the observed discrepancies a probabilistic model is developed by linking physical mechanisms of cleavage with continuum mechanics analysis applied to fracture. The model assumes that cleavage is nucleated and triggered in pearlite within the blunting zone of the crack tip by the mechanism of Miller and Smith. Once the model is calibrated for the steel, its predictions are shown to agree with the experimental results.     

Correlation of fracture behavior in high pressure pipelines with axial flaws using constraint designed test specimens--Part I: Plane-strain analyses

This work presents a numerical investigation of crack-tip constraint for SE(T) specimens and axially surface cracked pipes using plane-strain, nonlinear computations. The primary objective is to gain some understanding of the potential applicability of constraint designed fracture specimens in defect assessments of pressurized pipelines and cylindrical vessels. The present study builds upon the J-Q approach using plane-strain solutions to characterize effects of constraint on cleavage fracture behavior for the analyzed fracture specimens and cracked pipes. Under increased loading, each cracked configuration follows a characteristic J-Q trajectory which enables comparison of the corresponding driving force curve in the present context. A key outcome of this investigation is that toughness data measured using SE(T) specimens appear more applicable for cleavage fracture predictions of pressurized pipelines and cylindrical vessels than standard, deep notch fracture specimens under bend loading. The results provide a strong support for use of constraint-designed SE(T) specimens in fracture assessments of pressurized pipes and cylindrical vessels.     

Influence of T-stress, cohesive strength and yield strength on the competition between decohesion and plastic flow in a crack edge vicinity

The discussion is based on the cell model of materials. Each cell contains one dominant kernel of micro-separation, for instance a particle. A cell is either in a cohesive or a decohesive state, the latter implying instability at load control. The process region consists of cells which have reached the decohesive state. The cells are characterized by their linear size and their cohesion-decohesion relation. The process region develops either in an elastic or in a plastic environment. In the latter case, it may be more or less deeply embedded in the plastic region. In some cases there will be no process region, only plastic flow. The ratio between the cohesive strength and the yield strength is an important parameter for describing this competition between decohesion and plastic flow, but the T-stress also plays a part. The fracture toughness depends on the area under the cohesion-decohesion curve and on the embedment of the process region.     

Mode I DCB testing of composite laminates reinforced with z-direction pins: a simple model for the investigation of data reduction strategies

The applicability of standard composite mode I interlaminar toughness data reduction schemes is investigated for composites reinforced by inserting straight pins in the z-direction. A simple model of a z-pinned mode I test is used to generate load, displacement and crack length data. Data reduction schemes are applied to the model data and the resulting toughness is compared to the true toughness. These schemes are shown to be inaccurate for the initial stage of crack growth, during which the zone bridged by z-pins is developing and an improved scheme is proposed. Beyond this stage, the toughness is most accurately predicted if the ‘developing’ data is omitted.     

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.     

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.