A review of the CTOA/CTOD fracture criterion

The crack-tip-opening angle or displacement (CTOA/CTOD) fracture criterion is one of the oldest fracture criteria applied to fracture of metallic materials with cracks. During the past two decades, the use of elastic–plastic finite-element analyses to simulate fracture of laboratory specimens and structural components using the CTOA criterion has expanded rapidly. But the early applications were restricted to two-dimensional analyses, assuming either plane-stress or plane-strain behavior, which lead to generally non-constant values of CTOA, especially in the early stages of crack extension. Later, the non-constant CTOA values were traced to inappropriate state-of-stress (or constraint) assumptions in the crack-front region and severe crack tunneling in thin-sheet materials. More recently, the CTOA fracture criterion has been used with three-dimensional analyses to study constraint effects, crack tunneling, and the fracture process. The constant CTOA criterion (from crack initiation to failure) has been successfully applied to numerous structural applications, such as aircraft fuselages and pipelines. But why does the “constant CTOA” fracture criterion work so well? This paper reviews the results from several studies, discusses the issues of why CTOA works, and discusses its limitations.     

Constraint-corrected fracture failure criterion based on CTOD/CTOA

In engineering design, a difficulty has always existed in those standard laboratory tests that cannot accurately predict the behavior of large structures like pipelines due to the different constraint levels. At present, extensive work has been done to characterize the constraint effects on fracture toughness by introducing a second parameter, while the systematic research on constrained transformation is inadequate. To address this issue, the ductile fracture process of X65 SENB specimen is simulated through the finite-element method coupled with the Gurson–Tvergaard–Needelman model. The parameters crack tip opening displacement (CTOD) and crack tip opening angle (CTOA) are chosen to characterize the fracture behaviors. The effects of specimen thickness on fracture toughness based on CTOD/CTOA and constraints ahead of crack tips in SENB specimen are studied. The results indicate that the critical values of CTOD/CTOA decrease with the increase of specimen thickness, but the constraint parameters are opposite. Furthermore, it finds that there is a near linear relationship between critical values of CTOD/CTOA and the stress constraint ahead of the crack tip. Thus, a constraint-corrected fracture failure criterion based on CTOD/CTOA is proposed, which can be used for the prediction and simulation of stable tearing crack growth in specimens and structures, made of this steel with any thickness value.     

Comparison of the measured and predicted crack propagation behaviour of Zr–2.5Nb pressure tube material

The crack propagation behaviour of Zr–2.5Nb pressure tube material was studied through comparison of the measured and predicted behaviour. Three-dimensional finite element simulations of compact tension and burst specimens were performed using the crack tip opening angle (CTOA) as a fracture criterion to allow for crack propagation. To obtain reasonable agreement with the measured force, crack extension, displacement, and crack tunnelling behaviour from compact tension simulations, a non-constant CTOA profile was required. This CTOA profile was then used as the crack propagation criterion in simulations of burst specimens where it was again found that there was reasonable agreement obtained with experiment.     

Ductile tearing resistance of metal sheets

The concept of R-curves has been adopted to characterise stable crack extension and predict residual strength of thin-walled structures particularly in the aircraft industry. The present contribution uses results of FE simulations of crack extension in panels by the cohesive model to validate analytical procedures for determining J-integral values at large crack extension from measurable quantities, namely the force vs. displacement records. The numerically determined J-integral is taken as the benchmark for the outcome of the analytical formulas. The geometry dependence of J and CTOD based R-curves is investigated and alternative concepts like CTOA and dissipation rate at crack extension are discussed.     

Determination of the CTOA and J integral by fracture surface topography analysis

In order to investigate the causes of material fracture, a new method is proposed that uses only the fracture surfaces for determining the fracture parameters in terms of the Cract-tip opening angle (CTOA) and the J integral. This method is based on the principle of fracture-surface topography analysis (FRASTA). In FRASTA, the fracture surfaces are scanned by laser microscope and the elevation data is recorded. Based on this recorded elevation data, the J integral can be calculated. The J integral calculated by the new method agrees well with that calculated by the elastic compliance method. FRASTA allows easy determination of the crack opening deformation (CTOA and COA) and the variation in CTOA and COA through specimen thickness.     

A crack propagation criterion based on ΔCTOD measured with 2D‐digital image correlation technique

The fatigue cracks growth rate of a forged HSLA steel (AISI 4130) was investigated using thin single edge notch tensile specimen to simulate the crack development on a diesel train crankshafts. The effect of load ratio, R, was investigated at room temperature. Fatigue fracture surfaces were examined by scanning electron microscopy. An approach based on the crack tip opening displacement range (ΔCTOD) was proposed as fatigue crack propagation criterion. ΔCTOD measurements were carried out using 2D‐digital image correlation techniques. J‐integral values were estimated using ΔCTOD. Under test conditions investigated, it was found that the use of ΔCTOD as a fatigue crack growth driving force parameter is relevant and could describe the crack propagation behaviour, under different load ratio R.     

A study of fracture criteria for ductile materials

In order to establish a ductile fracture criterion, several potential fracture parameters were investigated by comparing numerical simulations of crack extension with available experimental data. Based on the comparison, a fracture criterion, the Global-Local Fracture Criterion (GLFC), was proposed. The J-integral is employed as a global parameter to characterize the initial stage of crack extension. In the subsequent steady state crack growth, the fracture criterion is switched to a local parameter characterizing the crack tip stress or strain. The accuracy of the proposed fracture criterion in predicting ductile fracture behavior was verified.     

J and CTOD estimation formulas for C(T) fracture specimens including effects of weld strength overmatch

This work focuses on an evaluation procedure to determine the elastic?Cplastic J-integral and Crack Tip Opening Displacement (CTOD) fracture toughness based upon the ??-method for C(T) fracture specimens made of homogeneous and welded steels. The primary objective of this investigation is to enlarge on previous developments of J and CTOD estimation procedures for this crack configuration while, at the same time, addressing effects of strength mismatch on the plastic ??-factors. The present analyses enable the introduction of a larger set of factors ?? for a wide range of crack sizes (as measured by the a/W-ratio) and material properties, including different levels of weld strength mismatch, applicable to pipeline and pressure vessel steels. Very detailed non-linear finite element analyses for plane-strain and 3-D models of C(T) fracture specimens with centerline-cracked welds provide the evolution of load with increased load-line and crack mouth opening displacement required for the estimation procedure. Overall, the present study, when taken together with previous investigations, provides a fairly extensive body of results to determine parameters J and CTOD for different materials using C(T) specimens with varying overmatch conditions.     

Crack tip opening angle measurement through a girth weld in an X100 steel pipeline section*

Crack tip opening angle (CTOA) is becoming one of the most accepted methods for characterizing fully plastic fracture. It provides a measure of the resistance to fracture for a material in cases where there is a large degree of stable‐tearing crack extension during the fracture process. Our current pipeline research uses the CTOA test as an alternative, or addition, to the CTOD (crack tip opening displacement) and the fracture energy characterization provided by the J‐integral approach. A test technique was developed for measurement of CTOA that uses a modified double cantilever beam (MDCB) specimen. A digital camera and image analysis software were used to record the progression of the crack tip and to estimate the CTOA. In this article, CTOA data on crack growth orientations perpendicular to pipeline girth welds are presented. The CTOA for X100 high strength bainitic gas pipeline steel is reported. Two different specimen gauge sections, 3 mm and 8 mm, were used and the effect of the specimen thickness on the CTOA is discussed. The results show a change in the CTOA as the crack grows into the heat affected zone (HAZ). A slight improvement in the fracture resistance is measured, and through the weld, a slight decrease in fracture resistance is observed.     

Crack-opening angle and dissipation-rate analysis of R-curves for side-grooved pieces of HY130 steel in bending

The behaviour of side-grooved deep-notch three-point bend test pieces of 20 mm thick HY130 steel has been studied for large amounts of crack growth in three different widths. Growth occurs at limit load and the conventional R-curves follow the pattern that wider pieces give lower R-curves. Analysis of this behaviour is made in terms of the crack-tip opening angle, (CTOA) and the energy dissipation rate, dW _dis/Bda, or D, from which a particular R-curve, J _dis, can be formed. After an initial transient regime of about 2 mm growth, a steady-state region develops in terms of both CTOA and D. The steady state CTOA reduces with increase of initial width. The energy rate, D, is split into areal and volumetric components, γ and ρ, and, with neglect of the elastic components, ρ is related to the steady-state CTOA. The cumulative dissipation defined by J _dis is compared to several conventional R-curves. It is concluded that the interpretation of steady-state crack growth in deep-notch three-point bend pieces can be expressed in terms of either CTOA or D, but that transference of data even from one size of a side-grooved piece to another, let alone to another configuration, cannot yet be made except on a lower bound basis.     

The Use of Crack-Tip Opening Displacement for Testing of the Hydrogen Embrittlement of High-Strength Steels

Stress-corrosion cracks are, as a rule, brittle and often encountered under stresses much lower than the yield strength. For this reason, the methods of linear elastic fracture mechanics (LEFM) can be used for the investigation of stress-corrosion cracking (SCC). However, in some cases, these methods are inapplicable, and it is necessary to use the methods of so-called elastoplastic fracture mechanics (EPFM). In the EPFM approach, the J-integral is the most commonly used parameter for correlating crack initiation and propagation but the crack-tip opening displacement (CTOD) and crack-tip opening angle (CTOA) prove to be promising alternatives, especially for thin-sheet materials. Since both these parameters are connected with the crack geometry and, hence, reflect the level of strain at the crack tip, they appear to be useful correlation parameters for the cases of SCC, where the level of strain in the vicinity of the crack tip and, in particular, the strain rate, are the determining variables of the process. The hydrogen embrittlement of a higher-strength structural steel and welded joints of a C-Mn steel is assessed by using the CTOA and CTOD methodologies. In constant-extension-rate tests (CERT), fatigue precracked specimens were loaded with various low strain rates and electrolytically charged with hydrogen. It was discovered that hydrogen embrittlement significantly affects the crack-growth resistance curves (R-curves) thus generated and the opening angle for which the crack propagates into the material.__________Published in Fizyko-Khimichna Mekhanika Materialiv, Vol. 40, No. 6, pp. 35–40, November–December, 2004.     

The crack tip opening angle (CTOA) of the plane stress moving crack

Recently, Crack Tip Opening Angle (CTOA) was proposed by C.F. Shih et al. to describe the instability criterion of ductile crack propagation during plane strain (flat crack) conditions, and was derived by J. R. Rice analytically by means of the slip line field theory and the incremental theory of plasticity. CTOA appears to be applicable in (some or most) cases, but does not accurately describe the plane stress growing crack (slant crack).Unstable ductile crack propagation of the plane stress crack is widely studied for the safe design of highly pressurized gas pipelines. The impact absorption energy of the Charpy test is well correlated to the fracture arresting properties of the structures, but the mechanics of the fracture are not yet well established.In this paper, CTOA of the plane stress growing crack is derived from the plane stress plasticity of perfectly plastic materials by Sokolovsky's approach. Our proposed modification of CTOA expressed as follows: $\text{CTOA}\text{= (α/δ}{}_0\text{)(}\text{d}\text{J/}\text{d}\text{l) + β(δ}{}_0\text{/E)}\text{ln}\text{(eR/r)}$ where $\text{β = 1.40}$ under the plane stress conditions.CTOA in the Dugdale model is also defined and compared with the results of laboratory test. The results show that $\text{α = 0.5}$, and $\text{β = 1.27}$ for plane stress crack growth. These analyses give similar results to those obtained by Rice et al. for CTOA under plane strain conditions, that is, $\text{α = 0.65}$ from the experimental results and $\text{β = 5.08}$ from the slip line theory.The CTOA obtained for plane stress ductile crack growth is applied to the wide plate tensile crack growth test. The results of the present analysis coincide well with those of the plane stress finite element method (FEM) computed by T. Kanazawa et al. The phenomena of plane stress ductile crack propagation are also explained by the CTOA criterion under plane stress conditions.     

Mechanisms and modeling of low cycle fatigue crack propagation in a pressure vessel steel Q345

The fatigue process near crack is governed by highly concentrated strain and stress in the crack tip region. Based on the theory of elastic–plastic fracture mechanics, we explore the cyclic J-integral as breakthrough point, an analytical model is presented in this paper to determine the CTOD for cracked component subjected to cyclic axial in-plane loading. A simple fracture mechanism based model for fatigue crack growth assumes a linear correlation between the cyclic crack tip opening displacement (ΔCTOD) and the crack growth rate (da/dN). In order to validate the model and to calibrate the model parameters, the low cycle fatigue crack propagation experiment was carried out for CT specimen made of Q345 steel. The effects of stress ratio and crack closure on fatigue crack growth were investigated by elastic–plastic finite element stress–strain analysis of a cracked component. A good comparison has been found between predictions and experimental results, which shows that the crack opening displacement is able to characterize the crack tip state at large scale yielding constant amplitude fatigue crack growth.     

THE EFFECTS OF CRACK DEPTH ON THE J-INTEGRAL AND CTOD FRACTURE TOUGHNESS FOR WELDED BEND SPECIMENS

This work deals with the influence of crack depth on the fracture toughness at initiation of crack growth and the constraint factor in relationship between the J-integral and the crack tip opening displacement (CTOD). A series of tests were performed on high strength low alloyed HT80 steel welds, and the critical J-integral and CTOD were determined using the load versus load point displacement record from three-point bend specimens with 0.05 < a/W < 0.5. It was found that the fracture toughness for shallow cracks at the onset of crack growth was larger than that for deep cracks for the steel welds tested, but it is felt that there is no fixed relationship between these values in the welds tested. The constraint factor is also a function of crack depth, and values of the factor increase from 0.5 to 1.5 when a/W increases from about 0.05 to 0.5. The factors are not very sensitive to the crack tip materials (HAZ or weld metal) in the welds tested.     

Residual strength analyses of stiffened and un-stiffened panels––Part I: laboratory specimens

This paper presents the results of residual strength analyses on stiffened and un-stiffened panels using the STructural Analysis of General Shells (STAGS) finite-element shell code and the critical crack-tip-opening angle (CTOA) fracture criterion. Previous analyses of wide, flat panels have shown that high-constraint conditions around a crack front must be modeled in order for the critical CTOA fracture criterion to predict wide panel failures from small laboratory tests. Thus, the STAGS code with the “plane-strain” core option was used in all analyses. In the present study, the critical CTOA (Ψ_c) value and the plane-strain core height were determined from a fit to the experimental load-against-crack-extension results from a series of middle-crack tension specimens (76–1016 mm wide) tested with anti-buckling guides. In the residual strength analyses of the 305-mm wide stiffened panels with a single crack, modeling of the sheet, stiffeners, rivet flexibility and buckling were based on methods and criteria, like that currently used in industry. STAGS and the CTOA criterion were used to predict load-against-crack extension for the single stiffened panels for both intact and cut stiffeners. Analyses were able to predict stable crack growth and residual strength of the single stiffened panels within about ±5% of the test failure loads.     

Prediction of mode I crack growth resistance based on a comparative investigation of J-integral and energy dissipation rate concept

An energy dissipation rate concept is employed in conjunction with the J-integral to calculate crack growth resistance of elastic-plastic fracture. Different from Rice’s J-integral, the free energy density is employed in place of the stress working density to define an energy-momentum tensor, which yields that the slightly changed J-integral is path dependent regardless of incremental plasticity and deformational plasticity. The J-integral over the remote contour is split into the plastic influence term and the J _FPZ-integral over the fracture process zone which is an appropriate estimate of the separation work of fracture. Finite element simulations are carried out to predict the plane strain mode I crack growth behavior by an embedded fracture process zone. It can be concluded that J-integral characterization is in essence a stress intensity-based fracture resistance similar to the K criterion of linear elastic fracture, and energy dissipation rate fracture resistance can be taken as an extension of the Griffith criterion to the elastic-plastic fracture.     

Effects of weld strength mismatch on <Emphasis Type="Italic">J</Emphasis> and CTOD estimation procedure for SE(B) specimens

This work examines the effect of weld strength mismatch on fracture toughness measurements defined by J and CTOD fracture parameters using single edge notch bend (SE(B)) specimens. A central objective of the present study is to enlarge on previous developments of J and CTOD estimation procedures for welded bend specimens based upon plastic eta factors (η) and plastic rotational factors (r _p ). Very detailed non-linear finite element analyses for plane-strain models of standard SE(B) fracture specimens with a notch located at the center of square groove welds and in the heat affected zone provide the evolution of load with increased crack mouth opening displacement required for the estimation procedure. One key result emerging from the analyses is that levels of weld strength mismatch within the range ±20% mismatch do not affect significantly J and CTOD estimation expressions applicable to homogeneous materials, particularly for deeply cracked fracture specimens with relatively large weld grooves. The present study provides additional understanding on the effect of weld strength mismatch on J and CTOD toughness measurements while, at the same time, adding a fairly extensive body of results to determine parameters J and CTOD for different materials using bend specimens with varying geometries and mismatch levels.     

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.     

The effect of specimen thickness on the experimental and finite element characterization of CTOD in extra deep drawn steel sheets

Recent experimental results by us have indicated that the load-drop technique can serve as a valid fracture criterion for predicting elastic-plastic fracture in extra deep drawn (EDD) steel sheets or in predominantly plane stress conditions. The purpose of this investigation is to examine the validity of aJ-integral as a fracture parameter and theJ-CTOD relation for the determination of critical CTOD in predominantly plane stress fracture (CTOD-crack tip opening displacement). Fracture tests were performed and experimental results were generated on fracture behaviour of EDD (0·06%C) steel sheets with CT specimens and using ‘load-drop’ as a fracture criterion. Critical CTOD was determined by using theJ-CTOD relation in addition to several existing techniques. A full 3-D finite element model was formulated to verify the critical load, critical CTOD and plastic-zone size. The critical CTOD was shown to increase with increasing specimen thickness and appeared to be approaching a higher limiting value. The characteristic features of predominantly plane stress fracture or general yielding fracture mechanics are summarized in conclusion     

Constraint loss correction for assessment of CTOD fracture toughness under welding residual stress. Part I: Methodology using the equivalent CTOD ratio

This part I of a two-part paper presents a method of assessing the effects of welding residual stress and constraint loss on the cleavage fracture of a wide plate subjected to membrane stress based on the Weibull stress criterion. It has been found that the Weibull stress criterion is efficient for evaluating the fracture instability of wide plates with and without a welding residual stress field. The concept of an equivalent crack-tip opening displacement (CTOD) ratio β_r under a welding residual stress field is introduced for assessment of constraint loss effects on CTOD fracture toughness of wide plates. The equivalent CTOD ratio β_r is defined as the ratio of the CTOD in the standard fracture toughness specimen to the CTOD in a wide plate with a welding residual stress at the same level of the Weibull stress. Fracture assessment procedures using β_r for wide plates are shown within the framework of the failure assessment diagram. It has been found that the excessive conservatism observed in the conventional procedure can be reduced reasonably by applying the proposed method. The companion part II of the paper presents applications of the CTOD toughness correction method using β_r to the fracture test data of welded joints and verifies the proposed method in the ductile-brittle transition temperature region.