Unified correlation of in‐plane and out‐of‐plane constraints with fracture toughness

In this paper, the specimens with different geometries and loading configurations were used to study the unified correlation of in‐plane and out‐of‐plane constraints with fracture toughness by using numerical simulation method. The results show that the unified constraint parameter A_p which was defined on the basis of the areas surrounded by the equivalent plastic strain isolines ahead of crack tip can characterise both in‐plane and out‐of‐plane constraints induced by different specimen geometries and loading configurations. A sole linear relation between the normalised fracture toughness J_IC/J_ref and was obtained. The J_IC/J_ref ‐ line is a unified correlation line of in‐plane and out‐of‐plane constraints with fracture toughness of a material, and the constraint dependent fracture toughness of a material can be determined from the unified correlation line. The results also demonstrate that the out‐of‐plane constraint effect is related to the in‐plane constraint effect, and there exists interaction between them. The higher in‐plane constraint strengthens the out‐of‐plane constraint effect, whereas the lower in‐plane constraint is not sensitive to the out‐of‐plane constraint effect.     

Unified characterisation of in‐plane and out‐of‐plane constraint based on crack‐tip equivalent plastic strain

Constraint can be divided into two conditions of in‐plane and out‐of‐plane, and each of them has its own parameter to characterize. However, in most cases, there exists a compound change of both in‐plane and out‐of‐plane constraint in structures, a unified measure that can reflect both of them is needed. In this paper, the finite element method (FEM) was used to calculate the equivalent plastic strain (ɛ_p) distribution ahead of crack tips for specimens with different in‐plane and out‐of‐plane constraints, and the FEM simulations based on Gurson–Tvergaard–Needleman (GTN) damage model and a small number of tests were used to obtain fracture toughness for the specimens with different constraints. Unified measure and characterisation parameter of in‐plane and out‐of‐plane constraints based on crack‐tip equivalent plastic strain has been investigated. The results show that the area A_PEEQ surrounded by the ɛ_p isoline ahead of crack tips can characterize both in‐plane and out‐of‐plane constraints. Based on the area A_PEEQ, a unified constraint characterisation parameter A_p was defined. It was found that there exists a sole linear relation between the normalised fracture toughness J_IC/J_ref and regardless of the in‐plane constraint, out‐of‐plane constraint and the selection of the ɛ_p isolines. The unified J_IC/J_ref−reference line can be used to determine constraint‐dependent fracture toughness of materials. The FEM simulations with the GTN damage model (local approach) can be used in obtaining the unified J_IC/J_ref−reference line for materials with ductile fracture.     

Three‐dimensional analyses of unified characterization parameter of in‐plane and out‐of‐plane creep constraint

Based on extensive three‐dimensional finite element analyses, the unified characterization parameter A_c of in‐plane and out‐of‐plane creep constraint based on crack‐tip equivalent creep strain for three specimen geometries (C(T), SEN(T) and M(T)) were quantified for 316H steel at 550 °C and steady‐state creep. The distributions of the parameter A_c along crack fronts (specimen thickness) were calculated, and its capability and applicability for characterizing a wide range of in‐plane and out‐of‐plane creep constraints in different specimen geometries have been comparatively analysed with the constraint parameters based on crack‐tip stress fields (namely R*, h and T_Z). The results show that the parameter A_c in the centre region of all specimens appears uniform distribution and lower value (higher constraint), and in the region near free surface it shows protuberant distribution and higher value (lower constraint). The parameter A_c can simultaneously and effectively characterize a wide range of in‐plane and out‐of‐plane creep constraints, while the parameters R*, h and T_Z based on crack‐tip stress fields cannot achieve this. The different capabilities of these parameters for characterizing in‐plane and out‐of‐plane creep constraints originate from their underlying theories. The parameter A_c may be useful for accurately characterizing the overall constraint level composed of in‐plane and out‐of‐plane constraints in actual high‐temperature components, and it may be used in creep life assessments for improving accuracy.     

Correlation of material constraint with fracture toughness of interface regions in a dissimilar metal welded joint

In this work, the constraint parameter A_p based on crack‐tip equivalent plastic strain was calculated by finite element analyses for the cracks located at different locations in two interface regions in a dissimilar metal weld joint (DMWJ). The capabilities of the parameter A_p for characterizing material constraint and establishing correlation of material constraint with fracture toughness of the interface region cracks have been examined. The results show that the parameter A_p can characterize material constraint effect caused by material mismatch and initial crack positions in the interface regions. Based on the A_p, the correlation lines and formulae of material constraint with fracture toughness of the interface region cracks in the DMWJ can be established, and they may be used for obtaining material constraint‐dependent fracture toughness for the interface region cracks. The results in this work combining with those in the previous studies indicate that the parameter A_p may be a unified constraint parameter that can characterize both geometry constraint (including in‐plane and out‐of‐plane constraints) and material constraint, and it may be used in accurate fracture assessments of welded components with different geometry and material constraints.     

In‐plane and out‐of‐plane constraint parameters along a three‐dimensional crack‐front stress field under creep loading

Full‐field three‐dimensional (3D) numerical analyses was performed to determine in‐plane and out‐of‐plane constraint effect on crack‐front stress fields under creep conditions of finite thickness boundary layer models and different specimen geometries. Several parameters are used to characterize constraint effects including the non‐singular T‐stresses, the local triaxiality parameter, the T_z ‐factor of the stress‐state in a 3D cracked body and the second‐order‐term amplitude factor. The constraint parameters are determined for centre‐cracked plate, three‐point bend specimen and compact tension specimen. Discrepancies in constraint parameter distribution on the line of crack extension and along crack front depending on the thickness of the specimens have been observed under different loading conditions of creeping power law hardening material for various configurations of specimens.     

In‐plane and out‐of‐plane constraint for single edge notched bending specimen and cruciform specimen under uniaxial and biaxial loading

Considering fracture constraint is an efficient way to describe stress–strain field and fracture toughness more accurately, so it is necessary to realise the relationship with in‐plane and out‐of‐plane constraint for different standard specimens. In this paper, three‐dimensional finite element method is applied to study the in‐plane and out‐of‐plane constraint for both cruciform specimen and single edge notched bending specimen made from commercial pure titanium. Crack length and in‐plane loading as the factors affecting in‐plane constraint, and thickness as the factor affecting the out‐of‐plane constraint are used to study the effect on both in‐plane and out‐of‐plane constraint in this paper. From the results, in‐plane and out‐of‐plane constraint are both related to specimen geometries and loading styles. And there exist relationships with in‐plane and out‐of‐plane constraint because of factors for different specimens. Depending on crack length, out‐of‐plane constraint increases with in‐plane constraint. While depending on transverse loading, out‐of‐plane constraint decreases with in‐plane constraint. In addition, when the in‐plane constraint of a specimen is higher, in‐plane constraint increases with out‐of‐plane constraint (thickness). When the in‐plane constraint is lower, in‐plane constraint almost remains unchanged with out‐of‐plane constraint.     

Constraint‐corrected J–R curve based on three‐dimensional finite element analyses

Three‐dimensional (3D) finite element analyses are carried out on single‐edge bend [SE(B)] specimens for which the J‐integral resistance curves (J–R curves) have been experimentally determined to develop the constraint‐corrected J–R curves for the X80 grade pipe steel. The constraint parameters considered in this study include Q_HRR, Q_SSY, Q_{SSY_m}, Q_LM, Q_BM1, Q_BM2, A₂, h and T_z. The constraint‐corrected J–R curves were developed on the basis of the constraint parameters obtained from finite element analysis and experimentally determined J–R curves associated with deeply cracked and medium‐cracked SE(B) specimens and validated against shallow‐cracked SE(B) specimens. The analysis results indicate that all the constraint parameters considered in this study except Q_HRR, Q_SSY, Q_{SSY_m} and Q_LM lead to reasonably accurate constraint‐corrected J–R curves if the crack extensions are relatively small (≤0.7 mm). For larger crack extensions (≤1.5 mm), the Q_BM1‐based constraint‐corrected J–R curve leads to the most accurate predictions of J among all the constraint parameters considered.     

The effect of T-stress on plane strain dynamic crack growth in elastic–plastic materials

In this paper, the effects of T‐stress on steady, dynamic crack growth in an elastic–plastic material are examined using a modified boundary layer formulation. The analyses are carried out under mode I, plane strain conditions by employing a special finite element procedure based on moving crack tip coordinates. The material is assumed to obey the J₂ flow theory of plasticity with isotropic power law hardening. The results show that the crack opening profile as well as the opening stress at a finite distance from the tip are strongly affected by the magnitude and sign of the T‐stress at any given crack speed. Further, it is found that the fracture toughness predicted by the analyses enhances significantly with negative T‐stress for both ductile and cleavage mode of crack growth.     

Reduction of measured toughness due to out‐of‐plane constraint in ductile fracture of aluminium alloy specimens

It is generally believed that a lower bound on the fracture toughness of a material is obtained from a standard test, particularly in metals where yielding occurs prior to fracture. The understanding is that in such a test the material around the crack tip is highly constrained hence reducing the extent of yielding. In this paper, we report the results of fracture tests where a tensile load is applied to a biaxial aluminium alloy specimen in the direction parallel to the crack front in addition to the fracturing load normal to the crack surface. We show that in this case a lower fracture toughness is measured than that obtained from a standard test. Indeed, for the highest value of tensile load used in our tests the J‐integral at fracture was half the value measured in a standard test. It is also shown that the volume of the plastic region can be used to measure the effect of constraint, irrespective of the manner in which the constraint arises. This approach suggests an even lower fracture toughness may be obtained than that measured here in certain loading conditions.     

Effect of loading rate on dynamic fracture initiation toughness of brittle materials

Numerical simulation is carried out to investigate the effect of loading rate on dynamic fracture initiation toughness including the crack-tip constraint. Finite element analyses are performed for a single edge cracked plate whose crack surface is subjected to uniform pressure with various loading rate. The first three terms in the Williams’ asymptotic series solution is utilized to characterize the crack-tip stress field under dynamic loads. The coefficient of the third term in Williams’ solution, A ₃, was utilized as a crack tip constraint parameter. Numerical results demonstrate that (a) the dynamic crack tip opening stress field is well represented by the three term solution at various loading rate, (b) the loading rate can be reflected by the constraint, and (c) the constraint A ₃ decreases with increasing loading rate. To predict the dynamic fracture initiation toughness, a failure criterion based on the attainment of a critical opening stress at a critical distance ahead of the crack tip is assumed. Using this failure criterion with the constraint parameter, A ₃, fracture initiation toughness is determined and in agreement with available experimental data for Homalite-100 material at various loading rate.     

Effect of residual stresses on crack behaviour in single edge bending specimens

Residual stresses due to manufacturing processes, such as welding, change the load bearing capacity of cracked components. The effects of residual stresses on crack behaviour in single edge bending specimens were investigated using Finite element analyses. Three parameters (J, Q and R) were used to study the crack behaviour. The J‐integral predicts the size scale over which large stresses and strains exist, the constraint parameter Q describes the crack‐tip constraint as a result of geometry, loading mode and crack depth and the constraint parameter R is used to describe the constraint resulting from residual stresses. To carry out a systematic investigation on the effect of residual stresses on the J‐integral and crack‐tip constraints, models under different combinations of residual stresses and external loads with different crack depths were analysed. It has been shown that the crack‐tip constraint R increased by tensile residual stresses around the crack‐tip. On the other hand, the constraint parameter R decreased and tended to zero at high external load levels.     

Effects of divider orientation split‐out on fracture toughness

The Split‐out phenomenon is a sudden instability, which takes place near the crack tip, and is generated by a rapid growth and arrest of a crack at the divider orientation. This phenomenon is related to stress triaxiality in the front of the crack tip when the structure is under plane strain conditions, plus the existence of weak interfaces oriented normal to the thickness direction. The weak interfaces can be generated as a result of some metallurgical process such as hot‐rolling lamination in materials with a high level of impurities, as well as steels where the structure results in a strong banding of ferrite and pearlite. When a cracked structure is loaded, the high σ_z stresses related to the plain strain state can lead to a split‐out brought about by delamination of the weak interfaces. During a fracture toughness test, this is noticed as a drop in the load‐displacement record, which is similar to that produced by the well‐known pop‐in instability in welded joints. Although the load drops are very similar in both phenomena, their etiologies are completely different, since the pop‐in is produced by an unstable crack growth and its ulterior arrest, due to local brittle zones located generally at the heat affected zone (HAZ) at or near a welded joint. As the split‐out has been less studied than the pop‐in, and due to the similarities between both load‐displacement records, it is very common to consider both instabilities under the same failure acceptance criteria. In this way, the split‐out is usually treated as a critical event, and therefore, many materials are rejected when split‐outs occur in fracture testings. However, when a split‐out occurs, the σ_z stress is drastically reduced, leading to a condition close to a plane stress situation (in fact, the σ_z stress reduces to 0 at the crack surfaces of the split‐out). The aim of this work was to study the effect of the split‐out in the fracture toughness of hot laminated steels working in the upper‐shelf region. In order to achieve this, the fracture toughness was evaluated by means of the resistance curve J‐R of the material, using a single specimen test method. The occurrence of load drop and displacement increase in the test record are explained based on a constant main plane crack length hypothesis.     

Effect of lattice orientation on crack tip constraint in ductile single crystals

In this work, the effect of lattice orientation on the fields prevailing near a notch tip is investigated pertaining to various constraint levels in FCC single crystals. A modified boundary layer formulation is employed and numerical solutions under mode I, plane strain conditions are generated by assuming an elastic–perfectly plastic FCC single crystal. The analysis is carried out corresponding to different lattice orientations with respect to the notch line. It is found that the near‐tip deformation field, especially the development of kink or slip shear bands is sensitive to the constraint level. The stress distribution and the size and shape of the plastic zone near the notch tip are also strongly influenced by the level of T ‐stress. The present results clearly establish that ductile single crystal fracture geometries would progressively lose crack tip constraint as the T ‐stress becomes more negative irrespective of lattice orientation. Also, the near‐tip field for a range of constraint levels can be characterized by two‐parameters such as K–T or J–Q as in isotropic plastic solids.     

The effect of T‐stress on crack‐tip plastic zones under mixed‐mode loading conditions

In this paper, the influence of T‐stress on crack‐tip plastic zones under mixed‐mode I and II loading conditions is examined. The crack‐tip stress field is defined in terms of the mixed‐mode stress intensity factors and the T‐stress using William's series expansion. The crack‐tip stress field is incorporated into the Von Mises yield criteria to develop an expression that determines the crack‐tip plastic zone. Using the resultant expression, the plastic zone is plotted for various combinations of mode II to mode I stress intensity factor ratios and levels of T‐stress. The properties of the plastic zone affected by T‐stress and mixed‐mode phase angle are discussed. The observations obtained on plastic zones variations are important for further fatigue and fracture analyses for defects in engineering structures under mixed‐mode loading conditions.     

A node release approach to estimate J‐R curve for single‐edge‐notched tension specimen under reversed loading

This paper proposes a node release approach to estimate the fracture resistance curve, often known as the J‐R curve, for monotonic and cyclic fracture tests. The node release approach simulates the crack extension by releasing the constraints imposed on the node at the crack tip and estimates the J‐R curve by coupling the domain integral value with the corresponding crack extension. This proposed node release approach estimates closely the J‐R curve for SE(B) and SE(T) specimens subjected to monotonic loading. For SE(T) specimen under cyclic loading, this study implements the node release analysis in two approaches: (1) an equivalent monotonic analysis corresponding to the envelope of the cyclic load‐CMOD response and (2) a direct cyclic simulation. Both approaches lead to close estimations of the experimentally measured J‐R curve. The numerical analysis also confirms the path independence of the domain integral values in the direct cyclic simulation.     

A numerical study on the crack tip constraint of pipelines subject to extreme plastic bending

This study investigates the fracture response and crack tip constraint of thick wall pipelines subject to large plastic bending. Such a circumstance frequently occurs during the installation of offshore pipelines (such as the reeling method), and accidental overloading, both inducing inelastic bending. The near-tip stress and strain fields are obtained through the fully nonlinear 3D finite element models constructed to examine the response of a practical range of cracked pipeline geometries and material properties. It is observed that throughout the loading history (up to the large scale yielding of the pipeline), by incorporation of the J–Q two parameter fracture theory, the near crack tip fields do indeed resemble those obtained from a K–T modified boundary layer formulation. This analogy provides sufficient proof for the applicability of the similitude concept inherent and fundamental to any fracture assessment procedure. All the pipelines considered in this study, which had realistic crack sizes, exhibited low constraint behavior (i.e. −1.4 < Q < −0.4). Additionally, Q was observed to decrease as a linear function of the global bending strain. Based on this correlation, simplified design equations are presented by which the constraint of such pipelines could be effectively estimated. The equations would be suitable for incorporation in the constraint-matched integrity assessment procedures that would in turn overcome the overt conservatism produced by the use of single parameter fracture mechanics approaches. Suitability of the low constraint laboratory specimens for fracture toughness measurements is also confirmed.     

A review of fracture toughness transferability with constraint and stress gradient

In this review paper, only constraint and stress gradient approaches to transferability of fracture toughness are examined. The different constraint parameters are defined and discussed, and one example is given in each case. Factors that influence the constraint are studied. Special attention is given to the actual trends to use the plastic constraint in the material failure master curve and the material transition temperature master curve. The paper also deals on the influence of T stress on the crack path and out‐of‐plane constraint and on the influence of thickness on fracture toughness. The uses of plasticity with gradient and the relative stress gradient in local fracture approaches are also examined.     

Crack-tip constraint in mode II deformation

Studies of cracked specimens loaded in mode I have shown that the stresses near the crack tip depend significantly on the level of constraint. The stresses can be determined near the crack tip using the HRR solution, but only for high constraint specimens. For other levels of constraint, O'Dowd and Shih's Q parameter may be used to adjust the stresses derived from the HRR solution. Only limited research has been carried out to study the effect of constraint in mode II. In this paper a mode II boundary layer formulation is used to study the effect of far field elastic stresses on the size and shape of the plastic zone around the crack tip and on the stresses inside the plastic zone. It is shown that in mode II, both positive and negative values of remote T-stress influence the tangential stress along the direction of maximum tangential stress. In the spirit of O'Dowd and Shih, a dimensionless parameter Q _II is introduced to quantify the constraint for mode II specimens failing by brittle fracture. The relation between Q _II and T/₀ is determined for different values of the strain hardening coefficient n. To investigate the range of validity of the Q–T diagram for real specimens, the constraint parameter Q _II is calculated directly from finite element analysis for three mode II specimens and compared with the evaluation using the Q–T diagram.     

Effects of side‐groove depth on creep crack‐tip constraint and creep crack growth rate in C(T) specimens

The effects of side‐groove depth on creep crack‐tip constraint and creep crack growth (CCG) rate in C(T) specimens have been quantitatively studied. The results indicate that with increasing side‐groove depth, the constraint level and CCG rate increase and constraint distribution along crack front (specimen thickness) becomes more uniform. The constraint and CCG rate of thinner specimen are more sensitive to side‐groove depth. Two new creep constraint parameters (namely R^* and A_c) both can quantify constraint levels of the specimens with and without side‐grooves, and the quantitative correlations of CCG rate with constraint have been established. The mechanism of the side‐groove depth effect on the CCG rate has also been analyzed.     

Effect of Constraint on the Fracture Behaviour of a Simulated Heat‐Affected Zone of an X‐70 Steel Used in Pipelines

Abstract: The typical heat‐affected zone developed in an X‐70 steel, usually used in pipeline manufacture, has been simulated via thermal treatment. A non‐equilibrium microstructure consisting of bainite, with ferrite and some martensite, has been produced and mechanically characterised, resulting in much greater hardness and strength and a lower ductility and toughness than the corresponding base metal. This also has a lower ductility. Different single‐edge notched bending specimens with different crack lengths (a/W between 0.1 and 0.5) were experimentally tested to assess the fracture behaviour of this product under different degrees of constraint. The J–a resistance curves at room temperature were determined, and the obtained results were explained by the effect of constraint on ductile crack growth.