Two-parameter Fracture Criterion Research Articles

The Two-Parameter Fracture Criterion Taking into Account Two-Dimensional Deformation Constraints at the Front of a Mixed-Type Crack

In this article, a new fracture criterion for a mixed-type crack (type I + type II) is proposed. It is based on the assumption that the maximum tangential stresses in the prefracture region are equal to the local strength of the material. In this case, the size of the prefracture area and the local strength are determined taking into account nonsingular Тхx and Тzz stresses included in the expansion of the Williams stress function. The use of Тхx and Тzz stresses in the calculations allows one to describe the two-dimensional local constraint of the deformation along the crack front in three-dimensional bodies. In addition to KI and KII, the expression for the effective stress intensity factor (SIF) includes the ratios of the Тхx and Тzz stresses to the yield strength. This makes it possible to take into account the constraint of deformations in the transverse and longitudinal directions of the crack front, respectively. An example of the implementation of the developed criterion in relation to the determination of the fracture load of a tensile-stressed plate with a through inclined crack is given. Dependences of the Тxx and Тzz stresses on the plate thickness are presented for various crack slopes and plate thicknesses. It is shown that an increase in the plate thickness and a decrease in the crack slope lead to a decrease in the fracture load.

Assessment of failure probabilities and the allowable size of defects in structural elements using the criteria of fracture mechanics

The possibility of assessing the safe size of the defects of metal continuity using risk criteria are considered. Such defects occur at all stages of the life cycle of structures. Assessment of their hazard and determination of their allowable size becomes important when the defects can lead to brittle or quasi-brittle fracture. In this case, models of linear and nonlinear fracture mechanics are used. In these models, defects are considered as internal elliptical or surface semi-elliptical cracks. The stochastic variety of shapes, sizes, locations, and orientations of defects has a significant effect on the destruction mechanisms. In this regard, the relevant probabilistic problem of assessing the permissible size of defects according to the criteria of the risk of destruction comes to the fore. We consider a general approach to assessing hazard of defects by risk criteria. Two settings of the probabilistic problem of risk assessment based on of one- and two-parameter fracture criteria are presented. The risk function in the form of the probability of fracture according to a given criterion is used as the main calculated characteristic. The expression of the risk function based on one-parameter criteria of fracture mechanics is presented. The main attention is paid to the probabilistic model based on the two-parameter fracture criterion by E. M. Morozov. This criterion provides ample opportunities for analyzing fracture mechanisms with a change in the size of defects. An expression for the risk function based on the family of two-dimensional probability distributions of Lu – Bhattacharya of the Weibull type is obtained. It is shown that the correlations of the fracture mechanisms can significantly affect the values of the fracture probabilities, and, consequently, the allowable size of defects.

Evaluation of the two‐parameter fracture criterion for various crack configurations made of 7075‐T6 aluminium alloy using finite‐element fracture simulations

AbstractFor many years, a two‐parameter fracture criterion (TPFC) has been used to correlate and predict failure loads on cracked metallic fracture specimens. The current study was conducted to evaluate the use of the TPFC on a high‐strength aluminium alloy, using elastic‐plastic finite‐element (FE) analyses with the critical crack‐tip‐opening angle (CTOA) fracture criterion. In 1966, Forman generated fracture data on middle‐crack tension, M(T), specimens made of thin‐sheet 7075‐T6 aluminium alloy, which is a quasi‐brittle material. The fracture data included a wide range of specimen half‐widths (w) ranging from 38 to 305 mm. A two‐dimensional FE analysis code (ZIP2D) with a “plane‐strain core” option was used to model the fracture process with a critical CTOA chosen to fit the M(T) test data. Fracture simulations were then conducted on other M(T), single‐edge‐crack tension, SE(T), and bend, SE(B), specimens over a wide range in widths (w = 19‐610 mm). No test data were available on the SE‐type specimens. The results supported the TPFC equation for net‐section stresses less than the material proportional limit. However, some discrepancies in the FE fracture simulations results were observed among the numerical analyses made on the three specimen types. Thus, more research is needed to improve the transferability of the TPFC from the M(T) specimen to both the SE(T) and SE(B) specimens for quasi‐brittle materials.

Validation of the two-parameter fracture criterion for various crack configurations made of 2014-T6 aluminum alloy using finite-element fracture simulations – Part 2 TL-Orientation

In the present paper, the two-parameter-fracture-criterion (TPFC) validity and transferability are analysed using two-dimensional (2D) elastic-plastic finite-element fracture simulations with the critical crack-tip-opening-angle (CTOA) fracture criterion and plane-strain-core concept. Fracture simulations were performed on three crack configurations: (1) middle-crack-tension, M(T), (2) single-edge-crack-tension, SE(T), and (3) single-edge crack-bend, SE(B), specimens to predict failure loads and moments. They were made of 2014-T6 (TL) aluminum alloy. Fracture test data from Orange (NASA) were only available on M(T) specimens (one-half width, w = 38–152 mm), which were all tested at cryogenic (−200 °C) temperature. The critical crack-tip-opening-angle (CTOA) and a plane-strain-core height, hc, were selected to match the failure loads on the M(T) specimens. Fracture test data on the M(T) specimens were also used to determine the two fracture parameters (KF and m). All crack configurations were analysed over a very wide range of widths (w = 19–610 mm) and initial crack-length-to-width (ci/w) ratios ranged from 0.05–0.95. The TPFC was shown to fit the M(T) test data (±5%) and the simulated fracture data fairly well (within 6.5%) for all crack configurations for net-section stresses (Sn) less than the material proportional limit (σplσ). For M(T) specimens, a simple approximation was shown to work well for Sn > σpl. Further study is needed for Sn > σpl, for the SE(T) and SE(B) specimens.

Validation of the two-parameter fracture criterion for various crack configurations made of 2014-T6 aluminum alloy using finite-element fracture simulations – Part 1 LT-Orientation

A two-parameter fracture criterion (TPFC), derived by Newman, related the linear-elastic stress intensity factor, KIe, and net-section stress, Sn, at failure with two material fracture parameters, KF and m. The objective of this work was to validate the TPFC using two-dimensional elastic–plastic finite-element analyses (FEA). The critical crack-tip-opening-angle (CTOA) fracture criterion and a plane-strain core were used to match the failure loads on middle-crack-tension, M(T), specimens made of thin-sheet 2014-T6 (LT) aluminum alloy. The same analyses were then used on other M(T), single-edge-crack-tension, SE(T), and bend, SE(B), specimens to predict failure loads and moments. Test data was not available on SE-type specimens. A wide range in specimen widths (w = 19–610 mm) were considered. The finite-element code, ZIP2D, was used to calculate critical stress-intensity factors at failure corresponding to all crack configurations for crack-length-to-width (ci/w) ratios of 0.05–0.95. The FEA results agreed very well with the TPFC predictions for net-section stresses less than the material proportional limit. Further studies are needed for net-section stresses greater than the proportional limit on both tension and bend specimens.

Comparison of constraint analyses with global and local approaches under uniaxial and biaxial loadings

Abstract Fracture toughness is an important material property used to perform the integrity assessment of engineering components containing cracks. Due to the difference in crack tip constraint, specimens may show different fracture toughness. The constraint difference for cruciform specimen with shallow crack, compact tension (CT) specimen and three point bending specimen with shallow and deep cracks are investigated. Both linear elastic and elastic-plastic fracture mechanics are applied to study the constraint effect based on two-parameter fracture criterion . Crack tip constraint depends on the applied loading. J-A 2 method is used to precisely capture the crack tip constraint and crack tip stress distributions. Local approach to fracture can be applied to transfer the fracture toughness among different specimens under uniaxial and biaxial loadings . In case of positive T -stress, T-stress increases with KI. In the case of negative T-stress, T-stress decreases with KI. Q-stress generally decreases with applied loading for both deep crack and shallow crack cases. Loss of constraint occurs for the single-edged bending (SEB) specimen with deep crack and thus raises the question whether the SEB specimen is proper to be used to obtain material toughness. For the cruciform bending (CRB) specimen, the constraint at the crack tip surface shows a least constraint while the deepest point has a relatively higher constraint. At a fracture probability of 10%, the fracture toughness difference between CT specimen and CRB specimen is about 50 MPa m0.5, i.e 200% of the fracture toughness. This big difference demonstrates the importance of considering the constraint effects in the integrity analysis.

A two-parameter criterion for predicting the fracture location along a surface crack

The fracture prediction for materials containing surface cracks is not well established. In this paper, a new two-parameter fracture criterion which can predict the location of maximum crack extension along a surface crack front was presented. The surface-cracked specimens with different crack sizes and loads were selected, and the finite element method (FEM) was used to model the J-integral and the equivalent plastic strain (εp) distributions along the crack front for different specimens. And then, in succession, the areas APEEQ surrounded by the εp isoline ahead of crack tips under different J-integrals and the unified constraint parameter Ap were calculated. Based on the parameter Ap, J-integral and the fracture toughness Jref measured in a standard test, the J/(JrefAp), which considered the combined effects of crack driving force and material resistance force, was determined. The critical fracture location is the location for which the J/(JrefAp) is a maximum. The results show that for all the specimens, the prediction results obtained by two-parameter criterion are consistent with the measured results obtained by experiments. The J/(JrefAp) is a suitable prediction criterion to model the location of the maximum crack extension of surface crack. For comparison, similar predictions were also made using a single-parameter J-integral and Ap.

Inhibition of a curvilinear bridged crack by induced thermoelastic stress field

ABSTRACTTemperature changes near the end of a curvilinear crack in the presence of end areas with cohesive forces of the material are considered. It is assumed that these areas adjoin to the crack tip and their sizes are comparable with the crack size. The goal of local changes of temperature is to delay or inhibit the growth of a curvilinear crack. A boundary value problem on equilibrium of a bridged curvilinear crack under the action of external tensile loads, induced thermoelastic temperature field, and tractions in the bonds preventing its disclosure is reduced to the system of nonlinear singular differential equations with a Cauchy-type kernel. Normal and tangential tractions in the bonds are found from the solution of this system. The stress intensity factors are calculated. The condition of limit equilibrium of a curvilinear crack with an end area is formed based on two-parameter fracture criterion.

Validation of the Two-Parameter Fracture Criterion using 3D finite-element analyses with the critical CTOA fracture criterion

A Two-Parameter Fracture Criterion (TPFC) was derived from Inglis’ elastic-stress-concentration equation for an elliptical hole under remote uniform stress and Neuber’s equation relating the plastic stress and strain concentration factors to the elastic stress-concentration factor at the maximum stress location. Elastic–plastic 3D finite-element analyses (FEA) employing the critical crack-tip-opening-angle (CTOA) fracture criterion was used to simulate fracture on thin-sheet middle-crack-tension, M(T), and compact, C(T), specimens made of 7249-T76511 aluminum alloy over a wide range in crack-length-to-width ratios to validate the TPFC. A critical CTOA (7.6°) and the fracture parameters (KF and m) were determined from M(T) test specimens, and these values were then used to predict the failure on C(T) specimens. The FEA predicted failure loads on C(T) specimens were within a few percent; whereas, the TPFC generally over predicted the failure loads (5–15%). The TPFC over prediction on the C(T) specimens is likely related to net-section stress (Sn) being greater than the yield stress of the material for all test data, while the M(T) specimens all had Sn less than the yield stress.

Transferability of the two‐parameter fracture criterion for 2219 aluminium alloy cracked configurations

AbstractTwo‐dimensional elastic–plastic finite‐element fracture simulations with the critical crack‐tip‐opening‐angle fracture criterion were used to evaluate the two‐parameter fracture criterion (TPFC). Three different crack configurations under tension and bending loads made of thin‐sheet 2219‐T87 aluminium alloy were analysed. A very wide range of widths (w = 76 to 2440 mm) and initial crack‐length‐to‐width ratios (ci/w = 0.05 to 0.95) were considered. A relation from the original TPFC was shown to fit the simulated fracture behaviour fairly well for the three different specimen types for net‐section stresses less than the yield stress (σy) of the material. Comparisons were also made on measured and simulated fracture tests on middle‐crack‐tension specimens. A relation between the elastic stress‐intensity factor, KIe, and net‐section stress, Sn, at failure was found to be linear for Sn < σy. The results demonstrated the transferability of the TPFC for different crack configurations for Sn < σy, but further study is needed for Sn > σy.

A two-parameter approach to assessing notch fracture behaviour in clay/epoxy nanocomposites

Fracture toughness and other mechanical properties of epoxy are known to be affected by the addition of nanoclays. Fracture toughness has been shown by many researchers to depend on the nanocomposite structure with well-dispersed and distributed nanoparticles resulting in improvements in this property by up to 50%. Notch fracture toughness depends on the acuity of the notch as well as on the structure of the nanocomposite. In the present work, a two-parameter fracture criterion based on a critical notch stress intensity factor, Kρ,c, and effective T-stress, Tef, was used to study the effect of notch severity and nanoclay addition on the fracture toughness of the epoxy matrix. The results show that the average value of Kρ,c for neat epoxy increased with increasing notch radius while the absolute value of Tef decreased with notch radius. The addition of nanoclay to pristine epoxy reduced the average value of Kρ,c and increased the absolute value of Tef. The critical notch radius was found to be around 1.0 mm and the notch sensitivity was higher for neat epoxy. SEM analysis of the fractured surfaces revealed that the lower Kρ,c for nanocomposites in both mode I and mixed mode fractures was due to early crack initiation at clay clusters or voids at the notch root.

Validation of the Two-Parameter Fracture Criterion using finite-element analyses with the critical CTOA fracture criterion

Elastic–plastic finite-element analyses employing the critical crack-tip-opening-angle (CTOA) fracture criterion were used to simulate fracture on thin-sheet middle-crack-tension, M(T), specimens made of several aluminum alloys over an extremely large range of widths (18–5700-mm) for a wide range in crack-length-to-width ratios. A linear relation between elastic stress-intensity factor (KIe) and net-section stress (Sn) at failure was observed for Sn less than yield stress of the material. Test data on two aluminum alloys (2024-T3 and 2219-T87) over a wide range of M(T) specimen widths (76–600-mm and 76–1200-mm, respectively) supported the linear relationship that validated the Two-Parameter Fracture Criterion.

Study on Fracture Model of the Blunt Crack in Rock-Type Material under Compression

The stress field around the tip of the blunt crack in rock-type materials under compressive loading is analyzed, and the relation of the stress field is set up. Compared to the stress field of the ideal mathematic crack, the effect on the distribution of the stress field by the thickness or the curvature radius of the blunt crack is considered. Based on the stress field, the two-parameter fracture criterion for the blunt cracks under compressive loading is set up. The two parameters in the criterion are related to the material property, and can be determined by the experiments or theoretical analysis. By the fracture model in the paper, the fracture analysis of the blunt crack under compression are carried out, and the theoretical results are in concordance with the experimental results, which shows the fracture criteria in the paper is available to the fracture analysis of the blunt crack.

Variation of material failure curve with constraint

A two-parameter fracture criterion has been proposed to predict fracture conditions of notched components. This criterion includes the critical notch stress intensity factor Kρ,c, which represents fracture toughness of a material with a notch of radius ρ, and the second non-vanishing parameter from the Williams equation, A3 term. The effective A3 –term A3,efhas been estimated as the average value of the A3 term distribution in the region ahead of the notch tip at the effective distance Xef. These parameters were derived from the volumetric method of notch fracture mechanics. The master curve Kρ,c=f(A3,ef) has been established as a result of the notched specimen. It was shown that the notch fracture toughness is a linear decreasing function of the A3ef,c-stress. The use of the master curve to predict fracture conditions was demonstrated on gas pipes with the longitudinal surface notch.

The Effect of Hydrogen on the Master Failure Curve of APL 5L Gas Pipe Steels

The effect of hydrogen on the material failure curves of APL 5L gas pipe steels was analysed from viewpoint of the notch fracture mechanics. The material failure curves based on two-parameter fracture criterion have been determined for API 5L X52, X70 and X100 gas pipe steels. The notch fracture toughness Kp,c and the effective T-stress were employed to describe the material failure curve. Tests were carried out on electrolytic hydrogen charged SENT, CT, RT (Roman Tile) and DCB specimens with a notch. Fracture initiation was detected by acoustic emission. Material failure curves for hydrogen charged steels and steels without hydrogen were constructed. It was shown that there is critical hydrogen concentration, which causes significant reduction of local fracture resistance of pipe steels.

Two-parameter fracture criterion (K ρ,c -T ef,c ) based on notch fracture mechanics

A two-parameter fracture criterion has been proposed to predict fracture conditions of notched components. This criterion includes the critical notch stress intensity factor K ρ,c, which represents fracture toughness of a material with a notch of radius ρ, and the effective T-stress. The effective T-stress T ef has been estimated as the average value of the T-stress distribution in the region ahead of the notch tip at the effective distance X ef . These parameters were derived from the volumetric method of notch fracture mechanics. The results of numerical T ef,c-stress estimation are compared to the T ef,c-stress results obtained from experimental analysis. The material failure curve or master curve K ρ,c = f(T ef,c) has been established as a result of the notched specimen tests. A large T ef,c range was covered from −0.80 σ Y to +0.19 σ Y using SENT, CT, RT (roman tile) and DCB specimens. It was shown that the notch fracture toughness is a linear decreasing function of the T ef,c-stress. The use of the material failure curve to predict fracture conditions was demonstrated on gas pipes with the surface notch.

Constraint Parameter for a Longitudinal Surface Notch in a Pipe Submitted to Internal Pressure

The use of two parameters fracture mechanics criterion as a tool for structural design and analysis has increased significantly in recent years. First, we discuse the elastic solution for the stress distribution at crack tip for two dimensional geometries and particularly constraint as T-stress under various loading conditions. Secondly, using notch fracture mechanics and particularly the Volumetric Method approach, we study the stress distribution at the tip of a notch in pipes submitted to internal pressure. The Notch Stress Intensity Factor Kρ and the effective T-stress are combined into a two-parameter fracture criterion (KIρ-Tef). This approach is then used to quantify the constraint of notch-tip fields for various pipe geometry and loading conditions.

Correlating cryogenic fracture strength using a modified two-parameter method

Damage tolerant approaches have been employed increasingly in the design of critical engineering components. In these approaches, one has to assess the residual strength of a component with an assumed pre-existing crack. In other cases, cracks may be detected during service. Then, there is a need to evaluate the residual strength of the cracked components in order to decide whether they can be continued safely or repair and replacement are imperative. A modified two-parameter fracture criterion is applied to correlate fracture data from tensile cracked plates made of aluminium alloys, titanium alloys and stainless steel at cryogenic temperatures. Fracture parameters to generate the failure assessment diagram are determined for the materials considered in the present study. An attempt is made to correlate the fracture data on pressure vessels tested at cryogenic temperatures. The fracture parameters obtained from the fracture data of tensile cracked plates and pressure vessels were found to be closely matching with each other. Failure pressure estimates were found to be in good agreement with test results. This study indicates that the failure assessment diagram of a material generated from tensile fracture plate configurations can be applied for failure pressure estimations of flawed pressure vessels.

On the temperature dependence of fracture toughness in the brittle-to-ductile transition region

A method for calculation of fracture toughness in the brittle-to-ductile transition region is proposed for a known fracture toughness at a given temperature and a known temperature dependence of the yield point. A two-parameter quasi-brittle fracture criterion proposed earlier is employed. This criterion contains in explicit form a plastic constraint factor that is sensitive to temperature variation under plane strain conditions. The obtained calculation dependence is compared with experimental data for four structural steels.

A concept of ultimate crack resistance

The ultimate crack resistance is regarded as a two-parameter fracture criterion. The concept of ultimate crack resistance enables one to estimate the state of a material and to compute the crack resistance of machine parts in the processes of brittle, transient, and viscous fracture. In relations used for numerical analysis, the stress intensity factors are preserved but the corresponding characteristic of the material is made more complicated.