A Gurson–Tvergaard based model to simulate the fracture of aged duplex stainless steels

In order to use duplex stainless steels for components designed to withstand temperatures up to 475 °C, the knowledge of the effect of ageing on the fracture mechanisms is required. For the unaged steel, the ductile fracture takes place due to the nucleation of microvoids initiated on precipitates and inclusions present in the austenitic phase. However, the ageing of the material leads to a global loss of ductility. The ferrite phase embrittlement causes it to break by cleavage at lower levels of plastic strain. These changes, both in mechanical properties and in fracture micromechanisms, can be simulated by means of variation in the parameters that govern the constitutive equation of the material. The local approach model developed by Gurson–Tvergaard has been used to simulate, by FE analysis, the elasto‐plastic behaviour and the fracture of these materials at various stages of ageing at 400 °C. An extensive experimental program has been undertaken in order to obtain the Gurson–Tvergaard model parameters needed for numerical simulations, which considers the effect of ageing conditions (time and temperature), the ferrite content of the steel that causes the global embrittlement and the geometry of the component from the stress triaxiality by using axisymmetric notched specimens. The numerical model developed allows predictions to be made for the damage constitutive parameters for duplex steels subjected to any ageing conditions and, as a consequence, can be directly used for designing components.     

Identification of Gurson–Tvergaard material model parameters via Kalman filtering technique. I. Theory

In this paper quasi-static ductile fracture processes are simulated within the framework of the finite element method by means of the Gurson–Tvergaard isotropic constitutive model for progressively cavitating elastoplastic solids. The progressive degradation of the material strength properties in the fracture process zone due to micro-void growth to coalescence is modeled through the computational cell concept. Among the several model parameters to be calibrated in the computations, attention is restricted to the Tvergaard coefficients q ₁ and q ₂ and to the initial porosity f ₀ in the unstressed configuration. To identify these model parameters the inverse problem is solved via the extended Kalman filter for nonlinear systems coupled to a numerical methodology for the sensitivity analysis. In part I of this work the theory of Kalman filtering and sensitivity analysis is presented. First results concerning the identification of the Tvergaard parameters for a whole crack growth in single edge notched bend specimens made of a pressure vessel steel are presented. In order to enhance the convergence towards the final solution of the identification procedure, during the tests measurements are made of the displacements of points located in the central portion of the notched specimens, where model parameters highly affect the system state variables. In part II of this work a numerical validation of the proposed procedure in terms of uniqueness of the final identified solution, requirements of accuracy for the Bayesian initialization of the model parameters and sensitivity to the experimental measurement errors will be presented and discussed.     

Gurson model parameters for ductile fracture simulation in ASTM A992 steels

This study is concerned with the modelling the ductile fracture in ASTM A992 steels using the Gurson‐Tvergaard‐Needleman (GTN) model for high stress triaxiality regime. The GTN model for ASTM A992 structural steels is calibrated from the experiments performed on axisymmetrically notched tensile specimens. The experiments are designed to obtain a range of stress triaxiality and different fracture initiation locations. The non‐uniqueness in the constitutive parameters of the GTN model is illustrated in this study. The choice of a unique set of GTN constitutive parameters is made by choosing the nucleation strain (?_N) as a material constant. The process of estimating this material specific nucleation strain is provided. All the other GTN model parameters corresponding to the material specific nucleation strain (?_N) are evaluated to best fit the experimental results. The calibrated GTN model is shown to predict the load displacement behaviour, ductility and fracture initiation locations in the notched specimens. The calibrated GTN parameters are used to successfully predict the ductility of structural components: (a) bars with a hole; (b) plate with reduced section and (c) plate with holes; that are typically found in structural engineering applications.     

Effect of strength mis-match and dynamic loading on ductile fracture initiation

It has been well known that ductile fracture of steels is accelerated by triaxial stresses. The characteristics of ductile crack initiation in steels are evaluated quantitatively using a two-parameter criterion based on equivalent plastic strain and stress triaxiality.The present study focuses on the effects of geometrical discontinuity, strength mis-match, which can elevate plastic constraint due to heterogeneous plastic straining, and loading rate on the critical condition for ductile fracture initiation using a two-parameter criterion. Fracture initiation testing has been conducted under static and dynamic loading using circumferentially notched round-bar specimens. In order to evaluate the stress/strain state in the specimens, especially under dynamic loading, a thermal elastic-plastic dynamic finite element (FE) analysis considering the temperature rise due to plastic deformation has been carried out.The tensile tests on specimens with an undermatching interlayer showed that the relationship between the critical equivalent plastic strain to initiate ductile fracture and stress triaxiality was equivalent to that obtained on homogeneous specimens under static loading. Moreover, the two-parameter criterion for ductile fracture initiation is shown to be independent of the loading rate. It was demonstrated that the critical global strain to initiate ductile fracture in specimens with strength mis-match under various loading rate can be estimated based on the local criterion, that is two-parameter criterion obtained on homogeneous specimens under static tension, by mean of FE-analysis taken into account accurately both strength mis-match and dynamic loading effects on stress/strain behaviors.     

Prediction of ductile fracture initiation using micromechanical analysis

In the paper ductile fracture initiation analysis of low-alloyed ferritic steel has been made by application of two micromechanical models: the Rice–Tracey void growth model and the Gurson–Tvergaard–Needleman (GTN) model. The aim of the study was to analyse transferability of micromechanical parameters determined on specimens without initial crack to pre-cracked specimens. A significant part of the research has been carried out through participation in the round robin project organised by the European Structural Integrity Society (ESIS). Tensile tests have been performed on cylindrical smooth specimens and CT specimens. Critical values of micromechanical parameters determined on smooth specimen for both applied models, have been used for prediction of the crack growth initiation in CT specimen. Modelling of the first phase of ductile fracture––void nucleation––has been carried out using quantitative metallographic analysis of non-metallic inclusion content in tested steel. For determination of critical values of model parameters corresponding to ductile fracture initiation a simple procedure has been applied based on a combination of experimental and numerical results. Evaluated J-integral values corresponding to onset of crack growth, J_i, are in good agreement with experimental result and both models have proved to be suitable for determination of the ductile fracture initiation in tested steel. The effect of FE size at a crack tip on J_i-value has been particularly analysed: it has been established that the calculation with FE size corresponding to the mean free path λ between inclusions in steel gives results that are in accordance with the experimental ones.     

Fracture behaviour and cleavage initiation in hypoeutectoid pearlitic steel

The paper reports on an investigation of the micromechanism of cleavage fracture in hypoeutectoid pearlitic R7T steel, commonly used for producing railway wheels. The steel possesses extensive Lüders deformation, which somewhat complicates finite element (FE) modelling and analyses of fracture behaviour. Standard Charpy V-notch specimens were used in order to analyse the fracture behaviour at quasistatic and impact loading. Finite element 3D calculations were performed and the elastic-plastic behaviour of notched bars up to the fracture was simulated. Detailed fractographic analysis was carried out on a number of Charpy V-notch specimens in order to investigate the origin site of cleavage fracture initiation and its distance from the notch root. The suitability of the three-criterion micromechanical model (Chen et al. Acta Materialia 51:1841–1855, 2003) for cleavage initiation was verified. The R7T steel under investigation exhibited a cleavage fracture stress of 1,837 MPa. Its independence on temperature evidenced the micromechanism of cleavage fracture to be microcrack propagation-controlled. For the investigated blunt-notched bend bars, an active volume exists ahead of the notch root in which pearlite colony-associated initiation sites are located. The cleavage fracture initiation of the steel is thus governed by the sites lying in the active volume. The active volume is determined by the values of three parameters. A plastic strain lying in interval from to (for the steel investigated from 0.033 to 0.108) is necessary to create a cleavage crack nucleus at any location within the active volume depending on the local pearlite properties. A stress triaxiality parameter ranging from h _min to h _max (from 0.93 to 1.39) is supposed to prevent the blunting process at the site of the cleavage nucleus. Once the main principal stress σ ₁ exceeds the local cleavage fracture stress σ _CFmin, an unstable global cleavage fracture occurs in a blunt-notched bar.     

Comparison of cohesive zone parameters and crack tip stress states between two different specimen types

The cohesive zone parameters (separation energy and cohesive strength) and the crack tip triaxialities are compared between a compact tension (CT) and a double edge notched tension (DENT) specimen with smooth side-surfaces. The material is a pressure vessel steel 20MnMoNi55. The cohesive zone parameters are determined by fitting the simulated crack extensions near the midsection to the experimental data. The purpose of the study is to understand the relationship between the cohesive zone parameters and the crack tip stress triaxiality. The results show that for the same cohesive zone parameters the crack tip triaxiality near the midsection is lower in DENT specimens than in CT specimens. When the separation energy is set constant for CT and DENT specimens, the cohesive strength for the DENT specimens should be significantly lower than that for the CT specimens in order to make the simulated crack extensions near the midsection fit to the experimental data. Near the midsection, the cohesive strength and crack tip triaxiality influence each other: the specimen with a higher stress triaxiality has a higher cohesive strength; an increase of cohesive strength results in an increase of the crack tip triaxiality.     

Investigating creep rupture and damage behaviour in notched P92 steel specimen using a microscale modelling approach

Idealized random grains separated by pseudo grain boundaries were generated by using Voronoi tessellation to simulate the polycrystalline microstructure. Combined with finite element analyses, this approach made it possible to addressing crack initiation and progressive failure due to crack growth in notched bar geometries of P92 steel at high temperature. The calculations provided good predictions for creep rupture lives of notched specimen with different notch radii and external stress. Simultaneously, irregular crack growth shape, intergranular crack mode, and wedge cracks at triple grain interaction were captured in the model. The crack initiation positions were found to be influenced by notch radius and applied stress causing high stress triaxiality at the subgrain level. Furthermore, the preferential crack growth directions were changed as the notch varied from sharp to blunt.     

Modelling of ductile fracture initiation in strength mismatched welded joint

In this paper, the effect of strength mismatch and width of the welded joints on the stress–strain distribution in the crack tip region has been discussed. The single-edge notched bend (SENB) specimens (precrack length a₀/W = 0.32) were experimentally and numerically analysed. The model of local approach to fracture, proposed by Gurson, Tvergaard and Needleman, was used. High-strength low-alloyed (HSLA) steel was used as a base metal in quenched and tempered condition. The flux-cored arc-welding process in shielding gas was used. Two different fillers were selected to make over- and undermatched weld metal. The experimental analysis of fracture behaviour of the over- and undermatched welded joints was followed by numerical computations of void volume fraction in front of the crack tip. The critical void volume fraction, f_c, used in prediction of the crack growth initiation on the SENB specimen had been previously determined on a round smooth specimen. Three widths of weld metal were considered: 6, 12 and 18 mm. A comparison of the crack tip opening displacement (CTOD) values corresponding to crack initiation in the SENB specimens is given, as determined both experimentally and using the GTN model.     

Fracture of austenitic steel subject to a wide range of stress triaxiality ratios and crack deformation modes

The stress triaxiality ratio (hydrostatic pressure divided by von Mises equivalent stress) strongly affects the fracture behaviour of materials. Various fracture criteria take this effect into consideration in their effort to predict failure. The dependency of the fracture locus on the stress triaxiality ratio has to be investigated in order to evaluate these criteria and improve the understanding of ductile fracture.This was done by comparing the experimental results of austenitic steel specimens with a strong variation in their stress triaxiality ratios. The specimens had cracks with varying depths and crack tip deformation modes; tension, in-plane shear, and out-of-plane shear. The crack growth in fracture mechanics specimens was compared with the failure of standard testing specimens for tension, upsetting and torsion. By associating the experimental results with finite element simulations it was possible to compare the critical plastic equivalent strain and stress triaxiality ratio values at fracture. In the investigated triaxiality regime an exponential dependency of the fracture locus on the stress triaxiality ratio was found.     

Tensile tests and analyses of notched specimens fabricated from high strength steels using a generalized yield model

This article summarises the uniaxial tension tests for 20 notched bars fabricated from high strength steel Q345 specified in Chinese National Standards. The effects of the notch radius, r, and that of the notch depth ratio, d/D , on the ductility and fracture resistance of this high strength steel are examined. The experimental data are further analyzed using a generalized yield model together with an elliptical fracture stress envelope originally proposed by the first author. The experimental results demonstrate that cracks initiate at the notched section, with the fracture surface filled with many dimples and shearing marks. Specimens with a sharper notch radius (a smaller r) and a larger notch depth (a smaller d/D ratio) show poor ductility, but high fracture strength. The stress field computed from the numerical procedure including the generalized yield model indicates that the crack initiation occurs at the centre of the notched section which experiences the highest stress triaxiality ratio (σ_m/σ_seq) . As the stresses at the notched section reach the limiting values determined from the elliptical fracture criterion, macroscopic fracture failure in the notched bar occurs.     

Fracture study of a polyacetal resin

Fracture experiments have been carried out with unnotched and notched tensile specimens of a polyacetal resin at room temperature at various rates of extension. An increase of approximately 13% in yield stress was observed in the unnotched tests with increased deformation rates from 1–1000 mm min^–1, whilst strain to failure was reduced from about 85% to approximately 0.05%. In all specimens, failure appeared to have taken place by initiation of a microscopic flaw upon yielding, and this flaw then slowly grew into a critical size when catastrophic fracture set in. In the slow-growth region, the mechanism of failure was by void growth whereby the lamellar texture was transformed into a fibrillar one. However, on the rapid fracture surface, there was evidence of lamellar texture being retained, but with small voids in the stacks of lamellae, In notched fracture specimens containing sharp razor cut, a fracture toughness, K _lc, of approximately 5 MPa m^1/2 was obtained between crosshead speeds of 0.5 and 50 mm min^–1. At speeds of 500 and 1000 mm min^–1, the K _lc was reduced to 4 MPa m^1/2. In the absence of a starter crack, blunt notch fracture toughness of about 6.3 MPa m^1/2 was observed at all test speeds. In specimens with a razor cut, the slow-growth region decreased as test speed increased; this can be used to construct an R-curve.     

Experimental investigation on low cycle fatigue and fracture behaviour of a notched Ni‐based superalloy at elevated temperature

In order to investigate the effects of stress concentration on low cycle fatigue properties and fracture behaviour of a nickel‐based powder metallurgy superalloy, FGH97, at elevated temperature, the low cycle fatigue tests have been conducted with semi‐circular and semi‐elliptical single‐edge notched plate specimens at 550 and 700 °C. The results show that the fatigue life of the notched specimen decreases with the increase of stress concentration factor and the fatigue crack initiation life evidently decreases because of the defect located in the stress concentration zone. Moreover, the plastic deformation induced by notch stress concentration affects the initial crack occurrence zone. The angle α of the crack occurrence zone is within ±10° of notch bisector for semi‐circular notched specimens and ±20° for semi‐elliptical notched specimens. The crack propagation rate decreases to a minimum at a certain length, D, and then increases with the growth of the crack. The crack propagation rate of the semi‐elliptical notched specimen decelerates at a faster rate than that of the semi‐circular notched specimen because of the increase of the notch plasticity gradient. The crack length, D, is affected by both the applied load and the notch plasticity gradient. In addition, the fracture mechanism is shown to transition from transgranular to intergranular as temperature increases from 550 to 700 °C, which would accelerate crack propagation and reduce the fatigue life.     

On the influence of state of stress on ductile failure initiation in high strength steels

The effect of stress state on the effective plastic strain to initiate ductile failure in three high strength steels is investigated. Circumferentially notched tension specimens were used, and failure initiation strains were correlated with a parameter which is a measure of the “triaxiality” of the stress state. Results are given for both in-plane and through-thickness directions in rolled plate.The failure initiation data, together with appropriate stress-strain fields, and estimates of characteristic lengths over which failure initiates, have been used to predict failure initiation at notches and at crack tips; conventional fracture mechanics parameters, such as K_I at crack extension in small scale yielding are estimated for one of the high strength steels.     

Stress state dependent fracture behaviour of porous PM steels

Ductile fracture of metals is a result of void nucleation, growth and coalescence. Various criteria have been proposed to model the ductile fracture strain as a function of the stress triaxiality that greatly influence the fracture process. In the present investigation, the well-known Rice and Tracey approach (with a re-evaluation conducted by Huang) was used to model the ductile fracture behaviour of two porous steels, produced by Powder Metallurgy (PM): a ferritic–pearlitic Fe–0.4%C PM steel and a high-strength steel produced by using diffusion-alloyed Fe–4%Ni–1.5%Cu–0.5%Mo–0.5%C powder. Tensile, compressive and bending tests were carried on un-notched and notched specimens. The experimental curves were used as a reference for the Finite Element (FE) modeling of the tests aimed at evaluating the equivalent fracture strain at fracture and the correspondent stress triaxiality for each geometry. The results obtained for the Fe–0.4%C PM steel proved the suitability of the modified Rice and Tracey relationship to successfully obtain a simple fracture criterion. However, in the case of high-strength steel, a mixed ductile/brittle fracture behaviour was observed because of the microstructural heterogeneity of the alloy. Because of this, the Rice and Tracey model overestimates the experimental equivalent fracture strains and has to be accordingly corrected.     

Experimental and numerical analysis of delamination growth in double cantilever laminated beams

Delamination crack growth in laminated composites is investigated using experiments and finite element (FE) models. Tests are performed on cross-ply graphite/epoxy specimens under static conditions. The load-displacement response is monitored in the tested coupons along with crack length. The FE models employ a cohesive layer that is used to simulate the debonding and crack propagation. The cohesive parameters are calibrated from the experimental load-displacement curves. Crack growth and strain measurements are compared with those from the FE models. The predicted results from the FE models are in good agreement with the test results. The same modeling approach is also used to simulate crack propagation in the transverse direction of a notched laminate. The proposed FE analysis with cohesive layers can simplify fracture toughness assessment in multilayered specimens.     

考虑罗德角参数的钢材薄板延性断裂标定方法

在钢结构中钢梁腹板的断裂表现出很强的剪切断裂特征,因此在预测其断裂行为时除了考虑应力三轴度以外,还应考虑与剪切状态有关的罗德角参数的影响。该文提出了一种适用于钢材薄板延性断裂预测的标定方法。共设计了5组试件,分别为平板试件、开孔平板、开槽平板、90°剪切平板和45°剪切平板。利用平板试件和停机平板试件单向拉伸试验,结合有限元反演得到钢材的全过程真实应力-应变曲线。然后将其代入各试件有限元模型进行计算,标定出各试件对应的等效断裂应变、平均应力三轴度和平均罗德角参数。最后利用Matlab优化工具箱对断裂模型进行优化拟合,确定模型中的各项待定参数。     

Prediction of ductile fracture in axisymmetric tension by void coalescence

Fracture strains are predicted for ductile materials undergoing void growth and coalescence. The calculation scheme is based on Gurson–Tvergaard yield function and its associated flow rule. Fracture condition is identified by vanishing stress-carrying capacity of the material. The plastic flow parameters are all determined from experimental evidences for a variety of alloys. Comparison among predicted fracture strains and experimental ones is given for a wide range of conventional and superplastic materials as well as powder compacts. Finally an approximate fracture criterion is proposed.     

Micromechanical modeling of fracture initiation in 7050 aluminum

Mechanical testing and finite element calculations have been carried out to characterize the fracture initiation behavior of the high-strength aluminum alloy 7050-T7451. Results show that fracture initiation is well predicted for two specimen types of differing constraint using the stress-modified, critical plastic strain micromechanical model. The relation between stress triaxiality and critical plastic strain was found from a series of notched tensile specimens. Data from these tests are interpreted using both companion finite element modeling and common, semi-empirical relations, and these two approaches are compared. Multiple, interrupted tests of standard, highly constrained single edge notched bend specimens are used to obtain the J–R curve in 7050 for small amounts of tearing to experimentally identify initiation. Companion modeling and the stress-modified, critical plastic strain relation are used to find the length scale for fracture, l^*, needed for initiation predictions. The calibrated stress-modified, critical plastic strain relation and length scale are then used to predict fracture initiation of a low-constraint specimen. The prediction is within 5% of the experimental measurements. Finally, various aspects of the procedure followed in the present work are compared to previous efforts using similar approaches.     

Local stress–strain field intensity approach to fatigue life prediction under random cyclic loading

According to the characteristic of the local behavior of fatigue damage, on the basis of stress field intensity approach, a theory of local stress–strain field intensity for fatigue damage at the notch is developed in this paper, which can take account of the effects of the local stress–strain gradient on fatigue damage at the notch. In order to calculate the local stress–strain field intensity parameters, an incremental elastic-plastic finite element analysis under random cyclic loading is used to determine the local stress–strain response. A local stress–strain field intensity approach to fatigue life prediction is proposed by means of elastic-plastic finite element method for notched specimens. This approach is used to predict fatigue crack initiation life, and good correlation was observed with U-shape notched specimens for normalized 45 steel.