A VIEW ON DUCTILE-FRACTURE MODELLING

Theoretical models of ductile fracture are reviewed in terms of experimental results from metallurgical studies of ductile fracture in metals and alloys. It is shown that the plastic limit-load model, which is based on a criterion of void coalescence by internal microscopic necking of the intervoid matrix, is fully consistent with scanning electron microscope (SEM) observations of both the ductile-fracture surface and the microstructure immediately adjacent to the fracture surface. On the other hand, the dilational-plastic models of ductile fracture, which are based on the dilational-growth of spherical voids to some arbitrary critical void-volume fraction, are inconsistent with the microstructural observations of ductile fracture. This inconsistency between the dilational-plastic models and experimental results is shown to be the combined effect of neglecting the controlling influence of extensional void-growth and the failure to incorporate a physically realistic criterion of void coalescence.
The problems of modelling the ductile crack-growth process by both analytical and numerical (finite element) studies, where problems of uniqueness of the plastic velocity field may occur, are also considered. The limitations of the finite-element method in modelling void-coalescence problems, where the equations of plasticity are of second-order hyperbolic form, are also discussed.     

Anisotropic ductile fracture of Al 2024 alloys

The anisotropic fracture of the 2024-T351 aluminium alloy is investigated using a micromechanics-based damage model accounting for the effect of the void aspect ratio and void distribution. The 2024-T351 Al alloy contains precipitation free bands in which most void nucleating particles are located. The presence of these bands, which are parallel to the rolling direction, primarily controls the distribution of damage and overall fracture anisotropy. The primary void nucleating particles also present a preferential elongation in the rolling direction. These key microstructural features have been determined using quantitative characterisation methods. The effects of void shape and void spacing on the fracture behaviour are elucidated by means of FE cell calculations. FE simulations of cylindrical notched round bars loaded in different orientations are made and compared with experimental data, allowing a better understanding of the damage process as well as the limitations of the modelling approach.     

A unified damage factor model for ductile fracture of steels with different void growth and shrinkage rates

Micromechanical fracture modelling is an effective method to predict ductile fracture in steel structures. This paper aims to establish a simplified and general fracture model for various loading conditions, which is convenient to calculate the instantaneous damage index. With the concept of the ductile damage factor, a unified ductile damage factor model considering the difference of rates between void growth and shrinkage has been proposed. Based on experiment results and finite element analysis, the model parameters for Q235B Chinese structural steel were calibrated. The ductile damage factor and equivalent plastic strain at fracture initiation were investigated. Comparison among the proposed model, experimental results, and the cyclic void growth model demonstrated the effectiveness and accuracy of the proposed model. A parametric study was conducted to investigate the influence of cyclic constitutive parameters on the accuracy of fracture prediction. The predicted results are acceptable while reducing those calibrated cyclic constitutive parameters by 20%.     

The effect of particle distribution on damage formation in particulate reinforced metal matrix composites deformed in compression

Image analysis results are reported on the generation of damage in particulate reinforced metal matrix composites during compressive deformation. The technique allows the automated collection of data on the incidence of particle fracture and void formation in the matrix as a function of important microstructural parameters such as local particle volume fraction and particle size. There is a strong relationship between damage and the local volume fraction of the reinforcement proving that damage formation is accentuated in regions of particle clustering. With the SiC reinforced materials examined, there was observed to be a change in dominance of damage mechanism from particle fracture at low local volume fractions to void formation in the matrix within strongly clustered regions. The results are compared with finite element (FE) modelling of the compressive deformation of clustered particles using a simple cluster of equi-spaced particles. The FE results suggest that plastic flow is generally inhibited in clustered regions. In certain highly clustered configurations shielding is such that flow does not occur in the heart of the cluster even at high levels of average plastic strain. The modelling suggests that the change in dominance of damage mechanism is related to the dramatic increase in tensile hydrostatic stresses in the matrix with higher levels of particle clustering.     

Stress triaxiality effect on void nucleation in ductile metals

The stress triaxiality effect on the strain required for void nucleation by particle‐matrix debonding has been investigated by means of micromechanical modelling. A unit‐cell model considering an elastic spherical particle embedded in an elastic‐plastic matrix was developed to the purpose. Particle‐matrix decohesion was simulated through the progressive failure of a cohesive interface. It has been shown that the parameters of matrix‐particle cohesive interface are correlated with macroscopic material properties. Here, a simple relationship for the maximum cohesive opening at interface failure as a function of material fracture toughness and yield stress has been derived. Results seem to confirm that, increasing stress triaxiality, the strain at which void nucleation is predicted to occur decreases exponentially in a similar way as for fracture strain. This result has substantial implications in modelling of ductile damage because it indicates that if the stress triaxiality is high enough, ductile fracture can occur at plastic strain lower than that necessary to nucleate damage for moderate or low stress triaxiality regime.     

Micro- and macroscopic damage and fracture behaviour of welding coarse grained heat affected zone of a low alloy steel: Mechanisms and modelling

Micro- and macroscopic damage and fracture behaviour of the simulated coarse grained heat affected zone (CGHAZ) of a high strength low alloy steel weld are studied through experimental and continuum damage mechanics (CDM) approaches. In order to study the damage and fracture behaviour of the CGHAZ carefully, weld thermal simulation technique is used to enlarge and to generate the region. The dynamic microprocesses of damage and fracture in the CGHAZ are observed through in situ techniques in conjunction with a scanning electron microscope (SEM) equipped with a tensile platform. Several mechanisms of void initiation and crack propagation are observed. A criterion for void initiation by cracking of the inclusion itself and/or debonding at the inclusion-matrix interface is derived. The macroscopic damage evolution law in the CGHAZ is measured through a new a.c. potential system and modelled by use of CDM. A damage evolution equation for the CGHAZ is presented. Comparison of experimental and modelling results is presented and good agreement is found. The effects of stress triaxiality on void initiation, damage evolution and failure in the CGHAZ are also discussed in the framework of CDM.     

Micromechanics of room and high temperature fracture in 6xxx Al alloys

The micromechanics of ductile fracture has made enormous progress in recent years. This approach, which was mostly developed in the context of structural integrity analysis, is becoming a key tool for materials scientists to optimize materials fracture properties and forming operations. Micromechanical models allow quantitatively linking fracture properties, microstructure features at multiple lengths scales, and manufacturing conditions. After briefly reviewing the state of the art, this paper illustrates the application of the micromechanics-based methodology by presenting the results of an investigation on the damage resistance of 6xxx Al produced by extrusion.The presence of coarse, elongated, particles is the key microstructural feature affecting the fracture behaviour of 6xxx Al. The detrimental elongated β-type particles are transformed into rounded α-type particles by heat treatment. In situ tensile tests revealed that, at ambient temperature, the α particles and the β particles oriented with the long axis perpendicular to the main loading direction undergo interface decohesion, while the β particles oriented perpendicular to the loading direction break into several fragments. At high temperatures, only interface decohesion is observed. Uniaxial tensile tests on notched and smooth round bars were performed on two different alloys, at different temperatures ranging between 20 °C and 600 °C, under different loading rates, while systematically varying the content in β versus α particles. The ductility increases with decreasing amount of β particles, increasing temperature and strain rates, and decreasing stress triaxiality.A viscoplastic extension of the Gurson model has been developed for capturing the complex hierarchy of damage mechanisms, coupled with viscoplastic and stress state effects. Three populations of voids are modelled while accounting for the different void nucleation mechanisms leading to different initial void aspect ratio. Proper modelling of the initial void aspect ratio and of its evolution with void growth was the key to predict the effect of the β → α conversion on ductility. The void coalescence criterion takes into account the presence of secondary voids resulting from particle fragmentation. The characteristics of particles entering the model were all measured experimentally. The temperature and rate dependent flow properties of the matrix material have been obtained by inverse modelling. The only fitting parameters are the critical stresses for void nucleation. The model is validated by comparing the predictions to the experimental data involving different relative proportion of α and β particles, temperature, loading rate and stress triaxiality. This type of model opens the path for an “alloy by design” strategy which relates end-use properties to upstream manufacturing operations.     

Investigation on fracture mechanisms of metals under various stress states

Fracture mechanisms for widely used metal materials are investigated under various loading conditions. Several specimens and different loading methods are deliberately designed to produce various stress states. The stress triaxiality is used to rate the level of tension and compression under various stress states. The stress triaxiality increases with adding a notch in the specimen under tension loading and decreases by changing the loading from tension to compression. Scanning electron microscopes are used to observe the microscopic features on the fracture surfaces. The fracture surfaces observed in the tests indicate that with the decreasing stress triaxiality the fracture mechanism for a given metal material includes intergranular cleavage, nucleation, growth, void coalescence, and local shear band expansion. With the fracture mechanisms changing from intergranular cleavage to nucleation, growth, and coalescence of voids, and expansion of a local shear band, four possible fracture modes can be observed, which are quasi-cleavage brittle fracture, normal fracture with void, shear fracture with void, and shear fracture without void. Quasi-cleavage brittle fracture and normal fracture with void are both normal stress-dominated fracture modes; however, their mechanisms are different. Shear fracture with and without void are both shear stress-dominated fracture, and shear fracture with void is also influenced by the normal stress. To a certain metal material, under high stress triaxiality, quasi-cleavage brittle fracture and normal fracture with void tend to occur, and under low stress triaxiality, shear fracture with and without void tend to occur. In addition, the critical positions and fracture criteria adapted to each fracture mode will also be different.     

Effect of pre-strain on fracture toughness of ductile structural steels under static and dynamic loading

The ductile fracture process consists of void nucleation, growth and coalescence. The whole ductile process can be divided into two successive steps: (I) the initial state to void nucleation, followed by (II) void growth up to void coalescence. Based on this suggestion, resistance to ductile fracture could be divided into the resistance to stage I and stage II, and accordingly the whole fracture toughness could be regarded to be due to contributions from stages I and II. The fracture toughness contributed from the two steps is, respectively, denoted as void nucleation-contributed fracture toughness and void growth-contributed fracture toughness. The effect of plastic pre-strain on the fracture toughness of ductile structural steels under static and dynamic loading (4.9 m/s) within the ductile fracture range was evaluated by summing contributions due to void nucleation-contributed and void growth-contributed fracture toughness. The effect of strain rate on fracture toughness was also investigated by the same means. The results show that both plastic pre-strain and high-speed loading decrease the void nucleation-contributed fracture toughness while their effects on the void growth-contributed fracture toughness depend on the variations in strength and ductility. Moreover, fracture toughness of structural steels generally decreases with increasing strain rate.     

Continuum damage mechanics modelling incorporating stress triaxiality effect on ductile damage initiation

Micromechanical modelling of void nucleation in ductile metals indicates that strain required for damage initiation reduces exponentially with increasing stress triaxiality. This feature has been incorporated in a continuum damage mechanics (CDM) model, providing a phenomenological relationship for the damage threshold strain dependence on the stress triaxiality. The main consequences of this model modification are that the failure locus is predicted to change as function of stress triaxiality sensitivity of the material damage threshold strain and that high triaxial fracture strain is expected to be even lower than the threshold strain at which the damage processes initiate at triaxiality as low as 1/3. The proposed damage model formulation has been used to predict ductile fracture in unnotched and notched bars in tension for two commercially pure α‐iron grades (Swedish and ARMCO iron). Finally, the model has been validated, predicting spall fracture in a plate‐impact experiment and confirming the capability to capture the effect of the stress state on material fracture ductility at very high stress triaxiality.     

Ductile fracture of materials with randomly distributed voids

International Journal of Fracture - A reliable determination of the onset of void coalescence is critical to the modelling of ductile fracture. Numerical models have been developed but rely mostly...     

Growth and coalescence of penny-shaped voids in metallic alloys

The growth and coalescence of penny-shaped voids resulting from particle fracture is a common damage process for many metallic alloys. A three steps modeling strategy has been followed to investigate this specific failure process. Finite element cell calculations involving very flat voids shielded or not by a particle have been performed in order to enlighten the specific features of a damage mechanism starting with initially flat voids with respect to more rounded voids. An extended Gurson-type constitutive model supplemented by micromechanics-based criteria for both void nucleation and void coalescence is assessed for the limit of very flat voids towards the FE calculations. The constitutive model is then used to generate a parametric study of the effects of the stress state, the microstructure and the mechanical properties on the ductility. Based on these results, a simple closed-form model for the ductility is finally proposed. The main outcomes of this study are that (i) the ductility of metal alloys involving penny-shaped voids is primarily controlled by the relative void spacing; (ii) the definition of an effective porosity in terms of an equivalent population of spherical voids is valid for low particle volume fraction; (iii) the presence of a particle shielding the void does not significantly affect the void growth rates and void aspect evolution; (iv) early fracture by void coalescence can occur under very low stress triaxiality conditions if the particle volume fraction is large enough, explaining that some alloys and composites can fail through a transgranular ductile fracture mode under uniaxial tension condition before the onset of necking; (v) the fracture mechanism moves from void growth controlled to void nucleation controlled when increasing the void nucleation stress, lowering the stress triaxiality, and increasing the initial void aspect ratio.     

Some contribution to process zone fracture mechanics

The mechanism of the ductile fracture is studied theoretically for the Al Alloy 7075-T6 specimens. A model for the interaction of a crack tip with a void nearby is analyzed by using the Modified Gurson's Model. Taking fracture criterion into consideration, the analysis of a crack propagation is carried out and besides the distribution of the equivalent plastic strain, the void volume fraction f and the localization are obtained. Microcracks nucleate on the ligament between crack and void, and grow and coalesce each other, and at last the main crack thus formed coalesces with the void and the coalescence of the crack and void is completed. And these phenomenon occurs in the localized region.The initiation of the microcrack of 7075 occurs at small J and the microcrack penetration between crack and void occurs at larger J, and the propagation does not occur smoothly. These results coincide with the results of the experiments by FRASTA (FRActure Surface Topographic Analysis) and Fractography.     

MODELLING FOR TRANSCRYSTALLINE AND INTERCRYSTALLINE FRACTURE BY VOID NUCLEATION AND GROWTH

A criterion has been formulated for transcrystalline and intercrystalline fracture caused by the evolution of voids located both in a grain and on grain boundaries. The criterion is based on the idea of plastic collapse for a unit cell that is a regular structural mezovolume of polycrystalline material. The criterion does not require the introduction of any empirical parameters, such as critical void size, critical size of ligament between voids and critical void volume fraction, which are used in most models.
Modelling has been performed for void nucleation and growth in a grain and on grain boundaries for elastic–plastic deformation and under creep conditions. A scheme is proposed to describe the transition from transcrystalline to intercrystalline cavitation fracture as a function of strain rate and temperature.
The effect of stress triaxiality on the critical strain and the lifetime for both transcrystalline and intercrystalline fracture has been investigated. A comparison of the results predicted by the suggested criterion with available empirical data has been performed.     

The effect of carbon nanotubes on the fracture toughness and fatigue performance of a thermosetting epoxy polymer

Multi-walled carbon nanotubes, with a typical length of 140 μm and a diameter of 120 nm, have been used to modify an anhydride-cured epoxy polymer. The modulus, fracture energy and the fatigue performance of the modified polymers have been investigated. Microscopy showed that these long nanotubes were agglomerated, and that increasing the nanotube content increased the severity of the agglomeration. The addition of nanotubes increased the modulus of the epoxy, but the glass transition temperature was unaffected. The measured fracture energy was also increased, from 133 to 223 J/m² with the addition of 0.5 wt% of nanotubes. The addition of the carbon nanotubes also resulted in an increase in the fatigue performance. The threshold strain-energy release-rate, G _th, increased from 24 J/m² for the unmodified material to 73 J/m² for the epoxy with 0.5 wt% of nanotubes. Electron microscopy of the fracture surfaces showed clear evidence of nanotube debonding and pull-out, plus void growth around the nanotubes, in both the fracture and fatigue tests. The modelling study showed that the modified Halpin–Tsai equation can fit very well with the measured values of the Young’s modulus, when the orientation and agglomeration of the nanotubes are considered. The fracture energy of the nanotube-modified epoxies was predicted, by considering the contributions of the toughening mechanisms of nanotube debonding, nanotube pull-out and plastic void growth of the epoxy. This indicated that debonding and pull-out contribute to the toughening effect, but the contribution of void growth is not significant. There was excellent agreement between the predictions and the experimental results.     

Numerical modelling of fracture initiation in large steel specimens at impact

Dynamic mode I fracture initiation in impact loaded single edge bend specimens with a quarter notch is investigated by numerical modelling and the results are compared with sets of experimental data from two different steel qualities. The finite element analysis include 2D (two-dimensional) plane strain, 2D plane stress and 3D models. No crack growth is included in the calculations. The impact velocities are approximately 15, 30 and 45 m/s and the specimen size is 320×75 mm² with a thickness of 20 or 40 mm. Some specimens have side grooves. Details of the deflection of the specimens are accurately reproduced prior to crack initiation both by the 2D plane strain model and by the 3D model.The experiments were performed in the ductile to brittle transition region. It is assumed that cleavage fracture initiation can be predicted by the Ritchie-Knott-Rice (RKR) model, i.e. cleavage fracture initiates when the opening stress exceeds the macroscopic cleavage stress over a fixed, critical distance. At an impact velocity of 15 m/s, fracture initiation by void nucleation and growth is observed, though the RKR-conditions is seemingly fulfilled according to the computational results. Possible limitations in the use of the RKR model are discussed.     

Void configuration under constrained deformation in ductile matrix materials

SiC particle reinforced Al 2025 aluminum alloy composite is used for tensile tests. The ductile fracture by nucleation, growth and coalescence of micro voids, particle cracking and the interfacial debonding under the different constraint conditions, which are obtained by changing the notch radius, is analyzed. The effect of the local constraint on the respective damage phenomenon is analyzed using the axi-symmetric unit cell FE model by changing the local stress triaxiality and side constraint. The results show that the fracture process of the notched tensile bar is simulated well and the damage phenomena agree qualitatively with the experimental ones. Finally the effect of constraint on the void configuration and coalescence is investigated experimentally using three-dimensional fracture surface observations using the SEM (scanning electron microscope) and three-dimensional imaging analysis method. The constraint effect is analyzed by changing the specimen’s shape. The experimental results show that the void aspect ratio is decreased with increases of SiC particle volume fraction in aluminum alloy. The final void volume fraction at fracture also changes by the specimen shape and the material’s structure.     

The prediction of damage accumulation during upsetting tests based on McClintock's model

The finite element modelling of various upsetting tests permits the evaluation of the stress and strain history at any point within the specimen. The present work highlights the predictive capability of McClintock's void growth model, which is applied to evaluate the rate of damage and the damage accumulation at selected points in AISI 1045 steel specimens during upsetting tests. In spite of certain shortcomings inherent in the model, it does seem to be of assistance in explaining the lower levels of fracture strains and subsurface cracking observed in collar test samples. An attempt has been made to predict a fracture locus based on linear straining paths, and to compare experimental, observations with the predicted locus.     

基于微孔贯通细观损伤模型的金属韧性断裂分析

韧性材料断裂过程通常可看作是材料内部微孔洞的形核、扩展及相互贯通的积累。经典的Gurson- Tvergaard (GT)模型能够很好地模拟具有变形均匀、各向同性的孔洞的萌生及扩展过程;但无法模拟由孔洞贯通而引起的局部变形过程,因此需要对其修正,引入相应的孔洞贯通准则。该文采用两种贯通准则对经典GT模型进行修正,即Thomason的塑性极限载荷准则和临界等效塑性应变准则。借助用户自定义程序UMAT将采用这两种贯通准则修正的GT本构关系嵌入至商用有限元软件ABAQUS中,从而可通过对金属材料应力状态和断裂机理的分析控制孔洞的贯通。以一组含有不同缺口根半径的圆棒拉伸试验件为例,分析了该类金属构件自孔洞萌生至最终断裂的整个损伤演化过程,并与试验数据进行了对比,验证了该模型的有效性。该文还讨论了金属断裂过程中应力三轴度对微裂纹萌生与扩展的影响。     

Modelling macroscopic imperfections for the prediction of flow localization and fracture

Ductile materials subjected to plastic deformation experience the different stages of void nucleation, growth and coalescence that eventually lead to ductile fracture. Several models have been proposed to assess the influence of this damage on flow localization and fracture. In general, the plastic behaviour is represented by a constitutive model for porous or damaged materials. It is typical to introduce a material imperfection, with porosity higher than average, which evolves up to localization and fracture. However, the void volume fraction in the imperfection is chosen more or less arbitrarily. In the present work, a model that evaluates this void volume fraction more rigorously is developed. The forming limit diagram (FLD) for a dual phase‐steel is calculated using the damage‐based imperfection calculation and validated with experimental results. The effect of void shape on the imperfection porosity level and limit strains in sheet forming is also assessed with the present method.