Ductile shear failure or plug failure of spot welds modelled by modified Gurson model

For resistance spot welded shear-lab specimens, interfacial failure under ductile shearing or ductile plug failure are analyzed numerically, using a shear modified Gurson model. The interfacial shear failure occurs under very low stress triaxiality, where the original Gurson model would predict void nucleation and very limited void growth. Void coalescence would therefore be largely postponed. However, using the shear modification of the Gurson model, recently introduced by Nahshon and Hutchinson (2008) [1], failure prediction is possible at zero or even negative mean stress. Since, this shear modification has too large effect in some cases where the stress triaxiality is rather high, an extension is proposed in the present study to better represent the damage development at moderate to high stress triaxiality, which is known to be well described by the Gurson model. Failure prediction and tensile response curves for an interfacial shear failure or a ductile plug failure, are here compared when using either the original Gurson model, the shear modified model, or the extension to the shear modified model. The suggested extension makes it possible to use the shear modified model as a simple way of accounting for damage development under low triaxiality shearing, without further increasing the damage rate in regions of moderate to high stress triaxiality.     

A modified Gurson model and its application to punch-out experiments

Recent experimental evidence has reiterated that ductile fracture is a strong function of stress triaxiality. Under high stress triaxiality loading, failure occurs as a result of void growth and subsequent necking of inter-void ligaments while under low stress triaxiality failure is driven by shear localization of plastic strain in these ligaments due to void rotation and distortion. The original Gurson model is unable to capture localization and fracture for low triaxiality, shear-dominated deformations unless void nucleation is invoked. A phenomenological modification to the Gurson model that incorporates damage accumulation under shearing has been proposed. Here we further extend the model and develop the corresponding numerical implementation method. Several benchmark tests are performed in order to verify the code. Finally, the model is utilized to model quasi-static punch-out experiments on DH36 steel. It is shown that the proposed modified Gurson model, in contrast to the original model, is able to capture the through-thickness development of cracks as well as the punch response. Thus, the computational fracture approaches based on the modified Gurson model may be applied to shear-dominated failures.     

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.     

Void growth and coalescence during high velocity impact

The extent of void growth and cracking due to ductile fracture occurring during symmetric Taylor cylinder impact tests on leaded brass has been determined experimentally. Void growth occurs within these predominantly compressively-loaded specimens through the development of large tensile hydrostatic stresses along the specimen axis near the impact face during expansion of the cylinder, termed “mushrooming”.

The measured porosities have been compared to predictions using a constitutive model based on the Gurson (1975, Ph.D. Thesis, Brown University) yield function, implemented within the DYNA2D finite element code. The initiation of void coalescence and subsequent crack development was also predicted using the approach of Tvergaard and Needleman (1984, Acta Metall. 32, 157) based on a critical porosity criterion.

The calculations were able to qualitatively predict the development of the porous zone and void coalescence within the impact specimens; however, the predicted void growth exceeded that observed experimentally and the predicted extent of void coalescence was too large. It is suggested that the primary source of error lies in excessively high predicted void growth rates using the Gurson yield function at high stress triaxiality levels.     

APPLICATION OF LOCAL DAMAGE MODELS TO THE NUMERICAL ANALYSIS OF DUCTILE RUPTURE

Abstract— Two damage models were implemented into the finite element program ADINA to study the correlation between microscopical damage and macroscopical material failure. In the first model, based on the Gurson yield function the nucleation, growth and the coalescence of voids were incorporated into the constitutive relations. In the second model the void growth was determined according to the Rice and Tracey model using the von Mises yield function, and material failure was simulated by eliminating the elements where the critical void growth ratio was exceeded. The numerical results for the local and global behaviour of the specimens were compared with experiments. The generality of the damage parameters was checked by investigating several specimen geometries. Both damage models deliver qualitatively consistent results with regard to the influence of the stress triaxiality on the void growth and on the beginning of the material failure. However, the Gurson model gives a more accurate numerical simulation because the damage development and the stress drop continue after the onset of void coalescence while the critical void growth model causes less convergence problems in the simulation of large crack extension. The J _n-curve was estimated on the basis of both models.     

Constitutive modeling of void shearing effect in ductile fracture of porous materials

Many solids, including geomaterials and commercially available metallic alloys, can be considered as a porous media. The Gurson-like model has been proposed to describe plastic deformation for such type of materials. It has attracted a great deal of attention and various modifications to this model have been proposed. The constitutive equations of Gurson-like model are governed by the first and second stress invariants and the current void volume fraction of the material. Tvergaard and Needleman included void nucleation, growth and coalescence to Gurson model in a phenomenological way [Tvergaard V, Needleman A. Analysis of the cup-cone fracture in a round tensile bar. Acta Metall 1984;32(1):157–69] – thus suggesting the so called GTN model. Meanwhile, little attention was given to the dependence of the damage evolution on the third stress invariant. McClintock et al. [McClintock FA, Kaplan SM, Berg CA. Ductile fracture by hole growth in shear bands. Int J Fract Mechan 1966;2(4):614–27] proposed damage model based on the void evolution in localized shear banding. In the present paper, a separate internal damage variable which differs from the conventional void volume fraction is introduced. The GTN model is further extended to incorporate the void shearing mechanism of damage, which depends on the third stress invariant. Numerical aspects are addressed concerning the integration of the proposed constitutive relations. A unit cell is studied to illustrate the intrinsic mechanical behavior of the modified model. Computations of the deformation in axisymmetric and transverse plane strain tension are also performed. Realistic crack modes in these simulations are achieved for the modified GTN model.     

On the numerical implementation of a shear modified GTN damage model and its application to small punch test

Gurson-type models have been widely used to predict failure during sheet metal forming process. However, a significant limitation of the original GTN model is that it is unable to capture fracture under relatively low stress triaxiality. This paper focused on the fracture prediction under this circumstance, which means shear-dominated stress state. Recently, a phenomenological modification to the Gurson model that incorporates damage accumulation under shearing has been proposed by Nahshon and Hutchinson. We further calibrated new parameters based on this model in 22MnB5 tensile process and developed the corresponding numerical implementation method. Lower stress triaxiality were realized by new-designed specimens. Subsequently, the related shear parameters were calibrated by means of reverse finite element method and the influences of new introduced parameters were also discussed. Finally, this shear modified model was utilized to model the small punch test (SPT) on 22MnB5 high strength steel. It is shown that the shear modification of GTN model is able to predict failure of sheet metal forming under wide range of stress state.     

Numerical modeling of elasto-plastic porous materials with void shape effects at finite deformations

A new constitutive model for elasto-plastic (rate-independent) porous materials subjected to general three-dimensional finite deformations is presented. The new model results from simple modifications of an earlier model of Kailasam and Ponte Castañeda (1997, 1998) [40], [41] so that it reproduces the exact spherical and cylindrical shell solution (composite sphere and composite cylinder assemblage) under purely hydrostatic loadings, while predicting (by calibration) accurately the void shape evolution according to the recent “second-order” model of Danas and Ponte Castañeda [17]. Furthermore, the present model is based on a rigorous homogenization method which is capable of predicting both the constitutive behavior and the microstructure evolution of porous materials. The microstructure is described by voids of arbitrary ellipsoidal shapes and orientations and as a result the material exhibits deformation-induced (or morphological) anisotropy at finite deformations. This is in contrast with the well-known Gurson [32] model which assumes that the voids remain spherical during the deformation process and thus the material remains always isotropic. The present model is implemented numerically in a finite element program where a three-dimensional thin-sheet (butterfly) specimen is subjected to a combination of shear and traction loading conditions in order to examine the effect of stress triaxiality and shearing upon material failure. The ability of the present model to take into account the nontrivial evolution of the microstructure and especially void shape effects leads to the prediction of material failure even at low stress triaxialities and small porosities without the use of additional phenomenological damage criteria. At high stress triaxialities, the present model gives similar predictions as the Gurson model.     

Use of a modified Gurson model for the failure behaviour of the clinched joint on Al6061 sheet

This paper discusses the ductile fracture behaviour of the clinched joint on Alloy 6061 sheets. Failure behaviour of the clinched joint is associated with the nucleation, growth and coalescence of voids within the microstructure. Various corrections to the original Gurson model are proposed to allow for instability and final fracture of the material. This paper is concerned with the application of the modified Gurson–Tvergaard–Needleman damage developed by Ken Nahshon and Zhenyu Xue. The results of the tension tests are compared with those of numerical analysis to obtain the initial void volume fraction f₀ and the shear damage coefficients k_w. The modified Gurson–Tvergaard–Needleman model is used to describe the failure behaviour and the shear strength of the clinched joint. Good agreement is shown between the experiment results and the numerical results on the location of fracture and the maximum failure load.     

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。     

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.     

An exponential ductile continuum damage model for metals

In the frame of continuum damage mechanics an isotropic ductile plastic damage model is derived. The model is based on void damage variable, defined using effective stress concept and thermodynamics. The damage evolution from this model is exponential with equivalent plastic strain as experienced in some low carbon steel like AISI 1015. The damage model is sensitive to stress triaxiality and emulates the damage evolution as recorded in experiments conducted on such metals. The model is validated by comparison with the Rice-Tracey model and other experimental results published in the literature. This model can be used to study the growth and coalescence of micro voids, influence of stress triaxiality on strain to rupture and crack initiation phenomena.     

Influence of Triaxial Stress State on Ductile Fracture Strength of Polycrystalline Nickel

In this contribution, the effect associated with stress triaxiality on ductile damage evolution in high purity nickel has been investigated from both experimental and theoretical points of view. Tensile tests on smooth and notched round bar specimens were performed to calibrate the fracture strain in a wide range of stress triaxiality. The capability of the Gurson model to reproduce and predict physical failure behaviour was examined. It was shown that stress triaxiality played a major role on damage evolution as demonstrated by the progressive reduction of material ductility under increasing triaxial states of stress.     

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.     

Collapse and coalescence of spherical voids subject to intense shearing: studied in full 3D

Micro-mechanical 2D cell model studies have revealed ductile failure during intense shearing to be governed by the interaction of neighbouring voids, which collapse to micro-cracks and continuously rotate and elongate until coalescence occurs. For a three-dimensional void structure, this implies significant straining of the matrix material located on the axis of rotation. In particular, the void surface material is severely deformed during shearing and void surface contact is established early in the deformation process. This 3D effect intensifies with decreasing stress triaxiality and complicates the numerical analysis, which is also reflected in published literature. Rather than moving towards very low triaxiality shearing, work has focused on extracting wide-ranging results for moderate stress triaxiality (T ~ 1), in order to achieve sufficient understanding of the influence of initial porosity, void shape, void orientation etc. The objective of this work is to expand the range of stress triaxiality usually faced in 3D cell model studies, such that intense shearing is covered, and to bring forward details on the porosity and void shape evolution. The overall material response is presented for a range of initial material configurations and loading conditions. In addition, a direct comparison to corresponding 2D cell model predictions for circular cylindrical voids under plane strain shearing is presented. A quantitatively good agreement of the two model configurations (2D vs. 3D) is obtained and similar trends are predicted. However, the additional layer of matrix material, connecting voids in the transverse direction, is concluded to significantly influence the void shape evolution and to give rise to higher overall ductility. This 3D effect is demonstrated for various periodic distributions of voids.     

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.     

On the Effect of Stress Triaxiality on Void Coalescence

Failure of ductile materials is often related to coalescence of microscopic voids. The stress triaxiality is one of the primary factors that influence the coalescence. In the present work, a 3D unit cell model is employed to investigate this effect. The cell model contains two aligned voids. A coalescence criterion is proposed in which the critical void volume fraction is expressed in terms of stress triaxiality.     

Three-dimensional cell model analyses of void growth in ductile materials

Three-dimensional micromechanical models were developed to study the damage by void growth in ductile materials. Special emphasis is given to the influence of the spatial arrangement of the voids. Therefore, periodical void arrays of cubic primitive, body centered cubic and hexagonal structure are investigated by analyzing representative unit cells. The isotropic behaviour of the matrix material is modelled using either v. Mises plasticity or the modified Gurson-Tvergaard constitutive law. The cell models are analyzed by the large strain finite element method under monotonic loading while keeping the stress triaxiality constant. The obtained mesoscopic deformation response and the void growth of the unit cells show a high dependence on the value of triaxiality. The spatial arrangement has only a weak influence on the deformation behaviour, whereas the type and onset of the plastic collapse behaviour are strongly affected. The parameters of the Gurson-Tvergaard model can be calibrated to the cell model results even for large porosity, emphasizing its usefulness and justifying its broad applicability.