Modification of the Gurson Model for shear failure

Recent experimental evidence points to limitations in characterizing the critical strain in ductile fracture solely on the basis of stress triaxiality. A second measure of stress state, such as the Lode parameter, is required to discriminate between axisymmetric and shear-dominated stress states. This is brought into the sharpest relief by the fact that many structural metals have a fracture strain in shear, at zero stress triaxiality, that can be well below fracture strains under axisymmetric stressing at significantly higher triaxiality. Moreover, recent theoretical studies of void growth reveal that triaxiality alone is insufficient to characterize important growth and coalescence features. As currently formulated, the Gurson Model of metal plasticity predicts no damage change with strain under zero mean stress, except when voids are nucleated. Consequently, the model excludes shear softening due to void distortion and inter-void linking. As it stands, the model effectively excludes the possibility of shear localization and fracture under conditions of low triaxiality if void nucleation is not invoked. In this paper, an extension of the Gurson model is proposed that incorporates damage growth under low triaxiality straining for shear-dominated states. The extension retains the isotropy of the original Gurson Model by making use of the third invariant of stress to distinguish shear dominated states. The importance of the extension is illustrated by a study of shear localization over the complete range of applied stress states, clarifying recently reported experimental trends. The extension opens the possibility for computational fracture approaches based on the Gurson Model to be extended to shear-dominated failures such as projectile penetration and shear-off phenomena under impulsive loadings.     

Modelling of plastic flow localisation and damage development in friction stir welded 6005A aluminium alloy using physics based strain hardening law

Plastic flow localisation and ductile failure during tensile testing of friction stir welded aluminium specimens are investigated with a specific focus on modelling the local, finite strain, hardening response. In the experimental part, friction stir welds in a 6005A-T6 aluminium alloy were prepared and analysed using digital image correlation (DIC) during tensile testing as well as scanning electron microscopy (SEM) on polished samples and on fracture surfaces. The locations of the various regions of the weld were determined based on hardness measurements, while the flow behaviour of these zones was extracted from micro-tensile specimens cut parallel to the welding direction. The measured material properties and weld topology were introduced into a 3D finite element model, fully coupled with the damage model. A Voce law hardening model involving a constant stage IV is used within an enhanced Gurson type micro-mechanical damage model, accounting for void nucleation, growth and coalescence, as well as void shape evolution. The stage IV hardening, observed in Simar et al. (2010), was found to increase the stiffness during plastic flow localisation as well as to postpone the onset of fracture as determined by the void coalescence criterion. Furthermore, the presence of a second population of voids was concluded to strongly affect the fracture strain of the high strength regions of the welds. This modelling effort links the microstructure and process parameters to macroscopic parameters relevant to the optimisation of the welds.     

不同应力三维度条件下孔洞的演变及修正的Gurson模型

本文对含不同形状孔洞的幂硬化材料的圆柱体胞模型，运用控制宏观应力三维工的方法进行了有限元分析。计算结果表明：１．孔洞初始形状，应力三维度对孔洞的长大有重要影响；２．Ｇｕｓｏｎ模型对孔洞长大规律的描述是不准确的，不准确度与孔洞初始形状，应力三维度有关，修正后的Ｇｕｒｓｏｎ模型与有限元结果吻合较好；３．在低应力三维度区，孔洞以及形状改变为主，在高应力三维度区，孔洞以扩张为主；     

Tension–torsion fracture experiments – Part II: Simulations with the extended Gurson model and a ductile fracture criterion based on plastic strain

An extension of the Gurson model that incorporates damage development in shear is used to simulate the tension–torsion test fracture data presented in Faleskog and Barsoum (2013) (Part I) for two steels, Weldox 420 and 960. Two parameters characterize damage in the constitutive model: the effective void volume fraction and a shear damage coefficient. For each of the steels, the initial effective void volume fraction is calibrated against data for fracture of notched round tensile bars and the shear damage coefficient is calibrated against fracture in shear. The calibrated constitutive model reproduces the full range of data in the tension–torsion tests thereby providing a convincing demonstration of the effectiveness of the extended Gurson model. The model reinforces the experiments by highlighting that for ductile alloys the effective plastic strain at fracture cannot be based solely on stress triaxiality. For nominally isotropic alloys, a ductile fracture criterion is proposed for engineering purposes that depends on stress triaxiality and a second stress invariant that discriminates between axisymmetric stressing and shear dominated stressing.     

Relations between a micro-mechanical model and a damage model for ductile failure in shear

Gurson type constitutive models that account for void growth to coalescence are not able to describe ductile fracture in simple shear, where there is no hydrostatic tension in the material. But recent micro-mechanical studies have shown that in shear the voids are flattened out to micro-cracks, which rotate and elongate until interaction with neighbouring micro-cracks gives coalescence. Thus, the failure mechanism is very different from that under tensile loading. Also, the Gurson model has recently been extended to describe failure in shear, by adding a damage term to the expression for the growth of the void volume fraction, and it has been shown that this extended model can represent experimental observations. Here, numerical studies are carried out to compare predictions of the shear-extended Gurson model with the shear failures predicted by the micro-mechanical cell model. Both models show a strong dependence on the level of hydrostatic tension. Even though the reason for this pressure dependence is different in the two models, as the shear-extended Gurson model does not describe voids flattening out and the associated failure mechanism by micro-cracks interacting with neighbouring micro-cracks, it is shown that the trends of the predictions are in good agreement.     

Experimental investigations and modeling of volume change induced by void growth in polyamide 11

Polymers are known to be sensitive to hydrostatic pressure. The influence of stress triaxiality ratio on cavitation and damage has been highlighted in numerous studies. This paper proposes experimental investigations allowing the control of both the stress triaxiality ratio and the void distribution via microscopic observations of microtome-cut surfaces from interrupted tests. With the help of a finite element code, the Gurson–Tvergaard–Needleman model was calibrated by using these multi-scale experimental data. Then comparison between both numerical and analytical models and experimental data was performed. Bridgman formulae were reported to be valid up to the peak load. Moreover, a better understanding of the time evolution of significant parameters such as the porosity (volume change) and the stress triaxiality ratio (hydrostatic pressure), was highlighted.     

Evaluation of Ductile Damage Parameters under High Stress Triaxiality

The GTN model proposed by Gurson, Tvergaard and Needleman has been widely applied to predict ductile fracture. However, the evaluation of the GTN model under high stress triaxiality has only been reported in a few studies. In this paper, a series of tensile tests on round notched specimens were performed to evaluate the applicability of the GTN model parameters under high stress triaxiality. The evaluation was carried out by comparing the predicted load-displacement curves with experimental results. It was observed the GTN model parameters only depend on the material except the critical void volume fraction. The influence of stress triaxiality on the critical void volume fraction was discussed. A further discussion about the construction of a new void coalescence criterion for the GTN model was also presented in this paper.     

Verification of the transferability of micromechanical parameters by cell model calculations with visco-plastic materials

Finite element (FE) calculations of a cylindrical cell containing a spherical hole have been performed under large strain conditions for varying triaxiality with three different constitutive models for the matrix material, i.e. rate independent plastic material with isotropic hardening, visco-plastic material under both isothermal and adiabatic conditions, and porous plastic material with a second population of voids nucleating strain controlled. The “mesoscopic” stress-strain and void growth responses of the cell are compared with predictions of the modified Gurson model in order to study the effects of varying triaxiality and strain rate on the critical void volume fraction. The interaction of two different sizes of voids was modelled by changing the strain level for nucleation and the stress triaxiality. The study confirms that the void volume fraction at void coalescence does not depend significantly on the triaxiality if the initial volume fraction of the primary voids is small and if there are no secondary voids. The strain rate does not affect f_c either. The results also indicate that a single internal variable, f, is not sufficient to characterize the fracture processes in materials containing two different size-scales of void nucleating particles.     

The modified Gurson model accounting for the void size effect

The Gurson model [J. Engrg. Mater. Technol. 99 (1977) 2] has been widely used to study the deformation and failure of metallic materials containing microvoids. The void volume fraction is the only parameter representing voids since the void size does not come into play in the Gurson model. Based on the Taylor dislocation model [Proc. R. Soc. (Lond.) A145 (1934) 362; J. Int. Metals 62 (1938) 307], we extend the Gurson model to account for the void size effect. It is shown that the yield surfaces for micron- and submicron-sized voids are significantly larger than that given by the Gurson model. For a voided, dilating material subject to uniaxial tension, the void size has essentially no effect on the stress–strain curve at small initial void volume fraction. However, as the initial void volume fraction increases, the void size effect may become significant.     

The evolution of voids in the plasticity strain hardening gradient materials

The main aim of this paper is to opens out the meso-mechanism of void growth and coalescence in the matrix materials with graded strain-hardening exponent distribution. For this end, detailed finite element computations of a representative cylindrical cell containing a spherical void have been carried out. According to the FE analyses, significant effects of the strain-hardening exponent gradient (SEG) in the matrix on the void growth and coalescence are revealed: (1) In the homogeneous materials, the void growth and coalescence are slightly dependent on the strain-hardening exponent, however, the SEG distribution in the matrix can increase remarkably the void growth rate and decrease seriously the void coalescence strain. (2) The critical void shapes in the homogeneous materials are mainly governed by the macroscopic stress triaxiality, but due to earlier plastic flow localization in the softer matrix layer, the SEG distribution in the matrix has very significant effects on the deformed void shapes, especially when the stress triaxiality is lower. (3) When the triaxial stress levels are lower, in the homogeneous materials, the shape change mode of the void evolution is dominate so the void growth rate is very low; however, the SEG distribution in the matrix can bring the volume change mode out, as a result of increasing the void growth rate. (4) Comparisons of the numerical results with the existing damage model indicate that the classic damage model cannot give satisfying prediction to the void growth in both the homogeneous strain-hardening matrix and the SEG materials. On the basis of large numbers of numerical computations, a new damage model, which can uniformly describe the void growing in the homogeneous and plasticity gradient materials, is suggested. A mass of element computations have validated that the new damage model can give satisfying agreement with the FE results of cell model.     

A selfconsistent formulation for the prediction of the anisotropic behavior of viscoplastic polycrystals with voids

In this work we consider the presence of ellipsoidal voids inside polycrystals subjected to large strain deformation. For this purpose, the originally incompressible viscoplastic selfconsistent (VPSC) formulation of Lebensohn and Tomé (Acta Metall. Mater. 41 (1993) 2611) has been extended to deal with compressible polycrystals. In doing this, both the deviatoric and the spherical components of strain-rate and stress are accounted for. Such an extended model allows us to account for the void and for porosity evolution, while preserving the anisotropy and crystallographic capabilities of the VPSC model. The formulation can be adjusted to match the Gurson model, in the limit of rate-independent isotropic media and spherical voids. We present several applications of this extended VPSC model, which address the coupling between texture, plastic anisotropy, void shape, triaxiality, and porosity evolution.     

Tension–torsion fracture experiments—Part I: Experiments and a procedure to evaluate the equivalent plastic strain

Ductile failure experiments on a double notched tube (DNT) specimen subjected to a combination of tensile load and torque that was applied at a fixed ratio is presented. The experimental results extend those in Barsoum and Faleskog (2007a) down to zero stress triaxiality. A new and robust evaluation procedure for such tests is proposed, and a simple relation for the equivalent plastic strain at failure for combined normal and shear deformation, respectively, is developed. Tests were carried out on the medium strength medium hardening steel Weldox 420, and the high strength low hardening steel Weldox 960. The experimental results unanimously show that ductile failure not only depends on stress triaxiality, but is also strongly affected by the type of deviatoric stress state that prevails, which can be quantified by a stress invariant that discriminates between axisymmetric stressing and shear dominated stressing, e.g., the Lode parameter. Additional experiments on round notch bar (RNB) specimens are recapitulated in order to give a comprehensive account on how ductile failure depends on stress triaxiality, ranging from zero to more than 1.6, and the type of stress state for the two materials tested. This provides an extensive experimental data base that will be used to explore an extension of the Gurson model that incorporates damage development in shear presented in Xue et al. (2013) (Part II).     

半结晶聚合物损伤演化的实验表征与数值模拟

损伤本构模型对研究材料的断裂失效行为有重要意义, 但聚合物材料损伤演化的定量表征实验研究相对匮乏. 通过4种高密度聚乙烯(high density polythylene, HDPE)缺口圆棒试样的单轴拉伸实验获得了各类试样的载荷-位移曲线和真应力-应变曲线, 采用实验和有限元模拟相结合的方法确定了HDPE材料不同应力状态下的本构关系, 并建立了缺口半径与应力三轴度之间的关系;采用两阶段实验法定量描述了4种HDPE试样单轴拉伸过程中的弹性模量变化, 并建立了基于弹性模量衰减的损伤演化方程, 结合中断实验和扫描电子显微镜分析了应力状态对HDPE材料微观结构演化的影响. 结果表明缺口半径越小, 应力三轴度越大, 损伤起始越早、演化越快; 微观表现为: 高应力三轴度促进孔洞的萌生和发展, 但抑制纤维状结构的产生;基于实验和有限元模拟获得的断裂应变、应力三轴度、损伤演化方程等信息提出了一种适用于聚合物的损伤模型参数确定方法, 最后将本文获得的本构关系和损伤模型用于HDPE平板的冲压成形模拟, 模拟结果与实验结果吻合良好.      

考虑三轴约束时孔洞的聚合机理及有效能量准则

通过体胞分析方法,对不同状孔洞在从光滑试样到裂纹试样的三轴应力场中的聚合机理进行了较精解的有限元分析,计算结果表明：（１）孔洞的相互靠近和横向扩展是导致相邻孔洞发生内颈缩聚合的两种基本机制,在应力三维度Ｒσ等于１．２５附近,这两种机制发生较明显的变化。（２）单纯以孔洞体积分数ｆＣ概念为基础的材料破坏参数一般敏感于应力三维度,不能很好地预报不同三轴应力场中材料的破坏,在此基础上,提出了描述孔洞聚合的     

Atomistic insights into dislocation-based mechanisms of void growth and coalescence

One of the low-temperature failure mechanisms in ductile metallic alloys is the growth of voids and their coalescence. In the present work we attempt to obtain atomistic insights into the mechanisms underpinning cavitation in a representative metal, namely Aluminum. Often the pre-existing voids in metallic alloys such as Al have complex shapes (e.g. corrosion pits) and the defromation/damage mechanisms exhibit a rich size-dependent behavior across various material length scales. We focus on these two issues in this paper through large-scale calculations on specimens of sizes ranging from 18 thousand to 1.08 million atoms. In addition to the elucidation of the dislocation propagation based void growth mechanism we highlight the observed length scale effect reflected in the effective stress-strain response, stress triaxiality and void fraction evolution. Furthermore, as expected, the conventionally used Gurson's model fails to capture the observed size-effects calling for a mechanistic modification that incorporates the mechanisms observed in our (and other researchers') simulation. Finally, in our multi-void simulations, we find that, the splitting of a big void into a distribution of small ones increases the load-carrying capacity of specimens. However, no obvious dependence of the void fraction evolution on void coalescence is observed.     

On the predictive capabilities of the shear modified Gurson and the modified Mohr-Coulomb fracture models over a wide range of stress triaxialities and Lode angles

The predictive capabilities of the shear-modified Gurson model [Nielsen and Tvergaard, Eng. Fract. Mech. 77, 2010] and the Modified Mohr-Coulomb (MMC) fracture model [Bai and Wierzbicki, Int. J. Fract. 161, 2010] are evaluated. Both phenomenological fracture models are physics-inspired and take the effect of the first and third stress tensor invariants into account in predicting the onset of ductile fracture. The MMC model is based on the assumption that the initiation of fracture is determined by a critical stress state, while the shear-modified Gurson model assumes void growth as the governing mechanism. Fracture experiments on TRIP-assisted steel sheets covering a wide range of stress states (from shear to equibiaxial tension) are used to calibrate and validate these models. The model accuracy is quantified based on the predictions of the displacement to fracture for experiments which have not been used for calibration. It is found that the MMC model predictions agree well with all experiments (less than 4% error), while less accurate predictions are observed for the shear-modified Gurson model. A comparison of plots of the strain to fracture as a function of the stress triaxiality and the normalized third invariant reveals significant differences between the two models except within the vicinity of stress states that have been used for calibration.     

Modeling of ductile fracture: Application of the mechanism-based concepts

This paper summarizes our recent studies on modeling ductile fracture in structural materials using the mechanism-based concepts. We describe two numerical approaches to model the material failure process by void growth and coalescence. In the first approach, voids are considered explicitly and modeled using refined finite elements. In order to predict crack initiation and propagation, a void coalescence criterion is established by conducting a series of systematic finite element analyses of the void-containing, representative material volume (RMV) subjected to different macroscopic stress states and expressed as a function of the stress triaxiality ratio and the Lode angle. The discrete void approach provides a straightforward way for studying the effects of microstructure on fracture toughness. In the second approach, the void-containing material is considered as a homogenized continuum governed by porous plasticity models. This makes it possible to simulate large amount of crack extension because only one element is needed for a representative material volume. As an example, a numerical approach is proposed to predict ductile crack growth in thin panels of a 2024-T3 aluminum alloy, where a modified Gologanu–Leblond–Devaux model [Gologanu, M., Leblond, J.B., Devaux, J., 1993. Approximate models for ductile metals containing nonspherical voids – Case of axisymmetric prolate ellipsoidal cavities. J. Mech. Phys. Solids 41, 1723–1754; Gologanu, M., Leblond, J.B., Devaux, J., 1994. Approximate models for ductile metals containing nonspherical voids – Case of axisymmetric oblate ellipsoidal cavities. J. Eng. Mater. Tech. 116, 290–297; Gologanu, M., Leblond, J.B., Perrin, G., Devaux, J., 1995. Recent extensions of Gurson’s model for porous ductile metals. In: Suquet, P. (Ed.) Continuum Micromechanics. Springer-Verlag, pp. 61–130] is used to describe the evolution of void shape and void volume fraction and the associated material softening, and the material failure criterion is calibrated using experimental data. The calibrated computational model successfully predicts crack extension in various fracture specimens, including the compact tension specimen, middle crack tension specimens, multi-site damage specimens and the pressurized cylindrical shell specimen.     

On the extension of the Gurson-type porous plasticity models for prediction of ductile fracture under shear-dominated conditions

One of the major drawbacks of the Gurson-type of porous plasticity models is the inability of these models to predict material failure under low stress triaxiality, shear dominated conditions. This study addresses this issue by combining the damage mechanics concept with the porous plasticity model that accounts for void nucleation, growth and coalescence. In particular, the widely adopted Gurson–Tvergaard–Needleman (GTN) model is extended by coupling two damage parameters, representing the volumetric damage (void volume fraction) and the shear damage, respectively, into the yield function and flow potential. The effectiveness of the new model is illustrated through a series of numerical tests comparing its performance with existing models. The current model not only is capable of predicting damage and fracture under low (even negative) triaxiality conditions but also suppresses spurious damage that has been shown to develop in earlier modifications of the GTN model for moderate to high triaxiality regimes. Finally the modified GTN model is applied to predict the ductile fracture behavior of a beta-treated Zircaloy-4 by coupling the proposed damage modeling framework with a recently developed J₂–J₃ plasticity model for the matrix material. Model parameters are calibrated using experimental data, and the calibrated model predicts failure initiation and propagation in various specimens experiencing a wide range of triaxiality and Lode parameter combinations.     

Rupture mechanisms in combined tension and shear—Micromechanics

A micromechanics model based on the theoretical framework of plastic localization into a band introduced by Rice is developed. The model consists of a planar band with a square array of equally sized cells, with a spherical void located in the centre of each cell. The periodic arrangement of the cells allows the study of a single unit cell for which fully periodic boundary conditions are applied. The micromechanics model is applied to analyze failure by ductile rupture in experiments on double notched tube specimens subjected to combined tension and torsion carried out by the present authors. The stress state is characterized in terms of the stress triaxiality and the Lode parameter. Two rupture mechanisms can be identified, void coalescence by internal necking at high triaxiality and void coalescence by internal shearing at low triaxiality. For the internal necking mechanism, failure is assumed to occur when the deformation localizes into a planar band and is closely associated with extensive void growth until impingement of voids. For the internal shearing mechanism, a simple criterion based on the attainment of a critical value of shear deformation is utilized. The two failure criteria capture the transition between the two rupture mechanisms successfully and are in good agreement with the experimental result.