Two mechanisms of ductile fracture: void by void growth versus multiple void interaction

Two distinct mechanisms of crack initiation and advance by void growth have been identified in the literature on the mechanics of ductile fracture. One is the interaction a single void with the crack tip characterizing initiation and the subsequent void by void advance of the tip. This mechanism is represented by the early model of Rice and Johnson and the subsequent more detailed numerical computations of McMeeking and coworkers on a single void interacting with a crack tip. The second mechanism involves the simultaneous interaction of multiple voids on the plane ahead of the crack tip both during initiation and in subsequent crack growth. This mechanism is revealed by models with an embedded fracture process zone, such as those developed by Tvergaard and Hutchinson. While both mechanisms are based on void nucleation, growth and coalescence, the inferences from them with regard to crack growth initiation and growth are quantitatively different. The present paper provides a formulation and numerical analysis of a two-dimensional plane strain model with multiple discrete voids located ahead of a pre-existing crack tip. At initial void volume fractions that are sufficiently low, initiation and growth is approximately represented by the void by void mechanism. At somewhat higher initial void volume fractions, a transition in behavior occurs whereby many voids ahead of the tip grow at comparable rates and their interaction determines initiation toughness and crack growth resistance. The study demonstrates that improvements to be expected in fracture toughness by reducing the population of second phase particles responsible for nucleating voids cannot be understood in terms of trends of one mechanism alone. The transition from one mechanism to the other must be taken into account.     

Ductile fracture of materials with high void volumefraction

In the present paper, axisymmetric cell models containing one or two voids and athree-dimensional cell model containing two voids have been used to investigate void size andspacing effect on the ductile fracture in materials with high initial void volume fraction. They areperformed for round smooth and round notched specimens under uniaxial tension. The examplematerial used for comparison is a nodular cast iron material GGG-40 with initial void volumefraction of 7.7%. The parameters considered in this paper are void size and shape foraxisymmetric cell models containing a single void, and void distribution pattern foraxisymmetric and 3D cell models containing two voids of different sizes. The results obtainedfrom these cell models by using FEM calculations are compared with the Gurson model, theGurson–Tvergaard–Needleman model, the Rice–Tracey model and the modified Rice–Traceymodel. It can be stated that the influence of void size and void spacing on the growth in volumeof voids is very large, and it is dependent on the distribution of voids. Using non-uniform voiddistribution, the results of axisymmetric cell models can explain how a void can grow in anunstable state under very low stress triaxiality at very small strain as observed in experiments.Calculations using cell models containing two voids give very different results about the stableand unstable growth of voids which are strongly dependent on the configuration of cell model.     

A molecular dynamics study of void growth and coalescence in single crystal nickel

Molecular dynamics simulations using Modified Embedded Atom Method (MEAM) potentials were performed to analyze material length scale influences on damage progression of single crystal nickel. Damage evolution by void growth and coalescence was simulated at very high strain rates (10⁸–10¹⁰/s) involving four specimen sizes ranging from ≈5000 to 170,000 atoms with the same initial void volume fraction. 3D rectangular specimens with uniform thickness were provided with one and two embedded cylindrical voids and were subjected to remote uniaxial tension at a constant strain rate. Void volume fraction evolution and the corresponding stress–strain responses were monitored as the voids grew under the increasing applied tractions.The results showed that the specimen length scale changes the dislocation pattern, the evolving void aspect ratio, and the stress–strain response. At small strain levels (0–20%), a damage evolution size scale effect can be observed from the damage-strain and stress–strain curves, which is consistent with dislocation nucleation argument of Horstemeyer et al. [Horstemeyer, M.F., Baskes, M.I., Plimpton, S.J., 2001a. Length scale and time scale effects on the plastic flow of FCC metals. Acta Mater. 49, pp. 4363–4374] playing a dominant role. However, when the void volume fraction evolution is plotted versus the applied true strain at large plastic strains (>20%), minimal size scale differences were observed, even with very different dislocation patterns occurring in the specimen. At this larger strain level, the size scale differences cease to be relevant, because the effects of dislocation nucleation were overcome by dislocation interaction.This study provides fodder for bridging material length scales from the nanoscale to the larger scales by examining plasticity and damage quantities from a continuum perspective that were generated from atomistic results.     

A criterion for the onset of void coalescence under combined tension and shear

Depending on the relative positions of voids and on the loading conditions, shear loading components can play an important role in the void coalescence process leading to ductile fracture. Yet, most void coalescence criteria including the original criterion of Thomason, and its various extensions/improvements, take only normal loads into account and neglect the contribution from shear loads to coalescence. Shear can affect both the stress/strain at the onset of coalescence and the direction of deformation localization. In this paper, first, the predictive capabilities of different coalescence criteria without shear effect are critically assessed and the expressions involved in the original Thomason criterion are fine-tuned by comparing with 3D finite element calculations performed on a unit cell containing a spheroidal void. Then, the improved Thomason criterion is theoretically extended—by using limit load analysis—to incorporate the effect of shear. The predictions of this new coalescence criterion are in good agreement with the results produced by 3D finite element calculations, for both loadings involving or not a shear component.     

FCC晶体中不均匀变形对空洞长大的影响

通过编制率相关有限元用户子程序,采用一个单胞模型研究了FCC晶体中孔洞在单晶及晶界的长大行为,分析了由于晶体取向及变形失配对孔洞长大和聚合的影响。研究结果表明：孔洞的形状和长大方向与晶体取向密切相关;晶界上孔洞的长大速度大于单晶中孔洞的长大速度;晶粒间的变形失配加速了晶界上孔洞的长大趋势,因而使材料易发生沿晶断裂,随着晶粒间取向因子差异的增加,孔洞越易沿着晶界长大。     

Analysis of a model for void growth and coalescence ahead of a moving crack tip

One mechanism for the slow, steady motion of a crack tip in plane strain is void growth and coalescence. A model of this process is formulated by isolating a geometrical cell containing a single void and extending the cell in uniaxial strain. A rigid-plastic material is assumed and the velocity field is analysed by means of a finite element procedure. From the results of the analysis, the dissipation of energy in the creation of crack surfaces is estimated as a function of the original void-size and the volume fraction of voids. Further, the shallow shape of dimples found experimentally is verified by the results of the analysis. A check on the accuracy is obtained by application of the numerical method to cases where analytical solutions are known.     

On ductile fracture initiation toughness: Effects of void volume fraction,void shape and void distribution

This paper studies the effects of the initial relative void spacing, void pattern, void shape and void volume fraction on ductile fracture toughness using three-dimensional, small scale yielding models, where voids are assumed to pre-exist in the material and are explicitly modeled using refined finite elements. Results of this study can be used to explain the observed fracture toughness anisotropy in industrial alloys. Our analyses suggest that simplified models containing a single row of voids ahead of the crack tip is sufficient when the initial void volume fraction remains small. When the initial void volume fraction becomes large, these simplified models can predict the fracture initiation toughness (J_Ic) with adequate accuracy but cannot predict the correct J–R curve because they over-predict the interaction among growing voids on the plane of crack propagation. Consequently, finite element models containing multiple rows of voids should be used when the material has large initial void volume fraction.     

Lattice orientation effects on void growth and coalescence in fcc single crystals

Void growth and coalescence in fcc single crystals were studied using crystal plasticity under uniaxial and biaxial loading conditions and various orientations of the crystalline lattice. A 2D plane strain unit cell with one and two cylindrical voids was employed using three-dimensional 12 potentially active slip systems. The results were compared to five representative orientations of the tensile axis on the stereographic triangle. For uniaxial tension conditions, the void volume fraction increase under the applied load is strongly dependent on the crystallographic orientation with respect to the tensile axis. For some orientations of the tensile axis, such as [1 0 0] or [1 1 0], the voids exhibited a growth rate twice as fast compared with other orientations ([1 0 0], [2 1 1]). Void growth and coalescence simulations under uniaxial loading indicated that during deformation along some orientations with asymmetry of the slip systems, the voids experienced rotation and shape distortion, due mainly to lattice reorientation. Coalescence effects are shown to diminish the influence of lattice orientation on the void volume fraction increase, but noteworthy differences are still present. Under biaxial loading conditions, practically all differences in the void volume fraction for different orientations of the tensile axes during void growth vanish. These results lead to the conclusion that at microstructural length scales in regions under intense biaxiality/triaxiality conditions, such as crack tip or notched regions, the plastic anisotropy due to the initial lattice orientation has only a minor role in influencing the void growth rate. In such situations, void growth and coalescence are mainly determined by the stress triaxiality, the magnitude of accumulated strain, and the spatial localization of such plastic strains.     

Formulation of a damage internal state variable model for amorphous glassy polymers

The following article proposes a damage model that is implemented into a glassy, amorphous thermoplastic thermomechanical inelastic internal state variable framework. Internal state variable evolution equations are defined through thermodynamics, kinematics, and kinetics for isotropic damage arising from two different inclusion types: pores and particles. The damage arising from the particles and crazing is accounted for by three processes of damage: nucleation, growth, and coalescence. Nucleation is defined as the number density of voids/crazes with an associated internal state variable rate equation and is a function of stress state, molecular weight, fracture toughness, particle size, particle volume fraction, temperature, and strain rate. The damage growth is based upon a single void growing as an internal state variable rate equation that is a function of stress state, rate sensitivity, and strain rate. The coalescence internal state variable rate equation is an interactive term between voids and crazes and is a function of the nearest neighbor distance of voids/crazes and size of voids/crazes, temperature, and strain rate. The damage arising from the pre-existing voids employs the Cocks–Ashby void growth rule. The total damage progression is a summation of the damage volume fraction arising from particles and pores and subsequent crazing. The modeling results compare well to experimental findings garnered from the literature. Finally, this formulation can be readily implemented into a finite element analysis.     

Modeling of ductile fracture: Significance of void coalescence

In this paper void coalescence is regarded as the result of localization of plastic flow between enlarged voids. We obtain the failure criterion for a representative material volume (RMV) in terms of the macroscopic equivalent strain (E_c) as a function of the stress triaxiality parameter (T) and the Lode angle (θ) by conducting systematic finite element analyses of the void-containing RMV subjected to different macroscopic stress states. A series of parameter studies are conducted to examine the effects of the initial shape and volume fraction of the primary void and nucleation, growth, and coalescence of secondary voids on the predicted failure surface E_c(T, θ). As an application, a numerical approach is proposed to predict ductile crack growth in thin panels of a 2024-T3 aluminum alloy, where a porous plasticity model is used to describe the void growth process and the expression for E_c is calibrated using experimental data. The calibrated computational model is applied to predict crack extension in fracture specimens having various initial crack configurations and the numerical predictions agree very well with experimental measurements.     

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.     

Size effects on void growth in single crystals with distributed voids

The effect of void size on void growth in single crystals with uniformly distributed cylindrical voids is studied numerically using a finite deformation strain gradient crystal plasticity theory with an intrinsic length parameter. A plane strain cell model is analyzed for a single crystal with three in-plane slip systems. It is observed that small voids allow much larger overall stress levels than larger voids for all the stress triaxialities considered. The amount of void growth is found to be suppressed for smaller voids at low stress triaxialities. Significant differences are observed in the distribution of slips and on the shape of the deformed voids for different void sizes. Furthermore, the orientation of the crystalline lattice is found to have a pronounced effect on the results, especially for the smaller void sizes.     

A constitutive model for elastoplastic solids containing primary and secondary voids

In many ductile metallic alloys, the damage process controlled by the growth and coalescence of primary voids nucleated on particles with a size varying typically between 1 and 100 μm, is affected by the growth of much smaller secondary voids nucleated on inclusions with a size varying typically between 0.1 and 3 μm. The goal of this work is first to quantify the potential effect of the growth of these secondary voids on the coalescence of primary voids using finite element (FE) unit cell calculations and second to formulate a new constitutive model incorporating this effect. The nucleation and growth of secondary voids do essentially not affect the growth of the primary voids but mainly accelerate the void coalescence process. The drop of the ductility caused by the presence of secondary voids increases if the nucleation strain decreases and/or if their volume fraction increases and/or if the primary voids are flat. A strong coupling is indeed observed between the shape of the primary voids and the growth of the second population enhancing the anisotropy of the ductility induced by void shape effects. The new micromechanics-based coalescence condition for internal necking introduces the softening induced by secondary voids growing in the ligament between two primary voids. The FE cell calculations were used to guide and assess the development of this model. The use of the coalescence condition relies on a closed-form model for estimating the evolution of the secondary voids in the vicinity of a primary cavity. This coalescence criterion is connected to an extended Gurson model for the first population including the effect of the void aspect ratio. With respect to classical models for single void population, this new constitutive model improves the predictive potential of damage constitutive models devoted to ductile metal while requiring only two new parameters, i.e. the initial porosity of second population and a void nucleation stress, without any additional adjustment.     

RVE-based studies on the coupled effects of void size and void shape on yield behavior and void growth at micron scales

The present paper extends the Gurson and GLD models [Gurson, A.L., 1977. Continuum theory of ductile rupture by void nucleation and growth, Part I—yield criteria and flow rules for porous ductile media. J. Mech. Phys. Solids 99, 2–15; Gologanu, M., Leblond, J.B., Devaux, J., 1993. Approximate models for ductile metals containing non-spherical 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 non-spherical voids—case of axisymmetric oblate ellipsoidal cavities. J. Eng. Mater. Technol. 116, 290–297] to involve the coupled effects of void size and void shape on the macroscopic yield behavior of non-linear porous materials and on the void growth. A spheroidal representative volume element (RVE) under a remote axisymmetric homogenous strain boundary condition is carefully analyzed. A wide range of void aspect ratios covering the oblate spheroidal, spherical and prolate spheroidal void are taken into account to reflect the shape effect. The size effect is captured by the Fleck–Hutchinson phenomenological strain gradient plasticity theory [Fleck, N.A., Hutchinson, J.W., 1997. Strain gradient plasticity. In: Hutchinson, J.W., Wu, T.Y. (Eds.), Advance in Applied Mechanics, vol. 33, Academic Press, New York, pp. 295–361]. A new size-dependent damage model like the Gurson and GLD models is developed based on the traditional minimum plasticity potential principle. Consequently, the coupled effects of void size and void shape on yield behavior of porous materials and void growth are discussed in detail. The results indicate that the void shape effect on the yield behavior of porous materials and on the void growth can be modified dramatically by the void size effect and vice versa. The applied stress triaxiality plays an important role in these coupled effects. Moreover, there exists a cut-off void radius r_c, which depends only on the intrinsic length l₁ associated with the stretch strain gradient. Voids of effective radius smaller than the critical radius r_c are less susceptible to grow. These findings are helpful to our further understanding to some impenetrable micrographs of the ductile fracture surfaces.     

Cylindrical void in a rigid-ideally plastic single crystal. Part I: Anisotropic slip line theory solution for face-centered cubic crystals

The fracture toughness of ductile materials depends upon the ability of the material to resist the growth of microscale voids near a crack tip. Mechanics analyses of the elastic–plastic deformation state around such voids typically assume the surrounding material to be isotropic. However, the voids exist predominantly within a single grain of a polycrystalline material, so it is necessary to account for the anisotropic nature of the surrounding material. In the present work, anisotropic slip line theory is employed to derive the stress and deformation state around a cylindrical void in a single crystal oriented so that plane strain conditions are admitted from three effective in-plane slip systems. The deformation state takes the form of angular sectors around the circumference of the void. Only one of the three effective slip systems is active within each sector. Each slip sector is further subdivided into smaller sectors inside of which it is possible to derive the stress state. Thus the theory predicts a highly heterogeneous stress and deformation state. In addition, it is shown that the in-plane pressure necessary to activate plastic deformation around a cylindrical void in an anisotropic material is significantly higher than that necessary for an isotropic material. Experiments and single crystal plasticity finite element simulations of cylindrical voids in single crystals, both of which exhibit a close correspondence to the analytical theory, are discussed in a companion 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.     

强动载作用下孔洞汇合对延性金属层裂损伤演化过程的影响

针对强动载作用下延性金属的层裂问题,在分析孔洞之间几何关联的基础上,定义了一个新的耦合损伤及孔洞几何信息的孔洞汇合判定方法,同时,基于能量守恒原理,解析了孔洞汇合对损伤快速增长影响的物理机理.通过分析数值计算结果和对比相关文献的实验可知:孔洞汇合后不仅引起损伤增长,而且导致了损伤材料内部微孔洞数目的减少、孔洞平均尺寸的增加。     

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     

Cylindrical void in a rigid-ideally plastic single crystal II: Experiments and simulations

Experimental results and finite element simulations of plastic deformation around a cylindrical void in single crystals are presented to compare with the analytical solutions in a companion paper: Cylindrical void in a rigid-ideally plastic single crystal I: Anisotropic slip line theory solution for face-centered cubic crystals [Kysar, J.W., Gan, Y.X., Mendez-Arzuza, G., 2005. Cylindrical void in a rigid-ideally plastic single crystal I: Anisotropic slip line theory solution for face-centered cubic crystals, International Journal of Plasticity, 21, 1481–1520]. In the first part of the present paper, the theoretical predictions of the stress and deformation field around a cylindrical void in face-centered cubic (FCC) single crystals are briefly reviewed. Secondly, electron backscatter diffraction results are presented to show the lattice rotation discontinuities at boundaries between regions of single slip around the void as predicted in the companion paper. In the third part of the paper, the finite element method has been employed to simulate the anisotropic plastic deformation behavior of FCC single crystals which contain cylindrical voids under plane strain condition. The results of the simulation are in good agreement with the prediction by the anisotropic slip line theory.