An analysis of ductile rupture in notched bars

Ductile fracture in axisymmetric and plane strain notched tensile specimens is analyzed numerically, based on a set of elastic-plastic constitutive relations that account for the nucleation and growth of microvoids. Final material failure by void coalescence is incorporated into the constitutive model via the dependence of the yield function on the void volume fraction. In the analyses the material has no voids initially; but as the voids nucleate and grow, the resultant dilatancy and pressure sensitivity of the macroscopic plastic flow influence the solution significantly. Considering both a blunt notch geometry and a sharp notch geometry in the computations permits a study of the relative roles of high strain and high triaxiality on failure. Comparison is made with published experimental results for notched tensile specimens of high-strength steels. All axisymmetric specimens analyzed fail at the center of the notched section, whereas failure initiation at the surface is found in plane strain specimens with sharp notches, in agreement with the experiments. The results for different specimens are used to investigate the circumstances under which fracture initiation can be represented by a single failure locus in a plot of stress triaxiality vs effective plastic strain.     

Three-dimensional micromechanical modeling of voided polymeric materials

A three-dimensional micromechanical unit cell model for particle-filled materials is presented. The cell model is based on a Voronoi tessellation of particles arranged on a body-centered cubic (BCC) array. The three-dimensionality of the present cell model enables the study of several deformation modes, including uniaxial, plane strain and simple shear deformations, as well as arbitrary principal stress states.The unit cell model is applied to studies on the micromechanical and macromechanical behavior of rubber-toughened polycarbonate. Different load cases are examined, including plane strain deformation, simple shear deformation and principal stress states. For a constant macroscopic strain rate, the different load cases show that the macroscopic flow strength of the blend decreases with an increase in void volume fraction, as expected. The main mechanism for plastic deformation is broad shear banding across inter-particle ligaments. The distributed nature of plastic straining acts to reduce the amount of macroscopic strain softening in the blend as the initial void volume fraction is increased. In the case of plane strain deformation, the plastic flow is observed to initiate across inter-particle ligaments in the direction of constraint. This particular mode of deformation could not have been captured using a two-dimensional, plane strain idealization of cylindrical voids in a matrix.The potential for localized crazing and/or cavitation in the matrix is addressed. It is observed that the introduction of voids acts to relieve hydrostatic stress in the matrix material, compared to the homopolymer. It is also seen that the predicted peak hydrostatic stress in the matrix is higher under plane strain deformation than under triaxial tension (with equal lateral stresses), for the same macroscopic stress triaxiality.The effect of void volume fraction on the macroscopic uniaxial tension behavior of the different blends is examined using a Considère construction for dilatant materials. The natural draw ratio was predicted to decrease with an increase in void volume fraction.     

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.     

Experimental study and modeling of damage of Al alloys using tensor theory

This article presents and evaluates experiments for the characterization and modeling of damage of structural aluminum and aluminum–magnesium alloys. Tensile tests were performed for specimens with artificial defects (voids) represented by different arrangements of pre-drilled micro-holes. The corresponding stress–strain curves were experimentally obtained. Plastic dilatation and deviatoric strain were determined both for the local zones with artificial defects and directly for meso-elements (i.e., material cells with artificial defects). A symmetric second-rank order tensor of damage was applied for a quantitative estimate of the material damage connected with the volume fraction and shape of micro-defects. The definition of this tensor is physically motivated, since its hydrostatic and deviatoric parts describe the evolution of damage connected with a change in volume fraction and shape of micro-defects, respectively. Such a representation of damage kinetics allows us to use two integral measures for the calculation of damage in deformed materials. The first measure determines damage related to an increase in void volume fraction (i.e., plastic dilatation). A critical amount of plastic dilatation corresponds to the moment of macro-fracture of the deformed metal. By means of experimental analysis, we can determine the function of plastic dilatation which depends on the strain accumulated by material particles under various stress and temperature-rate conditions of forming. The second measure accounts for the deviatoric strain of meso-elements, and is related to the change in their shapes. The critical deformation of ellipsoidal voids corresponds to the onset of their coalescence and to the formation of large cavernous defects. The second measure is considered as a criterion of micro-destruction due to formation of cavities in the deformed material. Based on the experimental data, some numerical modeling is realized for the investigated Al alloys to taken a change in stress triaxiality into account. It shows that a change in triaxiality toward smaller values results in an appreciable decrease of damage induced by strain. Both damage measures are important for the prediction of the meso-structure quality of metalware produced by metal forming techniques.     

Modeling of crack growth in ductile solids: a three-dimensional analysis

A population of several spherical voids is included in a three-dimensional, small scale yielding model. Two distinct void growth mechanisms, put forth by [Int. J. Solids Struct. 39 (2002) 3581] for the case of a two-dimensional model containing cylindrical voids, are well contained in the model developed in this study for spherical voids. A material failure criterion, based on the occurrence of void coalescence in the unit cell model, is established. The critical ligament reduction ratio, which varies with stress triaxiality and initial porosity, is used to determine ligament failure between the crack tip and the nearest void. A comparison of crack initiation toughness of the model containing cylindrical voids with the model containing spherical voids reveals that the material having a sizeable fraction of spherical voids is tougher than the material having cylindrical voids. The proposed material failure determination method is then used to establish the fracture resistance curve (J–R curve) of the material. For a ductile material containing a small volume fraction of microscopic voids initially, the void by void growth mechanism prevails, which results in a J–R curve having steep slope. On the other hand, for a ductile material containing a large volume fraction of initial voids, the multiple voids interaction mechanism prevails, which results in a flat J–R curve. Next, the effect of T-stress on fracture resistance is examined. Finally, nucleation and growth of secondary microvoids and their effects on void coalescence are briefly discussed.     

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     

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.     

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.     

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；     

The effect of straining mode on void growth

Large strain finite element method is employed to investigate the effect of straining mode on void growth. Axisymmetric cell model embedded with spherical void is controlled by constant triaxiality loading, while plane-stress model containing a circular void is loaded by constant ratio of straining. Elastic-plastic material is used for the matrix in both cases. It is concluded that, besides the known effect of triaxiality, the straining mode which intensifies the plastic concentration around the void is also a void growth stimulator. Experimental results are cited to justify the computation results. This paper is jointly supported by the National Natural Science Foundation of China (19872064), the Chinese Academy of Sciences (KJ951-1-201) and the Laboratory for Nonlinear mechanics of Continuous Media of the Institute of Mechanics     

Void growth to coalescence in a non-local material

The size-effect in metals containing distributed spherical voids is analyzed numerically using a finite strain generalization of a length scale dependent plasticity theory. Results are obtained for stress-triaxialities relevant in front of a crack tip in an elastic-plastic metal. The influence of different material length parameters in a multi-parameter theory is studied, and it is shown that the important length parameter is the same as under purely hydrostatic loading. It is quantified how micron scale voids grow less rapidly than larger voids, and the implications of this in the overall strength of the material is emphasized. The size effect on the onset of coalescence is studied, and results for the void volume fraction and the strain at the onset of coalescence are presented. It is concluded that for cracked specimens not only the void volume fraction, but also the typical void size is of importance to the fracture strength of ductile materials.     

The size effect on void growth in ductile materials

We have extended the Rice-Tracey model (J. Mech. Phys. Solids 17 (1969) 201) of void growth to account for the void size effect based on the Taylor dislocation model, and have found that small voids tend to grow slower than large voids. For a perfectly plastic solid, the void size effect comes into play through the ratio εl/R₀, where l is the intrinsic material length on the order of microns, ε the remote effective strain, and R₀ the void size. For micron-sized voids and small remote effective strain such that εl/R₀?0.02, the void size influences the void growth rate only at high stress triaxialities. However, for sub-micron-sized voids and relatively large effective strain such that εl/R₀>0.2, the void size has a significant effect on the void growth rate at all levels of stress triaxiality. We have also obtained the asymptotic solutions of void growth rate at high stress triaxialities accounting for the void size effect. For εl/R₀>0.2, the void growth rate scales with the square of mean stress, rather than the exponential function in the Rice-Tracey model (1969). The void size effect in a power-law hardening solid has also been studied.     

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.     

Theoretical study of void closure in nonlinear plastic materials

Void closing from a spherical shape to a crack is investigated quantitatively in the present study. The constitutive relation of the Void-free matrix is assumed to obey the Norton power law. A representative volume element （RVE） which includes matrix and void is employed and a Rayleigh-Ritz procedure is developed to study the deformation-rates of a spherical void and a penny-shaped crack. Based on an approximate interpolation scheme, an analytical model for void closure in nonlinear plastic materials is established. It is found that the local plastic flows of the matrix material are the main mechanism of void deformation. It is also shown that the relative void volume during the deformation depends on the Norton exponent, on the far-field stress triaxiality, as well as on the far-field effective strain. The predictions of void closure using the present model are compared with the corresponding results in the literature, showing good agreement. The model for void closure provides a novel way for process design and optimization in terms of elimination of voids in billets because the model for void closure can easily be applied in the CAE analysis.     

A microscopic damage model considering the change of void shape and application in the void closing

A microscopic damage model of ellipsoidal body containing ellipsoidal void for nonlinear matrix materials is developed under a particular coordinate. The change of void shape is considered in this model. The viscous restrained equation obtained from the model is affected by stress ∑_(ij), void volume fraction f, material strain rate exponent m as well as the void shape. Gurson's equation is modified from the numerical solution. The modified equation is suitable for the case of nonlinear matrix materials and changeable voids. Lastly, the model is used to analyze the closing process of voids.     

Void growth and coalescence in anisotropic plastic solids

Large strain finite element calculations of unit cells subjected to triaxial axisymmetric loadings are presented for plastically orthotropic materials containing a periodic distribution of aligned spheroidal voids. The spatial distribution of voids and the plastic flow properties of the matrix are assumed to respect transverse isotropy about the axis of symmetry of the imposed loading so that a two-dimensional axisymmetric analysis is adequate. The parameters varied pertain to load triaxiality, matrix anisotropy, initial porosity and initial void shape so as to include the limiting case of penny-shaped cracks. Attention is focussed on comparing the individual and coupled effects of void shape and material anisotropy on the effective stress–strain response and on the evolution of microstructural variables. In addition, the effect of matrix anisotropy on the mode of plastic flow localization is discussed. From the results, two distinct regimes of behavior are identified: (i) at high triaxialities, the effect of material anisotropy is found to be persistent, unlike that of initial void shape and (ii) at moderate triaxialities the influence of void shape is found to depend strongly on matrix anisotropy. The findings are interpreted in light of recent, microscopically informed models of porous metal plasticity. Conversely, observations are made in relation to the relevance of these results in the development and calibration of a broader set of continuum damage mechanics models.     

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.     

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.     

A constitutive model for plastically anisotropic solids with non-spherical voids

Plastic constitutive relations are derived for a class of anisotropic porous materials consisting of coaxial spheroidal voids, arbitrarily oriented relative to the embedding orthotropic matrix. The derivations are based on nonlinear homogenization, limit analysis and micromechanics. A variational principle is formulated for the yield criterion of the effective medium and specialized to a spheroidal representative volume element containing a confocal spheroidal void and subjected to uniform boundary deformation. To obtain closed form equations for the effective yield locus, approximations are introduced in the limit-analysis based on a restricted set of admissible microscopic velocity fields. Evolution laws are also derived for the microstructure, defined in terms of void volume fraction, aspect ratio and orientation, using material incompressibility and Eshelby-like concentration tensors. The new yield criterion is an extension of the well known isotropic Gurson model. It also extends previous analyses of uncoupled effects of void shape and material anisotropy on the effective plastic behavior of solids containing voids. Preliminary comparisons with finite element calculations of voided cells show that the model captures non-trivial effects of anisotropy heretofore not picked up by void growth models.