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.     

The coupled effects of plastic strain gradient and thermal softening on the dynamic growth of voids

This paper examines the combined effects of temperature, strain gradient and inertia on the growth of voids in ductile fracture. A dislocation-based gradient plasticity theory [J. Mech. Phys. Solids 47 (1999) 1239, J. Mech. Phys. Solids 48 (2000) 99] is applied, and temperature effects are incorporated. Since a strong size-dependence is introduced into the dynamic growth of voids through gradient plasticity, a cut-off size is then set by the stress level of the applied loading. Only those voids that are initially larger than the cut-off size can grow rapidly. At the early stages of void growth, the effects of strain gradients greatly increase the stress level. Therefore, thermal softening has a strong effect in lowering the threshold stress for the unstable growth of voids. Once the voids start rapid growth, however, the influence of strain gradients will decrease, and the rate of dynamic void growth predicted by strain gradient plasticity approaches that predicted by classical plasticity theories.     

Effect of a nonuniform distribution of voids on the plastic response of voided materials: a computational and statistical analysis

This study investigates the overall and local response of porous media composed of a perfectly plastic matrix weakened by stress-free voids. Attention is focused on the specific role played by porosity fluctuations inside a representative volume element. To this end, numerical simulations using the Fast Fourier Transform (FFT) are performed on different classes of microstructure corresponding to different spatial distributions of voids. Three types of microstructures are investigated: random microstructures with no void clustering, microstructures with a connected cluster of voids and microstructures with disconnected void clusters. These numerical simulations show that the porosity fluctuations can have a strong effect on the overall yield surface of porous materials. Random microstructures without clusters and microstructures with a connected cluster are the hardest and the softest configurations, respectively, whereas microstructures with disconnected clusters lead to intermediate responses. At a more local scale, the salient feature of the fields is the tendency for the strain fields to concentrate in specific bands. Finally, an image analysis tool is proposed for the statistical characterization of the porosity distribution. It relies on the distribution of the ‘distance function’, the width of which increases when clusters are present. An additional connectedness analysis allows us to discriminate between clustered microstructures.     

受有两级空洞损伤时韧性材料的力学行为

本文利用大应变有限元方法研究了两级空洞对韧性材料的损伤作用.模型是以轴对称圆柱基体作为胞元,内含一初始的球型空洞.基体内的应力/应变随胞元外载的增大而达到临界状态,从而在围绕初级空洞的基体内将萌生次级空洞.后者是由空单元实现的.两级空洞的交互作用被证明将促进材料中的空洞化现象从而加速损伤并导至材料的总体弹性模量值在临近破断时急剧下降.     

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.     

Ductile fracture by cavity nucleation between larger voids

A mechanism of ductile fracture involving the interaction of relatively large voids with small-scale voids is studied by a computational model. The larger voids are described as circular cylindrical holes arranged in a doubly periodic array in the initial state. In the matrix material between these voids the nucleation and growth of much smaller voids is accounted for by using approximate constitutive equations for a ductile, porous medium. The computations show bands of highly localized straining and void growth, initiating at the surfaces of larger voids and growing into the matrix material, until the bands connect two neighbouring voids. The materials are analysed both under plane strain conditions and under conditions approximating those in a round tensile bar. The failure strains obtained under different principal stress ratios show rather good agreement when plotted against a measure of the stress-triaxiality.     

Experimental investigation into plastic damage of microvoids with interaction

A mechanism of plastic flow localization in ductile matter near microvoids is studied. The voids with the size-scale of micromillimeter exist in sheet specimens under tensile loading, and the plastic strain field around voids is obtained by digital image processing of deformed grids. The size growth of the microvoids, the spacing change of the neighboring voids, and the development of shear bands in the ligament between the voids, are presented by experimental results accompanied with the plastic strain distribution, that gives good interpretation to the process of void growth and coalescence with the flow localization in the ligaments. The project supported by the National Natural Science Foundation of China     

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.     

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.     

Void growth and coalescence in single crystals

Void growth and coalescence in single crystals are investigated using crystal plasticity based 3D finite element calculations. A unit cell involving a single spherical void and fully periodic boundary conditions is deformed under constant macroscopic stress triaxiality. Simulations are performed for different values of the stress triaxiality, for different crystal orientations, and for low and high work-hardening capacity. Under low stress triaxiality, the void shape evolution, void growth, and strain at the onset of coalescence are strongly dependent on the crystal orientation, while under high stress triaxiality, only the void growth rate is affected by the crystal orientation. These effects lead to significant variations in the ductility defined as the strain at the onset of coalescence. An attempt is made to predict the onset of coalescence using two different versions of the Thomason void coalescence criterion, initially developed in the framework of isotropic perfect plasticity. The first version is based on a mean effective yield stress of the matrix and involves a fitting parameter to properly take into account material strain hardening. The second version of the Thomason criterion is based on a local value of the effective yield stress in the ligament between the voids, with no fitting parameter. The first version is accurate to within 20% relative error for most cases, and often more accurate. The second version provides the same level of accuracy except for one crystal orientation. Such a predictive coalescence criterion constitutes an important ingredient towards the development of a full constitutive model for porous single crystals.     

Influences of initial porosity,stress triaxiality and Lode parameter on plastic deformation and ductile fracture

Local mechanical properties in aluminum cast components are inhomogeneous as a consequence of spatial distribution of microstructure,e.g.,porosity,inclusions,grain size and arm spacing of secondary dendrites.In this work,the effect of porosity is investigated.Cast components contain voids with different sizes,forms,orientations and distributions.This is approximated by a porosity distribution in the following.The aim of this paper is to investigate the influence of initial porosity,stress triaxiality and Lode parameter on plastic deformation and ductile fracture.A micromechanical model with a spherical void located at the center of the matrix material,called the representative volume element(RVE),is developed.Fully periodic boundary conditions are applied to the RVE and the values of stress triaxiality and Lode parameter are kept constant during the entire course of loading.For this purpose,a multi-point constraint(MPC)user subroutine is developed to prescribe the loading.The results of the RVE model are used to establish the constitutive equations and to further investigate the influences of initial porosity,stress triaxiality and Lode parameter on elastic constant,plastic deformation and ductile fracture of an aluminum die casting alloy.     

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.     

Plastic flow localization due to non-uniform void distribution

Infinite band calculations indicate that the process of flow localization in voided solids is highly sensitive to non-uniformity in void distribution. In this paper, a model is proposed for an elastic-plastic solid with an excess of voids in a disk-shaped cluster embedded in a uniform background distribution. The model is used to study the effect of a void cluster on plastic flow localization. Substantial reductions in ductility due to nonuniformity only occur for quite large clusters when the triaxiality of the overall stresses is low, as in uniaxial tension. At higher stress triaxialities, a small cluster can be severely deleterious.     

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 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.     

Pressure-sensitive ductile layers – II. 3D models of extensive damage

The mechanisms of void growth and coalescence in ductile polymeric layers, taking into account the effects of pressure-sensitivity, α, and plastic dilatancy, β, are explored in this two-part paper. In Part I, a two-dimensional model containing discrete cylindrical voids was used to simulate void growth and coalescence ahead of a crack. This paper extends the previous work by explicitly modeling initially spherical voids in a three-dimensional configuration. Damage predictions from the present 3D model for low yield strain adhesives are found to be in good agreement with both the 2D model in Part I and the computational cell element model. Significant discrepancies in the damage predictions, however, exist among all three models for high yield strain adhesives (e.g. polymers). The present 3D study also discusses the increasing damage level and its spatial extent with pressure-sensitivity, as well as the exacerbation of these effects arising from the deviation from an associated flow rule. In fact, both high porosity and high pressure-sensitivity promote void interaction. In addition, pressure-sensitivity increases the oblacity of the voids and reduces the intervoid ligament spacing over a wide range of load levels. These effects are compounded as the fracture process zone thickness decreases relative to the adhesive thickness. Results further show that both the adhesive toughness levels and the critical porosity governing the onset of void coalescence are significantly lowered with increasing pressure-sensitivity.     

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.     

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.     

Effect of stress-state and spacing on voids in a shear-field

Unit cell model analyses are carried out for a material with a periodic array of voids, subject to shear loading. Thus the focus is on ductile fracture in conditions of low stress triaxiality. It has been shown recently that voids in shear are flattened out to micro-cracks, which rotate and elongate until interaction with neighboring micro-cracks gives coalescence, so that the failure mechanism is very different from that under tensile loading. In the present studies the plane strain unit cell has fully periodic boundary conditions, so that any combination of the stress components in the overall average stress state can be prescribed. This also allows for studies of the effect of different initial void spacing in the two in-plane coordinate directions. The stress states considered are essentially simple shear, with various levels of tensile stresses or compressive stresses superposed, i.e. low positive stress triaxiality or even negative stress triaxiality. For high aspect ratio unit cells a clear localization band is found inside the cell, which actually represents several parallel bands, due to periodicity. In the materials represented by a low aspect ratio unit cell localization would also occur after that the maximum shear stress has been passed, but this is not shown when periodicity is enforced. The effect of superposed tensile or compressive stresses is found to be bigger for high aspect ratio unit cells than for low aspect ratios.