Coupling effects of void size and void shape on the growth of prolate ellipsoidal microvoid

The combined effects of void size and void shape on the void growth are studied by using the classical spectrum method. An infinite solid containing an isolated prolate spheroidal void is considered to depict the void shape effect and the Fleck-Hutchinson phenomenological strain gradient plasticity theory is employed to capture the size effects. It is found that the combined effects of void size and void shape are mainly controlled by the remote stress triaxiality. Based on this, a new size-dependent void growth model similar to the Rice-Tracey model is proposed and an important conclusion about the size-dependent void growth is drawn: the growth rate of the void with radius smaller than a critical radius r_c may be ignored. It is interesting that r_c is a material constant independent of the initial void shape and the remote stress triaxiality.The project supported by the National Natural Science Foundation of China (A10102006) and the New Century Excellent Talents in Universities of China. The English text was polished by Keren Wang.     

A physically motivated anisotropic tensorial representation of damage with separate functions for void nucleation,growth, and coalescence

A phenomenological anisotropic damage progression formulation for porous ductile metals with second phases is described through mechanisms of void nucleation, growth and coalescence. The model is motivated from fracture mechanisms and microscale physical observations. To describe the creation of new pores, the decohesion at the particle–matrix interface and the fragmentation of second phase particles, the void-crack nucleation equation is related to several microstructural parameters (fracture toughness, length scale parameter, particle size, volume and fraction of second phase), the plastic strain level, and the stress state. Nucleation is represented by a general symmetric second rank tensor, and its components are proportional to the absolute value of the plastic strain rate components. Based on the Rice and Tracey model, void growth is a scalar function of the trace of damage tensor and the positive triaxiality. Like nucleation, coalescence is a second rank tensor governed by the plastic strain rate tensor and the stress state. The coalescence threshold is related to the void length scale for void impingement and void sheet mechanisms. The coupling of damage with the Bammann–Chiesa–Johnson (BCJ) plasticity model is written in the thermodynamic framework and derives from the concept of effective stress assuming the hypothesis of energy equivalence. A full-implicit algorithm is used for the stress integration and the determination of the consistent tangent operator. Finally, macroscale correlations to cast A356 AL alloy and wrought 6061-T6 AL alloy experimental data are completed with predictive void-crack evolution to illustrate the applicability of the anisotropic damage model.     

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.     

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.     

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.     

A void–crack nucleation model for ductile metals

A phenomenological void–crack nucleation model for ductile metals with secondphases is described which is motivated from fracture mechanics and microscale physicalobservations. The void–crack nucleation model is a function of the fracture toughness of theaggregate material, length scale parameter (taken to be the average size of the second phaseparticles in the examples shown in this writing) , the volume fraction of the second phase, strainlevel, and stress state. These parameters are varied to explore their effects upon the nucleationand damage rates. Examples of correlating the void–crack nucleation model to tension data in theliterature illustrate the utility of the model for several ductile metals. Furthermore, compression,tension, and torsion experiments on a cast Al–Si–Mg alloy were conducted to determinevoid–crack nucleation rates under different loading conditions. The nucleation model was thencorrelated to the cast Al–Si–Mg data as well.     

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.     

Micromechanics of coalescence in ductile fracture

Significant progress has been recently made in modelling the onset of void coalescence by internal necking in ductile materials. The aim of this paper is to develop a micro-mechanical framework for the whole coalescence regime, suitable for finite-element implementation. The model is defined by a set of constitutive equations including a closed form of the yield surface along with appropriate evolution laws for void shape and ligament size. Normality is still obeyed during coalescence. The derivation of the evolution laws is carefully guided by coalescence phenomenology inferred from micromechanical unit-cell calculations. The major implication of the model is that the stress carrying capacity of the elementary volume vanishes as a natural outcome of ligament size reduction. Moreover, the drop in the macroscopic stress accompanying coalescence can be quantified for many initial microstructures provided that the microstructure state is known at incipient coalescence. The second part of the paper addresses a more practical issue, that is the prediction of the acceleration rate δ in the Tvergaard-Needleman phenomenological approach to coalescence. For that purpose, a Gurson-like model including void shape effects is used. Results are presented and discussed in the limiting case of a non-hardening material for different initial microstructures and various stress states. Predicted values of δ are extremely sensitive to stress triaxiality and initial spacing ratio. The effect of initial porosity is significant at low triaxiality whereas the effect of initial void shape is emphasized at high triaxiality.     

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.     

伴随微孔洞生长的裂纹弹塑性定常扩展

裂纹在韧性材料中扩展时,将们随着微孔洞的萌生和生长,孔洞的萌生和深化将直接影响着材料的总体断裂韧性和强度,以往的研究主要集中在将裂纹的扩展刻划为微孔洞的萌生、生长和汇合这样一个过程。从传统的断裂过程区模型出发研究微孔洞的萌生和生长对材料总体断裂韧性的影响,通过采用Ｇｕｒｓｏｎ模型,建立塑性增量本构关系,然后针对定常扩展情况直接进行分析,孔洞对材料断裂韧性的影响由本构关系刻划,而在孔洞汇合模型中,上     

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。     

Size effects on stress concentration induced by a prolate ellipsoidal particle and void nucleation mechanism

There generally exist two void nucleation mechanisms in materials, i.e. the breakage of hard second-phase particle and the separation of particle–matrix interface. The role of particle shape in governing the void nucleation mechanism has already been investigated carefully in the literatures. In this study, the coupled effects of particle size and shape on the void nucleation mechanisms, which have not yet been carefully addressed, have been paid to special attention. To this end, a wide range of particle aspect ratios (but limited to the prolate spheroidal particle) is considered to reflect the shape effect; and the size effect is captured by the Fleck–Hutchinson phenomenological strain plasticity constitutive theory (Advance in Applied Mechanics, vol. 33, Academic Press, New York, 1997, p. 295). Detailed theoretical analyses and computations on an infinite block containing an isolated elastic prolate spheroidal particle are carried out to light the features of stress concentrations and their distributions at the matrix–particle interface and within the particle. Some results different from the scale-independent case are obtained as: (1) the maximum stress concentration factor (SCF) at the particle–matrix interface is dramatically increased by the size effect especially for the slender particle. This is likely to trigger the void nucleation at the matrix–particle interface by cleavage or atomic separation. (2) At a given overall effective strain, the particle size effect significantly elevates the stress level at the matrix–particle interface. This means that the size effect is likely to advance the interface separation at a smaller overall strain. (3) For scale-independent cases, the elongated particle fracture usually takes place before the interface debonding occurs. For scale-dependent cases, although the SCF within the particle is also accentuated by the particle size effect, the SCF at the interface rises at a much faster rate. It indicates that the probability of void nucleation by the interface separation would increase.     

Modeling stress state dependent damage evolution in a cast Al–Si–Mg aluminum alloy

Internal state variable rate equations are cast in a continuum framework to model void nucleation, growth, and coalescence in a cast Al–Si–Mg aluminum alloy. The kinematics and constitutive relations for damage resulting from void nucleation, growth, and coalescence are discussed. Because damage evolution is intimately coupled with the stress state, internal state variable hardening rate equations are developed to distinguish between compression, tension, and torsion straining conditions. The scalar isotropic hardening equation and second rank tensorial kinematic hardening equation from the Bammann–Chiesa–Johnson (BCJ) Plasticity model are modified to account for hardening rate differences under tension, compression, and torsion. A method for determining the material constants for the plasticity and damage equations is presented. Parameter determination for the proposed phenomenological nucleation rate equation, motivated from fracture mechanics and microscale physical observations, involves counting nucleation sites as a function of strain from optical micrographs. Although different void growth models can be included, the McClintock void growth model is used in this study. A coalescence model is also introduced. The damage framework is then evaluated with respect to experimental tensile data of notched Al–Si–Mg cast aluminum alloy specimens. Finite element results employing the damage framework are shown to illustrate its usefulness.     

Plasticity size effects in voided crystals

The shear and equi-biaxial straining responses of periodic voided single crystals are analysed using discrete dislocation plasticity and a continuum strain gradient crystal plasticity theory. In the discrete dislocation formulation, the dislocations are all of edge character and are modelled as line singularities in an elastic material. The lattice resistance to dislocation motion, dislocation nucleation, dislocation interaction with obstacles and annihilation are incorporated through a set of constitutive rules. Over the range of length scales investigated, both the discrete dislocation and strain gradient plasticity formulations predict a negligible size effect under shear loading. By contrast, under equi-biaxial loading both plasticity formulations predict a strong size dependence with the flow strength approximately scaling inversely with the void spacing. Excellent agreement is obtained between predictions of the two formulations for all crystal types and void volume fractions considered when the material length scale in the non-local plasticity model is chosen to be (about 10 times the slip plane spacing in the discrete dislocation models).     

A stochastic approach to the internal damage in a structure solid: Part 1: Theoretical model

The material system is considered as heterogenous medium of actual microstructural elements. These elements exhibit random geometric and physical characteristics and are further disturbed by a latitude of randomly oriented, second phase particles. A stochastic model is presented for the occurring damage process due to the nucleation and growth of microvoids under external loading. From a micromechanical point of view, the nucleation of a void at a partile-matrix interface is considered to be associated with the cut-off of the interfacial binding potential. The growth of an elemental void is seen, then, to follow a random walk of the discrete Markov type. The latter is associated with the build-up of strain in front of the tip of the advancing void and the redistribution of local stress. As the void reaches the boundary between neighbouring elements, a dicrete inter-elemental fracture process is examined in relation to the intensities of transformation within the elemental boundary.     

材料三维微结构表征及其晶体塑性有限元模拟

材料的力学性能，尤其是在有限变形下所呈现的宏观各向异性，是材料结构设计和服役寿命考虑的关键因素。由于宏观模型不能较好地反映材料微观结构（晶粒的形貌和取向等）对宏观塑性各向异性的影响，因此，本文建立了能实际反映晶粒形貌的三维Voronoi模型，并基于晶体塑性理论对铝合金在有限变形下的响应进行计算。首先，建立反映材料微结构的代表性体积单元RVE模型进行计算，并与实验结果进行对比验证。然后，以单向拉伸为例，分析了有限变形过程中试件的晶粒形貌和取向分布等微观因素对宏观各向异性演化的影响，并从材料和结构两个层面讨论了微观结构对宏观力学性能的影响。结果表明，本文模型能够反映微观结构对宏观力学性能的影响，为实际生产制造领域构件的力学性能提供可靠的预测。     

Dynamic void nucleation and growth in solids: A self-consistent statistical theory

We present a framework for a self-consistent theory of spall fracture in ductile materials, based on the dynamics of void nucleation and growth. The constitutive model for the material is divided into elastic and “plastic” parts, where the elastic part represents the volumetric response of a porous elastic material, and the “plastic” part is generated by a collection of representative volume elements (RVEs) of incompressible material. Each RVE is a thick-walled spherical shell, whose average porosity is the same as that of the surrounding porous continuum, thus simulating void interaction through the resulting lowered resistance to further void growth. All voids nucleate and grow according to the appropriate dynamics for a thick-walled sphere made of incompressible material. The macroscopic spherical stress in the material drives the response in all volume elements, which have a distribution of critical stresses for void nucleation, and the statistically weighted sum of the void volumes of all RVEs generates the global porosity. Thus, macroscopic pressure, porosity, and a distribution of growing microscopic voids are fully coupled dynamically. An example is given for a rate-independent, perfectly plastic material. The dynamics of void growth gives rise to a rate effect in the macroscopic material even though the parent material is rate independent.     

A micromechanics based damage model for composite materials

The predictive capacity of ductile fracture models when applied to composite and multiphase materials is related to the accuracy of the estimated stress/strain level in the second phases or reinforcements, which defines the condition for damage nucleation. Second phase particles contribute to the overall hardening of the composite before void nucleation, as well as to its softening after their fracture or decohesion. If the volume fraction of reinforcement is larger than a couple of percents, this softening can significantly affect the resistance to plastic localization and cannot be neglected. In order to explicitly account for the effect of second phase particles on the ductile fracture process, this study integrates a damage model based on the Gologanu–Leblond–Devaux constitutive behavior with a mean-field homogenization scheme. Even though the model is more general, the present study focuses on elastic particles dispersed in an elasto-plastic matrix. After assessing the mean-field homogenization scheme through comparison with two-dimensional axisymmetric finite element calculations, an extensive parametric study is performed using the integrated homogenization-damage model. The predictions of the integrated homogenization-damage model are also compared with experimental results on cast aluminum alloys, in terms of both the fracture strain and overall stress–strain curves. The study demonstrates the complex couplings among the load transfer to second phase particles, their resistance to fracture, the void nucleation mode, and the overall ductility.     

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.     

Void fraction for random loose packing of the cylindrical particles considering filling rate,material and shape

Particle packing is widely applied in organic pollutant adsorption, catalytic reaction, biomass combustion, nuclear cooling, and other scenarios. Due to the complexity of the shape, the studies on the void fraction of the cylindrical particles are not as thorough as the spherical particles. This study investigated the influence of the filling rate, material properties and sphericity on the void fraction of cylinders through experiments and simulation. DEM (discrete element method) was validated by the internal structures of the packing obtained by CT (computed tomography). Based on the logarithmic correlation between the void fraction and filling rate, an ingenious framework for predicting the void fraction of cylindrical particles was presented with two intermediate coefficients. By correlating the coefficients with the material property and sphericity, a novel void-fraction prediction model was established with R-squared of 0.996. The mechanism of void fraction under random loose packing for cylinders was eventually found in this study.