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.     

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     

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.     

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.     

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.     

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.     

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.     

Rupture mechanisms in combined tension and shear—Micromechanics

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

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；     

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.     

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.     

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.     

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 growth and coalescence in ductile solids with stage III and stage IV strain hardening

State of the art ductile fracture models often rely on simple power laws to describe the strain hardening of the matrix material. Power laws do not distinguish between the two main stages of hardening observed in polycrystals, referred to as stage III and stage IV hardening, and which emerge from the evolution of the dislocation substructure. The aim of this study is to couple a physics based strain hardening law including these two stages to a micromechanics based ductile damage model. One of the main motivations is that, the stage IV constant hardening rate stage, occurring only at large strain, will be attained in most ductile failure problems if not at the overall level of deformation, at least locally around the growing voids. Furthermore, proper modelling of the stage III involving dislocation storage and recovery terms and the transition to stage IV provides a link with the underlying physical mechanisms of deformation and with the microstructure. First, in order to evaluate the effects of the stage III and stage IV hardening on void growth and coalescence, an extensive parametric study is performed on two-dimensional (2D) axisymmetric finite element (FE) unit cell calculations, using a Kocks-Mecking type hardening law. The cell calculations demonstrate that accounting for the stage IV hardening can have a profound effect on delaying void coalescence and increasing the ductility. The magnitude of the recovery term during stage III has also a significant effect on the void growth rate. Then, the Kocks-Mecking law is incorporated into the Gologanu-Leblond-Devaux (GLD) porous plasticity model supplemented by two different versions of the Thomason void coalescence criterion. The predictions of the damage model are in good agreement with the results of the FE calculations in terms of the stress-strain curves, the evolution of void shape and porosity, as well as the strain value at the onset of void coalescence.     

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

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

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.     

Effects of void band orientation and crystallographic anisotropy on void growth and coalescence

The effects of void band orientation and crystallographic anisotropy on void growth and linkage have been investigated. 2D model materials were fabricated by laser drilling a band of holes into the gage section of sheet tensile samples using various orientation angles with respect to the tensile axis normal. Both copper and magnesium sheets have been studied in order to examine the role of crystallographic anisotropy on the void growth and linkage processes. The samples were pulled in uniaxial tension inside the chamber of an SEM, enabling a quantitative assessment of the growth and linkage processes. The void band orientation angle has a significant impact on the growth and linkage of the holes in copper. As the void band orientation angle is increased from 0° to 45°, the processes of coalescence and linkage are delayed to higher strain values. Furthermore, the mechanism of linkage changes from internal necking to one dominated by shear localization. In contrast, the void band orientation does not have a significant impact on the void growth and linkage processes in magnesium. Void growth in these materials occurs non-uniformly due to interactions between the holes and the microstructure. The heterogeneous nature of deformation in magnesium makes it difficult to apply a coalescence criterion based on the void dimensions. Furthermore, the strain at failure does not show a relationship with the void band orientation angle. Failure associated with twin and grain boundaries interrupts the plastic growth of the holes and causes rapid fracture. Therefore, the impact of the local microstructure outweighs the effects of the void band orientation angle in this material.     

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.