The evolution of voids in the plasticity strain hardening gradient materials

The main aim of this paper is to opens out the meso-mechanism of void growth and coalescence in the matrix materials with graded strain-hardening exponent distribution. For this end, detailed finite element computations of a representative cylindrical cell containing a spherical void have been carried out. According to the FE analyses, significant effects of the strain-hardening exponent gradient (SEG) in the matrix on the void growth and coalescence are revealed: (1) In the homogeneous materials, the void growth and coalescence are slightly dependent on the strain-hardening exponent, however, the SEG distribution in the matrix can increase remarkably the void growth rate and decrease seriously the void coalescence strain. (2) The critical void shapes in the homogeneous materials are mainly governed by the macroscopic stress triaxiality, but due to earlier plastic flow localization in the softer matrix layer, the SEG distribution in the matrix has very significant effects on the deformed void shapes, especially when the stress triaxiality is lower. (3) When the triaxial stress levels are lower, in the homogeneous materials, the shape change mode of the void evolution is dominate so the void growth rate is very low; however, the SEG distribution in the matrix can bring the volume change mode out, as a result of increasing the void growth rate. (4) Comparisons of the numerical results with the existing damage model indicate that the classic damage model cannot give satisfying prediction to the void growth in both the homogeneous strain-hardening matrix and the SEG materials. On the basis of large numbers of numerical computations, a new damage model, which can uniformly describe the void growing in the homogeneous and plasticity gradient materials, is suggested. A mass of element computations have validated that the new damage model can give satisfying agreement with the FE results of cell model.     

Effects of pressure-sensitivity and plastic dilatancy on void growth and interaction

Hydrostatic stress can affect the non-elastic deformation and flow stress of polymeric materials and certain metallic alloys. This sensitivity to hydrostatic stress can also influence the fracture toughness of ductile materials, which fail by void growth and coalescence. These materials typically contain a non-uniform distribution of voids of varying size-scales and void shapes. In this work, the effects of void shape and microvoid interaction in pressure-sensitive materials are examined via a two-prong approach: (i) an axisymmetric unit-cell containing a single ellipsoidal void and (ii) a plane-strain unit-cell consisting of a single large void and a population of discrete microvoids. The representative material volume in both cases is subjected to physical stress states similar to highly stressed regions ahead of a crack. Results show that oblate voids and microvoid cavitation can severely reduce the critical stress of the material. These effects can be compounded under high levels of pressure-sensitivity. In some cases, the critical stress responsible for rapid void growth is reduced to levels comparable to the yield strength of the material. The contribution of void shape and pressure-sensitivity to the thermal- and moisture-induced voiding phenomenon in IC packages is also discussed.     

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.     

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.     

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.     

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.     

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     

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.     

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.     

Micromechanical finite element calculations of temperature and void configuration effects on void growth and coalescence

We present micromechanical finite element results that quantify coalescence effects based upon temperature and different spatial arrangements of voids. We propose a critical intervoid ligament distance (ILD) to define void coalescence that is derived from micromechanical simulations in which void volume fraction evolves as a function of strain. Several parameters were varied using the temperature and strain rate internal variable plasticity model of Bammann–Chiesa–Johnson to determine the coalescence effects. The parameters include two types of materials with different work hardening rates (304L stainless steel and 6061T6 aluminum), three different temperatures (298, 400, and 600 K), several boundary conditions (force and displacement: uniaxial, plane strain, and biaxial), type of element used (plane strain and axisymmetric), different ILDs, and the number of voids (one and two void configurations). The present study provides a basis for macroscale modeling of coalescence which is briefly discussed.     

The mechanical behaviour of ductile materials damaged by two generations of voids

Large strain finite element method is employed to investigate the damaging effect of two generations of voids in ductile materials. An axisymmetric cylinder embedding an initially spherical void is chosen as the model cell. Secondary voids will initiate around the initial void when the local stress/strain in the matrix increases to certain critical conditions. This event is numerically simulated through an empty element technique. The interaction between these two generations of voids has been proved to be favourable to the voiding condition, thus accelerating the material damage, characterized by the value of the overall elastic modulus which may undergo drastic drop when nearing final fracture.     

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.     

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.     

Finite deformation analysis of crack-tip opening in elastic-plastic materials and implications for fracture

Analyses of the stress and strain fields around smoothly-blunting crack tips in both non-hardening and hardening elastic-plastic materials, under contained plane-strain yielding and subject to mode I opening loads, have been carried out by use of a finite element method suitably formulated to admit large geometry changes. The results include the crack-tip shape and near-tip deformation field, and the crack-tip opening displacement has been related to a parameter of the applied load, the J-integral. The hydrostatic stresses near the crack tip are limited due to the lack of constraint on the blunted tip, limiting achievable stress levels except in a very small region around the crack tip in power-law hardening materials. The J-integral is found to be path-independent except very close to the crack tip in the region affected by the blunted tip. Models for fracture are discussed in the light of these results including one based on the growth of voids. The rate of void-growth near the tip in hardening materials seems to be little different from the rate in non-hardening ones when measured in terms of crack-tip opening displacement, which leads to a prediction of higher toughness in hardening materials. It is suggested that improvement of this model would follow from better understanding of void-void and void-crack coalescence and void nucleation, and some criteria and models for these effects are discussed. The implications of the finite element results for fracture criteria based on critical stress or strain, or both, is discussed with respect to transition of fracture mode and the angle of initial crack-growth. Localization of flow is discussed as a possible fracture model and as a model for void-crack coalescence.     

Applications of planar anisotropic yield criteria to porous sheet metal forming simulations

Planar anisotropic yield functions, with rounded vertexes especially near the equal-biaxial direction of the corresponding yield loci, appropriate for some structure metals are employed for the matrix surrounding voids in the present study. The widely adopted Hill anisotropic yield functions are also implemented into the matrix for comparisons. Mechanisms of the void growth, void nucleation, and void coalescence are simultaneously considered here. Effects of the yield function of the corresponding matrix on the sheet metal under two typical sheet forming operations, a hemispherical punch stretching operation and a cup drawing operation, are investigated via a finite element analysis. Thickness strains in various orientations of the sheet are then evaluated. Simulation results show that the yield function of the corresponding matrix plays important roles on the strain distribution and the strain localization as well. Early localization would be found for the sheet with relatively small initial void volume fraction in two operations. Yield functions of the matrix rather influence the earing phenomenon under the cup drawing procedure even similar displacement profiles of the outer boundary could be observed.     

Porous materials with two populations of voids under internal pressure: II. Growth and coalescence of voids

This study is devoted to the mechanical behaviour of polycrystalline materials with two populations of voids, small spherical voids located inside the grains and larger spheroidal voids located at the grain boundaries. In part I of the work, instantaneous effective stress–strain relations were derived for fixed microstructure. In this second part, the evolution of the microstructure is addressed. Differential equations governing the evolution of the microstructural parameters in terms of the applied loading are derived and their integration in time is discussed. Void growth results in a global softening of the stress–strain response of the material. A simple model for the prediction of void coalescence is proposed which can serve to predict the overall ductility of polycrystalline porous materials under the combined action of thermal dilatation and internal pressure in the voids.     

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.     

Void collapse/growth in solid materials under overall shear loading

The present work aims at studying numerically the influence of void concentration, number of voids and absence/presence of inclusion on void collapse/growth and coalescence in materials submitted to shear loading. Starting from the experimental observation that voiding mostly forms within bands of localised deformation in the form of void sheets, the geometrical configuration retained to that purpose is a layer of periodic cells with 1–5, empty or particle-containing voids, subject to simple shearing.