The modified Gurson model accounting for the void size effect

The Gurson model [J. Engrg. Mater. Technol. 99 (1977) 2] has been widely used to study the deformation and failure of metallic materials containing microvoids. The void volume fraction is the only parameter representing voids since the void size does not come into play in the Gurson model. Based on the Taylor dislocation model [Proc. R. Soc. (Lond.) A145 (1934) 362; J. Int. Metals 62 (1938) 307], we extend the Gurson model to account for the void size effect. It is shown that the yield surfaces for micron- and submicron-sized voids are significantly larger than that given by the Gurson model. For a voided, dilating material subject to uniaxial tension, the void size has essentially no effect on the stress–strain curve at small initial void volume fraction. However, as the initial void volume fraction increases, the void size effect may become significant.     

A note on the effect of surface energy and void size to void growth

Recent studies revealed that rapid void growth is the dominant failure mechanism in an elasto-plastic solid under high mean tensile stress. This paper studies the effect of the surface energy and void size to the void growth. The models of a thick spherical shell and a thick spherical column in void growth are analyzed and numerically estimated. The main conclusion from this study is that, for typical metals, the surface energy effect is negligible for voids larger than 100 nm in size, but it may become significant when the void size is on the order of 10 nm.     

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.     

COUPLING EFFECTS OF VOID SHAPE AND VOID SIZE ON THE GROWTH OF AN ELLIPTIC VOID IN A FIBER-REINFORCED HYPER-ELASTIC THIN PLATE

The growth of a prolate or oblate elliptic micro-void in a fiber reinforced anisotropic incompressible hyper-elastic rectangular thin plate subjected to uniaxial extensions is studied within the framework of finite elasticity. Coupling effects of void shape and void size on the growth of the void are paid special attention to. The deformation function of the plate with an isolated elliptic void is given, which is expressed by two parameters to solve the differential equation. The solution is approximately obtained from the minimum potential energy principle. Deformation curves for the void with a wide range of void aspect ratios and the stress distributions on the surface of the void have been obtained by numerical computation. The growth behavior of the void and the characteristics of stress distributions on the surface of the void are captured. The combined effects of void size and void shape on the growth of the void in the thin plate are discussed. The maximum stresses for the void with different sizes and different void aspect ratios are compared.     

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.     

Impact of surface stress on stress concentration

Effect of surface stress on stress concentration near a spherical void in an elastic medium is examined in the framework of continuum surface elasticity. It is assumed that the void is of spherical shape, and the elastic medium is elastically isotropic and infinitely extended. By using the Papkovitch–Neuber displacement potentials, the elastic field caused by a unidirectional remote load is obtained explicitly. Numerical results show that the influence of surface stress becomes remarkable when the size of the void is reduced to nanometer scale, leading to that stress concentration near the void depends not only on the void size but also on the remote load.     

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.     

Effect of bubble size on void fraction fluctuations in dispersed bubble flows

It is known that bubble size affects seriously the average void fraction in bubbly flows where buoyant velocities vary considerably with bubble size. On the contrary, there is no systematic literature report about bubble size effects on the intensity and frequency of void fraction fluctuations around the average void fraction. This work aims to provide such information. An electrical impedance technique is employed along with non-intrusive ring electrodes to register void fraction fluctuations down to 10⁻⁵. Bubble size fluctuations are estimated from high resolution optical images. Experiments are conducted in co-current upward dispersed bubble flow inside a 21 mm tube with average bubble size between ∼50 and ∼700 μm. Water and blood simulant are used as test liquids with velocity from ∼3 to ∼30 cm s⁻¹. The above resemble conditions of Decompression Sickness (DCS) in the bloodstream of human vena cava. It is found that the intensity and frequency of void fraction fluctuations vary appreciably with bubble size at constant gas and liquid flow rates. Moreover, these variations are not random but scale with bubble size. As a first step to quantify this effect, an empirical expression is derived that relates average bubble size to the ratio standard deviation/average value of void fraction.     

强动载作用下孔洞汇合对延性金属层裂损伤演化过程的影响

针对强动载作用下延性金属的层裂问题,在分析孔洞之间几何关联的基础上,定义了一个新的耦合损伤及孔洞几何信息的孔洞汇合判定方法,同时,基于能量守恒原理,解析了孔洞汇合对损伤快速增长影响的物理机理.通过分析数值计算结果和对比相关文献的实验可知:孔洞汇合后不仅引起损伤增长,而且导致了损伤材料内部微孔洞数目的减少、孔洞平均尺寸的增加。     

Relating cellular structure of open solid food foams to their Young’s modulus: Finite element calculation

This paper investigates the role of structure on Young’s modulus of open cell materials of relative densities between 0.1 and 0.3. The cellular solid is obtained by generating mixture size of spherical voids using the Random Sequential Addition – RSA algorithm. The relative density of the material is controlled by increasing void number and overlap. Structural effects consider mainly a Gaussian distribution of spherical void size of varying width, distribution centre and void overlap distance. Finite element method is used to calculate effective Young’s modulus using a regular meshing scheme of 3D typical cellular solids and Conjugate Gradient solver. It is found that sphere overlap has the largest effect compared to sphere distribution width for a given density. A large scatter in the wall thickness distribution is predicted when overlapping is increased or when the width of sphere size distribution is decreased. Increased rigidity is found to be correlated to particular arrangement of mixture size spheres which is pointed out using the Pair Correlation Function. Experimental evidence of the role of void overlapping is treated in the case of bread crumbs structures determined using X-ray tomography. The scatter of effective Young’s modulus for a given relative density is sensitive to void overlapping.     

The dynamic growth of a single void in a viscoplastic material under transient hydrostatic loading

We have examined the problem of the dynamic growth of a single spherical void in an elastic-viscoplastic medium, with a view towards addressing a number of problems that arise during the dynamic failure of metals. Particular attention is paid to inertial, thermal and rate-dependent effects, which have not previously been thoroughly studied in a combined setting. It is shown that the critical stress for unstable growth of the void in the quasistatic case is strongly affected by the thermal softening of the material (in adiabatic calculations). Thermal softening has the effect of lowering the critical stress, and has a stronger influence at high strain hardening exponents. It is shown that the thermally diffusive case for quasistatic void growth in rate-dependent materials is strongly affected by the initial void size, because of the length scale introduced by the thermal diffusion. The effects of inertia are quantified, and it is demonstrated that inertial effects are small in the early stages of void growth and are strongly dependent on the initial size of the void and the rate of loading. Under supercritical loading for the inertial problem, voids of all sizes achieve a constant absolute void growth rate in the long term. Inertia first impedes, but finally promotes dynamic void growth under a subcritical loading. For dynamic void growth, the effect of rate-hardening is to reduce the rate of void growth in comparison to the rate-independent case, and to reduce the final relative void growth achieved.     

The size dependence of micro-toughness in ductile fracture

Micro-toughness in ductile fracture is defined as the plastic work dissipated per unit fracture surface area in the material separation processes of void growth and coalescence. A micromechanics model for the estimation of the size dependence of micro-toughness in ductile fracture is presented. Size effects are incorporated in the model using the conventional mechanism-based strain gradient plasticity (CMSG) theory. A finite element model of an axisymmetric representative unit cell with an initial spherical void is used to validate model predictions. Two characteristic length scales emerge from the model. The initial void radius sets the scale for the initial spherical void growth. For the subsequent void coalescence, the scale is set by the width of the intervoid ligament. Energy dissipation in ductile fracture is found to be dominated by the mechanisms of coalescence, and the micro-toughness in ductile fracture is found to be size dependent for dimple sizes approximately one order of magnitude larger than the material length scale.     

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.     

The void-size effect on plastic flow localization in the Gurson model

Recent studies have shown that the size of microvoids has a significant effect on the void growth rate. The purpose of this paper is to explore whether the void size effect can influence the plastic flow localization in ductile materials. We have used the extended Gurson‘s dilatational plasticity theory, which accounts for the void size effect, to study the plastic flow localization in porous solids with long cylindrical voids. The localization model of Rice is adopted, in which the material inside the band may display a different response from that outside the band at the incipient plastic flow localization. The present study shows that it has little effect on the shear band angle.     

A parametric-experimental study of void growth in superplastic deformation

Substantial void growth in metals constitutes a problem in many industrial operations that utilize superplastic deformation. This is because of the likelihood of material failure due to such growth. Hence, there is a need to study void growth mechanisms in an effort to understand the parameters governing it. In this work, numerical and experimental studies of void growth, and the parameters that affect it, in a superplastically deforming (SPD) metal have been performed. In the numerical studies, using the finite-element method, a 1×2 sized thin plate (i.e. plane stress conditions) of a viscoplastic material with pre-existing holes has been subjected to a constant extension rate. The experimental studies were performed under similar conditions to the numerical ones and provided for qualitative comparison. The parameters affecting void growth in SPD are: m (the strain-rate sensitivity), void size (i.e. diameter) and the number (density) of existing voids. The results showed that increased m values produced strengthening and decreased the rate of void growth. In addition, larger initial void size (or, equivalently, a larger initial void fraction) had the effect of weakening the specimen through causing accelerated void growth. Finally, multiple holes had the effect of increasing the metal ductility by reducing the extent of necking and its onset. This was realized through diffusing the plastic deformation at the different hole sites and reducing the stress concentration. The numerical results were in good qualitative agreement with the experiment and suggested the need to refine existing phenomenological void growth models to include the dependence on the void 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.     

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.     

Computer simulations of nanoindentation on Cu (1 1 1) with a void

This paper employs static atomistic simulations to investigate the effect of a void on the nanoindentation of Cu (1 1 1). The simulations minimize the potential energy of the complete system via finite element formulation to identify the equilibrium configuration of any deformed state. The size and depth of the void are treated as two variable parameters. The numerical results reveal that the void disappears when the indentation depth is sufficiently large. A stress concentration is observed at the internal surface of the void in all simulations cases. The results indicate that the presence of a void has a significant influence on the nanohardness extracted from the nanoindentation tests.     

Void fraction,bubble velocity and bubble size in two-phase flow

Experiments were performed in atmosphereic vertical air-water flows, for void fractions between 0.25 and 0.75 (cross-sectional averages) and superficial liquid velocities of 1.3, 1.7 and 2.1 m/s. Local values of void fraction and bubble velocity as well as the bubble diameter were measured by means of a resistivity probe technique. Reliable values were obtained for the local void fraction over the entire range $\text{0 ≤ α ≤ 1}$. The void fraction profiles appeared to have a local maximum at the pipe center, local maxima close to the wall were obviously absent. The resistivity probes are shown to measure the velocity of the interface between the conducting and nonconducting phases, which equals the gas velocity only for low void fractions. The measured data for void fraction and bubble velocity were correlated by means of power law distribution functions, with exponents given by a function of the cross-sectionally averaged void fraction. The Sauter mean diameters for the bubble size spectra found, agree reasonably well with diameters predicted by a theoretical model based on the energy dissipation in the flow.