Approximate yield criteria for anisotropic metals with prolate or oblate voidsCritères macroscopiques pour des métaux plastiques anisotropes contenant des cavités non sphériques

Following the study of Gologanu et al. (1997) which has extended the well-known approach of Gurson (1975), we propose approximate yield criteria for anisotropic plastic voided metals containing non spherical cavities. The plastic anisotropy of the matrix is described by means of Hill's quadratic criterion. The procedure to establish the closed form expression of approximate macroscopic criteria, in which void shape and plastic anisotropic effects are included, is detailed. The new criteria allow us to recover existing results in the cases of spherical and cylindrical voids in an Hill type plastic matrix. Moreover, they agree with previous criteria for non spherical voids in an isotropic plastic matrix. Finally, for validation purposes, we provide, in the general case of non spherical cavities in the anisotropic matrix, a comparison with the numerical exact two field criteria. To cite this article: V. Monchiet et al., C. R. Mecanique 334 (2006).     

Plasticity and ductile fracture of IF steels: experiments and micromechanical modeling

If one aims at the simulation of plasticity and failure of multiphase materials, the choice of an appropriate material law is of major importance. Plasticity models for porous metals contain, in addition to the yield surface and the flow potential, also functions describing the void nucleation, dependent on some macroscopically observable quantities, and the growth of these voids. In this paper, a micromechanically based method to develop a void nucleation function for porous plasticity models is proposed which is valid for all possible microstructures as long as the amount of second phase particles is low (i.e. the particles do not interact with respect to the stress and strain fields), and as long as the particles are large enough (above 0.1 μm) justifying a continuum mechanical approach. The method described consists of two stages: In the first stage, the microstructure is investigated via a finite element model. The FE model implicitly contains the effects of the shape of the precipitates, of the material parameters of both the matrix and the precipitates, of the void nucleation hypothesis (by the assumption of “nucleation limits” for characteristic damage-related quantities), and of the applied stress state. In the second stage, during postprocessing, the volume fraction of precipitates as well as the influences of the particle orientation distribution, size distribution, and size dependence of the damage-related quantities are taken into account. The model is applied to the microstructure of IF (Interstitially Free) steel, a material with a ductile matrix and rigid second phase particles of cubical shape. This microstructure is particularly suited for investigating shape and size effects. The model shows that either the size effect or the shape effect dominate the void nucleation behavior: in the case of particles of roughly the same size, the size distribution will hardly alter the nucleation strain distribution obtained by taking into account only the shape and orientation effects. For particles of very different sizes, the size effect will completely override the rather “sharp” original distribution regarding particle shape and orientation.     

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.     

A method of plasticity for general aligned spheroidal void or fiber-reinforced composites

Based on the general concept of the secant moduli method, together with a new way of evaluating the average matrix effective stress originally proposed by Qiu and Weng (“A Theory of Plasticity for Porous Materials and Particle-Reinforced Composites”, ASME J. Appl. Mech. (1992), 59, 261.), a method for nonlinear effective properties of general aligned fiber or void composites is proposed. The method is capable of predicting composite (especially for porous materials) yielding under a hydrostatic load. Compared to the Tandon and Weng (“A Theory of Particle-Reinforced Plasticity,” ASME J. Appl. Mech. (1988), 55, 126.), model the proposed method always gives softer prediction in the uniaxial tension. The proposed method will predict the same nonlinear stress and strain relation as the Ponte Castaneda (“The Effective Mechanical Properties of Nonlinear Isotropic Composite,” J. Mech. Phys. Solids (1991), 39, 45.) variational model if the same estimates or bounds for the linear comparison composite are adopted.     

Axisymmetric deformation of power-law solids containing a dilute concentration of aligned spheroidal voids

T response of an incompressible power-law matrix containing a dispersion of aligned, spheroidal voids is investigated. Attention is restricted to dilute concentrations of voids and to axisymmetric deformation of the solid. The essential step in the analysis is the solution of a kernel problem for an isolated void, and this solution is obtained accurately and efficiently using a Ritz procedure developed for this purpose. Results for macroscopic strain-rates are presented for void shapes ranging from penny-shaped cracks to infinitely long circular cylinders and for a wide range of triaxialities and matrix hardening exponents. These results are used to assess the role of void shape on the overall response of porous solids.     

On ductile fracture initiation toughness: Effects of void volume fraction,void shape and void distribution

This paper studies the effects of the initial relative void spacing, void pattern, void shape and void volume fraction on ductile fracture toughness using three-dimensional, small scale yielding models, where voids are assumed to pre-exist in the material and are explicitly modeled using refined finite elements. Results of this study can be used to explain the observed fracture toughness anisotropy in industrial alloys. Our analyses suggest that simplified models containing a single row of voids ahead of the crack tip is sufficient when the initial void volume fraction remains small. When the initial void volume fraction becomes large, these simplified models can predict the fracture initiation toughness (J_Ic) with adequate accuracy but cannot predict the correct J–R curve because they over-predict the interaction among growing voids on the plane of crack propagation. Consequently, finite element models containing multiple rows of voids should be used when the material has large initial void volume fraction.     

Modeling of crack growth in ductile solids: a three-dimensional analysis

A population of several spherical voids is included in a three-dimensional, small scale yielding model. Two distinct void growth mechanisms, put forth by [Int. J. Solids Struct. 39 (2002) 3581] for the case of a two-dimensional model containing cylindrical voids, are well contained in the model developed in this study for spherical voids. A material failure criterion, based on the occurrence of void coalescence in the unit cell model, is established. The critical ligament reduction ratio, which varies with stress triaxiality and initial porosity, is used to determine ligament failure between the crack tip and the nearest void. A comparison of crack initiation toughness of the model containing cylindrical voids with the model containing spherical voids reveals that the material having a sizeable fraction of spherical voids is tougher than the material having cylindrical voids. The proposed material failure determination method is then used to establish the fracture resistance curve (J–R curve) of the material. For a ductile material containing a small volume fraction of microscopic voids initially, the void by void growth mechanism prevails, which results in a J–R curve having steep slope. On the other hand, for a ductile material containing a large volume fraction of initial voids, the multiple voids interaction mechanism prevails, which results in a flat J–R curve. Next, the effect of T-stress on fracture resistance is examined. Finally, nucleation and growth of secondary microvoids and their effects on void coalescence are briefly discussed.     

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.     

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.     

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.     

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

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

A new model for porous nonlinear viscous solids incorporating void shape effects – II: Numerical validation

The aim of this work is to critically assess the new model for porous, nonlinear viscous solids incorporating void shape effects proposed in Part I, by comparing its predictions with the results of some numerical micromechanical simulations. Two kinds of simulations are performed. First, the gauge surface of spheroidal representative volume elements, as considered in Part I, is determined for various values of the porosity, the aspect ratio of the void and the Norton exponent. This is done through minimization of the macroscopic viscous potential over a family of trial velocity fields especially adapted to the spheroidal geometry, which was proposed by Lee and Mear. Such simulations allow not only for satisfactory validation of the approximate analytical gauge surface proposed, but also for adjustment of the heuristic coefficients involved in the evolution equation for the void shape parameter. Second, the evolution in time of cylindrical cells subjected to various mechanical loads is determined by the finite element method. The quasi-periodicity of this new geometry is intended to approximately represent interactions between neighbouring voids. These simulations also reveal very good agreement between model predictions and numerical calculations, provided that the effect of the new geometry considered is accounted for by using a non-unity value for the analog of Tvergaard's famous “$q_1$” parameter for porous plastic solids.     

Simulation of the behaviour of FCC polycrystals during reversed torsion

Taylor-type polycrystal plasticity models with various single slip hardening laws are evaluated by studying the large strain behaviour of FCC polycrystals during reversed torsion. The hardening laws considered include the model of Asaro and Needleman (“Texture Development and Strain Hardening in Rate Dependent Polycrystals,” Acta Metall. (1985), 34, 1553) as well as a power-law and an exponential version of that, and a more recent model by Bassani and Wu (“Latent Hardening in Single Crystals II. Analytical Characterization and Predictions,” Proc. R. Soc. Lond. (1991), A435, 21). The material parameters for the various hardening laws are fitted to experimental compression data for copper and then used to predict reversed large strain torsion of tubes. Differences under “free-end” (axially unconstrained) or “fixed-end” (axially constrained) conditions between predictions and experimental observations are discussed in detail. In addition to the torque response, the Swift effects upon twist reversal are studied.     

A simplified “polycrystalline” model for viscoplastic and damage finite element analyses

In the framework of classical polycrystalline models, drastic reductions of the numbers of slip systems and of “grains” are proposed. With a number of “grains” representing the texture of the material smaller than 10, good results are obtained either for initially isotropic fcc steel or anisotropic hcp zirconium alloy, with some predictive capacity despite the partial loss of physical relevance. Finite element analyses CPU times are not significantly increased as compared to advanced macroscopic models. Novel extensions of the polycrystalline model are developed for intergranular creep or void growth damage. This methodology increases the field of application of the polycrystalline approach in plastic anisotropy, cyclic plasticity, plastic instability and fracture, and in corresponding industrial problems.     

A comparative study of hardening theories in torsion using the Taylor polycrystal model

A study of five rate-independent hardening rules (from Taylor and Elam (“The Distortion of an Aluminium Crystal during a Tensile Test”, Proc. R. Soc. Lond. (1923), A102, 643) to Bassani and Wu (“Latent Hardening in Single Crystals II. Analytical Characterization and Predictions,” Proc. R. Soc. Lond. (1991), A435, 21)) is presented based upon the classic Taylor polycrystal model in finite strain torsion. Comparisons of aggregate shear stress-strain curves, evolving crystallographic textures, yield loci, and axial stresses among the theories are made; and results are assessed against experiments on polycrystalline copper from the literature.     

Évaluation de l'influence des changements de géométrie sur les déformations de séchage d'un milieu poreux fissuréInfluence of geometry changes on drying-induced strains in a cracked solid

The macroscopic response of a cracked solid subjected to drying is investigated within the framework of micromechanics. The originality of this contribution lies in the fact that the variations of the aspect ratios of cracks induced by the capillary pressure increase are accounted for. When the initial aspect ratio is small enough, it is shown that neglecting the geometrical changes yields an erroneous prediction of the sign of the macroscopic volume strain rate. To cite this article: X. Chateau et al., C. R. Mecanique 331 (2003).     

Dynamics of void collapse in shocked energetic materials: physics of void–void interactions

This work presents the response of a porous energetic material subjected to severe transient loading conditions. The porosities, represented by voids, entirely change the response of an otherwise homogeneous material. The variations in terms of energy distribution and maximum temperature reached in the material in the presence of heterogeneities (voids) but in the absence of chemical reactions are studied. This study also accounts for void–void interactions to enhance the understanding of the localization of energy in the material. It is observed that relative position of voids can have important consequence on energy distribution as well as rise in temperature of the energetic material. The relative position of voids further influences the interaction of secondary shock waves generated during the collapse of one void with the downstream voids. This interaction can either enhance or diminish the strength of the shock depending on the location of downstream voids. This work also reveals that the findings from mutual void–void interactions can be used to study systems with multiple voids. This is shown by analyzing systems with 10–25 % void volume fraction. The effect of void–void interactions are connected to the overall response of a chemically inert porous material to imposed transient loads.     

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.