Void Growth Model Research Articles

Numerical modelling of ductile spall fracture

In this paper, a macroscopic constitutive equation for porous materials, which is based on a void growth model under the combined action of hydrostatic and deviatoric stresses, is presented. The formulation follows previous work by Gurson [Continuum theory of ductile rupture by void nucleation and growth: Part I—Yield criteria and flow rules for porous ductile media. J. Engng Mater. Tech. 99, 2–15 (1977)] for perfectly plastic materials, which is here expanded to include a non-linear viscous term. Dynamic effects associated to void growth under a general stress state have also been incorporated in the model. In this manner, at each time increment of the numerical analysis, the rate of porosity increase may be computed and the local porosity values evaluated. The model is applied to spall experiments in aluminium. Spall fracture in aluminium was experimentally studied by loading aluminium plates by use of an explosive. The damage mechanism leading to fracture of the tested specimens was the nucleation and growth of microvoids. From the numerical analysis of the above experiments, the free surface velocity histories were compared with the corresponding experimental records, showing that in the cases studied the inclusion of a non-linear viscous term is essential to obtain good agreement between analysis and experiment.

Microstructure and fracture behaviour of particle‐reinforced metal–matrix composites

SummaryThe fracture behaviour of particle‐reinforced metal‐matrix composites is shown to be controlled by a period of damage nucleation and evolution prior to final failure. The nucleation of damage can be by reinforcement fracture or decohesion and the mode of damage is shown to be controlled by the size of the reinforcement and segregation of alloying elements from the matrix. The nucleation and growth of damage can be monitored by a number of techniques. Acoustic emission and tomography are used here and the results are found to be consistent with simple models of void growth.

Damage evolution in ductile materials: from micro- to macro-damage

This research presents a new simulation concept of damage evolution for metallic materials under large displacements and deformations. The complete damage range is subdivided into both the micro-damage and the macro-damage range. The micro-damage phase is described by the Cocks/Ashby void-growth model for isotropic, ductile materials under isothermal conditions. After having reached a critical void-volume fraction, a macro-crack is introduced into the model. With such a concept the damage evolution from nucleation and growth of first micro-voids to initiation of macro-cracks and complete failure of the material can be simulated. Applying the Finite Element Method for the numerical formulation, at every incremental macro-crack step the Finite Element mesh is adapted such that the crack path remains independent of the initial mesh.

Modeling of void growth on grain boundaries in materials under high-temperature loading

Within the framework of nonlinear nonequilibrium thermodynamics, we construct a model of void formation at elevated temperatures. On the basis of a linear analysis of the stability of solutions of a formulated boundary-value problem, we estimate a conservative value of the void-formation threshold and conditions for the development of dissipative damage structures. It is shown that, for each level of loading above the threshold, void formation can be characterized by a corresponding average density, in agreement with experimental data. The distribution of concentrations of vacancies and microvoids (and, hence, the sites of macrovoids) depends on the initial conditions only implicitly and is governed by the properties of the dissipative system; this is the principal difference between the model suggested in the present work and traditional theories of creep damage.

Subcooled void growth mechanisms and prediction at low pressure and low velocity

Measurements were made of void growth and wall temperatures for two circular annular geometries and forced-convective subcooled nucleate boiling conditions, with flow rates between 0.02 and 0.20 kg/s, pressures of 1.05, 2 and 3 bar, and inlet temperatures between 30 and 90°C. Void growth results at low pressure show that the highly subcooled void region is important, and void fraction at the onset of significant void may be as high as 10%. Observations of bubble ebullition and detachment mechanisms obtained from a high speed photographic study were compared to assumptions generally used in void growth modeling. The photographic results show that bubbles do not travel far downstream after nucleating, that there is no region of attached void and that bubbles slide along the wall before being ejected into the flow after the onset of nucleate boiling. Unlike findings for high pressure systems reported in the literature, the onset of significant void was found to be independent of (i) bubble detachment, (ii) the location where the bubble is first ejected from a bubble layer and (iii) the transition from partial nucleate boiling to fully developed boiling. A phenomenological void growth model is presented which accounts for the vapor volume inside a heated channel at atmospheric pressure, and includes the bubble ebullition cycle, formulated on the basis of information obtained from the high speed photographic study.

Ductile fracture by the growth and coalescence of microvoids of non-uniform size and spacing

A two-dimensional plane strain model of void growth and coalescence in a rigid/plastic solid, containing void sizes and spacings which can be highly non-uniform, is developed to investigate the effects of non-uniform distributions of void-nucleating particles on the ductility of a metal. The theoretical void-growth strains to ductile fracture for a wide variation in void diameters and spacings show that, for a given volume fraction of voids, the minimum ductile-fracture strain occurs when the voids are of uniform size and spacing. For the same volume fraction of voids, greatly increased ductility is likely to be achieved when the void sizes and spacings are highly non-uniform and the sub-cell volume fractions are also non-uniform.

A Combined Kinematic-Isotropic Hardening Theory for Porous Inelasticity of Ductile Metals

A general treatment of multiaxial inelastic deformation for arbitrarily large strain necessitates consideration of effects of material porosity and its evolution, combined kinematic-isotropic hardening, dislocation substructure stress-state effects and textural anisotropy, rate- and temperature-dependence, void anisotropy, and failure criteria, among other things. The full set of three invariants of the stress (overstress) tensor is essential for a satisfactory first-order description of these effects. A framework is set forth in this article that is sufficiently general to include arbitrary void growth laws and various state variable inelasticity theories. The framework is applied, within the context of rate-independent plasticity, to predict evolution of void growth during tensile loading of pressurized, cir cumferentially notched specimens using two different void growth models.

Modeling ductile void growth in viscoplastic materials—Part I: Void growth model

In many forming processes, the mechanical properties of the workpiece material are deteriorated by the growth of voids during plastic deformation. In this study we present a new constitutive relation for void growth based on mechanical analyses of an isolated void growing in metallic media under various conditions of stress triaxiality. The incompressible, strain hardening, viscoplastic behavior of the matrix material is characterized by an isotropic state variable model. Systematic variations of the shear strength, mean stress, porosity and deviatoric deformation rate provided families of growth rate curves that were collapsed by appropriate scaling into two master curves. The forms of these curves and the scaling factors needed to obtain two master curves motivated the mathematical structure of the constitutive relation for void growth.

A continuum model for finite void growth around spherical inclusion

Within the bounds of large deformation theory the necessary mathematical analysis for incompressible and nearly incompressible elastic materials is briefly stated and some results of structural stress and deformation analysis, simulating void formation in filled elastomers, are given. Void development around spherical inclusion in stretched elastomeric matrix under external pressure, and without it, is analysed.

Analysis of Thermal Stress Induced Void Growth During Thermal Cycling

AbstractRecent work in thermal stress induced voiding in passivated lines has focussed on the isothermal, quasi-steady state growth of existing voids. In this work, the transient growth and shrinkage of voids during the thermal cycling of narrow, passivated lines was studied. Analysis of experimental data for passivated lines demonstrates the quick buildup and evolution of a backstress at the grain boundary during heating and cooling. This backstress greatly reduces the driving force for void growth at higher temperatures and must be accounted for in stress induced void growth models for the case of appreciably varying temperatures in short time scales.

Analysis of Thermal Stress Induced Void Growth During Thermal Cycling

ABSTRACTRecent work in thermal stress induced voiding in passivated lines has focussed on the isothermal, quasi-steady state growth of existing voids. In this work, the transient growth and shrinkage of voids during the thermal cycling of narrow, passivated lines was studied. Analysis of experimental data for passivated lines demonstrates the quick buildup and evolution of a backstress at the grain boundary during heating and cooling. This backstress greatly reduces the driving force for void growth at higher temperatures and must be accounted for in stress induced void growth models for the case of appreciably varying temperatures in short time scales.

Void growth in drawing and extension of ductile metals

Two applications of a new void growth model are presented to illustrate the ability of the model to predict porosity increases in metal forming. The model was developed from analyses of void growth under conditions of macroscopically isotropic and deviatoric stress states. Special attention was given to scaling of the mean stress by a measure of the matrix material strength rather than the effective stress. The advantage of this is evident in the applications in regions where the macroscopic stress state approaches purely hydrostatic.

Influence of superimposed hydrostatic tension on void growth in the neck of a metal sheet in biaxial stress fields. Part - II - Plastic instability

The growth of a void in the neck of a metal sheet under superimposed hydrostatic tension as modelled in part-I is used in this paper to study its influence on the sheet metal forming limits using the M-K analysis incorporating the correction for triaxial stress state in the neck, and void growth equivalent of Marciniak inhomogeneity index as proposed by Rao and Chaturvedi. It has been observed that consideration of the triaxial stress state enhances the forming limits as compared to the M-K model but simultaneous incorporation of the void growth and coalescence criteria in the neck reduces them in the equibiaxial strain region. This analysis is also used to study the effect of sheet thickness on forming limits. The predictions based on this model show relatively faster increase in forming limits with higher sheet thickness as compared to the predictions based on the void growth and coalescence model assuming biaxial stress state in the neck. This is in accordance with the experimental observations of Haberfield and Boyles.

Effects of prestrain on the fracture properties of pressure vessel steel

In this paper, various methods have been used to measure the effects of prestrain on the fracture toughness of 20 g steel experimentally. Based on elastic-plastic FEM analysis, the relationships between crack tip opening displacement δT and the maximum strain emaxc have been obtained and the effects of prestrain ep on the critical fracture strain emaxc have been examined. At the same time, based on the Rice-Tracey's void growth model [4], the cause of the void formation ahead of the crack tip, the condition of the principal crack propagation and the effects of prestrain ep on the fracture properties of 20 g steel were investigated. For ductile material there was a maximum value of V G ahead of the crack tip that was the cause of the void formation in that place. It is believed that when the maximum value of V G ahead of the crack tip reaches a critical value, it may be the principal crack propagating condition for high ductile material.

Model of Elliptic Void Growth in Ductile Fracture.

A simple void growth model in ductile fracture is proposed. The model is that an elliptic void grows under the plane strain condition in an infinite material, which has a rate-dependent constitutive equation of the exponential type. Assuming that the void maintains elliptic shape during the growth process, a simple analytical relationship is derived through the use of kinematical relationships of the elements. On this basis, the void growth process is simulated by simple calculations. The applied parameters are the triaxiality of uniform remote stress, the initial shape of the void (aspect ratio of the ellipse) and the exponent of the constitutive equation. It is shown that there is a very strong dependence of void elongation on the exponent of the constitutive equation, and that the initial circular void elongates in the perpendicular direction to the loading axis when the exponent is large enough. In general the growth rate and shape are determined by the combinations of stress triaxiality, exponent of constitutive equation and initial shape of void. These relationships are quantitatively obtained by the analysis.

Dynamic void growth in rate-sensitive plastic solids

A dynamic void growth model in rate-sensitive plastic materials is derived. The constitutive relation of the matrix is in the overstress form proposed by Perzyna. When the rate of deformation sensitive parameter tends to zero, the Gurson model is retrieved. When the porous material element is under triaxial tension, the Carroll-Holt and Johnson models are retrieved. The normality condition of the plastic rate of deformation to the dynamic loading surface at constant equivalent rates of deformation (with the volumetric part also taken into account) is discussed, and it is shown that the normality rule no longer exists in general. Finally, an approximate expression of the dynamic loading surface that may be convenient for engineering applications is suggested.

Stress-driven void growth in narrow interconnection lines

We consider a model of stress-driven void growth in interconnection lines that are narrower than the metal grain size. The metal line is imbedded in an SiO2 matrix at a temperature T0, and incurs tensile stress after being brought down to an annealing temperature Tann. A void nucleates at a particular grain boundary, along which the diffusion is assumed to be so fast that the stress relaxation at that grain boundary is nearly complete in a very short time. This is assumed to occur before the void spans the width of the line. Thereafter, the void continues to grow by matter transport through bulk diffusion, cascading down sequential tiers of grain boundaries, resulting in a layer of extra matter at each grain boundary for strain relaxation. The kinetics of the void growth is given analytically. The consideration of climbing dislocations is deferred for future analysis.

On the relationship between crack tip opening displacement at the initiation of a ductile tear in low carbon steel, hydrostatic stress, and void growth

The effect of hydrostatic stress on the growth of voids from inclusions located near the tips of deep and shallow notches in three-point bend specimens is analysed using slip-line field theory and the Rice-Tracey void growth model. The analysis explains qualitatively the observed dependence of the crack tip opening displacement δi at the initiation of a ductile tear in low carbon steel on the hydrostatic stress. Although a plot of the experimental values of δi normalised by the inclusion spacing against the inclusion spacing/size ratio do not agree well with the theory, the plot is a promising means of determining experimentally the dependence of δi on hydrostatic stress for any particular steel.

Optimization of geometry and strain gauge techniques in fatigue testing of hourglass profile sheet steel specimens. (Dissertation): Stawiarski, T.T.Diss. Abstr. Int. Aug. 1990 51, (2), 276 pp

In order to study the ultra-low cycle fatigue (UCLF) behavior of austenitic stainless steel, uniaxial tensile tests and cyclic loading tests of round bar specimens and notched specimens for S30408 austenitic stainless steel were carried out respectively. In terms of the experimental results of smooth round bars, monotonic and cyclic constitutive models of austenitic stainless steel S30408 were calibrated. Furthermore, the parameters of stress modified critical strain (SMCS) model, void growth model (VGM) and cyclic void growth model (CVGM) were calibrated through comparing the notched round bar tests with the corresponding finite element analyses (FEA). Finally, the influence of element size on the fracture prediction of the micromechanical models was analyzed. Results indicated that the cyclic hardening responses of austenitic stainless steel were obvious. Meanwhile, the calibrated numerical simulation was able to well capture the key characteristics observed in the tests. In addition, as for VGM and SMCS models, the fracture prediction values increased with the element size. Last but not least, in terms of the degraded parameter and the fracture toughness parameters, it was proved that ULCF fracture resistance of austenitic stainless steel was better than common structural steels. Therefore, from this perspective, austenitic stainless steel structures have potential advantages to be applied in seismic areas.

The critical void growth ratio and crack initiation condition in the crack tip zone of 20g steel

In this paper, the elastic-plastic FEM method with a kinematic hardening model has been used to analyse a three-point-bend specimen. The relation between the maximum strain and the crack opening displacement has been investigated. Based on the Rice-Tracey void growth model, the V G value has been calculated with the numerical integration method. For ductile materials, the cause of void formation and the condition of the void coalescence with the main crack have been discussed and, the results agree well with those of Li Changchun et al. [ J. Huazhong Univ. Sci. Technol. 17, 139–144 (1989)] and our experiments in this paper.