Void Interaction Research Articles

On Void Growth in Elastic-Nonlinear Viscous Solids Under Creep and Cyclic Creep Conditions

The growth of initially spherical periodic grain boundary voids in an elastic-nonlinear viscous material is investigated. Large geometry changes are taken into account to correctly capture the trend of void growth rate and evolution of void shape. To this end, an axisymmetric unit cell model of the damaged material is considered. The temporal characteristics of the void growth rate and shape changes and void interaction effects were determined for several triaxial stress states, under constant stress conditions. The results are compared with existing approximate analytical models for void growth rate in a purely nonlinear viscous solid. The finite element calculations indicate that while the effect of elasticity is minimal in determining the transient time for void growth, the growth rates after the initial transient may be more rapid in elastic-nonlinear viscous materials compared with the rates for purely nonlinear viscous materials, especially so under high triaxial stress states. In addition, the approximate analytical expression for void growth rates significantly under predicts the rates for high triaxial stress states and predicts a wrong trend for low triaxial stress states. Calculations of void growth under balanced strain cycling conditions are also performed. The results indicate that void growth can occur, consistent with experimental observations under balanced cycling loading conditions, provided nonlinear shape changes are taken into account. In addition, the analyses show that the cavity growth rate is constant under balanced cyclic loading conditions. This observation is in agreement with the experimental findings of Barker and Weertman (1990, Scr. Metal. Mater. 24, pp. 227–232). That material elasticity does not play any role in void growth or shrinkage under balanced cyclic creep conditions is also clearly demonstrated by the results. [S0094-4289(00)00803-3]

Experimental and numerical comparison of void growth models and void coalescence criteria for the prediction of ductile fracture in copper bars

Ductile fracture of round copper bars is investigated within the scope of the local approach methodology. Two damage models are analysed: the Rice and Tracey model and the Gurson–Leblond–Perrin model. A comparative study of four coalescence criteria is conducted: a critical value of the damage parameter, the Brown and Embury criterion, the Thomason criterion and a criterion based on the reaching of the maximum von Mises equivalent stress in a Gurson type simulation. Ellipsoidal void growth and void interaction are accounted for. As far as possible, all the parameters of the models are identified from experiments and physical observations. The effect of stress triaxiality is studied using specimens presenting a wide range of notch radii. The effect of strain-hardening is analysed by comparing the behaviour of the material in the cold drawn state and in the annealed state.

Modelling fracture process in a ductile solid under intense impulsive loading using a model of the void growth with temperature-dependency

A model of void growth under intense impulsive loading, in which the thermal effect due to high rate of deformation, the inertial effect and rate-sensitive effect are taken into account, is incorporated in a Lagrangian finite-difference dynamic code to model the fracture process in a ductile solid, copper. Plate impact tests are performed for modelling the dynamic damage and fracture process of copper. Microscopic observations of mechanisms of growth and coalescence of voids in a softly recovered specimen are carried out. Computations of fracture process include the nucleation, growth and coalescence of voids. Numerical results show that in the damage cells, voids nucleate unceasingly as long as the tensile stress is greater than the threshold tensile stress σ n in the matrix material as larger voids are growing. Interaction of voids is considered by void nucleation and thermal effect indirectly. One of the major results of temperature-dependency in the model of the void dynamic growth is that the experimentally measured value of the viscosity coefficient η can be used to simulate the behaviour of damage and fracture in ductile solids. A few parameters in the model need to be determined, hence making the model of dynamic damage used in the present work easy to be applied to practical problems. Some computational details are discussed. A comparison of predicted location, and shapes of one and two dimensional spall surfaces in the targets with the experimental data seems reasonably good.

An Experimental Study Evaluating Macroscopic Cavity Interactions as a Model of Microvoid Ductility

An experimental study of the interactions between macroscopic cavities was conducted. The goal was to determine simple cavity arrangements producing deformations similar to commonly observed features of ductile fracture. The production of cavity-centered fracture surface features and ligament thinning, representative of the type observed around and between microvoids on ductile fracture surfaces, were the specific objectives of this macroscopic study. The specimens employed were cylindrical tension specimens containing from one to six coplanar radial holes, with spherical end profiles, located at the center of the specimen and either zero or one cylindrical holelocated on the longitudinal axis of the specimen. The results of this parametric study identified the two specimens that best produced the desired features. The load-displacement results of these experiments also illustrate the effect of cavity geometry on the net strength and ductility of the array system. Specimens of these designs provide a future basis for simple and economical model studies of void interactions.

An experimental model of the growth of neighboring voids during ductile fracture

We have examined aspects of the strain-induced growth behavior of pairs of neighboring voids using a macroscopic model consisting of a uniaxial tensile specimen containing pairs of opposing blind-end holes. Experimental measurements have been made of the thinning behavior of the inter-hole ligament for hole-ends spaced initially 0.5 D, 1.0 D, and 2.0 D apart, where D is the hole diameter. These results indicate that deformation causes the interhole ligament to thin at a rate which increases with increasing strain and decreasing hole spacing. When compared to predictions of the growth of isolated voids (based on the Rice-Tracey analysis [1,5]), our results indicate void interaction effects accelerate ligament thinning and become pronounced with decreasing void spacing (≤ 1.0 D) and increasing applied strain (ε 1≥ 0.2) even in uniaxial tension.

On the electronic structures and spectra of NiCl2 and CuCl2

The geometry and ground state of NiCl2 have been studied using local density-functional (LDF) calculations within the linear combination of Gaussian-type orbitals (LCGTO) framework. The molecule is found to be linear with a 3Σg– ground state in agreement with recent experimental results but contrary to previous theoretical studies. The calculated bond length and vibrational frequencies are in excellent agreement with experiment. A cellular ligand-field (CLF) analysis of the spectrum has been performed. The LDF and CLF calculations provide a consistent description of the bonding and suggest that the chloride ions are acting as good σ and π donors in this molecule with both roles enhanced compared to systems with higher co-ordination numbers. A proper account of the important interaction of the dσ orbital with the antibonding σg+ orbital is given using a cylindrically symmetric ‘void’ cell. A recent CLF analysis of CuCl2 is shown to be incorrect because of the treatment of the void interaction. A new study of the ligand-field spectrum of CuCl2 is presented, the results of which are consistent with the LDF and CLF studies on NiCl2 and with recent molecular orbital and LDF calculations on CuCl2.

Evolution of Void Shape and Size in Creeping Solids

A boundary value problem for a periodic array of initially spherical voids in a power law creeping solid is analyzed. An axisymmetric cell model relevant for simulating grain boundary void growth is used. The rate boundary value problem is solved by means of a finite element method. Void growth histories accounting for void shape changes and, within the cell model context, void interaction effects are computed for various remote stress triaxiality states. An automatic remeshing algorithm permits computations for large changes in void size and shape. The computed void growth rates are compared with predictions from available analytical formulas that neglect shape change effects.

On the strain-induced growth of neighboring voids

The purpose of this communication is to report preliminary experimental results which describe the strain-induced growth of a pair of spherical cavities, spaced about one cavity diameter apart, and embedded in material (titanium) subjected to far-field uniaxial tension. Ultrasonic imaging is used to monitor cavity dimensions during the incremental straining of the specimens, and data are presented for both longitudinal and transverse cavity growth for extensional strains up to [approximately] 0.5. As will be evident, the data indicate a significant degree of void interaction even in this uniaxial tension case.

Void evolution and the large-scale structure

We investigate the evolution and interaction of voids in an Einstein-de Sitter universe. Using N-body simulations, we study the growth of different configurations of negative top-hat density perturbations. We find that adjacent voids merge to form larger voids. Thin planar walls form between the merging voids. The peculiar velocity is tangential to a given wall and rises linearly from the wall center. The density contrast of the walls steadily declines as the walls are stretched by the greater rate of expansion within the new void resulting from the merger. A simple hierarchical configuration of negative perturbations demonstrates that smaller scale voids effectively disappear within larger voids as larger scales become nonlinear. The remnants of previous structure formation are frozen in within evolving voids

Viscoplastic analysis of plate-impact spallation

A model to predict material damage and spall fracture under high strain-rate conditions is applied to plate impact experiments. The model uses the Perzyna viscoplastic constitutive theory, appropriately modified to include a nonlinear isotropic hardening law that allows for saturation of the hardening with increase of strain. The constitutive equation contains a scalar variable for description of the material damage, expressed as the void volume fraction of the polycrystalline solid with microvoids. Incorporation of the damage parameter into the completely phenomenological elasto-viscoplastic constitutive equations permits description of rate-dependent, compressible, inelastic deformation, and ductile fracture. The evolution equation for the parameter describes microvoid nucleation and growth. The model for microvoid growth is based upon a random distribution of microvoids, idealized as spherical holes of arbitrary size. Microvoid coalescence is considered by incorporation of void interaction functions to describe enhanced nucleation and growth rates. A local spall fracture criterion based upon the attainment of a critical microvoid volume is utilized. The constitutive equations are specialized to uniaxial deformation with multiaxial stress, which is appropriate for the planar impact experiments. A finite-difference wave propagation computer code is used to solve the equation of motion. The computed stress wave profiles demonstrate the effect of using a viscoplastic material description. Calculations predicting the rear-surface velocity-time profile and the stress-time profiles of the OFHC copper target are compared with measured profiles. The damage (void volume) distribution across the plate thickness is also calculated and compared with experimental data.

Finite element analysis of void growth in elastic-plastic materials

Three-dimensional finite element computations have been carried out for the growth of initially spherical voids in periodic cubic arrays and for initially spherical voids ahead of a blunting mode I plane strain crack tip. The numerical method is based on finite strain theory and the computations are three-dimensional. The void cubic arrays are subjected to macroscopically uniform fields of uniaxial tension, pure shear and high triaxial stress. The macroscopic stress-strain behavior and the change in void volume were obtained for two initial void volume fractions. The calculations show that void shape, void interaction and loss of load carrying capacity depend strongly on the triaxiality of the stress field. The results of the finite element computation were compared with several dilatant plasticity continuum models for porous materials. None of the models agrees completely with the finite element calculations. Agreement of the finite element results with any particular constitutive model depended on the level of macroscopic strain and the triaxiality of the remote uniform stress field. For the problem of the initial spherical voids directly ahead of a blunting mode I plane strain crack tip, conditions of small scale yielding were assumed. The near tip stress and deformation fields were obtained for different void-size-to-spacing ratios for perfectly plastic materials. The calculations show that the holes spread towards the crack tip and towards each other at a faster rate than they elongate in the tensile direction. The computed void growth rates are compared with previous models for void growth.

A damage model for ductile metals

A physically-based theory of damage for ductile metals is outlined. It rests upon a direct extension of the authors recently proposed viscoplastic model for finite deformations to include the effects of dislocation—void interactions as they manifest themselves in void nucleation, growth, and coalescence. Emphasis is put on illustrating the general structure of the present framework within which coupling effects of texture development, void formation, and adiabatic heating can be considered and their role to the localization of deformation and failure can be evaluated. No special attention is placed on justifying the various growth laws and simplifying assumptions pertaining to the detailed structure of the model, for example the manner that spatial gradients of the damage variable enter into the theory. Such simplifications, however, facilitate the solution of the relevant equations for a case of homogeneous triaxial state of stress permitting a qualitative comparison with experimental data obtained for Bridgeman-notch specimens.

Void Growth in Elastic-Plastic Materials

Three-dimensional finite element computations have been done to study the growth of initially spherical voids in periodic cubic arrays. The numerical method is based on finite strain theory and the computations account for the interaction between neighboring voids. The void arrays are subjected to macroscopically uniform fields of uniaxial tension, pure shear, and high triaxial stress. The macroscopic stress-strain behavior and the change in void volume were obtained for two initial void volume fractions. The calculations show that void shape, void interaction, and loss of load carrying capacity depend strongly on the triaxiality of the stress field. The results of the finite element computation were compared with several dilatant plasticity continuum models for porous materials. None of the models agrees completely with the finite element calculations. Agreement of the finite element results with any particular constitutive model depended on the level of macroscopic strain and the triaxiality of the remote uniform stress field.

A mechanism of formation and properties of the void lattice in metals under irradiation

We propose a new mechanism of interaction between voids, which is based on the elastically induced absorption of self-interstitial dislocation loops (SIA-loops) from the matrix. The main peculiarity of this dislocation interaction is its rigid relation to the crystal structure, namely, it arises only in the crystals, where SIA-loops can glide, and exists only between voids lying along the same loop-gliding directions. Void ordering is shown to originate from competition between the dislocation interaction and radiation-induced diffusion coarsening of voids, which leads to shrinkage of unfavourably positioned voids, if the void density exceeds the critical value. Voids in the superlattice thus formed have immediate neighbours along the loop-gliding directions, copying the host lattice due to the coincidence of the loop-gliding directions with close-packed directions of the matrix. The proposed model explains why the superlattice forms more readily in bcc than fee metals and gives analytical expressions for the void lattice parameter and stationary size, being in good agreement with experimental data over the whole temperature range of superlattice formation.

The dislocation void interaction

In terms of the isotropic elasticity theory the elastic displacement fields in the straight edge dislocation void system are determined. An exact solution for the dislocation void interaction energy is obtained. The interaction energy and the drift current lines of point defects are calculated. An application of the obtained results in a radiation swelling study is discussed.

A constitutive description of nonlinear materials containing voids

The material containing voids is characterized by two state variables, the void volume fraction ϱ and the equivalent matrix stress σ . Void interaction becomes significant in nonlinear materials at very low void volume fractions. This interaction is modeled by a cell consisting of a single void within a thick shell of nonlinear matrix material, the dimensions of which are chosen to give a desired void fraction. The cell, loaded at the outer surface, is analyzed numerically to obtain an approximate expression for the equivalent matrix stress, from which a flow rule is derived.

On effective moduli of an elastic body containing periodically distributed voids

For an elastic body containing periodically distributed voids, several effective techniques are presented which can be used to obtain the effective elastic moduli with any desired degree of accuracy. The results include the effects of void geometry as well as void interactions. For a body containing spherical voids, numerical results are presented and compared with those obtained by other methods.

Effect of additional sinks of point defects on the diffusion interaction of voids. Stability of the void lattice under irradiation

Abstract A theory of the diffusion interaction of vacancy voids in a many-particle system, i.e. that consisting of N-voids is constructed. The void interaction results from the diffusion exchange of point defects, which are present in the concentration cloud around every void. For a metal containing impurities, dislocations etc. an expression for the growth rate of an individual void of an arbitrary size is obtained within the framework of this model, taking into account the effect of the other voids in it. It is shown that the diffusion interaction in the void system has a certain screening radius, which depends on the total number of voids and additional sinks for point defects in a metal. Using the diffusion interaction of the vacancy voids we have studied the kinetic stability of the lattice void distribution both under continuous irradiation and ordinary temperature annealing. The criterion for the existence of a stable lattice of vacancy voids is obtained in a real crystal containing dislocations an...

Diffusion interaction in the void system and stability of the void lattice under annealing

Abstract A void interaction mechanism based on the diffusion exchange of point defects is studied for a system of N-voids. In the absence of additional sinks for point defects, an expression is derived for the growth rate of a void, taking into account the influence of the other voids on its growth. It is shown that the average size of the interacting voids is constant while the dispersion of void size distribution increases with the characteristic time close to the growth rate of noninteracting voids at the coalescence state. This quantity determines the incubation time of the lattice under annealing.

Voids in ion-implanted silicon

Abstract The vacancy defect subsystem is studied by means of electron microscopy in ion-implanted silicon. It has been discovered that, unlike the interstitial atoms which form rod-like defects and two-dimensional defects (dislocation loops, stacking faults), vacancies condense during annealing into three-dimensional associations—voids. The formation temperature of a void consisting of ∼103 vacancies in 400–500°C. Voids have the tendency to associate into more complicated complexes; as a result of void interaction, disk-like clusters are formed in P+-implanted silicon. At temperatures 700–750°C, disk interaction is likely to take place which is accompanied by extended defect formation—chains. At annealing temperatures 800–850°C voids do not collapse in vacancy dislocation loops and chains do not rearrange in dislocations; all the defects dissolve in matrix. It has been shown that the postannealing defect distribution is heterogeneous. The experimental data are discussed in detail.