Dynamic Void Growth Research Articles

A micromechanics-based framework to predict transitions between dimple and cup-cone fracture modes in shocked metallic glasses

In an earlier work (Tang et al., 2020), we derived evolution equations governing dynamic void growth in amorphous materials with a number of idealizing assumptions. Here, we extend and further generalize the constitutive theory to better account for general stress states, strain-softening, stable and unstable void growth modes, as well as viscous and micro-inertial retarding effects on void growth rates. The enhanced theory is implemented into a commercial finite element software package via a user-defined material subroutine to understand transitions in fracture morphologies. In particular, metallic glasses exhibit a dimple type fracture mode at low impact velocities, which transitions to a cup-cone type fracture mode at higher impact velocities. Our theory reveals that two competing processes drive this transition: (i) strain-softening behavior that is inherent to many amorphous materials and (ii) a so-called stress plateau effect that arises due to bursts of stable and unstable void growth. Our simulation results also suggest that strain-softening during the dynamic compression phase, which precedes subsequent tensile failure, is essential to the formation of cup-cone fracture morphology.

A cavitation and dynamic void growth model for a general class of strain-softening amorphous materials

In this work, we derive a homogenized framework for cavitation and dynamic void growth in a general class of strain-softening materials with particular emphasis on amorphous materials. Analytic solutions are derived by considering an assemblage of spherical shells subject to dynamic hydrostatic tensile pressure. The isochoric response of the matrix material is captured by a free volume theory based amorphous viscoplasticity model. The framework accounts for the coupled rate-dependent, strain-softening behavior typically exhibited in amorphous materials. Validation of the homogenized theory is carried out against numerous direct numerical simulations with excellent agreement. We find that a reduction in cavitation strength and significant acceleration of dynamic void growth rates is induced as a consequence of strain-softening in the amorphous matrix material. Lastly, the model is utilized to interpret some non-intuitive experimental observations of amorphous materials subject to shock compression and subsequent spall failure.

A viscoplastic model for void growth under dynamic loading conditions

We present a dynamic theoretical viscoplastic model of voids subjected to external dynamic tensile loadings. The material viscosity and temperature are factors that are considered in the dynamic evolution of voids in viscoplastic materials. We focus on the thermal effect; the temperature affects the thermal softening of the material strength and also the material viscosity. Viscous flow is the dominant growth mechanism under high stresses and rates based on the results of the calculations for dynamic void growth predictions. The factors are independently studied, and the correlations are systematically analyzed.

On the micromechanics of void dynamics at extreme rates

In an earlier work, we derived evolution equations governing the dynamic growth of voids considering the combined retarding effects of micro-inertia and dislocation kinetics. Here we derive closed-form approximate solutions to these governing equations that accurately capture the transient evolution of void size history. In addition to being invaluable to the development of computationally-efficient dynamic ductile failure frameworks, these closed-form models help reveal deeper insights into the underlying physics governing complex dynamic void growth. In particular, we show that the transient effects of inertia are important for families of voids that nucleate from coarse, micron-sized second phase particles. For nano-sized voids the finite kinetics of dislocation substructure evolution towards a saturated state play a vital role. In spite of the significant complexity associated with capturing an evolving substructure, an elegant closed-form approximation presents itself. These closed-form models are further bolstered by good agreement with molecular dynamics calculations. Lastly, we derive closed-form models of void size distribution evolution to confirm an experimentally observed transition in fracture surface characteristics above a critical loading rate.

Cavitation instability in bulk metallic glasses

Recent experiments have shown that fracture surfaces of bulk metallic glasses (BMGs) usually exhibit an intriguing nanoscale corrugation like fractographic feature mediated by nanoscale void formation. We attribute the onset of this nanoscale corrugation to TTZs (tension transformation zones) mediated cavitation. In our recent study, the spall experiments of Zr-based BMG using a single-stage light gas gun were performed. To uncover the mechanisms of the spallation damage nucleation and evolution, the samples were designed to be subjected to dynamic tensile loadings of identical amplitude but with different durations by making use of the multi-stress pulse and the double-flyer techniques. It is clearly revealed that the macroscopic spall fracture in BMGs originates from the nucleation, growth and coalescence of micro-voids. Then, a microvoid nucleation model of BMGs based on free volume theory is proposed, which indicates that the nucleation of microvoids at the early stage of spallation in BMGs is resulted from diffusion and coalescence of free volume. Furthermore, a theoretical model of void growth in BMGs undergoing remote dynamic hydrostatic tension is developed. The critical condition of cavitation instability is obtained. It is found that dynamic void growth in BMGs can be well controlled by a dimensionless inertial number characterizing the competition between intrinsic and extrinsic time scales. To unveil the atomic-level mechanism of cavitation, a systematic molecular dynamics (MD) simulation of spallation behaviour of a binary metallic glass with different impact velocities was performed. It is found that micro-void nucleation is determined TTZs while the growth is controlled by shear transformation zones (STZs) at atomic scale.

A dynamic void growth model governed by dislocation kinetics

Here we examine the role of dislocation kinetics and substructure evolution on the dynamic growth of voids under very high strain rates, and develop a methodology for accounting for these effects in a computationally efficient manner. In particular, we account for the combined effects of relativistic dislocation drag and an evolving mobile dislocation density on the dynamics of void growth. We compare these effects to the constraints imposed by micro-inertia and discuss the conditions under which each mechanism governs the rate of void growth. The consequences of these constraints may be seen in a number of experimental observations associated with dynamic tensile failure, including the extreme rate-sensitivity of spall strength observed in laser shock experiments, an apparent anomalous temperate dependence of spall strength, and some particular features of void size distributions on spall surfaces.

Cavitation instabilities in bulk metallic glasses

To reveal the mechanism of a void-dominated fracture process in bulk metallic glasses, a theoretical description of void growth undergoing remote hydrostatic tension is presented. Special attention is focused on cavitation instabilities and dynamics of a dynamic void growth process. The critical stress for cavitation instabilities is derived theoretically, which is validated by numerical simulations with a finite difference method. To characterize the dynamic void growth process, a dimensionless number is proposed, which embodies the competition of inertial effects, loading rate effects and viscous effects. It is found that inertial effects can induce vibration of the void growth rate at the rise stage of loading history and impede the growth at the steady stage. In addition, to study the void growth at the early stage of a void-dominated fracture process, quasistatic cases without inertial effects are examined. It is shown that the void growth rate strongly depends on the evolution of free volume concentration.

A Visualized Investigation of Void Growth and Coalescence in Pure Copper Sheets under Impact Tension

The multi-tension loading in the optimized tensile split Hopkinson bar tests for pure copper sheets was used to investigating growth and coalescence of drilled voids in pure copper sheets, recorded by a high-speed camera. The results of scanning electron microscopical investigation of the microvoid evolution in recovered pure copper sheets showed void coalescence mechanisms which are similar to that of visualized drilled voids. The semi-empirical relation [8] for void shape evolution under quasi-static tension was compared with our computed results revealing the dynamic and clustering effects on void growth. The possibility of application of Thomason model[9] and Considere’s condition[10] for void coalescence to thermoviscoplatic constitutive model was explored under impact tension. The main effects affecting dynamic growth and coalescence of voids were presented including the geometry (void size, shape, orientation, spacing), the material properties (dynamic constitutive model) and the stress state (impact tension condition).

Quasistatic and dynamic growth of sub-microscale spherical voids

This study examines the quasistatic and dynamic growth of sub-microscale spherical voids. A general dynamic model of void growth is first developed by considering conservation of local energy, including external work, surface energy, kinetic energy, elastic strain energy, and plastic dissipation associated with growing void driven by hydrostatic tensile stress. It properly accounts for material compressibility and limited size of elastic deformation in dynamic growth of spherical voids. A closed form of pressure-void size relation is obtained for quasistatic growth of spherical voids embedded in non-strain hardening materials. Critical condition is identified for unbounded growth, which depends strongly on initial void size. For subcritical loading, a void can only grow to a very limited size. An interesting crossover phenomenon is identified pertaining to the influences of yield stress on void growth rate.

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 coupled effects of plastic strain gradient and thermal softening on the dynamic growth of voids

This paper examines the combined effects of temperature, strain gradient and inertia on the growth of voids in ductile fracture. A dislocation-based gradient plasticity theory [J. Mech. Phys. Solids 47 (1999) 1239, J. Mech. Phys. Solids 48 (2000) 99] is applied, and temperature effects are incorporated. Since a strong size-dependence is introduced into the dynamic growth of voids through gradient plasticity, a cut-off size is then set by the stress level of the applied loading. Only those voids that are initially larger than the cut-off size can grow rapidly. At the early stages of void growth, the effects of strain gradients greatly increase the stress level. Therefore, thermal softening has a strong effect in lowering the threshold stress for the unstable growth of voids. Once the voids start rapid growth, however, the influence of strain gradients will decrease, and the rate of dynamic void growth predicted by strain gradient plasticity approaches that predicted by classical plasticity theories.

The effects of thermal softening and heat conduction on the dynamic growth of voids

This paper seeks to examine the dynamic growth of a single void in an elastic–plastic medium through analytical and numerical approaches. Particular attention is paid to the instability of void growth, and to the effects of inertia, thermal softening and heat conduction. A critical stress is known to exist for the unstable growth of voids. The dependence of this critical stress on material properties is examined, and this critical stress is demonstrated to correspond to the lower limit for the ductile spall strength in many materials. The effects of heat conduction on the dynamic growth of voids strongly depend on the time and length scales in the early stages of the dynamic void growth.

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.

Void-containing nonlinear materials subject to high-rate loading

A new concept of dynamic potential (macrostress potential or macrostrain-rate potential), which has two components, i.e., deformation and kinetic potentials for heterogeneous materials under intense dynamic loading, is proposed using a micromechanical approach. In particular, for voided nonlinear materials, the approximate expression of the macrostress potential is derived through an upper bound approach. Simplified physical models for ductile porous materials (aggregates of voids and ductile matrix) are employed with the matrix material taken as power-law thermal viscoplastic and incompressible. Approximate velocity fields for the matrix are adopted to derive the dynamic loading criterion and an approximate functional form of the dynamic loading criterion is developed. The inertial, rate sensitivity, and thermal effects are directly included in the expressions of the macrostress potential and the dynamic loading function. Various features of the loading function are numerically investigated in detail. Numerical analysis reveals that both the dynamic growth of voids and the loading surface have strong rate dependence and temperature dependence. Experimental data, theoretical analysis, and numerical modeling indicate that the inertial and thermal effects play a very important role in the dynamic behavior of the voided nonlinear materials.

Dynamic damage and failure in visco-plastic materials

In this paper, a dynamic damage model in ductile solids under the application of a dynamic mean tensile stress is developed. The proposed model considers void nucleation and growth as parts of the damage process under intense dynamic loading (strain rates ϵ ⩾ 10 3 s −1). The evolution equation of the ductile void has the closed form, in which work-hardening behavior, rate-dependent contribution and inertial effects are taken into account. Meanwhile, a plate impact test is performed for simulating the dynamic fracture process in LY12 aluminum alloy. The damage model is incorporated in a hydrodynamic computer code, to simulate the first few stress reverberations in the target as it spalls and postimpact porosity in the specimen. Fair agreement between computed and experimental results is obtained. Numerical analysis shows that the influence of inertial resistance on the initial void growth in the case of high loading rate can not be neglected. It is also indicated that the dynamic growth of voids is highly sensitive to the strain rates.

Inertial Effects on Void Growth in Porous Viscoplastic Materials

The present work examines the inertial effects on void growth in viscoplastic materials which have been largely neglected in analyses of dynamic crack growth and spallation phenomena using existing continuum porous material models. The dynamic void growth in porous materials is investigated by analyzing the finite deformation of an elastic/viscoplastic spherical shell under intense hydrostatic tensile loading. Under typical dynamic loading conditions, inertia is found to have a strong stabilizing effect on void growth process and consequently to delay coalescence even when the high rate-sensitivity of materials at very high strain rates is taken into account. Effects of strain hardening and thermal softening are found to be relatively small. Approximate relations are suggested to incorporate inertial effects and rate sensitivity of matrix materials into the porous viscoplastic material constitutive models for dynamic ductile fracture analyses for certain loading conditions.

Void growth and compaction relations for ductile porous materials under intense dynamic general loading conditions

Void growth and compaction relations for ductile porous materials under intense dynamic general loading conditions are developed. The mathematical models include the influence of inertial effects, material rate sensitivity, as well as the contribution of surface energy of a void and material work-hardening. Numerical analysis shows that inertia appears to resist the growth of voids. The inertial effects increase quickly as the strain rates become higher. The theoretical analysis suggests that the inertial effects cannot be neglected at high strain rates. It is also indicated that the dynamic growth of voids is highly sensitive to the strain rates. The threshold applied external stresses for the void growth and compaction, which are the functions of the distention, are derived from void growth and compaction relations.

Growth of voids in porous ductile materials at high strain rate

A hollow-sphere model, with temperature-dependent viscoplastic material response, is developed to investigate the inertial and thermal effects on dynamic growth of voids in ductile materials. Theoretical analysis indicates that the inertial effect (kinetic energy of void growth) mainly dominates the behavior of the void growth in temperature-dependent and high-strain-rate cases. Otherwise, the viscoplastic effect dominates and the inertial effect can be neglected. The rate of the dynamic growth of voids increases when the thermal effect is considered. An expression of the threshold stress for the void growth is obtained, which depends on the initial porosity, the porosity, the yield strength, the density of surface energy of voids, the initial temperature, and the melting temperature.

Influence of inertial and thermal effects on the dynamic growth of voids in porous ductile materials

The influence of inertial, thermal and rate - sensitive effects on the void growth at high strain rate in a thermal - viscoplastic solid is investigated by means of a theoretical model presented in the present paper. Numerical analysis of the model suggests that inertial, thermal and rate - sensitive effects are three major factors which greatly influence the behavior of void growth in the high strain rate case. Comparison of the mathematical model proposed in the present work and Johnson's model shows that if the temperature - dependence is considered, material viscosity eta can take the experimentally measured values.

The influence of inertial effects on dynamic growth of voids in ductile solids under intense dynamic loading

Summary In this paper, a dynamic damage model in ductile solids under the application of a dynamic mean tensile stress is developed. The proposed model considers void nucleation and growth as part of the damage process under the condition of intense dynamic loading (strain rates ge > 10 3 s −1 ). In the evolution of porosity, work-hardening behavior, rate-dependent contribution and inertial effects are taken into account. Meanwhile, a plate impact test is performed for simulating a dynamic fracture process in pure copper. The damage model is incorporated in a hydrodynamic computer code, to simulate the first few stress reverberations in the target as it spalls and post-impact porosity in the specimen. Fair agreement between computed and experimental results is obtained. Numerical analysis shows that the influence of inertial resistance on initial void growth in the case of high loading rate cannot be neglected.