Plasticity driven growth of nanovoids and strength of aluminum at high rate tension: Molecular dynamics simulations and continuum modeling

Highlights

• Mechanism of void growth in aluminum is studied with the help of molecular dynamics.

• It is shown, that the void growth is connected with the plastic flow around a void.

• Continuum model of the dislocation driven void growth is proposed.

• Arrhenius-type law applied for description of the dislocation nucleation rate.

• Nucleation energy and activation volume are determined from the atomistic simulations.

Abstract

In this paper the initial stage of nanovoids growth in monocrystal aluminum is studied with molecular dynamics simulations. The dependencies of critical negative pressure in simulated system on void radius, system size and temperature under high rate tension are obtained. It is shown that decrease of the ratio of void radius to system size causes an increase of system tensile strength, temperature rise leads to an almost linear drop of the tensile strength. We propose a continuum dislocation based model of nanovoids growth that is intended to describe the critical negative pressure in systems. Nucleation of dislocations near a growing void is taken into account through the equation for the probability of critical thermal fluctuation. Parameters of the model are determined in comparison with the molecular dynamics data.

Introduction

Fracture of materials under dynamical loading attracts substantial attention of researchers through last decades (Staudhammer et al., 2001, Trivedi et al., 2007, Vogler, 2009, Colvin et al., 2009, Kanel et al., 2014). Nowadays the tensile strength of heated metals (de Resseguier et al., 2012, Zaretsky and Kanel, 2012) and metals deformed at extremely high rate up to 10⁹ s⁻¹ (Ashitkov et al., 2010, Ashitkov et al., 2013) are becoming increasingly the subject of study. Theoretical interest in the behavior of matter at such high strain rates in a wide temperature range is stimulated with the ability to observe the macroscopic effects of relaxation processes on nanometer space and picosecond time scale, for example, the phonon friction and homogeneous nucleation of dislocations. From a practical point of view, the results of studies in this area can be for a construction of targets for inertial fusion and interpretation of fusion ignition experiments, also, for a development of spacecrafts with thin multi-layer shields designed to protect a craft from collisions with typical velocities of several kilometers per second.

It is well known that fracture of metals involves two mechanisms: 1) ductile fracture, with voids nucleation, growth and coalescence, and 2) brittle fracture, with cracks generation and growth as dominant process (Dodd and Bai, 1987, Nemat-Nasser and Hori, 1987, Koplik and Needleman, 1988, Tvergaard and Hutchinson, 1992, Pardoen and Hutchinson, 2003). Also experiments demonstrated that fracture and plasticity are coupled processes: Rice and Thomson (1974) observed the dislocation emission from growing crack, Christy (1986) registered the deformation hardening and Meyers (1994) reported about the plasticity zone around a void in regime of ductile fracture. The attention to plastic deformation near voids is connected with assumption about dislocation stimulated growth of voids. It was Stevens et al. (1972), who first suggested this point, when they conjectured the absorption of dislocations by growing voids. Later Wolfer (1988) proposed a model of dislocation emission from gas bubble in solid metal and performed the estimation of interaction energy. Analogous work based on interaction energy was done by Lubarda et al. (2004), but they analyzed the dislocation formation at void surface, the results of this work are widely used for comparison at present. The following papers of several groups (Fischer and Antretter, 2009, Lubarda, 2011, Wang et al., 2011, Pohjonen et al., 2012) presented more detailed results of continuum approach study of stress field around a void and generation of dislocation from the void surface.

Another mechanism of void growth presented in papers Rosi and Abrahams, 1960, Raj and Ashby, 1975, Brager et al., 1977, Hirth and Nix, 1985 is vacancy condensation under action of driving force from external stress field. But diffusion of vacancies, even through the most rapid diffusion mechanism, i.e. migration along the dislocations, cannot provide fracture in experiments with high strain rates as it was reported by Cuitino and Ortiz (1996) and Bringa et al. (2010).

Bulatov et al. (2010) criticized the point of view about dislocation stimulated nanovoid growth and proved the impossibility of atoms structure rearrangement on void surface after closed dislocation loop detaches from a void. At the same time, many molecular dynamics (MD) works (Seppala et al., 2005, Ahn et al., 2007, Traiviratana et al., 2008, Rudd, 2009, Bringa et al., 2010, Xu et al., 2011, Tang et al., 2012, Ariza et al., 2012, Nguyen and Warner, 2012) demonstrated that a void growth during the high rate tension is connected with emission of dislocations, ends of which remain attached to the void providing thereby displacements of surface atoms. Seppala et al. (2005) with the help of MD studied the emission of dislocations from nanovoid in copper; detached dislocations were identified as prismatic dislocation loops; leading and trailing partial dislocations were fixed. In work Ahn et al. (2007) the estimation of additional volume caused by dislocation removing was offered. Rudd (2009) investigated the evolution of void volume for such metals as V, Nb, Mo, Ta, and W. The author demonstrated that Peierls barrier is an important factor for dislocation emission in the case of BCC crystals. Traiviratana et al. (2008) and Bringa et al. (2010) validated the assumption about dislocation stimulated void growth. They stated that emitted dislocation is shear loop, ends of which are connected with a void, dislocation formation occurs at one atom step at a void surface; the detailed research of dislocation structure and slip system activity was performed. Xu et al. (2011) carried out the research of a void growth in vanadium and of related emitted dislocation structure. They reported about increase of the hydrostatic yield stress in systems with fixed ratio of a void radius to system size simultaneously with increase of system size. Tang et al. (2012) studied the tension of tantalum; they supported the viewpoint of shear loop formation and of related void volume increase. It was shown that plastically deformed annulus is formed around the void, formation of a dislocation in the annulus leads to increase of the void volume. Authors suggested a model for estimation of critical shear stress for systems with various voids. The model takes into account elastic interaction of emitted dislocation with a void, self energy of dislocation, energy of additional surface of a void formed after detachment of dislocation and, in case of FCC crystal, stacking fault energy. The results, which are predicted by the model, are in good agreement with MD data for wide range of void radius. Nguyen and Warner (2012) investigated the generation of dislocations near the void surface in aluminum. The estimation of nucleation time was based on Arrhenius-type law; the energy change connected with the emission of dislocations considers self-energy of dislocation, stacking fault energy and energy of external stress field interaction with dislocation.

The rate of dislocation nucleation in the ideal lattice or in the areas weakened due to presence of other defects can be described with the Boltzmann distribution of the dislocation nuclei (Hirth and Lothe, 1968, Zhu et al., 2004, Li, 2007, Gutkin and Ovid'ko, 2008, Nguyen and Warner, 2012), where the energy of critical nucleus defines the probability of the thermal fluctuation leading to the formation of stable dislocation nucleus. Norman and Yanilkin (2011) used the Arrhenius law, which is the particular case of the Boltzmann distribution approach with the fixed nucleation energy and activation volume. Ngan et al. (2006) and Zuo and Ngan (2006) offered the approach to the estimation of shear strength of Ni₃Al nano-sized samples, based on the usage of the Weibull-type distribution for the critical relative displacement of atoms.

Today there is a sufficiently large number of continuum fracture models in literature (Gurson, 1977, Tvergaard and Needleman, 1984, Tvergaard, 1990, Ortiz and Molinari, 1992, Cuitino and Ortiz, 1996, Thomason, 1999, Benzerga et al., 2004, Ikkurthi and Chaturvedi, 2004, Wen et al., 2005, Chen et al., 2006, Czarnota et al., 2008, Wright and Ramesh, 2009, Mayer and Krasnikov, 2011, Wilkerson and Ramesh, 2014). The problem of void growth in plastically deformed material was stated by McClintock (1968). Gurson (1977) offered the plastic potential function to describe stress relaxation in volume containing a void; in this model yield criterion depends on von Mises effective stress, hydro-static stress and a void volume fraction. This model was widely applied by many authors with some modifications (Tvergaard and Needleman, 1984, Tvergaard, 1990, Wen et al., 2005) to study the void growth in materials. Benzerga et al. (2004) offered a model, taking into account the following: shape of inclusions and voids, void distribution and plastic anisotropy as an effect of material texture. If the fracture of metals is considered in conditions of high strain rates, then an inertial effect associated with the accelerated motion of material close to a void becomes the significant role (Ortiz and Molinari, 1992, Tong and Ravichandran, 1995, Molinari and Mercier, 2001). In the papers by Molinari and co-authors (Molinari and Mercier, 2001, Molinari and Wright, 2005, Jacques et al., 2012) the effect of inertia was taken into account through the decomposition of macroscopic stress tensor onto quasi-static and inertial parts which depend on the volume of voids in the material. Since the works by Tong and Ravichandran (1995) and Wright and Ramesh (2009) it is clear demonstrated that sensitivity of shear strength on the deformation rate can provide the effect comparable with inertia one under high rate tension. Mayer and Krasnikov (2011) starting from the Lagrange formalism obtained the equation for crack growth similar in form to the Rayleigh–Plesset equation. The offered approach inherently dealt with the inertia effect and surface tension, also the plastic relaxation of shear stress around a crack was introduced through the equations of dislocation motion and kinetics. It was stated that the influence of dislocation dynamics and kinetics on crack growth depends both on the crack size and deformation rate. Wilkerson and Ramesh (2014) offered the model combining the equations of dislocation motion and kinetics with the equation of radial momentum balance considering acceleration potential (Carroll and Holt, 1972). The authors showed that motion and multiplication of dislocations are more important in comparison with inertia effect for nanovoid growth under high rate tension. Both the above cited works by Mayer and Krasnikov (2011) and Wilkerson and Ramesh (2014) considered the phonon friction, relativistic effect and multiplication of dislocations within the parts of the models that are devoted to plastic reaction.

In this paper we offer two-level model of dislocation stimulated growth of nanovoids in aluminum in order to predict critical tensile pressure in substance volume containing nanovoid. It combines MD simulations of void growth and continuum equations which capture basic micromechanical processes observed at MD level: we interpret the growth of nanovoids as a result of plastic deformation in a zone close to a void and corresponding atom rearrangement on a void surface. This atom rearrangement is possible because ends of emitted dislocations are bonded with a void. The continuum model is formulated in a way similar to papers by Mayer and Krasnikov (2011) and Wilkerson and Ramesh (2014); the main advantage of this work consists in accounting of emission of dislocations from a void. Process of dislocation generation near the void surface is described with the relation based on the Boltzmann–Gibbs distribution for rate of dislocation nucleation analogous with Norman and Yanilkin (2011) and Nguyen and Warner (2012). The parameters of this relation are defined in comparison with MD data.