Molecular dynamics simulations of crack growth behavior in Al in the presence of vacancies

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Abstract

Fracture properties of a material are strongly influenced by the presence of defects. In view of this, molecular dynamics simulations were performed within the framework of embedded atom method to investigate the crack tip behavior under mode I loading in FCC Al with and without defects. Defects consisting of a random distribution of vacancies were introduced in the precracked matrix. The results reveal a significant change in the fracture behavior of Al with defects. The interactions between the crack tip and vacancies were found to modify the stress concentrations at the crack tip. Increase in vacancy content lead to increased crack tip blunting and reduced crack propagation speeds as compared to the vacancy-free material, thereby toughening the material to fracture. The significance of such studies towards gaining deeper insights into the fracture behavior at higher scales is highlighted.

Introduction

Fracture mechanics is a useful tool for the prediction of failure properties of crystalline materials in the field of materials science and engineering. The propagation and growth of cracks are the important processes which outline the mechanical properties of a material in fracture mechanics. Such processes involve breaking of individual bonds between the atoms at the atomic scale. Accurate modeling of such processes inherently involves coupling of various length scales, ranging from atomistic domains of cracks, dislocations to those of grain boundaries in a polycrystalline material. To this end, considerable efforts have been made at the macroscopic scale by implementing cohesive zone modeling with crystal plasticity finite element method (CPFEM) to describe fracture process using phenomenological traction-displacement relations [1], [2]. While such formulations enjoy the ease of numerical implementation, they lack physical descriptions of microstructure dependent crack growth and the associated mechanisms at the micro/nano scales.

Molecular dynamics (MD) is a powerful tool in characterizing the inception and evolution of plastic deformation and associated failure mechanisms at the atomic scale. Applied to the problem of crack propagation and growth, the method has been applied extensively to investigate the fundamental mechanisms of material’s fracture behavior in the past [3], [4], [5], [6], [7], [8], [9]. These studies display the formation of a heterogeneous network of dislocations near the crack tip, which is consistent with previous transmission electron microscopy (TEM) observations [10], [11]. Such experimental observations and numerical predictions represent a qualitative nanoscale understanding of the fracture processes, providing valuable insights into the crack tip plasticity and associated deformation mechanisms-dislocation nucleation and multiplication, dislocation junction formation, deformation twinning, etc.

Defects like dislocations influence the dynamics of crack propagation through their long range stress field [12], [13]. Cracks also interact with other crystalline defects like vacancies, interstitials, their clusters and secondary phase particles which alter the stress field around the crack tip, thereby reducing or amplifying the crack tip stress intensity factor [14]. While the studies of interactions between cracks and microstructural constituents, especially vacancies/voids, have gained increased momentum by a continuum formulation [13], [14], [15], [16], [17], only a relatively few studies exist at the atomistic level. For instance, the influence of cylindrical voids on a (0 1¯ 1)[0 1¯ 1¯] crack in α-Fe was studied by Liu and Groh [18]. They found that the location of void influences the dislocation nucleation stress at the crack tip. Machova et al. [19] simulated the interaction of a through thickness bcc-Cu precipitate with (0 1¯ 1)[0 1¯ 1¯] crack in α-Fe. Depending upon the distance between crack tip and the precipitate, variations in the dislocation nucleation stress at the crack tip were observed. Petucci et al. [20] studied the interaction of rows of Cu, Pd, Pt,Ag and Au as substitutional atoms with (0 0 1)[1 0 0] cracks in FCC Ni. The presence of substitutional atoms led to an increase in the critical load required for crack propagation as compared to defect free Ni. This was, again, attributed to the change in stress filed at crack tip by the impurity atoms. Very recently, Wang et al. [21] found that the presence of pre-existing voids can significantly affect the crack propagation speeds in single crystal silicon. Different crack-retarding mechanisms were observed depending upon the crack-void distance and different orientation of voids.

Although the aforementioned studies have provided deep insights on crack growth phenomena in the presence of inclusions like voids, a majority of them are focused on studying the general features of brittle fracture using ductile materials by crack orientations in such a way that brittle failure takes place (no dislocation activity at crack tip) [20]. Moreover, only two type of setups are usually modeled: either the voids are situated directly at the crack tip or within the path of the propagating crack, or the voids are situated either above or below the crack tip. While the former condition allows one to investigate direct crack-obstacle interaction, the latter setup also allows to study the change in stress field around the crack tip. Either of the two cases are, however, an oversimplification of the actual phenomena, wherein the inclusions like voids may be distributed randomly within the material. Since voids can be considered as a cluster of vacancies, the first step in developing a nanoscale understanding in this regard should be to study the fracture processes in presence of random distribution of vacancies. To this end, very few atomistic studies are present in the literature for BCC [22], [23] and FCC [24], [25] metals. These studies have investigated the change in mechanical response of the precracked system due to inclusions like vacancies/interstitials by either introducing a limited No. of point defects (maximum 20 in the work of Saraev et al. [23]), or by investigating the crack tip behavior by putting only a few lines of vacancies in front of the moving crack [24]. This makes the results predicted by such simulations configuration specific, and may also sometimes lead to contrasting observations. For instance, while Saraev et al. [23] predicted a decrease in system strength with increasing number of Frenkel defects in the fracture behavior of α-Fe, Borodin and Vladimirov [22] found almost no effect of vacancies on the crack propagation kinetics in Fe. In addition, while these studies have provided important insights into the fracture behavior of investigated metals, the analysis was, however, predominantly limited to a qualitative perspective. A quantitative description and correlation between the plastic evolution and mechanical response was mostly unresolved.

In view of this, we have performed molecular dynamics simulations to investigate the effect of a random distribution of vacancies on the crack propagation behavior in FCC Al. We begin by examining the fracture process in vacancy-free Al using an EAM (Embedded Atom Method) potential. Successively, vacancies were introduced corresponding to different vacancy concentrations in the system to analyze the crack-tip behavior by monitoring and quantifying the plastic evolution ahead of the crack front. This will help in improving our understanding on the fracture behavior of defective systems, and will also inform advanced simulations at nano-, micro- or meso-scales aiming to gain more insights in fracture processes that may occur in a real aggregate.

This work is organized as follows: Section 2 details the computational methodology adopted in this study, with the results representation in Section 3. Discussion of results is presented in Section 4. Finally, we provide our conclusions in Section 5.

Section snippets

Interatomic potential

The interaction between Al atoms was simulated within the framework of Embedded Atom Method (EAM) [26]. In this framework, an atom is considered to be embedded in a host lattice consisting of many other atoms. The energy of an atom is, therefore, calculated as a sum of effect of local environment on each atom, and the pairwise contribution of each pair of atoms in the ensemble. This leads to the following functional form of the EAM potentialEtotal=iFi(ρh,i)+12ijiϕijRijThe first term is the

Results

The stress–strain curve for vacancy-free Al single crystal is shown in Fig. 2. For this system, the material remains elastic till 5.49 % strain. At 5.49% strain, first dislocation is nucleated at the crack tip (in the (1 1¯1) slip plane). The subsequent plasticity caused by gliding of dislocations resulted in crack tip blunting. Thus, no evidence of brittle crack propagation was found for this orientation, and crack growth was mainly governed by crystallographic slip, followed by void growth and

Discussion

Fracture and associated plastic deformation processes in the materials are obvious cases where the material properties at the macroscopic scale are entirely determined by the events occurring at the nanoscale. Given such separation of length scales, KI>KIC is a general criteria for crack growth under mode-I loading conditions, KI being the stress intensity factor which characterizes the strength of stress singularity at the crack tip. By using the Griffith’s widely used analytical model [38],

Concluding remarks

Assembling all pieces of this work, we hypothesize that the presence of vacancies in a single crystal Al containing an edge crack, under mode-I loading, significantly alters the crack growth behavior. We base this hypothesis by monitoring the stress field and slip vector magnitude at the crack tip for the vacancy-free model and systems containing 0.5%, 1.0% and 5.0% vacancies. For the system containing no vacancies, a one-to-one relationship between the tensile stress and microstructure

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