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Nanofragmentation as a relaxation mechanism in post deformed solids:
molecular-dynamics investigation
A.I. Dmitriev and S.G. Psakhie
Institute of Strength Physics and Materials Science SB RAS, Tomsk, 634021, Russia
In the paper the possibility of nanofragmentation of the material in layers near to a free surface at initial stages of relaxation was investigated. Investigations have been carried out by a method of molecular dynamics. It was shown the possibility of off-oriented nanoblocks formation at the initial stage of relaxation. Fragmented structure initially formed in the area of localized deformation near to stress concentrators and then distributed into the material. It was shown that in the area of localized deformation the radial distribution function of atomic density has broad peaks corresponding to peaks of ideal FCC structure and in other part of crystallite there are splitting peaks of FCC structure caused by symmetry disorder due to deformation. With propagation of the area of localized deformation deep into material peak broadness significantly decreases and their location close to corresponded peaks of ideal FCC lattice. The obtained results enable to assume that a possible mechanism of internal stress relaxation in the post-loaded solids is the effect of material nanofragmentation and some off-oriented blocks are separated in the fragmented area.
1. Introduction
The understanding of elementary acts and mechanisms of development of plastic deformation is one of key problems of the modern materials science, physics of strength and plasticity. Along with traditional mechanisms within the framework of physical mesomechanics the particular influence of processes of fragmentation with involving a rotary mode of deformation [1-3] is outlined. In certain papers it is emphasized that the particular influence of surface layer is related not only to the initial defect nature of the surface or the influence of surrounding medium, but also to a reduced stability of the surface layers with respect to shear [4]. The particular influence of the surface as a special condition of the loaded solid is caused by the fact that it is a source of the free volume and lead to an essential increase in a possibility of formation and development of various kinds of collective mechanisms at the atomic level.
As a rule, similar phenomena are studied at the stage of active loading [5]. At the same time, characteristic velocities of such processes are small enough, that makes possible of their realization at the initial stages of relaxation or changes deformation mode, for example, at fatigue loading. Therefore a study of the possibility of material fragmentation at initial stages of relaxation is essential. Experimental investigations of similar phenomena meet significant difficulties due to their extremely small time and spatial parameters.
In order to realize the features of deformation of solids with free boundaries analytical methods need to be developed. Among these methods the molecular dynamics simulation should be emphasized. Computer simulation is valuable due to its applicability to a precisely defined model for the investigated material. So, significant interest represents modeling of solids with free boundaries under loading to reveal and study elementary processes in deformed solids and the processes of formation and development of plastic deformation.
In this paper a molecular-dynamics scheme in a threedimensional approximation is proposed by which the simulation of uniaxial compression of copper crystallite and further relaxation is carried out. Features of material plastic deformation near the free surface at the initial stage of material relaxation (immediately on removal of active loading) were investigated.
2. Theoretical backgrounds
In the paper the response of a copper FCC single crystal preliminary subjected to mechanical loading was investigated. The crystallite has the shape of parallelepiped which edges are co-directional with the crystallographic directions [ 100], [010] and [001 ]. There are a total of about 21000 atoms with sizes in atomic units of length 150 x 75 x 150, respectively. Atomic unit of length is equal to 0.529177-10-10 m
© A.I. Dmitriev and S.G. Psakhie, 2004
[6]. The model crystallite is schematically depicted in Fig. 1, where region I represents a deformed part of the crystal, while regions II realized the so-called string boundary conditions [7] used for simulation of external action. In the case under consideration, the projections of atomic velocities onto the direction [001] in regions II were fixed, while the projections onto other directions were determined by the corresponding atomic environment.
In order to take into account the extension of the model fragment, we used periodic boundary conditions in the [010] direction and simulated free boundaries in the [100] direction. The interatomic interactions were described within the framework of the method of embedded atom [8, 9]. In order to avoid the induced effects related to the symmetry of the ideal crystal lattice, the model copper crystal was heated to 20 K. The final temperature in the crystal was set based on the Maxwell-Boltzmann distribution of atomic velocities [10]. Due to simulation truncation errors the real temperature of the simulated system can deviates from the predetermined value. A modification of velocity fields was carried out on the basis of the following equation:
Vm = v/yl t d/t r,
(1)
where Vm and V* represent the modified and real velocities of atom i, respectively, Td represents the predetermined temperature and Tr is the real temperature of the system.
At the first stage, the model crystal was subjected to compression by displacing regions II with velocities -50 and 50 m/s, respectively. Loading was continued before reaching the strain degree corresponding to plastic flow, after which the crystal was allowed to relax. Detailed investigation of the relaxation process was performed by studying the time evolution of atomic configurations, atomic displacements and radial distribution functions of atomic density (RDF) [11] for various areas of the crystallite. The following relationship was used for evaluation of RDF:
g (r) = n(r )/(p4nr 2Ar ). (2)
[001]
[100]
Here n(r) is the mean number of atoms in a shell of width Ar at distance r, p is the mean atom density.
3. Results of simulations
The results of our investigation showed that relaxation of the modeled crystallite is accompanied by the formation and development of bands of localized deformation. The detailed analysis has shown that they initially occur on the free surface. Thus, sources of the formation of deformation localization bands are in zones of stress concentrators, namely in the middle part of free surfaces and in the zones of contact of the deformable area I with areas II simulating external loading. It is well visible in Fig. 2(a) where the
[001]
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Fig. 1. Initial structure of the simulated crystallite
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Fig. 2. Structure (a) and RDF of “A” (b) and “B” (c) fragments of the simulated crystallite at the moment of the formation of localized deformation bands. Dotted lines mark peaks of the ideal FCC lattice
structure of a fragment of the simulated crystallite is shown at the instant corresponding to the beginning of the formation of localized deformation bands. It is clear to see that localization of deformation arising on surfaces then distributed deep into the material. The detailed analysis of the radial distribution function of atomic density for various fragments of the simulated crystallite has shown that the formation of localized deformation bands is accompanied by the rearrangement of atomic structure. It can be seen in Fig. 2(b and c) RDF peaks transform from strongly “split” basic peaks in the case of deformed lattice (fragment “B”) to broad separate peaks corresponding to peaks of FCC lattice in the areas of localized deformation (fragment ‘A”).
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In Fig. 3(a) the structure of the simulated crystallite at the subsequent moment of time is depicted. It is clear to see that localized deformation bands having distributed inside cover already practically all simulated crystallite. The detailed analysis of the radial distribution function of atomic density for various fragments of the crystallite has shown that in area with rearranged atomic structure peak broadness noticeably decreases and their locations close to corresponded peaks of ideal FCC lattice (cf. Fig. 2(b) and Fig. 3(b)). At the same time, for the area without rearrangement of atomic structure the kind of RDF is still the same (cf. Fig. 2(c) and Fig. 3(c)).
The further analysis of relaxation development has allowed one to reveal mechanisms of atomic structure rearrangement. In Fig. 4 a fragment of the central part of the simulated crystal is depicted. It can be seen that the formation of separate off-oriented nanoblocks results from structural rearrangement. So, Fig. 4(a) differs from Fig. 4(b) only by the orientation of the fragment around axis OX (initial direction [100]) on 6 degrees and around axis OZ (initial direction [001]) on 3 degrees. Radial distribution functions of atomic density for the marked areas correspond to FCC
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Fig. 3. Structure (a) and RDF of “A” (b) and “B” (c) fragments at the instant when the fragmented area distributed deep into the crystallite
Fig. 4. Various orientation of the central fragment of the crystallite after the formation of nanoblocks. Rotation in the Cartesian coordinate system
lattice. Similar nanoblocks with FCC packing are possible to reveal in other parts of the simulated crystallite. The analysis of atomic structure shows that separated nanoblocks are connected by an intermediate layer. Due to mutual influence of blocks the structure of these layers essentially differs from FCC packing. Thus, broadness of RDF peaks, constructed for the fragmented area of the crystallite (Fig. 2(b) and Fig. 3(b)), can be explained by the existence of interblock boundaries and mutual influence of blocks.
4. Summary
The results of the simulation show that plastic deformation at the initial stage of relaxation can develop by the formation of off-oriented crystalline nanoblocks. Fragmented structure initially formed in the area of localized deformation near to the free surface and stress concentrators and then distributed deep into the material. It is also confirm a particular influence of the surface as a special condition of a loaded solid. It was shown that in the area of localized deformation the radial distribution function of atomic density has broad peaks corresponding to peaks of ideal FCC structure and in other part of the crystallite there are splitting peaks of FCC structure caused by symmetry disorder due to deformation. Broadness of RDF peaks corresponding to ideal FCC lattice is explained by the existence of interblock boundaries and mutual influence of blocks. The results obtained allows us to suppose that the possible mechanism of internal stress relaxation in post-loaded solids is the effect of materil nanofragmentation.
Acknowledgments
This study was supported in part by the Presidential Program of Support for Leading Scientific Schools in Russia (project No. NSh-2324.2003.1), the Ministry of Education of the Russian Federation (project No. PD02-1.5-425), CRDF Grant No. T0-016-02 and Russian Science Support Foundation.
References
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