The role of the excess volume at the nucleation of plastic deformation in metals
XIII International Conference on Computational Plasticity. Fundamentals and Applications
COMPLAS XIII
E. Oñate, D.R.J. Owen, D. Peric and M. Chiumenti (Eds)
THE ROLE OF THE EXCESS VOLUME AT THE NUCLEATION OF
PLASTIC DEFORMATION IN METALS
DMITRIJ S. KRYZHEVICH1,2, ALEKSANDR V. KORCHUGANOV1,2,
KONSTANTIN P. ZOLNIKOV1,2, SERGEI G. PSAKHIE1,3,4
1
International Institute of Strength Physics and Materials Science (ISPMS)
2/4, pr. Akademicheskii, Tomsk, 634021, Russia
e-mail: [email protected], web page: http://www.ispms.ru/en/
2
Tomsk State University (TSU)
36, Lenina Avenue, Tomsk, 634050, Russia
e-mail: [email protected], web page: http://tsu.ru/english/
3
Tomsk Polytechnic University (TPU)
30, Lenin Avenue, Tomsk, 634050, Russia
e-mail: [email protected], web page: http://tpu.ru/en/
4
Skolkovo Institute of Science and Technology,
100, Novaya St., Skolkovo, 143025, Russia
web page: http://www.skoltech.ru/en/
Key words: Plastic Deformation, Molecular Dynamics, Defects Formation, Nanoindentation.
Abstract. The computer simulation results on the atomic structure of the copper crystallite
and its behavior in nanoindentation demonstrate the key role of local structural
transformations in nucleation of plasticity. The generation of local structural transformations
can be considered as an elementary event during the formation of higher scale defects,
including partial dislocations and stacking faults. The cause for local structural
transformations, both direct fcc-hcp and reverse hcp-fcc, is an abrupt local increase in atomic
volume. A characteristic feature is that the values of local volume jumps in direct and reverse
structural transformations are comparable with that in melting and lie in the range 5-7 %.
1
INTRODUCTION
The last decades the study of plastic deformation is subjected to close attention. However,
issues related to the peculiarities of the plastic deformation nucleation in metals at the lowest,
atomic, level still remain poorly understood. It is known that different types of external action
with attendant thermal fluctuations can give rise to local regions of excess volume in a
material. In this context, the question arises: What are the conditions under which a local
volume change will lead to a local structural change? The possibility of such structural
changes is supported by studies showing that mechanically loaded fcc metals experience
specific local structural distortions which correspond to a local structural transition of the fcc-
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Dmitrij S. Kryzhevich, Aleksandr V. Korchuganov, Konstantin P. Zolnikov, Sergei G. Psakhie
hcp type. It is demonstrated that these distortions are preceded by the generation of excess
volume. Effect of the atomic density, or, equivalently, the atomic volume is discussed for a
long time [1-4]. As shown in [4] atomic density is an important thermodynamic variable that
determines the phase state of the system. The introduction of this parameter allowed
substantiating a new type of phase diagrams. This improved the understanding of the
conditions of coexistence of phases at changing the atomic density, and also explained the
instability of the crystal lattice and its melting, when the solid expands up to a critical specific
volume. Molecular dynamics simulations [5-9] revealed the atomic mechanisms of this
process. For example, in [5] it was shown that the increase in volume is the cause of the
lattice instability, and, as follows from [1] no matter how the excess volume is achieved either
by heating or by mechanical loading. These studies and the results obtained in [3] illustrate
the validity of the conclusions made in [10,11] for case of homogeneous increase of the whole
sample volume. In [10, 12] it was shown that local structural changes (LSCs) generated under
mechanical loading may be considered as some protodefects.
In this connection, it is vital to answer the question, how the formation of the local
structural changes is connected with the local increase in volume. The aim of the present
study is molecular dynamics simulation to elucidate the role of local excess volume in
nucleation of plasticity in metals. Based on the calculations performed, it will be concluded
that the local increase in volume is necessary condition for local structural transformations of
different types and responsible for nucleation of plasticity in loaded materials.
2 FORMALISM
Nanoindentation is one of the most visual and effective methods for the study of physical
and mechanical properties of materials in contact interaction. It enables to influence the origin
and evolution of the source of plastic deformation by changing indentation parameters [1316]. As a rule, the purpose of the investigations related to computer simulation of material
behavior during indentation, is the study of the mechanisms of plastic deformation of the
material in the vicinity of the indenter, the analysis of defect structures and interpretation of
nanoindentation force–displacement curves [17-20]. Despite the informative value of such
studies, it is difficult to analyse the obtained results due to the complexity of deformation
picture.
An indenter of extended cylindrical shape was used in simulations for simplifying the
interpretation of the indentation results. The loading was realized by its side surface. The
indenter had the atomic structure of copper with atoms fixed in lattice sites. The objects of the
simulation are copper single crystals of the parallelepiped shape (Fig. 1). The size of their
edges is 160 Å, the initial temperature is 300°K. The simulated specimens contain about 400
thousand atoms. Loaded crystallite faces simulated as free surfaces. Crystallographic indexes
of loaded surfaces in different calculations were next ones: (011̅), (001) and (111). Periodic
boundary conditions were used in the direction parallel to the indenter axis. The rest of the
faces perpendicular to the direction of the indentation are simulated as free surfaces. Three
atomic planes of the face opposite to loading one were fixed for eliminating movement of
crystallite as a whole. The indenter is moved with a speed of 25 m/s.
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Dmitrij S. Kryzhevich, Aleksandr V. Korchuganov, Konstantin P. Zolnikov, Sergei G. Psakhie
Figure 1: Initial shape and structure of simulated copper crystallite and indenter
Interatomic interaction was described in the frame of the embedded atom method [21]. In
the framework of this approach the potentials of interatomic interaction allow to describe with
high degree of accuracy the elastic properties of copper, its surface properties, as well as the
energy of defect formation, the effects of contraction of the near-surface atomic planes, etc.
The calculation of the atomic volume was carried out by the following formula:
𝑉𝑉𝑖𝑖 = (
𝑁𝑁
𝑠𝑠 |𝑟𝑟̅ −𝑟𝑟̅ |
∑𝑗𝑗=1
𝑖𝑖 𝑗𝑗
𝑁𝑁𝑠𝑠
3
) ,
(1)
where Ns is the number of nearest neighbors of the i-th atom, 𝑟𝑟̅𝑖𝑖 and 𝑟𝑟̅𝑗𝑗 are corresponding
radius vectors.
Analysis of changes in the simulated structure and classification of generated defects was
carried out on the base of common neighbor analysis, taking into account the mutual
arrangement of the nearest neighbors for each of the atoms [22]. Force–displacement curves
were calculated in order to evaluate the system response on nanoindentation. The indentation
force (F) was defined as the total force acting on the indenter from crystallite atoms.
Indentation depth (d) calculated as the distance from the lower boundary surface of the
indenter to a level corresponding to the surface of the crystallite in the initial state (before
loading).
2 RESULTS AND DISCUSSION
The simulation results showed that LSCs similar to those found in [7] are nucleated in the
specimen at the mechanical loading. The analysis revealed that they are the result of specific
rearrangements in the first and second coordination spheres of atoms-centers of these changes.
The initial local fcc structure of these atoms is transformed into a hcp structure (Fig. 2). They
are realized when one of the atoms β leaves the first coordination sphere while another atom α
from the second coordination sphere comes into it. Further loading of specimen leads to the
formation of planar defects by LSCs. Analysis of alternating atomic planes showed that the
LSCs form the intrinsic stacking faults as well as extrinsic ones. They are bounded by partial
dislocations which move in {111} glide planes from contact area to lateral free surfaces. The
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Dmitrij S. Kryzhevich, Aleksandr V. Korchuganov, Konstantin P. Zolnikov, Sergei G. Psakhie
simulation results showed that local excess of the atomic volume always precedes formation
of LSCs in the crystal lattice. The magnitude of the jump in the volume of direct local fcc-hcp
and inverse hcp-fcc structural transformations is 6-8 % (Fig. 3).
a)
b)
c)
Figure 2: The scheme of fcc-hcp structural rearrangements in the first coordination sphere: a) the initial fcc
configuration of atoms in the first coordination sphere; b) an intermediate stage; c) the stage at which the
environment of the central atom (red coloured) already has hcp topology. The dashed lines indicate the position
of the central atom’s first coordination sphere
Figure 3: The dependence of the relative change in the local volume for one of the atoms, which nearest
neighbours undergo direct and reverse transformation. Colour of curves indicates the type of the local structure:
blue – fcc local environment, red – hcp local environment
The resulting indentation force-displacement curve for the loaded crystal surface
orientation (011̅) is shown in Fig. 4. Note that the interaction between the indenter and the
crystallite occurs as soon as the distance between them becomes smaller cutoff radius of the
interatomic interaction. As in [15], initially between the indenter and load surface an
attraction takes place (Fig. 4, this corresponds to the negative value indentation force). This
effect is called “jump-to-contact” [23]. After further motion of the indenter the attraction is
replaced by repulsion. Accordingly, the indentation force in Fig. 4 has a pronounced
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Dmitrij S. Kryzhevich, Aleksandr V. Korchuganov, Konstantin P. Zolnikov, Sergei G. Psakhie
minimum. The dependence between indentation force and penetration depth calculated for the
given conditions can be divided into four stages.
Figure 4: Indentation force versus penetration depth for loaded surface (011̅)
The first stage is characterized by a linear dependence of the indentation force from the
indentation depth and corresponds to the elastic response of the material. Generation of LSCs
takes place at the beginning of the second stage in the zone of contact between the indenter
and crystallite (Fig. 5a). Generation of LSCs in a perfect crystal leads to a partial relaxation of
excess stresses and reducing the curve slope in Fig. 4 to the zero value. Although at this stage
the classic structure defects such as dislocations and stacking faults are not yet formed the
stress relaxation is already carried out by generation of LSCs. It should be noted that change
in the behaviour of the indentation force directly during the transition from the first to the
second stage is also related to structural features of the indenter. In particular, as it moves next
atomic layers of indenter start interacting with the crystallite free surface. Initially this
interaction is attractive thus the value of indentation force is reduced. The number of defects
in the contact zone in the second stage quickly reaches saturation and after that their number
in the third stage do not changes significantly. This leads to the exhausting of this stress
relaxation mechanism, resulting in an increase of the loading curve slope at the third stage.
a)
b)
Figure 5: Fragment of the simulated crystallite at different indentation depths: a) -1,0 Å, b) 8,7 Å. Atoms having
nearest neighbours with fcc symmetry are not shown. Indenter atoms are marked as red spheres, atoms with hcp
symmetry of the nearest neighbours are blue, atoms with broken symmetry of nearest neighbours are green
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Dmitrij S. Kryzhevich, Aleksandr V. Korchuganov, Konstantin P. Zolnikov, Sergei G. Psakhie
Analysis of the simulation results shows that the further indentation (the beginning of the
fourth stage) leads to the intensive growth of the number of LSTs (Fig. 5b) and, as a
consequence, loading curve not only slows its growing, but also experiences drops. At this
stage generation of LSCs leads to the formation and development of structural defects of a
higher rank, in particular partial dislocations and stacking faults. Note that structural defects
are generated in the contact zone in {111} atomic planes. Formed partial dislocations and
stacking faults may move in gliding planes and reach the free surface. This leads to the
change in shape of the crystallite – formation of surface steps.
Calculations showed that the crystallite behaves differently at loading of (001) surface. At
the indenter position -3.4 Å attraction force between it and the surface reaches a maximum
value. The number of defects increases due to the bending of the free surface of the sample.
The indentation force and the number of atoms involved in LSCs start to grow upon further
loading. When the indentation depth becomes -0.8 Å a defective area of a size comparable to
the size of the indenter forms in contact zone. After that the size of defect region grows with
the penetration depth of the indenter. It should be noted that a more intense increase in the
number of defects at indentation depth of 7.0 Å reduces the indentation force. Simulation
results showed that, after stopping the indenter at a depth of 10.0 Å dislocation loops are
emitted to the lateral sides of specimen from the bottom of defect region (Figure 6a-d). They
escape at the free surfaces and form surface steps. Note that after that the percentage of LSCs
is reduced from 2.0 to 0.3 %. Development of defect structures in copper crystals with the
loaded surface (111) is presented in Figures 6e-h. Calculations have shown that the LSCs are
nucleated in the sample at the indenter penetration depth of -3.0 Å. At the indentation depth of
0.8 Å stacking fault starts to grow in the (011̅) plane, and the indentation force slightly
changes. During sample loading a group of partial dislocations a/6 <112> {111} is generated
and starts moving in the adjacent atomic planes (011̅). As a result a twin bounded by twin
boundaries, which are recognized as hcp atoms is formed in the specimen (Fig. 6h). Note that
a fragmented region consisting of hcp atoms is formed in the contact zone. This region
develops in a direction toward the right side edge of the specimen.
12 CONCLUSIONS
The results of the simulations showed that LSCs are formed during nanoindentation of the
copper crystallites. Their nucleation is always preceded by a local increase of the atomic
volume by 6-8 %. Generation and evolution of LSCs leads to the formation of structural
defects of a higher level: dislocations, stacking faults, etc. It is noted that the indentation force
and the number of atoms involved in the LSCs are well correlated. This behaviour is due to
the fact that the generation of structural defects is a mechanism of elastic relaxation.
It was found that the crystallographic orientation of the loaded specimen surface
substantially affects nanoindentation response of copper. For example, when loading surface
oriented in (011̅) plane, stacking faults are generated and move towards lateral free surfaces.
During indentation of (001) surface all LSCs are formed in the contact zone, and after
stopping the indenter they escape at lateral free surfaces of specimen. The loading of (111)
surface leads to the formation of twin bounded by LSCs.
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Dmitrij S. Kryzhevich, Aleksandr V. Korchuganov, Konstantin P. Zolnikov, Sergei G. Psakhie
a)
e)
b)
f)
c)
g)
d)
h)
Figure 6: Projection of crystallite structure with loaded surface (001) at different moments of time at the
indenter penetration depth of 10.0 Å: a) 0 ps, b) 25 ps, c) 50 ps, d) 120 ps. Projection of crystallite structure with
loaded surface (111) at different indenter penetration depths: e) -1.4 Å, f) 1.4 Å, g) 5 Å, h) 10 Å
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