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4
Chelyabinsk Physical and Mathematical Journal. 2023. Vol. 8, iss. 2. P. 292-304.
DOI: 10.47475/2500-0101-2023-18212
PLASTICITY INCIPIENCE IN ALUMINUM WITH COPPER INCLUSIONS
A.E. Mayer
Chelyabinsk State University, Chelyabinsk, Russia mayer@csu.ru
The dislocation activity controls the plastic deformation in the most of metallic materials. Mechanical loading with high strain rates or with high strain gradients can lead to either homogeneous nucleation of the dislocation or emission of dislocations from various heterogeneities, such as nanopores and phase precipitates. The dislocation nucleation and emission trigger plasticity, which relaxes the shear component of stresses. In this work, we study the threshold of dislocation emission from nanosized copper inclusions in an aluminum single crystal in comparison with the homogeneous nucleation of dislocations in pure metal. We consider different shapes of inclusions (spherical, cylindrical and cubic) and rather arbitrary axisymmetric deformations by means of molecular dynamics (MD) simulations. For most deformation paths, the copper inclusions substantially reduce the threshold of plasticity incipience, while the inclusions have no effect for some deformation paths with either axial or transverse extension. Depending on the deformation path, the shape of inclusion can either influence the emission threshold or not. Thus, there is a complex dependence of the threshold of plasticity incipience on the deformation path, the presence and the form of copper inclusions. This dependence is approximated by means of an artificial neural network (ANN) trained on the results of MD simulations. The trained ANN can be further applied as a constitutive equation at the level of continuum mechanics.
Keywords: aluminum, copper inclusion, plasticity incipience, emission of dislocations, molecular dynamics, artificial neural network.
Introduction
The dislocation activity controls the plastic deformation in the most of metallic materials. Dislocations can arise even in an ideal crystal lattice due to the process of homogeneous nucleation at strong enough elastic shear [1-6]. It is also possible the heterogeneous nucleation and emission of dislocations from various surface and bulk defects, such as grain boundaries [7-12], nanopores [13-15] and phase precipitates or inclusions [16-19]. Under normal circumstances, both homogeneous and heterogeneous nucleation is suppressed by slip and multiplication of the dislocations already existing in the material. Plasticity and relaxation of shear stresses are started by the pre-existing dislocations, and the nucleation threshold is not achieved. Contrariwise, mechanical loading with high strain rates or with high strain gradients can make pre-existing dislocations insufficient and lead to either homogeneous nucleation of dislocation or emission of dislocations from heterogeneities. Very strong elastic deformations are experimentally recorded in thin metal films subjected to ultra-short powerful laser irradiation [20-23]. Nanoindentation creates high stresses in a local area of material,
This research was funded by the Russian Science Foundation, grant number 20-11-20153.
which is typically devoid of dislocations [24-26]. In both these cases, the dislocation nucleation and emission trigger plasticity.
Various phase imperfections and precipitates are typical for metals, all the more so for the specially prepared alloys. For instance, copper addition to aluminum forms a variety of Al-Cu intermetalic phase precipitates [27-30], which obstruct the motion of dislocations [31; 32] and leads to the strengthening of alloys. Artificial aging of alloys is used to transform the initial solid solution of copper atoms into required localized precipitates and achieve a preferable size distribution of precipitates [30; 33]. Although Al-Cu intermetalics are thermodynamically more stable than pure Cu inclusions in Al matrix, the Cu-containing clusters can be precipitated in aluminum alloys by combination of room temperature aging and the ultrasound treatment [34]. Besides, the Cu inclusion in Al can be treated as a model system to study the main regularities of the influence of phase inclusions on the dislocation nucleation and emission. It is shown in [16; 17] by means of molecular dynamics (MD) simulations that Guinier — Preston (GP) zones, which are essentially parts of atomic planes in aluminum replaced by Cu atoms, reduce the dislocation nucleation stress. MD simulations performed in [18] show that spherical Cu inclusions reduce the threshold of nucleation and plastic grow of voids at tension due to the stress concentration near the inclusions. The Cu-containing precipitates can have an almost spherical shape for the case of solute clusters [34] or the cylindrical shape typical for 9 and 9' precipitates. Therefore, the study of various shapes of inclusions is relevant. Previous MD simulations reveal substantial dependence of the dislocation nucleation threshold in pure single crystal metals on the pressure [6; 35] and, in general, on the deformation path [36]. This fact motivates investigation of different lading conditions for the metals with inclusions as well.
In this work, we study the threshold of dislocation nucleation and emission from nanosized copper inclusions in an aluminum single crystal in comparison with the homogeneous nucleation of dislocations in pure metal. We consider different shapes of inclusions (spherical, cylindrical and cubic) and rather arbitrary axisymmetric deformations by means of MD simulations. The MD shows that there is a complex dependence of the threshold of plasticity incipience on the deformation path, the presence and the form of copper inclusions. This dependence is approximated by means of an artificial neural network (ANN) trained on the results of MD simulations.
1. MD problem statement
For classical molecular dynamics simulations, we use the widely applied code LAMMPS [37] together with the interatomic potential for Al-Al, Cu-Cu and Al-Cu interactions developed in [38]. This potential describes Al-Al and Cu-Cu interactions based on the EAM (embedded atom method) model [39], which takes into account the interaction of atoms with the density of free electrons important for metals. In addition, dipole and quadrupole corrections are taken into account for Al-Cu interactions based on the ADP (angular-dependent potential) model suggested in [40]. The used potential describes Al-Cu system with a high accuracy. The obtained atomic configurations are analyzed by means of OVITO program [41] equipped with the dislocation extraction algorithm (DXA) [42] for detecting of dislocations and the construct surface mesh algorithm [43] for detecting of pores. The deviation of the atom environment from the ideal crystal lattice is also characterized by the centrosymmetry parameter [44].
MD systems contain half a million atoms and have dimensions of about 20 nm x 20 nm x 20 nm. An aluminum single crystal with a face-centered cubic (FCC) lattice is oriented in such a way that the lattice directions [100], [010] and [001] coincide with
the x, y and z coordinate axes. In the case of Al-Cu system, a pore is cut out in the central part of the system, which is then filled with copper atoms arranged in the FCC lattice of the same orientation as for aluminum. We consider spherical Cu inclusions with the diameter of 4 nm, which provides the volume fraction of about 0.4% and the mass concentration of copper of about 1.4% typical for Al-Cu alloys. We also consider cubic and cylindrical copper inclusions of the same volume as the spherical inclusions of 4 nm in diameter. Thus, we examine the influence of the shape of inclusions at the same content of copper. Periodic boundary conditions are used for all boundaries of MD system, which makes it equivalent to a representative volume element (RVE) of an infinite periodically repeated material.
MD systems are initially relaxed at zero stresses and test temperature during 10 ps. After that, a uniform axial deformation is applied to the RVE by means of the scaling of atom coordinates. The engineering strain rates along the axial direction ¿n and along the transverse directions ¿22 = ¿33 fulfill the following requirement: |en| + 2|e22| = ¿, where the total strain rate is equal to e = 1010 s-1 in the main series of simulations. The deformed state is characterized by two independent diagonal components, E11 and E22 = E33, of the Green — Lagrange strain tensor defined as follows:
E11 = 1
Lx
L
1
x0
12
¿11 + 2 ¿11'
E22 = 1
L,
L
1
,0
12
= ¿22 + 2 ¿22'
where Lx, Ly are the current lengths and Lx0, Ly0 are the initial lengths of MD
system along the corresponding axes. The studied deformation paths in {E11, E22} space are plotted in Fig. 1. One can see that these deformation paths cover quite arbitrary deformed states, including the compressed ones {E11 < 0, E22 < 0}, stretched ones {E11 > 0, E22 > 0} and those with a predominantly shear deformation {E11, E22 : E11 • E22 < 0}.
r ^
0.1
-0.1
-0.2
#5 #6 /#4
#7 \ #8 \ #3 #2 #1
#10 // #11 / \ #15 #16
#12 #14 #13
-0.2
0.2
0.4
'11
Fig. 1. Deformation paths in {En, E22} space; the path numbers are shown
2
2
The uniform deformation of the RVE with the constant engineering strain rate is applied during 40 ps, which gives the total engineering strain of 0.4. The threshold of dislocation nucleation and emission is achieved for all considered deformation paths. The process of dislocation emission from the inclusions of different shapes is illustrated in Fig. 2 by an example of the compressive deformation path #11, see Fig. 1.
Fig. 2. Nucleation and emission of dislocations in aluminum on the copper nanoinclusion of: (a)-(c) spherical shape, (d)-(f) cylindrical shape and (g)-(i) cubic shape. Dislocation lines and atoms belonging to lattice defects are shown for 300 K and deformation path #11
MD simulations are performed for two test temperatures, the room temperature of 300 K and an elevated temperature of 700 K. The test temperature is maintained constant during each simulation by means of the Nose — Hoover thermostat [45]. The results for Al-Cu systems are compared with the case of pure single crystal aluminum. Thus, the total number of MD simulations of the main series is equal to 128 = 4 systems x 16 deformation paths x 2 temperatures. Besides it, MD simulations with an order of magnitude lower strain rate of £ = 109 s-1 are performed for spherical nanoinclusions at 300 K in order to study the strain rate effect, and MD simulations with two times larger spherical nanoinclusions with the diameter of 8 nm and the volume fraction of about 3.4% at 300 K are performed to study the size effect.
2. Training of ANN
The results of the main set of sim train an artificial neural network (AN state and the distance to the nucleat defined by two variables, which are t
at * if /_
0.1 IHh ^ * .V
|y L S> J i
The nucleation strain distance function Q is defined in [35; 36] as the difference between the current engineering strain £ along some deformation path and the threshold value sc for this deformation path leading to the dislocation nucleation and emission at a reference strain rate ¿0, which is ¿0 = 1010 s-1 in our case. Here we use a slightly corrected definition:
with the minimal level of -0.05 introduced in order to avoid the ambiguty of this function at small strains, where several deformation paths converge. This correction is admissible, because the behavior of Q near the nucleation threshold is only essential, but not in the vicinity of zero strains. In the MD, the function Q is defined for a discret set of deformed states, while after the training of the ANN it becomes a continous function of En and E22, because the ANN interpolates Q on all intermediate deformed states. The dislocation nucleation threshold at arbitrary strain rate s can be calculated from the condition [35]:
where kB is the Boltzmann constant and A is the strain rate parameter, which can be fitted by comparison of MD simulations for different strain rates.
During the processing of MD data, we determine ¿c for all MD systems and deformation paths as the strain value, at which a sharp increase in the dislocation length calculated by the DXA algorithm [42] takes place. One can see in Fig. 2 that dislocations initially exist around Cu precipitates due to the lattice mismatch, but the dislocation length rapidly increases after a certain strain level corresponding to the nucleation threshold. The value ¿c let us to calculate the function Q according to Eq. (1) and select the elastic part of the deformation path. Stresses and internal energy of the elastic part of each deformation path is approximated by a third-order polynomial, and then extrapolated up to 1.5 ¿c. The using of polynomial approximation is efficient for a single deformation path, but it is inapplicable for the whole range of considered systems and deformed states, where the training of an ANN is preferable to describe the deformation behavior. The training data are collected, which include about 14 thousands of input-output pairs reflecting the elastic behavior of the material and its extrapolation beyond the threshold of plasticity incipience.
We use a fully connected feed-forward ANN with four hidden layers and 40 artificial neurons in each hidden layer. This ANN has a total of 320 artificial neurons in hidden layers that use the "LeaklyReLU" transfer function. The output layer forms the output vector and consists of 4 artificial neurons with the "Sigmoid" transfer function. The input layer redistributes 5 input values to the artificial neurons of the first hidden layer. The adopted structure of the ANN is selected by a parametric study to provide high precision at minimal complexity. The selected ANN has about 5.3 thousands of parameters, which are the weights and biases of the artificial neurons. The training means fitting of this parameters to minimize an error of ANN prediction on the training dataset. The training procedure is described in detail elsewhere [6; 35; 47], it includes calculation of the gradient of the error in the ANN parameter space using successive matrix operations commonly referred to as the backpropagation algorithm. Each training step, the randomly selected weights and biases are varied opposite to the error gradient. The strategy for choosing the step of this variation significantly determines the efficiency of the training procedure. In contrast to other approaches, we vary all the randomly selected parameters by the same constant step during each round of training, and the variation step discretely decreases at the beginning of the next round. This procedure
Q = max (e — ec, -0.05}
(1)
(2)
provides very fast training for the most of datasets. In the present work, single-threaded training for 4 hours gives an average error of 0.12% and a maximum error of 2.7% on the training dataset. Fig. 3 shows that the trained ANN provides a very good reproduction of the MD data.
3. Threshold of plasticity incipience
Let us consider the results of MD study on the dislocation nucleation threshold and their ANN-based approximation collected in Figs. 4-7.
0.08
0.04
-0.04
-0.08
-0.12
- / * ^XN > t\ \
- a i ¿4 7/ ta
- t (' ■ to j fl A /i r. A ' \ ^ \
If *'1 / • / i / / // /i % 1 1 J
V' / - ^y ■ ^ i 1 1
Cu inclusions of different shapes at 300 K
♦ MD pure AI
-ANN pure AI
O MI) spherical - ANN spherical □ MD cylindrical
— — ANN cylindrical ■ MD cubic
— - - ANN cubic
-0.16
-0.08
0.08
0.16
E
ii
Fig. 4. Threshold of plasticity incipience in pure single crystal aluminum and in Al-Cu systems with different shapes of copper inclusions; the temperature is 300 K; the strain rate is 101 show MD data and lines show the ANN-based interpolation
110 s 1. Markers
Fig. 4 shows the influence of the shape of copper inclusions. One can see that this influence has a complex manner and strongly depends on the deformation path. The threshold of plasticity is almost insensitive to the presence of copper inclusions both under uniaxial tension {En > 0, E22 = 0} and under pure lateral tension {En = 0, E22 > 0}. Inclusions of all considered shapes equally decrease the nucleation threshold for the strain paths close to the hydrostatic tension {En = E22 > 0}. Under the hydrostatic compression {En = E22 < 0}, spherical and cylindrical inclusions do not influence, while the cubic inclusions reduces the threshold due to a more strong stress concentration in the corners. For the deformation paths close to the uniaxial compression {En < 0, E22 = 0} and for mostly shear deformations {En • E22 < 0}, the influence of copper inclusions is evident and depends on their shape. All this demonstrates the complex dependence, which is perfectly described by the ANN-based approach. The ANN not only coincides with the calculated MD points, but also interpolates MD results on the intermediate deformation paths.
Fig. 5 illustrates the temperature effect on the dislocation nucleation. For tension {En > 0, E22 > 0} and predominantly shear deformation {En • E22 < 0}, the raised
temperature leads to a decrease in the threshold strain of dislocation nucleation, while an opposite effect is observed under compression [En < 0, E22 < 0}. Previous MD simulations [6; 35] show that the dislocation nucleation stresses rapidly grow with pressure; thus, a difference in the pressure sensitivity between the room and elevated temperatures can explain this inverse temperature effect under compression. In the most cases, the temperature effect is stronger for pure aluminum than for the Al-Cu system.
0.08
0.04
0
fS r M
-0.04
-0.08
-0.12
-0.16 -0.08 0 0.08 0.16
En
Fig. 5. The influence of temperature on the dislocation nucleation threshold in pure aluminum and in
the aluminum with spherical copper inclusions of 4nm in diameter; the strain rate is 1010 s-1.
Markers show MD data and lines show the ANN-based interpolation
Fig. 6 shows the strain rate effect studied by the MD and described by the ANN with Eq. (2); the strain rate sensitivity parameter of A = 2.5eV is chosen from this comparison. One can see that a decrease in strain rate on the order of magnitude leads to a moderate decrease in the threshold strains of dislocation nucleation, and this decrease can be accurately taken into account using the approximate model of Eq. (2).
Fig. 7 shows a relatively small decrease in the threshold strains of plasticity incipience together with the eight-fold increase in the inclusion volume. We do not use MD data with larger inclusions to train the present ANN, but this additional dataset can be added in the future work.
Conclusions
We study the threshold of dislocation emission from nanosized copper inclusions in aluminum single crystal in comparison with the homogeneous nucleation of dislocations in pure metal. We consider different shapes of inclusions (spherical, cylindrical and cubic) and rather arbitrary axisymmetric deformations by means of MD simulations. For most deformation paths, the copper inclusions substantially reduce the threshold of plasticity incipience, while the inclusions have no effect for some deformation paths with either axial or transverse extension. Depending on the deformation path, the shape of inclusion
300 K k * \
- f . S^ 700K / V ► •. \
- J/ ! . / v n \ > V \ . / \ V / \ * \
i IT yf 1 1 1 y / 1
_ v. i ri s* 1
Influence of temperature
♦ MD pure Al, 300 K -ANN pure Al, 300 K
♦ MD pure Al, 700 K -ANN pure Al, 700 K
C MD spherical, 300 K - ANN spherical, 300 K
♦ MD spherical, 700 K - - - ANN spherical, 700 K
0.08
0.04
-0.04
-0.08
-0.12
-0.16
-0.08
0.08
E
ii
- SX ,<y f / « "V \
O " is * i \
u O' ! ' 6 A A I
r - ,/ // , A A A A • 9 • / ✓ / ✓
I i. / - v/- • m 1 ** 1 1
Strain rate effect for spherical Cu inclusions at 300 K
MD 10/ns ANN 10/ns MDl/ns
- ANN 1/ns
0.16
Fig. 6. The strain rate effect for the case of aluminum with spherical copper inclusions of 4 nm in diameter; the temperature is 300 K. Markers show MD data and the line shows the ANN-based interpolation. The strain rate sensitivity parameter of A = 2.5 eV is used in Eq. (2) to plot the ANN-based interpolation for the strain rate of 109 s-1
0.08
0.04
-0.04
-0.08
-0.12
- < / © t \
é à) ) e © \ <T> r i
w «fe A A © ©i ■
/ i i i i A A A m / / ✓
1 y ' S* ...A • s • i
Effect of inclusion size, spherical Cu inclusions in A1 at 300 K
Ml) 4 il m
ANN 4 nm Ml) 8 nm
-0.16
-0.08
0.08
0.16
E
ii
©
Fig. 7. The size effect for the case of aluminum with spherical copper inclusions at 300 K and 10 s
10 0-i
can either influence the emission threshold or not. Thus, there is a complex dependence of the threshold of plasticity incipience on the deformation path, the presence and the form of copper inclusions. This dependence is approximated by means of an ANN trained on the results of MD simulations. The trained ANN can be further applied as a constitutive equation at the level of continuum mechanics.
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Article received 12.11.2022.
Corrections received 11.12.2022.
Челябинский физико-математический журнал. 2023. Т. 8, вып. 2. С. 292-304.
УДК 539.3 Б01: 10.47475/2500-0101-2023-18212
ЗАРОЖДЕНИЕ ПЛАСТИЧНОСТИ В АЛЮМИНИИ С МЕДНЫМИ ВКЛЮЧЕНИЯМИ
А. Е. Майер
Челябинский государственный университет, Челябинск, Россия mayer@csu.ru
Дислокационная активность контролирует пластическую деформацию большинства металлических материалов. Механическое нагружение с высокими скоростями деформации или с большими градиентами деформации может привести как к гомогенному зарождению дислокаций, так и к эмиссии дислокаций из различных неоднород-ностей, таких как нанопоры и фазовые выделения. Зарождение и испускание дислокаций вызывают пластичность, релаксирующую сдвиговую составляющую напряжений. В данной работе исследуется порог испускания дислокаций из наноразмерных включений меди в монокристалле алюминия по сравнению с гомогенным зарождением дислокаций в чистом металле. Мы рассматриваем различные формы включений (сферические, цилиндрические и кубические) и довольно произвольные осесиммет-ричные деформации с помощью моделирования методом молекулярной динамики (МД). Для большинства траекторий деформации включения меди существенно снижают порог начала пластичности, тогда как для некоторых траекторий деформации с осевым или поперечным растяжением включения не влияют. В зависимости от пути деформации форма включения может как влиять на порог эмиссии, так и не влиять на него. Таким образом, существует сложная зависимость порога начала пластичности от пути деформации, наличия и формы медных включений. Эта зависимость аппроксимируется с помощью искусственной нейронной сети (ИНС), обученной на результатах МД-моделирования. Полученную ИНС можно в дальнейшем применять как определяющее уравнение на уровне механики сплошной среды.
Ключевые слова: алюминий, включения меди, зарождение пластичности, эмиссия дислокаций, молекулярная динамика, искусственная нейронная сеть.
Поступила в редакцию 12.11.2022. После переработки 11.12.2022.
Сведения об авторе
Майер Александр Евгеньевич, доктор физико-математических наук, доцент, заведующий кафедрой общей и теоретической физики, Челябинский государственный университет, Челябинск, Россия; mayer@csu.ru.
Работа поддержана Российским научным фондом, грант 20-11-20153.
