Inverse structural states of the stochastic deformation field of fractal dislocation
New structural states of fractal dislocation are investigated on the basis of fractional calculation theory and Hamilton operators. In order to describe the behaviour of the stochastic deformation field of a fractal dislocation within the framework of the statistical approach, average complex functi...
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Донецький фізико-технічний інститут ім. О.О. Галкіна НАН України
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irk-123456789-696352014-10-18T03:01:31Z Inverse structural states of the stochastic deformation field of fractal dislocation Abramov, V.S. New structural states of fractal dislocation are investigated on the basis of fractional calculation theory and Hamilton operators. In order to describe the behaviour of the stochastic deformation field of a fractal dislocation within the framework of the statistical approach, average complex functions are introduced. Numerical modelling of the complex deformation field behaviour is fulfilled on a rectangular discrete lattice. It is shown that for inverse (with a negative fractal index) states of a fractal dislocation, there is an interval of change of this index with anomalous behaviour of the deformation field: there is no effective attenuation within the interval. The introduced functions allow to educe the presence of quantum and unusual statistical properties of the deformation field. Исследуются новые структурные состояния фрактальной дислокации на основе теории дробного исчисления и операторов Гамильтона. В рамках статистического подхода для описания поведения стохастического поля деформации фрактальной дислокации вводятся усредненные комплексные функции. Выполнено численное моделирование поведения комплексного поля деформации на прямоугольной дискретной решетке. Показано, что для инверсных (с отрицательным фрактальным индексом) структурных состояний фрактальной дислокации существует интервал изменения этого индекса с аномальным поведением поля деформации: внутри интервала отсутствует эффективное затухание. Введенные усредненные функции позволяют выявить наличие квантовых и необычных статистических свойств поля деформации. Досліджуються нові структурні стани фрактальної дислокації на основі теорії дробового обчислення й операторів Гамільтона. У рамках статистичного підходу для опису поведінки стохастичного поля деформації фрактальної дислокації вводяться усереднені комплексні функції. Виконано чисельне моделювання поведінки комплексного поля деформації на прямокутній дискретній решітці. Показано, що для інверсних (з негативним фрактальним індексом) структурних станів фрактальної дислокації існує інтервал зміни цього індексу з аномальною поведінкою поля деформації: всередині інтервалу відсутнє ефективне затухання. Введені усереднені функції дозволяють виявити наявність квантових і незвичайних статистичних властивостей поля деформації. 2013 Article Inverse structural states of the stochastic deformation field of fractal dislocation / V.S. Abramov // Физика и техника высоких давлений. — 2013. — Т. 23, № 3. — С. 54-62. — Бібліогр.: 17 назв. — англ. 0868-5924 PACS: 81.40.Vw, 05.45.Df, 61.43.Hv, 62.20.F−, 61.72.Ff http://dspace.nbuv.gov.ua/handle/123456789/69635 en Физика и техника высоких давлений Донецький фізико-технічний інститут ім. О.О. Галкіна НАН України |
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| description |
New structural states of fractal dislocation are investigated on the basis of fractional calculation theory and Hamilton operators. In order to describe the behaviour of the stochastic deformation field of a fractal dislocation within the framework of the statistical approach, average complex functions are introduced. Numerical modelling of the complex deformation field behaviour is fulfilled on a rectangular discrete lattice. It is shown that for inverse (with a negative fractal index) states of a fractal dislocation, there is an interval of change of this index with anomalous behaviour of the deformation field: there is no effective attenuation within the interval. The introduced functions allow to educe the presence of quantum and unusual statistical properties of the deformation field. |
| format |
Article |
| author |
Abramov, V.S. |
| spellingShingle |
Abramov, V.S. Inverse structural states of the stochastic deformation field of fractal dislocation Физика и техника высоких давлений |
| author_facet |
Abramov, V.S. |
| author_sort |
Abramov, V.S. |
| title |
Inverse structural states of the stochastic deformation field of fractal dislocation |
| title_short |
Inverse structural states of the stochastic deformation field of fractal dislocation |
| title_full |
Inverse structural states of the stochastic deformation field of fractal dislocation |
| title_fullStr |
Inverse structural states of the stochastic deformation field of fractal dislocation |
| title_full_unstemmed |
Inverse structural states of the stochastic deformation field of fractal dislocation |
| title_sort |
inverse structural states of the stochastic deformation field of fractal dislocation |
| publisher |
Донецький фізико-технічний інститут ім. О.О. Галкіна НАН України |
| publishDate |
2013 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/69635 |
| citation_txt |
Inverse structural states of the stochastic deformation field of fractal dislocation / V.S. Abramov // Физика и техника высоких давлений. — 2013. — Т. 23, № 3. — С. 54-62. — Бібліогр.: 17 назв. — англ. |
| series |
Физика и техника высоких давлений |
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AT abramovvs inversestructuralstatesofthestochasticdeformationfieldoffractaldislocation |
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2025-07-05T19:07:54Z |
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1836835117246447616 |
| fulltext |
Физика и техника высоких давлений 2013, том 23, № 3
© V.S. Abramov, 2013
PACS: 81.40.Vw, 05.45.Df, 61.43.Hv, 62.20.F−, 61.72.Ff
V.S. Abramov
INVERSE STRUCTURAL STATES OF THE STOCHASTIC
DEFORMATION FIELD OF FRACTAL DISLOCATION
Donetsk Institute for Physics and Engineering named after A.A. Galkin,
National Academy of Sciences of Ukraine
72 R. Luxemburg St., Donetsk, 83114, Ukraine
Received April 4, 2012
Исследуются новые структурные состояния фрактальной дислокации на основе
теории дробного исчисления и операторов Гамильтона. В рамках статистическо-
го подхода для описания поведения стохастического поля деформации фракталь-
ной дислокации вводятся усредненные комплексные функции. Выполнено численное
моделирование поведения комплексного поля деформации на прямоугольной дис-
кретной решетке. Показано, что для инверсных (с отрицательным фрактальным
индексом) структурных состояний фрактальной дислокации существует интер-
вал изменения этого индекса с аномальным поведением поля деформации: внутри
интервала отсутствует эффективное затухание. Введенные усредненные функ-
ции позволяют выявить наличие квантовых и необычных статистических свойств
поля деформации.
Ключевые слова: фрактальная дислокация, стохастическое поле деформации, чис-
ленное моделирование, статистические свойства, инверсные структурные состояния
Досліджуються нові структурні стани фрактальної дислокації на основі теорії
дробового обчислення й операторів Гамільтона. У рамках статистичного підходу
для опису поведінки стохастичного поля деформації фрактальної дислокації вво-
дяться усереднені комплексні функції. Виконано чисельне моделювання поведінки
комплексного поля деформації на прямокутній дискретній решітці. Показано, що
для інверсних (з негативним фрактальним індексом) структурних станів фрак-
тальної дислокації існує інтервал зміни цього індексу з аномальною поведінкою по-
ля деформації: всередині інтервалу відсутнє ефективне затухання. Введені усеред-
нені функції дозволяють виявити наявність квантових і незвичайних статистич-
них властивостей поля деформації.
Ключові слова: фрактальна дислокація, стохастичне поле деформації, чисельне
моделювання, статистичні властивості, інверсні структурні стани
Физика и техника высоких давлений 2013, том 23, № 3
55
1. Introduction
Last time many actively try to explain different experimental anomalous
properties of a physical object on the basis of the definition of the fractal [1].
An adequate description of anomalous behavior of physical parameters near
phase transitions, real structure of the lattice in real samples of magnets, fer-
roelectrics, high-temperature superconductors, amorphous alloys [2] requires a
model of a nonlinear lattice with the spontaneous deformation to be further de-
veloped on the basis of qualitatively new representations of the nature of frac-
tal and stochastic properties of the lattice. Among the real nanomaterials, ac-
tive nanostructural elements are clusters, pore, quantum dots, wells, corrals,
surface superlattices (see [3,4]). Active nanostructural elements can find their
application in quantum nanoelectronics [5], quantum informations, quantum
optics [6]. The fractal dislocation [7,8] is one of non-classical active nanos-
tructural objects. For the theoretical descriptions of fractal objects, the theory
of fractional calculations [12] has been proposed [9−11]. The fractal string
model has been proposed. The equations with fractional space-time derivatives
have been introduced in order to describe plastic subsystem of a fractal string.
To solve the basic dynamic equation in fractional derivatives, two approaches
have been suggested: reduction to a system of equations and the use of compo-
sition formulas for fractional derivative operators. The obtained results have
been generalized to the solution of the Cauchy problem in the matrix form.
The fractal string model was used in order to construct a model of a fractal
dislocation [13−16]. The inverse structural states of fractal dislocation are in-
vestigated in this paper.
2. The model and simulations
Plasticity of materials is determined by the motion of an ensemble of disloca-
tions. In article [13] the dynamic equations for the anisotropic plastic subsystem
of a fractal medium are obtained explicitly on the basis of the fractional calculus
model. For the special case of isotropic fractal medium [9−11], the original equa-
tion for the fractal string is
( ) ( )t t z zD D D Dν ν α α
ν α α αρ Φ = μ Φ , (1)
where the function ( , )t zαΦ depends only on the dimensionless variables of time t
and coordinates z; ν, α are fractal indices of partial fractional derivatives of Rie-
mann−Liouville tDν , zDα on t, z, respectively; ρν and μα are dimensionless effect-
ive parameters that are associated with the mass density and power characteristics
of a fractal medium. Fractal indices ν, α have the meaning of fractal dimension
along the axes Ot, Oz. If ρν(z) and μα(t) are functions of z and t, respectively, then
(1) reduces to a system of equations for an unknown function ( , )t zαΦ with
varying eigenvalues λα(t), λν(z):
Физика и техника высоких давлений 2013, том 23, № 3
56
2 0t t z zD D w D Dν ν α α
α αΦ − Φ = , 2 ( , ) ( ) / ( )w t z t zα ν= μ ρ , (2)
( )zD tα
α α αΦ = λ Φ , ( )tD zν
α ν αΦ = λ Φ , 2 2 2( ) ( )z w tν αλ = λ . (3)
We will choose zDα , tDν , zIα in the following form
( , ) d / (1 )
c
z
z z z
D t z
−α
α
α αΦ = ∂ Φ ξ −ξ ξ Γ −α∫ , (4)
( , ) d / (1 )
a
t
z t t
D z t
−ν
ν
α αΦ = ∂ Φ τ − τ τ Γ −ν∫ , (5)
1
( , ) d / ( )
c
z
z z
I t z
α−
α
α αΦ = Φ ξ −ξ ξ Γ α∫ , (6)
where ∂z, ∂t are the operators of the usual partial derivatives; Γ is the gamma
function. When z > ξ, the operator zDα coincides with the operator of the left-
sided
czDα
+ and when z < ξ it coincides with the operator of the right-sided
czDα
−
that are fractional derivatives of Riemann−Liouville [12]. Using these operators al-
lows to describe the behavior of the function Φα when passing through the value z = zc.
Next, we find the functions Φα as solutions of equations (3) for the Cauchy-
type problem [11] in two representations
( ) 1
1 ,, ( )ah t z t t Eν−
α α ν ν νΦ = − ψ , ( ) az t t ν
ν νψ = λ − , (7)
( ) 1
2 ,, ( )c ch t z z z Eα−
α α α αΦ = − ψ , ( ) ct z z α
α αψ = λ − , (8)
1
1( , ) th t z I −ν
α= Φ , 1
2 ( , ) zh t z I −α
α= Φ , (9)
where , ( )Eν ν νψ , , ( )Eα α αψ are the Mittag−Leffler functions [12].
To find the eigenvalues λα (or λν), we use the Hamilton operator Ĥ of
[8,14,15] for the energy spectrum of the fractal dislocation
0
ˆ ˆ ˆ
bH H H= + , ( )0 2 1 2 3 3
ˆ ˆ ˆ ˆH n n n= ε + + ε , 3 0 3̂
ˆ
bH n b= ε , (10)
1
3̂ ˆ(1 ) ,z zb I D zα −α⎡ ⎤= −α = ⎣ ⎦ , ( )2
0 1 2 ,n sn u kα= − , 0u u uα = − . (11)
Here k, u − u0 are the module and the argument of the elliptic sine; 1̂n , 2n̂ , 3n̂ are
operators of occupation numbers of states of the dislocation with the dimension-
less self-energy ε1 = ε2, ε3; ẑ is the coordinate operator. Variable u is the dimen-
sionless displacement of the deformation field, which is related to the dimension-
less stress field λα on Hooke’s law u = λα/λ0, where λ0 is the force parameter.
Note that the operators (10), (11) allow to describe inverse (α < 0) states. This
is due to the fact that in the theory of fractional calculus [12] for the values α ∈
Физика и техника высоких давлений 2013, том 23, № 3
57
[−1, 0], it is possible to have a transition from operators zDα , zIα to operators
( )
z zD I− −α −α= , ( )
z zI D− −α −α= . In regards to such a transition, the commutator from
(11) retains its form. In this case, the old physical interpretation of the fractal di-
mension is preserved for the module |α|. Operators 0Ĥ , ˆ
bH do not depend ex-
plicitly on time.
Let us consider the state of fractal dislocations at t = tc. The function describing
this state Φ(z) = ΦεΦαc(z) depends only on z. Here Φε is the eigenfunction of the
operator 0Ĥ in the diagonal representation of the population numbers n1, n2, n3;
Φαc(z) = Φα(tc, z) is a function obtained from (8) at t = tc. Further, we find the to-
tal energy E of fractal dislocation
Ĥ EΦ = Φ , 3̂ ( ) ( )c c cb z zα α αΦ = χ Φ , (12)
3 0 ;cE E nε α= + ε χ ( )2 1 2 3 3.E n n nε = ε + + ε (13)
From (13), (2), (12) we receive the system of the nonlinear equations for g1, λac
( ) ( ) ( )1 3 01 / 1 /cg n E Eα ε= −α χ = −α ε − , (14)
( )2 2 2 2
1( ) / ,c cz w t z gα νλ = λ = , (15)
( ) ( )1 , ,2/c c cg z z E E−α
α α α α α ε= − ψ ψ , c c cz z α
α αψ = λ − . (16)
Modeling the behavior of the deformation field is made on a rectangular lattice
with discrete sizes. N1 × N2 Deviation of the sites of this lattice from the state
with u = 0 is described by the operator of displacement û . This operator is as-
signed to a rectangular matrix with the elements unm, where 11,n N= , 21,m N= .
Four nonlinear model equations for u(z, a) = λαc/λ0 = uεi(z,α) (i = 1, 2, 3, 4) are
obtained from (16)
1 1 2 42u g g gε = − + , 2 1 2 42u g g gε = − − , (17)
3 1 2 52u g g gε = − − + , 4 1 2 52u g g gε = − − − , (18)
( )
1/ 22
4 2 1 3g g g g⎡ ⎤= + −
⎣ ⎦
, ( )
1/ 22
5 2 1 3g g g g⎡ ⎤= + −
⎣ ⎦
. (19)
These elements unm are found by solving nonlinear equations (17), (18) by the it-
eration method. An iterative procedure itself simulates a stochastic process on a
rectangular discrete lattice. In general case, elements unm(z,α) are random com-
plex functions of two real variables z and α, and they also depend on a number of
other internal and external governing parameters. Functions g1, g2, g3 for numeri-
cal simulation are in the form
( ) ( ) ( )( ) ( )2
1 0 0 1 2 3, 1 1 2 , /g u sn u u k p p n p m p jα = −α − − − − − , (20)
Физика и техника высоких давлений 2013, том 23, № 3
58
( ) ( )2 3 1/ 2
2 , 2 3 / Г 1/ 2cg z z z −α− α α−
αα = − γ π α + , (21)
( ) 23 1/ 2
3 , 3 2 /cg z z z − αα−
αα = − γ π , ( ) ( )Г 1/ 3 Г 2 / 3αγ = α + α + , (22)
where zc is the critical value of the dimensionless coordinate z; p0, p1, p2, p3 are the
governing parameters. A distinctive feature of the behavior of the displacement field
of fractal dislocation for inverse states is the existence of an interval of fractal dimen-
sion, where the imaginary parts uεi(z,α) approach zero, which indicates the absence
of effective damping. Within this interval, there are singular points (attractors [17])
with the values α = −1/3 and α = −2/3. When going through these singular points, the
displacement field shows a different behavior. As an example, we give the depend-
ency of the function uε1 from the lattice nodes indices n, m for values α = −0.5, zc =
= 2.7531, z = 1.753 (Fig. 1), received by iterations method by an index m. In the
modeling we have assumed that: k = 0.5, j = 1, u0 = 29.537, p0 = 0.01, p2 = p3 = 0; the
initial condition is u1,1 = 0, 1;30n = , 1;40m = . Filling in the matrix was carried out
by rows. Changing the governing parameter results in different states of fractal dislo-
cation with stochastic behavior of the deformation field (Fig. 1).
a b
c d
Fig. 1. The behavior of the function uε1 depending on n, m for the fractal dislocation
Физика и техника высоких давлений 2013, том 23, № 3
59
On Fig. 1,a, the dislocation is located near the lower boundary (n = 1). The
localization region of the dislocation is limited to the value n < 13. When n >
13 the surface uε1(n, m) is close to a smooth one. Plane singularities of the
dislocation are located at nc = p0/p1 = 0.06579 and go out of the lower bound-
ary. Decrease in the parameter p1 (Fig. 1,b,c) results in an increase in value nc,
which is accompanied by a shift of fractal dislocation parallel to the axis m. In
this case, a regular deviation of the rectangular lattice nodes with uε1 ≠ 0 starts
to appear at the lower boundary, and the range of the randomization applies to
all other nodes in the lattice (Fig. 1,b). For the lattice nodes with n < 13, the
regular behavior is characteristic, and for n > 13, the stochastic behavior is ob-
served (Fig. 1,c). In this case there are deviations at the boundary sites (n = 1)
of the damping amplitude wave type. With further decrease in p1, the plane of
the singular points of the dislocation comes out of the specified region of the
lattice (Fig. 1,d). Moreover, for all the lattice sites, the regular behavior with
uε1 ≠ 0 is characteristic. The presence of the sites, with the deviations type
wave with damping amplitude along the axis m is clearly expressed at the
lower boundary.
The analysis of the behavior of the deformation field in the various nodal
planes z = zj is convenient to be made in terms of averaged functions Mi with the
operator of the density of states ρ̂ and matrix elements pnm
( ) ( )ˆ ˆ,i iM z Sp uεα = ρ ,
2 1 2 1
ˆ ˆˆ /T
N N N Nρ = ξ ξ ,
2 1
1 1
1
N N
mn
m n= =
ρ =∑∑ , (23)
where Sp, T are the operations of calculating the trace of square matrix, transposi-
tion;
1
ˆ
Nξ ,
2Nξ are the row-vectors of dimension 1 × N1, 1 × N2, respectively, and
the elements equal to unity. The averaging is performed over the nodes of a discrete
rectangular lattice (n, m), and in directions perpendicular to the plane of the lattice,
the averaging is absent. This makes it possible to obtain the dependency of the av-
eraged functions from z and identify their clearly expressed stochastic behavior for
all four branches of the dimensionless displacement function uεi (Fig. 2). The pre-
sence of the step-type behavior of a lattice is closely connected with the occurrence
of quantum properties and quantum chaos [3] at average functions of fractal dislo-
cation. In modeling, we assumed that z = zj = 0.053 + 0.1 (j − 1), where 1;67j = .
Values z were varied in the interval [0.053; 6.653] with the step zh = 0.1.
When z = z1 the values M3, M2 (Fig. 2,a) and M1, M4 (Fig. 2,c) are zero. Near
z = z28 the features of the global local minima and maxima type with a nonzero
gap between the curves 2, 4 (Fig. 2,d) and 1, 4 (Fig. 2,f) are observed. The effect
of mixing-up curves is observed when changing z. The behavior of the functions
M (of soft mode type) at z = z1 and near z = z28 agrees with the behavior of fractal
dislocation displacements (Fig. 1).
Физика и техника высоких давлений 2013, том 23, № 3
60
Физика и техника высоких давлений 2013, том 23, № 3
61
3. Conclusions
A model of fractal dislocation is constructed on the basis of the coupled system
of the following equations: the dynamics for fractal strings with the operators of
fractional derivatives, the Hamiltonian operator for the energy spectrum of fractal
dislocation and Hooke’s law, which describes the connection between stress and
strain of a fractal dislocation. Within the framework of this model, the simulation
of the deformation field of dislocation has been executed. For the inverse of the
structural states of the fractal dislocation, the soft-mode type behavior is observed.
The stochastic behavior, the change of the states of the dislocation, the absence of
the effective damping and unusual quantum properties are observed near singular
points (attractors).
Some material of this paper was reported at the 4-th Chaotic Modeling and
Simulation International Conference (CHAOS 2011), May 31−June 3, 2011, Ag-
ios Nikolaos, Crete, Greece.
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(2010) (in Russian).
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Физика и техника высоких давлений 2013, том 23, № 3
62
V.S. Abramov
INVERSE STRUCTURAL STATES OF THE STOCHASTIC
DEFORMATION FIELD OF FRACTAL DISLOCATION
New structural states of fractal dislocation are investigated on the basis of fractional
calculation theory and Hamilton operators. In order to describe the behaviour of the sto-
chastic deformation field of a fractal dislocation within the framework of the statistical
approach, average complex functions are introduced. Numerical modelling of the com-
plex deformation field behaviour is fulfilled on a rectangular discrete lattice. It is shown
that for inverse (with a negative fractal index) states of a fractal dislocation, there is an
interval of change of this index with anomalous behaviour of the deformation field: there
is no effective attenuation within the interval. The introduced functions allow to educe the
presence of quantum and unusual statistical properties of the deformation field.
Keywords: fractal dislocation, stochastic deformation field, numerical modeling, statisti-
cal properties, inverse structural states
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