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Вiсник Київського нацiонального унiверситету
iменi Тараса Шевченка
Серiя: фiзико-математичнi науки
2022, 2
Bulletin of Taras Shevchenko
National University of Kyiv
Series: Physics &Mathematics
УДК 539.376 + 519.642.2
Н.I. Затула1,к.ф.-м.н., доц.
Д.В. Затула2,к.ф.-м.н.
Математичне моделювання
напруженого стану в’язкопружної
напiвплощини з включеннями
1Нацiональний авiацiйний унiверситет, 03058,
м. Київ, пр-т. Гузара Любомира 1,
e-mail: nelli.zatula@npp.nau.edu.ua
2Київський нацiональний унiверситет iменi
Тараса Шевченка, 83000, м. Київ, пр-т. Глу-
шкова 4д,
e-mail: dm_zatula@univ.kiev.ua
N. I. Zatula1,Ph.D., Associate Prof.
D. V. Zatula2,Ph.D.
Mathematical modeling of the stressed
state of a viscoelastic half-plane with
inclusions
1National Aviation University, 03058, Kyiv,
1 Liubomyra Huzara ave.,
e-mail: nelli.zatula@npp.nau.edu.ua
2Taras Shevchenko National University of Kyiv,
83000, Kyiv, 4d Glushkova str.,
e-mail: dm_zatula@univ.kiev.ua
Розглянуто застосування методу граничних iнтегральних рiвнянь при дослiдженнi напру-
женого стану плоских в’язкопружних тiл iз включеннями. Метод базується на використаннi
комплексних потенцiалiв та апарату узагальнених функцiй. Отримано аналiтичний розв’язок
задачi для напiвплощини iз довiльними за формою включеннями. Для чисельного дослiдження
змiни напруженого стану залежно вiд часу та геометрiї включень було розроблено дискретний
аналог системи гранично-часових iнтегральних рiвнянь.
Ключовi слова: в’язкопружнiсть, плоске в’язкопружне тiло, комплекснi потенцiали, метод
граничних iнтегральних рiвнянь, в’язкопружнi характеристики областей, резольвентнi опера-
тори.
The application of the method of boundary integral equations is considered for studying the stress
state of flat viscoelastic bodies with inclusions. The method is based on the use of complex potentials and
the apparatus of generalized functions. An analytical solution of the problem is obtained for a half-plane
with inclusions of arbitrary shape. For a numerical study of the change in the stress state depending
on the time and geometry of the inclusions, a discrete analogue of the system of boundary-time integral
equations has been developed.
Key Words: viscoelasticity, flat viscoelastic body, complex potentials, method of boundary integral
equations, viscoelastic characteristics of regions, resolvent operators.
Communicated by Prof. Moklyachuk M.P.
The method of boundary integral equations
is one of the most popular methods for solving
various problems in the theory of elasticity and
viscoelasticity. Using this method, one can quite
simply and accurately take into account infinitely
distant boundaries, reduce the dimension of the
problem by one, and also reveal the interactions
of contacting areas in a natural way.
Let us consider a viscoelastic half-plane
occupying a region Dand containing a finite
number of inclusions Dpof arbitrary shape (p=
1, n, where nis the number of inclusions). The
outer contour goes to infinity, and the inclusions
are bounded by piecewise-smooth contours. The
half-plane is subjected to distributed tangential
and normal loads [8].
To determine the stress state of the gi-
ven region, we use the statement of the second
main problem of the theory of elasticity for
inhomogeneous bodies in movements [3], as well
as the approach proposed for studying the stress
state of a plane viscoelastic piecewise-isotropic
body [8].
Let us assume that there are no mass forces,
and also that the unknown densities of potenti-
als are the stresses on the contours of inclusions.
Then, based on the properties of the generali-
zed functions, and due to Maxwell’s theorem, the
expressions for the components of the vector of
movements can be written as:
c
N.I. Zatula, D.V. Zatula, 2022
DOI: https://doi.org/10.17721/1812-5409.2022/2.5
42
Вiсник Київського нацiонального унiверситету
iменi Тараса Шевченка
Серiя: фiзико-математичнi науки
2022, 2
Bulletin of Taras Shevchenko
National University of Kyiv
Series: Physics &Mathematics
uk(x, t) = Γ0
gi(ξ, t)Ui
k(x, ξ, t)dγ(ξ)−
−Ep−E0
EpΓp
σ(p)
ij (ξ, t)nj(ξ)Ui
k(x, ξ, t)dγ(ξ),
(1)
where E0and Epare viscoelastic operators of
homogeneous regions belonging to the class of
resolvent operators [5, 6];
Ui
k(x, ξ, t) = ˙
Ui
k(x, ξ, t) +
Ui
k(x, ξ, t),
˙
Ui
k(x, ξ, t)is the fundamental solution for
problems of two-dimensional flat-strain state of
a viscoelastic infinite body, and
Ui
k(x, ξ, t)is an
additional term, which provides that the conditi-
on gi
k(x, ξ, t) = 0 is satisfied ∀x∈Γand ∀ξ∈D;
gi(ξ, t)— given densities along the contour Γ0;
σ(p)
ij (ξ, t)nj(ξ)— unknown densities of potentials
along the contours Γp.
Using the Cauchy relations, Hooke’s law, and
formulas (1), the stress tensor components can be
expressed in the following form:
σij (x, t) = 1 + Eq−E0
E0
SDq×
×Γ0
gk(ξ, t)Uk
ij (x, ξ, t)dγ(ξ)−
−Ep−E0
EpΓp
σ(p)
kl (ξ, t)nl(ξ)Uk
ij (x, ξ, t)dγ(ξ)
,
(2)
where Eqis viscoelastic operator of homogeneous
region belonging to the class of resolvent
operators; SDqis the characteristic function of
the region Dqand Uk
ij (x, ξ, t)are stresses arising in
a viscoelastic homogeneous body occupying region
Dunder the action of unit concentrated forces at
the point ξ∈D.
To determine the unknown densities, multiply
both sides of (2) by nj(x)and obtain the system
of boundary-time integral equations:
σ(q)
ij (x, t)nj(x) = 2Eq
Eq+E0
×
×Γ0
gk(ξ, t)Uk
ij (x, ξ, t)nj(x)dγ(ξ)−
−Ep−E0
Ep
×
×Γp
σ(p)
kl (ξ, t)nl(ξ)Uk
ij (x, ξ, t)nj(x)dγ(ξ)
.(3)
In order to apply the integral representati-
ons (1) and (2) for the semi-infinite region,
it is necessary to set additional conditions for
the functions used. These conditions concern the
behavior of functions at infinity and are defined as
regularity conditions [1]. In this case, the functi-
ons uk(x, t)and σij (x, t)behave at infinity as
fundamental solutions:
uk(x, t)∼Ui
k(x, ξ, t) = o(ln ρ+ 1), i =k,
o(1), i =k;
σij (x, t)∼Uk
ij (x, ξ, t) = oρ−1.
Note that the expressions for fundamental
movements and corresponding stresses for the case
of an elastic half-plane are given in [1].
Based on this, we write integral representati-
ons for the components of the movement vector
and the stress tensor:
uk(x, t) = +∞
−∞ +∞
0
Xi(ξ, t)Ui
k(x, ξ, t)dξ2dξ1−
−Ep−E0
EpDp
Xi(ξ, t)Ui
k(x, ξ, t)dS(ξ) +
++∞
−∞
gi(ξ1, t)Ui
k(x, ξ, t)dξ1−
−Ep−E0
EpΓp
σ(p)
ij (ξ, t)nj(ξ)Ui
k(x, ξ, t)dγ(ξ),
(4)
σij (x, t) = 1 + Eq−E0
Ep
SDq×
×+∞
−∞ +∞
0
Xk(ξ, t)Uk
ij (x, ξ, t)dξ2dξ1−
−Ep−E0
EpDp
Xk(ξ, t)Uk
ij (x, ξ, t)dS(ξ) +
++∞
−∞
gk(ξ1, t)Uk
ij (x, ξ, t)dξ1−
−Ep−E0
EpΓp
σ(p)
kl (ξ, t)nl(ξ)Uk
ij (x, ξ, t)dγ(ξ)
.
(5)
In the absence of mass forces, the expressi-
ons for movements (4) and stresses (5) take the
43
Вiсник Київського нацiонального унiверситету
iменi Тараса Шевченка
Серiя: фiзико-математичнi науки
2022, 2
Bulletin of Taras Shevchenko
National University of Kyiv
Series: Physics &Mathematics
following form:
uk(x, t) = +∞
−∞
gi(ξ1, t)Ui
k(x, ξ, t)dξ1−
−Ep−E0
EpΓp
σ(p)
ij (ξ, t)nj(ξ)Ui
k(x, ξ, t)dγ(ξ),
(6)
σij (x, t) = 1 + Eq−E0
E0
SDq×
×+∞
−∞
gk(ξ1, t)Uk
ij (x, ξ, t)dξ1−
−Ep−E0
EpΓp
σ(p)
kl (ξ, t)nl(ξ)Uk
ij (x, ξ, t)dγ(ξ)
.
(7)
Using relations (3), we write down the equati-
ons that allow us to determine the stresses on the
contours of the inclusions:
σ(q)
ij (x, t)nj(x) = 2Eq
Eq+E0
×
×+∞
−∞
gk(ξ1, t)Uk
ij (x, ξ, t)dξ1−
−Ep−E0
EpΓp
σ(p)
kl (ξ, t)nl(ξ)Uk
ij (x, ξ, t)dγ(ξ)·
·nj(x), x ∈Γq.(8)
For the numerical solution of the system (8),
the contours of the inclusions were discretized by
linear elements, which are characterized by the
coordinates of their midpoints.
In accordance with [2, 8], unknown densiti-
es of potentials for n-th boundary element of the
p-th inclusion were approximated using the functi-
on fx(p)
n, ξ, t. For inclusions bounded by contours
that do not contain angular points, the densi-
ty of potential along each boundary element was
assumed to be constant.
Moreover, as it turned out, the function
fx(p)
n, ξ, t, which approximates the stresses on
each boundary element, is equal to 1 and does not
depend on time tif the nodal point x(p)
nbelongs to
an ordinary boundary element, and fx(p)
n, ξ, t=
r
ds1(t)−1if the nodal point x(p)
nbelongs to an
angular boundary element, where dis the length
of this element, ai(i= 1,2) are the coordinates
of the angular point, r= [(ξi−ai) (ξi−ai)]1
2.
The unknown densities of potentials for n-th
boundary element of the p-th inclusion can be
represented as follows:
σ(p)
kl (ξ, t)nl(ξ) =
=Aklx(p)
n, tfx(p)
n, ξ, tnlx(p)
n,(9)
where Aklx(p)
n, tare unknown constants;
nlx(p)
n— components of the outward normal
vector to the contour of the inclusion at the point
x(p)
n.
Unknown constants Aklx(p)
n, tare determi-
ned from a system of linear algebraic equations
for a given t=ts:
Aij x(q)
n, tsfx(q)
n, ξ, tsnjx(q)
n=2Eq
Eq+E0
×
×M
m=0
gkξ∗
m, tξ′
m+1
ξ′
m
Uk
ij x(q)
n, ξ, tsdγ(ξ)−
−
N
p=1
Ep−E0
Ep
Mp
m=1
Aklx(p)
m, tsnlx(p)
m×
×ξ(p)
m+1
ξ(p)
m
fx(p)
m, ξ, tsUk
ij x(q)
n, ξ, tsdγ(ξ)
×
×njx(q)
n,(10)
where x(p)
nare coordinates of the n-th nodal
point of the p-th inclusion; Mpis the number
of boundary elements on the p-th inclusion;
ξ(p)
n, ξ(p)
n+1 — coordinates of the n-th boundary
element of the p-th inclusion; ξ′
m, ξ′
m+1 — coordi-
nates of the m-th discrete element belonging to
contour Γ;ξ∗
mis the middle of this element (m=
1, M ).
Integrals over segments ξ(p)
m, ξ(p)
m+1should be
understood in the sense of Cauchy principal value
[4].
Discrete analogs for the stress tensor
components have the form
σij (x, ts) = 1 + Eq−E0
E0
SDq×
×M
m=0
gkξ∗
m, tξ′
m+1
ξ′
m
Uk
ij (x, ξ, ts)dγ(ξ)−
−
N
p=1
Ep−E0
Ep
Mp
m=1
Aklx(p)
m, tsnlx(p)
m×
44
Вiсник Київського нацiонального унiверситету
iменi Тараса Шевченка
Серiя: фiзико-математичнi науки
2022, 2
Bulletin of Taras Shevchenko
National University of Kyiv
Series: Physics &Mathematics
×ξ(p)
m+1
ξ(p)
m
fx(p)
m, ξ, tsUk
ij (x, ξ, ts)dγ(ξ)
, x ∈D.
Thus, the constructed mathematical model
with its subsequent numerical implementation
makes it possible to determine the stress state of
a viscoelastic half-plane with arbitrary inclusions,
taking into account the rheological parameters of
materials at any given time.
Список використаних джерел
1. Бреббиа К. Методы граничных элементов /
К. Бреббиа, Ж. Теллес, Л. Вроубел. — М.:
Мир., 1987.
2. Kaminskii A.A. Investigation of the stress-
strain state of viscoelastic piecewise-
homogeneous bodies by the method of
boundary integral equations / A.A. Kami-
nskii, N.I. Zatula, V.N. Dyakon // Mechanics
of composite materials. — 2002. — № 38 (3).
— С. 209-214.
3. Ломакин В.А. Теория упругости неодноро-
дных тел / В.А. Ломакин. — М.: Изд. МГУ.,
1976. — 368 с.
4. Мусхелишвили Н.И. Некоторые основные
задачи математической теории упругости:
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гости: Кручение и изгиб / Н.И. Мусхели-
швили. — М.: АН СССР, 1949.
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scous half-plane with circular inclusions / N.I.
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ed mechanics. — 1995. — № 31 (9). — С. 754-
760.
9. Zatula N.I. Approximation of density of
potentials for the flat viscoelastic bodi-
es with inclusions, bounded by a piecewi-
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Zatula // Bulletin of Taras Shevchenko Nati-
onal University of Kyiv. Series: Physics and
Mathematics. — 2021. — № 1. — С. 39-42.
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DOI: 10.17721/1812-5409.2021/1.4
Received: 04.05.2022
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