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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:

Основные уравнения: Плоская теория упру-

гости: Кручение и изгиб / Н.И. Мусхели-

швили. — М.: АН СССР, 1949.

5. Работнов Ю.Н. Ползучесть элементов кон-

струкций / Ю.Н. Работнов. — М.: Наука.

Гл. ред. физ.-мат. лит., 1966. — 752 с.

6. Савiн Г.М. Елементи механiки спадкових

середовищ / Г.М. Савiн, Я.Я. Рущицький.

— К.: Вища школа, 1976. — 252 с.

7. Wineman A. Nonlinear viscoelastic solids —

a review. / A. Wineman // Mathematics and

mechanics of solids. — 2009. — № 14 (3). — С.

300-366.

8. Zatula N.I. Stressed-strained state of a vi-

scous half-plane with circular inclusions / N.I.

Zatula, V.I. Lavrenyuk // International appli-

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-

se smooth contours / N.I. Zatula, D.V.

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

45