Математическое моделирование зарождения трещины в композите при изгибе
Проведено математическое описание модели зарождения трещины в связующем композита при изгибе. Определение неизвестных параметров, характеризующих зародышевую трещину, сводится к решению сингулярного интегрального уравнения. Построена замкнутая система нелинейных алгебраических уравнений, решение кот...
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Інстиут проблем машинобудування ім. А.М. Підгорного НАН України
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nasplib_isofts_kiev_ua-123456789-1419042025-02-09T20:31:48Z Математическое моделирование зарождения трещины в композите при изгибе Математичне моделювання зародження тріщини в композиті при вигині Modeling Crack Initiation in a Composite Under Bending Гасанов, Ш.Г. Динаміка та міцність машин Проведено математическое описание модели зарождения трещины в связующем композита при изгибе. Определение неизвестных параметров, характеризующих зародышевую трещину, сводится к решению сингулярного интегрального уравнения. Построена замкнутая система нелинейных алгебраических уравнений, решение которой позволяет прогнозировать трещинообразование в композите при изгибе в зависимости от геометрических и механических характеристик связующего и включений. Сформулирован критерий зарождения трещины в композите при действии изгибающих нагрузок. Відомо, що багатокомпонентні структури більш надійні та довговічні, ніж однорідні. На етапі проектування нових конструкцій з композиційних матеріалів необхідно враховувати випадки, коли у матеріалі можуть з'явитися тріщини. Метою цього дослідження є побудова розрахункової моделі для композитного тіла, що включає зв'язування, це дає змогу розрахувати граничні зовнішні згинальні навантаження, за яких відбувається розтріскування в композиті. Розглянуто тонку пластину із пружного ізотропного середовища (матриці) та розподілених в ній включень (волокон) з іншого пружного матеріалу в плиті під час згинання. Проведено математичний опис моделі зародження тріщини у зв'язувальному композиті під час згинання. Використовується теорія аналітичних функцій та метод степеневих рядів. Визначення невідомих параметрів, що характеризують зародкову тріщину, зводиться до розв’язання сингулярного інтегрального рівняння. Побудовано замкнуту систему нелінійних алгебраїчних рівнянь, розв'язок якої дозволяє прогнозувати тріщиноутворення в композиті під час згинання залежно від геометричних та механічних характеристик з’єднувального та включень. Сформульовано критерій зародження тріщини в композиті під впливом згинних навантажень. Розмір обмежувальної мінімальної зони попередньої фракції, за якої відбувається зародження тріщини, рекомендується розглядати як конструктивну характеристику з’єднувального матеріалу. It is known that multi-component structures are more reliable and durable than homogeneous ones. At the design stage of new structures from composite materials, it is necessary to take into account the cases when cracks may appear in the material. The purpose of this paper is to construct a computational model for a binder-inclusion composite body, which makes it possible to calculate the limiting external bending loads at which cracking occurs in a composite. A thin plate of elastic isotropic medium (matrix) and inclusions (fibers) from other elastic material, distributed in the plate under bending, is considered. A mathematical description of a crack initiation model in a binder composite under bending is carried out. The theory of analytic functions and the method of power series are used. The determination of the unknown parameters characterizing an initial crack reduces to solving a singular integral equation. A closed system of nonlinear algebraic equations is constructed, whose solution helps to predict cracks in a composite under bending, depending on the geometric and mechanical characteristics of both the binder and the inclusions. The criterion of crack initiation in a composite under the influence of bending loads is formulated. The size of the limiting minimum pre-fraction zone, at which crack initiation occurs is recommended to be considered as a design characteristic of a binder material. 2018 Article Математическое моделирование зарождения трещины в композите при изгибе / Ш.Г. Гасанов // Проблеми машинобудування. — 2018. — Т. 21, № 2. — С. 25-31. — Бібліогр.: 28 назв. — рос., англ. 0131-2928 https://nasplib.isofts.kiev.ua/handle/123456789/141904 539.375 ru Проблеми машинобудування application/pdf application/pdf Інстиут проблем машинобудування ім. А.М. Підгорного НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
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| language |
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| topic |
Динаміка та міцність машин Динаміка та міцність машин |
| spellingShingle |
Динаміка та міцність машин Динаміка та міцність машин Гасанов, Ш.Г. Математическое моделирование зарождения трещины в композите при изгибе Проблеми машинобудування |
| description |
Проведено математическое описание модели зарождения трещины в связующем композита при изгибе. Определение неизвестных параметров, характеризующих зародышевую трещину, сводится к решению сингулярного интегрального уравнения. Построена замкнутая система нелинейных алгебраических уравнений, решение которой позволяет прогнозировать трещинообразование в композите при изгибе в зависимости от геометрических и механических характеристик связующего и включений. Сформулирован критерий зарождения трещины в композите при действии изгибающих нагрузок. |
| format |
Article |
| author |
Гасанов, Ш.Г. |
| author_facet |
Гасанов, Ш.Г. |
| author_sort |
Гасанов, Ш.Г. |
| title |
Математическое моделирование зарождения трещины в композите при изгибе |
| title_short |
Математическое моделирование зарождения трещины в композите при изгибе |
| title_full |
Математическое моделирование зарождения трещины в композите при изгибе |
| title_fullStr |
Математическое моделирование зарождения трещины в композите при изгибе |
| title_full_unstemmed |
Математическое моделирование зарождения трещины в композите при изгибе |
| title_sort |
математическое моделирование зарождения трещины в композите при изгибе |
| publisher |
Інстиут проблем машинобудування ім. А.М. Підгорного НАН України |
| publishDate |
2018 |
| topic_facet |
Динаміка та міцність машин |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/141904 |
| citation_txt |
Математическое моделирование зарождения трещины в композите при изгибе / Ш.Г. Гасанов // Проблеми машинобудування. — 2018. — Т. 21, № 2. — С. 25-31. — Бібліогр.: 28 назв. — рос., англ. |
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Проблеми машинобудування |
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2025-11-30T12:56:22Z |
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2025-11-30T12:56:22Z |
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| fulltext |
DYNAMICS AND STRENGTH OF MACHINES
ISSN 0131–2928. Journal of Mechanical Engineering, 2018, Vol. 21, No. 2 25
Sh. G. Hasanov, D. Sc.
(Engineering)
Azerbaijan Technical University,
Baku, Azerbaijan
e-mail: hssh3883@gmail.com
UDC 539.375
MODELING CRACK INITIATION IN A
COMPOSITE UNDER BENDING
A mathematical description of a crack initiation model in a binder composite un-
der bending is carried out. The determination of the unknown parameters charac-
terizing an initial crack reduces to solving a singular integral equation. A closed
system of nonlinear algebraic equations is constructed, whose solution helps to
predict cracks in a composite under bending, depending on the geometric and
mechanical characteristics of both the binder and the inclusions. The criterion of
crack initiation in a composite under the influence of bending loads is formulated.
Keywords: binder, inclusion,
composite plate, bending, pre-fracture
zone, crack formation.
Introduction
The creation of new materials of high strength, rigidity and reliability opens up great opportunities
for their wide application in various fields of engineering and construction. These include, in particular, fi-
brous composite materials. Solving various engineering problems requires objective information about the
stress-strain state in the structural elements made from composite materials. Such information can only be
obtained by taking into account the main features of these materials. At the design stage of new structures
made from composite materials, it is necessary to take into account the cases when cracks may appear in the
material. In this regard, it is necessary to carry out a limit analysis in order to establish that the expected ini-
tial damage will not grow to critical dimensions and will not cause damage during the design life.
At present, one of the main places in the mechanics of composite materials is occupied by the problems
related to their structural peculiarities. One of the main features of composite materials, which is important to
take into account while studying various kinds of problems in the mechanics of composite bodies, are damages
to their structures. These damages can be caused by the construction of composite materials themselves, and
can result from various factors in technological processes. In practice, successful application of artificially cre-
ated composite materials is largely connected with the solution of the problems of determining their stress-
strain state, taking into account the their structural features, in particular, damages in both the binder and rein-
forcing elements. Therefore, studies of the stress-strain state in damaged composite materials should be recog-
nized as very relevant. A large number of papers [1–22 and others] have been devoted to the problems of the
stress-strained state and destruction of a fiber composite. It is important to develop a mathematical model that
allows predicting the stress-strain state of a composite at the pre-fracture stage (the formation of cracks).
The purpose of this paper is to construct a computational model for a binder-inclusion composite body,
which makes it possible to calculate the limiting external bending loads at which cracking occurs in a composite.
Problem formulation
Let the unbounded composite plate (composite) be subjected to bending by the mean moments
(bending at infinity) xx MM , yy MM , 0xyH . When a composite is loaded in a binder material, a
pre-fracture zone will appear, oriented in the direction of the maximum tensile stresses. A pre-fracture zone
is modeled as a region of weakened inter-particle bonding of the material. It is believed that when a compo-
site is loaded in it (a layer of over-stressed material), a zone of plastic flow is formed. The studies [23–25] of
the formation of regions with a disturbed structure of the material indicate that at the initial loading stage
pre-fracture zones represent a narrow elongated layer, and then, with an increase in the external load, a sec-
ondary system of zones of weakened inter-particle bonding of the material suddenly appears.
Let, for definiteness, the external bending load change so that plastic deformation occurs in the zone
of weakened inter-particle bonds of the binder material. After a certain number of loading cycles, the possi-
bility of plastic deformation in the zone of weakened inter-particle bonds of the material is exhausted and the
opening of plastic flow zone faces sharply increases. If the opening of the pre-fracture zone faces at the point
of maximum concentration reaches the limit value δc for a given binder material, then a crack (a break in the
material inter-particle bonding) arises at this point.
Sh. G. Hasanov, 2018
mailto:hssh3883@gmail.com
ДИНАМІКА ТА МІЦНІСТЬ МАШИН
ISSN 0131–2928. Проблеми машинобудування, 2018, Т. 21, № 2 26
In the process of a composite being subjected to bending moments, a pre-fracture zone will occur in the
binder material. For a mathematical description of the interaction of pre-fracture zone faces, it is accepted that in
this zone there are bonds between the faces, that restrict the opening of the weakened inter-particle bond zone fac-
es of the material. The interaction of the pre-fracture zone faces is modeled by inserting plastic slip lines (degener-
ate plastic deformation bands) between its faces. The position and dimensions of the plastic flow zones depend on
the type of material and loading. It is believed that in a pre-fracture zone there is plastic flow at constant voltage.
In this case, the location and size of the pre-fracture zone are not known in advance and should be determined in
the process of solving the problem. The interaction of the binder with the fibers is analyzed on the basis of a sin-
gle-fiber model. The remaining fibers get 'smeared', and the material outside the isolated fiber appears to
be homogeneous and isotropic with the corresponding effective
elastic constants (by the 'mixture' rule). The interaction between
other smeared fibers and pre-fracture zones is carried out through
the corresponding effective elastic constants. At the same time,
there are no restrictions on the relative position and relative sizes
of the fibers and pre-fracture zones. It is believed that pre-fracture
zones do not intersect each other and the fiber.
The origin of the Oxy coordinate system is compatible
with the geometric center of a fiber (Fig. 1) in the middle plane of
the composite plate. It is assumed that an elastic fiber from other
material is inserted into the circular hole of the binder.
It is believed that all over the joining region boundary L
(τ=Rexp(iθ)) there is a rigid adherence of various materials. In the
center of a rectilinear pre-fracture zone, we place the origin of the
local coordinate system O1x1y1, whose x1 axis coincides with the
pre-fracture zone line and forms the angle α1 by the axis x.
Fig. 1. Design diagram of cracking in a
composite binder under bending
On the media separation boundary, the following conditions must be fulfilled
0ww ,
n
w
n
w
0 ,
t
w
t
w
0 ,
2
0
2
2
2
n
w
n
w
,
2
0
2
2
2
t
w
t
w
,
tn
w
tn
w
0
22
, (1)
where w and w0 are the deflections of a binder and fibers, respectively, n and t are natural coordinates (nor-
mal and tangent to the boundary L).
These relations (1) are a consequence of the continuity of composite deflections, angles of tangent
inclination and values of bending moments.
When a composite is subjected to external bending moments in the bonds connecting the pre-fracture
zone faces, both normal sy
1
and tangential syx
11
stresses will arise. Here σs is the tensile yield
stress of a material, τs is the shear yield stress of a material.
The boundary conditions on pre-fracture zone faces will have the following form:
Sn MM , s
nt
n H
t
H
N
,
where Ms=σsh
2/4; Hs=τsh
2/4; h is the thickness of a composite plate (composite); Mn, Hn are the specific
bending and twisting moments; Nn is the specific transverse force.
To determine the values of the external bending load, at which cracking takes place in the binder, it
is necessary to supplement the formulation of the problem with a crack appearance criterion (rupture of the
material inter-particle bonds). As a condition, we take the criterion of the maximum opening of the weak-
ened inter-particle bond zone faces of the material
cuuivv
1111 , (2)
DYNAMICS AND STRENGTH OF MACHINES
ISSN 0131–2928. Journal of Mechanical Engineering, 2018, Vol. 21, No. 2 27
where δc is the resistance characteristic of the binder material to crack formation; 11 vv is the normal
component of the opening of a pre-fracture zone faces; 11 uu is the tangential component of the opening
of a pre-fracture zone faces.
This additional condition makes it possible to specify the parameters of the composite plate (compo-
site) at which a crack originates in the binder.
Solution method
Let in the binder there be one rectilinear zone of weakened inter-particle bonds in the state of plastic
flow at a constant voltage (Fig. 1). The moments Mx, My, Hxy, transverse forces Nx, Ny and deflection w in the
technical theory of plate bending can be represented using the Kolosov-Muskhelishvili complex potentials
[26]. On the media separation boundary
)()()()()()( 000 , (3)
)()()(
)1(
)1(
)()()( 0000
00
*
n
D
D
n . (4)
Here, φ(z), ψ(z) and φ0(z), ψ0(z) are the complex potentials for the binder and fiber, respectively; τ is
a variable point on the media separation boundary, τ=exp(iθ) n*= –(3+ν)/(1–ν); D and D0 are the cylindrical
rigidity of the binder and fiber, respectively; and 0 are the Poisson coefficients of the binder and fiber
material; n0= –(3+ν0)/(1–ν0).
On the rectilinear pre-fracture zone faces we have the condition
1
0
111111* )()()()( iCfxxxxxn , (5)
where; ss iHMf 0
1 ; x1 is the affix of the pre-fracture zone points; C1 is the real constant determined in the
course of solving the problem from the condition that the deflection jump at the top of the pre-fracture zone is zero.
Under the accepted assumptions of the Kirchhoff theory, the problem of determining the stress-strain
state of a composite plate reduces to finding two pairs of the functions Φ(z), Ψ(z), and z), Ψ0(z) of the complex
variable z = x + iy, analytical in the corresponding regions and satisfying the boundary conditions (3)–5).
We seek the complex potentials φ0(z) and ψ0(z) describing the stress-strain state of the fiber in the form
1
0 )(
k
k
k zaz ,
1
0 )(
k
k
k zbz . (6)
We denote the left-hand side of the boundary condition (3) as f1+if2 and assume that on the boundary
L this complex function can be expanded into a Fourier series
k
ik
keAiff 21 . (7)
Based on the boundary condition (3) and relations (6), (7), using the power series method [26], we
find the coefficients an, bn of the functions φ0(z) и ψ0(z)
)1( n
R
A
a
n
n
n ,
R
A
a
2
Re 1
1 , )0()2( 2 n
R
A
n
R
A
b
n
n
n
n
n .
We seek the values of the coefficients An in the course of solving the problem for a binder. Using the
complex potentials φ0(z) and ψ0(z), after some elementary transformations, we write the boundary conditions
on the media separation boundary τ=exp(iθ) as follows:
k
ik
k eA)()()( , (8)
00
22
21
1
0
00
* )2(Re
)1(
)1(
)()()(
k
ikk
k
k
ikk
k
i
k
ikk
k eRbeRakaeRan
D
D
n (9)
We seek the solution to the boundary value problem (5), (8), (9) in the form
ДИНАМІКА ТА МІЦНІСТЬ МАШИН
ISSN 0131–2928. Проблеми машинобудування, 2018, Т. 21, № 2 28
)()()( 21 zzz , )()()( 21 zzz , (10)
1
1
)1(4
)(
k
k
k
yx
zcz
D
MM
z ,
1
1
)1(2
)(
k
k
k
xy
zdz
D
MM
z , (11)
1
1
1
1
2
)(
)1(
1
)(
l
l
zt
dttg
i
z , dttg
zt
eT
zt
tg
e
i
z
l
l
i
i
1
1
1
1 )(
)(
)(
)1(
1
)( 12
1
1
1
12
2 , (12)
where 0
11
1 zteT
i
; 0
11
1 zzez
i
; κ=(3–ν)/(1+ν); g1(x1) is the sought-for function characterizing the
discontinuity of the rotation angles when the pre-fracture zone line is crossed
y
w
i
x
w
dt
d
tg1 .
Satisfying the boundary conditions (9), (10) by the functions (10)–(12) and comparing the coefficients
of equal powers exp(iθ), we obtain a system of algebraic equations for determining the coefficients ck, dk,, and
Ak.. These equations are such that we can explicitly find formulas for ak, bk, ck, dk,, Ak via the function g1(x1).
Satisfying the boundary conditions (5) on the faces of a rectilinear pre-fracture zone by complex poten-
tials (10)–(12), we obtain a complex singular integral equation with respect to the unknown function g1(x1)
111
*
1111
*
111 )(),(),(),(),(
1
1
lxxFdtxtgxtSxtgxtR
l
l
nn
, (13)
where
)1(
)(
)( 1*
i
tg
tgn , 1
0
11111*1 )()()()()( iCfxxxxxnxF ; x, t, 0
1z , l1 are dimension-
less values related to R; R11, S11 determined by the known relations ([27] of the formula (VI.62) at n=k=1).
The singular integral equation (13) requires adding the equality
0)(
1
1
1
l
l
dttg , (14)
providing uniqueness of the rotation angles of the plate medium plane when the pre-fracture zone
boundary is bypassed.
To determine the constant C1 (in the general case of a piecewise constant function) we have the relation [28]
0)(Re
1
1
1
l
l
dtttg ,
ensuring a zero deflection at the pre-fracture zone tips.
The complex singular integral equation (13) under the additional condition (14) reduces [23, 27] to
the system of algebraic equations with respect to the approximate values of the sought-for function g1(x1) at
the nodal points:
)(),()(),()(
1
1
1
1111
*
11111
*
11 r
M
m
rmmrmm xFxltlStgxltlRtgl
M
,
M
m
mtg
1
*
1 0)( , (15)
where r=1, 2, …, M – 1,
M
m
tm
2
12
cos ,
M
r
xr
cos .
If we go over to the complex conjugate quantities in the system (15), we obtain another the system of
algebraic equations algebraic equations.
For the closedness of the obtained algebraic equations, two complex equations determining the location
of a pre-fracture zone (coordinates of the pre-fracture zone tips) are lacking. Since the solution to the singular
integral equation (13) must be sought in the class of everywhere bounded functions (stresses), the system (15)
requires adding the conditions of bounded stresses at the ends of the pre-fracture zone x1=±l1. These are the
solvability conditions for a singular integral equation in the class of everywhere bounded functions.
DYNAMICS AND STRENGTH OF MACHINES
ISSN 0131–2928. Journal of Mechanical Engineering, 2018, Vol. 21, No. 2 29
The above mentioned additional conditions have the form
0
4
12
tg1
1
*
1
M
m
m
mM
M
m
tg , 0
4
12
ctg1
1
*
1
M
m
m
m
M
m
tg , (16)
Adding these two complex equations (16) to the equations obtained earlier, we obtain a closed com-
bined algebraic system. Because of the unknown length of the pre-fracture zone, the combined system of alge-
braic equations is nonlinear. The numerical solution of the combined system allows one to find the coordinates
of the pre-fracture zone tips (location) and its size, the values )(*
1 mtg (m=1, 2, …, M). It is obvious that, having
determined the coordinates of the pre-fracture zone tips, and, using the known formulas of analytic geometry,
we can find the said zone size, coordinates 0
1z of its center, and angle α1 with the axis x (Fig. 1).
The obtained systems of equations with respect to g1(tm) (m=1, 2, …, M) allow one, at a given exter-
nal bending load, to determine the stress-strain state of a composite when there is a zone of weakened inter-
particle bonds of the material in the binder. A numerical calculation was made for the fiber ν=0.30;
μ0=4.5·105 MPa and the binder ν=0.32; μ0=2.6·105 MPa.
Fig. 2 shows a graph of the dependence of a pre-
fracture zone length l1/R on the external bending load
sy MM . For this case, α1=42°, 8/0
1 17,1 ieRz was
found.
With the help of the solution obtained, we calcu-
late the displacements on the pre-fracture zone faces
and, using the crack formation criterion (2), we find
c
x
l
dxxg
0
1
1
11
*
1 )( , (17)
where
0
1x is the coordinate of the pre-fracture zone point
at which the inter-particle bonds of the material break.
Fig. 2. Dependence of a pre-fracture zone length on
the external bending load
The value of the external bending load causing the appearance of a crack is determined from the re-
lation (17). The combined resolving system of equations due to the unknown quantity l1 turned out to be non-
linear. To solve it, the method of successive approximations is used. A combined system of equations is
solved at some definite value *
1l with respect to the unknowns ck, dk, Ak, and )(*
1 mtg . The value
*
1l and the
obtained values kkk Adc ,, and kkk Adc ,, are substituted into (16), i.e. into the unused equations of the com-
bined system. The taken value of the parameter
*
1l and the corresponding values ck, dk, Ak, and )(*
1 mtg , gen-
erally speaking, will not satisfy the equations (16). Selecting the values of the parameter
*
1l , we will repeat
the calculations over and over again until the equations (16) of the combined system are satisfied with the
required accuracy. The combined system of equations in each approximation was solved by the Gauss meth-
od with the choice of the principal element for different values of M.
Thus, the joint solution of the combined algebraic system and the condition (17) allows one (at the re-
quired characteristics of the binder crack resistance) to determine the limiting value of the external bending load,
the location and size of a pre-fracture zone for the state of limiting equilibrium at which cracking takes place.
Fig. 3 shows the graphs of the distribution of the normal Rvv 11 and tangential Ruu 11
components of the displacement vector. Dimensional coordinates 111 lxx were used in the calculation.
Fig. 4 shows the dependence of the ultimate bending load s
c
y MM on the relative opening of the
faces 1* l in the center of the pre-fracture zone. Here α1=42°,
s
c
M)1(
*
.
The resulting combined algebraic system of equations of the problem allows one to obtain a solution
with any pre-assigned accuracy.
ДИНАМІКА ТА МІЦНІСТЬ МАШИН
ISSN 0131–2928. Проблеми машинобудування, 2018, Т. 21, № 2 30
The analysis of the crack formation model in a composite binder in the process of its being subjected
to a bending load reduces to a parametric joint study of the combined solving algebraic system of the prob-
lem and the criterion for crack appearance (17) at different values of the composite plate free parameters.
These are various geometric and mechanical characteristics of the binder and fiber materials.
a b
Fig. 3. Distribution of the displacement vector components:
a – normal; b –tangent
Fig. 4. Dependence of the limiting bending load
on the relative opening of the pre-fracture zone
center faces
Conclusions
The practice of using fibrous composites shows that at the design stage it is necessary to take into ac-
count the cases when cracks may appear in the binder. The existing methods of strength calculation of fibrous
composites, as a rule, ignore this circumstance. Such a situation makes it impossible to design a composite with
a minimum material consumption, with guaranteed reliability and durability. In this connection, it is necessary
to carry out an ultimate analysis of a composite in order to establish the limiting bending loads at which crack-
ing takes place in the binder. The size of the limiting minimum zone of the material weakened inter-particle
bonds, at which crack initiation occurs is recommended to be considered as a design characteristic of the binder
material. Based on the proposed design model, taking into account the presence of damage in a composite
(zones of weakened inter-particle bonds in the material), a method has been developed for calculating the pa-
rameters of a composite, under which cracking occurs. Knowing the basic values of the limiting parameters of
crack formation and the influence of the material properties on them, it is possible to reasonably control the
phenomenon of crack formation through design and technological solutions at the design stage.
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Received 11 March 2018
Introduction
Problem formulation
Solution method
Conclusions
References
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