Last edited: 2014.09.18. 15:46
Periodica Polytechnica
Transportation Engineering
x(y) pp. x-x, (year)
doi: 10.3311/pp.tr.201x­x.xx
web: http://www.pp.bme.hu/tr
© Periodica Polytechnica year
Fracture of a circular disk with mixed
conditions on the boundary
Vagif M. MIRSALIMOV 1, Nailya M. KALANTARLY2,
RESEARCH ARTICLE
Received 201x-xx-xx
Abstract
A model of a circular disk fracture based on consideration
of the fracture process zone near the curvilinear crack tip
is suggested. It is considered that mixed boundary
conditions are given on the boundary of the disk. It is
accepted that the fracture process zone is a finite length
layer with a material with partially broken bonds between
its separate structural elements (end zone). Analysis of
limit equilibrium of the curvilinear crack is made on the
basis of ultimate extension of the material bonds.
Keywords
circular disk  mixed boundary conditions  curvilinear
crack  crack with interfacial bonds  cohesive forces
1 Introduction
Circular disks are widely used in up-to-date machines.
The problems of disks strength are very urgent and
undoubtedly, interest to these problems will increase in
connection with development of machine-building and
power engineering. For analyzing the disks reliability it is
necessary to study their stress-strain state and fracture.
Simulation and analysis of stress-strain state in disks has a
special applied value in the first turn for tame choice of
their construction, optimal sizes and admissible value of
actuating loadings. The disks often work in highly
stressful conditions. There is a wide reference (see review
in the monographs [1, 2]) devoted to strength analysis of
disks. In most existing papers A. Griffith’s model of a
crack is used. In the present paper we use a bridged
crack’s model [3-5].
2 Formulation of the problem
1
Institute of Mathematics and Mechanics of NAS of Azerbaijan
AZ-1141 Baku, B.Vahabzade, 9, Azerbaijan (e­mail:
[email protected])
2
Institute of Mathematics and Mechanics of NAS of Azerbaijan,
AZ-1141 Baku, B.Vahabzade 9, Azerbaijan (e­mail:
[email protected])
Assume that on the boundary of a circular disk the
normal displacements u r t  and tangential component of
surface forces N t  are given, and the normal pressure
N r t  should be defined in the course of problem
solution. Refer the cross section of the disk to polar
system of coordinates r having chosen the origin of
coordinates at the centre of the circle L of radius R
(Fig.1).
Let the disk be weakened by a crack. In real materials,
because of structural and technological factors the crack
surfaces have irregularities and curvatures. In the disk’s
cross section, the crack with end zones is represented by a
slot of length 2  b  a , whose contour has small
deviations from the rectilinear form (Fig.1).
The crack is assumed to be close to the rectilinear
form allowing only small deviations of the crack line
from the straight line y  0 .
The crack contour equation is accepted in the form
y  f x  ,
a xb.
Based on the accepted assumption on the form of the
Begin of Paper Title
year Vol(No)
1
crack with end zones, the functions f(x) and f (x) are
small quantities. Let us consider some arbitrary
realization of the curved surface of the crack.
y

x
O
2 b
S+
Fig. 1
2
.
  q 2y  q xy
S–
Computatinal diagram of fracture mechanics problem for
circular disk
As the disk is loaded, in the crack vertices there will
arise the prefracture zones (end zones) that we model as
the areas of weakened interparticle bonds of the material.
Interaction of faces of these areas is simulated by
introducing between the prefracture zone faces the bonds
having the given deformation diagram. Physical nature of
such bonds and the sizes of prefracture zones where the
interaction of faces of weakened interparticle bonds is
realized, depends on the kind of the material. The bonds
between the crack faces at the end zones retard the
fracture development. This braking effect grows by
increasing the size of the end zone of the crack occupied
by the bonds [6 – 8]. In the case when the size of the
crack end zone is not small in comparison with the crack
length, the approximate methods of estimation of fracture
thoughness of disks based on consideration of a small end
zone crack, are not applicable. In these cases direct
simulation of stress state at the end zone of the crack with
regard to deformation characteristics of bonds is
necessary.
Distinguish a part of the crack d1  1  a and
d 2  b  2 (end areas), where the crack faces interact.
Interaction of the crack faces at the end zones is simulated
by introducing between the crack faces the bonds
(cohesive forces) with the given deformation diagram. As
the circular disk is loaded, in the bonds connecting the
crack faces there will arise normal q y  x  and tangential
q xy x  forces. These stresses are not known beforehand
and should be defined.
The boundary conditions on the faces of the crack
with end zones are of the form
 n  i nt  0
2
for
a  x  1 and 2  x  b .
y  f (x) , 1  x  2

(     )  i (u   u  )  x,   q y ( x)  iqxy ( x) ,
R
a 1
for y  f (x) ,
The main relations of the problem should be
complemented by the equation connecting the opening of
end zone faces and forces in bonds. Without loss of
generality, represent this equation in the form [4]
N  
L
 n  i nt  q y ( x)  iqxy ( x)
(1)
Per. Pol. Transp. Eng.
(2)
The function   x,   is the effective compliance of
the bonds dependent on their tension;  is a modulus of
a stress vector in bonds; (     ) is a normal,
(u   u  ) is a tangential component of the opening of
end zone faces of the crack. that the problem of definition
Denote the considered area enclosed between the
circle L of radius R and one slit L1  a, b  by S  , the
area complemented to the complete complex plane by
S .
Since the functions f(x) and f (x) are small quantities,
we can represent the function f(x) in the form
f x    H x  ,
a  x b,
where  is a small parameter.
3 The Method of
Problem Solution
the
Boundary-Value
Using the perturbations method, we get boundary
conditions for y  0 , a  x  b :
for a zero approximation
( 0)
 (y0)  i xy
0
y  0 , 10  x  02 ,
for
(3)
( 0)
( 0)
for y  0 ,
 (y0)  i xy
 q (y0)  iqxy
a  x  10 and 02  x  b .
for a first approximation
(1)
 (y1)  i xy
 q y  iq xy
for y  0 , 11  x  12 , (4)
(1)
(1)
 (y1)  i xy
 q (y1)  iqxy
 q y  iq xy for y  0 ,
a  x  11 и 12  x  b .
Here
q y x   q (y0) x    q (y1) x  ;
( 0)
(1)
x    qxy
x  ;
q xy x   q xy
First A. Author, Second B. Author
Last edited: 2014.09.18. 15:46
1  10   11 ;
( 0)
q y  2 xy
2  02   12 ;
 (y0)
dH
for y  0 ;
H
dx
y

q xy   (y0)   x( 0)

dH
H
dx
y
( 0)
xy
1
2
Vk 
(5)
.
Similarly, we get boundary conditions on the contour
L at each approximation, and also the relations connecting
the opening of the end zone faces of the crack and forces
in bonds.
Now construct the solution in a zero approximation.
Denote the area occupied by a disk, bounded by a circle L
and one rectilinear cut [a,b] by S  , the area
complemented to complete complex plane by S  .
Under these conjectures, the problem is reduced to
definition of two complex variable functions  0  z  and
0  z  , analytic in the area S  and satisfying on the
basis of [9] the following boundary conditions:

   2 u'r t 

R2

Re  0 t    0 t  
t  '0 t   0 t 

t2

1
2
Tk 

on L,
2
 ur  e
ik
d
k=0,1,2,…
0
2
 iN  e
ik
d .
0
The main relations of the stated problem in a zero
approximation should be complemented by the equation
connecting the opening of the faces of the prefracture end
zone and the forces in bonds.


0 ( x,0)  0 ( x,0)  i u0 ( x,0)  u0 ( x,0) 



( 0)
x 
  x, 0 q (y0) x   iqxy
Passing in relations (6) and (7) to conjugate values,
after some transformations on the contour L we get
boundary conditions in the following form
  1 0 t    0 t  R2
 t 0 t   0 t 
2
t


t 2  R2
0 t   0 t   4  u 'r t 

R 2  t

(11)
on L.
(6)


t2



Im  0 t    0 t  
t '0 t   0 t    N t 
2


R


on L.
(7)
and also the condition on the faces of the crack with end
zones
0 x   0 x   x 0 x   0 x   f 0 x  ,
(8)
where   3  v  1  v  ; v is the Poisson ratio of the
disk’s material;  is the shear modulus;
0
y  0, 10  x  02

.
f0 ( x)  
(0)
( x ) y  0, a  x  10 , 02  x  b
q (y0 ) ( x )  iq xy

t R

t 0 t   0 t 

2
2
t
R2
t
2

v
v

t 
t 
Tv   ,
Vv   , iN t  
R
R
v    
v    


(9)
where Vv , Tv (v=0,1,2,…) are generally speaking,
known complex coefficients and are determined by the
formulas
2
R 
 2iN t 

0 t   0 t  

on L
(12)
Based on (11) and (12), on the circle L we shall have
the following relation
  1 0 t    0 t  2t2
 R2

0 t   0 t  

R  t

2
 22  u r t   iN t  on L.
Now, substitute relation (9) into the last equality and
get:
  1 0 t    0 t  2 t 2  R
2
In the general case, on the circle L we take the
functions u r t  and N t  in the form of Fourier series:
u r t  
(10)

2
R  t

0 t   0 t  

v
  
2  v  1
 t 
 2 Tv 
Vv 1    
R
 R 
  0 


2  v  1

 R 
V v 1   
T v 
R

 t 
v 1


(13)
v



on L.
Introduce on L a new unknown auxiliary function
Begin of Paper Title
year Vol(No)
3
0 t  H (the Holder condition) in the form


*  z   

2t  R 2
20 t     1  0 t    0 t   2 
0 t   0 t 
R  t

2
on L.
(14)
Putting together relations (13) and (14), we get
 0 t  

0 t 
1 

  1   1 v 0

 2  v  1
 t 
Vv 1  Tv    

R

 R 
v
1  
2  v  1
 R 
T 
Vv 1   
  1 v 1  v
R
 t 

v
on L.
(15)
on L.

1 
v2
  2 1    1 T
v2
v 0

 v  1
R
R2 t   
Vv1 


v 2
v
  v  1
1
 R 
Vv 1  Tv 2    

R
2
 t 

1  v  2 

 v  3  v  2 
 2 1    1 T v  2  R 1    1 V v  3  




v 3  

1 R
R

      V1  T0 

t
R
2  t2
 

1 
1 
1

2  R
  1 
V1 
V
;
T1  T1 


2


1
2


1
R
R 2  t

 
2t
0 t   0 t  R 1t 0 t  .
2
2
Based on the theorem on analytic continuation, and
the properties of the Cauchy type integral, from relations
(15) and (16) we get:
*  z    0  z  

1
 1

v 0
1 1 0 ( t )
dt 
  1 2i t  z
L
 2  v  1
 z 
Vv 1  Tv   
R
 R 
 
2  v  1

 R 
V v 1   
T v 
R
 z 
v 1 

v
zS ;
1 Q t 

0  z   2i t  z dt  R1  z ,

L
*  z   
1 Q t 

dt  R2  z ,
 2i t  z
L



z S
zS

.
(18)
In relations (17) – (18) the functions *  z  and
*  z  are analytic in the complete complex plane cut
along the slit L=[a,b] and vanish at infinity, i.e.
*    0 .
v
  t v
R 
 v      v    ,
R
 t  
v 1   


 0  z   *  z  
0  z   *  z  
2
R2

v
(19)
where  v v  0,  1,  2, ...  are, generally speaking,
complex coefficients.
We substitute relation (19) in the first formulas of
(17), (18) and using the Cauchy integral theorem get
general formulas for the sought-for functions:
v

Q t   
tz

0 t    0 
 t 
v2
1 
Vv3    ;


1


 R 
v
0 t dt
We shall look for the unknown auxiliary function
0 t   H on L in the form of Fourier series
1
 Tv2 
2
 v  3 
R
L
(16)
Here we introduce the following denotation
R1 t  
1
 1
1
*    0 ;
Now, having substituted (15) into (14), we get
 t   Qt   R1 t   R2 t 

1
  1 2i
(17)
zS ;
Here J v 
1 

z
Jv  
R
v 0  

v

v
z
Wv  
R
v 0

 T 
  1  v v
zS ,
(20)
zS .
(21)
2  v  1

Vv 1  ;
R

1
1 v2
1 v2
Wv   

 v  2    v  2  1 
T
2
2   1  v 2
 2  1 
1
 v  3  v  2 
 v  1
 T v  2 

V  v 1 .
1 
V
2
R   1  v 3
R
For determining the functions *  z  and *  z  ,
following [9] we consider the function
* z    z   z  z    z  ,
(22)
that is analytic on the whole complex plane outside of the
rectilinear slit (i.e. the cracks with end zones).
For the stress vector components we have [9]:
( 0)
 (y0)  i xy
  0 z    0 z   z  0 z   0 z  .
4
Per. Pol. Transp. Eng.
First A. Author, Second B. Author
Last edited: 2014.09.18. 15:46
Taking into account the loading condition on the faces
of crack and on the end zones, based on (8) as z  t (t is
the affix of the points of crack and end zones) we get the
conditions
 0
 0
t 
 0
t 
 0
t 
 t0
t 
 0
t 
 t0
t 
 0
t   f 0 ;
t 
 *
t 

 t 0
t 
 *
t   f1 (t )  f 0 ;
 J v  (v  1) J v  W v   R 
t 
.
* z   * z   * z   z* z 
(25)
Having substituted (25) in relation (24), we get
* t   * t   f  (t ) ;
Hence we obtain that the problem of definition of the
functions *  z  and *  z  is reduced to the Riemann
linear conjugation problem [9]:
* t   * t   * t   * t   f t  ;
* t   * t   * t   * t   0 .
(26)
(27)
v
 v  pv  t   2 f 0 ;
f  t   2 f 0  2 f1 (t ) 
R
v 0
 1 v2
1
1
 v  2    
 
 v  2    v  2 
2
 1 v
  2  1 
*  x   *  x   f1 ( x)
*  x   *  x   0
10  x  02 ,
(29)


1  v  2 
1
1
pv  2  1 
T 
 Tv  2  T v  2 
2


1
2

1 v

 
a  x b.
(30)
Since the stresses in the disk are restricted, the
solution of the boundary value problem (29) should be
sought in the class of everywhere bounded functions.
As a particular solution of homogeneous problem (30)
we take the function
 z  a  z  b 
and mean the branch for which the following equality
holds
X  ( x)   X  ( x) on a  x  b .
Based on the last relation, we rewrite the conjugation
problem (30) as follows
   x 

   x 
X  ( x)
 0 on a  x  b .
From the last boundary condition it follows that the
solution of the homogeneous problem vanishing at
infinity equals zero.
We represent the inhomogeneous conjugation problem
(29) in the form
   x 

X ( x)

   x 

X ( x)
v  ;
Begin of Paper Title
(28)
Based on (26) and (28), for the function *  z  we get
a linear conjugation problem
X  ( x)
Here
v 1
*  z   *  z   0
X ( z) 
* t   * t   f  (t ) .
 1

v2
1 
V v  3  .


1



R
The corresponding homogeneous problem is of the
form
Having changed in formula (22) z by z and passing to
conjugated values, we get

 v  3 
a  x  10 and 02  x  b .
v
v 0

V v 1 
( 0)
x   f1 ( x)
* x   * x   q (y0) x   iqxy


R
(24)
* t   * t   t 0 t   * t   f1 (t )  f 0 ,
where f1 (t )  
 v  1
Since *    *    0 , then the general solution
of problem (27) will be
t   f 0 ,


v 1
2  v  1
2  v  12
Tv 
Vv 1 
V

  1R
  1R v 1
k0  1
(23)
( 0)
where f 0  0 on the crack faces and f 0  q (y0)  iq xy
in
the end zones.
Substituting formulas (20), (21) in relation (23), we
have
*

Denote  0  z  

F ( x)
X  ( x)
* z 

X ( z)
on a  x  b
; F* ( x ) 
F ( x)
X  ( x)
(31)
,
then boundary value problem (31) takes the form
year Vol(No)
5
 0  z    0  z   F* ( x ) on a  x  b .
F*  x  
Here
F*  x  
for a 
f1 ( x)
for 10  x  02 ,
x  b x  a 
( 0)
q (y0)  iq xy
 f1 ( x )

 x  b x  a 
x  10
02
and
 z  b  z  a  b F* ( x ) dx .

2i
(32)
xz
a
According to the behavior of the function *  z  at
infinity, the solvability condition of the boundary value
problem has the form
f  t dt
b

a
t  a b  t 
tf  t dt
b
0;

t  a b  t 
a
0
(34)


2   
 
u0  u0  i
  0

1    x
x 0
 .
b


a

6
i

x  a x  b 


(36)
q (y0) t dt
t  a b  t t  x 

(35)

.
t  a t  b t  x  
f  t dt
f y t dt







t

a
b

t
t

x
a

b




2 d
 x,  0 q (y0 )  x  ;
1   dx
1

b


a

 x  a b  x  
(0)
t dt
q xy
(37)
t  a b  t t  x 

f xy t dt







t

a
b

t
t

x
a

b




2 d
(0)
x  ;
 x,  0 q xy
1   dx
Here f y t   Re f1 t  ; f xy t   Im f1 t  ;
f1 t   

t
 J v  v  1J v  W v  R 
v
v 0
Each of the equations (36) and (37) is a non-linear
integro-differential equation with the Cauchy kernel and
may be solved only numerically. For solving them we can
use the collocational scheme with approximation of
unknown functions [10]. The obtained relations (20) –
(21) with regard to formulas (15), (16), (19) and equations
(36), (37) permit to get the final solution of the problem in
a zero approximation if the coefficients  v
v  0,  1,  2, ...  will be determined.
For composing the infinite system of algebraic
equations with respect to unknowns  k , substitute
relations (20), (21) into condition (13) with regard to (32)
and expansions

Using the Sokhotskii-Plemelj formula and taking into
account formula (32), we find
 0  x    0  x   


 0  x    0  x  

b


a
 x  a b  x  
(33)
These relations help to find the unknown parameters
10 and 02 determining the sizes of the end zones of the
crack at a zero approximation.
The obtained relation contains the unknown stresses at
the end zones of the crack.
Now we construct an integral equation for determining
( 0)
unknown forces q (y0) ( x)  iqxy
( x) . The additional
relation (10) is the condition that determines the unknown
stresses in the bonds between the faces at the end zones of
the crack in a zero approximation.
In the considered problem it is convenient to write this
additional condition for the derivative of the opening of
the faces of the crack’s end zones.
Using the Kolosov-Muskhelishvili relation [9] on
boundary values of the functions   z  and   z  ,on
the segment y=0, a  x  b we get the following equality

1


 xb.
The desired solution of the problem is written as [9]:
* z  
We substitute the found expression into the left hand
side of (34), and taking into account relation (10) after
some transformations we find the system of nonlinear
integro-differential equations with respect to the unknown
( 0)
functions q (y0 ) and q xy
:
t  a t  b   t  M r  R 
r 0
1
t  a t  b 


t 
R
M r  
t 
r 0

r
,
r 1
After some transformations, condition (13) is reduced
to the form
Per. Pol. Transp. Eng.
First A. Author, Second B. Author
Last edited: 2014.09.18. 15:46


t 
R
Am   
Am
  
R
t 
m0
m0
m



(m=1,2,…,M1),
m

(38)
M

t 
R
Cm   
Cm
  .
R
 
t 
m0
m0

m
m

Here
In view of some bulky form of expressions for Am ,


, Cm , Cm
Am
m  0, 1, 2, ...  they are not cited.
Comparing the coefficients at the same degrees t R
and R t in the both sides of the obtained relation (38),
we get infinite systems of linear algebraic equations
m  0 ;
A0  A0  C0  C0
Am  Cm ,


Am
 Cm
(39)
m  1, 2, ... .
Now let us pass to procedure for converting a system
to an algebraic system of integro-differential equations
(36) and (37) with additional conditions (33). At first in
integro-differential equations (36) and (37) and in
additional conditions (33) all integration intervals are
reduced to one interval [–1, 1]. By means of quadrature
formulas all integrals are replaced by finite sums, and the
derivatives in the right sides of equations (36) and (37)
are replaced by finite-difference approximations.
Therewith the following boundary conditions are taken
into account: q y a   q y b   0 ; q xy a   q xy b   0 (this
corresponds to the conditions  0 a, 0    0 a, 0   0 ;
0 b, 0   0 b, 0   0 ;
u0 a , 0   u0 a, 0   0 ;


u0 b, 0   u0 b, 0   0 ). As a result, instead of each
integral equation with corresponding additional
conditions, we get M1+2 algebraic equations for
determining the stresses at the nodal points contained at
the end zone of the crack, and the sizes of the end zones.
 Am k q (y0, k)  f y , k  
M1
k 1

1  M

4 b  a


(40)

)
(0)
0
 С xm 1 ,  0  xm 1  q (y0, m
1  С x m 1 ,   x m 1  q y , m 1


f  y cos k   0 ,
k 1
M
 k f y  k   0 .
k 1
 Am k q xy(0), k  f xy, k  
M1
k 1


(41)

( 0)
( 0)
0
 С xm 1 , 0  xm 1  q xy
, m 1  С xm 1 ,  xm 1  q xy, m 1
Begin of Paper Title
f y ,k  f y  k  ,
xm1 
f xy,k  f xy  k  ,
 
ab ba
1

 , Amk   cot m k .
2
2 m1
M
2
The joint solution of the obtained equations permits at
the given characteristics of bonds to determine the forces
( 0)
in the bonds q (y0)  x  , q xy
x  and the sizes of the end
0
0
zones (parameters 1 and 2 ) at a zero approximation.
After solving the obtained algebraic systems, we can
pass to construction of basic resolving equations of the
problem in a first approximation. According to the found
solution we find the functions q y (x) and q xy (x) .
The succession of the solutions of the problem in a
first approximation is similar to a zero approximation. In
a first approximation the problem is reduced to
determination of two analytic functions 1 ( z ) and
1 ( z ) , analytic in the domain S  and satisfying the
following boundary conditions

  0

R2

Re  1 t   1 t  
t 1 t   1 t 

t2

(42)

on L,


t2



Im 1 t   1 t  
t 1 t   1 t   0 on L,
2


R


1 x   1 x   x 1 x   1 x   f1 x 
(43)
( a  x  b ),
where
Repeating the above mentioned method for solving the
boundary value problem of a zero approximation, we find
1  M

4 b  a

( 0)
( 0)
q (y0,k)  q (y0)  k  , q xy
,k  q xy  k  ,
q y  iq xy y  0,
y  0, 11  x  12

.
f1 ( x )  
(1)
(1)
1 1
q xy

iq

q

i
q
y

0
,
a

x


,


x

b
y
y
xy
1
2

(m=1,2,…,M1),
M
M
f xy  k   0 ,  k f xy  k   0 .

k 1
k 1

1 t    01 
v
  t v
R 
 v1     1 v    ,
R
 t  
v 1   


year Vol(No)
(44)
7
1  z    *  z  
1  z   *  z  
Here J v1 

z
J v1  
R
v 0  


v
z
Wv1  
R
v 0

 
v
zS ,
1
 v1
1 v2 1
; Wv1   
 v  2    v  2 .
2
 1
 2  1 
 ( z )    ( z ) 
 z  a  z  b 
2i

a
is the condition that determines the unknown stresses in
the bonds between the faces in the crack’s end zones in a
first approximation.
Behaving as in a zero approximation, after some
transformations we find the system of nonlinear integrodifferential equations with respect to the unknown
(1)
functions q (y1) and q xy
:
 b q (1) t dt b f 1 t dt 
y
y

 X ( x) 




X
(
x
)
t

x
X
(
x
)t  x  

a
a


1
The complex potentials have the form
b
F1 ( x ) dx
xz

(1)
x 
  x, 1 q (y1) x   iqxy
zS ,
,
(45)

 
(47)

2 d
 x,  1 q (y1)  x  ;
1   dx
Here
F1 ( x ) 
F1 ( x ) 
f11 ( x )  q y  iq xy
x  a x  b 

for 11  x  12 ;



v 0
v
t 
 ;
R
 
f11 t   
1
a
v
 J 1  v  1 J 1  W 1   t  .
v
v 
 v
R
v 0

1 t 
;
 1
1 t   
 t   1 t  R 1t 1 t 
2 1
R2
2t
2
b
 t F t  dt  0 .
1
(46)
a
These equations are used to determine the unknown
parameters 11 and 12 defining the sizes of the crack’s
end zones in a first approximation.
The obtained relations contain unknown stresses at the
crack’s end zones.
Now construct integral equations for finding the
(1)
unknown forces q (y1) ( x)  iqxy
( x) . The additional
relation


1  x,0   1  x,0   i u1  x,0   u1  x,0  
8

The obtained relations (44) – (46), allowing for
formulas
1 t  
The solvability conditions of the boundary value
problem in a first approximation have the form
 F t  dt  0 ;

1
t   q xy  Im f11 t  ;
f xy
v 1 1
 .
  1 v 
b
(48)
Here f y1 t   Re f11 t   q y (t ) ;
 1 v2 1
1
1
1v  2    
 v1 
 v  2   v 2 
2


1
2


1





 
x  a x  b 

1v 

2 d
(1)
x  ;
 x,  1 q xy
1   dx
(1)
q (y1)  iq xy
 q y  iq xy  f11 ( x )
for a  x  11 and 12  x  b ;
f11
 b q (1) t dt b f 1 t dt 
xy
xy

X ( x) 


X ( x )t  x  
 X ( x )t  x 
a
a

1
Per. Pol. Transp. Eng.
and equations (47), (48) permit to obtain the final solution
of the problem in a first approximation if the coefficients
 1v v  0,  1,  2, ...  will be determined.
For constructing the infinite system of linear algebraic
equations for the coefficients  1v of the auxiliary function
1 t  , we behave as in a zero approximation. As a result,
we get two infinite systems of linear algebraic equations
of type (39).
As a result of procedure for converting integrodifferential equations (47) and (48) with additional
conditions (46), instead each integral equation with
corresponding additional conditions, to an algebraic
system we get M1+2 algebraic equations for determining
the stresses at the nodal points contained at the crack’s
First A. Author, Second B. Author
Last edited: 2014.09.18. 15:46
end zones, and the sizes of the end zones in a first
approximation:
 Am k q (y1,)k  f y1, k  
M1
k 1

1  M

4 b  a


(49)

  xm1 , 1  xm1  q (y1,)m1  С xm1 , 1  xm1  q (y1,)m1

(m=1,2,…,M1),
M
M
k 1
k 1
 f y1 cos k   0 ,  k f y1  k   0 .
(1)
1
Am k q xy

, k  f xy , k  
k 1
M1

1  M

4 b  a
(50)

(1)
  xm 1 ,  1  xm 1  q xy
, m 1 


(1)
 С xm 1 ,  1  xm 1  q xy
, m 1
M
M
k 1
k 1

(m=1,2,…,M1),
 f xy1  k   0 ,  k f xy1  k   0 .
(1)
(1)
Here q (y1,)k  q (y1)  k  , q xy
, k  q xy  k  ,
f y1, k  f y1  k  ,
1
1
f xy
, k  f xy  k  .
The remaining denotations are the same as in a zero
approximation.
The joint solution of the obtained equations permits at
the given characteristics of bonds to determine the forces
(1)
in the bonds q (y1 x  and q xy
x  , the sizes of the end
1
zones (the parameters 1 and 12 ) and also to study the
influence of irregularities and curvatures of the crack
surface on the stress strain state of the circular disk.
4 Numerical Solution and Analysis
For formulation of the limit equilibrium criterion, we
use the criterion of critical opening of the crack. The
opening of the crack in the range of end zones may be
determined from the relations
   x , 0      x, 0     x,   q y  x  ,
a  x  1 and 2  x  b ,
edge of the end zone will be
 1 ,  1  1    c
2 , 2  2    c
for x  1 ,
for x  2 ,
where  c is the characteristics of the disk material
determined experimentally.
Even in the special case of linear-elastic bonds, the
obtained systems of equations become nonlinear because
of unknown sizes of the crack’s end zones. In this
connection for solving the obtained systems in the case of
linear bonds, the successive approximations method was
used. At each approximation the algebraic system was
solved numerically by the Gauss method with choice of
the main element. In the case of nonlinear law of
deformation of bonds, for determining the forces at the
end zones, an algorithm similar to the A.A. Il’yushin
method of elastic solutions [11] was also used. Analysis
of effective compliance is carried out similar to definition
of the secant modulus in the method of variables of
elasticity parameters [12]. The successive approximations
process ends as soon as the forces along the end zone,
obtained at two successive iterations will differ a little
from each other.
The nonlinear part of the curve of deformation of the
bonds was taken in the form of bilinear dependence
whose ascending portion corresponded to elastic
deformation of bonds ( 0 < V ( x) < V , V  u 2   2 )
with maximum tension of bonds. For V ( x) > V the
deformation law was described by a nonlinear relation
which is determined by two points (V ,  ) , and
 c ,  c  , and for  c    we have increasing linear
dependence (linear strengthening corresponding to elastoplastic deformation of the bonds). The algebraic system
with respect to tractions in bonds was solved numerically.
For numerical calculations it was assumed М=30 that
responds to partition of integration interval into 30
chebyshev nodal points.
The plots of dependence of dimensionless length of
the end zones of the right end d 2  b  2  b  a  on
dimensionless load N 0   for different sizes of cracks
   2  1  b  a  are depicted in Fig 2.
Fig. 3 represents the graphs of distribution of the
normal tractions q y N 0 in the bonds for the right end
zone of the crack (curve 1 for linear bonds, curve 2 for
bilinear curve of deformation of bonds).
The dependence of dimensionless opening modulus
    E 8N 0  for the right end of the crack on
dimensionless parameter N 0   is represented in fig. 4.
u  x, 0  u  x, 0  x,  q xy x  .
The condition of critical opening of the crack near the
Begin of Paper Title
year Vol(No)
9
0.6
Conclusions
d 2  b  2  b  a 
0.5
0.4
   0,75
0.3
   0,50
0.2
0.1
0
Fig. 2
N0
   0, 25
0.25
0.50

0.75
1.00
Dependence of dimensionless length of the end zones of the
right end d 2  b  2  b  a  on dimensionless load N0  
for different sizes of cracks    2  1  b  a 
4.0
q y N0
3.0
2
2.0
1
1.0
x

0
Fig. 3
0.5
0.6
0.7
0.8
0.9
1.0
Distribution of normal forces q y N 0 in the bonds for the
right end zone of the crack (curve 1 corresponds to linear law of
deformation of bonds, curve 2 to nonlinear deformation)
0.15
 E 8N 0 
0.10
0.05
N0
0
Fig. 4

0.2
0.4
0.6
0.8
Dependence of dimensionless opening modulus
    E 8N 0  for the right end of the crack on
dimensionless parameter N 0  
The obtained main resolving algebraic equations
permit to predict the ultimate admissible size of the crack
(technological defect) by means of numerical calculation
for a specific circular disk.
10
Per. Pol. Transp. Eng.
The obtained closed algebraic system of equations and
limit condition of the crack growth permits to determine
the admissible size of the crack for different laws of
deformation of interparticle bonds, elastic and
geometrical characteristics of the material and the circular
disk by means of numerical calculation for any specific
circular disk.
Distribution of forces in bonds and opening of crack
faces is determined directly from the solution of the
obtained algebraic systems. The model of a bridged crack
at the end zones enables to study main regularities of
forces distributions at the end zones, to make analysis of
ultimate equailibrium state of the curved crack by means
of deformation criterion.
The developed calculation method permits to solve the
following practically important problems:
to estimate the guaranteed resource of the disk with
regard to expected deficiency and loading conditions;
to set admissible level of deficiency and maximum
value of actuating loads that provide sufficient safety
margin;
to select the disk’s material with a complex of
characteristics of fracture thoughness.
The model of a crack with end zones, permits to
consider the fracture process including crack initiation,
crack formation and crack development from unified
positions.
References
[1] Savruk, M.P. "Fracture mechanics and strength of
materials. Vol. 2. Stress intensity factors in cracked
bodies," Naukova dumka, Kiev. 1988.
[2] Savruk, M.P. Osiv, P.N., Prokopchuk, I.V.
"Numerical analysis in plane problems of cracks
theory," Naukova dumka, Kiev. 1989.
[3] The special issue: Cohesive models, Eng. Fract.
Mech.,
70(14),
pp.
1741-1987.
2003.
DOI: 10.1016/S0013-7944(03)00121-8
[4] Mirsalimov, V.M. "The solution of a problem in
contact fracture mechanics of the nucleation and
development of a bridged crack in the hut of a
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120-136.
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DOI:
10.1016/j.jappmathmech.2007.03.003
[5] Mirsalimov, V.M., Hasanov, F.F. "Interaction
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under Transverse Shear", Acta Polytechnica
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11(5),
pp.
161-176.
2014.
DOI: 10.12700/APH.11.05.2014.05.10
First A. Author, Second B. Author
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[6] Rose L.R.F. "Crack reinforcement by distributed
springs", J. Mech. Phys. Solids, 35(4), pp. 383-405.
1987. DOI: 10.1016/0022-5096(87)90044-5
[7] Cox, B.N., Marshall, D.B. "Concepts for bridged
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[8] Budianscky, B., Cui, Y.L. "On the tensile strength of
a fiber-reinforced ceramic composite containing a
crack like flaw," J. Mechanics and Physics of Solids,
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[9] Muskhelishvili, N.I. "Some Basic Problems of
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Amsterdam. 1977.
Begin of Paper Title
[10] Ladopoulos, E.G. "Singular integral equations,
linear and non-linear theory and its applications in
science and engineering," Springer Verlag, New
York, Berlin. 2000.
[11] Il’yushin, A.A. "Plasticity," Gostexhizdat, Moscow
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[12] Birger, I.A. "Расчет конструкций с учетом
пластичности и ползучести" (The design of
structures allowings for plasticity and creep), Izv.
Akad. Nauk SSSR, Mekhanika, 2, pp. 113-119. 1965.
(In Russian)
year Vol(No)
11