Complex Adaptive Systems: Energy, Information and Intelligence: Papers from the 2011 AAAI Fall Symposium (FS-11-03)
The Investigation of Problems of Healing of Cracks by Injection of Fluids
with Inclusions in Various Thermoelastic Media
Alexander G. Bagdoev, Gohar A. Manukyan
Institute of Mechanics of the National Academy of Sciences Armenia,
Yerevan, Armenia
bagdoev@mechins.sci.am, avetisus@yahoo.com
fracture. It was assumed that current and temperature of
fluid in point of entrance in fracture are constant on time.
In Gliko (2008) is considered mentioned problem when as
temperature in point of entrance in system as current of
fluid can be function in time.
These problems were solved in Lowell et.all (1993), in
Gliko (2008, Ghassemi et all (2005) and in numerous
works of other foreign authors, in investigation of healing
of cracks by thermo-diffusion effects of sedimentation of
crystalline from hydrothermal mixture, by solution of
thermo conductivity equations, in application to fracture
in geophysical environment, without taking into account of
elastic stresses.
Experimental curves of healing of strips, filled by
fluid with inclusions in earth environment are done in
Bhattacharia Subhamoy et.al. (2008). In Kukudzanov et.al
(2010) it is investigated the problem of healing of plane
cracks due to cyclic action of electrical current in metallic
media. As result in Kukudzanov et.al. (2010) is shown
increasing of strength of material under action of electrical
current. These results also can be applied in examination of
interrelation of nano, micro, meso and macro levels action
on defects dynamics.In Panin(2000), in Bagdoev et.al.
(2010) is carried out application of general method of
solution of mixed unsteady anti-plane, plane and space
problems for cracks by solution of system of Winner-Hopf
equations, derivation of formulae for stress-intensity
coefficients and numerical calculations of them, which
allow apply results by any way in fracture problem.
In present paper mathematical solution by mentioned
methods is obtained for important for practice problems of
healing of biological cracks with camphor- oil and for
geothermal cracks by treatment of the coupled thermo
elastic and diffusion mixed boundary problem for crack
with fluid-crystalline mixture. Solution of unsteady mixed
plane boundary value problem for thermo elasticity is
obtained by integral transforms Laplace on time and
Fourier on coordinate, is brought to Winner–Hopf equation
with continuous coefficients. After reverse integral
transformations solution for displacements is brought to
Abstract
The purpose of this paper is to investigate the problem of
healing of cracks in infinite thermo elastic surroundings by
currents of fluids with inclusions. The process of healing is result
of growing of layer of sediments on crack’s surfaces due to
transverse diffusion currents of crystalline inclusions which leads
to closing of crack. The obtained analytical formulae for vertical
displacements of boundary of crack and graphs of it allow
determine coordinates and time moments of zero width of crack,
i.e. process of healing of crack. The applications to biological and
technological problems when as mixture is used fluid with
camphor crystalline as well as to geothermal cracks with fluid
currents containing silicon crystalline with application to
geophysics, and results of calculations also can be used in meso
mechanics. The investigation of obtained curves of dependence of
times and coordinates of healing of cracks processes is done as
by deterministic treatment, as well as on account of possible
randomness, taking places for micro sizes of cracks by methods
of nonlinear wave dynamics.The main methods of investigation
of healing problem for cracks are taking into account of thermo
diffusion and thermo elasticity stresses and displacements effects
and trybological supplements in boundary condition on the
cracks., and solution of posed problem of thermo elasticity by
method of integral transformations of Laplace and Fourier,
method of Winner-Hopf. The analytical
solution for
displacements of mentioned unsteady plane problem is brought to
effective Smirnov-Sobolev form. These analytic and numerical
methods based on dynamic thermo-elasticity approximation on
account of diffusion currents of inclusions in fluid mixture in
cracks allow determine time and coordinate of closing of cracks.
Besides are examined obtained curves of healing processes by
methods of nonlinear waves dynamics. These methods can be
applied also to extremely processes of transition from meso level
defects to synchronized processes of generation of macro cracks
and fracture.
Introduction
Problems on building up of layer of inclusions,
containing in fluid entering in crack, which is in infinite
thermo elastic plane are considered. These problems were
investigated in Lowell et.all (1993), as the problems of
healing of fracture by injection of hot hydrothermal
mixture with silica cristallines and their sedimentation on
7
term of cristallines, one obtains equation on fracture
surface y b1 ( x, t ) , which since b1 is small can be
Smirnov-Sobolev integral form. The graphs of vertical
displacements on crack surface are calculated numerically
and the moments of healing of crack in some points are
determined for various values of typical constants for
biological, technological and seismological cases.
Obtained curves of healing processes of cracks are
examined by methods of nonlinear wave dynamics to
obtain the probabilities distribution for them.
1. Statement of problem of crack in thermo elastic
plane with fluid current.
The problem of semi–infinite crack in thermo elastic plane,
when there is current of fluid with inclusions, entering in
crack at initial moment t 0 , is considered. These
problems are especially important for study of process of
narrowing of crack due to action of crystallite inclusions in
camphor -oil and in analogically by mathematical
treatment problem of building up of cracks by inclusions in
camphor- oil in corresponding problems of technology in
Laboratory of Institute of Mashinovedenia, Pushkino,
Moscow region., 2007, in Bhattacharia Subhamoy et
al.(2008). In plane x, y equation of crack boundary is
y
rb1 x, t ,
written on y
0 axis of fracture
T T0
wb
U s 1 qJ
y 0 KV yy i0 H x H t (1.1)
wt
l
here H (t ) is Heaviside unit function , and initial
conditions for unsteady problem are zero. The crack
equation y b1 , y b0 U y x,0, t ,0 x f .
One must put and solve corresponding problem of thermo
elasticity by solution of the equations of motion of thermo
elastic media (Landau et.al.1954) on account of the
stresses components are (Landau et.al.1954) and vx , y and
temperature T T 0 relation
V yy
where thickness 2b1 is small and later is taken only upper
y t 0 , due to symmetry.
U x ,U y are components of displacements in elastic media.
Let the temperature T of elastic plane and fluid are
approximately the same, denoting by T0 constant initial
value of
T , by q
J
wU y
c U cv § wU x
wU y ·
(1.4)
¨¨
¸
w y ¸¹
D c v © wx
Placing (1.4) in equations of motion and (1.2) we can write
system of equations:
w2Uy
w2U
w2U
a2 2x b2 2x (a2 b2)
wx
wy
wxwy
(1.5)
2
w § wU wU · w Ux
G ¨ x y ¸
wx © wx wy ¹ wt 2
T T0
balance, which supposed known constant, approximately
0, J
a2
K0 2
D T0
(1.3)
U
where U density, c U , cv specific thermal capacities.
First we neglect thermo-conductivity, k 0 , and obtain
wc
, entering in simple equation of mass
wT
taken for moment t
x
c p cv
U f –density
of fluid, i0 –constant diffusion current of inclusions along
y axis, which is supposed known.
In present paper is used treatment based on
investigation of processes of healing of fracture, on
account also
of abovementioned one dimensional
investigations in Lowell et.al. (1993), Gliko (2008),
Ghassemi et.al. (2005), by taking into account both
unsteady two dimensional thermo elastic stresses effects
and thermo-diffusivity effects.
In Lowell et.all.(1993) is used thermodiffusion
coefficient
2b 2
2
et.al.1954). Besides there is thermo conductivity equation
(Landau et.al.1954)
c U c v w § wU x
wU y ·
§ w 2T
wT
w 2T · ,
¨
¸ k¨
¸
2
¨ wx
wt
w y ¸¹
D c v w t ¨© w x
w y 2 ¸¹
©
U f vb0 the current of entering fluid in
crack, v–fluid velocity along x axis of crack,
wwUx
K0
D (T T0 ) ,
wy
U
U
V
§ wU x wU y ·
K0
4
¸ (1.2)
a 2 b 2 , xy b 2 ¨¨
wx ¸¹
U
3
U
© wy
where U is density of elastic media, a , b velocities of
longitudinal and transverse waves, K 0 is volume modulus
of media, D is coefficient of thermal expansion (Landau
b1 x, t b0 U y x, y , t , y | 0 ,
sign, and is solved problem for
a
a2
wc 0
. By K let us denote
wT0
G
constant coefficient of building up of crack surface. It is
written couplex statement of problem of narrowing of
crack surface on
account of
thermodiffusivity,
tribological term with stresses and transverse diffusion
8
w 2U
wy
2
y
b2
w 2U
wx
2
y
(a 2 b 2 )
wU y ·
w § wU x
¨
¸
wy © wx
wy ¹
w 2U
wt 2
y
w 2U x
wxwy
K 0 c U cv
2
, K 0 O P - volume
3
U
cv
elastic modulus, O , P -Lame constants, and for stresses
One can make factorization by Nobell(), Martirosyan
G
where
V yy
U
2 b2
a
2
2b
wu y
wU y ·
§ wU x
¨
¸
w
wy ¹
x
©
wU y ·
wU x
¸
wx
wy ¹
2
(2007) and obtain solution for V 1 V 2 V
The inverse transformations Laplace and Fourier from
V V , gives solution in
Smirnov–Sobolev form
Martirosyan (2007), Bagdoev et.al. (2010)and for y 0
one obtains
§a ·
ii0 a 2
a 1D ¨ D¸
2
SUs a b
t w
t
at
©b ¹
Re
,D
V
2
x
a wt
ax
K
K K
D 3 1 2 3 DD1 DD2i G D
a
a
(1.6)
§
G ¨
wy
©
wu y ·
§
w
u
x
b2 ¨
¸
w
y
wx ¹
©
V xy
U
Let us denote U x
U, Uy
§ at ·§ a ·
ii0 a
at
c ¨ Dc ¨ Dc¸dDc
1
D
x
SUs a b
© x ¹© b ¹
(1.10)
V xRe
³
K3
K2 K3 0 Dc D1 Dc D2i DcG Dc
1
a
a
V.
One can consider further simplified boundary value
problem for half–plane: y 0 ,
wU dV
V
0, f x f
wt
wx
­° K a 2 b 2 U
½° wU
wV
®
[¾
wt
wx
Us
¯°
¿°
0, x 0
­
½
2
à 2
1 2
2
2 2
° a
°
] 1 4 K2 K3 ] ] 1 2 ]
°°1 b
d] °°
b
b2a
a
G D Exp® ³arctg
¾
]D°
§ 2 à2 ·
1
°S1
2
2
K3 ] 1 K2] ¨2] 2 ¸
4
°
°
¨
b ¸¹
a
°¯
°¿
©
(1.7)
§ Ka 2 U
· wV i0
¨
[ ¸
H x H t , x ! 0
© Us
¹ wy U s
where [
b
b0 V
c p c v vb 0 JU
.
c vl
DU s
U ;V of Laplace on t and U ;V Fourier on x . In plane
x, y one can write solution as
f
2
¦ ³e D
U ;V
i x i E n y
K2
K(a 2 2b 2 )U
[,
Us
C0
(K 2 K 3 )b 2
a 2
where
The solution is looked for by integral transformations
U n ;V n d D ,
The
calculations
of
(2.9)
K3
Ka 2 U
[,
Us
for
n 1 f
E1
2
Z2
D , c1 a, c 2 b
(1.8)
2
cn
iZ is parameter of Laplace transformation.
a
D1,3
where s
Placing (1.8) in (1.7) one can for functions : and V V
0,8, K 3
r2.736i give graphs of fig. 1
a ™ t2
2
y 0
e
iD x
dx
x
01
02
03
04
05
06
Obtain Winner-Hopf equation
: 0, 3
1
f
1
V
2S ³0
3, K 2
t ™2 V
2
a
b
Us a
1
,
2 ˜ 10 5
and fig. 2
1 0 ­°§ K a b U · wU § Ka2U · wV ½° iDx
:
³ ®¨¨ Us [¸¸ wx ¨© Us [¸¹ wy ¾ e dx
2S f
°¯©
°¿ y 0
¹
2
cm
sec
r0.9814; D 2,4
10 5
i0
i0
2S iD s Us
iR D E2C 0V 1
(1.9)
2
fig. 1 Dimensionless accelerations of vertical
9
07
at
displacements of crack’s surfaces
2
m
1
k
m2
| 10 4
. One
, a1 |
300 sec1 / 2
U m c pm 2
sec
a12
V
106
at
can for rate
of increasing of temperature in time on boundary fluidelastic massive C1 take two variants,
10
5
C1
x
0.2
0.4
0.6
at
0.8
i.e.
10 3 grad / sec, 10 5 grad / sec, ,
C1 ˜ a1 20 / 3 m ˜ grad / sec 3/2 ,
C1a1
5
.
2000 / 3 m ˜ grad / sec3/2
10
These two variants relates to biological and geothermal
cracks closing.
Using fig.2. one can for negative part of V values obtain
fig. 2
Dimensionless verticaldisplacements of crack’s surfaces
for any values of
x
10V
values of
, and since
at
t
a 10 5 equation b0 V 0 yields
10
10
b0 ˜ V
0
(1.10)
t
t
Approximate taking into account of thermoconductivity by
method of Gliko (2008) yields to additional term in (1.10)
3
8J C1k
minus
t 2 . Then Gliko (2008) condition of
3 U s c pf S a12
Then one, for example, can consider following variants in
case b0 0,1cm , using fig.2
For example one can consider case of thermal fracture
in soils [5] where one has constants
healing of crack yields b 0 and from (3.12) one obtains
equation for a t , which can be solved numerically for
i0 K 3
, J . Furthermore we
,
U sa a
are used formula (3.11) in calculations of time t and
coordinate x of crack’s healing.
given formerly constants
J
Us
For example one can consider case of thermal
cracks in soils Gliko (2008) where one has constants
J
10 5 K 1 , U m 3 ˜ 10 3
c pm
10 3 J / kgK ,
c Pf
4 ˜ 10 3
J
, a
kgK
kg
, Us
m3
105
2,5 ˜ 10 3
kg
,
m3
10 5 K 1 ,
U m 3 ˜ 10 3
kg
,
m3
kg
3
, c pm 10 J / kgK ,
3
m
J
5 cm
4 ˜ 10 3
, a 10
kgK
sec
2,5 ˜ 10 3
c Pf
for biological media and approximately for geothermal
crack in thermo-elastic media, J,K are Jowl and Kelvin
units.
There are carried out calculations for following
constants:
1. For water-like fluids, injecting into crack, thermoconductive
coefficient[5]
cm
for biological media
sec
and approximately for geothermal crack in thermoelastic
media, J,K are Jowl and Kelvin units.
There are carried out calculations for following
constants:
1. For water-like fluids, injecting into crack, thermoconductive coefficient Ebert(1963) p 316.
1
kkal
20
J
,
2 m˜ hour˜ grad 36m˜sec˜ grad
J
Um ˜ cpm 3˜106 3 , K | 300grad , then coefficient
mK
of
temperature
conductivity
is
a1
k
1
kkal
20
J
,
2 m ˜ hour ˜ grad 36 m ˜ sec˜ grad
J
U m ˜ c pm 3 ˜ 10 6 3 , K | 300grad , then
m K
coefficient of temperature conductivity a1 is
k
a12
2
m
1
k
m2
| 104
, a1 |
. One can
Umc pm 2
sec
300 sec1 / 2
for rate of increasing of temperature in time on boundary
fluid-elastic
massive
C1 take two variants,
C1 103 grad / sec, 105 grad / sec, i.e.
10
C1 ˜ a1 20 / 3 m ˜ grad / sec3 / 2 ,
C1a1 2000 / 3 m ˜ grad / sec 3 / 2 . These two variants
2. For metal-like fluids, as aluminum at high
cm
, and thermo-conductivity [5]
sec
4 m
kkal
k 200
, and anew
, a1
3 sec 1 / 2
m ˜ hour ˜ grad
C1 103 grad / sec, 105 grad / sec, there are two
temperatures, a | 10
relates to biological and geothermal cracks closing.
Using Fig.1. one can for negative part of V values
V
x
values of
, and since
at
at
equation b0 V 0 yields. Then one, for
obtain for any values of
a
10 5
variants for technological cracks.
5
For case 2. graphs for b0 10 ;0,1;1 cm are
almost same as case 1. Using method of nonlinear wave
dynamics [3] one can on mentioned graphs, representing
mean curves of healing processes and in the same time
shock waves of probabilities of stochastic processes, use
example, can consider following variants in case
b0 0,1cm , using Fig.1
x
0.064 ,
at
0.0000918856sec , x
x
b)
0.136 ,
at
0.000170554sec , x
x
c)
0.262 ,
at
0.000056268sec . , x
x
e)
0.397 ,
at
0.0000315441sec . , x
a)
t
t
t
t
V
0.0108831,
at
0.588068cm
V
0.00586324,
at
2.31954cm
V
0.0177721,
at
1.47422cm
V
0.0317016,
at
1.2523cm
dx J
| P c , where P c P P0 , P is
dt 2
1
probability and P0 | . Since for macroscopic cracks
2
considered in this paper one can assume processes x(t )
the equation
as almost deterministic one can assume on direct lines of
microscopic sizes of width of crack of fig.3 and
macroscopic sizes of width of crack of Fig.3
P 1 Pc
whence one can in plane x,t construct graph of process of
healing. More detail solution of (3.5), where b 0 , give
for case 1. following graphs x from t for different b0 for
C1
5
dx
dt
J
10 3 , b0=10-5
1
2
and from graphs one can obtain
0.397 ˜ 10 5 ,
1.588 ˜ 10 5
JP c
2
0.397 ˜ 10 5 ,
cm
. The same results are true for all
sec
mentioned other graphs with direct lines. And for curvex
linear part of line fig.3 approximately
3
dx
dt
0.136 ˜ 10 5
5
i.e. for mentioned values of x ,t on Fig.2
16
5
for microscopic crack curve where b0 10 cm,
2
and P c
1
0.00005 0.00010
t
process of healing of crack is more chaotic. The used here
method of shock waves of probability in examination of
curves of process of healing of crack can be applied to
inverse processes of generation of macroscopic crack by
examination of curves of process of transition from small
micro and mezo cracks to macro crack [6].
Let the edge of a semi infinite crack, which contains
the fluid-crystalline inclusions, moves with arbitrary
law in an isotropic thermo elastic medium. Consider the
following problem when the boundary conditions are
generalization of a little changed, written from left to
right, conditions for case of moving crack y 0 fig.3 Time t and coordinate x dependence curve of
crack healing for b0 105 cm
x
6000
5000
4000
3000
2000
1000
0.10 0.15 0.20 0.25 t
fig.4 Time t and coordinate x dependence curve
of crack healing for b0 0.1cm .
11
Vxy
D1,3
V
0, x ! l (t)
wV
wt
2
2
°­ K a b U °½ wU § Ka 2U · wV
®
[¾
¨
[¸
Us
wx © Us
¹ wy
¯°
¿°
Z
Z
and
D1,3 , D 2,4 r iD 2,4 ; D1 , D 2  R
a
a
R D o 1 at D o f . After selecting the branches
0, f x f
(1.11)
of the functions E n , we obtain
function
i0
H x [ H t W , x l (t), [ l(W)
Us
initial conditions u
0, v
0,
wu
wt
0 we have zero
wv
wt
Rr D
0 . Solution
R D as follows
Z ·§
Z
§
G D ¨ D r D ¸ ¨ D r iD
a ¹©
a
©
r
1
transforms from U , V on
form of
f
2
¦³e
U ;V
U , V Laplace
i Dx i En y
x , solution is
En
V2
2
Z2
D , c1
2
cn
a, c2
b,
the
:
V1
E1
U1,
D
boundary conditions (1.11),
P
we
P
(1.14)
where we use the notation
§ wV ­ K a 2 b2 U ½ wU ·
°
°
¨
¸
®
[¾
f f
t
¨
w
U
s
°¯
°¿ wx ¸
st i Dx
: ³ ³e
¨
¸dtdx (1.15)
¨ § Ka 2U · wV
¸
f 0
[¸
¨¨ ¨
¸¸
¹ wy
© © Us
¹
C0
2
2
2
2
2
2
2
Z3
E1 2 D K2 E2 D 2E1E2 K3 E1 E2 D 2D E1E2
b
Z2
E1 E2C0
b2
or
V
S S:
(1.17)
Z ·§
Z
§
·
D D1 ¸ ¨ D iD 2 ¸
¨
a ¹©
a
¹G D
iC0 R E2 ©
1
2
§Z
·§ Z
·
¨ b D ¸¨ D ¸
©
¹© a
¹
Z ·§
Z
§
·
D D1 ¸¨ D iD 2 ¸
¨
a ¹©
a
©
¹G D
R E
2
§Z
·§ Z
·
¨ b D ¸¨ D ¸
©
¹© a
¹
1
2
1
S
§Z
·§ Z
· 2
D
D
¨
¸¨
¸
©b
¹© a
¹
G1 D
Z
Z
§
·§
·
¨ D D1 ¸¨ D iD 2 ¸
a
a ¹©
©
¹
S
§Z
·§ Z
·2
D
D
¨
¸¨
¸
1
©b
¹© a
¹
G1 D
Z ·§
Z
iC0 §
·
¨ D D1 ¸¨ D iD 2 ¸
a
a ¹©
©
¹
1
,
2 3 b 2
a
P PV
Laplace
where
: iÑ0 R D E2V
RD
planes Im D ! 0 and
(1.12)
(1.13)
(1.12) in
2
Im D 0 . Then equations (1.14), (1.15) yield
D
U2
E2
Substituting
obtain
·
¸
¹ ,(1.16)
respectively, in the half
n 1 f
there is a parameter of
transform on t and we have the relations
2
are analytic and different from zero functions,
sought in the
U n ;V n d D
iZ s
3
,
where R D and R D
U , V and by U , V the Fourier
transforms on t from
1
§Z
· 2§Z
·
¨ r D¸ ¨ r D¸
©a
¹ ©b
¹
of (1.11) is sought by the integral transformations of
Laplace on t and of Fourier on x. Denoting
factorization of the
R D R D
R D
(1.11) where "t the law of motion of the crack, K the coefficient of vertical wear. When t
r
2
Function of R D
in the complex plane D has two
purely imaginary and
two real roots:
12
­
½
2
1 2
2
2
2 2
° a
°
1
1
]
]
]
]
K
K
2
3
°°1 b
4
b2
d] °°(1.18)
b2a
a
G D Exp® ³arctg
¾
§ 2 2 ·
aD°
°S1
1
2
2
]
K3 ] 1 K2] ¨2] 2 ¸
4
°
Z°
¨
b ¸¹
a
©
¯°
¿°
§a
·
§a ·
2i2 i ¨ D2i¸ D2i 1G1 D2i dD 2i i ¨ D1 ¸ 1D1G1 D1 dD
w
w
©b
¹
©b ¹
wt § 1·
D ix wt § 1·
Dx
*¨ ¸ a D2i D1 t 2
*¨ ¸ aD1 D2i t 1
a
a
© 2¹
© 2¹
f t, x f
Functions Gr D , Gr
1
ª
to cuts « r1; r
¬
Z
D
a
exp st1 1
dt1 , O
³
* O 0 t1O 1
f
1 3
,
2 2
Gr D 1 (1.20)
for S (t , x ); P (t , x ) one can write
f
a
Gr D 1 b
³
1
a
Gr1 D 1 b
³
1
F1 u du
,
urD
F2 u du
urD
(1.24)
where
form
§a
·
1
¨ D ¸ 1 DG D d D
b
©
¹
F1 u
i
(1.22)
S t , x 3
³ 1
2S f
§ ·
§ Dx · 2
* ¨ ¸ a D D1 D D2i ¨ t ¸
a ¹
© 2¹
©
Since for x ! 0 integrand in (1.22) is an analytic
function of the variable D in the upper half plane of D , so
that one can deform the integration path in (1.22) in
the lower half D plane and at x ! 0 we can write
S t, x
Gr1 ] 1
d]
2Si C³r ] D
Contours
of
Cr are
passed
counter
clockwise. Deformation of the contours on the real axis
brings to the new expressions for the functions given in
the following form
f
Dx
function of the argument t t1 . Calculating the
a
Dx ·
§
integral of
the G ¨ t t1 of
the
¸ function
a ¹
©
variable t1 the formula (1.21) can lead to the following
f
Gr ] 1
d] ,
³
2Si Cr ] D
Gr1 D 1 §a ·
1
¨ D¸ 1DG D
b
i
ª§
Dx·º
exp«s¨t t1 ¸»dt1 (1.21)
S t,x 2 ³ ds³dD³ © ¹
3
1
4Si Vf
§
·
a ¹¼
©
2
f 0 * 0 i
¨ ¸ a DD1 DD2i t1
© 2¹
the G Integral in (1.21) with
respect
to s gives
i
V0f
aº
» . If a closed lines C r include mentioned
b¼
cuts, the values of analytic functions in the outer region are
defined by their
boundary
values using
Cauchy's
formula for
unbounded domains
and using the following representation
sO
D are analytic throughout
the complex half-planes D except for the points belonging
1
ds ³ f s, D exp st iDx d D (1.19)
4S2i V0³if f
Performing the inner integral by substitution D
D can be represented in a slightly different form.
S r (t , x); Pr (t , x ) by the inversion formula one can obtain
V0 if
1
To facilitate computing, the functions Gr D , Gr
It is important to note that the singular points of P r are
respectively in the lower and upper
Computing
the originals
half
plane
D.
F2 u
§a ·
i i ¨ D¸ D1G1 D dD
2H(x) w § at ·
©b ¹
(1.23)
H¨¨ 1¸¸ ³
S wt © x ¹ 1 § 1·
Dx
*¨ ¸ a DD1 DD2i t a
© 2¹
at
x
§ * a2 K K 2 * ·
¨E1 2 2 2 3 u EE
1 2¸
¨ b
¸
a
©
¹
2
2
§ * a2 K K 2 * ·
2
2 2 § K3 * 2 K2 2 ·
S ¨E1 2 2 2 3 u EE
E1 u ¸
1 2 ¸ ] E2 ¨
¨ b
¸
a
a ¹
©a
©
¹
§ a2 K K * ·
¨E1* 2 2 2 3 u2EE
1 2¸
¨ b
¸
a
©
¹
2
§ a K K * ·
2
2 2 § K3 * 2 K2 2 ·
S ¨E1* 2 2 2 3 u2EE
E1 a u ¸¹
1 2 ¸ ] E2 ¨
¨ b
¸
a
a
©
©
¹
2
2
2
E u *
1
13
G u
u 1, E2 u 2
a
u2
b2
G1 u
In the formula (4.13) on substitution
y
:
D 1
,
Dx
t
a
From (4.1) we have that
:
and using Cauchy residue theorem, we can, after
several transformations (integration
by part
and differentiation on p parameter under
sign
of
the integral) to obtain
­
ª a §a ·
º½
°
« b ¨ u¸ u1F2 u du
»°
2i iH x w ° § at ·«
§1 t ·»°
©b ¹
S t, x
H
H
1
1
¸
® ¨
¨
¸¾
§ 1· wt ¨ x ¹¸« at³
©b x¹»°
*¨ ¸ x ° ©
« uD uD i uat
»
1
2
°
© 2¹
«¬ x
»¼°¿
x
¯
(1.25)
Similarly,
we
can obtain expression for
original
P t , x , and
the
functions
1
and iC0 . Since
iC0
V
2i0
U s iS
T
Where : (t , x) ; V (t , x ) , are unknown and must be
V
determined. For originals V t , x , :t , x one can write:
V(t,x) =v(t,x),
: t, x : t, x H x " t : t, x H " t x (1.27)
Here :(t , x)
unknown.
From formula (4.15) is easily seen that
S s, D is
such that
the
above
factorization leads to the
functions S r , P r , originals of
conditions
S t , x P t , x 0 at
x bt , S t , x P t , x 0 at
x ! bt , b " t d " dt b
which satisfy
the
(1.28)
Substituting (1.25) (1.26) in (1.18) and using (1.28), we
can as in Martirosyan(2007), to obtain a solution to the
problem in the form of convolutions of x, t in the form
V
i0
Us
f t
³ ³ S t c, xc H
f 0
(1.30)
H x [ x [ H T 1
a t W
x[
8i0 x [
Re A I11 I12 I21 I22 I23 I24 (1.31)
UsC0
Where
v t , x v t , x H x " t v t , x H " t x x ! " t , v
Substituting S : and S (t , x ) in (1.29) , using
(1.26), after lengthy calculations, we obtain
0
(1.26)
: (t , x) , for
v t, x when x "t is
we
­ a§a
½
·
° b ¨ u ¸ u 1 u T F2 u du
°,
b
§
·
a
°
°
©
¹
H ¨ T ¸¾
®1 ³
u D1 u D 2i u
©b
¹°
° T
°
°
¯
¿
transforms of functions
V
and
x xc [ H t t c W dt cdxc
S (t , x ); P (t , x) are
V V ,
i0
H x [ H t W , x l (t ) ,
Us
S :
:(t , x);V (t , x) are :(s, D);V (s, D) one can write
: : : ,
can obtain using (4.15), represent S : in the form
obtained, respectively, from S (t , x); P (t , x) replacing
x
by
–x
and
multiplying by the
constants
P ª¬ S : P **V H x l 0 º¼ (1.29)
S ª¬ S : P V H l x 0 º¼ ,
14
I11
§
l t0'' x M0 ·
¨ 1 ln M0 1 ¸ H L0 1 ¨ 2 M0 1
¸
x[
©
¹
1
ln T T 2 1 T 2 1 H 1 L0 2
§ u L0 l t0'' [
·
¨
¸
¨
¸
T
u
x
[
M
0
a
¨
¸
b
¨
¸H L 1 du u
1
I12 ³ F4 u 0
M0 ¨
L0
u L0 ¸
1
¨
¸
ln
¨ 2 u 1
u 1 ¸
M0 ¨
¸
u L0 ¹
©
I23
§ u 1 T 2 1
¨
T u
a
¨
b
¨
T 1
³ F4 u ¨
1
¨ 1 ln T 1
¨ 2 u 1
T 1
¨
T 1
©
u 1
u 1
u 1
u 1
I21
I22
§
·
l t1'' x M1 u L1
¨
¸
¨
§
h· ¸
'
¨ u x [ l t1 [ ¨ L1 2 ¸ ¸
a
a
© M1 ¹ ¸
¨
b
b
¸H L1 1 dudh
³ F3 h ³ F4 u ¨
u h
¨
¸
M1 L1
1
¨
¸
u L1
1
ln
¨
¸
¨ 2 u h M u h
¸
1
¨
¸
u
L
1
©
¹
§ u 1T h T 1
¨
¨
h 1T u a
a
b
b
¨
T h
I 24 ³ F3 h ³ F4 u ¨
¨
L1
1
1
T 1
ln
¨
T h
¨2 uh
¨
T 1
©
u H 1 L1 dudh
·
¸
¸
¸
¸
¸
¸H1L1 u
¸
¸
¸
¸
¸
¹
u D1 u D 2i u
a t W
T
x[
l t x ''
n
l tn'' [
hn
, h0 1, h1
0, Mn
h
§a
·
¨ u ¸ F2 u du
b
¹
A 1 ³ ©
D
u
1 u D 2i 1 a
b
,
,
a t tn'' a tn'' W 15
uh
u 1
uh
u 1
·
¸
¸
¸
¸u
¸
¸
¸
¸
¹
§§a
·
·
¨ ¨ u ¸ u 1F2 u ¸
d ¨©b
¹
¸ du
³h du ¨ u D1 u D 2i ¸ u h ,
¨
¸
¨
¸
©
¹
§a
·
¨ u ¸ u 1 u T F2 u ©b
¹
F4 u a
b
F3 h § a·
u¨T ¸dudh
© b¹
·
¸
¸
¸
¸H 1 L0 du
¸
¸
¸
¹
­§
½
l t1'' x M1 ·
°¨ 1 ln L1 h ¸ H L 1°
1
°¨ 2 L1 h
°
¸
x[
a
©
¹
°
°
b
°
°
§
·
F
h
¾ dh
T h
³1 3 ® ¨
1
¸
2
° 1
°
T 1 ¸
H 1 L1 °
° ¨ ln T 1
¨
h 1 ¸
T h
° 2
°
1
¸
° ¨
°
1
T
¹
¯ ©
¿
§ bua bT a T h
¨
¨
uhbT u
¨
a
a
¨
T h uh
b
b
¨
a
a
³1 F3 h L³ F4 u ¨
T
u
1
¨
1
b
b
ln
¨
T h u h
¨ 2 u h
¨
a
a
T
u
¨
b
b
©
l tn'' [
l tn'' x
, Ln
,
Gliko A.O. (2008),”Influence of process of sediment of
solid phase from hydrothermal mixture on healing of
system of plane-parallel cracks”, VI Intern. Conference
Problems of dynamics of interactions of deformable media,
Goris- Stepanakert, pp. 170-177 (in Russian).
Goriacheva I.G., Dobichin M.N. (1988), “The contact
problems in tribology”, Moscow,
p. 253 (in Russian).
Koshliakov N.S., Gliner E.B., Smirnov M.M. (1970),
“The equations in partial derivatives of mathematical
physics” Ed. “Vishaya Shkola” M. p.709 (in Russian).
Kukudzanov V.N., Kolomiets-Romanenko A.V. (2010),
“The investigation of mechanical properties of materials
with ordered structures of defects under dynamic action of
electrical current”, Materials of VIII International
conference on irreversible processes in nozzles and jets,
Alushta, Kryme, M.; pp. 503-506(in Russian).
Landau L.D., Lifshits E.M. (1970), “Mechanics of
continua” M.; Tech.-Theor. Lit., p.795
(in
Russian).
Laytfut T. (1976), “The mass-transfer phenomena in
biological systems”, M.: Ed. “Mir”, p.320 (in Russian).
Lighthill M.J.,Whitham G.B. (1955) ,,On kinematic waves
on long crowded roads’’.Procceed.Roy Soc. A. Vol.229.N
1178, pp150-280.
Lighthill M.J. (1956), “Viscousity effects in sound waves
of finite amplitudes”, Surveys in Mechanics
Martirosyan A.N. (2007), “Mathematical investigations of
insteady linear boundary problems for continua”, Yerevan,
Ed. “Zangak-97”, p.244 (in Russian).
Martirosyan G. A., Bagdoev A. G. (2008), “The
investigation of stochastic processes in biology by methods
of nonlinear wave dynamics”, Dokl. National Academy of
Sci. Armenia,
Vol. 108,No4. pp.325-334.
Mazur I.I. and. Moldavinov O. I. (1989), “Introduction in
engineering ecology”,
.: "Nauka",
p.375 (in Russian).
Mirzoev
F.Kh.,
Panchenko
V.Yu.,
Shelepin
L.A.(1996),,Laser managment of processes in solid’’
Uspekhi Phys.Nauk., vol 166, No 1, pp.42-62(in Russian)
Nobl B. (1962), “The application of Wienner-Hopf
method to solution of differential equations with partial
derivatives”, M.; “Inost.Lit.”, p.278 (in Russian).
Panin V.E.(2000) ,,Synergetic principles of physical
mesomechanics’’. Vol.3, N6, pp 15-36(in Russian)
Ghassemi A. and Kumar S. (2007) Changes in fracture
aperture and fluid pressure due to thermal stress and silica
dissolution (precipitation induced by heat extraction from
subsurface craks. Geothermics. vol.36. Issue 2., pp. 115140.
Lowell R.P., Capperren P.V. and Germanovich L.N.
(1993) , Silica precipitation in fracture and the evolution of
permeability in hydrothermal upflow zones. Science. vol.
260, pp. 192-194.
As seen from this formula, in difference from absence of
tribological effects of Goriacheva et al.(1988),when there
is concentration of stresses near edge of crack in
Martirosyan(2007), if x o l t has no singularities.
2. Conclusions. The study of healing of fractures is very
actual and practical geophysical, technological and
biological problem. Many of works on geophysics are
devoted to investigation of this problem, where there are
used some onedimensional models of description of
healing processes of fractures by precipitation of silica,
containing in hot fluid current, entering in fracture, mainly
on account of thermodiffusion effects, in any of works are
taken into account also contribution in solution from
stressses. There are obtained quantitive and qualitative
results, bassed on these onedimensional along line of
fracture
treatments,
in any works using also
onedimensional in transverse direction to fracture
termoconductivity equation, whose solution mainly used
near top of fracture. Therefore the full investigation of
healing problem using unsteady twodimentional equations
of thermoelastisity in surrounding coupled by boundary
conditions, where are used thermodiffusion effects,
currents of mixtures with crystallines, transverse diffusion
current of them and tribological effects of fracture surfaces
is necessary. In present paper is done solution of this
unsteady plane coupled mixed boundary value problem of
thermoelasticity by application of effective method of
integral transformations, solution of Wienner-Hopf
equation and bringing of solution for vertical displacement
of banks of fracture and temperature to analytical form of
Smirnov- Sobolev due to method in Bagdoev et.al.(2010).
The graphs are constructed and equating of width of
fracture to zero are obtained dependences of coordinate
from time for healing processes and are for various range
of parameters constructed their graphs.For relatively small
values of leading parameters taking influence on solution
to obtaining formula of vertical displacement on fracture is
added term, obtained by solution of only thermodiffusion
onedimentional effects and also are constructed
corresponding graphs. From obtained graphs for different,
macroscopic and microscopic cracks, as well as values of
main parameter determining influence on healing
processes, are obtained physical conclusions of processes
which are in qualitative accordance in investigations
obtained by other treatments formerly.
References
Bagdoev A.G., Vardanyan A.V., Vardanyan S.V. and
Martirosyan A.N. (2010), “ Stress intensity coefficients
determination of dynamical mixed boundary problems for
cracks in elastic space. Application to problems of fracture
mechanics”, Multidiscipline Modeling in Materials and
Structures, MMMS, Vol. 6, No1, pp. 92-119.
Ebert G. (1963), “Handbook on physics”, Phys.-math. lit.
M., p.552 (in Russian).
16