•ıbul Üniv. Fen Fak. Mat. Dergisi
5758 (19981999), 113161
ANALYTIC SOLUTION OF ONEDIMENSIONAL
PROBLEM FOR PARTIAL INTEGRODIFFERANTIAL
EQUATIONS WHICH HAVE PARTIAL CONTINUOUS
C O E F F I C I E N T S IN T H E R M O V I S C O E L A S T I C 1 T Y
THEORY
1
Mustafa Kul
Su miliary: in this paper, a nonstationary problem on thermomechanic wave propagation is solved
in an environment, which is, consists o f a finite thick plate connected with a semiinfinile space.
Materials o f the plate and the space are in conformity with linear viscoelaslicity laws. Mathematical
model o f (he problem consists of: linear equations o f viscoelasticity and heat transfer for each
environment independently, initial conditions and on the connection surface o f environments
conditions o f increasing temperature and normal stress, depending only on time which are given as
known functions, it is assumed that temperature and mechanical fields depend on each other. As a
system, parabolic type partial inlegrodifferential equation o f temperature and hyperbolic type partial
integrodifferential equation o f wave are solved, it is assumed that kernels o f integral operators are
difference kernels. Depending on boundary conditions, functions o f temperature and mechanical
magnitudes become only functions o f time and a space axis, which is perpendicular to free surface. In
this case the problem turns out to be a onedimensional one.
1.INTRODUCTION
Determination o f the reaction of the bodies changing from to dynamical forces is one o f
important problems in the field o f mechanical science. We face to these problems in daily practice
frequently. •
Although pioneering works o f Euler, Bernoulli, Sofi.lerman and others were initialized in
18 century, in the field , it is gained scientific form almost in 2 0 century. Dynamics o f elastic
bodies has important application in seismology and in other branches o f technology. Path breaking
research accelerated alter 40\s o f 2 0 century, reached important results many articles and book were
published. We w i l l mention some o f the published materials that are concerned with our research.
Heal is one o f the important magnitude which effects waves in bodies changing forms. Effect
of heat is high in Composite and Polymer materials. Solution o f the thermoelastic wave propagation
problem in homogenous bar became classical and exists in many books. These problems may be
classified into two; one is dependent and the other is 'independent. In the first, heat and stress
lh
th
th
1. This paper is an English translation o f the substance o f a doctoral dissertation accepted by the
Institute o f Science o f Karadeniz Technical University İn February 1995. I am grateful to Prof. Dr.
M.Sait EROĞLU and Prof. Dr. Musa İLYASOV for their valuable help and
encouragement in all stages o f this work.
deformation fields are mutually effected and can not be determined independently. Mathematically,
in this case, motion and heat equations constitute an interdependent system and solved commonly. In
independent dynamical problems field o f temperature is determined initially then temperature is
added to motion equation as a defined function. It is clear that, in this case, the effect o f stress
deformation field is neglected. As it is seen, in second type solution o f the problem became easier,
mathematically is reduced to the one o f one dimensional problem which is finding solution o f non
homogenous hyperbolic type equation with initial and boundary conditions. The most widely known
of analytical solution methods o f linear problems is integral transform method. Interesting and
difficult part o f this method is calculation o f inverse transforms. Evens in the most simple dependent
problem inverse Laplace transforms can not be find. Owing to this fact in the solution o f similar
problems, we consider asymptotically situations, that is very large and small values o f Laplace
transform parameter, or initially solution o f independent problem is taken and perturbation method is
applied.
In this paper we research one dependent problem o f propagation o f mechanical waves in an
environment which consist o f viscoelastical plate and semiinfmite space. Considering partial constant
coefficient in addition to what explained above makes even more complex. Problem is solved by
Laplace integral transform, and system which is obtained by substation o f boundary conditions is
solved analytically in Computer by using software "Mathematica 2.0". Inverse Laplace transforms
are calculated for very large and small values o f Laplace parameter.
Classical solutions o f thermodynamically problems for elastic and viscoclastie materials are
in [9] and in several books.
In [31] perturbation method is developed to investigate nonstationary temperature field and
stress field in viscoelastic bodies o f properties depend on temperature. In solutions o f thermoelastic
wave propagation in cylinder o f infinite length and in sphere, and viscoelastic plates are given as
infinite series and determined convergence conditions for these series.
In [21] termoviscoelastic problem is approached in a general form by using perturbation
method with temperaturetime analog, and same problems are investigated which are emphasizing
effects o f temperature field on properties o f viscoelastic bodies.
In [10] nonstationary oscillations o f viscoelastic bar are investigated under the pressure o f
periodical variable force considering.temperature effect on properties o f materials. In addition under
the effect o f harmonic load, lengthwise oscillations are studied for viscoelastic bars. Mechanical
properties o f bar depend on temperature. Because o f continuos transformation o f mechanical energy
to heat, temperature o f bar is increased and this in turn changes speed o f the propagation o f waves.
This problem is solved numerically.
In articles [12,13,14,15] and[23,24] effects o f temperature and stress fields on propagation o f
waves in homogenous halfspace are studied.
In articles [1,2,3,5,6,7,16,17,18,19,20,22,25,26,27,28,29,30,32,33,34] related studies are
performed.
2.FORMULATION OF P R O B L E M
Let us consider a nonstationary problem that resulted from mechanical and temperature
strokes in a semiinfmite stratified environment.
Let us take origin point on free surface and let us take xaxis perpendicular to the free surface
downward. Let us assume termomechanical effect, which is in the boundary (x=0) uniform for other
coordinates, only depends on the time. In this situation determination o f stress, deformation and
temperature fields is reduced to solution o f onedimensional problem. Magnitudes which belong to
plate are indexed by " 1 " and those which belong to semiinfinite space are indexed by "2", Thickness
of the plate is shown by " h " .
Let us consider the linear problem o f the related thcrmoviscoelasticity theory and assume that
conditions o f the environment are not depend on temperature . In this situation we obtain following
displacement, heal conduction and state equations.
d a , ( x j )
d
=
dx
p
2
i
f
i
dt
>
( x j )
1
(1)
y d 0 ( xj)
ô V
~ ~ T ^ T
T
dxdi J
dV
?
Ôi
0
' r
cy,(.v.t)J R VT)
"*
i
+ ~Q ((i)
l
tie,
i
fa(t-T)c/O +~\y,0-r)^
l
J
(2)
!
'
Ji// (/r)J0,
(3;
;
Above, i = l , 2 . o, (x.t) is stress, u, (x,t) is displacement, p, is density, Tj (x,t) is temperature, To is
initial temperature, %, is temperature diffusion coefficient, nij (tx), v|/(tT),Rj(tT) and Qj(tx) are
functions which denote mechanical properties o f materials. R, ( t i ) is called volumerelaxation
function, Qi(tT) is called sliderelaxation function. Integrations in (2) and
{3) are Stiljes type and
i k
,
=
•"'•'"</:.,/„ =^ i >
i
/
r
,
O.fx.DF, (x,t)T„
ÔT
and
is deformation.
du Ax J)
,,<*,/) = — ^
Ox
(4)
Q
When we examine the linear problem we assume that e, and — are infinitesimal and the
T
same order. Beside let us assume following relation between extension resulting from temperature
and mechanical properties o f materials.
MM/)=E,Ri(0
(5)
where c, is coefficient of extension resulting from temperature which is independent o f the time.
Firstly using (1) and (3) then (4) and then (2) and (4) we obtain
V
d u,(x,T)
K',</ r )
'\ ,
,,
dWX~
J
,
V
a 0,(x,r)
di - W,(t ~ r )
ih = p,
,f
ÔTÔX
2
}
}
/
J
d u(xj)
' V
of
2
(6)
T
0
= — \m,{tT)
df <
cfcr
— i/r + — J ^ ( / r ) —
dx
dt *
dxdx
(
c/r
(7)
where
G,(/)Ri(/)+|^(/)
and there is no summation according to repeating index. (6) and (7) are partial integrodiiTercntial
equations. u (x,t) and 0j(x,l) w i l l be solved from this equations. I f viscoelasticity properties o f
materials have instantaneous elasticity property then (6) is hyperbolic and (7) is parabolic.
Relaxation function above are continuos functions o f the variable x>0 and when T < 0
}
R,(T>sO,
Qi(T)^0,
mfapQ
V|/i(TH)
Initial conditions are,
u,(x,0)=0
(8)
du,(x,()
= 0
ej(x,o)=o
(9)
(io)
In case x=Q
<T|(0,t)=f(t)
G,(0,t)=q>(t)
(II)
(12)
In case jr=h,
U!(h,t)u (h,t)
(13)
a,(h,t)=cT2(h,t)
(14)
(15)
2
e,(h,t)e (h,t)
2
dû,
= 1:
Dx
dû.
dx
.V
rl,
and for x —» co
lima,(.v,/)<+co
(17)
\ —> >
limÖ,(.v,/)<+co
(18)
Conditions <13).( 14),( 15),(16) show displacement, stress, temperature and temperature diffusion
values are equal on the boundary o f the space and the plate.
3.SOLUTION O F T H E P R O B L E M B Y L A P L A C E TRANSFORM
We will solve the problem (6)(l8) by Laplace integral transform according to lime. I f we
apply Laplace integral transform, with the initial conditions (8),(9),(10),lo the equations (6),(7) we
obtain following differential equation system.
(19)
£ j L - M 7 - ^ =o
dx
a,
a, dx
2
d'O,
T p m,
2
n
dx
x,
2
~
du,
'
=
(
)
{ 2 ( ) )
X,
where u is the Laplace transform o f u.
(21)
f{p)=[f\t)e'"d(
(I
(21 ) is the Laplace transform o f f(t) function, p is the parameter f the transform..
In order to determine u (x,p) and 0, (x,p) functions, a system o f equations is found which is second
t
order and have constant coefficients. Characteristic equation o f the system is
K A,K +BrO
4
2
(22)
where.
Solutions o f the equation (22):
(23)
(24)
(25)
(26)
where, i 1,2. Since the numbers K,, p i . 2 . 3 , 4 , arc' functions o f the Laplace parameter p, they are
complex numbers. But for real p:
Re<K| Kn)<0
Re(K;„l<U)>0
(27)
(28)
l4
Following properties exist among K„ solutions.
KiiKijK.ijlC^B,
(29)
K K
(30)
h
Y ^ ^ J b ,
2 l
Ki,K ,
KirIti,
K|,+K r(K H<4i)
(31)
(32)
(33)
2
3
2|
General solution o f the system (I9).(20) is as follows.
u~ {.wp) =
t
Q,{x,p)
=
+£
2
/
v
+
+ /%/'" + / s /
v
+ E e^
(34)
x
4l
v
+ /V' ' "
A
4
T
(35)
When this values.are replaced to (19) or (20), following relations among F,, and E„ are obtained.
F,,=^-^-E
j
a
i
1.2,3,4,
i-1.2
(36)
P i P i = =?
¥,
where a. ^=r- h
%
From <17).(18).(27) and <28)
.
and in thai way solutions of the system (19),(20) which are limited in infinity as follows:
wjU, p) = E e
k,,ï
n
M A ' , / ? )
Ô^p)
-
+Ee'
+Ee"
K 2,x
K
2l
Eey
K
i2
M
x
(38)
+E e"
K
x
Al
(39)
+E e"
K
x
%2
=^ p - E
n
e ^
K
+
f - ^ E
2
]
e ^
+
(40)
7
As shown solution depends on six constants ( E n . E i i . H i i . L ^ . L i i H ^ ) and these constants w i l l be
calculated from (12),(13),(14),(15),(16). When we apply Laplace transform to (3)
a, (*,/>) = / G ^ / ; < / / , 0,
ax
(42)
?
is obtained. Using (42) from(38),(39).(40).(41)
A
{ X
^
P )
= RIILE
A
ll
U
e ^ + ^ E
K
ll
+
E^'»*
+ ^ E
4
l
e ^
(43)
PlP'
K„v , P2P
<r (.v,/7) = ! ^ A > " +
A',,
"
A',,
h
(44)
v
1
are obtained. When we apply (38),(39),(40).(41 ) and (43).(44) equations to Laplace transforms o f
conditions (11 ),( 12).( 13).( 14),( 15),( 16) we obtain following system o f linear equations, which consist
of six equations.
^
A
^
+
ll
K
^
+
2\
-a,
ii—_L/
K
+
M
^
m
(
A',,"-a, ,
_iL_L£
r
i > | +
î | +
A , ,
/V""" +AV
A',,
A V
A , ,
A'
)
„ . —
L t ^ ^ , , ^
(46)
A
V
(47)
p,A,,
4I
A\,
M
'=°
Pi
A,,
LA/7^M'
Aj!
g.,,^''* = ( )
(49)
( A , , ^ ) / ^ , ^ " " + ( A / ) £ V * " + ( * V ~ « , ) / * : „ t ^ ' + (A'.„ <#,
: i
: |
—
5
A.,,
A\,
M
_Li_L
K..~-a,
^lL_
' + AV*"'' +
A\,
4
P\P'
K-,, -a. ,,
_J
L/
i l | +
A. , ,
A
=
* l l
£
v
(
A f '•
——
p
^l"2
b^e
=0
,
(50)
Zl*>2
When this system is solved by Cramer method;
Eu=-r
A,
(51)
A,
(52)
Eu = ~^
A
A,
£jr=—
A
A,
£ii=—
A
(53)
(54)
£ 1 2 = ^
(55)
A
A,
¿
3
2
=
—
(56)
are obtained. Here , A is the determinant o f coefficients o f system A , /=! ,2,3,4,5,6 are determinant
which are obtained by deleting i' column replacing right side values instead. Right side:
;
h
/(/>)
f ,<?(/>)/), ,0,0.0,0
Pi/
(57)
7
Let us show the coefficient matrix o f the system (45).(46),(47),<48),(49),(50) by
a
\\^~7^ \2
= —;—.a = — , a
A\,
"
A\,
a
l 3
A
A
a., = ™
M
J J
"
„ .
.ct„ = —
L
e
.\\ '~~.— - v
A
'
a
=
„ ,
|
=
,c£| = a , = 0
5
=
a
M
«, —
c
;
A,,£/,
^ i|
e
e
~ — -n = ~ —
A
A
\
,
=
.«„. =
(
u
A
^21
" I I
a
N
M
2.3,4,5,6 where
A'
c "
=
A'j,
P^
=—e
Pi
A , , — <7,
A , ,
"
A ,
A',," i / ,
A ' i / 1
A
a
ft>A'
/ > , A
n
«„, = ( A V « , ) / ' " \ a
««,.»= (K\f
(
,
=(AVf,)
h :
k' "\a ^ U V «1)**
A
M
Zl 2
*l^2
/ ?
Coefficients determinant o f the system and the other determinants are calculated by the formula
A=S(l)
i , )
Aj, M,j
(58)
where Ay are subdeterminant o f 2 x 2 which are subdeterminants o f the matrix o f 2 x 6 which is
obtained from first two rows. While M , | are respecting 4 x 4 minor determinats. Minus sing in front
o f summation stems from calculation o f determinant with respect to first two row.
When we express A and Aj determinants by help o f Mj,"s
A,
( g V / - F , M , +t\M ).
| ;
A, =
| :
r
^ L _ e ' ^ ' ^ ' V / • i A/ +
A
A
i2
A , =-J^e '-''"' ' (F M
lK
-F M
i >l,
i
lx
t
/•",A/ ),
(61)
).
(62)
2)
- F \l
yt
v
2(
A , ^ ^ U ^ ^ ^ ' ' ^ i / v V / , , -F,M +h\M
14
A , =hj^e ^iF M,
x
A'
),
M
F , M » + F M^
K
s
<
(60)
n
(63)
F, A / ),
J S
y
,
A , = 7 = ^ ' " ( F M „ + / v W , / ' ' A / , + F A/ , ) .
(
T
1
2(
)
;
i(
x
l(
(64)
(65)
arc obtained, where
/ • > i
T
(
^
<
, , , f . ^ ^
(
g
2
: "
)
f
Solution ol' so slated problem by the help o f Laplace transform,
considering
consist
of
(51).(52).(53).(54),(55),(56)
and
(59).(60),(61 ),(62),(63).(64).(65)
only
(38),(39).(40),(4I }.(43),(44). Lssenlial solution is obtained by calculation o f inverse Laplace
transforms.
Since calculation o f inverse Laplace transform are very difficult we will find approximate
solutions are asymptotic solutions. These are solutions, for very small and large values o f Laplace
transform parameter p.
firstly, let us assume materials o f semi infinite space and the plate are elastic. In that case
Laplace transforms o f material functions verify following conditions, nij Gj. R, i = l , 2 are constants
and
pm
=m
t
t
(66)
p(T - O,
(67)
P~R,=R,
(68)
t
4.SOLUTIONS FOR V E R Y S M A L L V A L U E S OF L A P L A C E P A R A M E T E R
By using (66), (67), (68) and (22)
/?
r„/>/;;,
t
!
£ /g,7;,/j
f
x>
fl
P%m^
=
t>,x,
where £ ~ = —
P,
^ -4B
i
= PL+ « >P~
T
'
k
•5,
E
m
+
,
R
2
I
X
,,2
2
P
o, X,
2
_ ^ L ^ > <~ X
^ y- 2 1 ^ 2 ,
R
+
2
s,
x,
1
•
s,
T m,E, R,
2
.
o,*,
y
I
2
z
b
<>x
2
I
Since /? and jr are very small can be neglected and
4
A, ~4B =:p
1
2
i
v
is obtained. When we replaced this value in (23),(24),(25).(26) and (36)
p-
T„m,p ^ £, R, T„p
2
|
24,
2
/; T„m,
f
|
x,
2X,
eRT
2
t
2
{)
^
o,x,
and by neglecting / / since it is very small;
in. +
X,
a
and
7oP
X,
7
oP
X,
2 \
m. +
m. +
G
G
(69)
1
}
(70)
' J
are obtained. From which
A.'
P
=
p
are obtained.
(71)
(72)
h.K,
' «p
r
X,
o,
m. +
i
x,
<<V*.
G.
e.-R
m. + G.
2 \
E,
neglecting p"\
G.
F,
7'
in. +
(73)
G.
• J
I
G.
m. +
F,_, =
(74)
G.
(75)
I,,—
E
Fu
(76)
arc obtained. Replacing these values in (38).(39),(40).(41 ),(43),{44)
~^(x.p)
=e
''^E +e''^E +e^E +e^'E,
n
2l
M
(77)
(78)
(3,,, r
L
0,U./>)
I
£,R,
t
41
U^p)
=^
e
^
E
„
+
-
R
M - e ^ E ^
(79)
(80)
(81)
where
(
1
D
7
^
L.
a =
Xi
In thai case by necessary simplification in coefficient matrix
1
a,, =
M M
pa
pa
22
a
( a ) /,/'=!,2,3.4,5,6:
tl
2£,
a,, =
p
~4p<i^ i
=cc .\
2
2
=«
V 2 Si
i
2 i
p
=0
a,, = e
a „. = -c
«„ =
/>//
/—
V /'"'
p
c
2
P
e
,a,
Pi
pat
«52 = V /
PiP
a
S i
JpA
7 £ / t
= a,, = 0 ,
, V / ' "' '
1
X\ 2
h
are obtained. So simplifications o f solutions (51),(52),(53).(54),(55),(56) as follows.
p+2itlij p
:
=
X2
(s.^P, " i , P 2 >
W
"
+
(S V P,
(83)
\(Xi
+
£
X ) \<P
R
(84)
2
^ pM^2 \ 2XlPl^
+ ^ ^ 2
£ G
2
2
£
G
a i P 2
R
l ) ^
+
l 2xX^ Pl
ifAG G
Jp
]
+ Z2 P
2)^"
V2/?
(85)
^ 4» =
V P
e R (pe
2
2
(86)
(87)
AG,4P
•
(88)
feiSiPi
~ ^ |
P .
2
" i
2
2
P
2
§ i i
2
P
2
)P
5 , C A L C U L A T I O N S O F I N V E R S E TRANSFORMS
By using these let us obtain inverse transform. I f
uu{x,p)=E e-"^
u
p.X
Uu(x,p)=E^e
^'
_
_£L
M4l(*,p)=£
4 |
i^'
are defined accordingly then
4
_
f= l
4 _
1=1
Inverse Laplace transforms o f w,i (*,/?) are
2fG Xi\s,i p ||"P| C
1
„, . .,)
i (
v
2
l
£ i A ( ^ ) .
=
>
—
,/
. J „ ( , _
r
)
*
v
—
|/(.Vr
—
jy^ j
r
r 7
|
v2p,is,s2Pi 4 V P ,
• Vr
2
ft^H
l.vjJ———r~
t l
p ^ i S ^ P : o
2
<5>"P
2
^s P2
2
/
(
r
)
£
/
+
r
j <T
I f we define
^2(V,/)^^^''
l
/
,
^'
)
then
// 2
(v. p ) = //12
p ) + z/12 (.v, p )
(A \
//•,(x,/)=// (j:,/)+i/ (jr,/)
12
12
Inverse Laplace transforms o f
u {xj)
=-
t2
;
r f p g
2
AO^ K
4
^
r
Sl^PiSl
l
(
,
/
?
)
=
^
p) and io (.v, p ) functions are
2
^ ( / r ) i / r
t
, *
| / ( T V I
S
P|^2"P2S|S P2
2
If we define
2
1
V r
«.(.v,/) =
0
r
1 / 2 ( JC ,
£
2
i
0
then
M
*
^
)
p)
Z M . Y .
=
/ I
0,(.v,/) = £ ( > „ ( * . , )
Inverse Laplace transforms o f
0,\(\\ p ) / = ! ,2,3.4 are
^ i W , i P i 4,"Pi
2
~
2
M
*
-
'
^
^
1
r
^
'
2i,£|^iP \s 4 Pi
1
l
==
1
l
*«•(*./)=
l
l
/
(
p ^ (£ G x
>j2jiE R aG xM^2P\
]
l
2
1
/
\
AG T[P
;
i
(
s
'
p
^
,
A
iy
£,R,
0*(x,p)=
°
V24V *2
i
P
r
e~&>E„
2
then
O (x,p)^0u(x,
p)+0n(x,
0 (xj)
0 (xj)±0, (xj)
2
1
=
l2
e
,
P
!
2
!</>
U
v/to
>
P I 4 Y P
G
_ rj
2
/
s
+ 2 \X\ ?)
~^P\
"sSV:
2 ]
ii&Pl
2
£
R
2
£ , ' p , &<5 p> j , i
If we define
-
,/(,)+
;
24\ I \PA$£2P\ -%\
i
:
l
;
£ R
2
iii,P )
;
" SI'PI ~ s Y P
2
V2^4 £ /i £//i6\^ ^ ^p
2
P
^
/
1
/
< / > (< )
» „ M =
l
S
p)
R
2
^ ^ p j
'r , ^
£i4%pj >
i
\2\jx
u
)
2
where
:re, Lrf(x)=/erf(x) ; e r f ( x ) = = \e " ds
1
l ^ ' M s i ^ P ,
^ s V P t
S "P2
2
4il:P
3
If we define
an
<r«i{.v./i) = V2 ; p />c'
l
l
A ' ,
1
a„(A.p) = V 2 c p / ^ ' "
l
/ ( v
l
F,
l l
then
_
i
_
a , (v. />)=•£<T |(.V. p )
(
< I
I
CT,(A./)=^0 ,(A./)
f
Inverse Laplace transforms o f a,i
i= 1.2,3,4. are
^ o ' ^ f o ^ p , 5, p,
S1S2P:)
2
:
i
(
,
i
)
=
w k ± i A
i
?
1
/
)
«"('1^1
+ S2P2)
S,(s,P,
s,C Pi
2
^ O C'iX\
SiS:P
X
O" • (.v./)
t
" S i "Pi
4 M S IPI
S1S2P1
4YP
^ s T P i
"sVPl
~ S Y P 2
/(/)•
2
S1S2P2
"S
~ S i P : )
~4Vp2 S , S P 2
/<<)+
2
2 p p ^ , ^ , ( ^ | G \ ^ ^ + £,G,^,^ )
l
Jx \ '\xM\$iP\
aCj C
i f we define
:
Si'Pi
l
2
^ ' P :
S1S2P2)
A
ou(x<p)=-<j21; p pe
2
E
2
%2
then
g 2 (.v, p)-c7\2 (.v, p )+<y n (A\ p )
C7 (x,/ ) = 0
2
| 2
( . Y, / ) + CT
(.V,/ )
I 2
Inverse Laplace transforms o f <T\i(x,p)
l
i
( ^ . ' ) =
A 0
CT
1 2 V
functions are
(105)
^ H ^ r V " ( ' ^
£
\ 2V TT J
i
2
and o^(x,p)
(106)
/
Y, / ) =
4l&Pl
g i V ,
"£,4^2
^2 P2
2
' I
V2<g
2
6.SOLUTIONS F O R V E R Y L A R G E V A L U E S OF L A P L A C E P A R A M E T E R
A1
1 * .P
AR-P"
1
T
4tS,~j+
M 2
1 *, *X P
2
A
£
2
2T
m,
3
2
0
+—y—5
2
X,
<
G
4\
X,
, , 2e RXP
2
/? +
X,
£
—
,
IT^m^Rrp
,
1
+
G,xi,
G,x,
neglecting p
1
- m.
1
Ç,
x,
m.
2Z,
V
K
2
"
=
P
1 ° '
T
24,
2
M
P
2*,
1 >~ ' »P
£
1
R T
1
2G,* 2Ç,
2
(
neglecting / ;
P
A
"
So
~ 7 T
+
—
7 ;
=
,
e,KU,
X,
m.
+
2*.
/ Î7.
/7
A.'
(107)
(108)
Again by (25)
- :
A'
P
, >,P
2
J
•+
, £,R, vP
1
T
+
PU;
0"
X,
/ , 2,
\
+
2
neglecting /?"
2
x,
and so
A'
i7>,/?
(109)
x,
% ,p
m
(UO)
arc obtained.
. >x>
I
4V
(j
G.
J
XiP
e, R, T £,
2G.
2
+
s,
X,
Since for very large values o f p
/ • ; , = ^ ^ Z ( i )
Taking first two terms o f the series
2
0
\ +
2^,X,P
<1
(un
S , ' * , T ; S V
X,
(112)
sV
P .
(113)
7 > , P
V
x,
(114)
'/ 7
Replacing this values in the solutions (38),(39).(40).(41 ).(43).<44)
/<f' Y
(115)
A*
Il
(116)
^l^l'AiSi
l\\ \P
P
m
sY
i»»>\i'
f
si
Xi
r
V
c,R T Ç
2
a
r
C
l7>,/>
c,/?,
0, (*/>) =
p
X,
P
Xx
e'
I
* 2
2
/•;„
(117)
P
X:
PlP"
PiP
II
p
/"
c
*
J
(118)
/ Y
v
"
\
s
g
.
P.P
% \P
m
Xi
e
p
Si
\
C
Î
J
2I
7
>iP
y
(119)
P?P~ _
P:P
(7
p>7p
x^
P
(120)
* <* p
where.
£,/? 7;,g
3
2
2<',X,
Making necessary simplifications in coefficient matrix ( a „ ) . / , /
C
Si
P + ^S
Si
öp =0
, a.. =
7>iP
=a
'/'()'»|P
P
a „ = 2A ^ 1
P
o:^
1.2.3.4.5.6. for large values o f > .
:
V
=0
2(l
,
X
^i
a, i = 0
f
«35
7 7 "
P? t'
a,, =
p
I''' Hh
P, />
İ
V,
I.V "L''
p f
a„
2
Pi
s
V
Ko'^P
V
P
x\
7>,P_
P" _ > i P
7
Si
X|
7
"Si
ct , = 2A
=0
<l'"'
2
Sı
«SM =
/7
p
t
.V
[?>iP
Xl
X I
£|/i,C/
s^^Ji RJ\
„IL
* \P
l
2
m
s/ ,
Xi
{
P
X2
a,
«64
= (),a, = —
<'|X|
2
=
e <
7>'iP
P
\
c
# 2
sı y
e,RJ\,c^R,p
a
= —
<^X|
I '"'V.
I „in, i'
\
Cf,
Y
X\ 2 2 \
£
R
G
*
1. \.11
2
In thai case let us calculate solutions "of (51 ),(52),{53),(54).(55),(56).
I f we expend A , Aj , A ? . A Î , A 4 . As. A determinants with respect to first two rows
(l
.A
A, =
-a ,
-(pb a
2
{
]{
PiP"
/
tt i/>/>,
A,
~ 2\
a
M
PiP
A, =
PiP
A
t
=
P\P
a <pA, a , ,
A/,
u
PiP
A/,
PiP
are obtained. Since exponents are the same in each column .of the matrix 4x6 which consists o f last
four rows o f the matrix (a )
i,j= 1,2,3,4,5.6, using properties o f determinant
n
1*'-p7
V X\
-1
í*W/A
(
& p
Ji»'"><>
M
]2
=>
£
V
If *¿
V *
/ I
V
j'jf'lP
I 'z?
/'''
I'/ JI" '
-I
V *•
1/'
•
M =e
ltt
are obtained.
If we define
16
—
*J
M
M
XA
t
,a,,
a <pb,~
u
z =
p
/
r
«,,of|
aa,
n
|
2
3
after same simplifications, we obtain
X,(2/)^
Z
2
7G 2bJ'G p
I
i
x
+^,c p /f,<p/? )
1
1
l
So,
f
—
a ,
-W^v-
2
~ h\X\P
4\V\P\fo*S*Xi
E
Z
2b
J i U
e
~X\P*
2
+s, ^i'/'o/^)
2
^
A /
'•II
A /
A/13
E
= Ze *
n
Xl
v
*'
-—
A/
13
After calculating Mi,,y2,3,4,5,6,
& 4%
h
E
= ^2Xi M\Pj o iX
£
R
T
irt
and making some simplifications we have
2 \Pi 2 oylXiX2 \ i
(r
R
7
m m
+^iPiX \X\
m
l
+£iPi^T m x
0
2
2
+
^ 2 ^ ^ 2 X 2 )
+ 4; ''»ıPı V > ı * ı + 4ı '»'iPiXiJ^
Sib?i'»iP\>¡Í>\X\
x
7
x
+
1
x
p +
A/ =
ı:
4, "Î;,(V »,/ -> , / / ;
x¿¡
,
I
where
/'ı
V >ıXı
£
V >ı*ı
7
=hı~ \< :X;P\Rı
!
7
V
-Ş\~ \ ıXıPAJI\>MıX
t: (1
XiX..
V
X I Z Ï
1/
y;, m ,
Px=-4 ^z^('2 "?Pi i ^ n"hX
,
R r
'Z¡V;<¡;XiX2PiR*.
+4,
T
i
1 i / ' '
A
J
+ 4i"4 .2 r\ hX{
(
+
İ\ $: Zı \Pı ı ı>4» \"hX\Xı
2
1
G
R r
i
where
A
l = "Sl^l^'lXlXW ,^!
4,'.,^x,x^^J^— +
1
Í T
X
V
V X: y
n
V
X:
y
P\ \
R
Yi
=si *
:
-^ 'ZI ^ -IX,PA
£
X:
"
V
{ X\
Xt
v
Xi
)
and
2xM^
^
+
A</i
tf
f
^
P +
4, »' P 7»
t
V;
4 V ' V"
1
(
/,,'»is,
P
—
/~
V
P
7 V
Moreover.
1
4/''lPi
+
P
—
IL
' ''
7
(\21)
X \ P '
1
ihu£\x*j
/ 7 7
lib
—
+$\~£\PAX2<PP
\
i'
tx,
N>
c
i'.Pi/TiSi^i
2/>(;,c / + 4r> Pi V/ > P
1
!
!
< ' I P I
X,
/;/ 7'
t
!
1
1
//,f
1
r Pi + " i " P ^ + — ( P , + "
P"
/>'
P
0i
si"'",?,,
+
/>'"
XlP
(1
^ 'iVi/ ^ i''i> '| 1 '/ '
|
|
i
+
4
A,
f
Yi<t>
p"
(131)
pY
P
E» =
4W>',Si(siP,
+ S : P : ) /
+
2 4 |
^ I P I ^ , ( S I P ,
2
+
CP:
B
:
Pi4Y0i
02
p»
p'!\
7> '| S j ^
t
0i
Zl ^
7.CALCULATIONS O F IN I N V E R S E T R A N S F O R M
By using above formulas, let us obtain inverse Laplace transforms o f solutions.
uu(\\p)
=e
U2\(\\p)
=e
E
V
b, '
H . I( . Y. / > ) = C
~
£\,
}
'» *
<
(•v./>) = f
7
£.„
£.
arc defined accordingly then
_
w
i „
, ( . r , p) = £ i / , i ( . r , />)
iI
I _
By using inverse Laplace transform theory and Convolution theory, inverse Lapl
tt,\(.v, p)
as follows.
/'V
„„(.v,,)
/__
X ,
2*
1^
sY'»|71t
I
I
/(D
/•
x
J
0
<PiT)
J
v
t — T
x
/- —
V
si
dx +
x
4,
^
-x dx +
)
\<p(x)ci X
G,
4=
f
A"
K
'
2 ^
Si >»I
2*.
2
P.,
{
'
</><<) /
V
+
x
4,
r
*
,
P:
rl /
h-x
r
'I
A'i
<p (r)c/r
J
+
< ^ IXI0I
7
+ P< +
/7 .Y
0i y
s, • y
1
h-x
P:
S|
2/7 Si
f Xl
P I
'
e
j/(rX/r>/r
|
"*M
Si">»l n
'
"Xl
(1
7
+
Sl'^l^t
(I
X|
(l
where
I
F(()*<p(!)=\F(l-s)<p(s)ds
which is called convolution o f F(t) and (p (t)
2/) Si
M ,(.V,/ ).
A ,}/( X/rVr +
I
4
A ,
+
Ml
0i
P
,
J -
H
M,7„
.
L
\f(rli-r)\lr
A ,
+
A |02_
0i
I f we define
/
"
A , |i/)(r)c/r +
}<p(rX/0^
til'
then
//j ( . V.
p)=U\i(\\
U,(xj)
p)+u
n(x.
p)
U^(xj)
/ / | ?( . Y. / ) +
Inverse I.aplacc transforms o f
„, , ( . v . i ) =
and
// I ?( A \ p)
u M(X. p)
= ^ e
Pis
/
has heen calculated as follows.
/r +
I0 I
x
2
f
M
<
—
h
-
civ
r
Y\<t>i
+
0
I
/
)')/
r
\ [r\/
c/ r
(
ip
''i0i
r +
hi
"
1 9
*
(137)
(x / ' ) /
II
Pl4,0i
J/ ( r X/ r V r +
V
2i;, /?, (c, p,
+ s:P
;
0i
vi
K h ^
X
2
^ 4
m
\
n
h X \ X i
0,
1
\(p(v)t/T
+
02
'0")4Y '
0
1
y
i V
(138)
If we define
Ou(x,p)
=
M
2l
X
E,
P
X\
0 ( .p)
L
=
"si
1
X\
'^'.,"'1/'
0 „(*./>) =
p
?
Xi
<S,
X\
Si"
2
"
/
V
7
V x,
then
0.(*./>)=X0,i(*/>)
0, ( * , / ) = X 0 „ ( x . / )
/ - I
Inverse Laplace transforms o f 0,i(x,p)
/= 1,2,3,4, has been calculated as follows.
x+
dr
X\P\
0
V
S !
+
(139)
Xfo
0 (xj)
2i
=
X,
~2hx e R T
2
l
]
0
P\Xx fa
2
" "'%
f,
\
M
J
<
-
h—x
o
-x
\
.
dr +
,_C'~.v)/
{
p,
f w
Si
h-x
r
2
^ 1+ — : —
0,
A
dx
4iPi
0
V
Si
+
//.V
G l * I Vi
0
(140)
\
01
S,
J
T„m.
_ / \
0 3 i M =
2bG,
1
Si T m
dr +
Jf(tr)Erf
0
1
P|£|R|S,
24~r
V
J
T m,
(1
St^om, V
Jf(T)k*Erf
2VT
#1
v
T m
.
0
X
1
JV(tr)Erf
y
o ı
m
X
2VT
X i
(141)
«
2V7
v
T
2/>G,
T
•[A,
o \
m
X\
¡f(t-v)-Erf
2r
£|#|S|P|0|
2
y
(2hx)^|
A.
7>,
1
Jf(r)dT*Erf
0i
Zi
V
"(2hx),
T m,
0
Xi
M«>(tr).Erf
0. 0
drL
2Vr~
(2hx).
24r~
0,
A +
¿102
¿i
7
0i
Xi
T m,
fl
X\
J<p(tr)Erf
)
dx
(142)
If we define
B§
0, (x.p) =
'HH k
2
2
X:
•om,p_
p
l„in p
2
0«(x,p) =
/T m>p
t: R^
e '
V;
^
0
2
«
\
/
then
12 (x. p)+M*< p)
^ ( x , l ) = O (x,t)+0, (x,t)
i :
J
Inverse Laplace transforms o f 0 , , ( , p ) and 0 , ( x . p ) has been calculated as follows.
x
g ,
1
( j
r
, / )
=
2
g
^ ^ n ^ ,
g
^ ) .
2
h
\ "f(
x - h
r
xh!
r
t
xh
dr+
£2
«
v
t
hi
J
M l
0,
dr
G
i0i
>M/
+
,.[xM/
S2
(143)
'$1
T
. !b|
m
(
0i
T m^
2
0
^
2
2
Xl
X\
2£,R,6
I
Jf ( r ) d r * E r f
2
7>
Z
Erf
0
2
\, X,
2
I4~T
T„m
M l
a
2Vr
I f we define
PlP
3
Jt H JL,
e\
E
b+
1
1
J
( X*P ) =
b+^
PlP
' 31
X|
P,P
<J 41
2
7
v
4
i T |
( 'p) Z
i=l
x
=
< T i |
J
(
X i
«"'i/ >
Xl
then
p)
+
h
02
7>|5|
0i
X\
T m,
0
X,
j V( t r ) E r f
cTn(x,p)=
\<f>(tT)
{
01
E
41
dr
Inverse Laplace transforms o f <j,, (.v, p) has been calculated as follows.
(
\
X
a „ ( . v . / ) = <T"
- I
/____
Si )
/
X
A /
Zi
*1
, gr iPi*i
G,
g
X s,
c
c
/ ..v
+
Si J
v.
A"
—
X,
%\
o
V
- X
tlx
(145)
Si
l)\
CTI|
hx
( ),_^L i>0
e
XA
r
Xi0i
'
Pi0>
v
)
Si
,
t
} f ( r i t i ^ r k
0i
i
+
/
Si^Pi^iZ^
G
lX|0,
h x
, h ( h x
Si
<r (x,l)=2b§
ill
jf(tr)Erf
l
hx
P
+
I
*
•
x
JY(r)dr*ErT
+
Srm,T
()
0
2r
G,
j
T
(
2r
o
m
, /
X
i
~
0i
^
dr
2r
V
4\( < )=
(2/fx).
r
X
x
Jfl>(r)d
/
T ,m,
(J
0
jT m,
v
Erf
^1
s,
dr +
2r
dr
\f(t-x)Erf
(147)
)
7>,
2r
A, +
V
¿102 ^
0.
,
(146)
(2hx)^>i
I
Jf(r)dr*Erf
Al
<• +
2t
¿1 J ^ ( t r )
10i
G
(2hx)^p^
Erf
(2hx)J
dr + A
2r
2
+
^.|J (t.r)Erf
V
?
ri
y<i
T
o
m
/
2r
If we define
B+^ k
CTi (x,p)=
2
B
P
+
CT.12
p p
];rn,p
J
(x,p)
V XT'
2
E
then
(7 2 (x, p) = O 12 (x, p ) + <T M (x, p)
0 (x,t)^(T (x,t)+CT, (x,t)
2
|2
3
Inverse Laplace transforms o f cr 12 and on
has been calculated as follows.
x l i
h)
r,
PiS,0i
Jf(r)|t^rdr
0
V
Jf(rjt
'
S2
xh
—
«2
'
. V
+
Jf(r)d r +
r
2
¿,02
01
y ^ - ^ B y
T dr
G
i0i
xh
xh
r
t~
+
V s y
[Xt<P
dr
2
2
dr]
+
V
01
^
,
a
4 h ( i , n, > \ ,
x
i
i
(I
( x h ) j ' ^
I
Jr (tr).;rf
m,
dr +
2V7
(xh)fe h
Xi
V0i
,
+
jY(r)dr*l;rf
V
T
'
m
X,
;
'
*
2VT
X:
2^
dr +
0,
X,
(xhl
JV(tr)Eii
* 2
dr
(150)
8.SOLUTIONS FOR VISCOELASTIC MATERIALS FOR LARGE VALUES OF
TIME
We mentioned before how difficult it was finding solutions o f problems for viscoelastic
bodies. One o f difficulties stem from functions m, , i / / , , R, , G, which are included in solutions o f
Laplace integral transforms. In order to reduce difficulties partially it is assumed that ratios o f these
functions mutually do not depend Laplace transforms parameter. In that case for very smalt values o f
p, and taking m,n constants,
m.p = m ( l + pm)
;
C i . p ^ O ' ^ l + pn)
formulas are used. So using solutions in part 5 solutions in large values o f time for viscoelastic
materials are found as follows. Again symbols in part 5 w i l l be used.
u (x,l) =
l
X .i( ' 0
u
i I
x
•s/P
S ı S ^ P i + Sı~Pl
2
2 )«,
+ S1S2P2
dr
J<p(tr)^<p'(lr)
(151)
2i'^,X,ls,4SP
u
a
ı
( x , t ) ^ '
R
< f :
^ j [
+
dr
( t r ) H y ( t r )
y
(152)
dr
aAO^i.Xilsi^Pi Sı'Pı " S :
P2S1S2P2
j V / r „L
2
+
(153)
\
/
u (xj)
M
=
P2£ı4Sfeı°2#2P
7
"ÜlOzXté&Pl
11
+^2^1X1^2)
,
>
/
X
X
m
I W r ) — <P ( t r )
" İ Y P l ~h{ Pl 5 l 5 l P 2 j VÎT „L,
2
2
"SVP,
2
u (x,t)=
+
(154)
J/(rVrf/(/)
V2plki4 Pl
dr
" « S I^PÍ )
" ^ P j
u (x,t)+u,,(x,t)
2
1 3
u (x,t) =
1 2
VI
e,R,
"
A
r
r
(155)
<p(tr)™<p ( l r ) | d r
AG V^ or
2
u„(x,t)
(156)
t
Sı^Pı 4ı"Pı 4 Y p
S1S2P2
2
0 . ( ^ 0 1 0 , (x,t)
1=1
ft
(
x
^
t
fol ^ ^ i f e P l
=
H
2^ (<5,<S P ^
2
1
|
2
2
P l
'
^
P
;
+
^ ' ^ ^ )
"^2 P2 H ^ P ?
p|
2
f
p (
t
)
(157)
)
(158)
2*ı
[f(t)nf"(t)nf(0)]
ö.„(x,t) =
2
^£| lPlfel^2Pl
R
^| P,
^ Pi(A^ ¿ G ^ p R,
2
2
t
2
2
2
2
S 2 V 2
+a^ £ G|XıP2^2)
2
2^^ £ R aAG p x ^ ^ Pı "S^Pı
l
1
i
2
!
1
1
3
"Él&pJ
2
S 2 V 2
S1S2P2)
J <p(tr)
+
i/r
(159)
— ç ) (/r)
'Jvr
,
ö„(x,l) =
°
^
:
^
4
—
[
r ( l )
.
n r
(t)]
I
t/r
—<P l ' r )
I <P
(160)
Ö (x,l) = ö„(x,t)+ö ,(x,t)
i
J
0 (x.t)=flîrr
,f
l UT
(161)
f(tr)dr
P
)
^
G
0 (x.l) =
nf
t
,
3 2
(162)
l
Pl ~S2 P2 ~~h\blP-
<M* -0 = I > i i M
(163;
a,
2aG /,(| £ p, £,~p, ~ í
1
¡
O", , ( x a ) = ' ^ ; ^
2a "G
p
(7
E
) R
2
2
2
(164)
k(t)m (t)]
y
g,fe,P,+g P )
2
„(x,t) =
4,S P.
2
S2 P Í1S2P2
2
~St"Pl
2
R
¡
1
1
2
S1S2P1
V2p,p
SıfeıP)
S2P2)
sTPı
İ Y P 2
£
G
R
2
f(l)
S1S2P2
£
G
R
+ £
G
•/
9
'
nm
X
t
_
r
'
+
»
/
X
t/r
^ r ~ ^ *'~ '
r
R
2
<P ( l r ) + y Ç ) ( / T )
c/r
(166)
V / T a A G ^ ^ ^ ^ p , S| P, S2 P "ÉiS^P:)
2
2
2
CT, (x, t ) = (T,, (x, t ) + C7 (x. t )
i 2
a
1 2
(M) = ^ j E r f
AG
¿
' Ax^
<p ( t r ) +
(p ( / r ) | d r
(167)
t
(168)
2
hÁlPl
2
<x, (x,t)=
2
" S l ^ P l < § ' P l " ^ f P i Sl^2P>
t
(165
+
Sı 4 ( l >X2 l 2 i X l 2)
2
;
S2 2 |Xlp2 2) 'l
S2 P2 S1S2P2) 0
a
+
V?TaAG , G X i ^ i ^ 2 P ı " 4 V P ı
C7 . (x,t) =
•f(t)
2
2^ ^P|(A^£ G X2P2 I
u
p ~íi^P: j
V2^2
9.SOLUTIONS FOR VISCOELASTIC MATERIALS FOR SMALL VALUES OF
TIME
In this case for very large values o f p. and taking M . N constants
mip=mj 1 +
G,p = G'
M/
Py
N
H
\
PJ
formulas are used. So using solutions in part 7 solutions in small values o f time for viscoelaslic
materials are found as follows. Again symbols in part 7 will be used.
u (x,t)=Xu„(x.l)
l
i
I
u,,(x.t) =
Jf(r)d
S,Pi
m,
Si
2*.
7
^ ^ t } f ( r i t f
X;
i '(ri
t
dr
r
£
{
£,R
SI^ -I
dr +
S,
jv(r)d T +
G
069)
ill -V) A .
/ï . Y
c/r +
Si
(hs
J
Pi0i ^
/
V
01
hx
)
dr +
o
V
P:
Si
f,R,^
G
2
lX,0|
Pi
J<p(r)dr +
1/
•Ut \
Sı
2b£,
u„(x,t) =
Jr (rXlr)J
^^|r(rXir) d
3
r +
^X\
11
(1
m,
J^(rXtr)d
J<p(r)d r +
X,
G
u,,(x,t)=—
2b,
A. +
A, j f ( r X t ) d r + •
r
|l(rXt
ı
/
m,
ı„n iıU i> \ ;
ı
+
G
I
A J<p(r*i r +
(
ı0ı
A, + M
1
Xi
Í
T,, m,
ı
u (x,l) = u (x,l)+u, (x,t)
2
l 2
2
x- h
{y M / iı
hi
J
ri
2
42
;
1
c/r + —
2
Y 3
+
2
+
J
X- h
M-
-(x-h)N/
D
ti T
G
$2
\ h
h)/
(v
A,
rifa
73 +
7r
(173)
Jf(rXlr)d r
u (xj)
i 2
f'"
'J
m
i
2\ ,
r
I I I f'»"'l
V 12
Jr ( r X t 0 d r
\ /A
2
01
2 g
+
/ri
II
ı ^ ( s ı P ı +S2P 2 ) n S 2 ^ " 2 2 A / ı : X ı 2 2
T
R
m
+
m
r +
,| HI|
^02
T
..™ısı
Xi
0i
2
V
(174)
7
Trt
0,(x,l)=¿O (x,l)
i I
il
fl„(x.l):
XlPl
-/M
sV™l o
T
Jf(r)d r
+
o
V
Si
dr +
v;V
Sl"
j<p(r)ch +
V.
W
Xi
lô
?
A
,
*S,
0
[rl /
v
£1
(175)
r
y
J±Lİİ
t
PıX,~0,
ı
'< r | /
î
{
+
T
/ S ı
VÍrVr + J ^ j
. (/< v ) . V
v/ı
J
p
V(r)j/
O
SI
•
1 " rir
(r
P3 + ^ f ^ ^ l P l
01
2bG,
Pıtısı
r
— t
GıZı 0ı
/Si
M
)
P.
;
;
2'
03 i
CİT +
T
Sı
iz
/ S ı
Pl
—
• •> ,,,
v
sı
h-x
t
CİT + — P , + ^ ^ , P ,
2\
. 0ı
/
/ 7 .Y
ATl
í
Pl
(176)
+
X\
X,
2Vr
ÚT +
|T m|
0
5|
Xl
O l jV(tr)Erf
Xi
T
m
2bG
0„(x,t) =
£
(
R
I
s
I 4 Í P I 0 I
A, J f ( t r ) E r f
(177)
Yx +
Y
(179)
dr
$ 2
.m,
0 (XA)
X,
Jf(tr)nrf
= -
X2
dr +
2Vr
PıSı0ı
i O _l_
T
01
2£|R,G X2(siPl
2
+
2V7
X2
XI
Xl
X2
r * Rif
m
+b2P2V 0 lX|
T
m
01
(xh)
Xl
X
i
j V (tr)*Krf
dr
7
02
'V 'i s Y
01
Xl
'h ihi
m
J(/)(/r)A>/
V X2
V Xl
X2
o"i(x,t)=X n(x.t)
< 7
(T j i ( x ,
t)
e
X (
\
X
r t — +
l
Si J
4ı'^ı o
T
¡f(r)dr
X,
x¡
0'»l
b
dx
(180)
f
<p
\
{</>(r)dr
X
Xl
t —
\
Xi
Si )
J ^ r j t f ~ r dr
V
0
a (xj)=
Si
t -e
2l
.
-
1
P.
P
X,0,
(I
c/r +
si
l
;
0i
V
(hx
SI
Mir
r dr
G
lX|0I
i
o, (x.t)2b^
i
Xl
Jl'(tr)Erf
l
V\<P(f-
2r
dr +
h-
Jf(r)dr*
X I
o
' <> I
T
Erl*
m
Xl
2t
Si
£i iP|
R
G i
JV(tr)Erf
Xl
2r
dr
T
x
s. "m ,
)
m
V x.
2r
JV( l r ) E r f
XI
«
o
dr
V
J
(2h xX
Ai
< T | ( x,t ) =
4
( 2h x)
V
1rf
(183)
ÍT n i|
0
Xl
JY (tr)Eif
d r + A? +
+
b1 £|P,R|
G,0|
(2h x)t
i
A, jV(tr)Erf
0
I
jV( t r ) E r f
+
0i
V
y
( 2h x) J^
V X,
2r
dr
l2
12
xh
I
f—
——
Ci
Í '"fır
.V -
«
h p
V
r i/ r +
V
01
II
irJ
?2
/
É/ T
—•
V
+
J yin
42
,
H
F
R
y F n
l iX2S2P>
fc A
G
"i0i
r_i02
¡
(r)d r *
,' o i ^
Vr
V
0i
Vi
1
2r
m
Xl
dr +
(184)
<y (x,t ) = e r ( x, l ) +<x ( x, t )
2
J7
T
lom.1
x\
2t
A, Ml
A ¡02
-ix'
ii
1
v
2Í,
2
V /l
j
v
( (,£Z*_ |
r
r
(185)
i / r
CT
V
Jf ( / r ) E r f
T m,
0
dr +
+ h
2
+
Xi
•
Xl
,
0
>+
{«Pv ^)
1
7>,
h
\
^02
X,
2Vr
(xh)
m
0 i Sl
0,
^ ^ W ^ l Z l ^ ^ R * '
Q
T
T m,
2s/t
Tm
2 \
02
I 01
Xi
jf(r)dr*Erf
Jp ( t r ) E r f
Xi
2V7
(xh)
Wit
V
I :
^'"liV
V0I
T m,
n
;
i olll,
+h
X2
X,
N
)
X,
dr
(186)
2V7
[1] A C H E N B A C H ,J.D.,SUN , C . T . and H E R R M A N , G . : On the
of the a laminated body ,LAp\?\. Mech., 35 (1968), 467475.
[2] B A R K E R , L . M : :A model for stress wave propagation
J.of Comp. Mater, 5 (1971), 140162.
vibration
in composite
materials ,
[3| B R I L L O I J I N , L . : Wave propagation
182(1952)
in periodic
[4j C A R S L A W ,H.S. and J A E G E R , J . C :
structures .Dover publishing,
Operational
Methods in Applied
Mechanics, Oxford University Press. 1941.
|5j C H E EN, C.C. and C L I F T O N , R . J : Asymptotic bilaminates, Proc.
14th Midwestern Mechanics Conf. University o f Oklahoma , 1975,399417
| 6 | C H E N , P . J . and G U R T 1 N , M . E . : On the propagation
deceleration
waves in laminated
od
one-dimensional
composites. J.Appl.Mech. N40 {1973), 1055¬
1060.
|7|
C H R I S T I N S E N , K.M.-.Wave
Mech.42 (1975).
propagation
in layered
elastic media, J.Appl.
153158.
|8]
C H R I S T E N S E N , R . M . : T h e o r y of Viscoelasticiiy,Academic press. 1971.
|9j
C R I S T E S C U , N . : D y n a m i k ' p l a s t i c i t y , NorthHolland Publishers, 1967.
.JlOj C R A C H , S . '.Wave propagation in a viscoelastic material with
properties and thermomechanical
coupling. Transactions o f A S M E ,J
temperature-dependent
O f Appl . Mech ., 86 (1964) .423429.
fll|
G U T E R M A N , M . M and N I T E C K I , : Z . H . : Differential Equations The
Saunders Series,1984.
112] H E G E M I E R ,G.A. and N A Y F E H , A . H . : A Continuum
propagation
in laminated
composites,J.Appl.
1 n j ^ H E T N A R S K S , R.B.: Solution
of series of function.
of Coupled
theory for
wave
Mech.,40 (1973), 503510..8
Thermae last ic Problem
in the
form
Arch. Mech. Stos.,6, N4 (1964), 3239.
[ 1 4 | . . H E T N A R S K I , R.B.: Coupled
Thermoelastic
Problem for the
halfspace^uW.
Acad. Polan. ScuSer. Sci. Techn., 12, N l (1964)J20128.
115] I G N A C Z A G , J . : Thermal displacement in a non-homogeneous elastic semifmite space,
by sudden heating of the boundary, Arch.Meech. Stos.,10 N2 (1958), 152159.
[16| K A R M A N , T h . : ^/ 7 the propagation
OSRDNo.365 (I942).
of elastic deformation
caused
in solid, NDRC report No,A29,
[17] L E E , E . H . : D y n a m i c s of Composite Materials, A S M E Appl.Mech.Divisions Series, Book
No.H0078(1972).
[18] L E E , E . H . a n d KANTER^J.: Wave propagation
o f Appl. Physics, 24, N9 (1953) 1115.
in finite roads of Viscoelastic Materials, Journal
[19]
L E E , E . H . and ROGERS J".G.-.Solution of viscoelastic stress analysis problems
using
measured creep or relaxation functions. Brown University Technical Report D A G 5 4 / I , J. Appl.
Mech, 30 Trans.ASME,85 (1963), 127133,Series E.
[20]
M O R L A N D , L . W . a n d LEE,E.H .:Slress
analysis for linear viscoelastic
temperature variation, Transactions o f the Society o f Rheology, 4 (1960), 223.
materials
with
[21] M U K I , R . and S T E R N B E R G ^ .-.On transient thermal stresses in viscoelastic materials with
temperature dependent properties. Journal o f Applied Mechanics,28,Trans. A S M E ,83.( 1961). Scries
E.
[22] P E C K , J . C and G U R T M A N , G . A . \Dispertive pulse propagation
laminated Composite, J.Appl. Mech., 36 (1969),479484.
[23] S N E D D O N , I . N . : The propagation
Sec. A , 9,65 (1959), 115121.
parallel to the interfaces of a
of thermal stresses in thin Metallic rods. Proc. Roy. Soe.
[24] SOOS,E.:77;ii Green 'functions (for short time) in the linear theory of Coupled
Arch. Mech. Stos.,18, N l , 1218.
thermoelastieitv.
[25] S V E , C :Stress wave attenuation in Composite Materials, J.Appl. Mech., 39 (1972),11511153.
[26] T A Y L O R , G . I . : Propagation o f earth waves from an explosion, British official Report .R.C5570.
(1940)21.
[27] T I N G , T . C . T . : Simple waves in an extensible String
[28] T I N G , T , C . T . - . D y n a m i c a l response of Composites,
J . Appl.Mech.,36 (1969),893896.
Applied. Mech. reviews, 33 ,NI2 (1980).
[29] T I N G , T . C . T and M U K U N O K I , I :A theory of viscoelastic analogy for wave propagation
normal to the layering of a layered medium, J. Appl. Mech. reviews, 46, N3 (1979),329336.
[30] T I N G , T . C . T and M U K U N O K I , I -.Transient wave propagation
finite layered medium, Int. J. Solids Structures, 16 (1980), 239251.
normal to the layering of
[31] V A L A N I S , K . C and L1AN1S, G.A.: Method of analysis of transient -thermal stresses in
thermorheologically
simple viscoelastic solids,Transactions
o f the A S M E . J o f Appl. Mech.,86
(1964), 4753.
[32] V A L A N I S , K . C . : Method of analysis of transient thermal
simple viscoelastic solids,Mechanical Engineering, 86 (1964), 64.
stresses
[33] V A L A N I S , K . C and L I A N I S , G.'.Error analysis of approximate
viscoelastic stresses, Purdue University Report A and ES 6213, (1962).
H d l V A I . A N I S . K.C:
in
thermorheoiogiea/ly
solutions
of
Studies in Stress Analvsis o f Viscoelastic Solids Under NonSteadv
thermal
Temperature Gravitational and Inertial Loads, PhD thesis, Purdue University, 1963.
Mustafa Kul
Department of Mathematics,
Faculty of Science,
Istanbul University
34459 Veznecileristanbul
TURKEY
Keep reading this paper — and 50 million others — with a free Academia account
Used by leading Academics
Nikos Mantzakouras
National & Kapodistrian University of Athens
Bissanga Gabriel
Marien Ngouabi Brazzaville
Florentin Smarandache
University of New Mexico
Nasreen Kausar
Yildiz Technical University
