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1
Introduction
) (Theory of Elasticity
[1] (Timoshenko and Goodier
8761(R. Hook)
(Elasticity)
(Elastic)(Fung) [2]
2
(Cauchy) (Theory of Linear
Elasticity) ً [3] (Thermally Isolated)
(Thermoelasticity)
(Stresses) (Strains)
(Sokolnikoff) [4]
Duhamel) [5] )[6] (Neumann)
[5] (Duhamel)
8111[6] (Neumann)
[5] (Duhamel)
(Uncoupled) .
(Governing Equations )
3
8517[7] (Biot)
(Coupled)
(Thermal Continuity) ,
8576[8] (Lord and Shulman)
) (Generalized
(Isotropic )8511 [9] (Dhaliwal and Shereif)
(Anisotropic)[10].
(Thermal diffusion)
4
(The elastic modulii) (Thermal conductivity
modulii)
51
511
8571[51] (Chen and Gurtin)
8561[53] , [52] (Warren and Chen)
0117 [12] (Youssef)
(conduction temperature The heat) ا
(The thermo-dynamic temperature)
(Heat supply)
6
Generalized thermoelasticity of an infinite body with a cylindrical cavity and
variable material properties
Two-temperature generalized thermoelasticity with variable thermal
conductivity
0117
9
11
Theory of linear elasticity
(perfectly elastic)
(Homogeneous (
(Isotropic)
[13], [2]
(Displacement Vector)iu
(Tensor)ije
ij
(Contineous Medium)V
SiF
i ij jp n in
(1.1.1) ii ji j
V S V
F d V n dS d V .u
10
ijij,k
V (Gauss Divergence Theorem)
(1.1.2) i j i i j
V
+ - d V = 0 ., uF
V
(1.1.3) i j i i j + = ., uF
(1.1.4)
i j i j j i j k i i j k j i
V S
= - d V + d S = 0 .uM x F x n
(1.1.5)ijk ji = 0 , i , j, k = 1 , 2 , 3 .
11
(1.1.6) i j j i = , i , j = 1 , 2 , 3 .
ijk
ijk
1 2 31
i j k
1 2 3.1
i j k
0
[13]:
i j i j j i
1 = ( + ) ., ,e u u
2
ije
(1.1.7) ij jie e .
زوجي إذا كان عدد التبديالت
فردي إذا كان عدد التبديالت
تساوى على االقل اثنين منهمإذا
12
: [4]
(1.1.8) ij ijkl klC e .
ijklC
ijkl ijlk klij jiklC C C C .
1708
W
(1.1.9)ijkl ij kl
1W C e e .
2
ijklC
ijkl ij kl ik jl il jkC ,
ij(Kronecker delta function)
ij
1 if i = j
= .
0 if i j
13
(Hook’s law)
(1.1.10) ij kk ij ije 2 e .
Lamé’s
Coefficients)
(1.1.11) i j
i j k k i j
- 1 = + .e
2 ( 3 + 2 ) 2
ijeij
0 3 2 0
| | < , | 3 + 2 | < .
14
12
Fundamentals of thermodynamics
(Open) ( Close)
(state)(boundary)
(walls)
( Heat exchange)(Guggenheim) [15] , (Andrews) [14]
(Adiabatic)
(Isolated)
(Equilibrium) (Andrews) [14]
(Reversible)
(Irreversible)
15
(Andrews)
[14]
(1.2.1)dK dE dQ dW .
WQ
E K(Andrews) [14]
(Entropy (
(1.2.2) e id d d .
ded
id
ed Q
(1.2.3) e
Q = ,d
T
T
16
id
id 0 (Irreversible )
id 0 (Reversible)
Qd
T
Qd
T
[15] , [14] .
13
Thermodynamics of deformed elastic body
V
(Conservation law of mass)
(1.3.1) V
ddV 0 ,
dt
17
(Continuity equation )
(Nowinski)[16]
(1.3.2) d
dV 0 .dt
(1.3.3)
i i i i i j j i
V V V S
i i
V S
d d 1d V u u d V F u d V n u dS
dt dt 2
Qd V q n dS .
Q(Heat source )iq
ix
iuin
(1.3.2)
(1.3.4) V V V
d d d dd V d V d V d V .
dt dt dt dt
(1.1.3)
18
(1.3.5) i i i i ji , j i
V V V
d 1u u d V u F d V u d V .
dt 2
(1.3.3) (1.3.4)
(1.3.5)
(1.3.6)i j i, j i, i
V
du q Q d V 0 .
d t
(1.3.7) i j i , j i, i
du q Q .
d t
(1.3.7) [16]
[16]
(1.3.8) i j i j i j ,ie ,T,T ,
(1.3.9) i j ,ie ,T,T ,
(1.3.10) i j ,ie ,T,T ,
(1.3.11) i j ,ie ,T,T ,
19
(1.3.12) i j ,iq q e ,T,T ,
( Helmholtz’ function)[16] , (Hetnarski) [11]
(Nowinski)
(1.3.13) T .
(1.3.12)
i i i j , j i j l j lq k k T k e .
ikijkijlk
iq,iT
i i i j , j i j l j lq k k T k e .
ijeT
(1.3.14) i i j , jq k T , i , j 1,2,3 .
ijk
20
84
The theory of uncoupled thermoelasticity
( Isotropic)
0
iz
iz
1 2 3x , x , x
(Nowinski) [16]
[17] (Nowacki),:
0 i i T = ( 1 + ) ,Z Z
21
T(Thermal Strain Coefficient)
ije
(1.4.1) i j i jT = .e
(Elastic strain)ije
(1.4.2) i j ij ij = e + e ,e
i j i j j i
1 = ( + ) ., ,e u u
2
(Neumann)8158ije
ij
(1.4.3) i j k k i jij
1e = - .
2 2 ( 3 + 2 )
22
, ,
(1.4.1)- (1.4.3)
(1.4.4) ij i j k k ijT
1 = - .e
2 2 ( 3 + 2 )
(1.4.5) i j k k i j i j ij = + 2 - ,e e
T = ( 3 + 2 ) .
(1.4.5) (Duhamel-Neumann law)
8111
8111 [16](Nowinski) , (Nowacki) [17].
(1.4.5)
(1.4.6) i j i i j + = ,, uF
23
(1.4.7) i i j i j j j i i + ( + ) + - = ., , ,u u uF
(Anisotropic)(1.4.5)
(1.4.8) i j i j k l k l i j = - .C e
ijklC
(1.4.8) (1.4.6)
(1.4.9) i i j k l k l i i j
j j
( ) - ( ) + = .C e uF x x
(Homogeneous )
(1.4.10) i i j k l k l i j j i j - + = ., ,C e uF
V
S
ixiq [16]
(Nowinski) , (Nowacki) [17] .
24
(1.4.11) i i = - k .q ,
k
tdS
(1.4.12) i iW = k d S t ., n
(Heat Equilibrium ) 1V
1S
t
1
i i
S
W = k d S t ., n
(Heat Source)Q
1V
W" = Q d V t .
25
W = W + W" .
1V
t
1
E
V
W = C d V t ,
EC(the specific heat)
W = W + W" .
1 1
iE i
V S
( C - Q ) d V - k d S = 0 ., n
(1.4.13) , i i Ek = C - Q .
(Thermal Diffusion Equation)
26
(Fourier’s law)
(1.4.14) i j j i = - .q ,k
(1.4.13)
(1.4.15) i j E j
i
( ) = C - Q .,k x
(1.4.7)(1.4.13) (1.4.9)
(1.4.15)
15
(1.4.13)
(1.4.15)
(1.4.7)(1.4.9)
28
0T T
[11]
0
1 ,T
(1.3.7)(1.3.13)
(1.6.1) i j i j i,ie q Q T T .
(Chain Rule) (1.3.11)
(1.6.2)i j ,i
i j ,i
e T T .e T T
(1.6.3)i j
i j
,e
(1.6.4),T
(1.6.5) , i
0 ,T
(1.6.6)i,iq Q T 0 .
29
(1.6.4)(1.6.6)
(1.6.7)
2 2
i,i i j 2
i j
q T e T Q 0 .e T T
(Potential Function )
(1.6.8) 2
i j 0 i j i j i jk l i j k l i j i j
1e ,T C e C e e e d .
2
0
ij ijkl ijC , C , , d
ij ij, e ,
ij 0C 0
(1.6.9) 2
i jk l i j k l i j i j
1C e e e d .
2
(1.6.3)
30
(1.6.10) i j i jkl k l i jC e .
(Stress equations)
(Duhamel-Neumann equation)ijklC
(The elastic modulii )ij
(The thermal modulii ) [11]
i jk l jik l i jlk jilk ij jiC C C C , .
(1.6.7) (1.6.9)
(1.6.11) i j , j o i j i j,ik T 2d e Q 0 .
T0T
(1.6.5) (1.3.13),iT
31
(1.6.12)i j
i j
d d T d e .T e
(1.3.13), (1.6.4), (1.6.9)(1.6.12)
(1.6.13)
2
2T T ,
T T T
(1.6.14) E
2dC T ,
T
oT T
(1.6.15) E
o
d C .2T
(1.6.11)(1.6.15)
(1.6.16) i j , j o i j i j E,ik T e C Q 0 .
i j i j . i j i jk k
32
(1.6.17) ,i E o ii,ik C T e Q 0 .
0k k
(1.6.18)o , i i E o iik C T e Q 0 .
(1.6.10)
(1.6.19) i j i j k k i j2 e e .
T = ( 3 + 2 ) T
(1.6.10)(1.1.3)
(1.6.20) i i j k L k iL j j i j - + = ., ,C u uF
(1.6.21) i i j i j j j i i + ( + ) + = + ., , ,u u uF
(1.6.20)
(1.6.16)(1.6.17)(1.6.21)
33
17
(1.5.16) , (1.5.17)
(parabolic partial differential equation )
[18](Weiner) [19] (Nickell and
Sackman)[20] (Hetnarski)
[21][22].
(Ignaczak)
[23] (Nowacki) [24](Takeuti)(Tanigawa) ( [25]
34
(Bahar)
(Hetnarski) [26]
18
Theory of generalized thermoelasticity
8576[8] (Lord and Shulman)
(2nd sound)
[10] (Dhaliwal and Sherief )
Anisotropic))
35
(1.8.1) i j i j k l k l i j = - .C e
ij (Coupling parameters )
(1.8.2)
i i i i
V V
j j i i j
S
d [ ½ + U ] d V = d V v v vF
d t
+ ( - ) d S .qv n
VSi iv u
U
iFiq
in
(1.8.3) i j i i j + = ., vF
36
(1.8.4) i j ji .
(1.8.2)
(1.8.5) i j i j i i- = U- .q , e
(1.8.6) i i = - T .q ,
T0T 0T
(1.8.5)(1.8.6)
i j i j
1d d U d e
T T
(1.8.7) i j i j
i j
U 1 U d = d T + - d .e
T T T e
37
dT
ije
(1.8.8) U
= , T T T
(1.8.9) i j
i j i j
1 U = - .
T e e
2 2
ij ij
= .
T Te e
(1.8.8)(1.8.9)(1.8.1)
(1.8.10)
i j ij
i j
1 U - = .
T e
(1.8.7)
(1.8.11) ij i j
U d = d T + d .e
T T
38
(1.8.12)E
UC = .
T
EC
0T T
(1.8.12)(1.8.11)
(1.8.13)ijE i j = C logT + + constant .e
00T Tije 0
(1.8.13)
(1.8.14) i jE i j
o
= C log 1 + + .eT
0
log(1 )T
0T
(1.8.15) o o i j E i j = C + .eT T
39
(1.8.6)
(1.8.16) o i i = - .q , T
(1.8.15)
(1.8.17) o i jE i i j i = - C - .q , eT
(1.8.18) i j o j i i
+ = - .q q ,k
o
(1.8.17)
(1.8.19)
o i j o i j o i jE j i j
i
C ( + ) + ( + ) = .,e eT k x
iu(1.8.1)
(1.1.3)
(1.8.20) i j i j j i = ½ ( + ) ,, ,e u u
40
(1.8.21) i i j k l k i l i j
j j
- + = .,C u uF x x
(1.8.19)(1.8.21) :
(1.8.22) 0 0 k k 0 k kE i ik T = C ( T + T ) + ( + ) ,, e eT
(1.8.23) i i j i i j j j i = ( + ) + - T + ,, , ,u u u F
t = ( 3 + 2 ) .
(1.8.24) ij ij kk ij2 e e .
(1.8.19)
(1.8.21)(1.8.22)(1.8.23)
41
19
(Hyperbolic partial
differential equation)
(parabolic partial differential
equation.. )
[8] (Lord and Shulman)8576
(Ignaczak)
[27] (Ignaczak)[28] (Sherief)[29] (Sherief and Dhaliwal)
(Sherief )[30]
[31] (Anwar and Sherief)
[32] (Sherief)
[33] (Sherief and Anwar)
(Anwar and Sherief )[34]
[35] (Sherief) (Sherief and Anwar ) [41]–[36]
(Sherief and Ezzat ) [42](Sherief and Hamza) [43]
42
(Youssef) (El-Magraby)(El-Bary)
[50] –[44] .
811
Theory of two-temperature generalized thermoelasticity
8571[51] (Chen and Gurtin)
T
8561[53] , [52] (Warren and Chen)
0117[12]
(Youssef)
43
(1.8.18)
(1.10.1) i j o j i i + = - .q q ,k
ijk o
(1.10.2) ,iiT a .
a 0(The two- temperature parameter)
(1.8.19)
(1.10.3) o i j o i j o i jE j i j
i
C ( + ) + ( + ) = .,e eT k x
(1.8.21)
44
(1.10.4) i i j k l k i l i j
j j
- + = .,C u uF x x
0T T .
(1.10.5) 0E 0 k k 0 k k i ik = ( + ) + ( + ) ,, C e eT
(1.10.6) i i j i i j j j i = ( + ) + - + ,, , ,u u u F
t = ( 3 + 2 ) .
(1.10.7) ij ij kk ij2 e e .
(Youssef and Al-Lehaibi)[54]
46
Generalized Thermoelasticity of an Infinite Body
with a Cylindrical Cavity and Variable Material
Properties
48
(2.3) i j i j i jkk = 2 + e - ,e
ij i, j j,i
1e u u .
2
o
o o
T T1
T T
[58]
(2.4)0 0 0 0f (T) , f (T) , K K f (T) , f (T) .
0 0 0 0, , K , f (T)
f (T) 1
(2.4)(2.1)(2.2) (2.3)
(2.5) i j i j i jo o kk o = f T 2 + e - ,e
49
(2.6)
i j i ji o o kk o , j
i j i jo o kk o, j
u = f T 2 + e - e,
f T 2 + e - e
(2.7)
o
o , i o o o i,i, i
K f TK f T T 1 T T f T u .
t
23
Problem formulation
R
(r , ,z)z
rt
r zu u(r , t ) , u (r , t ) u (r , t ) 0 .
50
f (T)[58]:
*f (T) 1 T .
*(Empirical Material )
0
0
T T1
T
f (T)
*
0 0f (T) f (T ) 1 T .
(2.8)
22 o o
o 2
o
Te ,
t t K
22
2
1.
r r r
(2.9a) o o o o
eu f T 2 ,
r r
51
(2.9b) r u1
e .r r
(2.10) r r o o o o
uf T 2 e ,
r
(2.11)
o o o o
uf T 2 e ,
r
(2.12)
zz o o of T e ,
(2.13)
zr r z 0 .
[58])
12
o o2 uu
12
o o2 rr
o o oo
2
o o2 tt
o
o
'T
12
o o2 RR
52
2
ba .
12
o o
o
2
o
o
gK
o oT
b
(2.8)
(2.12)
(2.14) 2
2
o 2ge ,
t t
(2.15) 2
2 2
o 2
ef T e a ,
t
(2.16) 2 2
r r o
u uf T 2 b ,
r r
(2.17) 2 2
o
u uf T 2 b ,
r r
(2.18) 2
zz of T 2 e b .
53
24
Formulation in the Laplace transforms domain
(2.14) (2.18)
st
0f s f t e dt .
(2.19) 2 2 2
o os s g s s e ,
(2.20) 2 2 2s e a ,
(2.21) 2 2
r r
u u2 b ,
r r
(2.22) 2 2u u2 b ,
r r
(2.23) 2
zz 2 e b ,
54
*
o
1.
1 T
e(2.19) (2.20)
(2.24) 4 2 2 2 2 2
o os 1 s s s s s 0 ,
a g .
e
(2.25) 4 2 2 2 2 2
o os 1 s s s s s e 0 .
(2.19)(2.20)
(2.26) 2
2 2
i i 0 i
i 1
A p s K p r ,
(2.27) 2
i 0 i
i 1
e B K p r .
0K (.)
55
1 2 1 2A ,A ,B , Bs
1 2p , p
(2.28) 4 2 2 2 2 2
o op s 1 s s p s s s 0 .
(2.26) (2.27)(2.20)
iB
(2.29) 2
i i iB ap A , i 1,2 .
(2.27) iA
(2.30) 2
2
i i 0 i
i 1
e a A p K p r .
(2.30)(2.9b)
r
(2.31) 2
i i 1 i
i 1
u a A p K p r .
1K (.)
56
(2.31)
0 1z K z d z z K z .
(2.26)(2.31)(2.21) - (2.23)
(2.32) 2
2 2
rr i 0 i i 1 i
i 1
2a A s K p r p K p r ,
r
(2.33) 2
2 2 2
i i 0 i i 1 i
i 1
2a A s 2p K p r p K p r ,
r
(2.34) 2
2 2 2
zz i i 0 i
i 1
a s 2p A K p r .
r R
1
Thermal boundary condition
r R
oR, t H t .
H(t)(Heaviside unit step function)
57
1 H(t) 0
H(t)
0 H(t) 0
(2.35) oR,s .s
0
Mechanical boundary condition
r R
r r oR, t H t .
(2.36) or r R,s .
s
(2.26)(2.32)(2.35),(2.36)
1 2A , A
(2.37) 2
2 2 oi 0 i i
i 1
p s K p R A ,s
58
(2.38) 2
2 2 oi0 i 1 i i
i 1
2 ps K p R K p R A .
R a s
25
Inversion of Laplace transforms
[59].
tf sf :
ct N
1
k 11 1 1
e 1 i k i k tf t f c R e f c exp , 0 t 2t.
t 2 t t
N
1
59
1
1 1
i N i N texp c t Re f c exp .
t t
c
sf[59]
FORTRAN
2-6
results and discussionNumerical
[58]
oK 386 N / K sec , 5 1
T1.78 10 K
, K/m1.383C 2
E ,
2sec/m73.8886 , 10 23.86 10 N / m , 10 2
7.76 10 N / m ,
38954kg / m
sec02.0o , K 293To , -1
1K -0.1 K ,
11.618 , 2 4 ,
a 0.01041 .
t 0.15
o 1 1o
1
71
31
Introduction
0117
32
The governing equations
[45]:
(3.1) ,i o o,i
KK 1 T e , i 1,2,3 .
t
K
e0T
72
oT T .
(3.2),iia , i 1,2,3 .
a 0
K
[45]
(3.3) o 1K K 1 K .
0K
1K 0
(3.2) (3.3)
o 1 1 ,iiK K 1 K aK ,
a1K
1 ,iiaK
73
(3.4) o 1K K 1 K .
(3.3)K
(3.5) o 1K K 1 K .
(3.4) (3.5)
K K .
(3.6) o 0
1K d ,
K
(3.7) o 0
1K d .
K
(3.6)ix
(3.8) o ,i ,iK K ,
ix
(3.9) o ,ii ,i ,iK K ,
74
(3.7)ix
(3.10) o ,i ,ik k ,
(3.7)t
(3.11) ok k ,
(3.9)(3.11)(3.1)
(3.12)
2
o,ii o 2
o
T1e .
t t K
(3.13) i j, j i i , j j ,iu u u , i 1,2,3 .
(3.10)
(3.14)
o
i j, j i i , j j ,i
Ku u u ,
K
0K1
K( )
[45]:
75
(3.15) i j, j i i, j j ,iu u u .
(3.16) i j i j k k i j = 2 + ( - ) , i 1, 2,3 .e e
(3.2)
(3.6)(3.7)
(3.2)ixK
(3.17) ,i ,i ,iiiK K a K , i 1,2,3 .
(3.8)(3.10)(3.17)
(3.18) o ,i o ,i ,iiiK K a K , i 1,2,3.
(3.9)
(3.19) o ,ii ,i ,i o 1 ,i ,iK K K 1 K , i 1,2,3.
(3.20) o ,ii i o 1 ,i ,ii ,iiiK 3K K K , i 1,2,3 .
76
o 1 ,i ,ii3K K
(3.21) o ,iii ,iiiK K , i 1,2,3 .
(3.21)(3.18)
(3.22),i ,i ,iiia , i 1,2,3 .
(3.23) ,iia , i 1,2,3 .
33
Formulation of the problem
(0 x ) x
(x 0)
tx
77
(3.24) x y zu ,u ,u u x, t ,0,0 ,
(3.25) u
e .x
(3.12)
(3.26)
2 2
oo2 2
o
T1 u,
x t t K x
(3.15)
(3.27) 2
2
uu 2 ,
x x
(3.28) xx
u = 2 ,
x
(3.2)
(3.29)
2
2a .
x
78
oT
2
2
o0 0
c
2
oct t
ocu u
ocx x
2
o
2c
oT
oT
.
(3.26),(3.27),(3.28)(3.29)
(3.30)
2 2
o 12 2
u,
x t t x
(3.31)
2
22
uu ,
x x
(3.32) 2
u = ,
x
(3.33)
2
2.
x
79
1
o
.K
o
2
T
2
2
o
2
ac
st
0
f s f t e dt .
(3.30)-(3.33)
(3.34) 2
2
o 12
ds s e ,
d x
(3.35)
2 22
22 2
d e ds e ,
d x d x
(3.36) 2 = e ,
(3.37)
2
2
d.
d x
(3.38)
2
1 1 12
de ,
dx
80
(3.39)
2
2 32
d ee ,
d x
2
o1 2
o
s s
1 s s
1 2 1
2
1 1 2
1
1
2
1 1 2 1
3
1 1 2
s 1.
1
4 3
State-space method
e
x(3.38)(3.39)
[60]:
(3.40)
2
2
d V(x,s)A(s)V(x,s) .
d x
1 11
32
A(s) .
x,sV(x,s)
e x,s
81
(3.40)
(3.41)V(x,s) exp[ A(s) x]V(0,s) .
x
exp[ A(s)x]
A(s)
(3.42) 2
1 3 1 3 1 1 2k k( ) 0 ,
1k2k
1 2 1 3k k ,
1 2 1 3 1 1 2k k .
(3.41)
(3.43)
n
n o
[ A(s)x]exp[ A(s)x] .
n!
2A
0 A II
82
(3.43)
(3.44)0 1exp[ A(s)x] a (x,s)I a (x,s)A(s) .
0a1asx
1k 2k
A(3.44)
(3.45)1 0 1 1exp( k x) a a k ,
(3.46)2 0 1 2exp( k x) a a k .
(3.47)
2 1k x k x
1 20
1 2
k e k ea ,
k k
(3.48)
1 2k x k x
1
1 2
e ea .
k k
83
(3.49) ijexp[ A(s)x] L x,s , i, j 1,2 ,
1 2k x k x
1 1
12
1 2
e eL
k k
2 1k x k x
1 1 2 111
1 2
e (k ) e (k )L
k k
1 2k x k x
2
21
1 2
e eL
k k
1 2k x k x
2 3 1 322
2 1
e (k ) e (k )L
k k
(3.41)
(3.50) ijV(x,s) L V(0,s) .
(3.51) 1 2k x k x
1 1 0 2 1 0 1 1 0 1 1 0
1 2
1e k e e k e ,
k k
(3.52) 1 2k x k x
2 0 2 3 0 2 0 1 3 0
1 2
1e k e e k e e ,
k k
(3.37)(3.51),
84
(3.53)
1
2
k x
1 1 0 2 1 0 1
k x1 2
1 1 0 1 1 0 2
e k 1 k e1
k k e k 1 k e
35
Application
x 0
1
The thermal boundary condition
x 0
(3.54)
10, t H t ,
1
(3.55) 10 , s .s
85
(3.5)(3.6) (3.55)
(3.56) 00,s ,s
211 1
K.
2
2
The mechanical boundary condition
x 0
(3.57) e 0, t 0 ,
(3.58) 0e 0,s e 0 .
(3.56) (3.58)(3.51)(3.53)
(3.59) 1 2k x k x
1 2 1 1
1 2
k e k e ,s k k
86
(3.60) 1 2k x k x2
1 2
e e e ,s k k
(3.61) 1 2k x k x
1 2 1 1 1 2
1 2
k 1 k e k 1 k e ,s k k
(3.25)(3.60)
(3.62)
1 2k x k x21 2
1 2
u e / k e / k ,s k k
(3.5)(3.6)(3.59)
(3.63)
1 2k x k x11 2 1 1
1 1 2
2K1x,s 1 k e k e 1 ,
K s k k
(3.4)(3.7)(3.61)
1 2k x k x11 2 1 1 1 2
1 1 2
2 K1x,s 1 k 1 k e k 1 k e 1 ,
K s k k
(3.64)
(3.60)(3.64)(3.36)
87
1 2
1 2
k x k x
2
1 2
k x k x2 11 2 1 1 1 2
1 1 2
e e =
s k k
2 K1 k 1 k e k 1 k e 1 .
K s k k
(3.65)
36
Inversion of Laplace transforms
37
Numerical Results and Discussion
[60] :
oK 386 N / K sec , 5 1
T1.78 10 K
, K/m1.383C 2
E , 2sec/m73.8886
, 10 23.86 10 N / m , 10 2
7.76 10 N / m , 38954kg / m , sec02.0o ,
K 293To , -1
1K -0.1 K ,
11.618 , 0.1 , 2 0.01041 .
94
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102
Summery
This work is concerning with elastic materials, i.e a material which
deformed when it is acted upon by an external forces. Almost all engineering
materials possess, To a certain extent, the property of elasticity. Thermoelasticity
is a branch of applied Mechanics that investigates the interaction between the
strain and the temperature fields using the thermodynamics of irreversible
processes.
In the introduction, we presents a review of the historical development of
the theory of thermoelasticity and give a literature survey on the main subject of
this theses.
This theses consists of three chapters as follows:
In the first chapter, we give the review of the classical theory of linear
elasticity and the basic principle of the theory of thermodynamics. Contains also
the derivation of the basic equations of the theory of uncoupled thermoelasticity
showing its basic shortcomings, the derivations of the basic equations of the
coupled theory of thermoelasticity showing how this theory has eliminated one of
the two shortcomings of the theory of uncoupled thermoelasticity and how this
theory still predicts infinite speeds of propagation of heat waves in contradiction
to physical observations, the derivations of the basic equations of the generalized
103
theory of thermoelasticity with one relaxation time and shows how this theory has
dealt with the shortcomings of both the uncoupled and the coupled theories of
thermoelasticity, the derivations of the basic equations of the two-temperature
generalized thermoelasticity with one relaxation time and finally, the basic
equation of magneto thermoelasticity.
In the second chapter, The equations of generalized thermoelasticity with
one relaxation time in an isotropic elastic medium with temperature-dependent
mechanical and thermal properties are established. The modulus of elasticity and
the thermal conductivity are taken as linear function of temperature. A problem of
an infinite body with a cylindrical cavity has been solved by using Laplace
transform techniques. The interior surface of the cavity is subjected to thermal and
mechanical shocks. The inverse of the Laplace transform is done numerically
using a method based on Fourier expansion techniques. The Temperature, the
displacement and the stress distributions are represented graphically. A
comparison was made with the results obtained in the case of temperature-
independent mechanical and thermal properties.
In the third chapter, the consideration of variable thermal conductivity
with linear function on temperature has been taken into account in the context of
two-temperature generalized thermoelasticity (Youssef model). The governing
equations have been derived and used to solve the problems of an elastic half-
space medium in one dimensional. The governing equations have been cast into
matrix form and state-space approach with Laplace transform techniques were
used to get the general solution for any set of boundary conditions. The solution
has been applied for thermally shocked medium which has no strain on its
104
bounding plane. The numerical Laplace transform has been calculated by using
the Riemann-sum approximation method. The distribution of the conductive heat,
the thermo-dynamical heat, the strain, the displacement and stress have been
shown graphically with some comparisons.