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)

5

Introduction to the theory of thermoelasticity

6

Generalized thermoelasticity of an infinite body with a cylindrical cavity and

variable material properties

Two-temperature generalized thermoelasticity with variable thermal

conductivity

0117

7

8

Introduction to The Theory of Thermoelasticity

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)

27

1-6

Theory of coupled thermoelasticity

V(Undeformation)

oT

(External Loading)

(Deformation)

iu

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]

45

[55]-[57].

46

Generalized Thermoelasticity of an Infinite Body

with a Cylindrical Cavity and Variable Material

Properties

47

21

Introduction

22

The basic equations

[58]

(2.1) ,i o o,i

KK T 1 T T e .

t

(2.2) i i j , ju .

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

60

95.1

85

0801

0505

61

62

63

64

65

66

67

68

69

70

TWO-TEMPERATURE GENERALIZED

THERMOELASTICITY WITH

VARIABLE THERMAL CONDUCTIVITY

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 .

88

t 0.2

11.0

x1K

1K 0.0

1K 0.1

0.1

1511

89

90

91

92

93

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.

105