R ECOMMENDATIONS FOR F UTURE W ORK

Ⅳ THREE DIMENSIONAL ELASTIC SOLUTIONS OF A

7.2 R ECOMMENDATIONS FOR F UTURE W ORK

The work accomplished in this thesis can be improved and enhanced by further research on the following points:

• By using the method of undetermined coefficients, it is possible to expand the approach to solve the multi-layer or multi-material problems.

• If the singular points can be completely found in the process of inverse transforms, then the proposed solutions could be easily extended to determine the displacements, strains, and stresses resulting from three-dimensional point loads in an inclined transversely isotropic half-space. The interesting results would be addressed in the

near future.

• Using the relation of the determinant of

[ ]

d , and the velocity of body waves, it is ij

possible to solve the loading problem for the cubic and orthotropic materials.

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APPENDIX A THE EXPRESSIONS OF aij (i, j=1-6)

The elastic constants in Eq. (3.06) can be expressed as:

11 a

a =

φ ₃ ₅ ²

2 4 1 21

12 a (a 2a )cos (a a )sin

a = = − + −

φ ₁ ₄ ²

2 5 3 31

13 a (a a )cos (a 2a )sin

a = = − + −

φ φsin cos ) 2

( ₁ ₃ ₄ ₅

14 a a a a a

a = = − − +

φ φ

φ ₃ ₅ ² ² ₂ ⁴

4 1

22 a cos 2(a a )cos sin a sin

a = + + +

} 4 cos )]

( 2 [

10 6 8{

5 3 2 1 5 3 2 1 32

23 a a a a a a a a a φ

a = = + + − − + − +

φ φ}sin2 2 cos )]

( 2 [

4{ 1

5 3 2 1 2 1 42

24 a a a a a a a

a = = − + + − +

φ φ

φ ₃ ₅ ² ² ₁ ⁴

4 2

33 a cos 2(a a )cos sin a sin

a = + + +

φ φ}sin2 2 cos )]

( 2 [

4{ 1

5 3 2 1 2 1 43

34 a a a a a a a

a = =− − + + + − +

} 4 cos )]

( 2 [

6 2 8{

5 3 2 1 5 3 2 1

44 a a a a a a a a φ

a = + − + − + − +

φ φ ₄ ²

2 5

55 a cos a sin

a = +

φ φsin cos ) ( ₄ ₅

56 a a a

a = = −

φ φ ₅ ²

2 4

66 a cos a sin

a = +

64 0

63 62 61 54 53 52 51 46 45 36 35 26 25 16

15 =a =a =a =a =a =a =a =a =a =a =a =a =a =a =a =

APPENDIX B THE DERIVATION OF THE CHARACTERISTIC EQUATION According to Eq. (3.02), the x’ and y’ axes are in the plane of transversely isotropy.

The generalized Hooke’s law for a transversely isotropic material can be expressed as:

⎥⎥

The strain-displacement relationship for small strain condition is:

[ ]

co-ordinate system.

The equation of force equilibrium is:

⎥⎥

For the elastic dynamic problem, an arbitrary time-harmonic body force in x’, y’, and z’ direction with angular frequency ω can be written as:

)

Further, Eqs. (B.5a)-(B.5c) are preformed the triple Fourier transforms as:

Substituting Eqs. (B.5a)-(B.5c) and Eqs. (B.6a)-(B.6c) into Eqs. (B.4a)-(B.4c), we have the triple Fourier-type integrals as:

*x'

Rearranging Eqs. (B.7a)-(B.7c) as:

⎥⎥

[

* ^*z'

]

* y '

x u u

u ≠

[

0 0 0

]

is called the eigenvalue of this matrix.

2 }

) (

a a ) (

{ a

2 }

) (

a a ) (

{ a )]}

( a a [ {

d d

2 2 5 2

2 2 2 1 2

2 2 5 2

2 2 2 1 2

2 2

4 2 5 2 2

33 2 32

2 23 22

13 2 12

− γ + β + α + γ + β +

− α ρω

Δ ⋅ + γ + β + α + γ + β +

− α ρω

⋅ β + α + γ

− ρω

−

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

ρω

− ρω

−

(B.9)

where Δ= [(a₁−a₅)(α²+β²)−(a₂−a₅)γ²]²+4a₃²(α²+β²)γ² .

In addition, the variables α, β, γ can be expressed in terms of k, θ_x_', θ_z_' as:

' x ' z cos sin

k θ θ

α ,

' x ' z sin sin

k θ θ

β ,

cos z

k θ

γ .

Hence, the relationships between α, β, γ, and k, θ_x_', θ_z_' are:

' 2 z 2 2

2 ) k sin

(α +β = θ ,

' 2 z 2 2=k cos θ

γ .

Moreover, by introducing ² ₂²

V =ωk (V denotes the body-wave velocity), the final results for the three body-wave velocities are identical with those in a transversely

isotropic medium, as follows:

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎟ −

⎟

⎠

⎞

⎜⎜

⎝

⎛ + −

− −

= +

2 1

2 1 2

5 2

32 2 2 1 5 5

2 33 2 2 1

3 5 a

4a a

a a a a a a

a a a a a 2

A 1 .

As depicted in Fig. B1, considering a new co-ordinate system x, y, z and the original system x’, y’, z’ with a common origin point be taken so that x and x’ axes lie in the xy plane (basic plane). The angle between the x and x’ axes is taken as one co ordinate, ϕ, and the angle between the y’ axis and the xy plane as the other co-ordinate, φ. We assign the cosines of the angles between the axes of the old and new co-ordinate system by Table B1.

Table B1 the cosines of the angles between the old and new axes

x’ y’ z’

x cosϕ sinϕ 0

y -cosφ sinϕ cosφ cosϕ sinφ z sinφ sinϕ -sinφ cosϕ cosφ

Then, the value of D in Eq. (B.11) can be presented as:

[

2 t

]

2 t i 3

1 i 6 52 2

t 2 , , SV t , , P t , , 6 SH 3

cos sin

A k a a

) V V

V ( k D

θ + θ

⋅

⋅ ρ

Π=

θ θ

θ (B.12)

where θ_t is the angle between the vector (α, β, γ) and the z axis, and it can be expressed in terms of α, β, γ, and φ as:

2 2 2

cos cos

sin sin

cos sin

γ β α

φ γ ϕ φ β ϕ φ θ α

+ +

= −

t (B.13)

Additionally, from Eq. (B.13), ₂ ₂ ₂

Hence, Eq. (B.12) can be rearranged as:

[ ]

Eventually, the six eigenroots can be generated by setting D=0 in Eq. (B.14) (also in Eq. (3.45)). They are respectively expressed as follow:

(j=1-3)(B.15a)

(j=4-6)(B.15b) As depicted in Fig. 1, considering a new co-ordinate system x, y, z is obtained from the original system x’, y’, z’ by rotation an angle φ about the x=x’ axis, ϕ =0. Then, the eigenvalue, γ of Eqs. (B.15a)-(B.15b), can be presented as:

φ φ

α β φ

φ β β

γ ₂ ₂

2 2

sin cos

)}

sin (cos

{ )

1 ( sin ) cos

, (

j j

j A

A A

i A

+ +

− +

−

=− (j=1-3) (B.16a)

φ φ

α β φ

φ β β

γ ₂ ₂

2 2

sin cos

)}

sin (cos

{ )

1 ( sin ) cos

, (

j j

j A

A A

i A

+ +

−

=− (j=4-6) (B.16b)

If we further set iγ=u_j, then, Eq. (B.16a)-Eq.(B.16b) can be respectively expressed in Eqs. (3.47a)-(3.47f).

APPENDIX C THE EXPRESSIONS OF Dij (i, j=1-3)

The complete expressions for Dij (i, j=1-3) in Eqs. (3.53a)-(3.53f) are presented as:

)

66 22 4 66 12 12 22 11 2 2 66 11 4

66 24 56 22 2 56 14 12 66 14 24 11 2

2 66 44 56 24 55 22 2 2 56 14 66 55 44 11 2

3 56 44 55 24

4 55 44 33

)) 2 ( (

))) (

)) (

( ( 2 (

))) 4

( ) ) (

( (

) (

2 ) ( ) (

a a a

a a a a a

u a a a a a

a a a a a a i

u a a a a a a a

a a a a a

u a a a a i

u a a u D

j j j

j j

β β

α α

β α

+ +

− +

+ +

−

− +

+ +

− +

−

− +

−

APPENDIX D THE EXPRESSIONS OF fij (i, j=1-6)

The expressions of fij (i, j=1-6) can be presented as:

)) (

) (

( ₅₆ ₁₁¹ ¹₂₁ ₅₅ ₃₁¹ ₁₁¹ ₁

11 i a D D a D iD u

f =− β +α + α − ,

)) (

) (

( ₅₆ ₁₁² ₂₁² ₅₅ ₃₁² ₁₁² ₂

12 i a D D a D iD u

f =− β +α + α − ,

)) (

) (

( ₅₆ ₁₁³ ₂₁³ ₅₅ ₃₁³ ₁₁³ ₃

13 ia D D a D iD u

f =− β +α + α − ,

)) (

) (

( ₅₆ ₁₁⁴ ₂₁⁴ ₅₅ ₃₁⁴ ₁₁⁴ ₄

14 i a D D a D iD u

f = β +α + α − ,

)) (

) (

( ₅₆ ₁₁⁵ ₂₁⁵ ₅₅ ₃₁⁵ ₁₁⁵ ₅

15 i a D D a D iD u

f = β +α + α − ,

)) (

) (

( ₅₆ ₁₁⁶ ₂₁⁶ ₅₅ ₃₁⁶ ₁₁⁶ ₆

16 i a D D a D iD u

f = β +α + α − ,

1 1 31 34 1 21 44 1 31 44 1 21 24 1 11 14

21 i( a D a D a D ) (a D a D )u

f =− α +β +β − + ,

2 2 31 34 2 21 44 2 31 44 2 21 24 2 11 14

22 i( a D a D a D ) (a D a D )u

f =− α +β +β − + ,

3 3 31 34 3 21 44 3 31 44 3 21 24 3 11 14

23 i( a D a D a D ) (a D a D )u

f =− α +β +β − + ,

4 4 31 34 4 21 44 4 31 44 4 21 24 4 11 14

24 i( a D a D a D ) (a D a D )u

f = α +β +β + + ,

5 5 31 34 5 21 44 5 31 44 5 21 24 5 11 14

25 i( a D a D a D ) (a D a D )u

f = α +β +β + + ,

6 6 31 34 6 21 44 6 31 44 6 21 24 6 11 14

26 i( a D a D a D ) (a D a D )u

f = α +β +β + + ,

1 1 31 33 1 21 34 1 31 34 1 21 23 1 11 13

31 i( a D a D a D ) (a D a D )u

f =− α +β +β − + ,

2 2 31 33 2 21 34 2 31 34 2 21 23 2 11 13

32 ( a D a D a D ) (a D a D )u

f =−α +β +β − + ,

3 3 31 33 3 21 34 3 31 34 3 21 23 3 11 13

33 i( a D a D a D ) (a D a D )u

f =− α +β +β − + ,

4 4 31 33 4 21 34 4 31 34 4 21 23 4 11 13

34 i( a D a D a D ) (a D a D )u

f = α +β +β + + ,

5 5 31 33 5 21 34 5 31 34 5 21 23 5 11 13

35 i( a D a D a D ) (a D a D )u

f = α +β +β + + ,

6 6 31 33 6 21 34 6 31 34 6 21 23 6 11 13

36 i( a D a D a D ) (a D a D )u

f = α +β +β + + ,

1 11

41 D

f = , f₄₂=D₁₁², f₄₃=D₁₁³, f₄₄=−D₁₁⁴, f₄₅=−D₁₁⁵, f₄₆=−D₁₁⁶,

1 21

51 D

f = , f₅₂=D₂₁² , f₅₃=D₂₁³, f₅₄=−D₂₁⁴ , f₅₅=−D₂₁⁵, f₅₆=−D₂₁⁶ ,

1 31

61 D

f = , f₆₂=D₃₁², f₆₃=D₃₁³, f₆₄=−D₃₁⁴, f₆₅=−D₃₁⁵, f₆₆=−D₃₁⁶ .

APPENDIX E BOUNDARY CONDITIONS

In order to demonstrate the Eq. (4.15), considering the pertinent continuity and discontinuity conditions at z=0:

In region Ι of Fig. 3.1, for z= 0⁺, the Cauchy′s theory of residues are utilized to integrate the contours, and a closed contour by adding a large semicirle on the upper γ-plane, is shown in Fig. E.1

γ

Fig. E.1 Path of integration for Eqs. (E.01a)-(E.01c)

Similarly, as depicted in Fig. 3.1, the region Ⅱ, for z= 0⁻, a closed contour by adding a large semicirle on the lower γ-plane is utilized.

)

C

γ

Imγ

Reγ

γ γ

γ

Fig. E.2 Path of integration for Eqs. (E.02a)-(E.02c)

From Eqs. (E.01a)-(E.01c), Eqs. (E.02a)-(E.02c), and Fig. E.3, we can find that:

0 ) ) , , ( )

, , ( 2 (

) 1 , , ( ) , , (

2 1

1 ^α ^β ^z −^u ^α ^β ^z = _π

∫

^U ^α ^β ^γ ^e^γ^d^γ −

∫

^U ^α ^β ^γ ^e^γ ^d^γ =

u _x ⁱ^z

C z

i C x

x S S

(E.03a) 0

) ) , , ( )

, , ( 2 (

) 1 , , ( ) , , (

2 1

1 ^α ^β ^z −^u ^α ^β ^z = _π

∫

^U ^α ^β ^γ ^e^γ^d^γ −

∫

^U ^α ^β ^γ ^e^γ^d^γ =

u _y ⁱ^z

C z i C y

y S S

(E.03b) 0

) ) , , ( )

, , ( 2 (

) 1 , , ( ) , , (

2 1

1 ^α ^β ^z −^u ^α ^β ^z = _π

∫

^U ^α ^β ^γ ^e^γ^d^γ −

∫

^U ^α ^β ^γ ^e^γ^d^γ =

u _z ⁱ^z

C z

i C z

z S S

(E.03c)

γ6 γ

Fig. E.3. Path of integration for Eqs. (E.03a)-(E.03c)

From Eqs. (4.16a)-(4.16c), Eqs. (4.17a)-(4.17c), and the relations of Eqs.

(E.03a)-(E.03c), we observe that:

when z=0, substituting Eqs. (E.04a)-(E.04c) into Eqs. (E.05a)-(E.05c) can obtain the following expressions:

)

By using the method of Laurent′s theorem, we find that:

Replacing Eqs. (E.07a)-(E.07c) into Eqs. (E.06a)-(E.06c), we prove that:

τ π According to the Cauchy′s theory of residues and the method of Laurent′s theorem,

we already accomplish the purpose desired and demonstrate the Eq. (4.15).

作者簡歷

姓名: 胡廷秉

籍貫:台灣省南投縣

出生年月日: 民國 59 年 04 月 15 日

學歷: 交通大學土木工程學士 (民國 78-82 年)

交通大學土木工程碩士 (民國 82-84 年)

交通大學土木工程博士 (民國 89-98 年)

經歷：工研院材料所機械性能實驗室副研究員台灣省石門水庫管理局薦任技士

長豐土木結構大地技師事務所負責人世合工程技術顧問股份有限公司負責人考試：土木工程技師高等考試及格(民國 82 年)

大地工程技師高等考試及格(民國 83 年) 結構工程技師高等考試及格(民國 84 年) 公務人員高等考試二級及格(民國 85 年) 消防設備師特考及格(民國 86 年)

著作清單:

期刊論文

1. Liao, J.J., Hu, T.-B., and Chang, C.-w. (1997), ”Determination of Dynamic Elastic Constants of Transversely Isotropic Rocks Using a Single Cylindrical Specmen”, Internation Journal of Rock mechanics and Mining Science, Vol. 34, No. 7, pp.1045-1054.(SCI)

NSC852211E009038

2. Tin-Bin Hu; Cheng-Der Wang; Jyh-Jong Liao (2007),“Elastic Solutions of Displacements for a Transversely Isotropic Full-Space with Inclined Planes of Symmetry

Subjected to a Point Load”, International Journal for Numerical and Analytical Methods in Geomechanics ,2007; 31:1401-1442( Accepted on 11 Augustl 2006, Published online 7 February 2007 in Wiley InterScience

(www.interscience.wiley.com). DOI:10.1002/nag.602.

3. Jyh-Jong Liao, Tin-Bin Hu , Cheng-Der Wang (2008) , “Elastic Solutions for an Inclined Transversely Isotropic Material Due to Three-Dimensional Point Load”, Journal of Mechanics of Materials and Structures, Vol. 3, No. 8,

pp.1521-1547.mathematical sciences publishers.

碩士論文:

胡廷秉，“ 超音波儀器之組裝及異向性岩石之超音波特性“，交大土木研究所，

1995

研討會論文:

1. 胡廷秉, 廖志中, 黃安斌 (1996). "壓電複合材料於岩石超音波量測的應用", 1996 岩盤工程研討會論文集, 民國 85 年 12 月, 台北,

325-332.NSC-84-2611-E009-004

2. 張峻維,廖志中,胡廷秉(1997)”受單軸壓力下異向性岩石之超音波及聲射特性,”

第七屆大地工程學術研討會論文集,第 1165-1170 頁.842611E009 004

3. 胡廷秉王承德廖志中,，” Elastic Solutions of Displacements for an Inclined Transversely Isotropic Full-Space Subjected to Three-Dimensional Point Loads”，

岩盤工程研討會 2004

4. 胡廷秉王承德廖志中, “承受三向度點荷重作用下之無限域傾斜橫向等向性岩石之應力解析解”, 岩盤工程研討會 2006.

在文檔中傾斜橫向等向性材料承受三向度點荷重的三維位移與應力基本解 (頁 144-175)

Ⅳ THREE DIMENSIONAL ELASTIC SOLUTIONS OF A

7.2 R ECOMMENDATIONS FOR F UTURE W ORK

[ ]

[ ]

[

]

[

]

[

]

[ ]

γ

C

γ

Imγ

Reγ

γ γ

γ

γ

γ

∫

∫

∫

∫

∫

∫

作 者 簡 歷

作者簡歷