S OLUTIONS FOR D ISPLACEMENTS AND S TRESSES BY I NVERSE F OURIER

−

−

−

−

−

−

=∂[( )( )( )( )( )( )], )

,

( ^{1 2 3 4 5 6}

(4.36d)

Suppose that u_j =iγ_j , u_x(α,β,z) , u_y(α,β,z) , and u_z(α,β,z) from Eqs.

(4.16a)-(4.16c) and Eqs. (4.17a)-(4.17c) have the same results as Eqs. (4.34a)-(4.34c) and Eqs. (4.35a)-(4.35c).

4.4 Solutions for Displacements and Stresses by Inverse Fourier Transforms

The displacements u_x(x,y,z), )u_y(x,y,z , and u_z(x,y,z) in Eqs. (4.16a)-(4.16c) and Eqs. (4.17a)-(4.17c) also can be solved by the following inverse double Fourier transforms as:

for z>0 (region 1, as shown in Fig. 3.1),

z u x z u x z u x

x z Be B e B e

u₁(α,β, )= ^{1 1} + ^{2 2} + ^{3 3} (4.37a)

z u y z u y z u y

y z B e B e B e

u ₁(α,β, )= ^{1 1} + ^{2 2} + ^{3 3} (4.37b)

z u z z u z z u z

z z Be B e B e

u₁(α,β, )= ^{1 1} + ^{2 2} + ^{3 3} (4.37c)

) , , ( )

, ,

( _{1( )}

1 z _{xx P} z

xx α β σ α β

σ =

(4.37d) )

, , ( )

, ,

( _{1( )}

1 z _{yy P} z

yy α β σ α β

σ =

(4.37e) )

, , ( )

, ,

( _{1( )}

1 z _{zz P} z

zz α β σ α β

σ =

(4.37f) )

, , ( )

, ,

( _{1( )}

1 z _{yz P} z

yz α β τ α β

τ =

(4.37g) )

, , ( )

, ,

( _{1( )}

1 z _{zx P} z

zx α β τ α β

τ =

(4.37h)

) , , ( )

, ,

( _{1( )}

1 z _{xy P} z

xy α β τ α β

τ = (4.37i)

and for z<0 (region 2, as also depicted in Fig. 3.1),

z u x z u x z u x

x z B e B e B e

u ₂(α,β, )=− ^{4 4} − ^{5 5} − ^{6 6} (4.38a)

z u y z u y z u y

y z B e B e B e

u ₂(α,β, )=− ^{4 4} − ^{5 5} − ^{6 6} (4.38b)

z u z z u z z u z

z z B e B e B e

u ₂(α,β, )=− ^{4 4} − ^{5 5} − ^{6 6} (4.38c)

) , , ( )

, ,

( _{2( )}

2 z _{xx P} z

xx α β σ α β

σ = (4.38d)

) , , ( )

, ,

( _{2( )}

2 z _{yy P} z

yy α β σ α β

σ = (4.38e)

) , , ( )

, ,

( _{2( )}

2 z _{zz P} z

zz α β σ α β

σ =

(4.38f) )

, , ( )

, ,

( _{2( )}

2 z _{yz P} z

yz α β τ α β

τ =

(4.38g) )

, , ( )

, ,

( _{2( )}

2 z _{zx P} z

zx α β τ α β

τ =

(4.38h) )

, , ( )

, ,

( _{2( )}

2 z _{xy P} z

xy α β τ α β

τ = (4.38i)

The desired displacements of u_x(x,y,z) , )u_y(x,y,z , and u_z(x,y,z) can be obtained by the double inverse Fourier transforms of Eqs. (4.37a)-(4.37c) and Eqs.

(4.38a)-(4.38c) as:

for z>0 (region 1),

β π ^{Be α β B e α β B e α β d}α^d z

y x

u_x { _x ^{i x y uz} _x ^{i x y u z} _x ^{i x y u z}} 2

) 1 , ,

( ^{1 ( ) 1 2 ( ) 2 3 ( ) 3}

1

+ + +

+ +

∞ +

∞

−

∞

∞

−

+ +

=

∫ ∫

^(4.39a)

β π ^{B e α β B e α β B e α β d}α^d z

y x

u_y { _y ^{i x y uz} _y ^{i x y u z} _y ^{i x y u z}} 2

) 1 , ,

( ^{1 ( ) 1 2 ( ) 2 3 ( ) 3}

1

+ + +

+ +

∞ +

∞

−

∞

∞

−

+ +

=

∫ ∫

^(4.39b)

β

In addition, the stress components of Eqs. (3.12a)-(3.12f) are performed by the double Fourier transforms expressed as Eqs. (3.42a)-(3.42f). The desired σ_xx(x,y,z), by the double inverse Fourier transforms as:

for z>0 (region 1),

β

j j x j

z j

y j

x j

xy(α,β)=−i(a₆₆βB +a₆₆αB +a₅₆αB )+ia₅₆B u

τ (4.43f)

In Eqs. (4.43a)-(4.43f), j=1-3 for z>0 (region 1), and j=4-6 for z<0 (region 2).

If we employe a spherical co-ordinate system, which is shown in Fig. 4.4, the variables α, β, and ui can be expressed in terms of (k, θx) as:

cos x

k θ

α = × (4.44a)

sin x

k θ

β= × (4.44b)

α

β

( k , ) θ

x

x

θ

Fig. 4.4 Spherical co-ordinate system (k, θx)

By substiting Eqs. (4.44a)-(4.44b) into Eqs. (3.47a)-(3.47f), we get

φ φ

φ φ

θ θ

φ φ β θ

α _{2 2}

2 2

2 2

sin cos

)) sin (cos

cos (sin

) 1 ( cos sin ) sin

, (

j

j x

x j j x

j A

A A

A k i

u +

+ +

− +

× −

= (j=1-3) (4.45a)

φ φ

φ φ

θ θ

φ φ β θ

α _{2 2}

2 2

2 2

sin cos

)) sin (cos

cos (sin

) 1 ( cos sin ) sin

, (

j

j x

x j j x

j A

A A

i A k i

u +

+ +

+ +

× −

= (j=4-6) (4.45b)

where 0<k<∞, and 0<θx<2π.

According to Eqs. (4.44a)-(4.44b) and Eqs. (4.45a)-(4.45b), D_mn(α,β,γ_j) (m, n=1-3) in Eqs. (3.53a)-(3.53f), B_x^j(α,β,u_j) , B_y^j(α,β,u_j) , B_z^j(α,β,u_j) (j=1-6) in Eqs.

(4.04a)-(4.04c), and U_j(α,β) in Eq. (4.09) can be expressed in terms of k and θx as:

) ( ) , ( ) ,

( _{j j x}

j u k ku

u α β = θ = θ (4.46a)

) ( )

, ( )

, ( )

, ,

( _{j mn mn x} ⁴ _{mn x}

mn u D D k k D

D α β = α β = θ = θ

(4.46b) )

( )

, ( ) ,

( _{j x} ⁵ _{j x}

j U k k U

U α β = θ = θ (4.46c)

) ( )

, ( ) ,

( _x^j _x ¹ _x^j _x

j

x B k k B

B α β = θ = ⁻ θ (4.46d)

) ( )

, ( ) ,

( _y^j _x ¹ _y^j _x

j

y B k k B

B α β = θ = ⁻ θ (4.46e)

) ( )

, ( ) ,

( _x^j _x ¹ _x^j _x

j

x B k k B

B α β = θ = ⁻ θ (4.46f)

Hence, we can obtain:

) ( )

, ( ) ,

( _{xj x} ¹ _{xj x}

xj u k k u

u α β = θ = ⁻ θ (4.47a)

) ( )

, ( ) ,

( _{yj x} ¹ _{yj x}

yj u k k u

u α β = θ = ⁻ θ (4.47b)

) ( )

, ( ) ,

( _{zj x} ¹ _{zj x}

zj u k k u

u α β = θ = ⁻ θ (4.47c)

Furthermore, Eqs. (4.43a)-(4.43f) can be rewritten as:

) ( ) , ( ) ,

( _xx^j _{x xx}^j _x

j

xx α β σ k θ σ θ

σ = = (4.48a)

) ( ) , ( ) ,

( _yy^j _{x yy}^j _x

j

yy α β σ k θ σ θ

σ = = (4.48b)

) ( ) , ( ) ,

( _zz^j _{x zz}^j _x

j

zz α β σ k θ σ θ

σ = = (4.48c)

) ( ) , ( ) ,

( _yz^j _{x yz}^j _x

j

yz α β τ k θ τ θ

τ = = (4.48d)

) ( ) , ( ) ,

( _zx^j _{x zx}^j _x

j

zx α β τ k θ τ θ

τ = = (4.48e)

) ( ) , ( ) ,

( _xy^j _{x xy}^j _x

j

zx α β τ k θ τ θ

τ = = (4.48f)

The exponential terms in Eqs. (4.41a)-(4.41c), Eqs. (4.42a)-(4.42c), Eqs.

(4.43a)-(4.43f) and Eqs. (4.44a)-(4.44f) can be expressed as:

) ( )

( x y u_jz k _{j x}

iα +β + = ×ψ θ (4.49)

Additionally, dαdβ can be replaced by dkdθ_x times the absolute value of Jacobian J as:

dkd x

J d

dα β = θ (4.50)

where:

cos k k sin

sin k cos

k k ) , k (

) , J (

x x

x x

x x x

− =

=

∂

∂

∂

∂ ∂

∂

∂

∂

∂ =

= ∂

θ θ

θ θ

θ β

β θ

α α θ

β

α (4.51)

and therefore, kdkd x

d

dα β= θ (4.52)

Based on Eqs. (4.46a)-(4.46f), Eqs. (4.47a)-(4.47c), Eq.(4.49) and Eq. (4.50), then Eqs. (4.39a)-(4.39c) and Eqs. (4.40a)-(4.40c) can be presented as:

For z>0 (region 1),

x

Also, from Eqs. (4.46a)-(4.46f), Eqs. (4.48a)-(4.48f), Eqs. (4.49) and Eq. (4.52), Eqs.

(4.41a)-(4.41f) and Eqs. (4.42a)-(4.42f) can then be written as:

For z>0 (region 1),

x

x

(4.53a)-(4.53c) and Eqs. (4.54a)-(4.54c) can be expressed as:

For z>0 (region 1),

ω ω

Besides, Eqs. (4.55a)-(4.55f) and Eqs. (4.56a)-(4.56f) can be represented as:

For z>0 (region 1),

ω ω

CHAPTER V

THREE DIMENSIONAL ELASTIC SOLUTIONS OF A

TRANSVERSELY ISOTROPIC HALF-SPACE SUBJECTED TO POINT LOADS

The analytical solutions for displacements and stresses in an transversely isotropic half-space subjected to a point load in the medium are derived in Chapters 5. Three distinct approaches as same as Chapters 4 were used to derive the solutions in Sec 5.1-5.3. In Sec 5.1, we consider the nonhomogeneous part of the ordinary differential equations and solve the homogeneous and particular solution of Eqs. (3.41a)-(3.41c).

In Sec.5.2, we separate the half-space into two regions of 0⁻< z<0⁺ (imaginary space) and 0⁺ <z<+∞ (region 1) , the point load force is in the region of 0⁻ <z<0⁺. In regions of 0⁺ <z<+∞, the right-hand side of Eqs. (3.41a)-(3.41c) does not exist, the equilibrium equations are homogeneous linear equations. Hence, we can solve the boundary-value problem consisting of the two regions. In Sec.5.3, the Laplace transform respect to variables of z can reduce the aforementioned ordinary differential equations (Eqs. (3.41a)-(3.41c)) to algebraic equations..

5.1 Traditional Method

The thesis aims to determine the distribution of stress and displacement in a semi-infinite elastic solid due to the application of an external point load to its surface analytically. In order to derive the homogenous solutions of Eqs. (3.41a)-(3.41c), we

redefine the three displacement functions of Eqs. (3.50a)-(3.50c) and Eqs.

(3.51a)-(3.51c) as follows:

for z>0 (region 1 and 2, as shown in Fig. 5.1),

z u x z u x z u x z u x z u x z u x H

x z A e A e A e A e A e A e

u_{1( )}(α,β, )= ¹₁ ¹ + ²₁ ² + ³₁ ³ + ⁴₁ ⁴ + ⁵₁ ⁵ + ⁶₁ ⁶

(5.01a)

z u y z u y z u y z u y z u y z u y H

y z A e A e A e A e A e A e

u _{1( )}(α,β, )= ¹₁ ¹ + ²₁ ² + ³₁ ³ + ⁴₁ ⁴ + ⁵₁ ⁵ + ⁶₁ ⁶ _(5.01b)

z u z z u z z u z z u z z u z z u z H

z z A e A e A e A e A e A e

u_{1( )}(α,β, )= ¹₁ ¹ + ²₁ ² + ³₁ ³ + ⁴₁ ⁴ + ⁵₁ ⁵ + ⁶₁ ⁶

(5.01c) and for z<0 (region 3, as also depicted in Fig. 5.1),

0 ) , ,

)(

(

2 z =

u_{x H} α β

(5.02a) 0

) , ,

)(

(

2 z =

u_{y H} α β

(5.02b) 0

) , ,

)(

(

2 z =

u_{z H} α β

(5.02c)

In Eqs. (5.01a)-(5.01c), the undetermined coefficients A_x^j₁, A_y^j₁, and A_z^j₁ (j=1-6) can be obtained by assuming the displacements in region 1, u_x1, u_y1, and u_z1 must be finite when z is approaching to ∞. Hence, A_x⁴₁= A_x⁵₁= A_x⁶₁ =0, A⁴_y1= A⁵_y1= A_y⁶₁=0, and 0A_z⁴₁= A_z⁵₁ = A⁶_z1= .

Now, let:

j d j j z j

j y

j j

x C

u D

A u

D A u

D A

2 31

1 21

1 11

1

) , , ( )

, , ( )

, ,

( = = =

β α β

α β

α (j=1-3) (5.03)

where Dij (i, j=1-3) can be written as the second order derterminants in the following, and the complete forms of Dij are presented in Appendix C.

Similar to the full space problem, the general solition of the homogeneous linear equation of Eqs.(3.41a)-(3.41c) can express as follow:

z u d z u d z u d H

x z C D e C D e C D e

u_{1( )}(α,β, )= ¹_{2 11}^{1 1} + ²_{2 11}^{2 2} + ³_{2 11}^{3 3} (5.04a)

z u d z u d z u d H

y z C D e C D e C D e

u _{1( )}(α,β, )= ¹₂ ¹₂₁ ¹ + ²_{2 21}^{2 2} + ³_{2 21}^{3 3} (5.04b)

z u d z u d z u d H

z z C D e C D e C D e

u_{1( )}(α,β, )= ¹₂ ¹₃₁ ¹ + ²_{2 31}^{2 2} + ³_{2 31}^{3 3} (5.04c) The point loads (Fx, Fy, Fz) acts at the point (0, 0, h) of the co-ordinate system can be

described as Eqs. (3.19a)-(3.19c). Hence, the three displacement functions of the particular solutions can be gained from those of full space case (Sec. 4.1.1):

for _z>h (region 1, as shown in Fig. 5.1),

) ( 3 ) ( 2 ) ( 1 )

(

1_P ( , , ) _x ^{u1 z h} _x ^{u2 z h} _x ^{u3 z h}

x z Be B e B e

u α β = ⁻ + ⁻ + ⁻

(5.05a)

) 3 ( ) ( 2 ) ( 1 )

(

1_P ( , , ) _y ^{u1z h} _y ^{u2 z h} _y ^{u3 z h}

y z B e B e B e

u α β = ⁻ + ⁻ + ⁻ _(5.05b)

) ( 3 ) ( 2 ) ( 1 )

(

1_P ( , , ) _z ^{u1 z h} _z ^{u2 z h} _z ^{u3 z h}

z z Be B e B e

u α β = ⁻ + ⁻ + ⁻

(5.05c) and for 0<z<h (region 2, as also depicted in Fig. 5.1),

) 6 ( ) 5 ( ) ( 4 )

(

2 _P ( , , ) _x ^{u4 z h} _x ^{u5 z h} _x ^{u6 z h}

x z B e B e B e

u α β =− ⁻ − ⁻ − ⁻

(5.06a)

) ( 6 ) ( 5 ) ( 4 )

(

2_P ( , , ) _y ^{u4 z h} _y ^{u5 z h} _y ^{u6 z h}

y z B e B e B e

u α β =− ⁻ − ⁻ − ⁻ _(5.06b)

) ( 6 ) ( 5 ) ( 4 )

(

2_P ( , , ) _z ^{u4 z h} _z ^{u5 z h} _z ^{u6 z h}

z z B e B e B e

u α β =− ⁻ − ⁻ − ⁻

(5.06c)

=0 half-space

When the half-space (z>h) and the strip-space (0<z<h) are ideally bonded at the interface z=h such that the material becomes continuous across the interface, we Consider the following boundary conditions:

)

Hence, the double Fourier transforms of Eqs. (5.07a)-(5.07c) can be obtained as:

β π

β π

The Eqs. (5.09a)-(5.09c) can be simplifed as follow:

β π

The system of three linear equations (Eqs. (5.10a)-(5.10c)) has three undetermined coefficients C , ¹_d2 C and _d²₂ C . These coefficients can be associated with _d³₂

[ ]

fij (i,

Eq. (5.11) can be separated into two parts as:

⎥⎥

⎥⎥ express as:

∑

=

By substiting Eqs. (5.14a)-(5.14b) into Eq. (5.12b), we get

)

The general solution of nonhomogeneous equation are the sum of the general solution

of homogeneous equation and particular solution.

Similarly,

z

5.2 Imaginary Space Method

In order to derive the solutions of Eqs. (3.15a)-(3.15c) in half-space, the Eqs.

(4.19a)-(4.19c) and Eqs. (4.20a)-(4.20c) can be rewritten as:

for z> 0⁺ (region 1, as shown in Fig. 5.2),

Introducing an imaginary plane along z=0, to separate the full-space into two half-spaces, one is 0⁺ <z<∞ (region 1 in Fig. 5.2), and the other is 0 z< <0⁺.

= 0+

z

= 0+

z

) 0 , ,

1(xy +

σzz

) ( ) (x y Pzδ δ

+∞

<

<

+ z

0

=0 z

= 0+

z

< +

< 0 0 z

x

y

z

=0 z

Fig. 5.2 Separate the half-space into two imaginary region of 0 z< <0⁺ and +∞

<

+ <z

0 .

Furthermore, considering the following pertinent continuity and discontinuity conditions at z=0 are:

) ( ) ( ) 0 , ,

(x y P_x x y

zx δ δ

τ ⁺ = (5.19a)

) ( ) ( )

0 , ,

(x y P_y x y

zy δ δ

τ ⁺ = (5.19b)

) ( ) ( ) 0 , ,

(x y P_z x y

zz δ δ

σ ⁺ = (5.19c)

where −∞< x<∞, −∞<y<∞. The subscripts 1 and 2 mean that z=0 plane is approaching to 0⁺ and 0⁻, respectively.

d1

C , C_d², C_d³.can be determined from the following system of three linear equations These coefficients can be associated with

[ ]

fij (i, j =1-3) as:

[ ]

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

=

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

=

⎥⎥

⎥

⎦

⎤

⎢⎢

⎢

⎣

⎡

z y x

d d d

d d d

ij

p p p C

C C f f f

f f f

f f f C C C

f 2π

1

3 2 1

33 32 31

23 22 21

13 12 11

3 2 1

(5.20) where fij (i, j =1-3) are presented in Appendix D.

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