ANALYTIC SOLUTIONS

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(1d) into the equilibrium equations with body forces neglected, the equilibrium equations are expressed in terms of displacements u i r zi



 ^,



and temperature change of the strata  as follows:

 

2 2 2

2 2 2

1 0

r r r r z

r

u u u u u

A L F L

r r r z r

r r z

 

           

       

  , (7a)

 

^{2 r 1 r 2}₂^{z 1 z 2}₂^z z ⁰

u u u u u

F L L C

r z r z r r r z z

 

       

              . (7b)

Let h be the heat flux vector and q be the heat sources, the law of conservation of energy is then listed as: _h

h 0

  h q  . (8)

The constants _tr and _tz are used to describe the behavior of the heat flow in a cross-anisotropic medium, in which _tr denotes the horizontal thermal conductivity of heat flow in the planes of isotropy and _tz is the corresponding vertical thermal conductivity in the plane perpendicular to isotropic plane. Assuming that the heat flow follows Fourier’s law, then

tr r tz z

r z

 

 ^  ^

  

 

h i i , (9)

in which i and _r i are unit vectors parallel to the radial and vertical directions, respectively. The point heat _z source of constant strength Q is considered at great depth of point  0,0 . Substituting (9) into (8) yields the third governing equation of temperature change  as below:

   

2 2

1 0

tr tz 2

Q r z

r r r

r z

  

   



 

      

   

  , (10)

where 

 

^r and 

 

^z are the Dirac delta functions.

For a linearly elastic medium with cross-anisotropic properties, the differential equations (7a), (7b) and (10) govern the steady state responses of the medium subjected to axisymmetric and thermoelastic disturbance.

2.2. Boundary Conditions

The point heat source at great depth is assumed no impact on the ground surface. This implies that the ground surface can be treated as a remote boundary, and the strata can be modeled as an infinite space as shown in Figure 1.

Thus, the effect of the deep thermally disturbance vanishes at the remote boundaries, z  . In other words, the displacements and the temperature change of the strata at remote boundaries should be vanished. Therefore, the remote boundary conditions are expressed as

 

^{, 0}

u r zr  , u r zz

 

^,  , and ⁰ 

 

^{r z,}  as z⁰   . (11) The thermoelastic responses are derived in this study by Hankel and Fourier transforms from the differential

equations (7a), (7b) and (10) corresponding with the remote boundary conditions at z  .

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performing appropriate Hankel transforms [17] with respect to the radial coordinate r of first, zeroth and zeroth orders in each equation, respectively. These equations then become:

 

2 2

2^{r z} 0

r r

d U dU

AU L F L

dz

 dz    

      , (12a)

 

^{r 2} z ² ₂^z z ⁰

dU d U d

F L LU C

dz dz dz

       , (12b)

2

 

2

2 2

tr tz

d Q

dz z

     

     , (12c)

where

     

1

; 0 ,

r r

U z 



^r u r z J r dr^{, (13a)}

     

0

0

; ,

z z

U z 



^r u r z J r dr^{, (13b)}

     

₀

; 0 ,

z r r z J r dr

  



^   ^{. (13c)}

In these equations, ^{J x}

 

represents the Bessel’s function of the first kind of order . The displacements and the temperature change of the strata are obtained by inverting the equations (13a) to (13c) as shown below:

     

1

, 0 ;

r r

u r z 



^U z J r d^{, (14a)}

     

0

0

, ;

z z

u r z 



^U z J r d^{, (14b)}

     

0

, 0 ;

r z z J r d

 



^   ^{. (14c)}

The Fourier transformations [17] are performed with respect to the axial coordinate z on equations (12a) to (12c). The results are expressed as



^{2A 2L U}



^{r i}

^

^{F L U}

^

^{z   r 0}

        , (15a)

 

r



^{2 2}



z z ⁰

i F L U  L  C U i 

        , (15b)



² tr ² tz



₂

     Q

   , (15c)

where



^,

  

^{; i z}

r r

U   ^ U z  e^ dz







 , (16a)



^,

  

^{; i z}

z z

U   ^ U z e^ dz







 , (16b)



,

  

z; e^{i z} dz

   ^ 







 . (16c)

The closed-form solutions of the long-term thermoelastic deformations and temperature change of the cross-anisotropic medium subjected to a deep point heat source are easily obtained in the Hankel-Fourier integral

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transformed domain



^{ ,}



by solving the simultaneous algebraic equations of (15a) to (15c). The results are shown as follows:

     

  ^{ }

2 2

2 2

, 2 ,

r z r

r

tr tz

L F L C

U Q      

        

   

  

 , (17a)

     

  ^{ }

2 2

2 2

, 2 ,

z r z

z

tr tz

i L F L A

U Q      

        

   

  

 , (17b)



^,



₂ ¹ ₂

2 _{tr tz}

   Q

    

 

 , (17c)

where 



 ^,



is defined as



 ^,



^CL^{4 AC F F}



^2L



 ^{2 2 AL}⁴

       . (18)

These solutions are also expressed in the domain ( ; )z by applying the following inverse Fourier transforms to equations (17a) to (17c):

 

; ¹



,



2

i z

r r

U z U   e ^ d



 





 ^{ , (19a)}

 

^{; 1}



^,



2

i z

z z

U z U   e ^ d



 





 ^{ , (19b)}

 

^{; 1}



^,



2

z ei z^ d

     



 





 ^{ . (19c)}

Then, we have

 

^{; 1}₂ ^{1 2}₂ ^{2 3}₂ ³

4

z z z

r

tz

a

a a

U z Q e ^{ } e ^{ } e^{ }

   

  

 

    

 , (20a)

 

^{; 1}₂ ^{1 2}₂ ^{2 3}₂ ³

4

z z z

z

tz

b b b

U z Q e^{ } e ^{ } e ^{ }

   

  

 

    

 

 , (20b)

 

³

3

; 1 4

z

tz

z Q e^{ }

 

  

  . (20c)

The upper and lower signs in equation (20b) are for the conditions of z and 0 z , respectively. Here, the 0 constants a_i



^{i1, 2, 3}



^and bi



^{i1, 2, 3}



are defined as

 

  

2 1

1 2 2 2 2

1 1 2 1 3

r z r

L F L C

a CL

   

    

   

   , (21a)

 

  

2 2

2 2 2 2 2

2 2 1 2 3

r z r

L F L C

a CL

   

    

   

   , (21b)

 

  

2 3

3 2 2 2 2

3 3 1 3 2

r z r

L F L C

a CL

   

    

   

   , (21c)

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 

  

2 1

1 2 2 2 2

1 2 1 3

z r z

L F L A

b CL

   

   

  

   , (21d)

 

  

2 2

2 2 2 2 2

2 1 2 3

z r z

L F L A

b CL

   

   

  

   , (21e)

 

  

2 3

3 2 2 2 2

3 1 3 2

z r z

L F L A

b CL

   

   

  

   . (21f)

In addition, the characteristic roots ₁ and ₂ must satisfy the following characteristic equation

 

4 2 2 0

CL AC F F  L  AL , (22)

and ₃  _{tr tz} .

Furthermore, using the inversions of Hankel transform formula [17-19] and the constitutive equations (1a) to (1d), the closed-form solutions of thermoelastic deformation, temperature increment and thermal stresses of the strata in real domain

 

r z, are obtained from the solutions of (20a) to (20c) in Hankel transformed domain

 

; z as below:

1 * 2 * 3 *

1 2 3

r 4

tz

Q r r r

u a a a

R R R



 

    

 , (23a)

1 1 1 2 1 3

1sinh 2sinh 3sinh

z 4

tz

z z z

u Q b b b

r r r



 

 ^{  }

 

    , (23b)

3 3

1 4 _tz

Q

 R

   , (23c)

1 2 3

1 2 3 1 * 2 * 3 * 1 2 3

1 2 3 1 2 3 1 2 3 3 3

1 1 1 1 1 1 1

4 2

rr r

tz

Q A a a a N a a a F b b b

R R R R R R R R R R



 

 

 

       

              

       

 , (23d)

1 2 3

1 2 3 1 * 2 * 3 *

1 2 3 1 1 2 2 3 3

1 1 1

4 _tz 2

z z z

Q A a a a N a a a

R R R R R R R R R



  

 

    

         

    



1 2 3

1 2 3

1 2 3 3 3

1

F b b b r

R R R R



  



  

     

  , (23e)

3

1 2

1 2 3 1 2 3

1 2 3 1 2 3 3 3

1 1 1 1

zz 4 z

tz

Q F a a a C b b b

R R R R R R R



 

 

 

     

          

     

 , (23f)

1 2 3

3

1 2

1 * 2 * 3 * 1 2 3

1 2 3

1 1 2 2 3 3

rz 4

tz

z z z

r

r r

Q L a a a b b b

rR rR rR

R R R R R R

  



 

 

  

       

  

 

 . (23g)

The upper and lower signs in equation (23g) are for conditions of z and 0 z , respectively. In equations 0 (23a) to (23g), R_i  r²_i^{2 2}z and R_i^*R_i_i z



i1, 2, 3



.

3.2. Cases of Isotropic Mechanical Behaviour and Cross-anisotropic Thermal Properties

The displacements, temperature change and thermal stresses for the strata with cross-anisotropic properties in mechanical and heat flows are analytically solved and expressed in equations (23a) to (23g) for the disturbance of a deep point heat source. For the special case of medium with isotropic mechanical properties, the associated

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closed-form solutions are obtained by the conditions of ₁ = ₂ = 1 for equations (23a) to (23g). This is carried out by using L’Hospital’s rule with careful calculations. The results are given as follows:

     



^{* 1 , 2 * 2 1 * 2 ,}



4

r sz r r z

tz

u Q   r z     r z

 ^{ }

     , (24a)

     



^{* 3 , 2 * 2 1 * 4 ,}



4

sz

z z z r

tz

u Q   r z     r z

 ^{ }

     , (24b)

5

 

, 4 _tz

Q r z

 

  , (24c)

             



^{2 1 * 6 , 2 1 * 7 , 2 2 1 * * 8 , * 9 ,}

4

sz

rr r z r z r

tz

QG r z r z r z r z

            

 ^{ }

        

      

* * *

10 , 2 2 1 11 ,

r r z r z r z

      

     , (24d)

             



^{2 1 * 6 , 2 1 * 7 , 2 2 1 * * 8 , * 9 ,}

4

sz

r z r z r

tz

QG r z r z r z r z



             

 ^{ }

        

      

* * *

10 , 2 2 1 11 ,

r r z r z r z

      

     , (24e)

           



^{2 1 * 6 , 2 * 7 , 2 * 2 3 1 * 8 , * 9 ,}



4

zz sz r z r z z

tz

QG r z r z r z r z

           

 ^{ }

         , (24f)

       



^{2 * 2 1 * 12 , 2 * 2 1 * 13 ,}



4

sz

rz r z z r

tz

QG r z r z

        

 ^{   }

         , (24g)

where 



1

 

12



. The parameters _r^{* and} _z^* are defined as follows:



  

 





_r^*2  _{sr sz} 12 , (25a)

   

* 2 1 2 1 2

z sr sz

           . (25b)

The definitions of functions i



i ^{1, ,13}



in equations (24a) to (24g) are listed below:

       

1 2 2 2 * 2 2 *

1 1 1

, 4 1 2 1 2 1

r r r

r z R R R_

 ^  ^{ }   ^    , (26a)

       

2 2 * * 2 2 * 2 2 *

1 1

, 4 1 2 1 2 1

r z r r r

r z RR R R R_

 

  

 

    

     , (26b)

       

2 2

1 1

3 2 2 2 2 2

, 1 sinh sinh

4 1 2 1 2 1

z z z

r z R r r

  

   

 

   

   , (26c)

     

¹

 

¹

4 2 2 2 2 2

1 1 1

, sinh sinh

4 1 2 1 2 1

z z z

r z R r r

 

  

 

  

   , (26d)

5

 

, 1

r z R_

 ^ ^{, (26e)}

       

2

6 2 3 2 2 2 2

1 1 1 1 1 1

, 4 1 2 1 2 1

r z z

R R R R_

    

 

    

     , (26f)

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       

2 3

7 2 3 2 2 2 2

1 3 1 1 1

, 4 1 2 1 2 1

r z z

R R R R_

 

  

 

    

     , (26g)

       

2

8 2 3 2 2 2 2

1 1 1 1

, 4 1 2 1 2 1

r z r

R R

R _

 

  

  

   , (26h)

9

 

, 1

r z R_

 ^{ } ^{, (26i)}

       

2 2 2

10 2 3 2 2 *2 2 2 *2

1 1 1

, 4 1 2 1 2 1

r r r

r z R RR R R_{ }

 ^{ }  ^{ }   ^    , (26j)

       

2 2 2 2

11 2 3 2 2 *2 2 2 *2

, 1

4 1 2 1 2 1

r r r

r z R RR R R_{ }

 

 ^  ^{ }   ^   , (26k)

       

2 2

12 2 3 2 2 * 2 2 *

, 1

4 1 2 1 2 1

rz r r

r z R RR R R_{ }

 

 ^  ^{ }   ^   , (26l)

       

13 2 3 2 2 * 2 2 *

1 1 1

, 4 1 2 1 2 1

rz r r

r z R RR R R_{ }

 ^{ }  ^{ }   ^   , (26m)

where the parameters   _{tr tz} , R r²z² , R^*  R z, R_  r²^{2 2}z and R_^* R_ z. 3.3. Cases of Isotropic Mechanical and Thermal Properties

Furthermore, the closed-form solutions for the special case of medium with isotropic mechanics and heat flows properties are acquired through the conditions of  for equations (24a) to (24g). Applying the L’Hospital’s 1 rule and careful calculations, the results are given as below:

 

 

1

8 1

s r

t

Q r

u R

 

 

 

 , (27a)

 

 

1

8 1

s z

t

Q z

u R

 

 

 

 , (27b)

1 4 _t

Q

 R

  , (27c)

 

 

2

3

1 1

4 1

s rr

t

QG r

R R

 

  

  

     , (27d)

 

 

1 1

4 1

s

t

QG

 R

 

  

  

 , (27e)

 

 

2

3

1 1

4 1

s zz

t

QG z

R R

 

  

  

     , (27f)

 

 

³

1

4 1

s rz

t

QG rz

R

 

  

  

 , (27g)

where _t denotes thermal conductivity of the isotropic soils or rocks.

The derived closed-form solutions, equations (27a) to (27g), illustrated that all field quantities are functions of the distance from the heat source and they are inversely proportional to the thermal conductivity. Besides, the shear modulus does not have influence on displacements and temperature increment of the homogeneous isotropic strata.

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在文檔中 典型抽水所引致三維壓密沉陷解析 (頁 57-63)