Analytic Solution

The closed-form solutions of thermoelastic consolidation deformation, temperature change of the soil mass, excess pore fluid pressure, and effective stresses due to a point heat source buried in a transversely isotropic elastic full space can be obtained by using Hankel transform and (1a)-(1d) as follows:

( )

[ ]

{

2

}

*

* 1

* 2 2 1

4Q f f

u _{r r z}

tz sz

r β ηβ η β

πηλ

α + − −

= , (11a)

( )

[ ]

{

* * 4

}

3

* 2 2 1

4Q f f

u _{z z r}

tz sz

z β ηβ η β

πηλ

α + − −

= , (11b)

=0

p , (11c)

4Q f5

πλtz

ϑ = , (11d)

( ) ( )

{

7

* 6

* 2 1

1

4 2 f f

QG

z r

tz sz

rr η β η β

πηλ

σ′ = α − + −

( )

[ ]

* 10

9

* 8

*

1 *

2

2 η− β_r −β_z f +ηβ_r f +β_r f +

+

[

2ηβ_r*−(2η−1)β_z*

]

f11

}

, (11e)

( ) ( )

{

7

* 6

* 2 1

1

4 2 f f

QG

z r

tz

sz η β η β

πηλ

σ_θθ′ = α − + −

( )

[ ]

* 10

9

* 8

*

1 *

2

2 η− β_r −β_z f +ηβ_r f −β_r f +

−

[

2ηβ_r*−(2η−1)β_z*

]

f11

}

, (11f)

( )

{

7

* 6

* 2

1

4 2 f f

QG

z r

tz sz

zz η β ηβ

πηλ

σ′ = α − +

( )

[ ]

* 9

}

8

*

* 23 1

2ηβ_r + η− β_z f +ηβ_z f

−

+ , (11g)

( )

[ ]

{

12

*

* 2 1

4 2 f

QG

z r

tz sz

rz ηβ η β

πηλ

σ′ = α − −

( )

[ ] }

13

*

* 2 1

2ηβ_z − η− β_r f

+ , (11h)

where the parameters ^β_r^*=2(^ν+^α_sr ^α_sz) (1−2^ν),

( )

[ ^{ν να α} ] ( ^ν)

β_z^*=2 1− +2 _{sr sz} 1−2 , and

( ^ν) ( ^ν)

η= 1− 1−2 . The functions f_i(i=1,_L,13)

in (11a)-(11h) are defined as:

(

²

) (

²

)

^{2 *}

1 2 1

1 1

4 1

R r R

f r

− −

= −

µ µ

(

² 1

)

^{2 *}

2 1 µ µ

µ ^R

r

+ − , (12a)

(

²

)

^{* *}

(

²

)

^{2 *}

2 2 1

1 1

4 1

R r R

r RR

z f r

+ −











− +

= −

µ µ

(

^{2 1}

)

^{2 *}

2µ µ

µ R

r

− − , (12b)

( )

^Rz

( )

^rz

f ₂ ¹

2 2

3 2 sinh

1 2 1 4

1 ₋

− −

− −

= µ

µ µ

(

^µµ2

)

^{2 1µrz}

2

1 sinh 2

−

+ − , (12c)

( )

^Rz

( )

^rz

f ₂ ¹

2

4 2 sinh

1 2

1 1

4

1 ₋

+ −

= −

µ µ

(

^{µ2 1}

)

^{2sinh 1µrz}

2

1 −

− − , (12d)

µRµ

f 1

5= , (12e)

( )

^{R Rz}

( )

^R

f 1

1 2

1 1

1 4

1

2 2 3

2 6 2

− −



 



 +

= −

µ µ

(

^µ

)

^µ

µ ^R

1 1 2

1

2− 2

+ , (12f)

15

( )

^{R Rz}

( )

^R

f 1

1 2

1 3

1 4

1

2 2 3

2 7 2

− −



 



− +

= −

µ µ

(

^{µµ 1}

)

^R1µ

2 ^{2 2}

3

+ − , (12g)

( )

^Rr

( )

^R

f 1

1 2

1 1

4 1

2 2 3 2 8 2

+ −

= −

µ µ

(

^{µµ 1}

)

^R1µ

2 ²− ²

− , (12h)

µRµ

f 1

9 =− , (12i)

( ) ( )

^*2

2 2 2

3 2 10 2

1 2

1 1

4 1

RR r R

f r

+ −

− −

= µ µ

( )

^*2

2 2 12

2 1

µ

µ µ

µ ^{R R}

r

− − , (12j)

( ) ( )

^*2

2 2 2

2 3

2 11 2

1 2 1

4 1

RR r R

f r

− −

= −

µ µ µ

( )

^*2

2 2 12

2µ µ µ

µ

R R

r

+ − , (12k)

( ) (

²

)

^{2 *}

2 2 3

12 4 1 2 1

1

RR r R

f rz

− −

= −

µ µ µ

(

²

)

^{2 *}

2

1 2µ µ µ

µ

R R

r

+ − , (12l)

(

²

)

³

(

²

)

^{2 *}

13 2 1

1 1

4 1

RR r R

f rz

+ −

− −

= µ µ

(

^{2 1}

)

^{2 *}

2 1

µ

µ ^Rµ^R

r

− − , (12m)

where the parameters µ= λ_tr λ_tz , R= r²+z² , z

z r

R^*= ²+ ² + , R_µ = r²+µ²z² , and z

z r

R_µ^* = ²+µ^{2 2} +µ , respectively. From these solutions, the excess pore fluid pressure disappears under the steady state thermoelastic consolidation.

Using L’Hospital’s rule and applying the limit

→1

µ and α_sr α_sz →₁ , the solutions of an isotropic soil mass with isotropic permeability and heat conductivity are obtained from (11a)-(11h).

Carrying out the procedure, we obtain

( ) ( )R

r u Q

t

r πλs ν

ν α

−

= + 1 8

1 , (13a)

( ) ( )R

z u Q

t s

z πλ ν

ν α

−

= + 1 8

1 , (13b)

=0

p , (13c)

R Q

t

1 4πλ

ϑ= , (13d)

( )

( ⁻⁺ ) ^_^{ + }_^

−

′ = 1 ²₃

1 4

1

R r R QG

t

rr πλs ν

ν

σ α , (13e)

( ) ( ) R QG

t

s 1

1 4

1 ν πλ

ν σ_θθ α

−

− +

′ = , (13f)

( )

( ⁻⁺ ) ^_^{ + }_^

−

′ = 1 ²₃

1 4

1

R z R QG

t s

zz πλ ν

ν

σ α , (13g)

( ) (1 ) ³

4 1

R rz QG

t s

rz πλ ν

ν σ α

−

− +

′ = , (13h)

where λ denotes thermal conductivity of the _t isotropic soils.

4.Numerical Results

Figure 1. Stresses on soil element due to a deep point heat source Q.

Referring to the soil element in Figure 1, the numerical results of the effective stress distributions

16 of σ ′ are summarized in Figures 2-5. The _ij assumed Poisson’s ratio of the soil mass is ν =0.3, and the effective stresses were normalized by the factor QGα_sz 4πλ_tz. As shown in the figures 2-5, the isobaric contours of effective stresses of soil mass σ ′ , ^rr σ ′ , _θθ σ ′ , and _zz σ ′ near the point heat _rz source are significantly affected by the ratio of thermal properties α^sr α^sz and λ^tr λ^tz . The rise

of the ratio of linear thermal expansion coefficients

sz sr α

α leads to corresponding rise of effective stress components with varying degrees of anisotropy.

Nevertheless, the effective stress components decrease with the increase of the ratio of thermal conductivities λ^tr λ^tz. All effective stress changes are compressive, and it helps to prevent thermal failure of soil.

Figure 2. Isobaric contours of normalized effective stress −σrr′

[

QGαsz 4πλtz

]

due to a point heat source for different ratios of α_sr α_sz and λ^tr λ^tz .

17

Figure 3. Isobaric contours of normalized effective stress −σ_θθ′

[

QGαsz 4πλtz

]

due to a point heat source for different ratios of α_sr α_sz and λ^tr λ^tz .

High-level radioactive waste generates heat, and it leads to temperature increase for the soil surrounding the canister. Hueckel and Peano (1987) indicated that European guidelines require that temperature increments in the soil close to the heat source should not exceed 80°C while the temperature increments at the ground surface is limited to less than 1°C. The heat outputs generated by canisters are assumed to be 224 W/m (Ma and Hueckel, 1993), 325 W/m (Ma and Hueckel, 1992), or 1000 W/m

(Smith and Booker, 1996).

On the basis of equations (13a)-(13h), Table 1 gives the normalized values of the derived analytical solutions. The selected representative parameters of isotropic soil, Boom clay, are listed in Table 2 to verify the proposed solutions. At the distance of r = 0~10 m and z = 0.3 m away from the heat source corresponding to the parameters listed in Table 2, the displacements, stresses and temperature increases of the stratum due to a deep point heat source are shown

18 in Table 3. The maximum radial displacement u_r, vertical displacement u_z, radial stress σ ′ , hoop _rr stress σ ′ , vertical stress _θθ σ ′ , shear stress _zz σ ′ _rz and temperature increase ϑ of the stratum shown in Table 3 are 1.86 mm, 1.86 mm, −1.35 MPa, −1.24 MPa, −2.48 MPa, −0.48 MPa and 51°C, respectively.

The temperature increase is below 1°C at the soil 16 m away from the point heat source. The ratio of the radial normal stress σ ′ to the vertical normal _rr stress σ ′ ranges from 0.5 to 2.0, and the ratio of _zz the hoop normal stress σ ′ to the vertical normal _θθ stress σ ′ ranges from 0.5 to 1.0. _zz

Figure 4. Isobaric contours of normalized effective stress −σzz′

[

QGαsz 4πλtz

]

due to a point heat source for different ratios of α_sr α_sz and λ^tr λ^tz .

19

Figure 5. Isobaric contours of normalized effective stress −σrz′

[

QGαsz 4πλtz

]

due to a point heat source for different ratios of α_sr α_sz and λ^tr λ^tz .

5. Conclusions

The closed-form solutions of thermoelastic consolidation due to a point heat source buried in a transversely isotropic elastic full space were obtained using the Hankel transformation. The results were examined by simplifying the solutions of transversely isotropic thermo-consolidation into the case of isotropic. All field quantities are functions of the distance from the heat source and are proportional to

the linear thermal expansion coefficient. However, they are inversely proportional to the thermal conductivity. The solutions show that the shear modulus does not have influence on displacements and temperature change of the soils.

Based on the numerical results obtained for the anisotropic thermo-consolidation, the effective stresses of soil mass are compressive and are significantly affected by the ratio of thermal

20 properties α^sr α^sz and λ^tr λ^tz . The rise of the ratio of linear thermal expansion coefficients

sz sr α

α leads to corresponding rise of effective stress components with varying degrees of anisotropy.

However, the effective stress components decrease with increase of the ratio of the thermal conductivities λ^tr λ^tz .

At the distance of r = 0~10 m and z = 0.3 m away from the heat source corresponding to the parameters listed in Table 2, the obtained maximum

radial displacement u_r, vertical displacement u_z, radial stress σ ′ , hoop stress _rr σ ′ , vertical stress _θθ

σ ′ , shear stress zz σ ′ and temperature increase ϑ _rz of the stratum are 1.86 mm, 1.86 mm, -1.35 MPa, -1.24 MPa, -2.48 MPa, -0.48 MPa and 51°C, respectively. The ratio of the radial normal stress σ ′ to the vertical normal stress rr σ ′ ranges from _zz 0.5 to 2.0, and the ratio of the hoop normal stress σ ′ to the vertical normal stress θθ σ ′ ranges from _zz 0.5 to 1.0.

Table 1. Normalized values of the displacements, effective stresses and temperature change of the isotropic soil due to a deep point heat source Q

z

r ( )

( ^ν)

πλ ν α

− + 1 8

1

t s r

Q

u ( )

( ^ν)

πλ ν α

− + 1 8

1

t s z

Q

u ( )

( )z QG

t s

rr

ν πλ

ν α

σ

− +

− ′

1 4

1 ( )

( )z QG

t s

ν πλ

ν α

σ_θθ

− +

− ′

1 4

1 ( )

( )z QG

t s

zz

ν πλ

ν α

σ

− +

− ′

1 4

1 ( )

( )z QG

t s

rz

ν πλ

ν α

σ

− +

− ′

1 4

1

z Q πλt

ϑ

4 0.0 0.0000 1.0000 1.0000 1.0000 2.0000 0.0000 1.0000 0.5 0.4472 0.8944 1.0733 0.8944 1.6100 0.3578 0.8944 1.0 0.7071 0.7071 1.0607 0.7071 1.0607 0.3536 0.7071 1.5 0.8321 0.5547 0.9387 0.5547 0.7254 0.2560 0.5547 2.0 0.8944 0.4472 0.8050 0.4472 0.5367 0.1789 0.4472 2.5 0.9285 0.3714 0.6916 0.3714 0.4226 0.1281 0.3714 3.0 0.9487 0.3162 0.6008 0.3162 0.3479 0.0949 0.3162 3.5 0.9615 0.2747 0.5287 0.2747 0.2955 0.0726 0.2747 4.0 0.9701 0.2425 0.4708 0.2425 0.2568 0.0571 0.2425 4.5 0.9762 0.2169 0.4237 0.2169 0.2271 0.0459 0.2169 5.0 0.9806 0.1961 0.3847 0.1961 0.2037 0.0377 0.1961 5.5 0.9839 0.1789 0.3520 0.1789 0.1846 0.0315 0.1789 6.0 0.9864 0.1644 0.3244 0.1644 0.1688 0.0267 0.1644 6.5 0.9884 0.1521 0.3006 0.1521 0.1556 0.0229 0.1521 7.0 0.9899 0.1414 0.2800 0.1414 0.1442 0.0198 0.1414 7.5 0.9912 0.1322 0.2620 0.1322 0.1345 0.0173 0.1322 8.0 0.9923 0.1240 0.2462 0.1240 0.1259 0.0153 0.1240 8.5 0.9932 0.1168 0.2321 0.1168 0.1184 0.0136 0.1168 9.0 0.9939 0.1104 0.2195 0.1104 0.1118 0.0121 0.1104 9.5 0.9945 0.1047 0.2082 0.1047 0.1058 0.0109 0.1047 10.0 0.9950 0.0995 0.1980 0.0995 0.1005 0.0099 0.0995

21

Table 2. Selected representative parameters (Ma and Hueckel, 1992)

Symbol Value Unit

G 100 MPa

Q 325 J/s

α s 1.17 × 10^{−4 o}_C⁻¹

λ t 1.69 J/(s⋅m⋅°C)

ν 0.35 Dimensionless

Note: The strength of the point heat source Q that simulates the high-level radioactive waste is obtained from an assumed 1 m length line heat source with the heat output strength of 325 W/m.

Table 3. Typical values of the displacements, stresses and temperature increase of the isotropic soil due to a deep point heat source Q at the distance of r = 0~10 m and z = 0.3 m

away from the heat source corresponding to the parameters listed in Table 2 r

(m) z

(m) ^r

u (m)

uz

(m)

σrr′ (Pa)

σθθ′ (Pa)

σzz′ (Pa)

σrz′ (Pa)

ϑ (°C) σ_zz^rr

σ

′

′ σzz

σ_θθ

′

′

0.0 0.3 0.000000 0.001859 -1,239,572 -1,239,572 -2,479,144 0 51.0 0.50 0.50 0.1 0.3 0.000588 0.001764 -1,293,558 -1,175,961 -2,234,327 -352,788 48.4 0.58 0.53 0.2 0.3 0.001031 0.001547 -1,348,736 -1,031,386 -1,745,423 -476,024 42.4 0.77 0.59 0.3 0.3 0.001315 0.001315 -1,314,765 -876,510 -1,314,765 -438,255 36.1 1.00 0.67 0.4 0.3 0.001487 0.001116 -1,219,739 -743,743 -1,011,491 -356,997 30.6 1.21 0.74 0.5 0.3 0.001594 0.000957 -1,106,692 -637,755 -806,572 -281,362 26.2 1.37 0.79 0.6 0.3 0.001663 0.000832 -997,836 -554,354 -665,224 -221,741 22.8 1.50 0.83 0.7 0.3 0.001709 0.000732 -900,813 -488,291 -564,061 -176,795 20.1 1.60 0.87 0.8 0.3 0.001741 0.000653 -816,826 -435,243 -488,903 -143,094 17.9 1.67 0.89 0.9 0.3 0.001764 0.000588 -744,776 -391,987 -431,186 -117,596 16.1 1.73 0.91 1.0 0.3 0.001781 0.000534 -682,967 -356,188 -385,598 -98,034 14.7 1.77 0.92 1.5 0.3 0.001823 0.000365 -476,850 -243,100 -252,450 -46,750 10.0 1.89 0.96 2.0 0.3 0.001839 0.000276 -363,711 -183,879 -187,925 -26,975 7.6 1.94 0.98 2.5 0.3 0.001846 0.000222 -293,282 -147,689 -149,786 -17,471 6.1 1.96 0.99 3.0 0.3 0.001850 0.000185 -245,463 -123,342 -124,563 -12,212 5.1 1.97 0.99 4.0 0.3 0.001854 0.000139 -184,897 -92,708 -93,226 -6,914 3.8 1.98 0.99 5.0 0.3 0.001856 0.000111 -148,215 -74,241 -74,507 -4,438 3.1 1.99 1.00 6.0 0.3 0.001857 0.000093 -123,648 -61,901 -62,056 -3,087 2.5 1.99 1.00 7.0 0.3 0.001858 0.000080 -106,054 -53,076 -53,173 -2,271 2.2 1.99 1.00 8.0 0.3 0.001858 0.000070 -92,837 -46,451 -46,517 -1,739 1.9 2.00 1.00 9.0 0.3 0.001858 0.000062 -82,546 -41,296 -41,342 -1,375 1.7 2.00 1.00 10.0 0.3 0.001859 0.000056 -74,307 -37,170 -37,204 -1,114 1.5 2.00 1.00

22

在文檔中 坡地災害潛勢、觀測與預警技術之研究（二）-子計畫： (頁 32-40)