Ishii Hiroshi

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Ishii, Hiroshi

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Theoretical Study on the Crustal Movements ?.Extension and Tilt Produced by a TemperatureDistribution Due to a Thermal Point Source ina Semi-infinite Elastic Medium著者 Ishii  Hiroshi雑誌名 Science reports of the Tohoku University. Ser.5, Geophysics巻 20号 3ページ 145-151発行年 1970-12URL http://hdl.handle.net/10097/44706The Science Reports of the  TOhoku University, Ser. 5, Geophysics, Vol. 20, No. 3, 1970 Theoretical Study on the Extension and Tilt Produced    Due to a  Thermal Point                 ElasticCrustal Movements III. by a Temperature Distribution Source in a Semi-infinite MediumGeophysical HIROSHI ISHII Observatory,  Thhoku University,  (Received December 15, 1970)Akita, Japan   Abstract: A problem of extension and tilt due to hyperbolically decreasing temperature distribution from a thermal point source in a semi-infinite elastic medium is theoretically investigated. An example of extension and tilt on a free surface is graphycally presented. It is clarified that tilt has a maximum  absolute value at a point of 1.3 times source depth and extension decreases more rapidly than tilt with increasing distance.1. Introduction     Extension and tilt are continuously observed by extensometer and tiltmeter in observatories of crustal movements. However, observed extension and tilt include various influences like temperature, atmospheric pressure, rainfall and so on. There-fore, we must know how various factors influence on extension and tilt, in order to take out a signal that we desire from observed data.  In a previous paper (Ishii and Takagi, 1967), the influence of horizontal discontinuity was studied. In this paper we will investigate a thermal influence on the crustal deformations. The effect of surface temperature on the crustal deformations was investigated by some workers (e.g. Nishimura (1930), Arakawa (1931), Matsuzawa (1942) and Nakano  (1963)  ). Shima (1958) solved displacement by application of potential theory in the case where the spheroidal or spherical region of constant temperature is embedded in a semi-infinite elastic body. However, a temperature distribution will be produced by a thermal source for long period. Then the temperature distribution may influence on extension and tilt being observed. Therefore, in the present paper we will investigate extension and tilt produced by a temperature distribution of hyperbolic type due to a thermal point source in a semi-infinite lastic medium. 2. Formulation    We take the cylindrical coordinates (r, 0, z) as in Fig. 1, and z=0 as free surface. We assume symmetrical displacement about a z-axis, namely that displacement may take place in planes through the axis and be same in all such planes. In this case the conditions would be expressed by reference to cylindrical coordinates r,  6, z by the equations146      Fig. 1. Cylindrical coordinate system used for  formulatio:                                 314r 514. o .                         o ,                      38 DOIt will be convenient to write U for u, and w for ux.  Th expressed by equations  8U  U  8w  e„—ar ,  e8,0 ,  ez,=—az,r38W                      e„=3U ,4,9-=ea.= 0  . Cubical dilatation and rotation are expressed by equations  U  U  a U              ++,0 — —            ar 8r8z  to,  0. We will write  w for  roe hereafter.    Then we have thefollowing forms of equations                 (x  2,a)  ad +2u ar5aaT              razar                                 ,a3,                  + 2y)az                        342r ar(rc) a where T is the  difference of temperature, and                            2                  a=qx+v)c, where c is the cubical expansion coefficient of the body. and (5), we have            H.  ISHII                 r     Source  (  0,H  )    z n. Free surface is  2.--=0.           au' aux (1)         i e strain components are thenaayaT   az Transforming equations(2)(3)(4)(5)(6) 3 (4)          THEORETICAL STUDY ON THE CRUSTAL MOVEMENTS III 147          (x 2it) [ 024 +1 adda[-r-ra2Tar,a2T                                            (7)    ar  r r z2 jar2 r ar az2'       a2cei as a2cii    + --= 0  .  (8)         ar2  r  ar  r2  az2 We take as a solution of equation (8)                       2tv=A                             aJo(kr)-k 1x-HI                                          (9)  ar where A is an arbitrary constant. To obtain an expression at a free surface, we use the case  aJo(kr) k  (x-H)          2W.  =  A   zSH (10) In order to determine a form of T in equations (4) and (5), we must first solve the equation of the conduction of heat such as  aT a2T1 aTa2T     K  (11)             at ar2 r  ar  az2 where  IC is the thermal conductivity. In this paper we deal with T satisfying the condition of the conduction at every moment and the conditions of the problem to be solved. Now we assume that the temperature distribution of the body is expressed by the following  form:  T  =  of  jo(kr)eh(z-H)  zSH, (12) where  es is an arbitrary constant. Inserting (12) into (7), we have  au 1  ad  a2d             ar2r araz2 = o  .  (13) We take as a solution of equation (13)  BA(kr)ek(1-11)  zSH (14) Displacement (U1, w1) satisfying  d expressed by equation (14) and  co =0, is expressed by                            (k                          = 2kz eh (z-H)                    a Jr)ar                                            (15)    (1  ±  kz)  Jo(kr)eh  (z-H)  2k Displacement (U2, w2) satisfying  CU expressed by equation (10) and  4=0, is expressed by148 H. ISHII                      A    U, = z aj  r)el ("I)  ,                2ar                        A                       w, =  ——2 (1—kz) J „(kr)eh-11) Displacement (U3, w5) answering to  4  =2w=0 is expressed by                         afo(kr)   U3 = Cek (x-H)  ar  w3  C  kJ  0(kr)ek  (z-H)  .Therefore, the general solution for displacement is U=  U  1+  U2+  U3  , w =  W1+  W2+1113  •Substituting (10), (12) and (14), we have the relation  (x+21.4)B + ,uAk =  act  .Stress components are denoted as follows         aW  aw          z zA,11+2— aT —(3UU++ (x+21.4)  — aT  ,       azarraz  ez  o  ,           auaw        rz=-72( +az  ar    The condition that free surface be free of forces requires  Z  Z  OZ  ==  r  z  =  0 at z  =  0  .Substituting equation (18) into the boundary condition (21), we obtain  (X+2,u)B+2juk2C  act  ,  B  +2k2C  =  0  . We have from equations (19), (21) and (22)                                 1 act                   A =-                         k  (X+µ)                                act                           B=  X-  I-                                         act —  2k2(x,+ From equations (18) and (23) we have an expression for extension and tilt as(16)(17)18)(19)(20)(21)(22)(23) follows          THEORETICAL STUDY ON THE CRUSTAL MOVEMENTS III 149  aU act82J(kr)                                       eh(g-11) °  k20r2                                             (24) ate) act.(kr)                                    ek(x-11)aj- 72(X+/2))  /ear Performing the operation                                                                        cc,  Idk  0 to the temperature distribution (12) and to the extension and tilt component (24), and using the identical formulae  eh  (x  --H)J  o(kr)dk  1   i/r2+ —                                             (25)                f ek-Hi(kr)dk                                    (r2+(H-42)312                            0 we have  et                T=  (26)                             14.2+ (H — z) 2  a  U  a  •  et  r  1/1/2+r2  H  —  z    =   Dr  2()..+µ)  L1/r2+  (H—z)2  r2  r2  1/42  +  Ha  1/r2+  (H  —  z)  2  I (27)  r2  are  aotr1  z  —  H (28)  BrV2(x+,(2) Lr r V72+ (H — 421 3. Numerical Computation and Discussion    The temperature distributiondue to a thermal point source is shown in Fig. 2 using equation (26). Fig. 2 representatively shows a form of the curve which moves depending on  a  . The temperature decreases hyperbolically with increasing distance from the heat source. Extension and tilt on free surface due to the temperature distribution are shown in Fig. 3. The ordinate is the value multiplied by  H  •  2 (X,-1- t)                                                                         c•at and the abscissa is  n=rIH the distance divided by H. H is the depth of the point source,  X. and  ,u Lame's constant,  et constant appeared in the temperature distribution, and c cubical expansion coefficient. We have known the relation c  =3a1 where  ai is the linear expansion coefficient. In this computation we employed the linear expansion coefficient of basalt which is equal to 8.3 x  10-6, and  cr  =0.25. Computed results are graphically presented  in Fig. 3. As can be seen in Fig. 3 extension decreases more rapidly than tilt with increasing distance. Tilt has a maximum absolute value at 1.3 times the source depth. Next, we computed A  =11(H .2(X+I14)) , for practical                                                                      C•C150  H. ISHIIa)  V  C 11! a)  0  ....,L       3              4.•  0            L.         (1,0     a E     A  ir-  >           Distance Fig. 2. Distribution of temperature difference due to a thermal point source .0.5 0 -  0.5 2(J.+P  ) n=iFig. 3. Extension and tilt on free surface due to the temperature distribution shown in Fig . 2.purpose. Namely, the value of Fig. 3 multiplied by the value A is the value of an extension and a tilt observed on the surface. Table 1 shows the value of A for various  a and H. We can estimate extension and tilt on the surface using Fig. 3 and Table 1. For example, in the case of  a  =104°C•cm and  H=10 km, the temperature difference at    THEORETICAL STUDY ON THE CRUSTAL MOVEMENTS III                                ),u Table  1. Value of A  =1  (H  •2(x-F • H is the depth of the point  soruce, A  c•cz    Lame's constant,  cz constant appeared in the temperature distribution,                     and c cubical expansion coefficient.and  ,u151 (°C•  cm)  103Temperature  Difference (°C) (at 1 km from center) 1    2(A.+,u)  H•   c  •  ati i I I H=lkm H=  10km 0.  01 1.  038  x  10-7 1,  038  x  10-8 10,0. 1 106 1,  0 1,  038  x  10-8  1,  038  x  10-8 1,  038  x  10-7  1  038  x  10-8a point of 1 km from the point source is  0.1°C and the value of A is  1.038x  10-7 for basalt, so that extension at  n—r/H=0.0  (just above the source) is  0.5x  1.038x  10-7=  5.19x  10-8. 4. Summary    Surface extension and tilt of semi-infinite elastic medium are theoretically investigated in the case where a hyperbolical temperature distribution from an internal thermal point source exists. It is found that extension diminishes more quickly than tilt with increasing distance. It is also found that the largest tilt appear at 1.3 times the source depth. For some situations values of extension and tilt are presented for practical estimation.  Acknowledgments: The author expresses his sincere thanks to Prof. A. Takagi for encouraging the study and for critically reading the manuscript. He also wishes to thank Miss N. Takahashi for preparing the manuscript.                                REFERENCES   Arakawa, H.,  1931: The effect of temperature on the deformation of infinite or semi-infinite          elastic body, Geophys. Mag., 4, 297-306.  Ishii, H and A. Takagi, 1967: Theoretical study on the crustal movements Part II. The          influence of horizontal  discontinuity, Sci. Rep.  TOhohu Univ. Ser.  5, Geophys.19,           95-106.   Matsuzawa, T., 1942: Temperatur verlauf an der  Bodenffache  and der Spannungszustand in           der Erdkruste, Bull. Earthq. Res. Inst., 20, 20-29.   Nakano, S.,  1963: The effect of surface temperature on the crustal deformations, Disaster           Prevension Res. Inst. Kyoto Univ. Bull., 60, 2-20.  Nishimura, G.,  1930: The effect of temperature distribution on the deformation of a semi-         infinite elastic body, Bull. Earthq. Res. Inst., 8, 91-141.   Shima, M.,  1958: On the thermoelasticity in the  semi-infinite elastic solid, Disaster Prevension           Res. Inst. Kyoto Univ. Bull., 25, 2-17.

Theoretical Study on the Crustal Movements III. Extension and Tilt Produced by a Temperature Distribution Due to a Thermal Point Source in a Semi-infinite Elastic Medium

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Theoretical Study on the Crustal Movements III. Extension and Tilt Produced by a Temperature Distribution Due to a Thermal Point Source in a Semi-infinite Elastic Medium

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