elastic configuration with initial static stress field (*)

Transcription

1 On the plane strain in a theory for self-gravitating elastic configuration with initial static stress field (*) E. BOSCHI (**) Received on March 22nd, 974 SUMMRY. This paper is concerned with the plane strain in a theory for an arbitrary, uniformly rotating, self-gravitating, perfectly elastic Earth model with a hydrostatic initial stress field. Using the associated matrices method, a representation of Galerkin type is given. This representation enables us to derive the solution of the vibration problem corresponding to concentrated body forces. RISSUNTO. Tn questo lavoro si tratta il problema della deformazione piana nell'ambito di un modello terrestre arbitrario, uniformemente ruotante, autogravitante, perfettamente elastico e soggetto ad un campo idrostatico di sforzo iniziale. Usando il metodo delle matrici associate, viene data una rappresentazione di tipo Galerkin. Questa rappresentazione permette la soluzione del problema delle vibrazioni corrispondenti a forze di massa concentrate. INTRODUCTION. Dahlen ( 3 ) has developed the linearized equations and linearized boundary and continuity conditions governing small elastic-gravitational disturbances away from equilibrium of an arbitrary uniformly rotating, self-gravitating, perfectly elastic Earth model with an arbitrary initial static stress field [see, also, Boschi (')]. (*) This work has been made during a tenure of a C.N.R. fellowship. Cavendish Laboratory, University of Cambridge. (**) On leave from istituto di Geofisica, Università di Bologna, and Istituto di Scienze della Terra, Università di ncona.

2 24 E. BOSCHI In this paper we consider this theory in the case of a hydrostatic initial stress field and derive the equations for the plane strain. The medium is assumed homogeneous and isotropic. By making use of the associated matrices method ( 5 ), we give a representation of Galerkin type. This representation enables us to obtain the solution corresponding to concentrated loads in an infinite medium for the case of stationary vibrations. nalogous problems have been studied in other fields. («2 ) BSIC EQUTIONS. In the following we employ a rectangular coordinate system Oxk and the usual indicial notations. The Greek indices are supposed to take the values, 2 and the Latin indices the values, 2, 3. Let 2 be a plane region occupied by the considered medium. We denote by sa the components of the displacement vector and by 0i the perturbation in the gravitational potential. In the case of plane strain, we have: sa = s a («, t), 0 = 0 («,»2, t), S 3 = 0. [] From the equations established by Dahlen ( 3 ), we can derive the following basic equations for the plane strain problem in the case of a hydrostatic initial stress tensor T j = p0su (pa = const.): the equations of motion: aa = [2] Q0 Si 2 Q0 Q3 S2 = go Zltl + p + ^ g0 Sj 2 g0 Q3 = go + T p».p + the constitutive equations: Tn = (2 + 2 ft 2 p0) si.i + ( po) S2.2 T22 = ( po) si.i + ( + 2 /i 2 p0) s2,2 [4] Tl2 = ( «2, + (fl Po) Si,2 T2I = Si,2 + (fl Po) 82, In the above equations, we have used the following notations: Tap - the components of the incremental pseudostress tensor; Fa - the components of body forces; Q = (0, 0, Q3) - the steady angular

3 ON THE PLNE STRIN IN THEORY FOR SELF-GRVITTING ETC. 25 velocity rotation;, i - the appropriate constants of the material; Qo - the mass density; G - the gravitational constant; a comma denotes partial differentiation with respect to space variables and a superposed dot denotes partial differentiation with respect to the time t. Using equations [4], the differential equations [2] and [3] may be written as: V- p + ( + p 2 p.) 3 %i 2 7) + 2 Qo Q 5 t <) 2 Qo «i 5 t 2 + ( + p-2p a) + 0X 0 X2 S2 Po 0! = F Dxi (X+fi 2 p0) t)x Qo DX2 2 2 go Qz 5 t Si a 2 at 2 S 2 p0 a X2 p + ( + p 2 Po) 0i = ^2 [5] 3 a 4 7 Qo G dici Si + 4 n Qo G - )X2 s2 + 0i = 0 GLERKIN REPRESENTTION. Using the associated matrices method ( 5 ), we obtain the following representation of Galerkin type: si = p L-'2 V2 2 DX2 2 V2 2 *X2 2 ( + p 2 po) i 2 cixi 3,»2 + 2Q 0 Q 3 a ai Po 2 G " c hxl DX2 ) + p Qo 2 a 2 Qz a 2 2 hxi V2 t)x2 i)t r3. (X + p 2p0) ^ 2QoQZ^r t>x2 ai po 2 G y- + p + p Qo N vi 2 a 2 + V2 2 ' Xi 2 D2 2 a^i 2 2 Qz a 2 n 2 V2 2 i>i [6]

4 2(5 E. BOSCHI 3 2 Q3 Ì>2 0 = Ì7I Qo G [X 2 ÛiCl V2 2 Ì>X2 ìli S 2 Q3 i 4 7 Qo G 2 2 òa;2 2 2 ûa;i ì)i 4 Q3 2 ì> 2 2 Di 2 «2 2 ~ì)t 2 r. L a2 V ' = ôa, a 2 2 = (X + 2 n 2 Po) I go [7] 2 2 = [X j Qo The functions = («i, ìc2, ì) satisfy to the following equations: D F a ~ fx (X + 2 t 2 p ) n n = o, [8] [9] > 2 M 2 t, 2 = 2 M fi 2 \ -i 7 QOG STTIONRY VIBRTIONS. In what follows, we assume that: Fa = Be jj * (an, x2) icot [0] In this case we seek the solution in the form: a = Be l S (Xi, x2) e icot ] cpl = Re 0* (x x2) e io)t []

5 ON TIIE PLNE STRIN IN THEORY FOR SELF-GRVITTING ETC. 27 From the Galerkin representation [6], by putting Pi = Be ITi icut (xi, xi) e we obtain «:=/* Vl 2 a2 4 7 Qo G a 2 V2 2 0X2 2 n V2 2 a#2 2 ) ( 2 Po) + 2 i Qo to ~òxi ÒX t Qo 2 G a2 ì)xi ÒX2 r2 Qo i)xi 2 i ü3 co a V2 2 ÒX2 a s* = P + fi Qo (X + /t 2p0) *2 i'l 2 Ò 2 *2 a 2 P..T5T + - a 2 tei ~òx2 + V2 ' i Qo Qs CO + 4 7t >o 2 G -! 0x2 4 7 Qo G a 2 ) r2 V2 2 a»i 2 j a òxi 3 r* [2], *2 () 0* = 4 7 Qo G fl n go G u a 2 i Q3 co a V 2 <X2 2 Ì Ü3 CO a a* V2 2 *2 *2 ih 2 m 2 Po) 2 Vi 2 V2 2 r; * 2 = zl ^ a Va2 [3] Tlie functions {xi, x2) satisfy the equations d * ra* = fi (X p0) Fn [4] d* a* = 0.

6 28 E. BOSCHI D',, *2, *2 2 4:7 Qo G ' [5] with " = + co 2 v3 2. The operator D can be written in the form: D* = ( -I 2 ) ( -2 2 ), [6] fti 2, 7i2 2 are the roots of the equation k" + fc 2 TO2 ft) TI G IH 2 " ^ Ï2 2 " + CO* / ft)2 4 TI OO G ft) 2 î)2 2 Vi- X>3 0 [7] EFFECT OF CONCENTRTED FORCES. Xi. Let us examine the effect of a body force acting along the axis In this case, we have: Ft = 0, [8] and from equations [4], we can take: r* = r3* = 0 The solution of the problem is given by: r " V2 2 V2 i>x Qo G 5 2 ) * ) )2 ~ÒX2 2 ' «2 = ( +/* 2Vo) ì)xi ÒX2 2 n0 D-3 CO 0Xl ÒX2 [9]. 3 co 2 3, 2 i Q3 co û \ _ * 0 = ± 7 Qo G [x\ \, r X ÏXl V2 2 ì)! V2 2 i)x2 '

7 ON THE PLNE STRIN IN THEORY FOR SELF-GRVITTING ETC. 29 \" is the solution of the equation: ( 7ci 2 ) ( -2 2 ) n* = [X (X + 2 x 2 p0) [20] We can write the solution of equation [20] in the form: r-r * = Gi & + 7ci 2 (fci 2 fe 2 (7.-i 2 /C2 2 ) ' 7ci 2 À-2 (?3. [2] the functions Gi satisfy the equations: ( V ) a = Zl (?3 = - IX (X + 2 ^ 2 p0) x (X + 2 /t 2 p ) Fi*, [22] Let us consider the concentrated body force Fi* = 6 (xi)d(x2). [23] In this case the functions Gi are given by: Ga = G3 = jfo (fc0 r) 2 n [x (X + 2 [x 2 p0) In r 2jr/i (2 + 2,«2 po) [24] K0 (z) is the modified Bessel function of third kind, and r 3 - xi~ Using equations [2] and [24], the solution of equation [20], for concentrated body force, is given by: r* 2 7 fx (X + 2 (X 2 p0) io (7ci r) iio (7c2 r) fcl 2 (fcl 2 fc2 2 ) fc2 2 (fcl 2 7i2 2 ) In r [25] d 2 fc2 2 Thus, the solution of the considered problem is given by [9], f\* has the expression [25]. In a similar way, we can obtain the solution for a concentrated body force acting along the axis Ox2.

8 220 E. BOSCHI CKNOWLEDGEMENTS. I wish to thank Professor. H. Cook F.R.S. for his kind hospitality in Cambridge, aud Professor M. Caputo for valuable discussions and suggestions. REFERENCES ( X ) BOSCHI, E., Reciprocity theorem and elastic dislocation theory for an earth model with an initial static stress field. "J. Geophys. Res.", 78, ( 2 ) BOSCIII, E., On the plane strain in a Generalized Theory of Thermoelasticity. Int. "J. Eng. Sci.," 2, 433. ( 3 ) T) H LEX, F.., Elastic Dislocation Theory for a Self-Gravitating Elastic Configuration with an Initial Static Stress Field. "Geophys. J. R. str. Soc.", 28, ( 4 ) IESN, D., On the plane coupled micropolar thermoelasticity. I, "Bull. cad. Polon. Sci.", Scr. sci. Teclin., 6, ( 5 ) MOISIL, Gr. C., Teoria preliminara, a sistemelor de ecuatii cu derivate partiale lineare cu coeficienti constanti, Boletinul Stiintific al cad. R. P. R., Seria, 4, ( 6 ) NOWCKI, W., Green functions for the thermoelastic medium. II, "Bull. cad. Polon. Sci.", Ser. sci. Teclin., 2,