SPIN AND TWIST MOTIONS IN A HOMOGENEOUS ELASTIC CONTINUUM AND CROSS-BAND GEOMETRY OF FRACTURING

Transcription

1 A C T A G E O P H Y I C A P O L O N I C A Vol. 5 No. 4 PIN AND TWIT MOTION IN A HOMOGENEOU ELATIC CONTINUUM AND CRO-BAND GEOMETRY OF FRACTURING Roman TEIEYRE Institute of Geophysics Polish Academy of ciences ul. Księcia Janusza Warszawa Poland rt@igf.edu.pl Abstract A uniform continuum with rotation motions of spin and twist type is sented; in this approach we supplement the ideal elasticity constitutive law the strain-stress relation by the rotation-asymmetric stress relation. In such a way we can evade an influence of the Hook law which when used as the unique law in the ideal elasticity rules out an existence of rotation waves. Thus in the ideal elastic continuum the rotation vibrations can propagate and are not attenuated. The asymmetric elastic rotation fields and their relation to asymmetric elastic stresses are proposed and discussed under the condition that the total fields with the elastic and self parts remain symmetric or antisymmetric as required by the compatibility conditions. The tensor of incompatibility splits into the symmetric or antisymmetric parts. The conservation and balance laws for spin and twist fields and the stress -related equations of motion for symmetric and antisymmetric parts of stresses are given. The relations obtained for elastic fields exssed by difference of the total and self-fields can be split into the self-parts vailing on the fracture plane and the total parts describing seismic radiation field in a surrounding space. The role of rotation processes in monitory and ound time domains is considered in estimating the most effective fracture patterns. Key words: asymmetric stresses spin and twist motions equations of motion rotation waves fracture pattern.

2 174 R. TEIEYRE 1. INTRODUCTION In a continuum with rotation nuclei the elastic asymmetric rotations and distortions can be exssed as a difference between the total and self-fields (Kröner 198): T E = E E T Τ ω = ω ω T β = β β = (1) where the total fields E T and T resent according to the compatibility conditions the symmetric fields; total rotations ω T are antisymmetric. The rotation motions of continuum points (in reality particles or grains) have weaker bonds as compared to the displacement motions and thus require less energy expenses than any other motions. The elastic and self deformations and rotations can be in general asymmetric; their antisymmetric/symmetric parts are mutually compensated: E E [] + [] = ω + ω = () () Also the related incompatibility tensors can be in general asymmetric. In such a way we can evade an influence of the Hook law which when used as the unique law in the ideal elasticity rules out an existence of rotation waves. Asymmetry of stresses may arise due to rotation effects related to friction processes at fracturing which usually has a non-symmetric geometrical pattern (fracture along the main fault). Homogeneous elastic continuum Our approach to the asymmetry of fields follows from the motion antisymmetric stresses introduced by himbo (1975; 1995) and related to grain motions under friction forces. Fracture processes develop usually along the main fault plane; hence there appears the initial asymmetry of the fracture pattern (Fig. 1); the rotation of grains adjacent to the main slip plane causes an appearance of antisymmetric stresses: A ω = µ [ lk] [ lk] ω = E = γ = A µ [ lk] [ lk ] [ lk ] [ lk ] (1 ) () where the factor A is introduced to indicate a difference between the constitutive parameter related to rotation motion and rigidity µ. This is himbo s constitutive law (himbo 1975; 1995) for rotation motion. The twist motion is another kind of rotational vibrations related to the shear deformation oscillating symmetrically around its zero value; such motions can be also sented as the opposite rotations counting from any perpendicular axes in a given quarter (Fig. ). However such deformations are already included in the classical strain-stress relation for the ideal elastic body:

3 PIN AND TWIT MOTION IN A HOMOGENEOU CONTINUUM 175 E 1 λ = δ µ µ (3λ + µ ) ( lk) ( lk ) lk ss E = ω = γ = 1 µ ( lk) ( lk ) ( lk) ( lk) (3) where the included bonds for a twist motion are compatible with results of our former papers (Teisseyre ; Teisseyre and Boratyński 3). Fig. 1. Anti-coincidence of the rotation pattern. Fig.. The case when the coincidence is not served: the twist pattern. Now using eqs. () and (3) we obtain the following formula for the asymmetric elastic stress field: T µ lk = lk lk = λδ lk Ess + µ Elk µγ ( lk) γ [ lk]. (4) A We will consider the equations of motion for symmetric and antisymmetric parts; for symmetric stresses we assume that the elastic velocity contributes to the elastic strain x k ( lk ) = ρ υl. (5) t Taking the space derivative of this relation and symmetrizing the result we arrive at = ρ x x t ( lk ) ( ls) k s ( ls) E. (6) The last term related usually to dislocation motion resents some deviations from the ideal elasticity as connected with self rotation fields; this term exssed by dislocation current and self fields (Teisseyre and Boratyński 3; 4) leads us with the help of eq. (3) to the following exssion for stresses with a given source field γ (sl) :

4 176 R. TEIEYRE x x t t T ( lk) = ρ E ( ls) ρ γ ( ls) k s ( ls) where the last term resents the self-field. In terms of displacement motion we obtain (7) uk u γ l ( lk) ( λ + µ ) + µ µ xl xs xk xk xk xs xk x s u u = ρ + t ( ls) l s ρ xs xl t γ ( ls). (8) This equation leads us to a classical displacement type equation with an additional source function. The associated waves sent propagation without attenuation like in the ideal elastic bodies. Considering a slip/fracture process along the tectonic plane we can try to split our equations of motion into two parts: the one describing the dynamic process with interaction of defects confined to very close vicinity to this plane and the other related to radiation in the surrounding elastic space. This method has been introduced by Teisseyre and Yamashita (1999). Antisymmetric stresses appear only on fracture plane in the dynamic processes with a counterpart containing defects and rotation nuclei. On the fracture plane the total rotation can now be assumed to be small in comparison to elastic and self-rotations. For the antisymmetric stress [in] we shall apply the balance law exssing on left side of the equation the rotation of force [ in] xn acting on a body element due to the antisymmetric stresses (rotational moment of forces per infinitesimall arm length) and on the right side the balancing term (the acceleration of rotation): 1 = ρ ω. (9) lki [ in] lki [ ki] xk xn t This correct approach to balance the antisymmetric stresses differs from that sented by Teisseyre and Boratyński (; 3) for a continuum with defects; nevertheless the final equation derived in the cited paper (eq. 18) is almost equivalent to the above equation (9). This is the equation for elastic spin ω [ki] at the given self-field γ [in] : or equivalently µ 1 γ = ρ ω Α lki [ in] lki [ ki] xkxn t (1) µ T 1 ( ω ω ) = ρ ω Α lki in [ in] lki [ ki] xkxn t. (1 )

5 PIN AND TWIT MOTION IN A HOMOGENEOU CONTINUUM 177 Its other form determines the relation of total rotations to the field γ [ki] : µ 1 T lki γ[ in] = ρ lki ( ω[ ki] γ[ ki] ) (11) Α xkxn t or 8µ ui u k lki γ[ in] = ρ lki γ[ ki]. (11 ) Α xkxn t xk xi When the displacement motions vanish we arrive respectively at the relations: γ ( lk) 1 µ = γ( sl) x x V t k s ( ls) T µ V T = (1) ρ 1 γ = γ x x V t [ in] [ ki] k n R µ = (13) Aρ 4 V R which are the wave type equations for twist γ (sl) and spin γ [ki] fields with the elastic bonds; note that we can have different velocities for spin and twist. The alternative way to introduce the rotation motions with a concept of continuum with dislocation and disclination densities and rotation nuclei was sented elsewhere (Teisseyre and Boratyński 3; 4); here we show that even in a uniform continuum such rotational vibrations and waves exist and are not attenuated as it happens for the displacement motions in ideal elasticity. Conservation laws for the γ fields For a spin vector we can write 1 γ[ s] = smn γ[ mn] = s mn γ[ mn ] (no permutation over bold indexes). (14) According to the spin conservation law the total spin at an internal source vanishes: x γ s [ s] = or smn γ[ mn ] =. (15) This is shown in Fig. 1 senting the pattern of rotations and directions of respective axes; the required anti-coincidence is served. The twist conservation law provides a new quantity named the twist charge Γ defined as follows: where γ (s) is a twist vector γ γ x s x s () s = smn ( mn) 4πΓ xs = (16)

6 178 R. TEIEYRE γ γ ( s) = smn (mn) (no permutation over bold indexes). (17) This is shown in Fig. with the pattern of rotations and directions of the respective axes; the required coincidence is not served hence there enters the twist charge Γ defined by relation (16). The diagonal elements of the generalized rotation resent stretching which may produce a scalar field γ = 13γ ss ; its time changes are opposite to the density changes & γ & ρ.. CRO-BAND FRACTURING MODEL OF EARTHQUAKE OURCE AND ROTATION PROCEE Comssion = 1 1 Under the assumption V = we reexamined the Dietrich (1978) comssion experiments which leads to the conclusion that for the cursory shear stress 1 > (as induced by the confining ssure) we arrive at the coseismic ound compensation shear stress < (signs are defined arbitrarily): (18) With the antisymmetric stresses induced by rotation processes we can propose the following reintertation: local induced shear stresses are asymmetric: 1 > and 1 < ; for stresses on the main (future) parallel plane we have > 1 < ; (19) 1 for rotations on the main (future) parallel plane we have γ clockwise γ anticlockwise ; () for stresses on an auxiliary perpendicular plane we have < 1 > ; (1) 1 for rotations on an auxiliary perpendicular plane we have γ clockwise γ anticlockwise. () Precursory rotations associated with slips or dislocations are opposite to that related to coseissmic process and both serve their sense on perpendicular planes. To relate the stress acumulation to defect densities we recall the basic relation = n for the dislocation array in equilibrium (supported by the external stress ). This relation comes from interaction of dislocations α i (resented here by their stress

7 PIN AND TWIT MOTION IN A HOMOGENEOU CONTINUUM 179 gradients i ) pushed by external field to the first blocking dislocation; is the resulting stress acumulation at the place of the first dislocation: n = di = x x dx (3) i In our case we have no initial shear field but only comssion: due to the lower value of shear resistance we have to assume that inside a body there appear regions with the induced shear stresses of opposite signs (as a result we get the induced antisymmetric shear stress). The shear stress according to the band-model is related to the concentration of induced dislocations (counted up to N ind or n ind ): the induced shear stressess are asymmetric = An hence the effects of processes at these planes are as follows: on the parallel plane ind ind AN α AN ind α ind and on the perpendicular plane (4) ind ind ind ind An α An α. The signs of dislocations of these planes can be different depending also on the position of their wedge but a common sense of rotations shall be served on the respective planes. Earthquake process and its energy release relates to coalescence of dislocation arrays of the opposite signs; also that part of energy release which depends on drop of moment of momentum depends on dislocation coalescence process; it can be exssed by rotation release. For a total shear release on the two planes we obtain (with signs as in eq. 19) + A A + while for an instantaneous shear stress drop and the total shear stress drop we get ind ind A N A n A N ind n ind. (7) ( ) Thus the total shear stress drop for comssion process can be in such a case small; the real stress drop relates to non-shear components. Coseismic rotations are opposite to the cursory rotations; it is the coseismic process that brings the release of rotation. The total rotation release at the plane and at the plane is R + R R + R. The final instantaneous rotation release for coalescence process on both planes and the total one are as follows: (5) (6) (8)

8 18 R. TEIEYRE ind ind R B N R B n R B( N ind + n ind ) (9) where B is an adequate proportionality coefficient. uch releases (eqs. 7 and 9) can be exssed as a release of momentum and that of moment of momentum for the D band fracture model for discs (Teisseyre et al. 1) on the main and auxiliary planes (πr D and πr d): External shear load ind ind M = Aπ( R D N r d n ) ω ind ind M B ( R D N r d n ). = π + = 1 1 and at the perpen- In a similar way as above we shall put at the main parallel plane dicular planes : > 1 > ; 1 > 1 > 1 For the regional shearing field ( 1 = 1 ) the effects of processes at the main (8 ) (9 ) (3) γ clockwise γ anticlockwise (31) γ clockwise γ anticlockwise. (3) shearing and at auxiliary perpendicular planes are (on both planes and ): + AN α + AN α (33) + An α + An α. The shear stress drops on both planes respectively: (34) and and the total shear stress drop is A N A n A( N + n). (35) The signs of rotation release on these planes are opposite; for instantaneous rotation release we get exssion with a relatively smaller value (comparing to the former case) comp R B( N n) (36) and for the momentum and the moment of momentum releases for the D band fracture model we get: M = Aπ ( R D N + r d n) ω M = Bπ ( R D N r d n). (35 ) (36 )

9 PIN AND TWIT MOTION IN A HOMOGENEOU CONTINUUM 181 Comssion and shear load < 11 Assuming 11 < we can sent the initial stress state as a combination of the two cases sented above: = and 11 = = 1 ( ) % uch a combination leads to: (a) bigger cursory phenomena (sum of the cursory shear and induced shear); and (b) smaller shears on perpendicular planes (difference of cursory and induced shears). Hence there follows a rule for earthquake asymmetry faulting (even independent of any vious stick-slips): the time interval between cursory and main processes (effect of confining ssure on shear processes); a combined shear stress drop becomes A[ N + N + n n ind ind an instantaneous rotation release becomes ] ; (37) ind ind R B[ N + N n + n ]. (38) Thus for momentum and moment of momentum releases for the D band fracture model we obtain ind M = Aπ[ R D( N + N ) + r d( n n ind )] (37 ) ω ind ind M = Bπ[ R D( N + N ) + r d( n + n )]. (38 ) Fig. 3. Antisymmetric stresses related to the opposite rotations on the main and auxiliary planes (induced stresses). Fig. 4. ymmetric stresses related to the opposite rotations on the main and auxiliary planes.

10 18 R. TEIEYRE Induced antisymmetric shear stresses and symmetric (global external) stresses may cause different slip motions along the perpendicular planes (Figs. 3 and 4); these slips form the antisymmetric stresses related to friction and the opposite rotations of grains adjacent to these planes. For a continuous process these antisymmetric stresses relate to perpendicular dislocations. 3. CONCLUION By applying the constitutive relation for antisymmetric stresses we can evade an influence of the Hook law which in the ideal elasticity when used as the unique constitutive law rules out the existence of rotation waves. Thus it is not true that in a homogeneous elastic continuum the rotation vibrations are automatically attenuated and ruled out. Contrary it seems that for such motions of particles the elastic bonds are not so strong as for displacements. The wave equations contain the source function depending on spin and twist processes in a seismic source. There is no attenuation for these waves as it is for ideal elastic waves but here such waves contain the part related to rotational vibration of the points of continuum. As the nature of the spin and twist motion is similar it seems justified to assume that the related velocities are equal V R = V T ; from eqs. (1) and (13) it follows that such a case requires to put A =. An alternative way to introduce the rotation motions which required the concept of continuum with dislocation and disclination densities and rotation nuclei was sented in our former paper; here we show that even in a uniform continuum such rotational vibrations and waves exist and are not attenuated as the displacement motion in ideal elasticity. Thus our important conclusion is that the influence of rotational processes in earthquake sources is not attenuated as it was believed according to classical ideal elasticity and can be recorded even at distant sites. Rotation processes at source zones help to understand geometry of fracturing and formation of the main fault; release of stresses and related rotation release processes are estimated for three cases of an external stress field for the D source band models. A c k n o w l e d g m e n t. The paper has been sponsored by the Polish Committee for cientific Research Grant No. 6 P4D 66. References Dietrich J.H.J Preseismic fault slip and earthquakes diction J. Geophys. Res. 83 B

11 PIN AND TWIT MOTION IN A HOMOGENEOU CONTINUUM 183 Kröner E. 198 Continuum theory of defects. In: Balian et al. (eds.) Les Houches ession XXXV 198 Physique des Defaults/Physics of Defects North Holland Publ. Comp. Dordrecht. himbo M A geometrical formulation of asymmetric features in plasticity Bull. Fac. Eng. Hokkaido Univ himbo M Non-Riemannian geometrical approach to deformation and friction. In: R. Teisseyre (ed.) Theory of Earthquake Premonitory and Fracture Processes PWN Warszawa Teisseyre R. Continuum with defect and self-rotation fields Acta Geophys. Pol Teisseyre R. and W. Boratyński Continuum with self-rotation nuclei: evolution of defect fields and equations of motion Acta Geophys. Pol Teisseyre R. and W. Boratyński 3 Continua with self-rotation nuclei: evolution of asymmetric fields Mech. Res. Commun Teisseyre R. and W. Boratyński 4 Generalized continuum with defects and asymmetric stresses Acta Geophys. Pol Teisseyre R. and T. Yamashita 1999 plitting stress motion equation into seismic wave and fault-related fields Acta Geophys. Pol Teisseyre R. K.P. Teisseyre and M. Górski 1 Earthquake fracture-band theory Acta Geophys. Pol Received 9 December 3 Accepted 19 January 4