Long-range stress re-distribution resulting from damage in heterogeneous media

Transcription

1 Long-ange stess e-distibution esulting fom damage in heteogeneous media Y.L.Bai (1), F.J.Ke (1,2), M.F.Xia (1,3) X.H.Zhang (1) and Z.K. Jia (1) (1) State Key Laboatoy fo Non-linea Mechanics (LNM), Institute of Mechanics, Chinese Academy of Sciences, Beijing , China. (2) Depatment of Applied Physics, Beijing Univesity of Aeonautics and Astonautics, Beijing , China. (3) Depatment of Physics, Peking Univesity, Beijing , China ( phone: ). Abstact It has been shown in cellula automata simulations and data analysis of eathquakes that declusteed o chaacteistic lage eathquakes may occu with a long ange stess e-distibution but afteshocks o powe law scaling events do not. In ode to undestand the long-ange stess e-distibution, we poposed a linea-elastic but heteogeneous-bittle model. The stess edistibution in the heteogeneous-bittle medium implies a longe-ange inteaction than that in elastic medium. Theefoe, it is supposed that the longe-ange stess e-distibution esulting fom damage in heteogeneous media may be a possible mechanism govening mainshocks. Intoduction Recently, the significance of long-ange stess e-distibution in undestanding of eathquake mechanism has dawn a geat deal of attention [1-5]. Vaious vesions of cellula automata (CA) with long-ange stess e-distibution fo eathquake faults wee widely used in studies. But undestanding what the natue of the long-ange inteaction and its significance is by no means an easy poblem. Diffeent eseach goups have diffeent undestandings. Fo instance, Klein et al [2] stated that linea elasticity yields long-ange stess tensos fo a vaiety of geological applications and fo a two-dimensional dislocation in a thee-dimensional homogeneous elastic medium, the magnitude of the stess tenso goes as 1/ 3. They noticed that while geophysicists do not know the actual stess tensos fo eal faults, they expect that long-ange stess tensos, which ae simila to the 1/ 3 inteaction, apply to faults. Moeove, they stessed that it is suspected that micocacks in a fault, as well as othe defects such as wate, sceen the 1/ 3 inteaction, leading to a poposed e -α / 3 inteaction, whee α<<1, implying a slow decay to the long-ange inteaction ove the fault s extent [2]. Contay to this, Weatheley et al [3] pointed out in thei cellula automata that the inteaction exponent (p in 1/ p ) detemines the effective ange fo stain e-distibution in the model. The effective ange deceases apidly as the exponent (p) inceases. The 9

2 event size-distibution illustates thee diffeent populations of events in the dissipative healing models (two-dimensional models): Chaacteistic lage events ( p < 1.5) Powe-law scaling events (1.5 < p 2.0) Ovedamped, no lage events ( p > 2.0). Physically, the existence of cacks and othe defects like wate may have two opposite effects on stess e-distibution. One is to sceen stess and then lead to a shoteange e-distibution, as claimed by Klein et al [2]. On the othe hand, the stess balance equies a compensational incease of the stess beyond the defects. This implies a longe-ange e-distibution of stess. Recently, Knopoff [5] investigated the magnitude distibution of declusteed eathquakes in Southen Califonia. He concluded that the pesumed univesality of the scale independence of the complete suite of local eathquakes is attibuted to the univesality of the popeties of afteshocks. All of the esults emind us that a declusteed o chaacteistic lage eathquake may occu elevant to a longe-ange stess e-distibution. But afteshocks do not. Then, instead of the commonly used homogeneous linea elastic theoy, could we find some possible altenative models with longe-ange stess e-distibution? Possible long-ange stess e-distibution in heteogeneous media It is well known that the main tems of stesses in a homogeneous linea elastic medium with ae 1/ 3 and 1/ 2 fo a spheical void in thee dimensions and a cylindical hole in two dimensions, espectively. Now, let us examine the stess edistibution owing to a void in a linea-elastic but heteogeneous-bittle medium to find out the effect of micodamage esulting fom the heteogeneity on stess e-distibution. It is supposed in the model that evey mesoscopic element has the same elastic moduli, like Young s modulus E and Poisson atio ν, but vaious beaking stengths σ c o c. Moeove, the stength of the element follows a distibution function, 0; when c < c h( c ) = q 1 ( c ) q. (1), q > 1; when c c c c We have to confess that we do not know the actual distibution of stength in geological media. Howeve, the simple distibution looks qualitatively easonable and povides some simple undestanding of the stess e-distibution in heteogeneous media. In fact, the heteogeneity must imply some intinsic length scales elevant to stuctues of geological media. But, as fist appoximation, we use local mean field to deal with the poblem. In this way, the intinsic length scales ae eliminated in the appoximation and the stess-stain elation in one-dimensional stess state is, as shown in Fig.1, E; when < c σ = 2 q Ec ( ) ; when c c (2) 10

3 0; when < c D = < 1 ( ) ; when c c In accod with the damage mechanics, the effect of damage can be descibed by the educed modulus, E ' = E(1 D) = E, (4) =. whee ( ) c In paticula, it is pesumed that the damage o the educed moduli ae govened by maximum stain, i.e. the cicumfeential stain. We call this the -model. Then, when > 1, the elastic-bittle constitutive elation in cylindical configuation (2-D plane stess), becomes σ = + ν ] (5) [ [ ν + ] σ =, (6) whee σ (1 ν 2 ) σ / E =. In the following we will ignoe the ba above all dimensionless vaiables. The balance equation in cylindical configuation (2-D) is σ σ σ + d d = 0. (7) Substitution of the stain definition and the elastic-bittle elation (5-6) into the balance equation (7) leads to a non-linea odinay diffeential equation of displacement u, u 2 ' ' ν u ν ν u 2 u'' + (1 q) + ( q+ vq) + ( q 1 ) = 0, (8) u whee u denotes diffeentiation with espect to adial distance. It is woth noticing that when q=1, Eq. (8) educes to the linea elastic vesion. But, when q is geate than unity, the non-linea second tem in Eq. (8) plays a significant ole. We use the following non-linea tansfomation to lineaize equation (8): u 1 q = v 2. (9) Then thee is powe type solution β v = A, (10) whee A is an abitay constant and β has two values q 2 β = 1 + ν qν. (11) Hence, the stesses, eithe cicumfeential o adial, will be in the fom q σ = [ O(1) + O( )] = [ O(1) + O( )][ A1 + A2 2 p whee A 1 and A 2 ae two abitay constants. One can veify that the tem ] (3), (12) 11

4 O( ) O(1) and p is. (13) 0 p= β + 2 q= 3 + ν (1 + ν ) q. (14) The powe index p in stess e-distibution appoaches 2 as q appoaches 1, and p deceases with inceasing q. That is to say, stess e-distibution in the heteogeneous elastic-bittle medium has longe inteaction ange with stonge heteogeneity. Similaly we obtained the solution fo the thee-dimensional (3-D) configuation and the coesponding exponent p, see Table 1: 0 p= + = β 2 q 2 ν 1+ ν 2 1 ν 1 ν q Discussion Thee ae two main assumptions made in the - model: that the Poisson atio ν emains invaiant and the cicumfeential stain govens the educed moduli. In ode to check the assumptions of the -model, we calculate an altenative model, the mixed model, which consists of elastic and damaged defomations. In the two-dimensional case, it is σ = [ + ν ] 1 = ν + ], (16) q σ. (17) [ In addition, finite element simulations in two-dimension wee made in accodance with the assumptions of the - model. The esults ae in good ageement, see Fig. 2. In summay, in ode to undestand long ange stess e-distibution, we poposed a linea-elastic but heteogeneous-bittle model. The stess e-distibution in the heteogeneous-bittle medium implies a longe-ange inteaction. Theefoe, it is supposed that the long-ange stess e-distibution esulting fom damage in heteogeneous media with intinsic length scales can quite possibly be a mechanism govening mainshocks. (15) Acknowledgments This eseach was funded by the National Natual Science Foundation of China (NSFC) unde the gant and Majo State Reseach Poject G

5 Refeences [1] Hill, D. et al 1993, Seismicity emotely tiggeed by the magnitude 7.3 landes, Califonia, eathquake, Science, 260, [2] Klein, W., M. Anghel, C.D. Feguson, J.B. Rundle and J.S. Sa Matins, 2000, Statistical analysis of a model fo eathquake faults with long-ange stess tansfe, in Geocomplexity and the physics of Eathquakes, eds Rundle JB, Tucotte D and Klein W, AGU [3] Weatheley, D., M.F. Xia and P. Moa, 2000, Dynamical Complexity in cellula automata with long-ange stess tansfe, AGU [4] Rundle, J.B. and W. Klein, 1995, Dynamical segmentation and uptue pattens in a toy slide block model fo eathquakes, Nonlinea Poc. In GeoPhysics, 2, [5] Knopoff, L., 2000, The magnitude distibution of declusteed eathquakes in Southen Califonia, PNAS, 97, [6] Bai, Y.L., M.F. Xia, F.J. Ke and H.L. Li, 2002, Non-Equilibium Evolution of Collective Micodamage and Its Coupling with Mesoscopic Heteogeneities and Stess Fluctuations, In: Shock Dynamics and Non-Equilibium Mesoscopic Fluctuations in Solids, eds. Hoie Y, Thadhani N and Davison L., Spinge-Velag (to appea) [7] Bai, Y.L., F.J. Ke, M.F. Xia, and H.L. Li, 2002, Closed tans-scale statistical micodamage mechanics, Acta Mechanica Sinica, 18, Table 1. The fomula and values of powe exponent p in appoximate powe law, in thee and two dimensional configuations. σ p Q P (3 D) P (2 D) 2 ν 1+ ν = + + p = q p 3 ν (1 ν ) q 2 1 ν 1 ν When ν = 1/

6 2 1.5 q = 1.5 stess 1 q = q = stain Fig.1 The one-dimensional stess and stain elation of the linea-elastic but heteogeneous-bittle model with diffeent q values. It shows that the geate the mesoscopic stength scatte is, the softe the model becomes Loading step 2 (pue elastic state) Loading step 6 (patial damaged state) Loading step 10 (damaged state) Analytic solution (pue elastic) Analytic solution (damaged elastic) 1.6 σ /p /a Fig. 2 The vaiations of cicumfeential stess with adial distance fo the elastic-heteogeneous bittle medium with paametes q=1.5 and ν=0.25 fo vaious loading steps. The points ae the FE esults. The two solid lines ae the analytical esults of elastic and the damaged elastic-bittle models espectively. The ageements between FE and analytic solutions ae vey good. The dotted line ( ) in between is the FE esult fo the state of patly elastic(oute pat) and patly damaged(inne pat). 14