1. Introduction

Occurrence of earthquake is usually a cyclic phenomena. About 90% of all earthquakes are natural which are the results of tectonic events primarily due to movement across the faults. Two major seismic events are separated by a long asesimic period in seismically active region. For better understanding of the earthquake processes it is necessary to study the stress accumulation pattern in the region during the aseismic period. Such studies can be done by developing suitable theoretical models with the essential features of the local geological structure and the earthquake faults situated in the lithosphere-asthenosphere system. The lithosphere-asthenosphere system may be represented by a multi-layered half-space model due to many inhomogeneities in it with respect to elastic properties. Many authors such as Maruyama [1], Rybicki [2, 3], Mukhopadhyay [4, 5], Debnath and Sen [6-9], Debnath and Sen [10-12] represented the lithosphere-asthenosphere system either by an elastic/ viscoelastic half-space or by a layered viscoelastic half-space. In our model we however consider a multi-layered elastic medium consisting of two elastic layers of finite depth overlying an elastic half-space to represent the lithosphere-asthenosphere system.

2. Formulation

We consider a theoretical model of lithosphere-asthenosphere system as three-layered elastic half-space with different rigidities. It is assumed that the medium is made up of two parallel, isotropic and homogeneous elastic layers lying over an isotropic homogeneous elastic half-space. The layers and half-space are in welded contact. The depth of the boundary dividing two elastic layers from free surface is and the depth of the boundary dividing second elastic layer and elastic half-space is . A buried vertical strike-slip fault whose length is large compared to its width l is taken to be situated in the first layer. The depth of upper edge of the fault below free surface is and the upper and lower edges of the fault are horizontal.

We now introduce a rectangular Cartesian co-ordinate system with the plane free surface as the plane is pointing downwards into the medium and is taken along the upper edge of the fault on the free surface. and give the boundary surfaces between two layers and second layer and half-space respectively. For convenient of analysis we introduce another set of Cartesian co-ordinate system with the upper edge of the fault as and the plane of the fault as the plane , so that the fault F is given by . The relations between two co-ordinate systems are given by

(1)

The first elastic layer occupies the region , the second elastic layer occupies the region and the elastic half-space occupies the region . The Figure 1 shows the section of the theoretical model by the plane .

Figure 1. Section of the model by the plane y1 = 0

Since we consider a long strike-slip fault then the displacements, stresses and strains are taken to be independent of and are functions of . Then the components of displacement, stress and strain will separated out into two distinct and independent groups, one associated with strike-slip movement and other associated with dip-slip movement of the fault. The components of displacements, stresses and strains associated with strike-slip movement of the fault are for first elastic layer, for second elastic layer and for elastic half-space respectively.

2.1. Stress-Strain Relations

The constitutive equations for the first elastic layer are taken as:

(2)

for second elastic layer are

(3)

and for elastic half-space are

(4)

where are rigidities of two layers and half-space respectively.

2.2. Stress Equation of Motion

For a slow, aseismic, quasi-static deformation of the system, the inertial terms in the stress equation of motion are very small and can be neglected as explained by Mukhopadhyay et. al. [13]. Then stress equation becomes

(5)

for first elastic layer

(6)

for second elastic layer

(7)

for elastic half-space .

From equation (2)-(7) we get

(8)

(9)

(10)

2.3. Boundary Conditions

We measure the time t from a suitable instant when the model is in aseismic state and there is no seismic disturbance in it. Since the free surface is stress free and the two layers and second layer and half-space are assumed to be in welded contact, then the boundary conditions are

(11)

2.4. Conditions at Infinity and Initial Conditions

are the values of at time t = 0 and they satisfy all the relations stated above. We assume that, at a large distance from fault plane there is a shear strain maintained by some tectonics forces (Kasahara [14]), such as mantle convection (Fowler [15]), etc. and then we have the following conditions

(12)

where , , , where are the values of at t = 0 and g(t) is a slowly increasing, continuous function of t with g(0) = 0. Same g(t) is taken for layers and half-space, since they are in welded contact, so that strains are continuous at the boundaries.

3. Displacements, Stresses and Strains in the Absence of Fault Movement

In this case the displacements, stresses and strains are continuous throughout the system. To obtain the solutions in the absence of fault movement we solve the boundary value problem given by (2)-(12) and get the solution in the following form

(13)

for first elastic layer,

(14)

for second elastic layer and

(15)

for half-space.

From these solution it is observed that there are accumulation of shear stress in the media. When this accumulated stress exceeds the total cohesive and frictional forces across the fault, then there will be a movement across it. Depending upon the nature of the fault and local geological conditions there may be two possible types of movements across the fault-either a sudden slip or an aseismic creeping movement across the fault. We consider the case of sudden movement across the fault for this model. Then the above solutions are no longer valid and required further modifications, after the fault slip.

4. Displacements, Stresses and Strains after Restoration of Aseismic State after a Sudden Movement across the Fault

It is to be noted that due to the sudden fault movement across the fault F, the accumulated stress will be released at least to some extent and the fault becomes locked again. For a comparatively short period of time during and after sudden fault movement when the seismic disturbances are present, inertial terms are not small and cannot be neglected. So we leave out this short period of time and consider the model after restoration of aseismic state. From this state we measure the time t afresh. In this second phase of aseismic state the equations (2)-(10), boundary conditions (11), (12) and initial conditions are valid.

The displacements, stresses and strains are continuous everywhere in the model except on the fault F across which the displacement component has a discontinuity which characterizes the sudden movement across the fault F.

The discontinuity of across F is defined as

(16)

where and is a continuous function of and U is constant, independent of . All the other components are continuous everywhere in the model.

To solve the above boundary value problem with new time origin for displacements, stresses and strains during aseismic state after commencement of sudden fault movement, we try to find the solutions in the following form

(17)

where represent the effect of displacements, stresses and strains present at t=0 and satisfy all the equations (2)-(12) and continuous throughout the medium while represent the effect of sudden fault movement across F and satisfy all the above relations (2)-(11) and also satisfy the dislocation condition (16) together with

(18)

Then,

(19)

(20)

(21)

where are the values of respectively at t=0 (i.e. new time origin).

Now to solve the boundary value problem containing we use Green’s function technique developed by Maruyama [1] and Rybicki [2, 3] as explained in Appendix. The required solutions are obtained as

(22)

for first elastic layer and

(23)

for second elastic layer and

(24)

for half-space.

where and analytical form of are given in Appendix.

It is found that the displacements, stresses and strains will be finite and single valued anywhere in the model if the following conditions are satisfied

(i) and are both continuous functions of for .

(ii) f(0) = 0, f(l) = 0 and

(iii) Either is continuous in is continuous except for a finite number of points of finite discontinuity in is continuous in except possibly for a finite number of points of finite discontinuity and for the end points of [0, l], there exist real constant m, n both < 1 such that or a finite limit as and or to a finite limit .

5. Numerical Computations

It is assumed that due to some tectonic reason there is a slow but steady accumulation of shear strain at a distance far away from the fault. Keeping this in view we take g(t) to be linearly increasing with time and g(0) = 0. With this assumption, we take g(t) = kt. From major earthquakes it has been observed that the stress release may be of the order of 400 bars. So we assume , noting also that the observed rate of strain accumulation in seismically active regions during the aseismic period is of the order of 10^-6 to 10^-8 per year.

For and the rigidities of the layers and half-space, We take , , as suggested by Aki [16], Bullen [17], Cathles [18], Chift [19], Karato [20] for lithosphere-asthenosphere. For the thickness of the layers we take depth of upper edge of the fault below free surface = 5 km. For l, the width of the fault, we use l = 10 km. noting that for San Andreas fault in North America, the value of l has been estimated to be in the range 5-15 kms. U = 40 cm corresponding to the observed relative displacements on the surface for moderate buried strike-slip fault movement. We take which satisfies the conditions as stated earlier for bounded stresses and strains everywhere in the model.

We now compute the following quantities:

(i) The residual/ additional surface shear strain due to fault movement near the fault after restoration of aseismic state

(ii) Change in shear stress in first layer due to fault movement after restoration of aseismic state.

The Figure 2 shows that the change of the residual/ additional surface shear strain with distance from the fault. The fault movement leads to release of the surface shear strain near the fault and the effect is symmetrical about the fault trace. The magnitude of the surface shear strain release due to fault movement is maximum near the fault trace and this strain release falls off rapidly as we move away from the fault on the surface and becomes very small for large values of . The magnitude of surface shear strain accumulation is found to be of the order of which is in conformity with the observed ground deformation during aseismic period in seismically active regions.

Figure 2. Additional surface shear strain due to fault movement

The Figure 3 shows the region of stress accumulation and release in first layer due to fault movement across F. The region coloured in pink (R) is the region of release of stress, while blue coloured region (A) is the region of accumulation of stress. This region is nearly symmetrical about the fault F. If a second fault is situated in the region ‘A’ then a movement across the fault F will enhance the rate of stress accumulation near the second fault and thereby a possible movement across the second fault will be advanced. But if a second fault is situated in the region ‘R’ then a movement across F will decrease the rate of stress accumulation near the second fault. Thus a possible movement across the second fault will be delayed due to movement across F.

Figure 3. Region indication for positive and negative accumulation of shear stress

The Figure 4 shows the change in shear stress with the depth for different values of . Here, for the first layer the region bounded by the upper and lower edges of , the shear stress is found to release in general irrespective of the distance from the strike of the fault. The magnitude of reduction is higher for point nearer to the fault. For gradually tend to zero.

Figure 4. Comparison of shear stress at different y2 in the first layer

The contour map of change in shear stress in the first layer due to the movement across the fault F situated in first layer has been shown in Figure 5.

Figure 5. Contour map of shear stresses in the first layer

6. Conclusions

It is observed that there are certain regions of stress accumulation in the first layer and certain other regions of stress release. The rate of stress accumulation/ release in the layer near the fault may be used to compute the happening of the next major event.

ACKNOWLEDGEMENTS

One of the authors Bula Mondal thanks the Head Mistress of Joka Bratachari Vidyasram Girls’ High School for allowing me to pursue the research, and also thanks the Geological Survey of India; Department of Applied Mathematics, University of Calcutta for providing the library facilities.

Appendix

Solutions of displacements, stress and strain in aseismic state after sudden movement across the fault:

The displacements, stress and strains for with new time origin after restoration of aseismic state followed by a sudden movement have been found in the form given by (17) where are given by (19)-(21) and satisfy (4.2)-(4.11), (16) and (18).

This boundary value problem involving can be solved by using modified Green’s function technique developed by Maruyama [1] and Rybicki [3]. According to them,

(A1)

(A2)

(A3)

where are the field points in the first layer, second layer and half-space respectively and is any point on the fault F and is the magnitude of discontinuity of across the fault F. According to Rybicki [3], the values of are given below

(A4)

(A5)

(A6)

(A7)

(A8)

(A9)

Where

(A10)

Continued equation (A10)

and

(A11)

and

Let is a point on the fault F with respect to the origin O and is any point on F with respect to the origin and a change of co-ordinate system from to is connected by the following relations ,. Then on the fault F, and . The discontinuity in is . Then from (A1), (A2), (A3) we get

(A12)

(A13)

(A14)

Now

(A15)

First part of (A15)

Now

Now second part of (A15)

where and . Putting the above values in (A15) we get,

(A16)

Where

Let

From Bullen and Bolt [21], we find that , , . Then .

Now,

So, as and and also .

Hence the function is monotonically decreasing. Since and is monotonically decreasing, then for all values of . Therefore, for our model. So, we can expand M as an infinite geometric series and neglecting the higher order terms and from (A16) we get

Now we assume in the above expressions and putting this value in (A4) and after integration we get

(A17)

Using similar process we obtain that

(A18)

and

(A19)

In change co-ordinate system . Putting it in (A17), (A18), (A19) and we get the new form of . Using these new form of in (A12), (A13), (A14), we get the following results

(A20)

(A21)

(A22)

(A23)

(A24)

(A25)

and

(A26)

(A27)

(A28)

where

(A29)

and

(A30)

(A31)

(A32)

and

(A33)

(A34)

(A35)

and

(A36)

(A37)

Where

(A38)

Continued equation (A38)