Radiation Damping as Approximation to Inertial Term for Dynamic Slip
Naoyuki Kato and Terry Tullis
An approximate way to deal with inertial effects in fault models is to include a "radiation damping" term (Rice, J. Geophys. Res., 98, 9885-9907, 1993) This term reduces the shear stress on the fault by an amount that is proportional to the slip velocity. The term is given by the following equation:
where t is the shear stress on the fault plane, b is the shear wave velocity, G is the shear modulus, and V is the velocity of slip of one side of the fault with respect to the other side.
The radiation damping term reduces the stress on the fault plane from what it would otherwise be. The other contribution to the total stress on the fault plane at the point of interest is the contribution to the stress at that point due to the slip on all the other parts of the fault (which in our situation are calculated using the appropriate Greenís functions in a boundary element model). Thus, the stress on the fault plane to be used in the friction constitutive law is the contribution via the Green's functions from slip on all other elements, minus the radiation damping term .
Origin of radiation damping term.
This radiation damping term can be derived from the full equation of motion for a simplified situation in which there is no variation in any quantities in the plane of the fault, i.e. and where x and z are in the plane of the fault and x is the slip direction. In this situation the only types of waves that can exist are S waves that travel as plane waves in the y direction away from the fault. Thus P waves do not exist and propagation of rupture is not directly included. However, if this term is used for determining the stress within one element in a boundary element model, it is found that the rupture will propagate in the xz plane at approximately the S wave velocity b . An explanation for this will be given below.
The following derivation shows that the radiation damping term is an exact representation of the equations of motion for the situation described above for which no variation of any quantities is allowed in the x and z directions. The equation of motion for forces and accelerations in the x direction is:
where r is density, u is displacement in the x direction and t ij is the shear stress in the j direction on the plane with normal in the i direction and t is equal to t yx, namely the shear stress on the xz fault plane in the x direction. Because the partial derivatives on the right hand side hold spatial derivatives fixed but not temporal ones, we can obtain an expression for t by integration of the partial derivative :
Evaluation of the integral can be done by noting that u(0,t) at the fault plane (y=0) propagates in the plus and minus y direction at speed b . We may use this to change the variable in . There is an equivalence between what occurs at the fault plane and what occurs at distance dy from the fault plane after time b dt and thus Substituting in to the above expression for t gives
where V is the slip velocity across the fault plane (y=0), noting that this velocity is twice the displacement rate of either side with respect to the middle and also remembering that and so where G is the shear modulus and r is density.
This stress t that causes the acceleration associated with the emission of S waves acts to retard the motion on the fault. Thus, the radiation damping term needs to be subtracted from the stresses applied to the fault, to give the frictional stress that is related to slip velocity by the fault friction law. Expressed differently, the sum of the frictional stress and the resistance to fault motion from inertia as given by the radiation damping term equals the total stress causing the fault to slip from the applied forces.
Rupture propagation speed due to radiation damping.
It is found in 2 dimensional numerical models that if the radiation damping term is included in the stresses acting on each element that the rupture propagates along the fault plane at about the shear wave velocity b . That this is to be expected from the inclusion of this term can be seen by the following argument.
In the absence of the radiation damping term there is nothing to stop the slip velocity or the rupture propagation velocity from becoming arbitrarily large. However, radiation damping limits the slip velocity and as shown below, this limits the rupture velocity as well. That this will happen can be seen by the following qualitative argument. In the radiation damping treatment, propagation of waves along the plane of the fault is not included and thus information about slip on one cell is transferred instantly to all other cells. However, the rupture will propagate sequentially from cell to adjacent cell because the stress increase due to slip on one cell is the highest in the adjacent cells. Thus as the rupture propagates, how fast the next cell in the sequence cell receives the information about slip in the previous cell is simply a function of how long it takes for that previous cell to slip. The following order of magnitude argument shows why in two dimensions this means that the rupture will propagate at about the shear wave speed.
Consider two cells adjacent to each other along the fault plane, where the dimension of each cell and the separation of their centers is D x. From 2D dislocation theory, the change in shear stress D t 2 on cell 2 due to slip on cell 1 of one side of the fault with respect to the other of magnitude u1 (pair of edge dislocations of opposite sign and magnitude u1 at boundaries of cell 1) is
when Poisson's ratio n = 0.25.
Now the change in stress on cell 1 is due to two contributions: the self stress on cell 1 due to slip on it and the radiation damping contribution. The self stress contribution from 2D dislocation theory is
and the radiation damping contribution is
so the total change in stress (a decrease) on cell 1 due to slip on it is the sum of these two terms, or
where D t is the time it takes for the displacement u1 to occur on cell 1 at slip velocity V1.
Now, without considering an exact rupture criterion (which can come from rate and state friction or some other assumption) we can say for an order of magnitude argument that rupture in cell 2 will take place when the stress increase in it equals some fraction F of the stress decrease in cell 1. From this we can approximately say that D t 2= -FD t 1 and from the above expressions we get
or or or
Thus we see that the distance D x from cell 1 to cell 2, divided by the time D t it takes for the stress to be transferred from cell 1 to cell 2 (a time that is equal to, and results from, the time it takes the slip on cell 1 to occur), gives what is best regarded as the rupture velocity and this proportional to the shear wave speed b . The value of the term in parenthesis depends on the value of F. For values of F less than 0.33 the term in parentheses is positive; larger values of F give a negative rupture velocity and thus are not physically plausible. That such values are not physically plausible can also be seen by noting that in the absence of radiation damping the stress drop on cell 1 due to the self stress term is 3 times larger than the stress increase on cell 2 due to slip on cell 1 (see above equations). Radiation damping results in even larger stress drops on cell 1. For values of F exactly equal to 1/3 the rupture propagates at infinite velocity and this corresponds to the case with no radiation damping. It is not possible to say what the correct value of F is in the absence of specifying more about the failure criterion. Given that this simple derivation only treats the interactions between two adjacent cells without regard for their initial states of stress or interactions with other cells, it should be regarded as an argument showing that radiation damping does influence the rupture propagation speed rather than as a rigorous derivation of the rupture velocity.
This derivation shows that although the radiation damping term results from assuming that plane waves propagate away from the fault plane in a situation where there is no variation in slip along the fault plane, its inclusion results in propagation of stress along the plane of the fault in proportion to the shear wave velocity.