2 .4 Model of Fault
Chapter 4 Chapter 4 Dynamic Energy Balance for Earthquakes
4.4 Change in Potential Energy
We define the change in potential energy as the energy released by the slip on the fault assuming that the slip occurs quasi-statically and that the domain behaves according to linear elasticity. Because both the radiated energy and the change in heat energy must be positive, conservation of energy dictates that the change in potential energy must be negative. This drop in the potential energy allows earthquakes to release energy as propagating waves and generate heat through frictional sliding.
We follow a procedure similar to that of Savage and Walsh (1978) and Dahlen (1977) to find the change in potential energy due to an earthquake. We start with the change in energy for an increment of slip,
dW -O'(D) dD dS, (4.11)
where dW is the incremental change in potential energy, er is the shear stress at a point on the fault, D is the slip at a point on the fault, dD is the increment of slip, and dS is the differential fault area.
The negative sign indicates the shear stress opposes slip. Assuming a linearly elastic medium, the stress follows
(4.12)
where <T0 is the shear stress just prior to the earthquake, u(D(t)) is the shear stress after the fault has slipped an amount D(t), D(t) is the slip at time t, Dis the final slip, and 6.<T is the final stress drop. We follow the convention that a decrease in stress gives a positive stress drop. Substituting the stress at slip D(t) into the expression for the incremental change in potential energy and integrating over both the slip and the fault area produces
(4.13)
39 Integrating over the slip and simplifying produces
AW=-~
{ (O"o+
0"1)DdS,2
ls
(4.14)where D and cr1 are the slip and stress at a point on the fault after the earthquake. After discretizing in time and space, equation ( 4.15) gives the change in potential energy caused by an earthquake, where (F(O)) is the friction force vector on the fault at zero slip and \Ft(D)) is the friction force vector on the fault at the completion of slip.
(4.15)
From the point of view of understanding the physics of the rupture, we would like to decompose the change in potential energy into the change in strain energy and change in gravitational potential energy. As shown by Savage and Walsh (1978) and Dahlen (1977), we cannot determine these changes in energy when we truncate the domain, because all points in the earth contribute equally to the computations; the domain must encompass the entire earth in order to compute the change in strain energy and the change in gravitational potential energy. Additionally, we neglect the change in Earth's rotational energy caused by earthquakes for the same reason.1
4.4.1 Topography and Changes in Gravitational Potential Energy
Tf WP rcmlcl rlPt.PrminP thP rhangP in gravitatiomi.l potPnthi.l PTIPrgy, it might ]pacJ 11~ t.o a hf1t.t.Pr im-
derstanding of the creation of mountains due to thrust earthquakes. The following simple thought experiment illustrates the general mechanism by which earthquakes change the gravitational poten- tial energy of the earth. Consider two containers of an incompressible fluid with widths b1 and b2
as shown in Figure 4.1. Figure 4.l(a) shows the containers filled with fluid to heights of h1 and h2 .
We may think of the two containers of fluid as the two sides of a thrust fault with the heights of the fluid corresponding to the level of the surface topography. The gravitational potential energy of this configuration is
(4.16)
We move the divider a distance d to the right. This represents a slip of d that generates the upward movement of the hanging wall and the subsidence of the footwall in a thrust earthquake.
1Chao and Gross (1995) computed the change in the rotational energy of the earth for a catalog of earthquakes using modal techniques and point sources.
(a) (b)
Figure 4.1: Configurations of the two fluid containers. (a) Original configuration (b) Configuration after the divider moves a distance d to the right.
The gravitational potential energy of the fluid.in the containers becomes
( 4.17)
The change in gravitational potential energy is
(4.18)
The movement of the divider increases the gravitational potential energy if
(4.19) which is nonlinear in the movement of the divider, d. Even for this simple analogy, the gravitational potential energy changes in a nonlinear fashion. If we start with equal heights and widths of the fluid containers (h1 = h2 and b1 = b2), the expression for the change in gravitational potential energy reduces to
( 4.20) and the change in gravitational energy is second order. This corresponds to no surface topography being present before the earthquake. On the other hand, when surface topography does the change in gravitational potential energy is first order.
For the same slip distribution on a given fault, the greater the differences in topographic fea- tures, the larger the change in gravitational potential energy. In other words, for mountain-building thrust fault earthquakes with the same amount of slip, each successive earthquake leads to greater changes in the gravitational potential energy. If the change in potential energy is the same for each
event, the change in strain energy must become more negative to balance the ever greater changes in gravitational potential energy. We do not know how the seismic behavior changes with these progressively larger changes in the strain energy and the gravitational potential energy, because we cannot compute the change in strain energy and change in gravitational potential energy. We must rely on the stress state as discussed in section 8.2.3 for insight into the roles of gravity and topography in seismic events.
on the fault according to some predetermined set of parameters. This method works well when we want to compute the ground motionG for Gceno.rioG with well known oouree parametero, such as the final distribution of slip on the fault, rupture speed, and maximum slip rate. On the other hand, we ignore the dynamics of the rupture process by not modeling the frictional sliding on the fault surface. Instead, we focus on the ground motions resulting from the choice of the earthquake source parameters.