An earthquake can be viewed as either an instability in frictional contact or the growth of a shear crack.  We model the crust of the earth as an elastic solid that can support an interesting variety of wave motions as well as static stress and strain fields.  Within this crust are fault planes, zones of weakness along which previous earthquakes have occurred.  Due to the large scale tectonic motion of plates, large stresses accumulate within the ground.  For example, here in Southern California, the Pacific and North American plate slide past each other along the San Andreas fault (you can imagine the fault as a vertical plane of contact extending from the surface down to about 10 or 15 km).  Below this depth, the increased temperature and pressure allow the rocks to move in a ductile manner that prevents the build-up of large stresses.  But closer to the surface, the rocks are brittle solids that are being compressed by the enormous weight of the overlying rocks (something you might be familiar with if you've been buried in the sand at the beach!).  This compressional stress will prevent the two sides of the earth from opening (which is what happens in a tensional crack).  Since the two sides of the fault are always in contact, there will be a frictional force between them. 

Now imagine trying to force the two sides to slide past each other.   For the San Andreas, the western Pacific plate (on which Santa Barbara sits) is moving northwest relative to the eastern North American plate (on which most of the country sits) at a rate of about 30-40 mm/yr.  This motion increases the shear stress on the fault over the course of decades or centuries, during which the fault is still locked.  At some critical level, the force on the fault will exceed the static frictional force, and the two sides will start to slide with some dynamic frictional resistance (less than the frictional value).  From a fracture point of view, there is a cohesive force that exists between the two sides of the fault, and the stress must be large enough to overcome this.  The discontinuity in the relative displacement is called slip.

At this point, even though there is slip, we still might not have an earthquake, in the sense that the crack might not continue to grow.  An insight from fracture mechanics tells us that at first, the growth of the crack is a stable process (like you hope the cracks in your windshield are!), and that in order to extend the crack you need to apply more force to it.  This is only true up to a certain critical length.  Once the crack grows beyond this length, it will become dynamically unstable and expand at speeds of up to 4-6 km/hr!  That's a good deal faster than the rate at which the plates move.  You can understand this instability in the following way.  The plate motion stores elastic energy in the earth that the earthquake is going to release.   Since slip allows the ground to return toward an undeformed configuration, it will release some of this elastic energy.  The amount of energy that is released when a crack grows by a certain amount is called the energy release rate, and increases with the length of the crack.  But there is a resistance to growing the crack since you have to break whatever cohesive force exists between the two unbroken sides - this is called the fracture energy and is a material property.  The instability occurs when the energy release rate exceeds the fracture energy; if you equate the two, you can solve for the minimum stable crack length or nucleation length.

From the above description, you can see than once an earthquake starts, it won't stop since the energy release rate keeps increasing as the crack grows!  Obviously this isn't true.  The earthquake can arrest for two main reasons: one, if it starts growing into an area that doesn't have any pre-existing stress field to relieve (the asperity model) or two, if it encounters a region that is impossible to break due to increased fracture energy (the barrier model).  Currently, we don't know which model is true, although recent work reveal different scaling laws present only in 3D.

During the dynamic expansion of the crack, much of the extra energy being released takes the form of elastic body waves.  These come in two varieties.  First, there is a longitudinal wave of alternating regions of compression and tension in the direction of propagation.  This is called a P-wave and is the same as a sound wave.  The second wave is a transverse shear wave, or S-wave, in which the material particles move side to side perpendicular to the the direction of propagation.  There are other types of wave motion that can occur when you apply a specific boundary condition, for example requiring that the surface of the earth have vanishing traction.  The most famous surface wave is the rolling Rayleigh wave, with a slower velocity than both the P- and S-waves.   Similar surface waves live on the fractured surface of the fault.  All of these varieties of elastic waves are responsible for the shaking we experience during earthquakes and for the destruction of buildings and roads.

Questions? E-mail Eric Dunham