ࡱ> bd_`a[@ bjbj44 6TViViJJJ8L+^@#(h#h#h#h#h#h#+ + + + + + +$t,R.h-+?%h#h#?%?%-+h#h#B+)))?%.h#h#+)?%+)*)))h#R SJm%)+X+0+)./g&./)./)Dh#L#6)#,$)h#h#h#-+-+F)FPurdue University EAS 557 Introduction to Seismology Robert L. Nowack Lecture 7 Conservation Principles The theorems of conservation are the physical laws that govern the deformation of the continuum and are derived from physical principles. The physical laws we are concerned with are: Conservation of linear momentum Conservation of mass Conservation of angular momentum Conservation of energy Conservation of Linear Momentum global derivation  Let the particle velocity be  EMBED Equation.3  and let dm be the mass of a particle at P, then the integrated momentum is  EMBED Equation.3 . This is the total momentum summed over all the particles of mass that build up the body. Conservation of linear momentum requires  EMBED Equation.3  where  EMBED Equation.3  is the total force applied on the body. Applied forces are of two types: a) Boundary forces These are external forces applied on the boundary such as  EMBED Equation.3 . b) Body forces These are forces applied in the interior of the body by external force fields. For gravity,  EMBED Equation.3  where  EMBED Equation.3  is body force and  EMBED Equation.3  is acceleration of gravity. Gravitational body forces are proportional to the mass of the particles P. We can then write the linear momentum equation as  EMBED Equation.3  Using conservation of mass (see below), we may write  EMBED Equation.3  In component form, the linear momentum equation then can be written as  EMBED Equation.3  We would like to have the surface integral transformed into an equivalent volume integral. This may be done using  EMBED Equation.3  From Cauchys formula  EMBED Equation.3 , we can write the boundary tractions in terms of the stress tensor. Then,  EMBED Equation.3  This surface integral may be transformed into a volume integral using  EMBED Equation.3  as  EMBED Equation.3  Putting this into the momentum equation gives  EMBED Equation.3  Define the density as  EMBED Equation.3 , then,  EMBED Equation.3  If the volume under consideration is arbitrary, we can equate the integrands to find  EMBED Equation.3  This is the general equation of motion in continuum mechanics. This equation is valid for elastic, viscoelastic, liquid, and plastic materials, since no assumptions about the behavior of the material were involved in its derivation. Linearization of the Equation of Motion The derivative  EMBED Equation.3  is usually called a total derivative since it includes both the time variability and the variability due to any flow of the material. In elasticity, we prefer to solve for the velocity at a given spatial position. That is, instead of following a particle, we solve for the velocity of the particle that at time t is at position x. The total derivative (or material derivative) is  EMBED Equation.3  where  EMBED Equation.3  is the total derivative,  EMBED Equation.3  is the local derivative, and  EMBED Equation.3  is the advection term. Note that the total derivative of the particle velocity is generally nonlinear because of the advection terms. To derive this for a small time increment,  EMBED Equation.3 , the particle will have moved  EMBED Equation.3 . Then, the change of an arbitrary function g will be  EMBED Equation.3  Expanding in a Taylor series,  EMBED Equation.3  Then,  EMBED Equation.3  Finally, the total derivative is  EMBED Equation.3  In the linear approximation, we ignore the advection term and make the approximation  EMBED Equation.3  In this approximation, the particles dont move very much from their initial position. This is true for the propagation of a seismic wave that we are interested in here. The linearized equation of motion is  EMBED Equation.3  then  EMBED Equation.3  Conservation of Mass To show conservation of mass, let the mass M =  EMBED Equation.3 . Then,  EMBED Equation.3  where the last term is the material flux through the boundary. This can be changed to a volume integral to obtain  EMBED Equation.3  where  EMBED Equation.3  is the flux of the material. Now for conservation of mass,  EMBED Equation.3  Since volume is arbitrary, then  EMBED Equation.3 , which can be written  EMBED Equation.3  In terms of the total derivative  EMBED Equation.3 , then  EMBED Equation.3  This is called the continuity equation. In terms of the dilatation,  EMBED Equation.3  and  EMBED Equation.3  Thus, the density change is proportional to the negative of the dilatation change. The negative sign implies that density decreases when  EMBED Equation.3  increases or volume dilates. This can be used for a function  EMBED Equation.3  where  EMBED Equation.3  is the density. The total derivative of G is  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Because the last two terms in the integrand are zero from the continuity equation, then  EMBED Equation.3  For example, for the total momentum equation  EMBED Equation.3  from conservation of linear momentum,  EMBED Equation.3  This was used in derivation of linear momentum equation above. Conservation of Angular Momentum We showed in an earlier lecture that conservation of angular momentum results in the symmetry of the stress tensor. Thus,  EMBED Equation.3  which reduces the stress matrix to six independent numbers. Conservation of Energy We will typically assume that heat transfer is small in elastic wave propagation problems (it is an adiabatic process). To first order, work done on system is stored as strain energy and is completely reversible upon unloading since the strain is on the order of 10-4 in seismic wave propagation problems. In general, however, there will be some heat dissipation which will result in attenuation of the seismic wave. PAGE 2 PAGE 3 4Phj" # C D X Y y z          3 C L O P c d e 簣{jhh:6hc EHUjB hc CJUVaJhc h:66h zjafh:6h:6EHUjB h:6CJUVaJjh:6U h@h9jh@Uh:6h:65 h:65h@h:6 h95h9h:65hc h905FPij# D Y z $a$gd9^gd9gd9$a$gd9gd9gd9xgd9e f i       ! 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