Stress-Strain Theory Under action of applied forces, solid bodies undergo deformation, i.e., they change shape and volume. The static mechanics of this deformations forms the theory of elasticity, and dynamic mechanics forms elastodynamic theory.

Displacement vector: u(x) = x’- x Strain Tensor After deformation u(x+dx) dx’ dx dx’ dx u(x) x x’ Displacement vector: u(x) = x’- x Length squared: dl = dx + dx + dx = dx dx 2 1 3 i dl = dx’ dx’ = (du +dx ) i 2 = du du + dx dx + 2 du dx i

Length squared: dl = dx + dx + dx = dx dx dl = dx’ dx’ = (du +dx ) Strain Tensor After deformation u(x+dx) dx’ dx dx’ dx u(x) x x’ Length squared: dl = dx + dx + dx = dx dx 2 1 3 i dl = dx’ dx’ = (du +dx ) i 2 = du du + dx dx + 2 du dx Length change: dl - dl = du du + 2du dx i 2 (1) du = du dx dx i j Substitute into equation (1)

= Length change: dl - dl = du du + 2du dx du = du dx Substitute dx Strain Tensor After deformation u(x+dx) dx’ dx dx’ dx u(x) x x’ Length change: dl - dl = du du + 2du dx i 2 (1) into equation (1) du = du dx dx i j Substitute Length change: dl - dl = U U i 2 Strain Tensor (2) (du + du + du du )dx dx i j dx k =

Problem V > C 1 light year

Problem V > C V < C 1 light year Elastic Strain Theory Elastodynamics

Acoustics = e xx = dL L’-L L L Length Change Length L’ L

Acoustics = e xx = dL L’-L L L Length Change Length L’ L

No Shear Resistance = No Shear Strength Acoustics = e xx = dL L’-L L L Length Change Length L’ No Shear Resistance = No Shear Strength L

Acoustics dw dz dw, du << dx, dz Tensional dx du

Acoustics e = = U dw dz dx du = + = = dxdz+dxdw+dzdu-dxdz dx dz really small big +small big +small = dxdz+dxdw+dzdu-dxdz dx dz + O(dudw) Area Change Area = (dz+dw)(dx+du)-dxdz dx dz dw du dz = dx + Infinitrsimal strain assumption: e<.00001 dw e zz xx = + dz = U Dilitation dx du

1D Hooke’s Law e k -k F/A = - P = ( ) dx du U Bulk Modulus pressure strain = U k -k du F/A = Infinitrsimal strain assumption: e<.00001 dx e zz xx + ( ) P = - Bulk Modulus Pressure is F/A of outside media acting on face of box

Hooke’s Law e e k k k F/A = F/A - P = ( ) ( ) U Bulk Modulus = + + zz xx + ( ) F/A = k = U k F/A Infinitrsimal strain assumption: e<.00001 e zz xx + ( ) P = k - Bulk Modulus Pressure is F/A of outside media acting on face of box

Hooke’s Law e e e e k k k F/A = - P = ( ) ( ) k U + S Bulk Modulus Dilation e zz xx + ( ) F/A = k e k U Infinitrsimal strain assumption: e<.00001 = e zz e k - P = ( ) + S + Bulk Modulus xx Source or Sink k Larger = Stiffer Rock Compressional

Newton’s Law .. .. .. ma = F - - r r ; P (x+dx,z,t) P (x,z,t) k dP dx u = r .. - dP dz w = r .. ; density u r .. Net force = [P(x,+dx,z,t)-P(x,z,t)]dz x, -dxdz k Larger = Stiffer Rock P (x+dx,z,t) P (x,z,t)

1st-Order Acoustic Wave Equation Newton’s Law 1st-Order Acoustic Wave Equation .. - P u = r u=(u,v,w) - dP dx u = r .. - dP dz w = r .. ; density k Larger = Stiffer Rock P (x+dx,z,t) P (x,z,t)

1st-Order Acoustic Wave Equation Newton’s Law 1st-Order Acoustic Wave Equation .. - P u = r (Newton’s Law) (1) (2) .. = - U k P (Hooke’s Law) Divide (1) by density and take Divergence: .. - P u = r 1 [ ] (3) Take double time deriv. of (2) & substitute (2) into (3) .. - P P = r 1 [ ] k (4)

2nd-Order Acoustic Wave Equation Newton’s Law 2nd-Order Acoustic Wave Equation .. - P P = r 1 [ ] k P P = k .. r Constant density assumption k r c = 2 Substitute velocity P P = .. c 2

.. .. .. ; P ] [ P P = c P P = c = k k k - - 1. Hooke’s Law: P Summary = - U k 1. Hooke’s Law: P 2. Newton’s Law: .. - P u = r .. - P P = r 1 [ ] k 3. Acoustic Wave Eqn: Constant density assumption P P = .. c 2 ; k r c = Body Force Term + F

Problems 1. Utah and California movingE-W apart at 1 cm/year. Calculate strain rate, where distance is 3000 km. Is it e or e ? xx xy 2. LA. coast andSacremento moving N-S apart at 10 cm/year. Calculate strain rate, where distance is 2000 km. Is is e or e ? xx xy 3. A plane wave soln to W.E. is u= cos (kx-wt) i. Compute divergence. Does the volume change as a function of time? Draw state of deformation boxes Along path

U n U dl A e k - P = n n Divergence = lim ( ) A 0 >> 0 = 0 + = U(x+dx,z)dz + U(x,z+dz)cos(90)dx dxdz - U(x,z)dz dxdz + U(x,z+dz)cos(90)dx dxdz >> 0 = 0 dxdz e zz xx + ( ) P = k - n Sources/sinks inside box. What goes in might not come out No sources/sinks inside box. What goes in must come out U(x,z) U(x+dx,z) (x+dx,z+dz) n (x,z)