Stress-Strain Theory

1. 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.

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

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

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

5. 1 light year Problem V > C

6. 1 light year Problem V > C V < C Elastic Strain Theory Elastodynamics

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

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

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

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

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

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

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

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

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

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

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

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

19. Problems 1. Utah and California movingE-W apart at 1cm/year. Calculate strain rate, where distance is 3000 km. Is it e or e ? 2. LA. coast andSacremento moving N-S apart at 10cm/year. Calculate strain rate, where distance is 2000 km. Is is e or e ? xx xx xy 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

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