Generalized Hooke’s Law and Anisotropic Wave Equation

Generalized Hooke’s Law

Outline 1.Generalized Hooke’s Law 2.Hooke+Newton =Anisotropic Wave Equation 3.Isotropic Wave Equation 4.P and S waves 5.Summary 6.Appendix

Generalized Hooke’s Law Cards are normal stiffer along 11 direction so stiffness c 11 > c 33 so  11 = c 11   1 and  33 = c 33   so  11 = c 11   1  c 13    c 13   x1 x2 x3x1x2x3  c 12    c 12    c 1?    c 1?   Not enough indices for c ij Normal deformation along 11 depends on 11 and 33 stiffnesses (Note buckled cards cause 33 stiff ness weak): What happens if we stress cards along horizontal? How about c ijkl ? Any stress induces any strain combination

Generalized Hooke’s Law Cards are normal strength stiffer along 11 direction so stiffness c 11 > c 33 so  11 = c 11   1 and  33 = c 33   Normal deformation along 11 depends on 11 and 33 stiffnesses (Note buckled cards cause 33 stiff ness weak): But cards might have cuts along 11 direction so buckling is easier (weaker stiffness) with 22 normal stress compared to 11 normal stress. Hence, stiffness values should depend on orientation of applied force. We made a mistake in 1a and 1b where we assumed stiffness c33 was same for traction applied to 11 plane and on 33 plane. Hence, x1x2x3(1a) (1b)  11 = c 1111   1  11 = c 1111   1  c 1133    c 1133   Cuts along 11 direction  11  33  22 We made a mistake, stiffness coefficient seems to depend on orientation of contact plane and normal traction component. so  11 = c 11   1  c 33    c 33  

Generalized Hooke’s Law Cards are normal stiffer along 11 direction so stiffness c 11 > c 33 so  11 = c 11   1 and  33 = c 33   Normal deformation along 11 depends on 11 and 33 stiffnesses (Note buckled cards cause 33 stiff ness weak): so  11 = c 11   1  c 33    c 33   x1 x2 x3 (1a) (1b)  11 = c 1111   1  11 = c 1111   1  c 1133    c 1133   Cuts along 11 direction Normal deformation along 11 plane depends on shear stiffness as well. Weak shear stiffness yields shear buckling (or breaking) from 11 normal stress. Hence,  c 1121    c 1121    11  33  22

Generalized Hooke’s Law Stiffness of rock depends on contact plane orientation and component of traction we impose; every type of deformation results along all three different planes. Strain tensor Stress tensor  = c  ijijijij ijkl kl Einstein notation (repeated indicies are summed) (repeated indicies are summed)  11  33  22

Generalized Hooke’s Law

81 constants (3x3x3x3=81) but 21 are independent constants. Strain tensor Stress tensor  = c  ij ijkl kl Stiffness tensor Generalized Hooke’s Law (Cauchy in 1800s figured this out) 1). Symmetry of strain:  kl lk 81->54 2). Symmetry of stress: kl lk 54->36  3). Symmetry of strain energy: W = 0.5  ij ij = ijkl c  kl 0.5  ij = klij kl 0.5  ij 36->21 Interchange ij with kl because kl and ij are dummy variables under summation c ijkl

Strain tensor Stress tensor  = c  ij ijkl kl Stiffness tensor Symmetry Strain: 81->54 (Cauchy in 1800s figured this out) 1). Symmetry of strain:  kl lk 81->54c ijkl  = c  ij ijkl kl = c  ijlk lk  subtracting both sides gives ijkl - c )  ijlk kl  0 = (c ijkl  Therefore c = c ijlk 81->54 kl 9x9 matrix C mn : 81 independent coefficents. We will show strain symmetry implies C mn has m=1...9 & n=1...6 indep. values 9x9 matrix: =54 independent values per row ij has 9 and kl only has 6 independent values  C mn only has 54 independent comp. Map ij  m 11  1 22>>2 33  3 12  4 13  5 21  6 23—7 31  8 32  9 c ijkl  C mn

 = c  ij ijkl kl Symmetry Stress: 54->36 (Cauchy in 1800s figured this out)  =  ijji  = c  jikl kl  0 = (c - c )  jikl kl ijkl Subtracting above gives  c = c jikl ijkl ij now has 6 and kl only has 6 independent values  C mn only has 36 indep. components Map ij  m 11  1 22>>2 33  3 12  4 13  5 21  6 23—7 31  8 32  9 c ijkl  C mn

Energy : 36->21 (Cauchy in 1800s figured this out) Subtracting above gives  c = c ijkl klij 3). Symmetry of strain energy: W = 0.5  ij ij = ijkl c  kl 0.5  ij = klij kl 0.5  ij 36->21 Interchange ij with kl because kl and ij are dummy variables under summation 6x636->21 c ijkl  C mn 6x6 matrix C nm =C mn 6x6 matrix

81 constants (3x3x3x3=81) but 21 are independent constants Strain tensor Stress tensor Isotropic assumption (rotational symmetry)  = c  ij ijkl kl ij ijkk  =  e 2  ij + Stiffness tensor Generalized Hooke’s Law (Cauchy in 1800s figured this out)

Summary

Voight Notation 36 components c ijkl can be reassembled into a 6x6 matrix C mn kl

Isotropic (2 parameters) Cubic Symmetry (3 param.) VTI Symmetry (5 param.) Orthotropy (9 param.)

Plane Waves & Christoffel Equation

Outline 1.Generalized Hooke’s Law 2.Hooke+Newton =Anisotropic Wave Equation 3.Isotropic Wave Equation 4.P and S Waves 5.Summary 6.Appendix

 = c  ij kl Anisotropic Wave Equation in Homogeneous Medium  ij,j =  u.. + F i i (1a) (1b) (u + u ) (u + u )k,l l,k ijkl Hooke’s Law 6 indep. eqs. Newton’s Law 3 indep. eqs. 9 unknowns: u 1, u 2, u 3,  11,   2,  33,  12,  13  23 /2 Substitute (1b) into (1a) yields u k,lj =  u.. + F i i c ijkl 3 indep. eqs. 3 unknowns ij,jk,lj l,kj Take derivative of (1a) w/r to x j (assume homogeneous medium) (1c) u ) l,kjc ijkl u k,lj ½(c + = u k,ljc u ) k,ljc ijlk u k,lj ½(c ijkl + = u k,ljc u ) k,ljc ijkl u k,lj ½(c + = u k,ljc

Outline 1.Generalized Hooke’s Law 2.Hooke+Newton =Anisotropic Wave Equation 3.Isotropic Wave Equation 4.P and S waves 5.Summary 6.Appendix

(u + u ) (u + u )k,l l,k Elastic Isotropic Wave Equation in Inhomogeneous Medium c =   j   kl +  ik  jl  il  jk  ijkl  =  ij ij kk ij u k,k = u.. + F i i (2a) (2c)  Substitute (2b) into W.E. (1b) to get  = c ijkl /2 ij General Hooke’s Law (2b) Substitute (2a) into (2b) to get ij u j,i  + u )] i,j j  u k,k  (u j,i  + u ) i,j,i,j + u k,ki u j,ij  + u ) i,jjor= u + F i i  (2d).. Isotropic Hooke’s Coefficients u k,lj =  u.. + F i i c ijkl

Elastic Isotropic Wave Equation in homogeneous Medium u k,k  (u j,i  + u ) i,j,i,j + u k,ki u j,ij  + u ) i,jj = u + F i i  (2d) Assume homogeneous medium u k,ki u j,ij  + u ) i,jj = u + F i i  (2e).... Explicit vector notation  u  = u + F  (2f)  u  2..  u = u - x x u 2  u  = u + F  (2g)  u ..  x x

Outline 1.Generalized Hooke’s Law 2.Hooke+Newton =Anisotropic Wave Equation 3.Isotropic Wave Equation 4.P and S Waves 5.Summary 6.Appendix

Pure P-wave Equation Pure S-wave Equation Helmholtz_decomposition ( ( ) ) + Motion parallel to k Motion perpendicular to k Displacement u = Sum of Potentials: Decomposition u into scalar and vector potentials

Pure P and S Wave Equations  u  = u + F  (2g)  u  x x   =  (2h)  ....  where 2 Plane P wave e propagates in k direction with particle motion parallel to propagation direction with propagation velocity ik r  Pure P ikikikik e ik rik rik rik r =  =  u  e ik rik rik rik r

Compressional Wave (P-Wave) Animation Deformation propagates. Particle motion consists of alternating compression and dilation. Particle motion is parallel to the direction of propagation (longitudinal). Material returns to its original shape after wave passes.

Pure P and S Wave Equations  u  = u + F  (2g)  u  x x   =  (2i)  .... x x x x  where 2 Plane S wave e propagates in k direction with particle motion perpendicular to propagation direction with propagation velocity ik r   S = u xzy

Pure SV and SH Plane Waves   =  (2i)  .. 2 Try plane wave propagating in x direction: (x) = e i ikx d/dx d/dy d/dz i j k e 00 ikxx     Particle motion is not parallel to propagation direction for a S wave x x x x  u= Particle motion parallel to propagation direction is not possible for a S plane wave in an isotropic medium  u  We relate solution to that of a Shear wave because it only depends on shear strength modulus only depends on shear strength modulus xzy

Pure SV and SH Plane Waves   =  (2i)  .. 2   Try plane wave propagating in x direction such that (x) = e j ikx d/dx d/dy d/dz i j k e 00 ikxx     0 0 Particle motion is || to vertical & perpendicular to propagation direction for a SV wave. Only true with a plane wave and a 2D medium (no variation of velocity in y-direction) ike ikx x x x x  u=  u  xzy SV

Pure SV and SH Plane Waves   =  (2i)  .. x ux ux ux u where 2   = Try plane wave propagating in x direction such that: (x) = e k ikx d/dx d/dy d/dz i j k e 00 ikx x ux ux ux u     0 0 Particle motion is horizontal&perpendicular to propagation direction for a SH wave ike ikx xzy  SH SH

Shear Wave (SV-Wave) Animation Deformation propagates. Particle motion consists of alternating transverse motion. Particle motion is perpendicular to the direction of propagation (transverse). Transverse particle motion shown here is vertical but can be in any direction. However, Earth’s layers tend to cause mostly vertical (SV; in the vertical plane) or horizontal (SH) shear motions. Material returns to its original shape after wave passes. xzy

Love Wave (L-Wave) Animation Deformation propagates. Particle motion consists of alternating transverse motions. Particle motion is horizontal and perpendicular to the direction of propagation (transverse). To aid in seeing that the particle motion is purely horizontal, focus on the Y axis (red line) as the wave propagates through it. Amplitude decreases with depth. Material returns to its original shape after wave passes.

P and S Waves “see” Different Things No volumetric strain  V p Extension Compression  bulk modulus: depends on fluids (low in gas)  shear modulus: no contribution from fluid/gas  density Shear waves can also be polarized

SH Wave Practical Uses 1. Near surface SH refraction -> V SH (z) 2. Earthquake studies: S body waves and Love waves S. America earthquake epicenters Cross-section of earth 3-component seismograms Polarization diagrams P-wave Polariz. V SH (z) (m/s) Harbor fill Resonance observed predicted

Outline 1.Generalized Hooke’s Law 2.Hooke+Newton =Anisotropic Wave Equation 3.Isotropic Wave Equation 4.P and S Waves 5.Summary 6.Appendix

Summary 1. Shear Strain: 2. Anisotropic Wave Equation  = c  ij kl  ij,j =  u.. + F i i (1a) (1b) (u + u ) (u + u )k,l l,k ijkl Hooke’s Law 6 indep. eqs. Newton’s Law 3 indep. eqs. /2ij,jk,lj l,kj  u  = u + F   u  x x.. 3. Isotropic Wave Equation (homogeneous) u k,lj =  u.. + F i i c ijkl 3 indep. eqs. 3 unknowns

Summary Direction prop. 4. P and S Waves “see” Different Things  bulk modulus: depends on fluids (low in gas)  shear modulus: no contribution from fluid/gas  density Direction prop. Love surface wave (free surface) P body wave S body wave (SV or SH or S) 5. Earthquakes rich in shear, explosions rich in compressional in compressional

Summary 2D modeling is for a line source and no variation of earth parameters in one of coordinates parameters in one of coordinates Line source creates an expanding cylinder, which increases area by r, so geometric spreading is 1/sqrt(r). This is what your 2D acoustic and elastic codes compute; also a tail and a pi/4 phase shift. xzy dP/dx + dP/dy + dP/dz = (1/c )dP/dt +f dP/dx + dP/dz = (1/c )dP/dt +f Line source f(x,z) and 2D medium 3D wave equation

Summary 2D modeling is for a line source and no variation of earth parameters in one of coordinates parameters in one of coordinates Line source creates an expanding cylinder, which increases area by r, so geometric spreading is 1/sqrt(r). This is what your 2D acoustic and elastic codes compute; also a tail and a pi/4 phase shift. xzy dP/dx + dP/dy + dP/dz = (1/c )dP/dt +f dP/dx + dP/dz = (1/c )dP/dt +f Line source f(x,z) and 2D medium 3D wave equation

Summary

Summary

Appendix

Compliance (or flexibility) matrix…larger values of S ij lead to more felxible materials where small stress leads to large strain

Can a shear stress induce a normal strain?

No shear stress Contribution to Normal strain An orthotropic material has three mutually orthogonal twofold axes of rotational symmetry so that its material properties are, in general, different along each axis.

No yx or yz shear stress contribution to xz shear strain