Part 5 ACOUSTIC WAVE PROPAGATION IN ANISOTROPIC MEDIA

Part 5 ACOUSTIC WAVE PROPAGATION IN ANISOTROPIC MEDIA

Review of Fundamentals displacement-strain relation stress-strain relation balance of momentum (deformation) (constitutive equation) (Newton's Law) equation of motion (Christoffel's equation) Notation: position vector x ( e11 x + e2x2 + e 3x3) displacement vector u strain matrix ε stress matrix τ stiffness tensor C

Displacement-strain relation: Three-Dimensional Problem vector notation ε su indicial notation ε ij ½ ( ui, j + uj, i) differential notation ε ij ½ u u j ( i + ) x j xi u1 u2 u ε 3 11, ε 22, ε 33, x1 x2 x3 ε 12 ε 21 ½ u ( 1 u + 2) x2 x1 ε 23 ε 32 ½ u ( 2 u + 3 ) x3 x2 ε 31 ε 13 ½ u ( 3 u + 1) x1 x3 Stress-strain relation: vector notation τ C : ε indicial notation τ ij Cijkl ε kl τ C ε + C ε + C ε ij ij11 11 ij12 12 ij13 13 + C ε + C ε + C ε ij21 21 ij22 22 ij23 23 + C ε + C ε + C ε ij31 31 ij32 32 ij33 33

Abbreviated Notation ε ε ε ε ε ε ε ε ε ε 11 12 13 21 22 23 31 32 33 τ τ τ τ τ τ τ τ τ τ 11 12 13 21 22 23 31 32 33 Stiffness matrix τ11 C11 C12 C13 C14 C15 C16 ε11 τ 22 C12 C22 C23 C24 C25 C 26 ε 22 τ33 C13 C23 C33 C34 C35 C36 ε33 τ23 C14 C24 C34 C44 C45 C46 2ε23 τ C C C C C C 2ε 31 15 25 35 45 55 56 31 τ12 C16 C26 C36 C46 C56 C66 2 ε12 Stress-strain relations for isotropic materials (Hooke's Law) In indicial notation, τ ij λεkk δ ij + 2με ij Kronecker delta δ 1ifi j andδ else ij ij τ11 λ+ 2μ λ λ ε11 τ 22 2 ε λ λ+ μ λ 22 τ33 λ λ λ+ 2μ ε33 τ23 μ 2ε23 τ 31 μ 2ε31 τ μ 2ε 12 12

Stress-Displacement Relation τ 11 λ ( + + ) + 2μ u1 u2 u3 u1 x1 x2 x3 x1 τ 22 λ ( + + ) + 2μ u1 u2 u3 u2 x1 x2 x3 x2 τ 33 λ ( + + ) + 2μ u1 u2 u3 u3 x1 x2 x3 x3 τ 12 τ 21 μ ( + ) u1 u2 x2 x1 τ 23 τ 32 μ ( + ) u2 u3 x3 x2 τ 31 τ 13 μ ( + ) Traction and Body Forces: x 3 B 3 dx 1 dx 2 dx 3 u3 u1 x1 x3 traction forces body forces τ 33 dx 1 dx 2 τ 31 dx 1 dx 2 τ 13 dx 2 dx 3 τ 32 dx 1 dx σ 2 x τ 23 dx 1 dx 3 τ 22 dx 1 dx 3 B 2 dx 1 dx 2 dx 3 τ 11 dx 2 dx 3 τ 12 dx 2 dx 3 τ 21 dx 1 dx 3 x 2 B 1 dx 1 dx 2 dx 3 x 1

Equilibrium Equations τ τ τ + + + B1 11 21 31 x1 x2 x3 τ τ τ + + + B2 12 22 32 x1 x2 x3 τ τ τ + + + B3 13 23 33 x1 x2 x3 x 1 -direction: [ τ 11( x1 + dx1) τ 11( x1 )] dx2dx3 + [ τ 21( x2 + dx2) τ21( x2)] dx1 dx3 + [ τ 31( x3 + dx3) τ 31( x3)] dx1 dx2 + B1 dx1 dx2 dx3 τ 11( x1 + dx1 ) τ11( x1 ) τ 21( x2 + dx2) τ21( x2) τ 31( x3 + dx3) τ31( x3) + + + B1 dx1 dx2 dx3 τ11 τ21 τ + + 31 + B1 x1 x2 x3 τ12 τ22 τ + + 32 + B2 x1 x2 x3 τ13 τ23 τ + + 33 + B3 x1 x2 x3 B ρu Balance of Momentum: τ ρu

Wave Equation ε s u τ C : ε C: u ρu s For isotropic materials: ( λ+μ) u + μ 2 u ρu indicial notation ( λ+μ ) uj ji+ μ ui jj ρ ui,, detailed differential equation form 2u 2 2 2 2 2 2 1 u2 u3 u1 u1 u1 u ( λ+μ )( + + ) + μ ( + + ) ρ 1 x2 x 2 2 2 2 1 1 x2 x1 x3 x1 x2 x3 t 2u 2 2 2 2 2 2 1 u2 u3 u2 u2 u2 u ( λ+μ )( + + ) + μ ( + + ) ρ 2 x 2 2 2 2 2 1 x2 x x 2 2 x3 x1 x2 x3 t 2u 2 2 2 2 2 2 1 u2 u3 u3 u3 u3 u ( λ+μ )( + + ) + μ ( + + ) ρ 3 x 2 2 2 2 2 1 x3 x2 x3 x3 x1 x2 x3 t

Plane Wave Solutions u Ap ei ( kx ωt) amplitude A angular frequency ω polarization unit vector p wave vector k d k propagation unit vector d wave number k k2 + k2 + k2 sound velocity c ω k 1 2 3 u i t ik( d11 x d2x2 d3x3) 1 Ap1e ω e + + u i t ik( d11 x d2x2 d3x3) 2 Ap2e ω e + + u i t ik( d11 x d2x2 d3x3) 3 Ap3e ω e + + ( λ+μ ) dd 2 1 1+ ( μ ρc) ( λ+μ) dd 1 2 ( λ+μ) dd 1 3 p1 ( λ+μ) dd 2 1 2 ( λ+μ ) d2d2+ ( μ ρc) ( λ+μ ) d2d3 p 2 ( ) dd 2 λ+μ 1 3 ( λ+μ) d2d3 ( λ+μ ) dd 3 3+ ( μ ρc) p3

Christoffel's Equation p1 Γ p 2 p 3 [ ] Γ Since the material is isotropic, d e 1 ( d1 1, d2 d3 ) can be assumed without loss of generality. λ+ 2μ ρc2 p1 μ ρ c2 p 2 c2 μ ρ p3 Longitudinal (or dilatational) wave cd λ+ 2μ ρ and p2 p3 Shear (or transverse) wave c s μ ρ and p 1

Symmetry Considerations lack of rotation Cijkl Cjikl Cijlk Cjilk reciprocity C ijkl C klij Independent elastic constants most general anisotropic 21 orthorhombic 9 cubic symmetry 3 isotropic 2 ABBREVIATED NOTATION ε11 ε12 ε13 ε ε21 ε22 ε23 ε31 ε32 ε33 τ τ11 τ12 τ13 τ21 τ22 τ23 τ31 τ32 τ33 Stiffness matrix: τ11 C11 C12 C13 C14 C15 C16 ε11 τ 22 C12 C22 C23 C24 C25 C 26 ε 22 τ33 C13 C23 C33 C34 C35 C36 ε33 τ23 C14 C24 C34 C44 C45 C46 2ε23 τ 31 C15 C25 C35 C45 C55 C56 2ε31 τ12 C16 C26 C36 C46 C56 C66 2 ε12

Simplest Anisotropy, Cubic Symmetry [1] [111] [1] [1] [11] τ11 C11 C12 C12 ε11 τ 22 C12 C11 C12 ε 22 τ33 C12 C12 C11 ε33 τ23 C44 2ε23 τ 31 C44 2ε31 τ12 C44 2ε12

Isotropic Material τ11 C11 C12 C12 ε11 τ 22 C12 C11 C12 ε 22 τ33 C12 C12 C11 ε33 τ23 C44 2ε23 τ 31 C44 2ε31 τ12 C44 2ε12 2C44 C11 C12 C11 λ+ 2 μ, C12 λ, C44 μ, λ and μ are Lame's constants τ11 λ+ 2μ λ λ ε11 τ 22 2 ε λ λ+ μ λ 22 τ33 λ λ λ+ 2μ ε33 τ23 μ 2ε23 τ 31 μ 2ε31 τ12 μ 2 ε12

Transformation of Tensors Rotation of Rectangular Coordinate Axes First-Rank Tensor [ u'] [ a][ u ] [a] denotes the transformation matrix cosθx ' x cosθxy ' cosθxz ' [ a] cosθy' x cosθyy ' cosθyz ' cosθzx ' cosθzy ' cosθzz ' z z' u z u' z u u' y u y θ y' y x u x θ x' u' x cosθ sin θ [ a] sinθ cosθ 1

Rotation of Second-Rank Tensors Symmetric strain tensor: [ du] [ ε ][ dx] [ du'] [ a][ du ] [ dx'] [ a][ dx ] [ du'] [ a][ ε ][ dx] [ dx][ a 1][ dx'][ a] T [ dx'] [ du '] [ a][ ε ][ a] T [ dx'] [ ε'] [ a][ ε ][ a] T Symmetric stress tensor: [ τ'] [ a][ τ ][ a] T

Bond Transformation [ C'] [ M][ C][ M ] T [M] is the so-called Bond transformation matrix [ M ] a2 2 2 11 a12 a13 2a12 a13 2a13 a11 2a11a12 a2 2 2 21 a22 a23 2a22 a23 2a23 a21 2a21a22 2 2 2 a31 a32 a33 2a32 a33 2a33 a31 2a31a32 a 21a31 a22 a32 a23 a33 a22 a33 + a23 a32 a21a33 + a23 a31 a22 a31+ a21a 32 a31a11 a32 a12 a33 a13 a12 a33 + a13 a32 a13 a31+ a1 1 a 33 a 11 a 32 + a 12 a 31 a11a21 a12 a22 a13 a23 a12 a23 a13 a22 a13 a21 a11a23 a11a22 a12 a + + + 21 The Bond method can be applied directly to elastic constants given in abbreviated notation!

Simple Rotation by angle θ around the z axis [ M ] cos2θ sin2θ sin 2θ sin2θ cos2θ sin 2θ 1 cosθ sinθ sinθ cosθ -½ sin 2θ ½ sin 2θ cos 2θ [ C'] [ M][ C][ M ] T ' C11 C C 12 2 11 C11 ( C44)sin 2θ 2 ' C11 C C 12 2 12 C12 + ( C44)sin 2θ 2 C C ' 13 C12 C C ( )sin2θcos2θ 2 ' 11 12 16 C44 C' 33 C11 C ' 44 C44 ' C11 C C 12 2 66 C44 + ( C44)sin 2θ 2 the other matrix elements are zero

Coupled Normal Stress and Shear Stain C C C ( )sin2θcos2θ 2 ' 11 12 16 C44 τ11 C11 C12 C13 C14 C15 C16 ε11 τ 22 C12 C22 C23 C24 C25 C 26 ε 22 τ33 C13 C23 C33 C34 C35 C36 ε33 τ23 C14 C24 C34 C44 C45 C46 2ε23 τ 31 C15 C25 C35 C45 C55 C56 2ε31 τ12 C16 C26 C36 C46 C56 C66 2 ε12 C16 symmetry direction (or isotropic) τ 11 1 τ11 2 ε12 off symmetry direction τ 11 ε 12 ε 12 /

Christoffel's Equation for an Anisotropic Solid ( C 2 ijk dj d c ρδ ik ) pk λ 2 11 ρc λ12 λ13 p1 λ 2 12 λ22 ρc λ 23 p 2 2 λ p 13 λ23 λ33 ρc 3 It is customary in the literature to denote the direction cosines d1, d2,and d3 by, m,and n. λ 2 2 2 11 C11 + m C66 + n C55 + 2mnC56 + 2n C15 + 2m C16 λ 2 2 2 22 C66 + m C22 + n C44 + 2mnC24 + 2n C46 + 2m C26 λ 2 2 2 33 C55 + m C44 + n C33 + 2mnC34 + 2n C35 + 2m C45 λ 2 2 2 12 C16 + m C26 + n C45 + mn( C46 + C25) + n ( C14 + C56) + m ( C12 + C66) λ 2 2 2 13 C15 + m C46 + n C35 + mn( C45 + C36) + n ( C13 + C55) + m ( C14 + C56) λ 2 2 2 23 C56 + m C24 + n C34 + mn( C44 + C23) + n ( C36 + C45) + m ( C25 + C46) Pure mode longitudinal waves: p d Pure mode shear waves: pi di (i 1,2,3) p d pd ( pd 1 1 + p 2 d 2 + pd 3 3 ) i i

Cubic Crystals Christoffel's equation: λ 2 11 ρc λ12 λ13 p1 λ 2 12 λ22 ρc λ 23 p 2 2 p 13 23 33 c 3 λ λ λ ρ λ 2 2 2 11 C11 + ( m + n ) C44 λ 22 m2c 2 2 11 + ( + n ) C44 λ 33 n2c 2 2 11 + ( + m ) C44 λ m ( C + C ) 12 12 44 λ 13 n ( C12 + C44) λ mn( C + C ) 23 12 44 three axes of symmetry: [1], [11] and, [111]

Pure Modes Along Symmetry Axes Sound Wave Propagating along the [1] Direction: 1, m n C 2 11 ρc p1 C 2 44 ρ c p 2 C 2 p 44 c 3 ρ Characteristic equation: ( C 2 2 2 11 ρc )( C44 ρ c ) Wave speeds (eigenvalues): c 1 C11 ρ c2 c3 C44 ρ (no birefringence) Polarizations (eigenvectors): p 2 1( C11 ρ c ) p 2 2( C44 ρ c )

p 2 3( C44 ρ c ) For c1 p1 1andp2 p3 (pure longitudinal wave) For c2 or 2 2 c3 p and p + p 1 (pure transverse waves) 1 2 3 [1] [1] [1] c 1 c 2 c 3 Sound Wave Propagating along the [11] Direction: m 1/ 2 and n 11 44 2 12 44 1 C12 C44 C11 C44 c2 p2 C 2 p 44 ρc 3 ½ ( C + C ) ρ c ½ ( C + C ) p ½ ( + ) ½ ( + ) ρ Characteristic equation: 2 2 2 2 11 44 12 44 44 [( C + C 2 ρc ) ( C + C ) ]( C ρ c )

Wave speeds (eigenvalues): c1 C11 + C12 + 2C44 2ρ c2 C11 C12 2ρ c 3 C44 ρ Polarizations (eigenvectors): Forc c1 : ½ ( C12 + C44) ½ ( C12 + C44) p1 ½ ( C12 + C44) ½ ( C12 + C44) p 2 ½ ( C11 + C12) p3 p1 p2 ( 1/ 2) and p3 (pure longitudinal wave) Forc c2 : ½ ( C12 + C44) ½ ( C12 + C44) p1 ½ ( C12 + C44) ½ ( C12 + C44) p 2 C44 ½ ( C11 C12) p3 C44 ½ ( C11 C12) / / p 3 and p1 p2 (. eg., p1 1 2and p2 1 2) pure shear wave polarized in the [11]

Forc c3 : ½ ( C11 C44) ½ ( C12 + C44) p1 ½ ( C12 + C44) ½ ( C11 C44) p 2 p 3 p1 p2 and p3 1 pure shear wave polarized in the [1] direction. [1] [1] c 2 c 3 c 1 [1] [11] Sound Wave Propagating along the [111] Direction: m n 1/ 3 λ 2 11 ρc λ12 λ12 p1 λ 2 12 λ11 ρc λ 12 p 2 2 λ p 12 λ12 λ11 ρc 3 1 11 C 11 C 44 λ 3 ( + 2 ) 1 12 C 12 C 44 λ 3 ( + )

Adding the three rows ( λ 2 11 + 2 λ12 ρ c1 )( p1 + p2 + p3) Characteristic equation: λ 2 11 + 2λ12 ρ c1 Wave speed (eigenvalue): c 1 λ + 2λ C + 2C + 4C ρ 3ρ 11 12 11 12 44 Polarization (eigenvector): p 1 p 2 p 3 ( 1/ 3) pure longitudinal mode Shear modes: p1 + p2 + p3 For example, p1 p2 and p3 ( p 1 is either 1/ 2 or 1/ 2). Characteristic equation: λ 2 11 λ12 ρ c

Wave speeds (eigenvalues): c C C C λ11 λ12 11 12 + 44 2 c3 ρ 3ρ (no birefringence) Polarization (eigenvector): p p p 1 2 and 3 in the (111) plane [1] [111] c 1c2 c 3 [1] [1] For Nickel, the pure longitudinal wave velocities are: [1] c 1 5,299 m/s [11] c 1 6,27 m/s [111] c 1 6,251 m/s isotropic c d 6,32 m/s

Anisotropy Factor for Cubic Crystals A 2C C 44 11 C12 Δ A 1 (zero for isotropic materials) For isotropic materials: A 1 3 Anisotropy Factor 2 1 Sodium Fluoride Yttrium Iron Garnet Fused Silica (Isotropic) Tungsten Aluminum Diamond Silicon Iron Nickel Gold Silver

Velocity Distributions in the (1) Plane Aluminum [1] longitudinal shear Nickel [1] [1] [1] (1 km/s per division)

Anisotropic Phenomena orientation-dependent acoustic velocity Specimen Longitudinal Transducer d A d B polarization-dependent transverse velocity (birefringence) Specimen Shear Transducer p B p A d skewed polarizations (quasi-modes) deviation between phase and energy directions (beam skewing), etc.

Birefringence "Fast" Mode "Slow" Mode o 22.5 o 45 o 67.5 o 9 o 2 2() cos ( θ2) ( ) c2 S t u t 2 3() cos ( θ3) ( ) c3 S t u t cos( θ 3) sin( θ 2) d St ut ut () cos 2( θ) ( ) + sin 2( θ) ( ) c2 c3 d d d

Quasi-Modes, Skewed Polarizations isotropic medium (no skewing, no birefringence) anisotropic medium (skewing, birefringence) y p L y p QL p QS2 p S p QS1 d x d x z z Particle orientations are always mutually orthogonal.

Huygens' Principle, Isotropic Case y θ P ( r, θ) x Time delay to P(r,θ) from source point at x: (,; r θ x) tr (,; θ x) (,; rθ xsr )(,; θ x) cr (,; θ x) tr (,; θ x) (,; rθ xs ) tr (,; θ x) ( r xsin θ ) s tr (,; θ x) x sin θ s θ (no skewing)

Huygens' Principle, Anisotropic Case xcosθ tr (,; θ x) (,; rθ xsr )(,; θ x) ( r xsin θ)( sθ ) r xcosθ s tr (,; θ x) ( r xsin θ)[() sθ ] r θ s tr (,; θ x) s() θ r xs()sin θ θ xcosθ θ tr (,; θ x) s s( θ)sinθ cosθ x θ 1 tan θ s s( θ ) θ θ (skewing) ray (group) direction Δθs( θ) d Δθ slowness curve s Δθ θ wave (phase) direction

Slowness Diagrams Velocity: c propagation distance propagation time Slowness: s propagation time propagation distance s c 1 Applications: wave direction is determined by the wave speed (group velocity, refraction, diffraction, scattering) velocity diagram Nickel slowness diagram [1] quasi-longitudinal true shear quasi-shear [1] [1] [1]

Wave (Phase) Direction vs Ray (Group) Direction 2D-case RQS2 wave direction RQS1 RQL In general, the three group directions (solid arrows) are different from the wave direction!

Ray Direction Analogy with dispersion: f f( ωt kx ) Phase velocity: c p ω k Group velocity: c g ω k x 3 c g c p γ slowness surface d x 2 x 1 Transmitter Beam Contour Phase Plane Phase Direction Ray Direction