Existence of extraordinary transonic states in monoclinic elastic

Existence of extraordinary transonic states in monoclinic elastic media Litian Wang and Kent Ryne Østfold University College 1757 Halden Norway Litian Wang Østfold University College

Main problems a) Existence of extraordinary transonic states associated with extraordinary zero-curvature slowness curve b) Existence of space of degeneracy c) Existence of generalized surface waves Litian Wang Østfold University College

Surface geometry of slowness surface Cubic (Cu) Litian Wang Østfold University College Monoclinic

Surface geometry of slowness surface Cubic (Cu) Litian Wang Østfold University College Monoclinic

Zero-curvature transonic states n m E 1 Litian Wang Østfold University College E 2 E 3 E 4 Barnett, Lothe & Gundersen

Surface geometry of slowness surface Cubic (Cu) Litian Wang Østfold University College Monoclinic

Problem 1 a) Can a slowness curve have zerocurvature locally? b) How flat a slowness curve can be? Litian Wang Østfold University College

Degree of freedom • Degree of freedom = 6 Litian Wang Østfold University College

Wave propagation in monoclinic media • Elastic stiffness matrix: Litian Wang Østfold University College

θ Litian Wang Østfold University College k

Christoffel equation Where d 13=c 13+c 55, ∆15=c 11 -c 55, ∆64=c 66 -c 44, ∆53=c 55 -c 33, Litian Wang Østfold University College

Curvature in slowness plot Let Curvature k and its second derivative k’’ in the neighborhood of zaxis are given by θ k Litian Wang Østfold University College

How to find the eigenvalue ? Where d 13=c 13+c 55, ∆15=c 11 -c 55, ∆64=c 66 -c 44, ∆53=c 55 -c 33, Litian Wang Østfold University College θ k

Perturbation method θ k Litian Wang Østfold University College

Where θ k Litian Wang Østfold University College

Results - 1 (a) Normal curvature of slowness curve along z-axis (See also Shuvalov et al) (b) Zero-Curvature along z-axis when d 132 = c 11∆35 or (c 13+c 55)2=c 11(c 33 -c 55) Litian Wang Østfold University College θ k

Results - 2 (a) The second derivative of curvature: (b) Extraordinary zero-curvature along z-axis when (c 11 c 36 -d 13 c 16)2=c 112 c 55∆45) θ k Litian Wang Østfold University College

Litian Wang Østfold University College

Litian Wang Østfold University College

Problem 2 a) Space of degeneracy in monoclinic media b) Generalized surface waves Litian Wang Østfold University College

Degeneracy of the Stroh eigenvalues E 1 zero-curvature transonic state: Litian Wang Østfold University College

Degeneracy of the Stroh eigenvalues E 4 zero-curvature transonic state: Litian Wang Østfold University College

Result 3 Space of degeneracy vs zero-curvature slowness curve: Litian Wang Østfold University College

Result 4 Space of degeneracy vs generalized surface waves • Subsonic surface waves • Supersonic surface waves Litian Wang Østfold University College

Conclusions a) Existence of extraordinary zero-curvature slowness curve b) Existence of space of degeneracy c) Existence of supersonic surface wave along the space of degeneracy d) Existence of generalized subsonic surface wave along the space of degeneracy Litian Wang Østfold University College