Hamiltonian formalism, seismic waves, and seismic barriers Sergey V. Kuznetsov Institute for Problems in Mechanics Russian Academy of Sciences web: www.ipmnet.ru/~svkuznec

A unified theory of surface acoustic waves based on the generalized Hamiltonian formalism

Equation of motion and representation for Surface Acoustic Waves with plane wave-front ν Representation for the wave Splitting the equation of motion

Matrix ordinary differential equation of motion of the second order Where (will be needed further)

6D Hamiltonian formalism New function Equation of motion System of two ODEs System in a matrix form

Exponential fundamental solution and boundary conditions (for Lamb waves) Traction-free: Clamped:

Dispersion equations for a single layer (conditions for existing non-trivial solutions) Traction-free plate: Clamped plate:

Generalization: Dispersion equation for a N-layered plate (Modified Transfer Matrix Method) Traction-free n-layered plate:

Numerical results Dispersion curves for SH waves in 31-layered composite plate (phase speed vs. circular frequency) Variation of the lower dispersion curve at varying depth of the median layer

Analytical results: A long-wave limit (soliton-like waves) A condition for obtaining the limiting wave speed where The limiting Lamb wave speed for a N-layered isotropic plate

The main types of seismic waves

The main types of Acoustic Waves initiated at earthquakes: I The main types of Acoustic Waves initiated at earthquakes: I. Bulk waves Definition: Bulk waves are either plane, spherical, or cylindrical (harmonic) waves that propagate in an infinite space. Described: Both longitudinal (L) and shear (S) waves were described by Siméon Denis Poisson (1828) Remarks In isotropic materials cL>cS . That is why longitudinal waves are also called as P-waves (Primary-waves) In anisotropic materials longitudinal waves can have smaller speed than shear waves (e.g. TeO2)

The main types of Acoustic Waves initiated at earthquakes: I The main types of Acoustic Waves initiated at earthquakes: I. Bulk waves Representation for bulk wave with plane wave-front: Christoffel equation: Another form of Christoffel equation: Remarks In isotropic medium:

The main types of Acoustic Waves initiated at earthquakes: II The main types of Acoustic Waves initiated at earthquakes: II. Rayleigh waves n ν Definition: A wave that (i) propagates on a homogeneous elastic halfspace with traction-free boundary conditions; (ii) attenuates exponentially with depth; is called the Rayleigh wave. Described: These waves were theoretically discovered by Lord Rayleigh in 1885

The main types of Acoustic Waves initiated at earthquakes: II The main types of Acoustic Waves initiated at earthquakes: II. Rayleigh waves Remark concerning “forbidden” directions For a long time it was supposed that there can be anisotropic materials (may be artificial), that possesses specific directions along which Rayleigh waves cannot propagate. These hypothetical directions (and materials) where called “forbidden”. Barnett and Lothe (1973-1976 ) proved a theorem on existence of Rayleigh wave propagating on a free surface of any anisotropic halfspace and any direction. Later on Ting (1998) and Kuznetsov (2001) considered a case of the non-semisimple degeneracy of the Jacobian resulting in a wave of the “non-Rayleigh” type.

The main types of Acoustic Waves initiated at earthquakes: II The main types of Acoustic Waves initiated at earthquakes: II. Rayleigh waves A principle for creating horizontal barriers against Rayleigh waves n ν Theorem of nonexistence for Rayleigh waves in a clamped anisotropic halfspace /Chadwick and Smith (1977)/. If the halfspace surface is clamped, then no Rayleigh wave can propagate

The main types of Acoustic Waves initiated at earthquakes: III The main types of Acoustic Waves initiated at earthquakes: III. Stoneley waves n ν Definition: A wave that (i) propagates on an interface between homogeneous elastic halfspaces in a contact; (ii) attenuates exponentially with depth in both halfspaces; is called the Stoneley wave. Described: These waves were theoretically discovered by Robert Stoneley in 1924

The main types of Acoustic Waves initiated at earthquakes: IV The main types of Acoustic Waves initiated at earthquakes: IV. Love waves n ν Definition: A wave that (i) propagates on a homogeneous elastic halfspace in a contact with a layer with the traction-free boundary conditions; (ii) attenuates exponentially with depth in a halfspace; is called the Love wave. Described: These waves were theoretically discovered by Augustus Love in 1911

The main types of Acoustic Waves initiated at earthquakes: IV The main types of Acoustic Waves initiated at earthquakes: IV. Love waves Theorem of non-existence for Love waves propagating in an isotropic layer and a halfspace /actually Love/. If speed of the transverse bulk wave in a layer is greater than in a halfspace: then Love wave cannot propagate. Corollary. Since the condition of nonexistence becomes:

Seismic waves: experimental data

Frequency range Wave nature Frequency range, Hz Natural seismic waves Most dangerous for civil structures: 10÷30Hz Anthropogenic seismic waves 10 - 120Hz Most dangerous for civil structures: 10-50Hz

Velocity range (Longitudinal bulk waves) Material Speed m/sec air 330 - 360 Bullet in the air ~800 soil 200 - 500 sand 150 - 450 water 1430 - 1590 slate (shale) 2000 - 5000 limestone 3000 - 6000 granite 4500 - 6500

Wavelength range (Rayleigh waves) Material Wavelength range, meters soil 5 - 30 sand 4 - 20 slate (shale) 35 - 270 limestone 45 - 300 granite 80 - 350 Remark

Seismic waves and barriers: FEM simulation

Seismic barriers, Where are they needed?

Two types of seismic barriers Longitudinal barrier (“thin and long”) The main principle: to prevent surface waves from propagation Transverse barrier (“thin and deep”) The main principle: To reflect most of the wave energy 2

2D FEM simulation of the seismic wave propagation near clamped surface (harmonic surface loading) Magnified:

2D FEM simulation of the seismic wave interaction with horizontal barrier (harmonic surface loading; a quarter of plane)

2D FEM simulation of the seismic wave interaction with composite vertical barrier (harmonic surface loading; quarter of a plane)

3D FEM simulation of seismic waves near the hypocenter

Longitudinal circular barrier against Rayleigh waves (Chadwick’s theorem)

Longitudinal circular barrier near the epicenter (Chadwick’s theorem)

Transverse vertical barrier against Rayleigh waves (principle of maximum reflection)

Another type of seismic barriers A barrier composed of piles optimized for best scattering A known result for scattering of bulk waves Hasimoto, 1959, Scattering of viscous fluid flow by spheres Datta, 1977, Scattering of elastic waves by ellipsoidal inclusions Bose, Mal, 1979, Scattering shear bulk waves by fibers Gubernatis, 1979, Scattering of elastic waves by inclusions Willis, 1980, Long-wave limit for scattering of elastic waves Kuznetsov, 1996, Scattering of elastic waves in anisotropic media ______________________________________________________ The best scattering (the largest scattering cross-section) is achieved by voids

Hollow piles

Clamped piles: single row

Clamped piles: two rows

Where to choose a building site?

Possible geological profile

FEM modeling of seismic wave propagation