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Wave Propagation in Discontinuous Media by the Scattering Matrix Method

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Abstract

Propagation of elastic waves in discontinuous media is studied in this paper by the scattering matrix method (SMM). An electromagnetic transmission line analogy is also used to set up the mathematical model. The SMM operates in the frequency domain and allows for all wave polarizations (P, SV and SH). Rock masses are examples of discontinuous media in which the discontinuities (fractures or joints) influence wave propagation. Both elastic and viscoelastic joints are considered and the latter are described by Kelvin–Voigt, Maxwell and Burgers models. Rock joints with Coulomb slip behavior are also analyzed, by applying the averaging principle of Caughy (J Appl Mech 27:640–643, 1960). The evaluation of the effects of periodic discontinuities in a homogeneous medium is presented by introducing the concept of Bloch waves. The dispersion curves of these waves are useful to explain the existence of frequency bands of strong attenuation, also in the case of lossless (perfectly elastic) structures. Simple expressions of transmission and reflection coefficients are obtained. Finally, the SMM results are compared with those computed via the distinct element method (DEM). The comparisons are performed on a medium with joints with Coulomb slip behavior and the agreement is satisfactory, although the SMM must be applied in conjunction with the equivalent linearization technique. Even if the DEM is much more general, the SMM in these simple cases is extremely faster and provides a higher physical insight.

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Abbreviations

A :

Transmission matrix

A N :

Absorption coefficient for N joints

A c :

Transmission matrix of the unit cell

B :

Joint compliance matrix

B :

Joint susceptance

C ij :

Elastic constants

c P±, c SV±, c SH± :

Amplitudes of the incident and scattered waves

c :

(3 × 1) matrix containing the above amplitudes

d :

Joint spacing

D :

Amplitude parameter

e SH±, e SV±, e P± :

Velocity vectors

E A :

Amount of energy absorbed by a frictional interface

E I , E R , E T :

Energy flux per unit area per cycle of oscillation associated with incident, reflected and transmitted waves

f SH±, f SV±, f P± :

Traction vectors

G :

Shear modulus

G f :

Shear modulus of the material filling the joint

h :

Joint thickness

I :

Identity matrix

I A± :

Electric current

i :

Cell number, generic index

Im:

Imaginary part of complex number

j :

Imaginary unit, generic index

K f :

Bulk modulus of the material filling the joint

k :

Wave vector

K :

Joint stiffness matrix

K nn, K ss :

Joint normalized normal and shear stiffness

k 2 :

Stiffness parameter

k eq :

Equivalent stiffness

k P, k S :

Wavenumber of P and S-waves

k nn, k ss :

Normal and shear joint stiffness

k zP, k zS :

Wave-vector longitudinal component of P and S-waves

M :

Modal matrix

N :

Number of discontinuities or joints

n :

Scalar parameter

P :

Propagation matrix

P :

Mechanical power

Re:

Real part of complex number

R N :

Reflection coefficient for N joints

S :

Scattering matrix

S ij :

Sub-matrix of the scattering matrix

T N :

Transmission coefficient for N joints

t :

Time

u :

Displacement vector

v :

Velocity

v :

Velocity vector

v*:

complex conjugate of v

V :

Matrix of the eigenvectors of the transmission matrix of the unit cell

V A± :

Electric voltage

V i :

Velocity of propagation of P-waves or S-waves

V S :

Shear wave velocity

W :

System matrix

x :

Relative displacement

X :

Maximum response value

x, y, z :

Spatial coordinates

\( {\mathbf{\hat{x},\hat{y},\hat{z}}} \) :

Unit vectors

x 0 :

Limit elastic displacement

y p :

Normalized admittance

Y :

Admittance matrix or compliance matrix

Y m :

Medium admittance or compliance

w :

Function of the phase-shift ψ between joints

Z m :

Medium or modal impedance

Z S1, Z S2 :

Shear impedances of the media 1 and 2

α :

Phase angle

Δζ sb :

Normalized width of the stop-band

η :

Specific viscosity matrix

η f :

Viscosity of the material filling the joint

η eq :

Equivalent damping

φ :

Phase angle

\({\overline{\Uptheta}}\) :

Parameter function of the limit elastic displacement and of the maximum response value

ϑ inc :

Angle of incidence

λ :

Eigenvalues; wavelength

λ f :

Wavelength of the material filling the joint

ρ :

Mass density

τ :

Stress tensor

τ:

Stress

τF :

Frictional stress

τ imax :

Maximum stress of the incident wave

τS :

Joint shear strength

τtrasm :

Maximum amplitude of the transmitted stress wave

ω :

Wave angular frequency

ξ :

Wave-vector transverse component

Ψ :

State vector

ψ:

Phase-shift between joints

ζ :

Normalized joint spacing ratio

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Authors and Affiliations

  1. Department of Structural, Geotechnical and Building Engineering, Politecnico di Torino, Turin, Italy

    A. Perino & G. Barla

  2. Department of Electronics and Telecommunications, Politecnico di Torino, Turin, Italy

    R. Orta

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  1. A. Perino
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Perino, A., Orta, R. & Barla, G. Wave Propagation in Discontinuous Media by the Scattering Matrix Method. Rock Mech Rock Eng 45, 901–918 (2012). https://doi.org/10.1007/s00603-012-0286-2

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