Seismic Waves in Pre-strained Media

Abstract

The equations of motion for infinitesimal vibrations due to a small perturbing force superimposed on an already existing state of finite strain in an elastic medium are established. They are similar in form to the equations for infinitesimal anisotropic elasticity. For the particular case of a homogeneous strain, these may be attacked by the standard procedure of Fourier-Laplace transformation and the solution written in integral form. This is followed by a consideration of the source term using the concept of body-force equivalents. However, the formulae given by Burridge & Knopoff may not be used in the case of a pre-strained medium and their derivation must be adapted slightly to give the correct formulae.The solution to the problem of a point source of the transverse slip type in a homogeneously pre-strained medium is then obtained and in the final section, the particular example of a Hadamard-Green material is considered in detail.It is well known that the region surrounding an earthquake source is in a state of strain prior to the occurrence of the earthquake and it is the purpose of the present work to set down the mathematical foundations in order to study the effects of this pre-strain. The geophysical applications of the work will be given in a further paper. * Received in original form 1972 June 14 373 3 374 K. WaltonAny permanent (irreversible or plastic) deformation which has occurred is assumed to have been accompanied by zero permanent stress. Furthermore, stress relaxation effects are assumed to be so slow as to be negligible in the static field, and sufficiently small to neglect damping in the dynamic (infinitesimal) equations.In this paper, certain results from the general theory of waves in pre-strained perfectly elastic media are presented. Of particular interest is the study of body-force equivalents (Burridge & Knopoff 1964) in a pre-strained medium. Here comparison can be made with the results of Dahlen (1972). In one aspect, the present work is more general than that of Dahlen as the initial body force causing the pre-stress is quite general whereas Dahlen considers a body force due only to gravity and rotation. On the other hand, the present work is more restrictive since it is assumed that the pre-stress and pre-strain are related by the constitutive law for finite elasticity whereas Dahlen has no such restriction on his pre-stress.The final section deals with the particular example of a Hadamard-Green material. This material is chosen because of its particularly simple form of strain energy function.The results from this paper will be used in a further paper on the geophysical applications of the present work. Examples of which are the effect of pre-strain on the Fault-Plane method of solution and the effect of pre-straining on the dispersion of S-waves. Pre-strained mediaThe problem to be considered is that of a perfectly elastic medium which is first subjected in some manner to a static finite strain and then to small vibrations superimposed on this existing state of strain by some sm...

Seismic Wave Propagation in Stratified Media

“…The analysis leading to equivalent forces for a dislocation in a prestressed medium is more complex but leads to comparable results (Dahlen, 1972;Walton, 1973). When we allow for the possibility of traction discontinuities, the force is once again given by (4.23).…”

Ray Theory for a Pre-strained Medium

“…For this to have a non-zero solution for A', we require Equation (13) is known as the characteristic equation and is directly analogous to the equation for the slowness surface as defined for the case of a homogeneous prestrain (Walton 1973). From equation (12), we conclude that A' must be a null-vector of Dpr(h, x).…”

“…From Walton (1973), equation (2.97)), we know that for the case of a homogeneous pre-strain, the first term in the ray expansion is given by where K m is the Gaussian curvature on the surface G" = 0 at the point q" and A" is the null-vector of DpmqO. The tensor K , depends on the particular source function.…”

“…In Walton (1973), equations of motion were established for infinitesimal vibrations due to a small perturbing force superimposed on an already existing state of finite strain in an elastic medium. Because of their similarity in form to the equations for infinitesimal anisotropic elasticity, one may expect that methods of analysis used in such problems may also be used in the present case.…”

“…Throughout the work, the perturbing force is assumed to be a point impulse with a view towards seismological applications. This also enables the result of Walton (1973) to be used in order to find initial values of the various unknown quantities. This is done in Section 5.…”