Elastic Wave Propagation
When a mechanical disturbance propagates in an isotropic elastic body inertial and elastic properties control the velocity of the advance and the form of the disturbance frequently changes with progression depending upon the initial character of the displ
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H.P. R o s s m a n i t h Institute of Mechanics Technicat University Vienna A-1040 Vienna, Austria
When a mechanical disturbance propagates in an isotropic elastic body inertial and elastic properties control the velocity of the advance and the form of the disturbance frequently changes with progression depending upon the initial character of the displacement and the history of propagation. Elastodynamic theory shows that only two types of waves can be propagated through an unbounded elastic solid: a longitudinal (dilatational,P-)wave, and a transverse (rotational, shear, S-)wave. Particle motion in a P-wave (S-wave) is parallel (normal) to the direction of wave propagation. When the solid has a free surface or an interface between two different media surface waves may also be propagated. In layered structures several types of·waves may be generated and propagated. In most practical engineering applications plane, cylindrical, anQ spherical waves form the majority of waves encountered. 1. Waves in unbounded media 1.1. Ptane waves
For a general mechanical disturbarice extending in a pre-stressed (Sx• SY) unbounded isotropic homogeneous elastic medium wave equations may be
H. P. Rossmanith (ed.), Rock Fracture Mechanics © Springer-Verlag Wien 1983
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H.P. Rossmanith
derived from the equations of motion for the displacement potentials ¢ and ~ in x- and y-directions, respectively: x-direction:
M
c1x
=
.t:.~
c2x
= ~'tt
(1)
y-direction:
M c21y
=
l:.~ c2y
= ~'tt
(2)
with
c1x= I(A+2~+S)/p c1y= I(A+2!J)/p c2x= l(p-S/2)/P c2y= I(!J+S/2)/p
velocity P-wave velocity s-wave velocity s-wave velocity P-~'lave
alona along along along
the the the the
x-axis y-axis x-axis y-axis
(3)
,
where A and p are Lame's constants, p is the material density, S = SX-sy is the difference of the pre-stress components, and .t:. denotes the Laplaceoperator. The relations between stresses ox, cry, Txy' strains sx, sy, sxy' displacements u and v and wave potentials ¢ and ~ for incremental isotropy are given in Ref./5/. The eqs.(1) and (2) differ in form and in number from those of the classical theory pertaining to zero initial stress. For non-pre-stressed or hydrostatic stress situations (S=O) classical theory applies. When the deformation is a function of only one coordinate, e.g. x, eqs.(1) become (X = ¢ , ~ ; i =1 ,2) (4) x,tt = ci x,xx with the general solution {5)
where the arbitrary functions f+(f_) correspond to a wave propagating in the positive (negative) x-direction. The shape f and f of a transient wave + depends upon the source of disturbance, the distance of travel and the material beh~vior. The dilatation wave propagates with speed c 1x and the shear wave propagates with speed c2x with c1;c 2 > 1 for S/E > -3 with E = Y~ung's modulus •. In a P-wave both the bulk modulus K and the shear modulus p control the velocity of propagation because A+21.l=K+4!1/3. The stress
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Elastic Wave Propagation
distribution in a plane P-wave follows from the plane strain condition Et=O, cry= crz = crxv/(1-v). Representat
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