FIFTH CANADIAN CONFERENCE ON NONDESTRUCTIVE ... - IAEA

pu = T XX + T Xy
tt x y
pV = T Xy + T yy
tt x y
- 347 -
where U and V are the displacements in the x and y directions respectively, the
T's are the stresses, and p is the density of the material. The stresses are
defined in the following way
T XX = \(U + V ) + 2uU
x y x
T yy = \(U + V ) + 2MV
x y J *" y
T Xy « u(U + V )
y x'
where X and p. are the lame elasticities. These two sets of equations can be
combined to give a pair of linear, second order, hyperbolic equations involving
only the displacements U and V.
Reformulating the equations in first order form was proposed by Clifton [1] and
later used by Smith [2] in the form given below. The equations are written in
terms of the 5-vector
We have
where
W T = (pUt,pVt,T xx ,T yy ,T xy ).
-t -x ° -y '
A =/0 0 1 0 0\ , B =/0 0 0 0
0 0 0 0 l \ /OOOIO
cOOOO) lOaOOO
aOOOO/ \ 0 c 0 0 0
b 0 0 0/ \b 0 0 O 0,
and a » A./p, b » u/p, c • (\+2(x)/p.
Again we have a linear, hyperbolic system of equations. The displacement
velocities and the stresses are calculated simultaneously in this first order
form of the equations. The displacement at any point in the space domain can
be found by integrating the velocity of displacement with respect to time.