FIFTH CANADIAN CONFERENCE ON NONDESTRUCTIVE ... - IAEA

1. INTRODUCTION
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We are concerned here with the computer simulation of ultrasonic testing techniques
using finite difference approximations of the equations of infinitesimal
elasticity. These equations are used to describe ultrasonic waves in metals and
in models of seismic waves in the Earth. Note that we are interested in the
dynamic (time dependent) form of the equations and that solution of these equations
is only superficially related to the solution of problems in elastostatics.
Another spurious connection is with the simpler Acoustic Wave
Equation; it does not describe the interplay between elastic waves of different
types.
Computer simulation is used in nondestructive testing (NDT) in a variety of
ways. One application is as an aid to the understanding of the physical
processes involved in, for example, wave interactions with reflecting surfaces,
cracks, inhomogeneities and other features. Computer models can be much more
informative than experiments in this work because the displacements and
stresses are predicted for all points in space and time. A more commercial
application is the modelling of specific transducers and testing configurations
where the model is already used to add detar'l to experimental work and will be
used to select transducers before experimental work is started.
We avoid the use of detailed mathematics in what follows and instead give a
general description of the techniques used with references for more detailed
study. The equations of elasticity are stated in two-dimensional form, but
approximation techniques are the same in any number of space dimensions. The
construction of finite difference schemes and the modelling of features such as
cracks and transducers are discussed. The use of the first order form of the
Elastic Wave Equation is highlighted, and its various advantages are explained.
The numerical approximation of the first order form that is used in the
computer model developed by the author is briefly described.
2. EQUATIONS OF INFINITESIMAL ELASTICITY
The equations of infinitesimal elasticity with plane strain can be written in a
number of different ways. They are most often expressed in Cartesian coordinates
in the second order form relating displacements and stresses as follows