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The Quarterly Journal of Mechanics and Applied Mathematics 1992 45(1):77-99; doi:10.1093/qjmam/45.1.77
© 1992 by Oxford University Press
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WAVE PROPAGATION IN SINGLY-CONSTRAINED AND NEARLY-CONSTRAINED ELASTIC MATERIALS

G. A. ROGERSON {dagger} and N. H. SCOTT

( Department of Theoretical Mechanics, The University Nottingham NG7 2RD
School of Mathematics, University of East Anglia Norwich NR4 7TJ )

The slowness surface of an unconstrained material has three sheets whilst that of a material subject to one arbitrary constraint has two sheets except that with some constraints there are certain exceptional directions in which three waves may propagate. This anomalous situation is resolved in this paper by considering a material in which the constraint is nearly satisfied. We find that such a material has a three-sheeted slowness surface and that between each pair of adjacent sheets is situated a sheet of the two-sheeted slowness surface of the constrained material which is obtained as a limiting form of the nearly-constrained material when the constraint is obeyed exactly. Furthermore, we find that the two outer sheets of the slowness surface of the nearly-constrained material collapse onto the two sheets corresponding to the constrained material when the constrained limit is taken away from exceptional directions, whilst the inner sheet collapses to the origin, again, away from exceptional directions. The transition from three waves travelling in each exceptional direction to two waves in neighbouring directions is accompanied by rapid variations in slowness close to the exceptional direction. The theory is applied, with graphical illustrations, to the examples of incompressibility and inextensibility.


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[Abstract]


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