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Doubly constrained elastic wave propagation. (English) Zbl 0943.74527
Summary: It is well known that an elastic material subject to \(N\) \((0\leq N\leq 3)\), internal constraints upon the deformation gradient admits the propagation of \(3-N\) distinct plane waves in most directions. Those directions in which more than \(3-N\) waves may propagate are termed exceptional. Here we investigate wave propagation in a material subject to two constraints by slightly relaxing both constraints and asymptotically expanding the wave speeds and the polarizations in inverse powers of the large elastic moduli associated with the slightly relaxed constraints. The limits in which (1) both constraints operate exactly, and (2) one constraint is exact and one slightly relaxed are both discussed and shown to confirm previous results. The theory and graphical illustrations of the slowness surface are presented for two examples of a practical nature: (1) an incompressible material reinforced by a set of parallel inextensible fibres; (2) a material reinforced by two sets of mutually orthogonal inextensible fibres.

MSC:
74J10 Bulk waves in solid mechanics
74A40 Random materials and composite materials
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