  What are the constraints placed on the frequency dependent constitutive
tensors of elastodynamics by the requirements that the linear elastodynamic
system under consideration be both causal (effects succeed causes) and passive
(system doesn't produce energy)? Compared to electromagnetism, elastodynamics
is complicated by its generally non diagonalizable constitutive tensors. In
this paper we clarify the constraints that causality and passivity place on
very general forms of elastodynamic constitutive relations. Specifically we
show that the satisfaction of passivity (and causality) directly requires that
the hermitian parts, as defined later, of the Fourier transforms of the time
derivatives of the elastodynamic constitutive tensors be positive semi-definite
at all frequencies. Additionally, we show that the conditions subsequently
require that the non-hermitian parts of the Fourier transforms of the
constitutive tensors, when divided by the imaginary number $i$, be positive
semi-definite for positive values of frequency and negative semi-definite for
negative values of frequency. We show that when major symmetries are assumed
these definiteness relations apply simply to the real and imaginary parts of
the relevant tensors. For diagonal and one-dimensional problems, these positive
semi-definiteness relationships reduce to simple inequality relations. Finally
we extend the results to highly general constitutive relations which include
the Willis inhomogeneous relations as a special case.
