Uniqueness Theorems in Elastodynamics. Relations with Existence, Stability, and Boundedness of Solutions

Abstract

There are comparatively few results in the literature on the uniqueness of elastodynamic solutions. Nevertheless, in spite of their small number, these results provide a level of comprehension not yet attained in the static theory. Most of these contributions have been concerned exclusively with solutions that are twice continuously differentiable and have imposed, in addition to the major symmetry on the elasticities, the extra (usually non-essential) minor symmetry

$$ {\text{C}}_{{\text{ijkl}}} {\text{ = C}}_{{\text{jikl}}} {\text{.}} $$

((8.0.1))

The classical result, due to Neumann [1885], states that the initial-mixed boundary value problem for finite regions has a unique solution provided

$$ {\text{C}}_{{\text{ijkl}}} \xi _{ij} \xi _{kl} \geqq 0 $$

(1)

for arbitrary tensors ξ_ij. Under the assumption of uniform density and elasticities, Gurtin and Toupin [1965], extending a result of Gurtin and Sternberg [1961b], proved that the displacement boundary value problem for finite regions has a unique solution provided

$$ {\text{C}}_{{\text{ijkl}}} \xi _{ij} \xi _{kl} \eta _j \eta _l \geqq 0,{\text{ }}for{\text{ }}\xi ,\eta \ne 0. $$

((8.0.3))

These conclusions were all derived by means of energy arguments. A completely different approach based on properties of analytic functions was used by Hayes and Knops [1968] to show that the result of Gurtin and Toupin remains true if

$$ {\text{C}}_{{\text{ijkl}}} \xi _{ij} \xi _{kl} \eta _j \eta _l < 0,{\text{ }}for{\text{ }}\xi ,\eta \ne 0. $$

((8.0.4))

Observe that condition (8.0.4) is necessary and sufficient for all plane waves to travel with purely imaginary speeds. Condition (8.0.3), implying that the equilibrium equations are semi-strongly elliptic, states that all plane waves travel with real speeds.¹ On the other hand, the classical condition (8.0.2) requires the elasticities to be positive-semi-definite, and makes no direct statement about the realness of wave-speeds.