This research addresses certain generalizations of the classical uniqueness theorem for the second boundary-initial value problem in the linearized dynamic theory of neither necessarily homogeneous nor isotropic elastic solids. An explanation of the traction problem applicable in these pathological conditions. necessitates the introduction of 'weak solutions' to the field equations, which are accompanied by correspondingly weakened boundary and initial conditions. In Section 2, after making the necessary preparations and providing an overview of the classical uniqueness theorem in Section 1, we address the uniqueness problem related to the traction problem of elastodynamics in the presence of finite stress discontinuities.
The proof of the uniqueness theorem given in this section is based on an extension of the usual energy identity to the class of singular fields. There is no difficulty in using this energy identity to arrive at a generalization of the uniqueness theorem established in Section 2 to the corresponding mixed problem which contains the first and second problems as. In this connection, it may be worth noting that the appropriate stress discontinuities at the wavefronts often appear to be derived from the solution rather than explicitly assumed in the problem formulation.
In Section 3 we proceed to a further relaxation of the usual smoothness hypotheses underlying the conventional uniqueness theorem. 2 This version of the divergence theorem follows easily from the broadest form proven by Kellogg [11 J (p. 119). The assumption that the second derivatives of the displacements have limits like t-0, although assumed in [121 and [13], is omitted here.
The uniqueness of the solution to the classical traction problem, which was first proven by Neumann [I J using the classical. energy identity, it is claimed in. In this section, the explanation of the traction problem of ... classical elastodynamics is generalized to include states that have continuous displacement fields but that possess stress fields that suffer from finite jump discontinuities. It is shown that these kinds of singular states obey an analogue of the classical energy identity {Theorem 1.1), which is then used to establish a corresponding extension of the classical uniqueness theorem {Theorem 1.2).
We will therefore require that the spring conditions hold on the regular parts of the surfaces that carry the discontinuities in tension. initial value problem to be considered, we introduce. The analogue to the classical tensile problem in the present context is the tensile problem for piecewise regular elastodynamic states, which can be stated as follows. An elementary calculation involving the use of the motion-stress equations ( 1.7 ), the stress-displacement relations ( 1.6 ) and the symmetry of c asserted in (1.5) yields.
An exposition of the theory of generalized derivatives and Sobolev spaces is also given, for example, in Smirnov [16]. Further, the concept of a weak elastodynamic state is an extension of the notion of a regular elastodynaric state. In preparation for a convenient generalization of the initial and boundary conditions (1.9), (l.10), we introduce three subclasses of tD1.
1 For a proof of the existence of a unit division subordinate to a given coverage, refer for example to Bremermann [19 J.
To see this, note that if R were star-shaped (rather than locally star-shaped), (A) and (B) would be sufficient for (C), since the necessary sequence1 of vector fields in ~(Rx[O, t ]) with trend q in norm e. Switching the order of integration in the right-hand member of (I), performing the time integration by parts, and then restoring the previous order of integration results in . Let R be a locally regular star-shaped region and assume:. t and ~ are square summed vector fields in R, Rand 8Rx[O,t. y) Sis positive semidefinite in R, so that c.
1we define the inner product of two nth_order tensor valued functions in ~2(Rx[O,t0]) to be the integral over Rx(O,t. 0) of their fully contracted outer product. On the other hand, the symmetry relations give ( 1. 5) for :::_together with the product rule for generalized time differentiation. However, the formulation of the traction problem given here excludes the possibility that the displacement field itself undergoes a finite discontinuity over some surface, since such a singularity cannot occur in a function belonging to 11,~ ( see for example Smirnov [16) ], Article 110).
It may be well to emphasize that a prior decision or a problem with a particular set of data covered by Theorem 3.2 will require an existence theorem suitable for the traction problem for weak elastodynamic conditions. Finally, we observe that the regularity hypotheses on p and c can be easily relaxed to accommodate discontinuities in material properties. The result just mentioned provides an alternative derivation of the classical jump relations (2.8). Uniqueness of solution to the traction problem for piecewise regular elastodynamic conditions).
3, which is free from the artificial assumption ((3), but which involves a small change in the hypothesis of the underlying spatial domain R. The reason for the presence of ((3) in Theorem 2. 3 and its absence in Theorem 3.3 is that while the difference in two weak elastodynamic states is obviously itself such a state, the analogous claim about piecewise regular elastodynamic states cannot be taken for granted.Finally, we note that a counterpart to Theorem 3. 2 for the first problem of homogeneous Boundary conditions can be established through a scheme similar to the one used here to solve the second problem 1.
If we then add to Definition 3.2 the requirement that the displacement field is continuous on R.X[O,t ], we get the analog- .. o . it is also possible to obtain a uniqueness theorem for data on inhomogeneous surfaces, provided that the region R satisfies some regularity conditions in addition to those assumed in this investigation. 1 Ladyzhenskaya [5 J considered the corresponding issue of uniqueness for a single hyperbolic equation of the second order. 2 see the article by Dafermos [9 J on linear dynamic thermoelasticity, in which the uniqueness of the "generalized" solution for the mixed boundary-initial value problem for homogeneous boundary data is proved. 3] M. E. Gurtin, Linear theory of elasticity, unpublished manuscript to be published in Handbuch der Physik, Springer, Berlin.
Dafermos, On the existence and asymptotic stability of solutions to the equations of linear thermoelasticity, Archives for Rational Mechanics and Analysis.