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On the uniqueness of singular solutions to boundary-initial value problems in linear elastodynamics


Brockway, George Samuel (1972) On the uniqueness of singular solutions to boundary-initial value problems in linear elastodynamics. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/WR6E-S243.


This investigation deals with certain generalizations of the classical uniqueness theorem for the second boundary-initial value problem in the linearized dynamical theory of not necessarily homogeneous nor isotropic elastic solids. First, the regularity assumptions underlying the foregoing theorem are relaxed by admitting stress fields with suitably restricted finite jump discontinuities. Such singularities are familiar from known solutions to dynamical elasticity problems involving discontinuous surface tractions or non-matching boundary and initial conditions. The proof of the appropriate uniqueness theorem given here rests on a generalization of the usual energy identity to the class of singular elastodynamic fields under consideration.

Following this extension of the conventional uniqueness theorem, we turn to a further relaxation of the customary smoothness hypotheses and allow the displacement field to be differentiable merely in a generalized sense, thereby admitting stress fields with square-integrable unbounded local singularities, such as those encountered in the presence of focusing of elastic waves. A statement of the traction problem applicable in these pathological circumstances necessitates the introduction of "weak solutions'' to the field equations that are accompanied by correspondingly weakened boundary and initial conditions. A uniqueness theorem pertaining to this weak formulation is then proved through an adaptation of an argument used by O. Ladyzhenskaya in connection with the first boundary-initial value problem for a second-order hyperbolic equation in a single dependent variable. Moreover, the second uniqueness theorem thus obtained contains, as a special case, a slight modification of the previously established uniqueness theorem covering solutions that exhibit only finite stress-discontinuities.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Applied Mechanics
Degree Grantor:California Institute of Technology
Division:Engineering and Applied Science
Major Option:Applied Mechanics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Sternberg, Eli
Thesis Committee:
  • Unknown, Unknown
Defense Date:19 May 1972
Funding AgencyGrant Number
Office of Naval ResearchUNSPECIFIED
Shell Companies FoundationUNSPECIFIED
Record Number:CaltechTHESIS:04012016-123848510
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:9640
Deposited On:01 Apr 2016 20:17
Last Modified:09 Nov 2022 19:20

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