The Emergence and Propagation of a Phase Boundary in an Elastic Bar
<p>This dissertation is concerned with the dynamical analysis of an elastic bar whose stress-strain relation is not monotonic. Sufficiently large applied loads then require the strain to jump from one ascending branch of the stress-strain curve to another such branch. For a special class of th...
| Summary: | <p>This dissertation is concerned with the dynamical analysis of an elastic bar whose stress-strain relation is not monotonic. Sufficiently large applied loads then require the strain to jump from one ascending branch of the stress-strain curve to another such branch. For a special class of these materials, a nonlinear initial-boundary value problem in one-dimensional elasticity is considered for a semi-infinite bar whose end is subjected to either a monotonically increasing prescribed traction or a monotonically increasing prescribed displacement. If the stress at the end of the bar exceeds the value of the stress at any local maximum of the stress-strain curve a strain discontinuity or "phase boundary" emerges at the end of the bar and subsequently propagates into the interior. For classically smooth solutions away from the phase boundary, the problem is reducible to a pair of differential-delay equations for two unknown functions of a single variable. The first of these two functions gives the location of the phase boundary, while the second characterizes the dynamical fields in the high-strain phase of the material. In these equations the former function occurs in the argument of the latter, so that the delays in the functional equations are unknown. A short-time analysis of this system provides an asymptotic description of the emergence and initial propagation of the phase boundary. For large-times, a different analysis indicates that the phase boundary velocity approaches a constant which depends on material properties and on the ultimate level reached by the applied load as well. Higher order corrections depend on the detailed way in which the load is applied. Estimates for the various dynamical field quantities are given and a priori conditions are deduced which determine whether the phase boundary eventually becomes the leading disturbance.</p> |
|---|