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Singularities and Phase Transitions in Elastic Solids: Numerical Studies and Stability Analysis


Silling, Stewart Andrew (1986) Singularities and Phase Transitions in Elastic Solids: Numerical Studies and Stability Analysis. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/0ytn-e775.


Numerical studies of the deformation near the tip of a crack are presented for a family of incompressible solids in the context of the theory of finite anti-plane shear of an elastic material. The numerical model computes the near-field and far-field solutions simultaneously, enabling observations of both small-scale and large-scale nonlinearity. The computed near-field solution is compared with a lowest-order asymptotic solution. An approximation for the J-integral under conditions of very large loads is discussed and compared with numerical results. The size of the region over which the lowest-order solution applies is observed.

Numerical solutions are presented for the same crack problem with materials for which the equilibrium equation changes in type from elliptic to hyperbolic as a result of deformation. These results show the emergence of surfaces of discontinuity in the displacement field in some cases. In other cases they show a chaotic mixture of elliptic phases near the crack tip.

Analysis of the stability of such coexistent phases is carried out for a specific material, the trilinear material. It is shown that the Maxwell relation, and therefore local stability, cannot in general be satisfied exactly for an arbitrary boundary value problem with this material. However, in those cases where it cannot be satisfied exactly, it may be satisfied in the sense of a limit of a certain sequence of deformations. This sequence produces a progressively chaotic pattern of two coexistent elliptic phases, as was observed numerically. The phases mix over a definite region in a given boundary value problem. This region may be computed using a constitutive relation which characterizes the mixture in the limit of the sequence.

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):
  • Knowles, James K.
Thesis Committee:
  • Knowles, James K. (chair)
  • Sternberg, Eli
  • Leonard, Anthony
  • Rosakis, Ares J.
  • Knauss, Wolfgang Gustav
Defense Date:21 May 1986
Additional Information:PDF is missing pp. ii.
Record Number:CaltechETD:etd-03082008-083510
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:900
Deposited By: Imported from ETD-db
Deposited On:14 Mar 2008
Last Modified:16 Apr 2021 23:11

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