The Mathematical Institute, University of Oxford, Eprints Archive

Well-posedness of dynamics of microstructure in solids

Sengul, Yasemin (2010) Well-posedness of dynamics of microstructure in solids. PhD thesis, University of Oxford.

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Abstract

In this thesis, the problem of well-posedness of nonlinear viscoelasticity under the assumptions allowing for phase transformations in solids is considered. In one space dimension we prove existence and uniqueness of the solutions for the quasistatic version of the model using approximating sequences corresponding to the case when initial data takes finitely many values. This special case also provides upper and lower bounds for the solutions which are interesting in their own rights. We also show equivalence of the existence theory we develop with that of gradient flows when the stored-energy function is assumed to be -convex. Asymptotic behaviour of the solutions as time goes to infinity is then investigated and stabilization results are obtained by means of a new argument. Finally, we look at the problem from the viewpoint of curves of maximal slope and follow a time-discretization approach. We introduce a three-dimensional method based on composition of time-increments, as a result of which we are able to deal with the physical requirement of frame-indifference. In order to test this method and distinguish the difficulties for possible generalizations, we look at the problem in a convex setting. At the end we are able to obtain convergence of the minimization scheme as time step goes to zero.

Item Type:Thesis (PhD)
Subjects:H - N > Mechanics of deformable solids
O - Z > Partial differential equations
D - G > Dynamical systems and ergodic theory
Research Groups:Oxford Centre for Nonlinear PDE
ID Code:1748
Deposited By: Eprints Administrator
Deposited On:20 Aug 2013 08:28
Last Modified:29 May 2015 19:26

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