Geometrical problems in the mathematical study of prestrained materials

Ochoa, Pablo (2015) Geometrical problems in the mathematical study of prestrained materials. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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Abstract

In my thesis, we derive a two dimensional energy model for deformations of unloaded elastic films as a result of the application of Gamma-convergence to appropriate three dimensional models. The limiting model obtained in this way constitutes a Von Karman type growth functional, attaining its minima at deformations v in W^{2, 2} satisfying a Monge-Ampere constraint of the form det nabla^{2}v = f, for some appropriate f. The main advantage of Gamma-convergence is that it connects 2d theories with 3d nonlinear theory in the sense
that minimizers of the 3d energy functionals converge to minimizers of 2d energy functionals.

Secondly, we study the variational behavior of discrete lattice energies associated with a pre-strained elastic body, as a mathematical justification of the theoretical non-Euclidean energy model employed in this theses. Via Gamma-convergence, we obtain asymptotic bounds on the Gamma-limiting energy and, in the context of near and next-to-near interactions, we identify exactly the integral form of the limiting energy, comparing it to the theoretical model.

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Details

Item Type:	University of Pittsburgh ETD
Status:	Unpublished
Creators/Authors:	Creators Email Pitt Username ORCID Ochoa, Pablo pdo2@pitt.edu PDO2
ETD Committee:	Title Member Email Address Pitt Username ORCID Committee Chair Lewicka, Marta lewicka@pitt.edu LEWICKA Committee Member Manfredi, Juan manfredi@pitt.edu MANFREDI Committee Member Pakzad, Mohammadreza pakzad@pitt.edu PAKZAD Committee Member Acharya, Amit acharyaamit@cmu.edu
Date:	14 January 2015
Date Type:	Publication
Defense Date:	4 November 2014
Approval Date:	14 January 2015
Submission Date:	5 November 2014
Access Restriction:	No restriction; Release the ETD for access worldwide immediately.
Number of Pages:	147
Institution:	University of Pittsburgh
Schools and Programs:	Dietrich School of Arts and Sciences > Mathematics
Degree:	PhD - Doctor of Philosophy
Thesis Type:	Doctoral Dissertation
Refereed:	Yes
Uncontrolled Keywords:	non-Euclidean plates; nonlinear elasticity; Gamma convergence; calculus of variations; isometric immersions; Monge-Ampere equation
Date Deposited:	14 Jan 2015 15:16
Last Modified:	19 Jul 2024 18:54
URI:	http://d-scholarship.pitt.edu/id/eprint/23457

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