City University of Hong Kong
DSpace
 

CityU Institutional Repository >
3_CityU Electronic Theses and Dissertations >
ETD - Dept. of Mathematics  >
MA - Doctor of Philosophy  >

Please use this identifier to cite or link to this item: http://hdl.handle.net/2031/5233

Title: Saint Venant compatibility conditions in curvilinear coordinates and applications to elasticity
Other Titles: Sheng wei nan xiang rong xing tiao jian ji ying yong
圣维南相容性条件及应用
Authors: Shen, Ming (沈鳴)
Department: Department of Mathematics
Degree: Doctor of Philosophy
Issue Date: 2008
Publisher: City University of Hong Kong
Subjects: Curvilinear coordinates.
Elasticity -- Mathematical models.
Notes: v, 81 leaves 30 cm.
Thesis (Ph.D.)--City University of Hong Kong, 2008.
Includes bibliographical references (leaves [79]-81)
CityU Call Number: QA556 .S54 2008
Type: thesis
Abstract: In this thesis, we study Saint Venant compatibility equations in curvilinear coordinates on a three-dimensional elastic body and on a surface. In the case of the three-dimensional elastic body, we establish that the linearized strains in curvilinear coordinates associated with a given displacement field necessarily satisfy compatibility conditions that constitute “Saint Venant equations in curvilinear coordinates”. Furthermore, we show that these equations are also sufficient, in the following sense: If a symmetric matrix field defined over a simply-connected open set satisfies the Saint Venant equations in curvilinear coordinates, then its coefficients are the linearized strains associated with a displacement field. In addition, our proof provides an explicit algorithm for recovering such a displacement field from its linear strains in curvilinear coordinates. We also show how these Saint Venant compatibility in curvilinear coordinates may be used for defining a new approach to existence theory in three-dimensional linearized elasticity in curvilinear coordinates. In the case of the surface, we establish that the linearized change of metric and linearized change of curvature tensors associated with a displacement field of a surface S immersed in R3 must satisfy compatibility conditions that may be viewed as the linear version of the Gauss and Codazzi-Mainardi equations. These compatibility conditions, which are the analog in two-dimensional shell theory of the Saint Venant equations in three-dimensional elasticity, constitute the Saint Venant equations on the surface S. We next show that these compatibility conditions are also sufficient in the following sense: If two symmetric matrix fields of order two defined over a simplyconnected surface S _ R3 satisfy the above compatibility conditions, then they are the linearized change of metric and linearized change of curvature tensors associated with a displacement field of the surface S, a field whose existence is thus established. The proof provides an explicit algorithm for recovering such a displacement field from the linearized change of metric and linearized change of curvature tensors. We also show how these Saint Venant compatibility on a surface may be used for defining a new approach to existence theory in linear shell theory.
Online Catalog Link: http://lib.cityu.edu.hk/record=b2268775
Appears in Collections:MA - Doctor of Philosophy

Files in This Item:

File Description SizeFormat
abstract.html132 BHTMLView/Open
fulltext.html132 BHTMLView/Open

Items in CityU IR are protected by copyright, with all rights reserved, unless otherwise indicated.

 

Valid XHTML 1.0!
DSpace Software © 2013 CityU Library - Send feedback to Library Systems
Privacy Policy · Copyright · Disclaimer