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In Chapter 1, for the convenience of the reader, we provide classical results on shape sensitivity analysis in smooth domains.

In Chapter 2, the results on the first order Eulerian semi-derivative in domains with cracks are presented. Of particular interest is the so-called structure theorem for the shape derivative.

In Chapter 3, the results on the Fréchet derivative in domains with cracks are presented as well, for first and second order derivatives, using a technique different from that in Chapter 2.

In Chapter 4, we extend those ideas to Banach spaces, and give some applications of this extended theory. The polyhedricity of convex sets is considered in the spirit of [71], [87], in the most general setting. These abstract results can be applied to sensitivity analysis of crack problems with non-linear boundary conditions. The results obtained use non-linear potential theory and are interesting on their own.

In Chapter 5, several techniques for the study of cracked domains with non-penetration conditions on the crack faces in elastic bodies are presented. The classical crack theory in elasticity is characterized by linear boundary conditions which do not correspond to the physical reality since the crack faces can penetrate each other in this model. In this chapter, non-penetration conditions on the crack faces are considered, which leads to a non-linear problem. The model is presented and the shape sensitivity analysis is performed.

Chapter 6 is devoted to the newly developed smooth domain method for cracks. In that chapter the problem on a domain with a crack is transformed into a new problem on a smooth domain. This approach is useful for numerical methods. In \cite{belh} this formulation is used combined with mixed finite elements, and some error estimates are derived for the finite element approximation of variational inequalities with non-linear condition on the crack faces. We give applications of this method to some classical problems.

Finally, in Chapter 7 we study integro-differential equations arising from bridged crack models. This is a classical technique, but we introduce a few modern approaches to it for completeness sake.

Kategoryzacja MSC:

Warszawa

Rozprawy Matematyczne
tom/nr w serii:
462

149

wydano

2009

autor

- Institut Elie Cartan, Laboratoire de Mathématiques, Université Henri Poincaré Nancy I, B.P. 239, 54506 Vandoeuvre lès Nancy Cedex, France

autor

- Department of Mathematics, 358 Little Hall, University of Florida, Gainesville, FL 32611, USA

autor

- Institut für Mathematik und wissenschaftliches Rechnen, Karl-Franzens-Universität Graz, Heinrichstraße 36, 8010 Graz, Austria

autor

- Department of Mathematics, 358 Little Hall, University of Florida, Gainesville, FL 32611, USA

autor

EN |

bwmeta1.element.bwnjournal-rm-doi-10_4064-dm462-0-1

DOI

10.4064/dm462-0-1

DML-PL

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