elasticity - Elasticity by Gordon C. Everstine 24 April...

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Unformatted text preview: Elasticity by Gordon C. Everstine 24 April 2010 Copyright c 1998–2010 by Gordon C. Everstine. All rights reserved. This book was typeset with L A T E X2 ε (MiKTeX). Preface These lecture notes are intended to supplement a one-semester graduate-level engineering course at The George Washington University in the theory of elasticity. Although the em- phasis is on the Cartesian tensor approach, the direct (vector-operator) approach is also used where appropriate. The main prerequisites are elementary strength of materials, a standard calculus sequence, and some exposure to linear algebra and matrices. In general, the mix of topics and level of presentation are aimed at graduate students in civil, aerospace, and mechanical engineering. Gordon Everstine Gaithersburg, Maryland April 2010 iii Contents 1 Mathematical Preliminaries 1 1.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Change of Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Symmetry and Skew-Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Derivatives and Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.5 The Divergence Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6 Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.7 Even and Odd Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 Strain 16 2.1 Admissible Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Affine Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.3 Geometrical Interpretations of Strain Components . . . . . . . . . . . . . . . 21 2.4 Strain as a Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.5 General Infinitesimal Deformation . . . . . . . . . . . . . . . . . . . . . . . . 23 2.6 Compatibility Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.7 Integrating the Strain-Displacement Equations . . . . . . . . . . . . . . . . . 27 2.8 Principal Axes of Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.9 Properties of the Real Symmetric Eigenvalue Problem . . . . . . . . . . . . . 30 2.10 Geometrical Interpretation of the First Invariant . . . . . . . . . . . . . . . . 31 2.11 Finite Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3 Stress 33 3.1 Momentum Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.2 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.3 Stress as a Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.4 Mean Stress in a Deformed Body . . . . . . . . . . . . . . . . . . . . . . . . 39 4 Equations of Elasticity 39 4.1 Hooke’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.2 Strain Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4....
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This note was uploaded on 11/10/2010 for the course AEROSPACE AE 1202 taught by Professor Dr.adib during the Spring '10 term at Sharif University of Technology.

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elasticity - Elasticity by Gordon C. Everstine 24 April...

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