ch1 - Notes for Introduction to Continuum Mechanics Lynn...

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Unformatted text preview: Notes for Introduction to Continuum Mechanics Lynn Schreyer Bennethum February 1, 2006 2 Contents 1 Vector Calculus, Tensors, and Indicial Notation 3 1.1 Dot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Special Matrices/Second-Order Tensors . . . . . . . . . . . . . . . . . . . 8 1.4 Contraction and Tensor Products . . . . . . . . . . . . . . . . . . . . . . 9 1.5 Cross Product and Determinant . . . . . . . . . . . . . . . . . . . . . . . 11 1.6 Review of Vector Calculus and More Indicial Notation . . . . . . . . . . 15 1.7 Divergence Theorem, Green’s Identity, and Stoke’s Theorem . . . . . . . 19 1.8 Fourth-Order Tensors with Minor Symmetries . . . . . . . . . . . . . . . 21 1.8.1 Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.8.2 6 × 6 Component Notation of Fourth-Order Tensors . . . . . . . . 22 1.9 Introduction to Indicial Notation in a Curvilinear Coordinate System . . 24 1.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2 Deformation and Strain 31 2.1 Two Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2 Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.2.2 Physical Interpretation . . . . . . . . . . . . . . . . . . . . . . . . 37 2.2.3 Strain and Displacement . . . . . . . . . . . . . . . . . . . . . . . 41 2.2.4 Small Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.2.5 Invariants of a Matrix and Principal Strains . . . . . . . . . . . . 45 2.3 Transformation of Area and Volume . . . . . . . . . . . . . . . . . . . . . 50 2.3.1 Area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.3.2 Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.4 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.4.1 Rate of Change of Arc Length and Strain . . . . . . . . . . . . . . 54 2.4.2 Rate of Change of Elemental Area and Elemental Volume . . . . 56 2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3 Balance Laws 63 3.1 Local Balance Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.2 Conservation of Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.3 Conservation of Momentum . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.3.1 Conservation of Linear Momentum in Eulerian Framework . . . . 67 3 CONTENTS 1 3.3.2 Surface Traction and the Stress Tensor . . . . . . . . . . . . . . . 67 3.3.3 Conservation of Linear Momentum . . . . . . . . . . . . . . . . . 70 3.3.4 Conservation of Angular Momentum . . . . . . . . . . . . . . . . 71 3.3.5 Conservation of Linear Momentum in Lagrangian Framework . . . 73 3.4 Conservation of Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . .Conservation of Energy ....
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ch1 - Notes for Introduction to Continuum Mechanics Lynn...

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