MAE 5100 - Continuum Mechanics Course Notes Prof. Brandon Runnels, University of Colorado Colorado Springs Contents LECTURE 1 0 Introduction 1.1 0.1 Motivation . 1.1 0.2 Notation . 1.2 0.2.1 Sets . 1.2 0.2.2 Proof notation . 1.2 1 Tensor Analysis 1.2 1.1 Index notation and the Einstein summation convention . 1.3 1.1.1 Vector equality . 1.4 1.1.2 Inner product . 1.4 1.1.3 Kronecker delta . 1.4 1.1.4 Components of a vector . 1.5 1.1.5 Norm of a vector . 1.5 1.1.6 Permutation tensor . 1.5 1.1.7 Cross product . 1.5 LECTURE 2 1.1.8 Notation . 2.1 1.2 Mappings and tensors . 2.1 1.2.1 Second order tensors . 2.1 1.2.2 Index notation . 2.2 1.2.3 Dyadic product . 2.2 1.2.4 Tensor components . 2.3 1.2.5 Higher order tensors . 2.3 1.2.6 Transpose . 2.3 1.2.7 Trace (first invariant) . 2.3 1.2.8 Determinant (third invariant) . 2.4 LECTURE 3 1.2.9 Inverse . 3.1 1.2.10 The special orthogonal group . 3.1 1.3 Tensor calculus . 3.1 1.3.1 Gradient . 3.2 1.3.2 Divergence . 3.2 1.3.3 Laplacian . 3.2 1.3.4 Curl . 3.3 1.3.5 Gateaux derivatives . 3.3 1.3.6 Notation . 3.3 1.3.7 Evaluating derivatives . 3.4 LECTURE 4 1.4 The divergence theorem . 4.1 1.5 Curvilinear coordinates . 4.2 All content © 2016-2018, Brandon Runnels 1 MAE 5100 - Continuum Mechanics Course Notes University of Colorado Colorado Springs https://canvas.uccs.edu/courses/22031 1.5.1 The metric tensor . 4.2 1.5.2 Orthonormalized basis . 4.3 1.5.3 Change of basis . 4.3 1.6 Calculus in curvilinear coordinates . 4.4 1.6.1 Gradient . 4.4 LECTURE 5 1.6.2 Divergence . 5.1 1.6.3 Curl . 5.2 1.7 Tensor transformation rules . 5.2 2 Kinematics of Deformation 5.3 LECTURE 6 2.1 Eulerian and Lagrangian frames . 6.1 2.2 Time-dependent deformation . 6.2 2.2.1 The material derivative . 6.3 2.3 Kinematics of local deformation . 6.4 LECTURE 7 2.4 Metric changes . 7.1 2.4.1 Change of length . 7.1 2.4.2 Change of angle . 7.1 2.4.3 Determinant identities . 7.2 2.4.4 Change of volume . 7.2 2.4.5 Change of area . 7.3 LECTURE 8 2.5 Tensor decomposition . 8.2 2.5.1 Eigenvalues and Eigenvectors . 8.2 2.5.2 Symmetric and positive definite tensors . 8.3 2.5.3 Spectral theorem (symmetric tensors) . 8.3 LECTURE 9 2.5.4 Spectral theorem (general case) . 9.2 2.5.5 Functions of tensors . 9.2 2.5.6 Polar decomposition . 9.3 2.6 Principal deformations . 9.3 LECTURE 10 2.7 Compatibility . 10.1 2.7.1 Continuous case . 10.1 2.7.2 Discontinuous case (Hadamard) . 10.3 2.8 Other deformation measures . 10.4 LECTURE 11 2.9 Linearized kinematics . 11.1 2.9.1 Linearized metric changes . 11.2 2.9.2 Small strain compatibility . 11.4 LECTURE 12 2.10 The spatial/Eulerian picture . 12.1 3 Conservation Laws 12.2 LECTURE 13 3.1 Conservation of Mass . 13.1 3.1.1 Control volume . 13.2 3.2 Conservation of linear momentum . 13.2 3.2.1 Forces, tractions, and stress tensors . 13.2 LECTURE 14 3.2.2 Balance laws . 14.1 3.2.3 Navier-Stokes momentum equations . 14.2 LECTURE 15 3.3 Conservation of angular momentum . 15.1 LECTURE 16 3.4 Conservation of energy . 16.1 3.4.1 Energetic quantities . 16.1 All content © 2016-2018, Brandon Runnels 2 MAE 5100 - Continuum Mechanics Course Notes University of Colorado Colorado Springs https://canvas.uccs.edu/courses/22031 3.4.2 Balance laws . 16.3 LECTURE 17 3.4.3 Power-conjugate pairs . 17.1 3.5 Second law of thermodynamics . 17.2 3.5.1 Introduction to statistical thermodynamics and entropy . 17.2 LECTURE 18 3.5.2 Internal entropy generation . 18.2 3.5.3 Continuum formulation . 18.3 3.6 Review and summary . 18.4 LECTURE 19 4 Constitutive Theory 19.1 4.1 Introduction to the calculus of variations . 19.1 LECTURE 20 4.1.1 Stationarity condition . 20.1 4.2 Variational formulation of linear momentum balance . 20.2 LECTURE 21 4.3 Material frame indifference . 21.1 4.4 Elastic modulus tensor . 21.1 4.5 Elastic material models . 21.2 4.5.1 Useful identities . 21.2 LECTURE 22 4.5.2 Pseudo-Linear . 22.1 4.5.3 Compressible neo-Hookean . 22.1 LECTURE 23 4.6 Internal constraints . 23.1 4.6.1 Review of Lagrange multipliers . 23.1 4.6.2 Examples of internal constraints . 23.2 4.6.3 Lagrange multipliers in the variational formulation of balance laws . 23.2 4.7 Linearized constitutive theory . 23.3 4.7.1 Major & minor symmetry and Voigt notation . 23.4 4.7.2 Material symmetry . 23.5 LECTURE 24 4.7.3 The Cauchy-Navier equation and linear elastodynamics . 24.1 4.8 Thermodynamics of solids and the Coleman-Noll framework . 24.1 LECTURE 25 4.8.1 Other thermodynamic potentials . ..