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Introduction to Continuum Mechanics

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Introduction to Continuum Mechanics ME185 Introduction to Continuum Mechanics Panayiotis Papadopoulos Department of Mechanical Engineering University of California, Berkeley 2020 edition Copyright c 2020 by Panayiotis Papadopoulos Introduction This is a set of notes written as part of teaching ME185, an elective senior-year under- graduate course on continuum mechanics in the Department of Mechanical Engineering at the University of California, Berkeley. Berkeley, California P. P. August 2020 i Contents 1 Introduction 1 1.1 Solidsandfluidsascontinuousmedia . ..... 1 1.2 Historyofcontinuummechanics . ... 2 2 Mathematical Preliminaries 4 2.1 Elementsofsettheory ............................. 4 2.2 Mappings ..................................... 5 2.3 Vectorspaces ................................... 6 2.4 Points, vectors and tensors in the Euclidean 3-space . ........... 10 2.5 Vectorandtensorcalculus . ... 24 2.6 ThedivergenceandStokestheorems . .... 29 2.7 Exercises...................................... 31 3 Kinematics of Deformation 37 3.1 Bodies,configurationsandmotions . ..... 37 3.2 The deformation gradient and other measures of deformation......... 48 3.3 Velocity gradient and other measures of deformation rate........... 72 3.4 Superposedrigid-bodymotions . .... 80 3.5 Exercises...................................... 88 4 Physical Principles 110 4.1 TheReynoldstransporttheorem . ... 110 4.2 Thelocalizationtheorem . ... 113 4.3 Massandmassdensity .............................. 114 4.4 Theprincipleofmassconservation . ..... 117 4.5 The principles of linear and angular momentum balance . .......... 119 ii 4.6 Stressvectorandstresstensor . ..... 123 4.7 The theorems of mechanical energy balance and virtual power ........ 138 4.8 Theprincipleofenergybalance . .... 142 4.9 Thesecondlawofthermodynamics . ... 146 4.10 The transformation of mechanical and thermal fields under superposed rigid- bodymotions ................................... 150 4.11 TheGreen-Naghdi-Rivlintheorem. ..... 155 4.12Exercises...................................... 158 5 Infinitesimal Deformations 173 5.1 TheGˆateauxdifferential . ... 174 5.2 Consistent linearization of kinematic and kinetic variables .......... 175 5.3 Exercises...................................... 182 6 Constitutive Theories 185 6.1 Generalrequirements . 186 6.2 Inviscidfluid.................................... 191 6.2.1 Initial/boundary-value problems of inviscid flow . ......... 195 6.2.1.1 Uniforminviscidflow. 195 6.2.1.2 Irrotationalflowofanidealfluid . 195 6.3 Viscousfluid.................................... 196 6.3.1 The Helmholtz-Hodge decomposition and projection methods in com- putationalfluidmechanics . 200 6.3.2 Initial/boundary-value problems of viscous flow . ........ 203 6.3.2.1 Gravity-driven flow down an inclined plane . 203 6.3.2.2 Couette flow between concentric cylinders . 206 6.3.2.3 Poiseuilleflow. 208 6.3.2.4 Stokes’secondproblem. 210 6.4 Non-linearlyelasticsolid . .... 212 6.4.1 Boundary-value problems of non-linear elasticity . .......... 223 6.4.1.1 Uniaxialstretching . 223 6.4.1.2 Rivlin’scube .......................... 225 6.5 Non-linearlythermoelasticsolid . ...... 228 6.6 Linearlyelasticsolid . ... 231 iii 6.6.1 Initial/boundary-value problems of linear elasticity .......... 235 6.6.1.1 Simpletensionandsimpleshear. 235 6.6.1.2 Uniform hydrostatic pressure and incompressibility ..... 236 6.6.1.3 Saint-Venant torsion of a circular cylinder . .... 237 6.6.1.4 Planewavesinaninfinitesolid . 238 6.7 Viscoelasticsolid ................................ 240 6.8 Exercises...................................... 245 7 Multiscale modeling 257 7.1 Thevirialtheorem ................................ 257 7.2 Exercises...................................... 259 Appendices 262 A Cylindricalpolarcoordinatesystem . ..... 262 iv List of Figures 1.1 From left to right: Portraits of Archimedes, da Vinci, Galileo, Newton, Euler andCauchy .................................... 2 2.1 Mapping between two sets ............................ 5 2.2 Composition mapping g f ............................ 6 ◦ 2.3 Schematic depiction of a set ........................... 6 2.4 Points and associated vectors in three dimensions ............... 11 2.5 The neighborhood (x) of a point x in E3................... 11 Nr 2.6 A surface bounded by the curve . ...................... 30 A C 3.1 A body B and its subset S . ........................... 37 3.2 Mapping of a body B to its configuration at time t. .............. 38 3.3 Mapping of a body B to its reference configuration at time t0 and its current configuration at time t. .............................. 39 3.4 Schematic depiction of referential and spatial mappings for the velocity v. 41 3.5 Pathline of a particle which occupies X in the reference configuration. .... 46 3.6 Streamline through point x at time t. ...................... 46 3.7 Mapping of an infinitesimal material line element represented by dX to the current configuration. ............................... 49 3.8 Vectors at point X in the reference configuration and at point x in the current configuration. ................................... 51 3.9 Application of the inverse function theorem to the motion χ at a fixed time t. 51 3.10 Infinitesimal material line elements along vectors M and m in the reference and current configuration, respectively. ..................... 53 3.11 Mapping of an infinitesimal material volume element dV to its image dv in the current configuration. ............................. 60 v 3.12 Mapping of an infinitesimal material surface element dA to its image da in the current configuration. ............................. 61 3.13 Interpretation of the right polar decomposition. ................. 64 3.14 Interpretation of the left polar decomposition. .................. 65 3.15 Interpretation of the right polar decomposition relative to the principal direc- tions M and associated principal stretches λ . .............. 68 { A} { A} 3.16 Interpretation of the left polar decomposition relative to the principal directions RM and associated principal stretches λ . ................ 69 { i} { i} 3.17 Geometric interpretation of the rotation tensor R by its action on a vector x. 72 3.18 A unit vector m and its rate m˙ . ......................... 76 3.19 A physical interpretation of the vorticity vector w as angular velocity of a unit eigenvector m¯ of D. ............................... 79 3.20 Configurations associated with motions χ and χ+ differing by a superposed rigid-body motion χ¯ +. .............................. 81 3.21 A rigid-motion motion superposed on the reference configuration ....... 86 + 3.22 Basis vectors in R0 and R and rigidly rotated basis vectors in R . ...... 88 4.1 A region with boundary ∂ and its image with boundary ∂ in the P P P0 P0 reference configuration. .............................. 110 4.2 The domain with a spherical subdomain centered at x . ......... 114 R Pδ 0 4.3 A limiting process used to define the mass density ρ at a point x in the current configuration. ................................... 116 4.4 Mass conservation in a domain with boundary ∂ . ............. 118 P P 4.5 Angular momentum of an infinitesimal volume element dv. .......... 120 4.6 External forces on body occupying region R with boundary ∂ (body force in R green, contact force in red). ........................... 121 4.7 A sequence of shrinking bodies under equilibrated point forces. ........ 123 4.8 Setting for the derivation of Cauchy’s lemma. ................. 124 4.9 Tractions at point x on opposite sides of a surface ∂ (surface is depicted P twice for clarity). ................................. 125 4.10 The Cauchy tetrahedron ............................. 126 4.11 Interpretation of the Cauchy stress components on an orthogonal parallelepiped aligned with the axes of the basis e . ..................... 130 { i} 4.12 Projection of the traction to its normal and shearing components. ...... 131 vi 4.13 A force df acting on a differential area on the boundary of a domain ∂ and P resolved over the geometry of the current and reference configuration ..... 134 4.14 Schematic depiction of the relation between the Cauchy stress T, the first Piola-Kirchhoff stress P, and the second Piola-Kirchhoff stress S. ...... 137 4.15 The relation between the traction vectors t and t+. .............. 151 5.1 Displacement vector u of a material point with position X in the reference configuration. ................................... 176 6.1 Schematic of tractions resisting the sliding of a continuum. .......... 191 6.2 Traction acting on a surface of an inviscid fluid. ................ 192 6.3 A ball of ideal fluid in equilibrium under uniform pressure. .......... 195 6.4 Shearing of a viscous fluid ............................ 197 6.5 Flow down an inclined plane ........................... 204 6.6 Couette flow between concentric cylinders .................... 206 6.7 Poiseuille flow ................................... 208 6.8 Semi-infinite domain for Stokes’ second problem ................ 210 6.9 Closed cycle for a part of a body containing particle P . ........... 215 P 6.10 Deformation relative to two reference configurations and ′ . ....... 217 P0 P0 6.11 An orthogonal transformation of the reference configuration. ......... 218 6.12 Homogeneous uniaxial stretching of a slender specimen ............ 223 6.13 Homogeneous uniaxial stretching of a slender specimen: Second Piola-Kirchhoff and Cauchy stress components along the stretch direction for λ =0 and µ =1. 225 6.14 Function f(λ) in Rivlin’s cube .......................... 228 6.15 A solution to Rivlin’s cube (λ = λ = λ , λ
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