Department of Mechanical Engineering University of California, BerkeleyFall Semester 2012 J. Casey ME 185: Introduction to Continuum Mechanics MWF 2-3, 3113 Etcheverry Hall [email protected] Office Hours: MW & Th 3-5, 6125 Etcheverry Hall SYLLABUS 0 INTRODUCTION 0.1 The continuum: a mathematical model. Phenomenological theories. Purely mechanical versus thermodynamical theories. Material behavior and constitutive equations: examples from elasticity, fluid mechanics, rigid body mechanics Historical remarks. 0.2 The six primitive concepts of the purely mechanical theory: body, space, time, mass, force, torque. One, two, and three-dimensional bodies and combinations of them. 0.3 Coordinate systems. Reference frames in classical mechanics. 0.4 The three balance laws: mass, linear momentum, angular momentum. Remark on Newton’s Laws versus Euler’s Laws. 0.5 Scalars, vectors, tensors. Euclidean vector spaces. Index notation. Tensor product and direct notation. Multilinear mappings and determinants. 0.6 Derivatives and differentials. 1 KINEMATICS 1.1 1.2 1.3 1.4 1.5 1.5 Body, configurations, motions. Convected curvilinear and Cartesian coordinates. Particle velocity. The deformation gradient. Material, referential, and spatial descriptions of fields. The material derivative. Material transport of lines, surfaces and volumes. Stretch of a line element. The Cauchy-Green deformation tensors and their properties. The stretch tensors. Maximal stretching. Method of Lagrange multipliers. Eulerian and Lagrangian strain tensors. 1.7 Deformation of area and volume elements. 1.8 The polar decomposition theorem. 1.9 Representation of the rotation tensor. Application to rigid body kinematics. Corotational frames, corotational derivatives. 1.10 The velocity gradient. The rate of deformation tensor, and the vorticity tensor and vector. Material derivative of deformation measures. 1.11 Streamlines and vortex-lines. Circulation. Vortex tubes. 1.12 Rigid motions superposed on a given motion of a deformable body. Objective fields. Objective rates of vectors and second-order tensors. Interpretation of 2.g. and on the first and second laws of thermodynamics. 2. Questions of uniqueness and stability of solutions. 3 CONSTITUTIVE RELATIONS 3. Examples. 2. “equipresence.14 Objectivity requirements. 2. Fréchet derivative.1 2. 2. and spatial integral statements. 2.3 Restrictions due to objectivity requirements. 2.2 Local action.5 Ideal fluids. The transport theorem. Bending and torsion of bars. Infinitesimal strain and rotation. Mass and mass density. Point forms of conservation law. 3.12 Kelvin’s kinematical theorem and Helmholtz’s vorticity theorems. 2. Piola-Kirchhoff stress tensors. Conservation of mass: Material.6 Viscous fluids.” materials with memory.1 Concept of a material in continuum mechanics. Examples of constitutive assumptions.4 BALANCE LAWS The divergence theorem. 1. Solids and fluids.5 Balance of linear momentum: Integral statements. Properties of the stress tensor. Constrained and unconstrained materials. Referential form of Euler’s laws and Cauchy’s laws.11 Application: the expansion of the universe under Newtonian gravitation. 3.3 2. 3. referential. Torques. Examples of simple deformations. 2.8 Applications of the balance laws in integral form (e. 2. Existence theorem. and some applications. 2.upper and lower convected rates.17 On the derivation of Euler’s Laws from the energy equation.4 Material symmetry. 3. Navier-Stokes equations.6 Balance of angular momentum: Integral statements. 3.9 Linear viscoelasticity. Saint Venant’s principle.7 Rigid body dynamics. .9 The Cauchy stress tensor.2 2. 3. 3.10 Cauchy’s first and second laws. Traction and body forces. Linear viscous fluids.16 Remarks on thermodynamical concepts. 2 2.13 Piola transforms.7 Elastic solids. 2.13 Linearization of kinematical measures.: material flowing through pipes).8 Linearly elastic solids.15 The work-energy theorem. 2. 3. Incompressibility. E. Stuart. 2006. R.J. CM12. Div Grad Curl and All That. Virginia. Gonzalez and A.C. Springer-Verlag New York. Berkeley.Oxford University Press. FM4. American Institute of Aeronautics and Astronautics. Mineola.E. Batra. CM10. Nonlinear Solid Mechanics. Berlin. 1989. CM6. New York.mit. 1981.. Ltd. Aerodynamics. Continuum Mechanics. 1980. Cambridge University Press. R. New York. Inc. Prandtl & O. Lai. Academic Press Inc. Elements of Continuum Mechanics. H. A. P. Inc. O.. 2005. Holzapfel. P. Dover Publications. Fluid Mechanics FM1. http://web. 1999. R. Papadopoulos. 2007. 1996. Introduction to Continuum Mechanics. A First Course in Continuum Mechanics. Halmos. Dover Publications. Chichester.: S. Abeyaratne. 1957. CM5. R. Oxford. P. and http://web. Courant & F.. The classical field theories.. Mathematics M1. Reston. III/1 (Ed. 4th ed.SELECTED REFERENCES Continuum Mechanics CM1. New York. Inc. Mineola.. John Wiley & Sons.. Vol. P.M. Dover Publications. CM8. Gurtin.edu/abeyaratne/Volumes/RCA_Vol_I_Math. J. SpringerVerlag New York. Chadwick. Lighthill. Schey. Meyer. Sommerfeld.M. FM2. 1974. 226-858.W. New York. Continuum Mechanics. L.. 2008. ME 185 (Classnotes). New York. M3. John. An Introduction to Continuum Mechanics. A. Vol. New York. W. Inc. Inc.edu/abeyaratne/Volumes/RCA_Vol_II. 1950. . G. Longman Group Limited. Inc. Truesdell & R. 2008 CM9. Inc. Introduction to Calculus and Analysis (2 Vols)..pdf. Finite-Dimensional Vector Spaces. Tietjens. Toupin.and Aerodynamics. Spencer. Berkeley. CM7.R.M. An Informal Introduction to Theoretical Fluid Mechanics. Th.pdf CM11. and E. D. Encyclopedia of Physics. Pergamon Press Inc. Naghdi. 2001. 1979.. Flügge). 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Cambridge. Encyclopedia of Physics. Love. M.. Ogden. Truesdell. Historical Introduction. SM4. Truesdell. History of classical mechanics: Part II. Bulletin of the American Mathematical Society. H6. SM2. 1973. Flügge and C. !956. 2nd ed.Vol. H. Inc. Non-Linear Elastic Deformations. New York. Springer-Verlag. H4. Batchelor. SM3. An Introduction to Fluid Dynamics. 1944. 63 (1976) 53-62. H2.W. Warsaw. Cambridge. Encyclopedia of Physics.N. R. Institute of Fundamental T.H. SM5. Gurtin. McGraw-Hill Book Company.K. C. 1-295.: S.FM5. R. Lamb. C. Naturwissenschaften. New York.E.. New York.S. 1-778. Mathematical Theory of Elasticity. The linear theory of elasticity. AMAS Lecture Notes. Sokolnikoff. 1968. Naturwissenschaften. G. SM6. the 19th and 20th centuries. Nonlinear Elasticity with Application to Material Modelling.Vol. Book Review. A. to 1800. Bell. Solid Mechanics SM1. Cambridge University Press. History of classical mechanics: Part I.. Berlin. 6th ed. 6.echnological Research. pp. A. 1972. Love. H3. I. J. Berlin. Springer-Verlag New York Inc. 1924. GRADING SCHEME Homework: 40% Midterm Exam: 20% Final Exam: 40% 22 August 2012 .E. Hydrodynamics. pp. H. Lamb. VIa/1 (Eds.