Fundamentals of Continuum Mechanics J. W. Rudnicki1 Department of Civil and Environmental Engineering and Department of Mechanical Engineering, Northwestern University, Evanston, IL Last update: November 3, 2011 Last printing: November 3, 2011 1 c John W. Rudnicki, 2006. Do not distribute without permission. ° 2 Do not distribute without permission Contents 1 Introduction 1 I Mathematical Preliminaries 3 2Vectors 5 2.1AdditonalReading.......................... 10 3Tensors 11 4 Coordinate Systems 17 4.1BaseVectors............................. 17 4.2IndexNotation............................ 18 4.3TensorComponents......................... 21 4.4AdditionalReading.......................... 21 5Dyads 23 5.1TensorandScalarProducts..................... 24 5.2Identitytensor............................ 25 5.3AdditionalReading.......................... 26 6 Vector (Cross) Product 27 6.1PropertiesoftheCross-Product................... 29 6.2UsesoftheCrossProduct...................... 29 6.2.1 VelocityDuetoRigidBodyRotation........... 30 6.2.2 Moment of a Force P about O ............... 31 6.3Triplescalarproduct......................... 31 6.4TripleVectorProduct........................ 32 6.5AdditionalReading.......................... 32 7 Determinants 33 7.1Inverse................................. 35 7.2ProductofTwoDeterminants.................... 36 7.3AdditionalReading.......................... 37 3 CONTENTS 8 Change of Orthonormal Basis 39 8.1ChangeofVectorComponents................... 41 8.2 Definitionofavector......................... 42 8.3ChangeofTensorComponents................... 42 8.4AdditionalReading.......................... 43 9 Principal Values and Principal Directions 45 9.1Example................................ 47 9.2AdditionalReading.......................... 49 10 Vector and Tensor Calculus 51 10.1Example:CylindricalCoordinates................. 55 10.2AdditionalReading.......................... 55 II Stress 57 11 Traction and Stress Tensor 59 11.1TypesofForces............................ 59 11.2 Traction on DifferentSurfaces.................... 62 11.3TractiononanArbitraryPlane(Cauchytetrahedron)...... 64 11.4Symmetryofthestresstensor.................... 65 11.5AdditionalReading.......................... 66 12 Principal Values of Stress 67 12.1DeviatoricStress........................... 68 12.2Reading................................ 69 13 Stationary Values of Shear Traction 71 13.1AdditionalReading.......................... 74 14 Mohr’s Circle 75 14.1ExampleProblem........................... 77 14.2AdditionalReading.......................... 79 III Motion and Deformation 81 15 Current and Reference Configurations 83 15.1AdditionalReading.......................... 86 16 Rate of Deformation 89 16.1Velocitygradients........................... 89 16.2 Meaning of D ............................. 90 16.3 Meaning of W ............................ 92 16.4AdditionalReading.......................... 93 4 Do not distribute without permission CONTENTS 17 Geometric Measures of Deformation 95 17.1DeformationGradient........................ 95 17.2ChangeinLengthofLines...................... 95 17.3ChangeinAngles........................... 97 17.4ChangeinArea............................ 98 17.5ChangeinVolume.......................... 99 17.6AdditionalReading..........................100 18 Polar Decomposition 101 19 Strain Tensors 105 19.1MaterialStrainTensors.......................105 19.1.1AdditionalReading......................107 19.2SpatialStrainMeasures.......................107 19.3 Relations between D and rates of EG and U ...........109 19.3.1 Relation Between E˙ G and D ................109 19.3.2 Relation Between D and U˙ .................111 19.3.3AdditionalReading......................111 20 Linearized Displacement Gradients 113 20.1LinearizedGeometricMeasures...................115 20.1.1 Stretch in direction N ....................115 20.1.2AngleChange.........................116 20.1.3Volumechange........................116 20.2LinearizedPolarDecomposition...................117 20.3 Small Strain Compatibility . ....................118 20.3.1AdditionalReading......................121 IV Balance of Mass, Momentum and Energy 123 21 Transformation of Integrals 125 21.1AdditionalReading..........................130 22 Conservation of Mass 131 22.1Reynolds’TransportTheorem....................134 22.2DerivativeofanIntegralOveraTime-DependentRegion....135 22.3AdditionalReading..........................137 23 Conservation of Momentum 139 23.1MomentumBalanceintheCurrentState.............139 23.1.1LinearMomentum......................139 23.1.2AngularMomentum.....................140 23.2MomentumBalanceintheReferenceState............141 23.2.1LinearMomentum......................141 23.2.2AngularMomentum.....................142 23.3AdditionalReading..........................143 5 Do not distribute without permission CONTENTS 24 Conservation of Energy 145 24.1WorkConjugateStresses.......................147 24.2AdditionalReading..........................149 V Ideal Constitutive Relations 151 25 Fluids 155 25.1IdealFrictionlessFluid........................155 25.2Isotropictensors...........................157 25.3LinearlyViscousFluid........................159 25.4AdditionalReading..........................161 26 Elasticity 163 26.1NonlinearElasticity.........................163 26.2LinearizationofElasticityEquations................165 26.3LinearizedElasticity.........................167 26.3.1MaterialSymmetry......................169 26.3.2LinearIsotropicElasticConstitutiveRelation.......171 26.4RestrictionsonElasticConstants..................172 26.5AdditionalReading..........................173 6 Do not distribute without permission Preface These notes are based on lectures given in C. E. 417-1, Mechanics of Continua, IatNorthwesternUniversity. 7 PREFACE 8 Do not distribute without permission Chapter 1 Introduction Continuum mechanics is a mathematical framework for studying the transmis- sion of force through and deformation of materials of all types. The goal is to construct a framework that is free of special assumptions about the type of material, the size of deformations, the geometry of the problem and so forth. Of course, no real materials are actually continuous. We know from physics and chemistry that all materials are formed of discrete atoms and molecules. Even at much larger size scales, materials may be composed of distinct grains, e.g., a sand, or of grains of different constituents, e.g., steel, or deformable particles such as blood. Nevertheless, treating material as continuous is a great advantage since it allows us to use the mathematical tools of continuous functions, such as differentiation. In addtion to being convenient, this approach works remarkably well. This is true even at size scales for which the justfication of treating the material as a continuum might be debatable. The ultimate justification is that predictions made using continuum mechanics are in accord with observations and measurements. Until recently, it was possible to solve a relatively small number of prob- lems without the assumptions of small deformations and linear elastic behavior. Now, however, modern computational techniques have made it possible to solve problems involving large deformation and complex material behavior. This pos- sibility has made it important to formulate these problems correctly and to be able to interpret the solutions. Continuum mechanics does this. The vocabulary of continuum mechanics involves mathematical objects called tensors. These can be thought of as following naturally from vectors. Therefore, we will begin by studying vectors. Although most students are acquainted with vectors in some form or another, we will reintroduce them in a way that leads naturally to tensors. 1 CHAPTER 1. INTRODUCTION 2 Do not distribute without permission Part I Mathematical Preliminaries 3 Chapter 2 Vectors Some physical quantities are described by scalars, e.g., density, temperature, kinetic energy. These are pure numbers, although they do have dimensions. It would make no physical sense to add a density, with dimensions of mass divided by length cubed, to kinetic energy, with dimensions of mass times length squared divided by time squared. Vectors are mathematical objects that are associated with both a magnitude, described by a number, and a direction. An important property of vectors is that they can be used to represent physical entities such as force, momentum and displacement. Consequently, the meaning of the vector is (in a sense we will make precise) independent of how it is represented. For example, if someone punches you in the nose, this is a physical action that could be described by a force vector. The physical action and its result (a sore nose) are, of course, independent of the particular coordinate system we use to represent the force vector. Hence, the meaning of the vector is not tied to any particular coordinate system or description. A vector u can be represented as a directed line segment, as shown in Figure 2.1. The length of the vector is denoted by u or by u . Multiplying a vector by a positive scalar α changes the length or magnitude| of| the vector but not its orientation. If α>1, the vector αu is longer than u;ifα<1, αu is shorter than u.Ifα is negative, the orientation of the vector is reversed. The addition of two vectors u and v can be written w = u + v (2.1) Although the same symbol is used as for ordinary addition, the meaning here is different. Vectors add according to the parallelogram law shown