Fundamentals of Continuum Mechanics Nov 25 2013

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1、FundamentalsofContinuumMechanicsJohnW.RudnickiDepartmentofCivilandEnvironmentalEngineeringandDepartmentofMechanicalEngineering,NorthwesternUniversity,Evanston,ILPrinted:November25,2013CONTENTSIntroductionviiPartOneMathematicalPreliminaries11Vectors3Exercises7References92Tensors112.1Inverse142

2、.2OrthogonalTensor142.3PrincipalValues152.4NthorderTensors16Exercises163CartesianCoordinates193.1BaseVectors203.2SummationConvention213.3TensorComponents233.4Dyads243.5TensorandScalarProducts25Exercises26References274Vector(Cross)Product294.1PropertiesoftheCross-Product314.2TripleScalarProduc

3、t314.3TripleVectorProduct324.4ApplicationsoftheCrossProduct324.4.1VelocityDuetoRigidBodyRotation324.4.2MomentofaForcePaboutO33Exercises345Determinants375.1Cofactor395.2Inverse39Exercises406ChangeofOrthonormalBasis436.1ChangeofVectorComponents456.2Definitionofavector466.3ChangeofTensorComponent

4、s466.4IsotropicTensors48Exercises49References517PrincipalValuesandPrincipalDirections537.1Example55Exercises578Gradient598.1Example:CylindricalCoordinates62Exercises63PartTwoStress679TractionandStressTensor699.1TypesofForces709.2TractiononDifferentSurfaces729.3TractiononanArbitraryPlane(Cauch

5、ytetrahedron)749.4Symmetryofthestresstensor75References7510PrincipalValuesofStress7710.1DeviatoricStress79Exercises8011StationaryValuesofShearTraction81Exercises8512Mohr'sCircle87Exercises91References92PartThreeMotionandDeformation9313CurrentandReferenceConfigurations95Exercises10014RateofDefo

6、rmation10314.1Velocitygradients10414.2MeaningofD10514.3MeaningofW106Exercises10715GeometricMeasuresofDeformation10915.1DeformationGradient11015.2ChangeinLengthofLines11015.3ChangeinAngles11215.4ChangeinArea11215.5ChangeinVolume11315.6PolarDecomposition114Exercises115References11716StrainTenso

7、rs11916.1MaterialStrainTensors12016.2SpatialStrainMeasures12116.3RelationsbetweenDandratesofEGandU12216.3.1RelationBetweenE˙andD12216.3.2RelationBetweenDandU˙123Exercises124References12517LinearizedDisplacementGradients12717.1LinearizedGeomet