Lecture 4 Introduction To Vectors And Tensors

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Spring, 2015ME 612 – Continuum MechanicsLecture 4Introduction to vectors and tensorsInstructor: Prof. Marcial Gonzalez

Lecture 4 – Introduction to tensors and vectorsdeformedconfigurationKINEMATICS OF DEFORMATIONSthermo-mechanical loadslaws of nature.CONSERVATION OF MASSBALANCE OF LINEAR MOMENTUMBALANCE OF ANGULAR MOMENTUMLAWS OF ATensor algebraTensor analysis16 unknown fields 5 equationscontinuously varying fields(time and space averages overthe underlying structure)atomic/micro/mesostructureis 11 equationsEmpiricalobservationExperimentalmechanics andthermodynamics2

Lecture 4 – Introduction to tensors and vectorsReview (tensor analysis in Cartesian coordinates)DIY1-order tensor (vector)2-order tensorSymmetric, positive-definite 2-order tensor,3

Tensor analysisTensor fields- In continuum mechanics we encounter tensors as spatially andtemporally varying fields over a give domain:- Partial differentiation of a tensor field (and the comma notation)DIYComma notation and summation convention4

Tensor analysis (in Cartesian coordinates)Tensor fields- GradientFrom tensor of rank mto tensor of rank m 1DIY5

Tensor analysis (in Cartesian coordinates)Tensor fields- CurlDIY6

Tensor analysis (in Cartesian coordinates)Tensor fields- DivergenceFrom tensor of rank mto tensor of rank m-1DIY7

Tensor analysis (in Cartesian coordinates)Divergence theorem- Given a vector fieldDIYCartesian components- Given a tensor fieldDIYCartesian components8

Curvilinear coordinate systemsCurvilinear coordinate systems- Two set of basis vectors at each position in spacetangent vectorsdefined throughreciprocal vectors- Contravariant components- Covariant components- Connection betweencomponentswith9

Curvilinear coordinate systemsCurvilinear coordinate systems- Two set of basis vectors at each position in spacetangent vectorsdefined throughreciprocal vectors- Covariant and contravariant components:10

Curvilinear coordinate systemsCurvilinear coordinate systems- Two set of basis vectorstangent vectorsreciprocal vectorsdefined through- Covariant, contravariant and mixed components:- Connection between components:- Metric tensor (it is the identity of a generalized coordinate system):(not equal to 1 in general) and symmetric ()11

Tensor algebra (in curvilinear coordinates)Inner product - Dot productCross productShaded areas are related by the determinant of the metric tensor (and are different in general).12

Tensor algebra (in curvilinear coordinates)TransposeContracted multiplicationScalar contractione.g.,13

Curvilinear coordinate systemsConstruction of tangent basis- Tangent vectors describe how the point in space changes as thecoordinates changewhereare the curvilinear coordinateswhereandare the Cartesiancoordinates and basis, resp.[usuallyare known]Inversely:with- Derivatives of tangent vectors:Christoffel symbol of the second kind14

Curvilinear coordinate systemsConstruction of metric tensor- By definitionNote:- The metric of the space is the following scalar invariant quadratic form(i.e., the metric is the elementary line element or arc length): . elementary volume element:15

Curvilinear coordinate systemsConstruction of reciprocal basis- Reciprocal vectors describe the coordinates change as the point inspace changesNotice that- Derivatives of reciprocal vectors:Christoffel symbol of the second kindNote 1:Note 2: Christoffel symbols are zero in Cartesian coordinates16

Tensor analysis (in curvilinear coordinates)Tensor fields- GradientCovariant derivative of a covariant componentCovariant derivate of a contravariant component17

Tensor analysis (in curvilinear coordinates)Tensor fields- Divergence18

Orthogonal curvilinear coordinatesPolar cylindrical coordinatesBasis vectorsare notunit vectors.Most of theChristoffel symbolsare zero.Only diagonalcomponentsare non-zero.19

Orthogonal curvilinear coordinatesSpherical coordinatesBasis vectorsare notunit vectors.Most of theChristoffel symbolsare zero.Only diagonalcomponentsare non-zero.20

Lecture 4 – Introduction to tensors and vectorsAny questions?21

Introduction to vectors and tensors Instructor: Prof. Marcial Gonzalez Spring, 2015 ME 612 -Continuum Mechanics. Lecture 4 -Introduction to tensors and vectors . (vector) 2-order tensor Symmetric, positive-definite 2-order tensor , 4 Tensor analysis Tensor fields-In continuum mechanics we encounter tensors as spatially and .

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