Multimedia course CONTINUUM MECHANICS FOR ENGINEERS By Prof. Xavier Oliver Technical University of Catalonia (UPC/BarcelonaTech) International Center for Numerical Methods in Engineering (CIMNE) http://oliver.rmee.upc.edu/xo First edition May 2017 This material is distributed under the terms of the Creative Commons Attribution Non- Commercial No-Derivatives (CC-BY-NC-ND) License, which permits any noncommercial use, DOI : distribution, and reproduction in any medium of the unmodified original material ,provided the original author(s) and source are credited. 10.13140/RG.2.2.22558.95046
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Acknowledgements Prof. Carlos Agelet (CIMNE/UPC) Dr. Manuel Caicedo (CIMNE/UPC) Dr. Eduardo Car (CIMNE) Prof. Eduardo Chaves (UCLM) Dr. Ester Comella s (CIMNE) Dr. Alex Ferrer (CIMNE/UPC) Prof. Alfredo Hues pe (CIMNE/UNL/UPC) Dr. Oriol Lloberas-Va lls (CIMNE/UPC) Dr. Julio Marti (CIMNE) … and the past students of my courses on Continuum Mechanics … Than anks for you our r con ontri tributi tion !!!! !!!!
TENSOR ALGEBRA Multimedia Course on Continuum Mechanics
Overview Introduction to tensors Lecture 1 Lecture 2 Indicial or (Index) notation Vector Operations Lecture 3 Lecture 4 Tensor Operations Lecture 5 Differential Operators Lecture 6 Integral Theorems Lecture 7 References 2
Introduction SCALAR ρ θ , , ... v VECTOR , , ... v f σ ε MATRIX , , ... ? , ... C 4
Concept of Tensor A TENSOR is an algebraic entity with various components which generalizes the concepts of scalar, vector and matrix. Many physical quantities are mathematically represented as tensors. Tensors are independent of any reference system but, by need, are commonly represented in one by means of their “component matrices”. The components of a tensor will depend on the reference system chosen and will vary with it. 5
Order of a Tensor The order of a tensor is given by the number of indexes needed to specify without ambiguity a component of a tensor. Scalar : zero dimension α = a 3.14 1.2 = , a a Vector : 1 dimension v i 0.3 0.8 0.1 0 1.3 2 nd order : 2 dimensions , A A = 0 2.4 0.5 E ij 3 rd order : 3 dimensions A , A 1.3 0.5 5.8 , A A 4 th order … 6
Cartesian Coordinate System Given an orthonormal basis formed by three mutually perpendicular unit vectors: ⊥ ⊥ ⊥ ˆ ˆ ˆ ˆ ˆ ˆ , , e e e e e e 1 2 2 3 3 1 Where: = = = ˆ ˆ ˆ 1 , 1 , 1 e e e 1 2 3 Note that = if 1 i j ⋅ = = δ ˆ ˆ e e ≠ i j ij if 0 i j 7
Indicial or (Index) Notation Tensor Algebra 10
Tensor Bases – VECTOR A vector can be written as a unique linear combination of the v { } ˆ i i ∈ three vector basis for . e 1,2,3 v = + + ˆ ˆ ˆ v v v v e e e 1 1 2 2 3 3 v 3 In matrix notation: v 1 [ ] = v v 2 v 1 v v 3 2 In index notation: = ∑ ˆ v i v e tensor as a physical entity i i [ ] component i of the tensor in the i = v i v { } i ∈ given basis 1,2,3 11
Tensor Bases – 2 nd ORDER TENSOR A 2 nd order tensor can be written as a unique linear combination A { } ⊗ ≡ i j ∈ of the nine dyads for . ˆ ˆ ˆ ˆ e e e e , 1,2,3 i j i j ( ) ( ) ( ) = ⊗ + ⊗ + ⊗ + ˆ ˆ ˆ ˆ ˆ ˆ A e e e e e e A A A 1 1 1 2 1 3 11 12 13 ( ) ( ) ( ) + ⊗ + ⊗ + ⊗ + ˆ ˆ ˆ ˆ ˆ ˆ A e e A e e A e e 2 1 2 2 2 3 21 22 23 ( ) ( ) ( ) + ⊗ + ⊗ + ⊗ ˆ ˆ ˆ ˆ ˆ ˆ e e e e e e A A A 31 3 1 32 3 2 33 3 3 Alternatively, this could have been written as: = + + + ˆ ˆ ˆ ˆ ˆ ˆ A e e e e e e A A A 11 1 1 12 1 2 13 1 3 + + + + ˆ ˆ ˆ ˆ ˆ ˆ e e e e e e A A A 2 1 2 2 2 3 21 22 23 + + + ˆ ˆ ˆ ˆ ˆ ˆ A e e A e e A e e 3 1 3 2 3 3 31 32 33 12
Tensor Bases – 2 nd ORDER TENSOR ( ) ( ) ( ) = ⊗ + ⊗ + ⊗ + ˆ ˆ ˆ ˆ ˆ ˆ A A e e A e e A e e 1 1 1 2 1 3 11 12 13 ( ) ( ) ( ) + ⊗ + ⊗ + ⊗ + ˆ ˆ ˆ ˆ ˆ ˆ e e e e e e A A A 2 1 2 2 2 3 21 22 23 ( ) ( ) ( ) + ⊗ + ⊗ + ⊗ ˆ ˆ ˆ ˆ ˆ ˆ e e e e e e A A A 3 1 3 2 3 3 31 32 33 In matrix notation: A A A 11 12 13 [ ] = A A A A 21 22 23 A A A 31 32 33 In index notation: ∑ ( ) tensor as a = ⊗ ˆ ˆ A ij A e e physical entity i j ij [ ] component ij of the tensor = A A { } i j ∈ , 1,2,3 in the given basis ij ij 13
Tensor Bases – 3 rd ORDER TENSOR A 3 rd order tensor can be written as a unique linear combination A { } ⊗ ⊗ ≡ i j k ∈ of the 27 tryads for . ˆ ˆ ˆ ˆ ˆ ˆ , , 1,2,3 e e e e e e i j k i j k ( ) ( ) ( ) = ⊗ ⊗ + ⊗ ⊗ + ⊗ ⊗ + ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ A A A A e e e e e e e e e 1 1 1 1 2 1 1 3 1 111 121 131 ( ) ( ) ( ) + ⊗ ⊗ + ⊗ ⊗ + ⊗ ⊗ + ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ A A A e e e e e e e e e 2 1 1 2 2 1 2 3 1 211 221 231 ( ) ( ) ( ) + ⊗ ⊗ + ⊗ ⊗ + ⊗ ⊗ + ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ A A A e e e e e e e e e 3 1 1 3 2 1 3 3 1 311 321 331 ( ) ( ) + ⊗ ⊗ + ⊗ ⊗ + ˆ ˆ ˆ ˆ ˆ ˆ ... A e e e A e e e 1 1 2 1 2 2 112 122 Alternatively, this could have been written as: = + + + ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ A A e e e A e e e A e e e 111 1 1 1 121 1 2 1 131 1 3 1 + + + + ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ A e e e A e e e A e e e 2 1 1 2 2 1 2 3 1 211 221 231 + + + + ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ A A A e e e e e e e e e 3 1 1 3 2 1 3 3 1 311 321 331 + + + ˆ ˆ ˆ ˆ ˆ ˆ ... A e e e A e e e 1 1 2 1 2 2 112 122 14
Tensor Bases – 3 rd ORDER TENSOR ( ) ( ) ( ) = ⊗ ⊗ + ⊗ ⊗ + ⊗ ⊗ + ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ A A e e e A e e e A e e e 111 1 1 1 121 1 2 1 131 1 3 1 ( ) ( ) ( ) + ⊗ ⊗ + ⊗ ⊗ + ⊗ ⊗ + ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ A A A e e e e e e e e e 2 1 1 2 2 1 2 3 1 211 221 231 ( ) ( ) ( ) + ⊗ ⊗ + ⊗ ⊗ + ⊗ ⊗ + ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ A A A e e e e e e e e e 3 1 1 3 2 1 3 3 1 311 321 331 ( ) ( ) + ⊗ ⊗ + ⊗ ⊗ + ˆ ˆ ˆ ˆ ˆ ˆ A A ... e e e e e e 1 1 2 1 2 2 112 122 In matrix notation: A A A 113 123 133 A A A A A A 112 122 132 111 121 131 A A A 213 223 233 A A A [ ] = A A A 212 222 232 A 211 221 231 A A A 313 323 333 A A A A A A 312 322 332 311 321 331 15
Tensor Bases – 3 rd ORDER TENSOR ( ) ( ) ( ) = ⊗ ⊗ + ⊗ ⊗ + ⊗ ⊗ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ A A A A e e e e e e e e e 1 1 1 1 2 1 1 3 1 111 121 131 ( ) ( ) ( ) + ⊗ ⊗ + ⊗ ⊗ + ⊗ ⊗ + ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ A A A e e e e e e e e e 2 1 1 2 2 1 2 3 1 211 221 231 ( ) ( ) ( ) + ⊗ ⊗ + ⊗ ⊗ + ⊗ ⊗ + ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ A A A e e e e e e e e e 3 1 1 3 2 1 3 3 1 311 321 331 ( ) ( ) + ⊗ ⊗ + ⊗ ⊗ + ˆ ˆ ˆ ˆ ˆ ˆ A A ... e e e e e e 1 1 2 1 2 2 112 122 In index notation: ( ) ∑ = ⊗ ⊗ = ˆ ˆ ˆ A A e e e ijk i j k ijk ( ) tensor as a = ⊗ ⊗ ≡ ˆ ˆ ˆ ˆ ˆ ˆ A A e e e e e e physical entity ijk i j k ijk i j k [ ] ijk = component ijk of the tensor A A { } ijk i j k ∈ in the given basis , , 1,2,3 16
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