Outline Outline 4 Basic Rules 4 Basic Rules 4 Vectors and Tensors - - PowerPoint PPT Presentation

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• G. Ahmadi

ME 637-Particle-II

• G. Ahmadi

ME 637-Particle-II

Outline Outline 4 4Basic Rules Basic Rules 4 4Vectors and Tensors Vectors and Tensors 4 4Tensor Operation Tensor Operation 4 4Isotropic Tensors Isotropic Tensors

• G. Ahmadi

ME 637-Particle-II

(Cartesian Tensor) (Cartesian Tensor) Basic Rules Basic Rules h h A free index appears only once in each A free index appears only once in each term of a tensor equation. The equation term of a tensor equation. The equation then holds for all possible values of that then holds for all possible values of that index. index. h h Summation is implied on an index, which Summation is implied on an index, which appears twice. appears twice. h h No index can appear more than twice in No index can appear more than twice in any term. any term.

• G. Ahmadi

ME 637-Particle-II

j ij * i

x Q x =

* i ij j

x Q x =

1 Q det

ij

± =

jk ik ijQ

Q δ =

ik kj ijQ

Q δ =

x y x* y*

θ

Change Change

• f Frame
• f Frame

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• G. Ahmadi

ME 637-Particle-II

x y x* y*

θ

r

* * * *

y x y x j i j i r + = + = θ + θ = sin cos

*

j i i θ + θ − = cos sin

*

j i j

i j i* j*

θ + θ = sin y cos x x* θ + θ − = cos y sin x y*

• G. Ahmadi

ME 637-Particle-II

[ ]

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ θ θ − θ θ = cos sin sin cos Q

Transformation in Transformation in Two Dimension Two Dimension Kronecker Kronecker Delta Delta

[ ]

⎥ ⎦ ⎤ ⎢ ⎣ ⎡ = δ 1 1

ij

• G. Ahmadi

ME 637-Particle-II

T T* =

v Q v ⋅ =

* T *

Q t Q τ ⋅ ⋅ =

Scalar Scalar Vector Vector Vector Second Order Second Order Tensor Tensor

• G. Ahmadi

ME 637-Particle-II

j ij * i

v Q v =

kl jl ik * ij

t Q Q t =

mnl kl jn im * ijk

Q Q Q λ = λ

Vector Vector Second Second Order Tensor Order Tensor Third Order Third Order Tensor Tensor

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• G. Ahmadi

ME 637-Particle-II

Alternating Alternating Symbol Symbol

ijk

ε

equal are indices two when , n permutatio

• dd

k , j , i for , 1 n permutatio even k , j , i for , 1

ijk ijk ijk

= ε − = ε = ε

• G. Ahmadi

ME 637-Particle-II

Gradient Gradient

i , j i j ij i , i i

v x v ) ( x ) ( = ∂ ∂ = ∇ ϕ = ∂ ϕ ∂ = ϕ ∇ v Divergence Divergence

i , i

v = ⋅ ∇ v

i , ij i ij j

x ) ( τ = ∂ τ ∂ = ⋅ ∇ τ

• G. Ahmadi

ME 637-Particle-II

Curl Curl

j , k ijk j k ijk i

U x U ) ( ε = ∂ ∂ ε = × ∇ U

k 3 j 2 i 1 ijk

A A A det ε = A

Determinant Determinant

km jn kn jm imn ijk

δ δ − δ δ = ε ε Identity Identity Laplacian Laplacian

ii , i i 2 2

x x ϕ = ∂ ∂ ϕ ∂ = ϕ ∇

• G. Ahmadi

ME 637-Particle-II

ij

αδ

Rank Two: Rank Two: Rank Three: Rank Three:

ijk

αε

Rank Zero: Rank Zero: Rank One: Rank One: All Scalars All Scalars All Scalars None None None

4

• G. Ahmadi

ME 637-Particle-II

Rank Four: Rank Four:

) ( ) (

jk il jl ik jk il jl ik kl ij

δ δ − δ δ γ + δ δ + δ δ β + δ αδ

• G. Ahmadi

ME 637-Particle-II

• Basic Rules
• Vectors and Tensors
• Tensor Operation
• Isotropic Tensors
• Basic Rules

Basic Rules

• Vectors and Tensors

Vectors and Tensors

• Tensor Operation

Tensor Operation

• Isotropic Tensors

Isotropic Tensors

• G. Ahmadi

ME 637-Particle-II