Nonlinear finite elements/Homework 7/Hints

Hints 1: Index notation

Index notation:

σ_{i j} = 2 μ ε_{i j} + λ ε_{k k} δ_{i j} .

If $j = i$

\begin{matrix} σ_{i i} & = 2 μ ε_{i i} + λ ε_{k k} δ_{i i} \\ = 2 μ ε_{k k} + 3 λ ε_{k k} \\ = (2 μ + 3 λ) ε_{k k} \end{matrix}

σ_{k k} = (2 μ + 3 λ) ε_{k k}

Dummy indices are replaceable.

Hint 2: Index notation

Index notation:

σ_{i j} = 2 μ ε_{i j} + λ ε_{k k} δ_{i j} .

Multiply by $δ_{i j}$ :

\begin{matrix} σ_{i j} δ_{i j} & = 2 μ ε_{i j} δ_{i j} + λ ε_{k k} δ_{i j} δ_{i j} \\ ⟹ σ_{j j} & = 2 μ ε_{i i} + λ ε_{k k} δ_{i i} \\ ⟹ σ_{k k} & = 2 μ ε_{k k} + 3 λ ε_{k k} \\ ⟹ σ_{k k} & = (2 μ + 3 λ) ε_{k k} \end{matrix}

Multiplication by $δ_{i j}$ leads to replacement of one index.

A_{i j} δ_{k l} = ? A_{i j} δ j l = ?

Hint 3: Index notation

Index notation:

\begin{matrix} σ & = σ_{i j} 𝐞_{i} \otimes 𝐞_{j} \\ ε & = ε_{i j} 𝐞_{i} \otimes 𝐞_{j} \end{matrix}

From the definition of dyadic product, we can show

\begin{matrix} (𝐚 \otimes 𝐛) : (𝐮 \otimes 𝐯) & = (𝐚 ∙ 𝐮) (𝐛 ∙ 𝐯) \end{matrix}

Contraction gives:

\begin{matrix} σ : ε & = (σ_{i j} 𝐞_{i} \otimes 𝐞_{j}) : (ε_{k l} 𝐞_{k} \otimes 𝐞_{l}) \\ = σ_{i j} ε_{k l} (𝐞_{i} \otimes 𝐞_{j}) : (𝐞_{k} \otimes 𝐞_{l}) \\ = σ_{i j} ε_{k l} (𝐞_{i} ∙ 𝐞_{k}) (𝐞_{j} ∙ 𝐞_{l}) \\ = σ_{i j} ε_{k l} δ_{i k} δ_{j l} \\ = σ_{i j} ε_{i j} \end{matrix}

Hint 4: Tensor product

Index notation:

\begin{matrix} ε & = ε_{i j} 𝐞_{i} \otimes 𝐞_{j} \\ 𝖢 & = C_{i j k l} 𝐞_{i} \otimes 𝐞_{j} \otimes 𝐞_{k} \otimes 𝐞_{l} \end{matrix}

Definition of dyadics products:

\begin{matrix} (𝐚 ∙ 𝐛) \otimes 𝐱 & = (𝐛 ∙ 𝐱) 𝐚 \\ (𝐚 ∙ 𝐛 \otimes 𝐜) \otimes 𝐱 & = (𝐜 ∙ 𝐱) (𝐚 \otimes 𝐛) \\ (𝐚 ∙ 𝐛 \otimes 𝐜 \otimes 𝐝) \otimes 𝐱 & = (𝐝 ∙ 𝐱) (𝐚 \otimes 𝐛 \otimes 𝐜) \end{matrix}

We can show that

\begin{matrix} (𝐚 \otimes 𝐛 \otimes 𝐜 \otimes 𝐝) : (𝐮 \otimes 𝐯) & = ((𝐚 ∙ 𝐛 \otimes 𝐜 \otimes 𝐝) \otimes 𝐯) ∙ 𝐮 \end{matrix}

Contraction gives:

\begin{matrix} 𝖢 : ε & = (C_{i j k l} 𝐞_{i} \otimes 𝐞_{j} \otimes 𝐞_{k} \otimes 𝐞_{l}) : (ε_{m n} 𝐞_{m} \otimes 𝐞_{n}) \\ = C_{i j k l} ε_{m n} (𝐞_{i} \otimes 𝐞_{j} \otimes 𝐞_{k} \otimes 𝐞_{l}) : (𝐞_{m} \otimes 𝐞_{n}) \\ = C_{i j k l} ε_{m n} ((𝐞_{i} ∙ 𝐞_{i} \otimes 𝐞_{k} \otimes 𝐞_{l}) \otimes 𝐞_{n}) ∙ 𝐞_{m} \\ = C_{i j k l} ε_{m n} (𝐞_{l} ∙ 𝐞_{n}) (𝐞_{i} \otimes 𝐞_{j} \otimes 𝐞_{k}) ∙ 𝐞_{m} \\ = C_{i j k l} ε_{m n} δ_{l n} (𝐞_{k} ∙ 𝐞_{m}) (𝐞_{i} \otimes 𝐞_{j}) = C_{i j k l} ε_{m n} δ_{l n} δ_{k m} 𝐞_{i} \otimes 𝐞_{j} \\ = C_{i j k l} ε_{k l} \end{matrix}

Hint 5 : Tensor product

Tensor Product of two tensors:

\begin{matrix} 𝑨 & = A_{i j} 𝐞_{i} \otimes 𝐞_{j} \\ 𝑩 & = B_{k l} 𝐞_{k} \otimes 𝐞_{l} \end{matrix}

Tensor product:

\begin{matrix} 𝑨 \otimes 𝑩 & = (A_{i j} 𝐞_{i} \otimes 𝐞_{j}) \otimes (B_{k l} 𝐞_{k} \otimes 𝐞_{l}) \\ = A_{i j} B_{k l} 𝐞_{i} \otimes 𝐞_{j} \otimes 𝐞_{k} \otimes 𝐞_{l} \end{matrix}

Hint 6: Vector transformations

Change of basis: Vector transformation rule

v_{i}^{'} = L_{i j} v_{j}

$L_{i j}$ are the direction cosines.

\begin{matrix} L_{11} & = 𝐞_{1}^{'} ∙ 𝐞_{1}; & L_{12} & = 𝐞_{1}^{'} ∙ 𝐞_{2}; & L_{13} & = 𝐞_{1}^{'} ∙ 𝐞_{3} \\ L_{21} & = 𝐞_{2}^{'} ∙ 𝐞_{1}; & L_{22} & = 𝐞_{2}^{'} ∙ 𝐞_{2}; & L_{23} & = 𝐞_{2}^{'} ∙ 𝐞_{3} \\ L_{31} & = 𝐞_{3}^{'} ∙ 𝐞_{1}; & L_{32} & = 𝐞_{3}^{'} ∙ 𝐞_{2}; & L_{33} & = 𝐞_{3}^{'} ∙ 𝐞_{3} \end{matrix}

In matrix form

𝐯^{'} = 𝐋 𝐯; 𝐯 = 𝐋^{T} 𝐯^{'}; 𝐋 𝐋^{T} = 𝐈 ⟹ 𝐋^{T} = 𝐋^{- 1}

Other common form: Vector transformation rule

v_{i}^{'} = Q_{j i} v_{j}

\begin{matrix} Q_{11} & = 𝐞_{1} ∙ 𝐞_{1}^{'}; & Q_{12} & = 𝐞_{1} ∙ 𝐞_{2}^{'}; & Q_{13} & = 𝐞_{1} ∙ 𝐞_{3}^{'} \\ Q_{21} & = 𝐞_{2} ∙ 𝐞_{1}^{'}; & Q_{22} & = 𝐞_{2} ∙ 𝐞_{2}^{'}; & Q_{23} & = 𝐞_{2} ∙ 𝐞_{3}^{'} \\ Q_{31} & = 𝐞_{3} ∙ 𝐞_{1}^{'}; & Q_{32} & = 𝐞_{3} ∙ 𝐞_{2}^{'}; & Q_{33} & = 𝐞_{3} ∙ 𝐞_{3}^{'} \end{matrix}

In matrix form

𝐯^{'} = 𝐐^{T} 𝐯; 𝐯 = 𝐐 𝐯^{'}; 𝐐 𝐐^{T} = 𝐈 ⟹ 𝐐^{T} = 𝐐^{- 1}

Hint 7: Tensor transformations

Change of basis: Tensor transformation rule

T_{i j}^{'} = L_{i p} L_{j q} T_{p q}

where $L_{i j}$ are the direction cosines.

In matrix form,

𝐓^{'} = 𝐋 𝐓 𝐋^{T}

Other common form

T_{i j}^{'} = Q_{p i} Q_{q j} T_{p q}

In matrix form,

𝐓^{'} = 𝐐^{T} 𝐓 𝐐

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Nonlinear finite elements/Homework 7/Hints

Contents

Hints 1: Index notation

Hint 2: Index notation

Hint 3: Index notation

Hint 4: Tensor product

Hint 5 : Tensor product

Hint 6: Vector transformations

Hint 7: Tensor transformations

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Nonlinear finite elements/Homework 7/Hints

Hints 1: Index notation

Hint 2: Index notation

Hint 3: Index notation

Hint 4: Tensor product

Hint 5 : Tensor product

Hint 6: Vector transformations

Hint 7: Tensor transformations

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