I. Vectors and Tensors

I. Vectors and Tensors

This section introduces the essential mathematical framework for continuum mechanics, tensor algebra, to describe behavior of material

A. Tensor Notation

Continuum mechanics seeks to provide a fundamental model for material response that does not depend on irrelevant details of a particular coordinate system
→write the theory in terms of variables that are unaffected by such changes
⇒Tensors ( or Tensor Fields )

1. Direct vs. Indicial Notation

• Tensors represent physical properties such as mass, velocity, and stress that do not depend on coordinate sys.
  →such notation ∃ and is called direct ( invariant ) notation
  →No indicies are attached to tensor symbol
• To perform operations on tensors, they must always be projected onto particular coordinate sys. →Indicial notation
• In some cases, operations defined by indicial notation can also be written using matrix ( m.x. ) notation.

$$ \text{Vector: } \underline{u} = \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix} \text{ (First-order tensor)} $$
$$ \text{2nd-Order Tensor: } \underline{T} = \begin{bmatrix} \sigma_{11} & \sigma_{12} & \sigma_{13} \newline \sigma_{21} & \sigma_{22} & \sigma_{23} \newline \sigma_{31} & \sigma_{32} & \sigma_{33} \end{bmatrix} $$

2. Summation Convention, Dummy, and Free Indices

$$ S = a_1 x_1 + \cdots + a_{n} x_{n} = \sum_{i=1}^{n} a_i x_i $$
Simplified Notation : A repeated twice index implies summation.
$$ S = a_i x_i = a_j x_j $$
where indices $i, j$ are dummy indices — the choice of index is irrelevant
→introduced by Albert Einstein in his famous 1916 paper on general relativity
⇒Einstein's Summation Convention

Ex. For $\ i, j = 1, 2, 3$
$$ A_{ij}x_{i}y_{j} = A_{11}x_{1}y_{1} + A_{12}x_{1}y_{2} + A_{13}x_{1}y_{3} + \cdots + A_{33}x_{3}y_{3} $$

An index that appears only once ⇒ free index
Ex. $A_{ij}x_{j} = b_{i}$ (Index $i$ is the free index, holding for each value of $i$)

3. Matrix Notation

• Indicial operations involving tensors of rank 2 or less can be represented as m.x. operations
  $$ A_{ij} x_j = \underline{A}\underline{x} = \begin{bmatrix}A_{11} & A_{12} & A_{13} \newline A_{21} & A_{22} & A_{23} \newline A_{31} & A_{32} & A_{33}\end{bmatrix} \begin{bmatrix}x_{1} \newline x_{2} \newline x_{3}\end{bmatrix} $$
  $$ A_{ji} x_j = \underline{A}^{T}\underline{x} $$
  $$ a_i x_i = \underline{a}^T \underline{x} = \begin{bmatrix}a_{1} & a_{2} & a_{3}\end{bmatrix} \begin{bmatrix}x_{1} \newline x_{2} \newline x_{3}\end{bmatrix} $$
• Another m.x. operation:
  $$ \mathrm{tr}(\underline{A}) = A_{ii} = A_{11} + A_{22} + A_{33} $$

4. Kronecker Delta ($\delta_{ij}$) and Permutation Symbol ($\epsilon_{ijk}$)

• Kronecker Delta ($\delta_{ij}$): It represents the identity m.x.
  $$ \delta_{ij} = \begin{cases} 1 & \text{if } i = j \newline 0 & \text{if } i \neq j \end{cases} \quad \rightarrow \quad \text{Identity m.x. } \underline{I} = \begin{bmatrix} 1 & & \newline & 1 & \newline & & 1 \end{bmatrix} $$
• An important property of $\delta_{ij}$ is index substitution ( or contraction ):
  $$ a_j = a_i \delta_{ij},\ \delta_{ij} \delta_{jk} = \delta_{ik} $$
• Permutation Symbol ($\epsilon_{ijk}$):
  $$ \epsilon_{ijk} = \begin{cases} 1 & \text{if } i, j, k \text{ form an even permutation of } 1, 2, 3 \newline -1 & \text{if } i, j, k \text{ form an odd permutation of } 1, 2, 3 \newline 0 & \text{if any two indices are the same (do not form a permutation)} \end{cases} $$
  Ex. $\epsilon_{123} = \epsilon_{231} = \epsilon_{312} = 1$; $\epsilon_{321} = \epsilon_{213} = \epsilon_{132} = -1$
  
  Interchanging any two indices changes the sign (e.g., $\epsilon_{\dots r \dots s \dots} = - \epsilon_{\dots s \dots r \dots}$). If $k$ order changes are made, $\epsilon_{\sigma'} = (-1)^k \epsilon_{\sigma}$ where $\sigma$ is arbitrary permutation.

• Properties of the permutation symbol
  ① useful identities : $\epsilon_{ijk} \delta_{ij} = \epsilon_{iik} = 0$, $\epsilon_{ijk} \epsilon_{mjk} = 2 \delta_{im}$, $\epsilon_{ijk} \epsilon_{ijk} = 6$
  ② provides expression for determinant ( det ) of m.x.
  $\quad \epsilon_{mnp}\det \underline{A} = \epsilon_{ijk} A_{im} A_{jn} A_{kp} = \epsilon_{ijk} A_{mi} A_{nj} A_{pk}$
  $\quad$→ demonstrates that $\det \underline{A} = \det \underline{A}^{T}$
  $\quad$→ multiply by $\frac{1}{6}\epsilon_{mnp}$: $\det \underline{A} = \frac{1}{6}\epsilon_{ijk}\epsilon_{mnp} A_{mi} A_{nj} A_{pk} = \epsilon_{ijk} A_{i1} A_{j2} A_{k3} $
  ③ $\frac{\partial \det \underline{A}}{\partial A_{ij}} = \det \underline{A} \ \underline{A}^{-T} = C$ ($C$ is the cofactor of $\underline{A}$)
  ④ $\epsilon-\delta$ Identity: $\epsilon_{ijk} \epsilon_{mnk} = \delta_{im} \delta_{jn} - \delta_{in} \delta_{jm}$