Index notation for tensors and vectors

Posted by: Pantelis Liolios | Sept. 17, 2020

Index notation is used extensively in literature when dealing with stresses, strains and constitutive equations. The reason is that it reduces drastically the number of terms in an equation and simplifies the expressions. We will use a right handed Cartesian coordinate system to describe the index notation (cf. Fig. 1). Moreover it is more convenient to name the axes \( x_{1} \), \( x_{2} \) and \( x_{3} \) instead of the more familiar notation \( x \), \( y \), \( z \).

Right handed Cartesian coordinate system — Figure 1: Illustration of a right handed Cartesian coordinate system and an arbitrary vector \( \vec{u} \).

Free Index

In the above coordinate system consider a vector \( \vec{u} \) pointing at point \( P \). This vector may be written as:

\[ u=\left(u_{1},u_{2},u_{3}\right)=u_{1}e_{1}+u_{2}e_{2}+u_{3}e_{3} \]

(1)

where \( \vec{e}_{1} \), \( \vec{e}_{2} \) and \( \vec{e}_{3} \) are the unit vectors of axes \( x_{1} \), \( x_{2} \) and \( x_{3} \), respectively. The vector components \( u_{1} \), \( u_{2} \) and \( u_{3} \) may be abbreviated by using an arbitrary index:

\[ u=\left(u_{1},u_{2},u_{3}\right)=u_{i} \]

(2)

where \( i \) is a freely chosen index taking the values \( 1 \), \( 2 \) and \( 3 \). It must be noted that the letter \( i \) has been chosen arbitrarily. Thus:

\[ u_{i}=u_{j}=u_{m}=\ldots \]

(3)

represent the same vector. A free index must occur precisely once on every term of an expression or equation.

Summation Convention (Dummy Index)

The summation convention (also known as Einstein's convention) helps to reduce the size and complexity of the equations even more. This rule says that whenever an index appears twice in a term then that index is to be summed from \( 1 \) to \( 3 \). For example, consider the dot product of two vectors \( u \) and \( v \):

\[ u\cdot v=u_{1}v_{1}+u_{2}v_{2}+u_{3}v_{3}=\sum_{i=1}^{n}u_{i}v_{i} \]

(4)

The above expression may be written as:

\[ u\cdot v=u_{i}v_{i} \]

(5)

where \( i \) ranges from \( 1 \) to \( 3 \). The summation symbol \( \sum \) is also omitted to reduce furthermore the complexity. Again the index letter \( i \) is freely chosen thus:

\[ u_{i}v_{i}=u_{j}v_{j} \]

(6)

are equal expressions. When an index appears twice in a term is called dummy index. The dummy index may or may not occur precisely twice in each other term of an expression or equation.

Finally, it should be noted that when an index appears thrice or more in a term, it makes no sense and it is considered as error.

Differentiation Notation

In index notation, a comma denotes partial derivative with respect to the index that follows. For example, consider \( \Psi \) to be a scalar quantity. Then:

\[ \Psi ,_{i}=\left(\frac{\partial \Psi}{\partial x_{1}},\frac{\partial \Psi}{\partial x_{2}},\frac{\partial \Psi}{\partial x_{3}}\right) \]

(7)

Kronecker Delta (\( \delta_{ij} \) )

The Kronecker delta (\( \delta_{ij} \) ) is a special matrix in mechanics:

\[ \delta_{ij}=\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right] \]

(8)

Kronecker delta is a useful operator and can be found in many expressions. For example:

\[ \delta_{ij}u_{j}=u_{i} \]

(9)

substitutes index \( j \) with \( i \).

Examples

System of linear equations

Consider the following system of linear equations:

\[ \begin{array}{c}a_{11}x_{1}+a_{12}x_{2}+a_{13}x_{3}=b_{1}\\a_{21}x_{1}+a_{22}x_{2}+a_{23}x_{3}=b_{2}\\a_{31}x_{1}+a_{32}x_{2}+a_{33}x_{3}=b_{3}\end{array} \]

(10)

Firstly the system takes the form:

\[ \begin{array}{c}a_{1j}x_{j}=b_{1}\\a_{2j}x_{j}=b_{2}\\a_{3j}x_{j}=b_{3}\end{array} \]

(11)

and finally:

\[ a_{ij}x_{j}=b_{i} \]

(12)

Stress equilibrium equations

When a body is in equilibrium the components of the stress tensor \( \sigma_{ij} \) in every point of the body must satisfy the equilibrium equations:

\[ \begin{array}{c}\dfrac{\partial \sigma_{11}}{\partial x_{1}}+\dfrac{\partial \sigma_{21}}{\partial x_{2}}+\dfrac{\partial \sigma_{31}}{\partial x_{3}}+f_{1}=0\\ \dfrac{\partial \sigma_{12}}{\partial x_{1}}+\dfrac{\partial \sigma_{22}}{\partial x_{2}}+\dfrac{\partial \sigma_{32}}{\partial x_{3}}+f_{2}=0\\ \dfrac{\partial \sigma_{13}}{\partial x_{1}}+\dfrac{\partial \sigma_{23}}{\partial x_{2}}+\dfrac{\partial \sigma_{33}}{\partial x_{3}}+f_{3}=0\end{array} \]

(13)

where \( f_{1} \), \( f_{2} \) and \( f_{3} \) are the three components of the body forces (e.g. gravity) that act on the body. The above equation firstly takes the form:

\[ \begin{array}{c}\sigma_{j1,j}+f_{1}=0\\ \sigma_{j2,j}+f_{2}=0\\ \sigma_{j3,j}+f_{3}=0\end{array} \]

(14)

and finally:

\[ \sigma_{ji,j}+f_{i}=0 \]

(15)

Tags: algebra | mechanics | tensors