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 , and instead of the more familiar notation , , .
In the above coordinate system consider a vector pointing at point . This vector may be written as:
where , and are the unit vectors of axes , and , respectively. The vector components , and may be abbreviated by using an arbitrary index:
where is a freely chosen index taking the values 1, 2 and 3. It must be noted that the letter has been chosen arbitrarily. Thus:
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 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 and :
The above expression may be written as:
where ranges from 1 to 3. The summation symbol is also omitted to reduce furthermore the complexity. Again the index letter is freely chosen thus:
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.
In index notation, a comma denotes partial derivative with respect to the index that follows. For example, consider to be a scalar quantity. Then:
Kronecker Delta ()
The Kronecker delta () is a special matrix in mechanics:
Kronecker delta is a useful operator and can be found in many expressions. For example:
substitutes index with .
System of linear equations
Consider the following system of linear equations:
Firstly the system takes the form:
Stress equilibrium equations
When a body is in equilibrium the components of the stress tensor in every point of the body must satisfy the equilibrium equations:
where , and are the three components of the body forces that act on the body. The above equation firstly takes the form:
S.P. Timoshenko and J.N. Goodier. Theory of Elasticity. McGraw-Hill, New York, 3rd ed., 1970.
Y.C. Fung. A First Course in Continuum Mechanics. Prentice Hall, Englewood Cliffs, New Jersey, 3rd ed., 1994.
L.E. Malvern. Introduction to the Mechanics of a Continuous Medium. Prentice Hall, Englewood Cliffs, New Jersey, 1969.