|
Appendix C
Index
Notation for Vector and Tensor Operations
Operations
on Cartesian components of vectors and tensors may be expressed very
efficiently and clearly using index
notation.
C.1. Vector and tensor components.
Let
x be a (three dimensional) vector
and let S be a second order
tensor.  Let   be a Cartesian basis. Denote the components
of x in this basis by ,
and denote the components of S by
Using
index notation, we would express x
and S as
C.2.
Conventions and special symbols for index notation
 Â Range
Convention: Lower case Latin subscripts (i, j, k…) have the range . The symbol  denotes three components of a vector  and . The symbol  denotes nine components of a second order
tensor,
 Summation
convention (Einstein convention): If an index is repeated in a product of
vectors or tensors, summation is implied over the repeated index. Thus
In
the last two equations, ,
 and  denote the  component matrices of A, B and C.
 The
Kronecker Delta: The symbol  is known as the Kronecker delta, and has the
properties
thus
You can also think of  as the components of the identity tensor, or
a  identity matrix. Observe the following useful results
 The
Permutation Symbol: The symbol  has properties
thus
Note that
C.3. Rules of index
notation
1. The same index (subscript) may
not appear more than twice in a product of two (or more) vectors or
tensors. Thus
are valid, but
are meaningless
2. Free indices on each term of an
equation must agree. Thus
are valid, but
are meaningless.
3. Free and
dummy indices may be changed without altering the meaning of an expression,
provided that rules 1 and 2 are not violated. Thus
C.4. Vector operations expressed using
index notation
 Addition. Â
 Dot
ProductÂ
 Vector
Product
 Dyadic
Product Â
 Change
of Basis. Let a be a vector. Let  be a Cartesian basis, and denote the
components of a in this basis by . Let  be a second basis, and denote the components
of a in this basis by . Then, define
where  denotes the angle between the unit vectors   and . Then
C.5. Tensor
operations expressed using index notation.
 Addition. Â
 TransposeÂ
 Scalar
Products
 Product
of a tensor and a vector
 Product
of two tensorsÂ
 Determinant
 Change
of Basis. Let A be a second order tensor. Let  be a Cartesian basis, and denote the
components of A in this basis by . Let  be a second basis, and denote the components
of A in this basis by . Then, define
where  denotes the angle between the unit vectors   and . Then
C.6. Calculus using index notation
The
derivative  can be deduced by noting that  and . Therefore
                                                                Â
The
same argument can be used for higher order tensors
                                                             Â
C.7. Examples of algebraic
manipulations using index notation
1. Let a, b, c, d be
vectors. Prove that
Express
the left hand side of the equation using index notation (check the rules for
cross products and dot products of vectors to see how this is done)
Recall
the identity
so
Multiply
out, and note that
(multiplying
by a Kronecker delta has the effect of switching indices…) so
Finally,
note that
and
similarly for other products with the same index, so that
2. The stressstrain
relation for linear elasticity may be expressed as
where  and  are the components of the stress and strain
tensor, and  and  denote Young’s modulus and Poisson’s
ratio. Find an expression for strain
in terms of stress.
Set i=j to see
that
Recall that ,
and notice that we can replace the remaining ii by kk
Now, substitute for  in the given stressstrain
relation
3. Solve the equation
for
 in terms of  and
Multiply
both sides by  to see that
Substitute
back into the equation given for  to see that
4. Let . Calculate
We
can just apply the usual chain and product rules of differentiation
5. Let . Calculate
Using
the product rule
|
