Page 54 - Intro to Tensor Calculus

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gives the dyad
1 1 1 2 1 3
ab =a 1 b 1 E E + a 1 b 2 E E + a 1 b 3E E
2 1 2 2 2 3
a 2 b 1 E E + a 2 b 2 E E + a 2 b 3E E
3 1 3 2 3 3
a 3 b 1 E E + a 3 b 2 E E + a 3 b 3E E .
In general, a dyad can be represented

i
A = A ij E E j i, j =1,... ,N

where the summation convention is in effect for the repeated indices. The coefficients A ij are called the
coefficients of the dyad. When the coefficients are written as an N × N array it is called a matrix. Every
second order tensor can be written as a linear combination of dyads. The dyads form a basis for the second
3
1
2
3
1
1
order tensors. As the example above illustrates, the nine dyads {E E , E E ,... , E E }, associated with
the outer products of three dimensional base vectors, constitute a basis for the second order tensor A = ab
having the components A ij = a i b j with i, j =1, 2, 3. Similarly, a triad has the form
j
i
T = T ijk E E E k Sum on repeated indices
k
i
j
where i, j, k have the range 1, 2,... ,N. The set of outer or direct products { E E E },with i, j, k =1,... ,N
constitutes a basis for all third order tensors. Tensor components with mixed suffixes like C i are associated
jk
with triad basis of the form
j
i
C = C E i E E k
jk
where i, j, k have the range 1, 2,...N. Dyads are associated with the outer product of two vectors, while triads,
tetrads,... are associated with higher-order outer products. These higher-order outer or direct products are
referred to as polyads.
The polyad notation is a generalization of the vector notation. The subject of how polyad components
transform between coordinate systems is the subject of tensor calculus.

i
In Cartesian coordinates we have E = E i = b e i and a dyadic with components called dyads is written
A = A ij b e i b e j or
A =A 11 b e 1 b e 1 + A 12 b e 1 b e 2 + A 13 b e 1 b e 3
A 21 b e 2 b e 1 + A 22 b e 2 b e 2 + A 23 b e 2 b e 3

A 31 b e 3 b e 1 + A 32 b e 3 b e 2 + A 33 b e 3 b e 3
where the terms b e i b e j are called unit dyads. Note that a dyadic has nine components as compared with a
vector which has only three components. The conjugate dyadic A c is defined by a transposition of the unit
vectors in A,to obtain
A c =A 11 b e 1 b e 1 + A 12 b e 2 b e 1 + A 13 b e 3 b e 1

A 21 b e 1 b e 2 + A 22 b e 2 b e 2 + A 23 b e 3 b e 2
A 31 b e 1 b e 3 + A 32 b e 2 b e 3 + A 33 b e 3 b e 3