2.3 Dyadics and Tensors
27
A much easier way to do this is to use the summation convention, i.e.,
ab = (a i e i )(b j e j ) = a i b j e i e j ,
(2.13)
which is equivalent to Eq. (2.12). Setting a i b j = T ij gives
ab = T ij e i e j ,
(2.14)
where the T ij can be identified as the scalar components of ab with respect
to the dyadic basis {e i e j }. In 3D space, therefore, a dyad contains nine scalar
components. Like a vector, these components depend on the particular coordinate
system, although the dyad itself does not. It is easy to show that these statements
apply to dyadics as well.
2.3.2 Tensors
A second-order tensor is defined as a linear vector function that transforms a vector
into another vector. The vector function F is linear if F(φa) = φF(a) and F(a+b) =
F(a) + F(b), where a and b are arbitrary vectors and φ is a scalar. The equation
ab · (c + φd) = a(b · c) + φ a(b · d) = [(b · c) + φ(b · d)]a
(2.15)
shows that a dyad satisfies these requirements, with the transformation operation
defined by the dot product. Here the dyad ab transforms the vector c + φd into the
vector [(b · c) + φ(b · d)]a. Hence, a dyad (as well as a dyadic) is a second-order
tensor.
More generally, any second-order tensor can be expressed in dyadic form. Like
writing a vector in terms of scalar components relative to the orthogonal unit basis
{e i }, we can write the tensor T in the form
T = T ij e i e j ,
(2.16)
where the T ij are scalar tensor components relative to the dyadic basis {e i e j }. 10
10 In this chapter, lowercase bold letters denote first-order tensors (vectors), and uppercase bold
letters denote second-order tensors (dyadics).
27
A much easier way to do this is to use the summation convention, i.e.,
ab = (a i e i )(b j e j ) = a i b j e i e j ,
(2.13)
which is equivalent to Eq. (2.12). Setting a i b j = T ij gives
ab = T ij e i e j ,
(2.14)
where the T ij can be identified as the scalar components of ab with respect
to the dyadic basis {e i e j }. In 3D space, therefore, a dyad contains nine scalar
components. Like a vector, these components depend on the particular coordinate
system, although the dyad itself does not. It is easy to show that these statements
apply to dyadics as well.
2.3.2 Tensors
A second-order tensor is defined as a linear vector function that transforms a vector
into another vector. The vector function F is linear if F(φa) = φF(a) and F(a+b) =
F(a) + F(b), where a and b are arbitrary vectors and φ is a scalar. The equation
ab · (c + φd) = a(b · c) + φ a(b · d) = [(b · c) + φ(b · d)]a
(2.15)
shows that a dyad satisfies these requirements, with the transformation operation
defined by the dot product. Here the dyad ab transforms the vector c + φd into the
vector [(b · c) + φ(b · d)]a. Hence, a dyad (as well as a dyadic) is a second-order
tensor.
More generally, any second-order tensor can be expressed in dyadic form. Like
writing a vector in terms of scalar components relative to the orthogonal unit basis
{e i }, we can write the tensor T in the form
T = T ij e i e j ,
(2.16)
where the T ij are scalar tensor components relative to the dyadic basis {e i e j }. 10
10 In this chapter, lowercase bold letters denote first-order tensors (vectors), and uppercase bold
letters denote second-order tensors (dyadics).