36
2 Vector and Tensor Analysis
e x
e y
e θ
M
e r
Fig. 2.4 Bending of a cantilever beam by end moment M. The bending-stress distribution is shown
for one section. Since the beam bends into a circular arc, beam geometry is described by Cartesian
and cylindrical coordinates before and after deformation, respectively
Before deriving the coordinate transformation equations, we note a convenient
way to extract components of vectors and tensors. Dotting the vector a with e i gives
a · e i = (a j e j ) · e i = a j δ ji = a i .
Similarly, double-dotting the tensor T with e i e j yields (see Eq. (2.11))
T : e i e j = (T kl e k e l ) : (e i e j )
= T kl (e k · e i )(e l · e j )
= T kl δ ki δ lj = T ij .
Thus, we have
a i = a · e i
T ij = T : e i e j = e i · T · e j .
(2.43)
These equations show that dotting a vector with e i gives the component of the vector
along e i , and double-dotting a second-order tensor with the dyad e i e j gives the
component of the tensor along e i e j . 12 This mechanism for obtaining components
holds for vectors and tensors written in terms of base vectors for any coordinate
system.
Example 2.5 Show that the components of the rotation tensor Q are the same in
both unrotated and rotated coordinate systems.
12 In Cartesian coordinates, the components of a vector are the orthogonal projections of the vector
onto the coordinate axes (Fig. 2.1). A physical interpretation for the components of a tensor cannot
be visualized as easily.
2 Vector and Tensor Analysis
e x
e y
e θ
M
e r
Fig. 2.4 Bending of a cantilever beam by end moment M. The bending-stress distribution is shown
for one section. Since the beam bends into a circular arc, beam geometry is described by Cartesian
and cylindrical coordinates before and after deformation, respectively
Before deriving the coordinate transformation equations, we note a convenient
way to extract components of vectors and tensors. Dotting the vector a with e i gives
a · e i = (a j e j ) · e i = a j δ ji = a i .
Similarly, double-dotting the tensor T with e i e j yields (see Eq. (2.11))
T : e i e j = (T kl e k e l ) : (e i e j )
= T kl (e k · e i )(e l · e j )
= T kl δ ki δ lj = T ij .
Thus, we have
a i = a · e i
T ij = T : e i e j = e i · T · e j .
(2.43)
These equations show that dotting a vector with e i gives the component of the vector
along e i , and double-dotting a second-order tensor with the dyad e i e j gives the
component of the tensor along e i e j . 12 This mechanism for obtaining components
holds for vectors and tensors written in terms of base vectors for any coordinate
system.
Example 2.5 Show that the components of the rotation tensor Q are the same in
both unrotated and rotated coordinate systems.
12 In Cartesian coordinates, the components of a vector are the orthogonal projections of the vector
onto the coordinate axes (Fig. 2.1). A physical interpretation for the components of a tensor cannot
be visualized as easily.