Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 379

a 3 /12, but any two perpendicular vectors in the plane could
have been selected. Select two other eigenvectors in the plane and show that
these two eigenvectors are given by e II ¼

eigenvalue 7 r

cos g e 2 þ sin g e 3 and e III ¼

e 3 . Let R be the orthogonal transformation between these
Latin and Greek systems,

sin g e 2 þ cos g

2 3

1 0 0

4 5 :

R ¼ 0 cos g sin g

0 sin g cos g

Show that when the Greek coordinate system is used rather than the Latin
one, the coordinate transformation that diagonalizes the I matrix is Q

R

rather than Q . Show that both R

Q and Q transform the I matrix into the
coordinate system in which it is diagonal,

2 3

400
070
007

a 3
12

¼ r

4 5 :

Q T Q T R T

Q I ¼ R Q I

Æ

A.10 Connection to Mohr's Circles

The material in the section before last, namely the transformation law (A.83) for
tensorial components and the eigenvalue problem for linear transformations, is
presented in standard textbooks on the mechanics of materials in a more elementary
fashion. In those presentations the second order tensor is taken to be the stress
tensor and a geometric analog calculator is used for the transformation law (A.83)
for tensorial components in two dimensions, and for the solution of the eigenvalue
problem in two dimensions. The geometric analog calculator is called the Mohr
circle. A discussion of the connection is included to aid in placing the material just
presented in perspective.
The special case of the first transformation law from (A.83), T ð L Þ ¼ T ð G Þ Q

Q T , is rewritten in two dimensions ( n

s 0 ¼

Q T ; thus,

¼ 2) in the form Q s

T (L) ¼ s 0 and T (G) ¼ s , where the matrix of stress tensor components , the

s

s 0 , and the orthogonal transforma-
tion Q representing the rotation of the Cartesian axes are given by

matrix of transformed stress tensor components

s x t xy s x 0 t x 0 y 0

cos y sin y

; s 0 ¼ s ¼ ; Q ¼ : (A.141)

t xy s y t x 0 y 0 s y 0

sin y cos y

s 0 ¼

Q T , Expansion of the matrix equation Q s

s x 0 t x 0 y 0 cos y sin y s x t xy cos y sin y

s 0 ¼

¼ ; (A.142)

t x 0 y 0 s y 0 sin y cos y t xy s y sin y cos y