Matrices and Tensors - Continuum Mechanics of Anisotropic Materials - page 406

2 3

1
r

2 2

1
r

@
@

r @
@

@ @

4 5

r r @ r @y @ r @ z

2 2 2

1
r

@

r 2 @

1 1

r

@

O ¼ ½r r ¼ ; (A.200)

2

@ r @y @ z @y

@y

2 2 2

@ 1

r

@ @

@ r @ z @ z @y @ z 2

and

;

T

p 1
r

p @ p 1

r

2 2 2 2 2

1
r

@
@

r @
@ r 2 @

1

2 ; @

@ @

O ¼

z 2 ; @y ; z ;

; (A.201)

r r @ @ z @ r @ @ r @y

@y

and the operation of O on a six-dimensional vector representation of a second order
tensor in three dimensions, O

T

¼ O : T ¼ tr O T , is given by

þ

2 T yy
@y

2 T zz
@

2 T z y 2 T rz 2 T r y

¼ @
@

r @

T rr
@ r 2 @

1

2 þ @

2 1

r @

2 @ 2 1

r @

O

T

z 2 þ @y þ z þ @y :

(A.202)

r r @ z @ r @ @ r

The divergence of a second order tensor T is defined in a similar fashion to the
divergence of a vector; it is a vector given by

e r þ e y

Þ¼ @

T rr
@ r @ @y þ @

T r y T rz

@

@ T r y

@ r @ @y þ @

T yy T y z

@

1 1

rTð r ; y; z ; t

r þ r þ z z

e z :

@ T zr

@

1

r @ @y þ @

T z y T zz

@

þ

r þ z

(A.203)

The strain-displacement relations (3.52) are written in cylindrical coordinates as

E rr ¼ @

u r
@

1

r @

u y
@y þ

u r
r ;

1
2

1

r @ @y þ @

u r u y

@

u r

r ;

E yy ¼ E r y ¼

r

E zz ¼ @

u z
@

1
2

@ u z

@ r þ @

u r
@

1
2

1

r @ @y þ @

u z u y

@

z ;

E rz ¼ ; E y z ¼ ; (A.204)

z z

and similar formulas apply for the rate of deformation-velocity tensor D , (3.33) if
the change of notation from E to D and u to v is accomplished. The stress equations
of motion (4.37) in cylindrical coordinates are