Basic Continuum Kinematics - Continuum Mechanics of Anisotropic Materials - page 50

and integrated by parts, thus

P 0

ð

P 0

ð

Y o x o x 0 Þþ x 0 Þ:

Y d x ¼ ð d x r Y ð x (2.61)

P o P o

Placing the result ( 2.61 ) into ( 2.59 ) it follows that

P 0

ð

x 0 Þ¼ u o Y o x o x 0 Þþ x 0 Þ

u ð ð

d x

½ E þr Y ð x

(2.62)

P o

or, in the indicial notation,

P 0

ð d x m :

E im @

Y ik

x 0 Þ¼ x 0 k Þþ x 0 k Þ u i Y ik ð x k u i ð

x m ð

x k (2.63)

@

P o

The relationship between the derivatives of the rotation and strain tensors,

@ Y ik

x m ¼ @

E im
@ x k @

E mk
@

; (2.64)

@ x i

may easily be verified by substituting the formulas ( 2.49 ) relating E and Y to the
displacement gradients. When the relationship ( 2.64 ) is substituted into ( 2.63 )it
becomes

P 0

ð

u i ðx 0 Þ¼ u i Y ik ð x k x 0 k Þþ

R im d x m ; (2.65)

P o

where

@ E im

@ x k @

E mk
@

x 0 k Þ:

R im ¼ E im ð x k (2.66)

x i

The condition that the integrand in the integral in ( 2.65 ) be an exact differential
is then expressed as the condition

@ R im

@ x k ¼ @

R ik

x m :

(2.67)

@

When ( 2.67 ) is substituted into ( 2.66 ), the result