Kinematics and Mechanics of Large Elastic Deformations - Continuum Mechanics of Anisotropic Materials - page 321

@ I C

; @

II C
@

; @

III C
@

III C C 1

C ¼

1

C ¼

I C 1 C

C ¼

: (11.81)

@

it follows that T 2 PK has the representation

1

@ W

I C @

W

@

W

þ @

W

T 2 PK

III C III C C 1

¼ 2

I C þ II C C

: (11.82)

@ @ II C @ @

This constitutive relation may also be written in a form which contains C 2 rather
than C 1 ,

1 C

@ W

I C @

II C þ @

W W @ W

I C @

W

þ @

W

T 2 PK

III C C 2

¼ 2

I C þ III C II C II C þ

@ @ @ @ @ III C @

(11.83)

by use of Cayley Hamilton theorem that states that a matrix satisfies its own
characteristic equation, thus C may replace

2

in ( 11.65 ), C 3 I C C 2

l þ II C C II

I C ¼ 0 . A term-by-term comparison of ( 11.78 ) with ( 11.69 ) shows that the

coefficients a o , a 1 , and a 2 in ( 11.69 ) are given in the case of hyperelasticity by

2 @

W

I C @

II C þ @

W W

2 @

W

I C @

W

a o ¼

I C þ III C II C

; a 1 ¼

II C þ

;

@ @ @ @ @ III C

2 @

W

a 2 ¼ (11.84)

@ III C

Also, in the case of an isotropic hyperelastic material, the coefficients h o ,h 1 , and
h 2 in the constitutive relation between the Cauchy stress T and the left
Cauchy-Green tensor B ,( 11.73 ), may be expressed in terms of the strain energy
function by the following formulas:

2
J

II C @

II C þ @

W W 2

J @

W 2

J III C @

W

h o ¼

III C III C

; h 1 ¼

I C ;

h 1 ¼

(11.85)

@ @ @ @ II C

Problems

11.9.1. Derive the result ð@ W =@ C Þ¼ð@ W =@ E Þ : ð@ E =@ C Þ¼ð 1 = 2 Þð@ W =@ E Þ ,

( 11.78 ), using the indicial notation and the formula 2 E ¼ C 1 . Hint: It

is useful to first derive the formula ð@ E gd =@ C ab Þ¼ð 1 = 2 Þd bd d ag from 2 E gd

C gd d gd .
11.9.2. Derive the result ( 11.76 ) using the indicial notation. Hint: It is useful to first
derive the formula

¼

ð@ C ab =@ F ig Þ¼ð@=@ F ig Þð F ka F kb Þ¼d ik d ag F kb þ F ka d ik

F T · F in the indicial notation C ab d bg beginning from the definition C ¼

¼ F ka F kb .