Kinematics and Mechanics of Large Elastic Deformations - Continuum Mechanics of Anisotropic Materials - page 324

constitutive behavior of rubber is the Mooney-Rivlin material with the constitutive
equation (Mooney 1940 ; Rivlin and Saunders 1976 )

B B 1

1
2 þ b

1
2 b

T ¼ p1 þ m m ;

(11.94)

where m and b are constants, which has the following strain energy function

ð ð

1
2 m

1
2 þ b

1
2 b

W ¼ I B 3 Þþ II B 3 Þ ; (11.95)

where the inequalities m > 0 and (1/2) b (1/2) are imposed upon the

constants so that the strain energy W is a positive semidefinite quantity.

The special case of the Mooney-Rivlin material when

m and b

b ¼ (1/2) is called the neo-

Hookian material

T ¼ p1 þ m B : (11.96)

Employing the assumption of incompressibility is the development of a consti-
tutive model for a soft biological tissue is quite easy to justify because soft tissues
contain so much water that their effective bulk compressibility is that of water,
2.3 GPa. When one compares the shear or deviatoric moduli of a soft biological
tissue with 2.3 GPa, it is usually orders of magnitude less. Only in the case of hard
tissues does the shear or deviatoric moduli approach and exceed (up to an order of
magnitude) the effective bulk compressibility of water.

Problems

B 1 } from 11.10.1. Derive ( 11.94 ){ T ¼ p1 þ mðð 1 = 2 ÞþbÞ B mðð 1 = 2 ÞbÞ

( 11.95 ){ W ¼ð 1 = 2 Þm½ðð 1 = 2 ÞþbÞð I B 3 Þþðð 1 = 2 ÞbÞð II B 3 Þ using

h 1 B 1 }and( 11.93 ){ h 1 ¼

( 11.92 ){ T

¼ p1 þ h 1 B þ ð@ W =@ I C Þ

, h 1

2

¼ ð@ W =@ II C Þ

}.
11.10.2. Calculate the components of the Cauchy stress T in a Mooney-Rivlin
material ( 11.94 ){ T

2

B 1 } when
the material is subjected to a simple shearing deformation given by x 1 ¼
X I þ

¼ p1 þ mðð 1 = 2 ÞþbÞ B mðð 1 = 2 ÞbÞ

X III . Require that the normal stress acting on the
surface whose normal is in the x 3 or X III direction be zero.
11.4.2. Calculate the components of the Cauchy stress T in a neo-Hookian material
( 11.96 ){ T

kX II , x 2 ¼ X II , x 3 ¼

B } when the material is subjected to a simple
shearing deformation given by x 1 ¼

¼ p1 þ m

X III . Require
that the normal stress acting on the surface whose normal is in the x 3 or X III
direction be zero.

X I þ kX II , x 2 ¼ X II , x 3 ¼