Kinematics and Mechanics of Large Elastic Deformations - Continuum Mechanics of Anisotropic Materials - page 319

T 2 PK a 1 F T a 2 F T F T ¼ a o 1 þ F þ F F ; (11.70)

1 and post multiplying by F 1 , thus

F T then premultiplying by ð Þ

1 1 1

F T T 2 PK F 1 F T F 1 F T F T F 1

ð Þ ¼ a o ð Þ þ a 1 ð Þ F þ;

1

F T F T F T F 1 a 2 ð Þ F F : (11.71)

This expression reduces to

a 1
J 1

a 2
J B

a o

J B 1

T ¼ þ þ ; (11.72)

when one takes note of ( 11.27 ), ( 11.28 ), and the second of ( 11.51 ). This expression
is rewritten in the form

h 1 B 1 T ¼ h o 1 þ h 1 B þ ; (11.73)

when it is observed that I C ¼ I B , II C ¼ II B , and III C ¼ III B ¼ J and the following

notation is introduced

a 1
J ð

h o ¼ h o ð I C ; II C ; III C Þ¼ h o ð I B ; II B ; III B Þ¼ I C ; II C ; III C Þ;

a o
J ð

h 1 ¼ h 1 ð I C ; II C ; III C Þ¼ h 1 ð I B ; II B ; III B Þ¼ I C ; II C ; III C Þ;

a 2
J ð

h 1 ¼ h 1 ð I C ; II C ; III C Þ¼ h 1 ð I B ; II B ; III B Þ¼ I C ; II C ; III C Þ: (11.74)

Problems

11.8.1. Setting A ¼ t ( C ) in the identity ( 11.60 ), verify the formula ( 11.63 ), tðCÞc
¼ð c .

11.8.2. Verify the result ( 11.60 ), then derive the result ( 11.61 ) from ( 11.60 ).
11.8.3. Determine the Cauchy stress tensor T and the second Piola-Kirchhoff stress
tensor T 2PK for an elastic isotropic material subjected to the deformation in
Example 11.2.1. Specify the numerical value of the functional dependencies
of the functions a o , a 1 , and a 2 as well as those of h o , h 1 , and h 1 .

c t ð C Þ c Þ

11.9 Finite Deformation Hyperelasticity

A hyperelastic material is an elastic material for which the stress is derivable from a
scalar potential called a strain energy function. Thus a hyperelastic material is
automatically a Cauchy elastic material, but not the reverse. In the case of small
deformation elasticity, a strain energy function always exists and therefore the