Kinematics and Mechanics of Large Elastic Deformations - Continuum Mechanics of Anisotropic Materials - page 315

11.6.2. Consider a unit cube subjected to a uniaxial extension. A net force P in the
x 1 or X I direction creates this uniaxial extension. The motion is described by
x 1 ¼ X III . Note that each face of the unit cube has

an initial area A o that is unity, A o ¼

(1 þ t ) X I , x 2 ¼ X II , x 3 ¼

1. Estimate the first Piola-Kirchhoff
stress T 1PK , then calculate the Cauchy stress T

J 1 T 1PK ·F T and the ¼

JF 1 ·T· ( F 1 ) T ).
11.6.3. Consider a unit cube subjected to a biaxial extension. A net force P 1 in the
x 1 or X I direction and a net force P 2 in the x 2 or X II direction create this
biaxial extension. The motion is described by x 1 ¼ (1 þ

second Piola-Kirchhoff stress ( T 2PK

F 1 ·T 1PK ¼

t ) X I , x 2 ¼ (1 þ 2

t ) X II , x 3 ¼ X III . Note that each face of the unit cube has an initial area A o

that is unity, A o ¼ 1. Estimate the first Piola-Kirchhoff stress T 1PK , then

J 1 T 1PK ·F T

calculate the Cauchy stress T ¼ and the second

JF 1 ·T· ( F 1 ) T ).
11.6.4. Consider an object that is the unit cube deformed by deformation gradient
tensor F given in Problem 11.3.1.

Piola-Kirchhoff stress ( T 2PK F 1 ·T 1PK

¼

If the homogeneous second

Piola-Kirchhoff stress tensor T 2PK is given by

2
4

3
5 ;

10 3 0
3 00
001

T 2 PK ¼

determine the Cauchy stress tensor T and the first Piola-Kirchhoff stress
tensor T 1PK .
11.6.5. Consider an object that is the unit cube deformed by deformation gradient
tensor F given in Problem 11.3.1 on page 511. If the homogeneous second
Piola-Kirchhoff stress tensor T 2PK is given by

2
4

3
5 ;

10 3 0
3 00
001

T 2 PK

¼

determine the stress vector acting on the sloping face whose normal is
(2/ √ 5, 1/ √ 5, 0) in the deformed configuration.

11.7 Finite Deformation Elasticity

An elastic material is a material characterized by a constitutive equation, which
specifies that stress is a function of strain only. It is also possible to represent an
elastic material by a constitutive equation that specifies stress as a function of the
deformation gradients F , provided one keeps in mind that, due to invariance under
rigid object rotations, the stress must be independent of the part of F that represents
rotational motion. In terms of the Cauchy stress T and the deformation gradient F
the constitutive equation for an elastic material can be written T

¼ g ( F ). Invariance

under rigid object rotations requires