Kinematics and Mechanics of Large Elastic Deformations - Continuum Mechanics of Anisotropic Materials - page 297

F

Undeformed

Deformed

Fig. 11.8 The deformation of a square by F

where R is an orthogonal tensor ( R T R T 1 ) representing the rotation and

called the rotation tensor and U and V are called the right and left stretch tensors,
respectively. Both U and V represent the same pure deformation, but in different
ways that will be demonstrated. The right and left stretch tensors, U and V , are
related to F by

R ¼ R ¼

¼ p F T ¼ p F F T U F ; V : (11.9)

In order to define the square root of a tensor involved in ( 11.9 ), the tensor must
be symmetric and positive definite. In that case the square root is constructed by
transforming the tensor to its principal axes where the eigenvalues are all positive,
then the square root of the tensor is the diagonalized tensor coincident with the
principal axes but containing the square roots of the eigenvalues along the diagonal.
To show that the definitions ( 11.9 ) are reasonable it should be shown that the
tensors U 2 and V 2 are positive definite,

U 2 F T V 2 F T

¼ F ; ¼ F : (11.10)

The positive definite character of U 2 may be seen by letting it operate on the
vector a , then taking the scalar product with a , thus

U 2 F T a a ¼ a F a ¼ð F a Þð F a Þ 0 ; (11.11)

where the fact that a·F T

a ) has been used. A similar proof will show the
positive definite character of V 2 . The fact that the tensor R is orthogonal follows
from the definitions of U and/or V . From ( 11.8 ) R is given by R

¼ ( F

U 1 ¼ F (or

V 1 F ) thus the calculation of R T R T ) yields R ¼ R (or R

R T U 1 F T U 1 U 1 U 2 U 1 R ¼ F ¼ ¼ 1 ;

where the fact that U and its inverse are symmetric and the definition ( 11.10 ) have
been employed. As an example of this decomposition consider the F associated
with a simple shearing deformation illustrated in Fig. 11.8 . The decomposition of
this simple shearing deformation is shown in Fig. 11.9