Kinematics and Mechanics of Large Elastic Deformations - Continuum Mechanics of Anisotropic Materials - page 296

Fig. 11.7 Langer lines on
a cadaver. From Danielson
( 1973 )

Problems
11.2.1 Draw a sketch of the set of parallel lines given by a ¼ [ 2, 1, 0] T and c ¼ 0

5, then draw a sketch of the set
of parallel lines after subjecting them to the homogeneous deformation of
Example 11.2.1.
11.2.2 Draw a sketch of the ellipse given by

and 5, 2 X I þ X II ¼ 0 and 2 X I þ X II ¼

X I = X II ¼

1, and then draw a
sketch of the same ellipse after it was subjected to the homogeneous
deformation of Example 11.2.1.
11.2.3 Show that the deformation x 1 ¼

ð 16 Þþ

(9/4) X I , x 2 ¼ X II , x 1 ¼ X II carries the

ellipse ( X I /4) 2 ( X II /9) 2 1 into the circle ( x 1 /9) 2 ( x 2 /9) 2

þ ¼ þ ¼ 1 and that

the inverse deformation carries the circle ( X I /4) 2 ( X II /4) 2

þ ¼ 1 into the

ellipse ( x 1 /9) 2 ( x 2 /4) 2

1. Provide a sketch of the undeformed and
deformed ellipses and circles.
11.2.4 Why is it not possible for an ellipse to deform into a hyperbola?

þ ¼

11.3 Polar Decomposition of the Deformation Gradients

It can be shown that the deformation gradient F can be algebraically decomposed in
two ways into a pure deformation and a pure rotation. This decomposition is
multiplicative and is written

F ¼ R U ¼ V R ; (11.8)