Kinematics and Mechanics of Large Elastic Deformations - Continuum Mechanics of Anisotropic Materials - page 294

Fig. 11.4 An illustration of
an ellipse being deformed
into an ellipse and of a circle
being deformed into an
ellipse

homogeneous deformation. In particular, spheres will deform into ellipsoids and, in
planar deformations, ellipses into ellipses (or circles into ellipses). This result is
illustrated in the planar case where ellipses deform into ellipses or circles into
ellipses in Fig. 11.4 .

Example Problem 11.2.2
Draw a sketch of the ellipse given by X I = X II =

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

ð 9 Þþð 4 Þ¼

X I = X II = Solution : A sketch of the ellipse given by 1 is shown in

Fig. 11.5a . Using the inverse of the homogeneous deformation F determined
in Example 11.2.1, the tensor A representing the ellipse

ð 9 Þþð 4 Þ¼

X I = X II = ð 9 Þþð 4 Þ¼ 1,

2
4

3
5 ;

1

9 00
0

A ¼

1

4 0
000

( L 1 ) T L 1 , is transformed into A* , A* ¼ A

2
4

3
5 ;

2 10

1
54

A* ¼

31

8 0
000

1

x 1 þð x 2

and the deformed ellipse is given by ð 1 = 27 Þð 31 = 16 Þ 2 x 1 x 2 Þ¼ 1 . A sketch

of the deformed ellipse is shown in Fig. 11.5b .
The geometric results of this example give an intuitive insight into a number of
tissue deformation situations. Consider the case when a circle is inscribed on a
tissue and the tissue is then greatly deformed. Fig. 11.6 illustrates the deformation
of a circle with a ratio of principal axes of 1:1 through ten steps to a ratio of
principal axes of 1:19; this is the deformation of a circle inscribed on a surface
subjected to a homogeneous deformation. At a ratio of principal axes of 1:19 the