Kinematics and Mechanics of Large Elastic Deformations - Continuum Mechanics of Anisotropic Materials - page 291

the deformation gradients will be constant. Thus a homogeneous deformation is
mathematically defined as a deformation that has a representation of the form

x ¼ L ð t Þ X ; (11.1)

where L (t) is a tensor independent of X . Recalling the definition (2.13) of the
deformation gradient tensor F , F

∇ O N x

( X ,t)] T , it follows that for a homo-

¼

[

geneous deformation,

F ¼ L: (11.2)

Thus, as would be expected for the definition of a homogeneous deformation, the
tensor of deformation gradients is independent of X . Lord Kelvin and P. G. Tait in
their Treatise on Natural Philosophy extensively developed the geometrically
interesting properties of homogeneous deformations. Their study of the effect of
homogeneous deformations upon simple geometric figures is briefly reviewed here.
Recall that a plane may be defined by its normal and the specification of one point in
the plane. Let a denote the vector normal to a plane and let X o denote a point in the
plane. Then all the other points in the plane are points X such that a ( X X o ) ¼ 0.

This is so because the condition a ( X X o ) ¼ 0 requires that the vector ( X X o )

be perpendicular to a , the normal to the plane. If we set a c , a constant, then a

material surface that forms a plane may be described in material coordinates by

X o ¼

a X ¼ c : (11.3)

L 1 x , into ( 11.3 ) yields Substituting the inverse of ( 11.2 ), X ¼

a* x ¼ c ;

(11.4)

L 1 has been defined. Eq. ( 11.3 ) is also
the equation of a plane, a plane in the spatial coordinate system, thus permitting one
to conclude that a plane material surface is deformed into a plane spatial surface by
a homogeneous deformation. More simply stated, homogeneous deformations map
planes into planes. Selecting different values for the constant c in ( 11.3 ) and ( 11.4 ),
it may be concluded that parallel planes will deform into parallel planes since the
normals to parallel planes have the same direction. Since the intersection of two
planes is a straight line (Fig. 11.1 ), it follows that parallel straight lines go into
parallel straight lines, parallelograms go into parallelograms, and parallelepipeds
deform into parallelepipeds. The results are illustrated in Fig. 11.2 .

where a new constant vector a* by a* ¼ a

Example Problem 11.2.1
Draw a sketch of the set of parallel lines given by the intersection of the planes
a [2 3 0] T with c

¼ ¼ 0 and 5, and X III ¼ 0. These lines have the representations

2 X I þ 5, respectively. Draw a sketch of the set of

parallel lines after subjecting them to the homogeneous deformation

3 X II ¼ 0 and 2 X I þ 3 X II ¼