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Kinematics of a Continuum 125

demonstrated in Example 3.22.1 and 3.23.1. The proof of the uniqueness of the decomposi-
tions will be given in Example 3.22.2.
For any material element dX at X, the deformation gradient transforms it (i.e., dX) into a
vector dfx:

Now, DdX describes a pure stretch deformation (Section 3.20) in which there are three
mutually perpendicular directions (the eigenvectors of U) along which the material element
dX stretches (i.e., becomes longer or shorter ) but does not rotate. Figure 3.10 depicts the
effect of U on a spherical volume \dX\ = constant; the spherical volume at X becomes an
ellipsoid at x. (See Example 3.20.2) The effect of R in R(U dX) is then simply to rotate this
ellipsoid through a rigid body rotation.(See Fig. 3.11)

Fig. 3.11

Similarly, the effect of the same deformation gradient can be viewed as a rigid body rotation
(described R) of the sphere followed by a pure stretch of the sphere resulting in the same
ellipsoid as described in the last paragraph.
From the polar decomposition theorem, F = RU = VR, it follows immediately that

Example 3.21.1

Show that if the eigenvector of U is n, then the eigenvector for V is Rn; the eigenvalues for
both U and V are the same
Solution. Let n be an eigenvector for U with eigenvalue A, then