Kinematics and Mechanics of Large Elastic Deformations - Continuum Mechanics of Anisotropic Materials - page 290

area in the reference configuration and the area in the instantaneous configuration in
the case of small deformations. In practice, in the case of small deformations, the
stress is generally calculated approximately using the original cross-sectional area,
even though the definition is for the instantaneous cross-sectional area. For large
deformations however, the cross-sectional area can change considerably. Consider
a rubber band in its unstressed state and measure or visualize the cross-sectional
area of the rubber band perpendicular to the long axis of the band, the long axis that
forms the closed loop of the band. When the rubber band is stretched, note how the
cross-sectional area decreases. When the band is stretched the force on the band is
increased and, since stress is defined as force per unit area and the area is decreas-
ing, the increasing stress is due not only to the increasing force, but also to the
decreasing cross-sectional area of the rubber band. Each incremental increase in
the force increases the stress, but it also reduces the cross-sectional area, thereby
further increasing the stress. This feature of the concept of stress at large
deformations, the fact that its change is not only due to changing force, but also
to changing area, is a characteristic feature of large deformations, namely that the
analysis of large deformations requires nonlinear mathematics. Nonlinear mathe-
matics is generally more difficult than linear.
A development of the kinematics of large deformations is presented in this
chapter. It begins in the following section with homogenous deformations and
continues with the polar decomposition theorem, strain tensors for large
deformations, and formulas for the calculation of volume and area change. Using
the formulas for the area change, the appropriate definitions of stress for large
deformations are then developed. These large deformation stress measures are
incorporated in the stress equations of motion. Constitutive equations for both
Cauchy elastic and hyperelastic materials capable of large deformations are then
considered along with the special cases of isotropic material models and incom-
pressible material models. Some solutions to large deformation anisotropic elastic
problems are then described. The chapter closes with a discussion of the literature on
this topic.

11.2 Large Homogeneous Deformations

In this section the easily understood and easily illustrated large class of
deformations called homogeneous deformations is described. It is most important
for the modeler to understand homogeneous deformations because most mechanical
testing of materials requires homogeneous deformations and many finite deforma-
tion problems for this class of deformations are easily solved. Homogeneous
deformations are deformations that are exactly the same for all particles, that is to
say all particles experience the same deformation, the same strain, and the same
rotation. Recalling the representation (2.2) for a motion, x ( X , t), and the fact

that the strain and rotation are derivatives of the motion with respect to X , means
that the motion must be linear in X so that the strain and rotation measures such as

¼ x