Kinematics and Mechanics of Large Elastic Deformations - Continuum Mechanics of Anisotropic Materials - page 311

Problems

11.5.1. Compute the volume of the parallelepiped whose three edges meeting at
one vertex are characterized by the vectors (1, 2, 3), (2, 3, 3), ( 1).

11.5.2. Consider the plane area that forms the top face of the deformed parallelo-
gram illustrated in Fig. 11.8 . Use the formulas of Nanson to determine the
magnitude and orientation of the original area if the deformed area was a
consequence of the deformation specified in Example Problem 11.3.1.
11.5.3. Beginning with the formula ( 11.37 ) for d a use the indicial notation and the
formulas for the Det F employing the alternator from the Appendix to derive
the formula

3, 2,

T

ðF 1 J 1 da: (11.42 alternate)

Þ dA ¼

11.6 Stress Measures

The stress equations of motion are the most useful form of Newton's second law in
continuummechanics. In Sect. 3.4 these field equations were shown to have the form

T T r€ x ¼r T þ r d ; T ¼ : ð 3 : 37 Þ repeated

These equations involve the most common stress measure, the Cauchy stress T
introduced in Chapter 2 . It was not called the Cauchy stress in Chapter 4 because
there was only one stress tensor under discussion there. The Cauchy stress is
referred to the instantaneous or spatial coordinate system; it is measured relative
to the instantaneous area. In mechanical testing the phrase “true stress” is used to
denote a stress calculated using the instantaneous rather than the original cross-
sectional area. Cauchy stress, which is sometimes called Eulerian stress, is therefore
a “true stress.”
There are a number of different stress measures used in the study of finite
deformations of materials; three are considered here. The first is the Cauchy stress.
The second stress measure is the first Piola-Kirchhoff stress tensor that is some-
times called the Lagrangian stress tensor. This stress tensor is referred to the
reference configuration. Consider an object in both its deformed and undeformed
configurations. Since this is the same object in the two configurations, it must have
the same total mass M in each configuration, and Eq. (3.1) maybe rewritten to
express that fact:

ð
O r R ð

ð
O rð

M ¼ X Þ dV ¼ x ; t Þ dv ; (11.43)

where (x,t ) is the density in the

instantaneous or deformed configuration; d V is an element of volume in the initial

r R (X ) is the density in the initial configuration and r