ABSTRACT
CHAPTER 3
KINETICS OF DEFORMATION
3.1. Cauchy Stress
Consider an internal surface S within a loaded deformable body. If the
resultant force across an innitesimal surface element dS with unit normal
n is df
n
, the corresponding traction vector is (Fig. 3.1)
t
n
=
df
n
dS
: (3.1.1)
The Cauchy or true stress is the second-order tensor related to the traction
vector t
n
by
t
n
= n : (3.1.2)
When is decomposed on an orthonormal basis in the deformed congura-
tion as
=
ij
e
i
e
j
; (3.1.3)
the traction vector over the area with the normal in the coordinate direction
e
i
can be written as
t
i
= e
i
=
ij
e
j
: (3.1.4)
From Eqs. (3.1.2) and (3.1.4) we conclude that the traction vector over the
surface element with unit normal n = n
i
e
i
can be expressed in terms of the
traction vectors t
i
as
t
n
= n
i
t
i
: (3.1.5)
Equation (3.1.5), known as the Cauchy relation, can also be derived directly
by applying the balance law of linear momentum to an innitesimal tetrahe-
dron around a point of the stressed body (e.g., Prager, 1961; Fung, 1965). In
Section 3.3 it will be shown that the Cauchy stress is a symmetric tensor,
provided that there are no distributed surface or body couples acting within
the body.
