Page 160 - The Combined Finite-Discrete Element Method

STRESS 143

strain tensor is best represented using a local frame, i.e. using configuration before any
rotation has taken place
(i, j, k) (4.55)

In the case of the left Cauchy–Green strain tensor B, stretch occurs after rotation. Thus,
the left Cauchy–Green strain tensor is best represented using a deformed local frame, i.e.
using configuration after the rotation has taken place:

(i, j, k) (4.56)
˜ ˜ ˜
Other strain tensors (also called strain measures) can be derived using stretch tensors, for
instance strain measure in the form
s − 1
 m
for m = 0

e = m (4.57)
ln(s) for m = 0

(where s is a stretch). Depending on the parameter m, the following strain tensors
are obtained:
• For m = 2 a Green–St. Venant strain tensor is obtained:

1
2
1
E 2 = (U − I) = (C − I) (4.58)
2 2
• For m =−2 a Almanasi–Hmel strain tensor is obtained:
−1
1
−2
1
E −2 = (I − V ) = (I − B ) (4.59)
˜
2 2
• For m = 0 a logarithmic strain tensor is obtained:
1
E 0 = ln U = ln C (4.60)
2
• For m = 1 a Biot strain tensor is obtained:
E 1 = U − I (4.61)

4.5 STRESS

4.5.1 Cauchy stress tensor

Cauchy’s theorem makes it possible for integral relations of momentum balance to be
replaced by partial differential equations. The necessary and sufficient condition for the
momentum balance law to be satisfied is the existence of a spatial tensor field T (also
called Cauchy stress) such that:
• for a vector m, the surface traction force is given by

s(m) = Tm (4.62)