Biomedical Engineering Reference
In-Depth Information
Fig. 3.17 Stress vectors acting on the volume element: a volume element cut from a body at any
arbitrary point of cutting plane A with the corresponding reaction measures in the form of C
AUCHY
stress vectors and, b stress coordinates at the positive cutting planes of the volume element
t
1
¼
r
11
e
1
þ
r
12
e
2
þ
r
13
e
3
¼
r
1j
e
j
t
2
¼
r
21
e
1
þ
r
22
e
2
þ
r
23
e
3
¼
r
2j
e
j
t
3
¼
r
31
e
1
þ
r
32
e
2
þ
r
33
e
3
¼
r
3j
e
j
ð
3
:
87
Þ
or for short
t
i
¼
r
ij
e
j
ð
i
¼
1
;
2
;
3
Þ:
ð
3
:
88
Þ
Due to the three different stress vectors t
i
nine coordinates r
ij
, namely r
11
, r
12
,
r
13
-r
33
, exist which (inevitably) must be denoted with a double index. The first
index refers to the direction of the normal vector of the corresponding area element
and the second index refers to the direction of the (stress) coordinate. The stress
coordinates with equal indices r
ii
(i = 1, 2, 3) are referred to as normal or direct
stress and, stress coordinates with mixed indices r
ij
(i = j) are referred to as
tangential or shear stress. The stress vectors on the opposite cutting planes of the
volume element, i.e. diametric to the stress vectors t
i
depicted in Fig.
3.17
b can be
derived from (
3.87
) by their increase via T
AYLOR
series expansion; they, however,
do not contribute any further information to the general stress state.
3.2.4.3 Stress Tensors
The introduction of a stress tensor is motivated by the following: according to
(
3.87
), the three stress vectors t
i
are constituted by nine stress coordinates r
ij
which
fully describe the stress state in a material point (volume element). A mathematical
object is now sought which ''economically'' combines the nine coordinates r
ij
into
one single object, such that using appropriate algebraic operations, the three
vectors t
i
result. Since the strain kinematics were organized by (second order)
tensors (cf.
Sect. 3.2.3.6
), it is obvious to again use an appropriate (second order)
tensor definition. The C
AUCHY
stress tensor S thus is defined by
