Biomedical Engineering Reference
In-Depth Information
8.7
Hydrostatic pressure and deviatoric stress
The hydrostatic pressure
p
is defined as the average of the normal stresses in
Fig.
8.8
. It is common practice to give the pressure the opposite sign of the average
normal stresses, so:
p
=−
(
σ
xx
+
σ
yy
+
σ
zz
)
/
3
=−
tr(
σ
)
/
3
=−
tr(
σ
)
/
3.
(8.69)
It can be shown that, if expressed in principal stresses, we find
p
=−
(
σ
1
+
σ
2
+
σ
3
)
/
3.
(8.70)
h
and defined accord-
The associated hydrostatic stress tensor is denoted with
σ
ing to
h
σ
=−
p
I
.
(8.71)
The difference of the (total) stress tensor
σ
and the hydrostatic stress tensor
σ
is
called the deviatoric stress tensor
σ
d
. Thus
h
d
d
h
d
d
.
σ
=
σ
+
σ
=−
p
I
+
σ
and also
σ
=
σ
+
σ
=−
p
I
+
σ
(8.72)
Splitting the stress state in a hydrostatic part and a deviatoric part appears to be
useful for the description of the material behaviour. This will be the theme of
Chapter
12
.
8.8
Equivalent stress
In general mechanical failure of materials is (among other items) determined by
the stresses that act on the material. In principle, this means that all components of
the stress tensor
in one way or another may contribute to failure. It is common
practice, to attribute a scalar property to the stress tensor
σ
that reflects the gravity
of the stress state with respect to failure. Such a scalar property is normalized,
based on a consideration of a uniaxial stress state (related to the extension of a
bar) and is called an equivalent stress
σ
. The equivalent stress is a scalar function
of the stress tensor and thus of the components of the stress matrix
σ
σ
=
σ
(
σ
) and also
σ
=
σ
(
σ
) .
(8.73)
The formal relationship given above has to be specified, based on physical under-
standings of the failure of the material. Only experimentally can it be assessed at
which stress combinations a certain material will reach the limits of its resistance
to failure. It is very well possible and obvious that one material fails by means
of a completely different mechanism than another material. For example, consid-
ering technical materials, it is known that for a metal the maximum shear strain
