Biomedical Engineering Reference
In-Depth Information
(
σ
s
)
max
is often normative, while a ceramic material might fail because of a too
high maximum extensional stress (
σ
n
)
max
. Such knowledge is important for the
design of hip and knee prostheses or tooth implants, where both types of materials
are used. But also biological materials may have different failure mechanisms. A
bone for example will often fail as a result of the maximum compression stress,
but a tendon will usually fail because it is overstretched, i.e. due to the maximum
extensional stress or maximum shear stress. The functional relationship
)is
primarily determined by the (micro) structure of the considered material. Thus,
depending on the material, different specifications of
σ
(
σ
σ
may be applied. We will
limit ourselves to a few examples.
According to the equivalent stress
σ
T
ascribed to Tresca (sometimes also called
Coulomb, Mohr, Guest) the maximum shear stress is held responsible for failure.
The definition is
σ
T
=
σ
s
)
max
=
σ
3
−
σ
1
,
2(
(8.74)
which is normalized in such a way that for a uniaxially loaded bar, with an
extensional stress
σ
ax
>
0, we find
σ
T
=
σ
ax
.
σ
M
according to von Mises (also Huber, Hencky) is based
on the deviatoric stress tensor:
The equivalent stress
3
2
tr(
σ
3
2
tr(
σ
d
d
)
=
d
d
).
σ
M
=
·
σ
σ
(8.75)
This can be elaborated to
1
3
yz
.
2
(
σ
zz
−
σ
xx
)
2
+
σ
xx
−
σ
yy
)
2
σ
yy
−
σ
zz
)
2
xy
xz
σ
M
=
+
(
+
(
σ
+
σ
+
σ
(8.76)
Here also the specification is chosen in such a way that for a uniaxially loaded bar
with axial stress
σ
ax
, the equivalent von Mises stress satisfies
σ
M
=
σ
ax
. It can be
proven that in terms of principal stresses:
1
2
(
σ
3
−
σ
1
)
2
.
σ
1
−
σ
2
)
2
σ
2
−
σ
3
)
2
σ
M
=
+
(
+
(
(8.77)
In general (for arbitrary
σ
) the difference between the equivalent stresses accord-
ing to Tresca and von Mises are relatively small.
The equivalent stress
σ
R
, according to Rankine (also Galilei) expresses that, in
absolute sense, the maximum principal stress determines failure. This means that
σ
R
=|
σ
3
|
if
|
σ
3
|≥|
σ
1
|
σ
R
=|
σ
1
|
if
|
σ
3
|
<
|
σ
1
|
.
(8.78)
