Biomedical Engineering Reference
In-Depth Information
s
punch
y
0
x
0
Figure 13.4
Rigid indenter impressing a deforming continuum.
Consider a plane stress continuum of which the midplane (coinciding with the
x
0
y
0
-plane) has a rectangular shape in the reference configuration. The current
state arises because the top edge is indented by means of a rigid punch. The
punch displacement is specified by
s
(Fig.
13.4
shows the deformation process,
the displacements are magnified). At the location of the contact between inden-
ter and continuum, the interaction is described with a friction model according to
Coulomb (which can be considered to be a constitutive description for the contact
interaction). For a material point at the top contour of the continuum the following
three distinguishable situations may arise:
(i) There is no contact between the point of the continuum and the indenter. In this case
the boundary conditions are
σ
yy
=
0,
σ
xy
=
0,
(13.22)
with as an extra constraint that in the current state the vector
x
0
+
u
does not cross
the edge of the indenter.
(ii) There is contact between the point of the continuum and the indenter, and that with
'stick' boundary conditions (no relative tangential displacement between continuum
and indenter):
u
x
=
0,
u
y
=−
s
,
(13.23)
with as extra constraints
σ
yy
≤
0and
|
σ
xy
|≤−
μσ
yy
with
μ
the friction coefficient.
(iii) There is contact between the continuum and the indenter, and that with 'slip'
boundary conditions:
u
x
|
u
x
|
σ
yy
,
u
y
=−
s
,
σ
xy
=
μ
(13.24)
0.
The principal problem in accounting for the interaction between the continuum
and the indenter is that it is not a priori known to which of the three categories
described above the points of the top layer of the continuum belong. In general an
with as additional constraint
σ
yy
≤
