Biomedical Engineering Reference
In-Depth Information
Fig. 6.2
An object denoted
by
O
whose total boundary is
divided into traction and
displacement boundaries
denoted by
∂
O
t
and
∂
O
u
,
∂
O
t
\
∂
O
u
¼
Ø,
∂
O
t
[
∂
O
∂
O
u
O
u
¼
∂
O
, respectively
∂
O
t
ρ
d
Fig.
6.2
is intended to suggest a boundary upon which the boundary condition on
the displacement is specified and the other surface is intended to suggest a boundary
on which the (zero or nonzero) surface tractions are specified.
In the
displacement boundary value problem
the continuous surface displace-
ment
u
(
x*
,
t
) is specified on the boundary
O
, where [0,
t
] is the
time interval for which a solution is desired, and the continuous initial displacement
field
u
o
(
x
) is specified for all
x
O
for [0,
t
],
x*
2
∂
∂
O
. The displacement boundary value problem is
the following; given a particular object
O
of density
2
C
r
and elastic coefficients
(or, in the case of isotropy,
) acted upon by an action-at-a-distance force
d
,
determine the fields
u
(
x
,
t
),
T
(
x
,
t
),
E
(
x
,
t
) which satisfy the system of equations
(2.49), (
6.18
) and some form of Hooke's law (5.7H), the initial conditions
l
and
m
Þ;
_
Þ¼
_
u
ð
x
;
0
Þ¼
u
ð
x
u
ð
x
;
0
u
ð
x
Þ;
x
2
O
;
(6.30)
and the displacement boundary condition
u
ð
x
;
x
2 @
u
ð
x
;
t
Þ¼
t
Þ;
O
;
t
2½
0
;
t
:
(6.31)
In the
traction boundary value problem
the specification of surface displacement
(
6.31
) is replaced by the specification of the surface traction
t
(
x*
,
t
) for all
x*
2
∂
O
and
t
2
[0,
t
], thus
tðx
;
Þ¼Tðx
;
; x
2 @
t
t
Þn; n?@
O
O
;
t
2½
0
;
t
;
(6.32)
where
n
is the unit exterior normal to the boundary. In a
mixed boundary value
problem
there is a portion of the boundary on which the displacements are specified
and a portion of the boundary on which the surface tractions are specified. These
portions of the boundary are denoted in Fig.
6.2
by
O
t
and they are non-
empty, non-intersecting portions of the boundary whose union is the entire bound-
ary
O
u
and
∂
∂
O
. The typical mixed-boundary value
problem must satisfy the condition (
6.31
)on
O,
O
t
\
∂
O
u
¼
Ø,
O
t
[
∂
O
u
¼
∂
∂
∂
∂
O
u
and the surface traction condition
∂
(
6.32
)on
O
t
. The mixed-mixed boundary value problem of elasticity is
characterized by boundaries where the two types of boundary conditions appear
on the same portion of the boundary. For example the normal displacement is
∂
Search WWH ::
Custom Search