Biomedical Engineering Reference
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in terms of the components of the displacement vector into the equations of motion
(
6.18
), thus
2
u
r€
u
¼ðl þ mÞrðr
u
Þþmr
þ r
d
:
(6.29)
This is the Navier equation. If the Navier equation is solved for the displacement
field
u
, then the strain field
E
can be determined from (2.49) and the stress field
T
from (
6.24
). The resulting stress field will satisfy the stress equations of motion
(
6.18
) because (
6.29
) is an alternate statement of the equations of motion.
Consider now the special case when one chooses the stress (or the strain) as the
unknown and the displacement is to be calculated last from the determined strain
tensor. In this case one must consider the strain-displacement relations (2.49) to be
a system of first-order partial differential equations to be solved for the components
of the displacement vector
u
given the components of strain tensor
E
(see Sect. 2.4).
The
conditions of compatibility
in terms of strain, (2.53), (2.54), or (2.55), are a set
of necessary and sufficient conditions that the first-order partial differential
equations (2.49) must satisfy in order that (2.49) have a single valued and continu-
ous solution
u
.
The general problem associated with the basic system of fifteen equations is to
find the fields
T
(
x
,
t
),
E
(
x
,
t
), and
u
(
x
,
t
) for all
x
2
O
and
t
2
[0,
t
] given a
and elastic coefficients
C
(or, in the case of isotropy,
particular object
O
of density
r
l
) acted upon by an action-at-a-distance force
d
and some surface loading or
specified displacements at the boundary during a specified time interval [0,
t
]. Such
problems are called the
initial-boundary value problems
of the theory of elasticity
and they are classified in several ways. First, they are classified as either
elastostatic, elastoquasi-static, or elastodynamic. The
elastostatic boundary value
problems
are those in which
T
(
x
),
E
(
x
), and
u
(
x
) are independent of time, and the
inertia term in the stress equations of motion,
and
m
ru
, is zero. The
elastoquasi-static
problems
are those in which
T
(
x
,
t
),
E
(
x
,
t
), and
u
(
x
,
t
) are time dependent, but the
inertia term in the stress equations of motion,
ru
, is small enough to be neglected.
The
elastodynamic initial-boundary value problems
are those in which
T
(
x
,
t
),
E
(
x
,
t
), and
u
(
x
,
t
) are time dependent and the inertia term in the stress equations of
motion,
ru
, is neither zero nor negligible.
The formulation of boundary value problems is considered next. The boundary
value problems are classified as displacement, traction, and mixed or mixed-mixed
boundary value problems. An object
O
with boundary
O
is illustrated in Fig.
6.2
.
The total boundary of the object
O
is divided into the sum of two boundaries, the
displacement boundary
∂
O
u
over which the boundary conditions are specified in
terms of displacement and the traction boundary
∂
O
t
over which the boundary
conditions are specified in terms of the surface tractions
t
. Note that
∂
O
t
\
∂
O
u
∂
¼
O
. It is required for some problems to further subdivide
the boundaries to include the situation in which the normal tractions and transverse
displacements, or transverse tractions and normal displacements, are specified over
portions of the object boundary, but that is not done here. The rigid wall indicated in
Ø and
O
t
[
∂
O
u
¼
∂
∂
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