Biomedical Engineering Reference
In-Depth Information
specified and the shear stress is specified on the same portion of the boundary, or the
normal stress and the tangential displacement are specified on the same portion of
the boundary. The mixed-mixed boundary value problems are complicated and are
not considered here.
Any solution to an elasticity problem is unique. That is to say that, for a specific
object of a specified material acted upon by a specified action-at-a-distance force
and subject to specific boundary conditions, there is one and only one solution to the
set of 15 equations in 15 unknowns. The common strategy for proving uniqueness
theorems is to assume non-uniqueness, that is to say assume there are two, and then
prove that the two must be equal. The uniqueness theorem for linear elasticity is
proved using this strategy. Assume that there are two solutions
u
(1)
(
x
,
t
),
T
(1)
(
x
,
t
),
E
(1)
(
x
,
t
), and
u
(2)
(
x
,
t
),
T
(2)
(
x
,
t
),
E
(2)
(
x
,
t
), to the same elasticity problem, that is to
say a problem in which the object, the material, the action-at-a-distance force, and
the boundary conditions to which the object is subjected, are all specified. The
linearity of the system of equations for linear elasticity permits one problem
solution for a specified object and material, action-at-a-distance force, traction
boundary conditions, and displacement boundary conditions, to be superposed
upon a second solution for the same specified object, material and displacement
boundary conditions, but for a different action-at-a-distance force, different traction
boundary conditions and different displacement boundary conditions. Thus, for
example, two solutions,
u
(1)
(
x
,
t
),
T
(1)
(
x
,
t
),
E
(1)
(
x
,
t
) and
u
(2)
(
x
,
t
),
T
(2)
(
x
,
t
),
E
(2)
(
x
,
t
), for the same specified object and material, but different traction boundary
conditions, displacement boundary conditions and action-at-a-distance forces, may
be added together,
u
(1)
(
x
,
t
)
E
(2)
(
x
,
t
),
to obtain the solution for specified object and material, for the traction boundary
conditions and displacement boundary conditions and action-at-a-distance force that
are the sum of the two sets of traction boundary conditions, displacement boundary
conditions and action-at-a-distance forces. In the proof of uniqueness the principle
of superposition is used to define the difference problem obtained by subtracting the
two (possibly different) solutions
u
(1)
(
x
,
t
),
T
(1)
(
x
,
t
),
E
(1)
(
x
,
t
), and
u
(2)
(
x
,
t
),
T
(2)
(
x
,
t
),
E
(2)
(
x
,
t
), to the same elasticity problem. The difference problem to
which the fields
u
(1)
(
x
,
t
)
u
(2)
(
x
,
t
),
T
(1)
(
x
,
t
)
T
(2)
(
x
,
t
), and
E
(1)
(
x
,
t
)
E
(2)
(
x
,
t
)
are a solution is thus a problem for the same specified object and material but for a
zero action-at-a-distance force and for zero stress boundary conditions on
u
(2)
(
x
,
t
),
T
(1)
(
x
,
t
)
T
(2)
(
x
,
t
),
E
(1)
(
x
,
t
)
þ
þ
þ
O
t
and
∂
zero displacement boundary conditions on
O
u
. The objective is to obtain the
solution to this difference problem by considering the work done on a linearly elastic
object by the surface tractions and the action-at-a-distance force does this most
efficiently. The relation between the work done by the surface tractions and by the
“action-at-a-distance force” on the object may be expressed as an integral over the
object of the local work done per unit volume, tr
T:E
(see (3.53) and (3.57))
:
Z
∂
Z
Z
Z
Z
O
T
E
d
v
t
u
d
a
þ
O
r
d
u
d
v
¼
tr
f
T
E
g
d
v
¼
T
:
E
d
v
¼
:
@O
O
O
(6.33)
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