Biomedical Engineering Reference
In-Depth Information
Fig. 1.6
Three lumped
parameter models. (
a
) the
Hookian spring element,
(
b
) the damping element or
viscous dashpot, and (
c
) the
Darcy or permeability
element
a
b
c
F
F
F
k
h
placing a constant tensile load on a specimen of the material and measuring the
strain as a function of time. The function of time obtained by dividing the resulting
strain against time data by a unit load is called the
creep
function.
Stress relaxation
is the reduction or decay of stress in a material under constant strain as the time
increases. In a typical stress relaxation experiment on a material a constant tensile
strain is applied to a material specimen and the resultant stress is recorded as a
function of time. The function of time obtained by dividing the resulting stress
against time data by a unit strain is called the
stress-relaxation
function. Equations
for representing creep and stress relaxation will be obtained in the discussion of the
standard linear solid model below. Materials that exhibit the time-dependent
behaviors of creep and stress relaxation are called
viscoelastic
, indicating that
they have some properties of both a viscous fluid and an elastic solid (Christensen
1971
; Lockett
1972
; Pipkin
1972
).
As an example of the use of a lumped parameter model, consider the question of
determining the stiffness of elastic compliance of a running track so that the runner
has the optimal advantage of the elastic rebound of the track surface. This question
was answered using the spring and dashpot model of the lower limb shown in
Fig.
1.7
. While running on soft surfaces like sod is easier on the body than running
on hard surfaces like concrete, runners know that they run faster on the harder
surface. The question of how hard the surface should be was answered (McMahon
& Greene
1978
; McMahon & Greene
1979
) by tuning the compliance of the track to
the compliance of the model of the runner's leg shown in Fig.
1.7
. The half dozen or
so running tracks that have been constructed on the basis of this model are known to
runners as “fast” tracks.
Each of these lumped parameter models is an ideogram for a constitutive idea,
e.g., elasticity, damping, or flow through porous media. The Darcy or permeability
element is a special lumped parameter model peculiar to porous media. It was
developed to explain the flow of fluids though porous media. Specifically, the city
engineer of Dijon, France (Henri Philibert Gaspard Darcy, 1803-1858) in the
middle of the 1800s developed the model to analyze the flow of water through a
packed sand layer in a city fountain (Darcy
1856
). A sketch of the type of
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