Biomedical Engineering Reference
In-Depth Information
especially useful in representing materials that exhibit instantaneous elastic response up to large
strains, such as rubber and bulk soft tissue. Given isotropy and additive decomposition of the devia-
toric and volumetric strain energy contributions in the presence of incompressible or almost incom-
pressible behavior, the strain energy potential can be represented with a polynomial expression.
A second-order polynomial strain energy potential (ABAQUS) was adopted with the form
2
2
1
__
__
∑
∑
U
=
ij
CI
(
−
3) (
i
I
−+
)
j
(
J
−
)
2i
(1.1)
1
2
el
D
ij1
+=
i1
=
i
where U is the strain energy per unit of reference volume;
C
ij
and
D
i
are material parameters;
__
I
and
__
I
are the first and second deviatoric strain invariants, defined as
2
=λ +λ +λ
2
2
__
__
__
__
(1.2)
I
1
1
2
3
(2)
−
(2)
−
(2)
−
__
__
__
__
(1.3)
I
=λ
+λ
+λ
2
1
2
3
with the deviatoric stretches λ
i
__
=
J
el
− 1/3 λ
i
and
J
el
and λ
i
are the elastic volume ratio and the prin-
cipal stretches, respectively.
The coefficients of the hyperelastic model extracted from the soft tissue experimental data are
0.08556, −0.05841, 0.03900, −0.02319, 0.00851, 3.65273, and 0.0 for C10, C01, C20, C11, C02, D1,
and D2, respectively.
Foot orthoses or insoles made of rubber-like elastomeric foam are elastic but compressible and
can be modeled as hyperelastic. Common examples of elastomeric foam materials are cellular poly-
mers for cushions, padding, and packaging materials that utilize the excellent energy absorption
properties of foams. Three distinct stages can be distinguished during compression of elastomeric
foam. Within small strains of about 5%, the foam deforms in a linear elastic manner due to cell wall
bending. The next stage is a plateau of deformation at almost constant or slowly increasing stress
caused by the elastic buckling of cell edges or walls. Finally, a region of densification occurs, where
the cell walls crush together, resulting in a rapid increase of compressive stress. The ultimate com-
pressive nominal strain usually ranges from 0.7 to 0.9.
1.2.3 l
oadinG
and
B
oundary
c
onditionS
In order to simulate the physiological loading, accurate ground reaction and muscle forces should
be applied. The major active muscles can be applied by adding force vectors with respect to their
lines of action. The muscle and joint forces can be determined using inverse dynamics based on
the kinematic and kinetic data obtained by any motion analysis system and force platform. Forces
from the major muscles can be obtained from either musculoskeletal models (Anybody, OpenSim)
or with the help of electromyography (EMG) and optimization (Rohrle et al. 1984; Glitsch and
Baumann 1997).
Simulating a subject with a body mass of 70 kg, a vertical force of approximately 350 N was
applied on each foot during balanced standing. The standing line of gravity was about 6 cm in front
of the ankle (Opila et al. 1988). Therefore, the plantar flexors act to balance the forward moment of
the body about the ankle in order to achieve an equilibrium balanced standing position. The triceps
surae provides the major stabilization of the foot during balanced standing, with minimal reaction
from all other intrinsic and extrinsic muscles. Therefore, only the Achilles tendon loading was con-
sidered during simulated balanced standing, while other intrinsic and extrinsic muscle forces were
neglected.
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