ARUP; ISBN: 978-0-9562121-5-3 - CMBBE 2012 - Cardiff University

associated with the volumetric change J = det F , F is the deformation gradient tensor
defined as F = ∂x ∂X
. The bulk modulus k plays the role of a parameter penalizing the
volumetric change quantified by J − 1.
In the limit of k → +∞ , any volumetric change is
strongly penalized, and the above formulation approaches total incompressibility. The
third term which arises from the use of the mixed type formulation is a function of both
the hydrostatic pressure p computed from the displacement field and the interpolated
pressure pɶ from the unknown nodal variables.
The term W is a function of the adopted hyperelastic constitutive model. We assumed
pas
the following transversely isotropic model W transversely for skeletal muscles
pas matrix fiber
W = W + W . (4)
tranversely
matrix
The term W corresponds to the connective tissue of muscles, assumed to behave as a
Mooney-Rivlin material,
2
matrix
i j
W = ∑ cij ( I1 −3) ( I2
−3)
(5)
i+ j=
1
where c are material constants. ij
I and 1 I are the reduced invariants of C defined as
2
fiber
I2 = 1/ 2 ⎡tr( C) − tr(
C ) ⎤ . The term
2 2
I1 = tr(
C ) and ⎣ ⎦
connects the normal Cauchy stress along fiber direction pas
fiber
pas where σ 0
fiber
∂W
λ
pas pas f pas
λf = σ fiber = σ0
f fiber
∂λ
λ0
is a material constant and 0
W corresponds to muscle fibers and
σ with fiber’s stretch λ f as
λ is the optimal fiber stretch. pas
f fiber is the
normalized passive force given by
⎧0,
*
for λ ≤1
pas ⎪
*
f fiber = ⎨γ
1[exp{ γ 2(
λ −1)} −1],
⎪
*
⎩γ
1γ 2 exp(0.4 γ 2)*( λ − 1.4) + γ1[exp(0.4 γ 2)
−1],
*
for 1< λ ≤1.4
*
for λ > 1.4
(7)
*
where γ 1 and γ 2 are material constants and λ is the normalized stretch defined as
*
λ = λ / λ [4].
f
0
3.2 Muscle contraction model
A Hill-type model was adopted to simulate muscle contraction [6]. The Cauchy stress
component along fiber direction (see Eq. (1)) is defined as
Fig.1 Schematic representation of (a) activation, (b) length and (b) velocity dependence
factors for the muscle contraction model used in the numerical simulations.
(6)