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

1 1 2
the volumetric part of the strain energy potential w : ( ) ( )
⎡ ⎤
f J : = J 1 ln J
D ⎢
− −
⎣ 2
⎥
⎦ .
Description of the viscoelastic behaviour of gluteal tissue is approached using the
linear-viscoelastic theory valid for finite deformations as introduced by Simo (1987).
Within the decomposition of the KIRCHHOFF stress tensor ττττ
D
( t)
= ττττ ( t)
ττττ ( t)
Η
ττττ +
the volumetric and deviatoric parts, H
ττττ and D
ττττ , read
H H
0
t
∫ κ
t′=
0
H
0
( ) : = ( ) + ( ) ( − )
ττ ττ t ττ ττ t t′ τ ττ
τ t t′ dt′
D D
0
(4) t
∫
t′=
0
t
D
0
t
T
−1 −
( ) ≡ ( ) + p⋅⋅ γ ( ) F ( − ) ⋅ ( − ) ⋅ F ( − )
ττ ττ t ττ ττ t t′ t t′ τ ττ
τ t t′ t t′ dt′
.
In (3) the KIRCHHOFF stresses H
0
In (3) and (4),
( 4)
⋅
( )
H ∂f
J
ττττ 0 = J I and
∂J
( 4)
whereby I ⋅A
= A
ττττ and D
ττττ at steady state are defined by
0
( C,H i )
(2)
(4) ⎡ ∂w
⎤
D T
ττττ 0 = p⋅⋅ ⎢2F ⋅ ⋅ F ⎥ . (4)
⎢⎣ ∂C
⎥⎦
p is the fourth-order spatial deviator operator with p : = I − ( 1/
3)II
1 and ⋅⋅ = ( tr )
−1
( t′
) : = F ( t′
) ⋅ F ( t)
II A A I apply for arbitrary second order tensors A, and
F t
is the relative deviatoric deformation gradient. The shear and bulk
relaxation functions are commonly based on a generalized MAXWELL-model and are
given by
⎡ NG
⎛ t
− ⎞⎤
G
i
( ) ⎢
0 1 ⎜ τ
G t = G − i 1− e ⎟⎥
⎢ ∑ g
⎥
i= 1 ⎜ ⎟
⎣ ⎝ ⎠⎦
Further, with regard to Gi G0g
i
γ read
and shear modulus ( t)
( )
0
and ( )
( 4)
( 4)
⎡ Nκ
⎛ t
− ⎞⎤
K
⎢ i
0 1 ⎜ τ
K t = K − i 1− e ⎟⎥
⎢ ∑ k
. (5)
⎥
i= 1 ⎜ ⎟
⎣ ⎝ ⎠⎦
:= (i = 1, 2,…..) the time-dependent bulk modulus κ ( t)
NK
i
t
−
K
τi
K
i= 1 τi
K̇ t k
κ ( t) : = = −∑ e
K
and γ ( )
( )
0
NG
i
t
−
G
τi
G
i= 1 τi
Ġ t g
t : = = −∑ e
G
1
. (6)
In (5)1 and (5)2, 0 G and K 0 are the instantaneous shear and bulk modulus,
respectively. The- i g , K
ki -and-τ i -and- G
τi -are material parameters to be calibrated using
creep or relaxation test data. Inserting (1) into (4) and with regard to (2) and (3), the
total KIRCHHOFF stress tensor for linear-viscoelastic isotropic materials derives to
(3)