This law uses a generalized viscoelastic Kelvin-Voigt model whereas the viscosity is
based on the Navier equations.
The effect of the enclosed air is taken into account via a separate pressure versus
compression function. For open cell foam, this function may be replaced by an equivalent
"removed air pressure" function. The model takes into account the relaxation (zero strain
rate), creep (zero stress rate), and unloading. It may be used for open cell foams,
polymers, elastomers, seat cushions, dummy paddings, and so on. In Radioss the law is compatible with shell and solid meshes.
The simple schematic model in Figure 1 describes the generalized Kelvin-Voigt material model where a
time-dependent spring working in parallel with a Navier dashpot is put in series with a
nonlinear rate-dependent spring. If
σ
m
=
I
1
3
is the mean stress, the deviatoric stresses
s
i
j
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9
vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr
0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGZbWaaS
baaSqaaiaadMgacaWGQbaabeaaaaa@3A69@
at steps
n
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36E9@
and
n + 1
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf
MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi
ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8
qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9
q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake
aacaWGUbGaey4kaSIaaGymaaaa@3B67@
are computed
by the expressions:
for
i
=
j
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9
vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr
0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGPbGaey ypa0JaamOAaaaa@3A4B@
else,
δ
i
j
=
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9
vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr
0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH0oazda
WgaaWcbaGaamyAaiaadQgaaeqaaOGaeyypa0JaaGimaaaa@3CE0@
with:
for
i
≠
j
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9
vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr
0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGPbGaey iyIKRaamOAaaaa@3B0C@
for
i
=
j
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9
vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr
0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGPbGaey ypa0JaamOAaaaa@3A4B@
Where,
G
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9
vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr
0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGhbaaaa@3834@
and
G
t
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9
vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr
0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGhbWaaS
baaSqaaiaadshaaeqaaaaa@3959@
are defined as:
G
=
max
(
E
2
(
1
+
ν
)
,
A
e
˙
+
B
2
(
1
+
ν
)
)
In
Equation 5 the coefficients
A
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36E9@
and
B
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@36E9@
are defined for Young's modulus updates (
E
=
E
1
ε
˙
+
E
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9
vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr
0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGfbGaey
ypa0JaamyramaaBaaaleaacaaIXaaabeaakiqbew7aLzaacaGaey4k
aSIaamyramaaBaaaleaacaaIYaaabeaaaaa@3F37@
).
Figure 1 . Generalized Kelvin-Voigt Model
The expressions used by default to compute the pressure is:
d
P
d
t
=
C
1
K
ε
˙
k
k
−
C
2
[
K
+
K
t
3
λ
+
2
η
0
σ
k
k
]
+
C
3
[
K
K
t
3
λ
+
2
η
0
ε
k
k
]
Where,
λ
and
η
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9
vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr
0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH3oaAda
WgaaWcbaGaaGimaaqabaaaaa@39FA@
are the Navier Stokes viscosity coefficients which can be
compared to Lame constants in elasticity.
λ
+
2
η
0
3
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9
vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr
0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaH7oaBcq
GHRaWkdaWcaaqaaiaaikdacqaH3oaAdaWgaaWcbaGaaGimaaqabaaa keaacaaIZaaaaaaa@3E23@
is called the volumetric coefficient of viscosity. For
incompressible model,
ε
k
k
v
=
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aa0
baaSqaaiaadUgacaWGRbaabaGaamODaaaakiabg2da9iaaicdaaaa@3C70@
and
λ
→
∞
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaey OKH4QaeyOhIukaaa@3B09@
and
μ
0
=
μ
3
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd02aaS
baaSqaaiaaicdaaeqaaOGaeyypa0ZaaSaaaeaacqaH8oqBaeaacaaI Zaaaaaaa@3C26@
. In Equation 11 , C 1 , C 2 and
C 3
are Boolean multipliers used to define different responses. For example, C 1 =1,
C 2 =C 3 =0 refers to a linear bulk model. Similarly, C 1 =C 2 =C 3 =1
corresponds to a visco-elastic bulk model.
For polyurethane foams with closed cells, the skeletal spherical stresses may be increased
by:
P
a
i
r
=
−
P
0
⋅
γ
1
+
γ
−
Φ
Where,
γ
Volumetric strain
Φ
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOPdyeaaa@3771@
Porosity
P
0
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa
aaleaacaaIWaaabeaaaaa@37B1@
Initial air pressure
In Radioss , the pressure may also be computed with the
P
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiuaaaa@36CB@
versus
μ
=
ρ
ρ
0
−
1
, by a user-defined function. Air pressure may be assumed as an
"equivalent air pressure" versus
μ
. You can define this function used for open cell foams or for closed
cell by defining a model identical to material LAW 33 (FOAM_PLAS) .