In Radioss material laws LAW32, LAW43, LAW72, LAW73,
LAW74, LAW78 and LAW93 use HILL criteria.
HILL Criteria
The typical HILL criteria is:
3D equivalent HILL stress:(6)
f =
F
(
σ
y y
−
σ
z z
)
2
+ G
(
σ
z z
−
σ
x x
)
2
+ H
(
σ
x x
−
σ
y y
)
2
+ 2 L
σ
y z
2
+ 2 M
σ
z x
2
+ 2 N
σ
x y
2
=
(
G + H
)
︸
σ
x x
2
+
(
F + H
)
︸
σ
y y
2
+
(
F + G
)
︸
σ
z z
2
−
2 H
︸
σ
x x
σ
y y
−
2 F
︸
σ
y y
σ
z z
−
2 G
︸
σ
z z
σ
x x
+
2 L
︸
σ
y z
2
+
2 M
︸
σ
z x
2
+
2 N
︸
σ
x y
2
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9
vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr
0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakqaabeqaaiaadA
gacqGH9aqpdaGcaaqaaiaadAeadaqadiqaaiabeo8aZnaaBaaaleaa
caWG5bGaamyEaaqabaGccqGHsislcqaHdpWCdaWgaaWcbaGaamOEai
aadQhaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGa
ey4kaSIaam4ramaabmGabaGaeq4Wdm3aaSbaaSqaaiaadQhacaWG6b
aabeaakiabgkHiTiabeo8aZnaaBaaaleaacaWG4bGaamiEaaqabaaa
kiaawIcacaGLPaaadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaWGib
WaaeWaceaacqaHdpWCdaWgaaWcbaGaamiEaiaadIhaaeqaaOGaeyOe
I0Iaeq4Wdm3aaSbaaSqaaiaadMhacaWG5baabeaaaOGaayjkaiaawM
caamaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaygW7caaIYaGaamit
aiabeo8aZnaaDaaaleaacaWG5bGaamOEaaqaaiaaikdaaaGccqGHRa
WkcaaIYaGaamytaiabeo8aZnaaDaaaleaacaWG6bGaamiEaaqaaiaa
ikdaaaGccqGHRaWkcaaIYaGaamOtaiabeo8aZnaaDaaaleaacaWG4b
GaamyEaaqaaiaaikdaaaaabeaaaOqaaiaabccacaqGGaGaaeiiaiaa
b2dadaGcaaqaamaayaaabaWaaeWaaeaacaWGhbGaey4kaSIaamisaa
GaayjkaiaawMcaaaWcbaaakiaawIJ=aiabeo8aZnaaDaaaleaacaWG
4bGaamiEaaqaaiaaikdaaaGccqGHRaWkdaagaaqaamaabmaabaGaam
OraiabgUcaRiaadIeaaiaawIcacaGLPaaaaSqaaaGccaGL44pacqaH
dpWCdaqhaaWcbaGaamyEaiaadMhaaeaacaaIYaaaaOGaey4kaSYaaG
baaeaadaqadaqaaiaadAeacqGHRaWkcaWGhbaacaGLOaGaayzkaaaa
leaaaOGaayjo+dGaeq4Wdm3aa0baaSqaaiaadQhacaWG6baabaGaaG
OmaaaakiabgkHiTmaayaaabaGaaGOmaiaadIeaaSqaaaGccaGL44pa
cqaHdpWCdaWgaaWcbaGaamiEaiaadIhaaeqaaOGaeq4Wdm3aaSbaaS
qaaiaadMhacaWG5baabeaakiabgkHiTmaayaaabaGaaGOmaiaadAea
aSqaaaGccaGL44pacqaHdpWCdaWgaaWcbaGaamyEaiaadMhaaeqaaO
Gaeq4Wdm3aaSbaaSqaaiaadQhacaWG6baabeaakiabgkHiTmaayaaa
baGaaGOmaiaadEeaaSqaaaGccaGL44pacqaHdpWCdaWgaaWcbaGaam
OEaiaadQhaaeqaaOGaeq4Wdm3aaSbaaSqaaiaadIhacaWG4baabeaa
kiabgUcaRmaayaaabaGaaGzaVlaaikdacaWGmbaaleaaaOGaayjo+d
Gaeq4Wdm3aa0baaSqaaiaadMhacaWG6baabaGaaGOmaaaakiabgUca
RmaayaaabaGaaGOmaiaad2eaaSqaaaGccaGL44pacqaHdpWCdaqhaa
WcbaGaamOEaiaadIhaaeaacaaIYaaaaOGaey4kaSYaaGbaaeaacaaI
YaGaamOtaaWcbaaakiaawIJ=aiabeo8aZnaaDaaaleaacaWG4bGaam
yEaaqaaiaaikdaaaaabeaaaaaa@DD25@
Shell element:(7)
f
=
F
σ
y
y
2
+
G
σ
x
x
2
+
H
(
σ
x
x
−
σ
y
y
)
2
+
2
N
σ
x
y
2
=
(
G
+
H
)
︸
σ
x
x
2
+
(
F
+
H
)
︸
σ
y
y
2
−
2
H
︸
σ
x
x
σ
y
y
+
2
N
︸
σ
x
y
2
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqGqFfpeea0xe9vq=Jb9
vqpeea0xd9q8qiYRWxGi6xij=hbba9q8aq0=yq=He9q8qiLsFr0=vr
0=vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGMbGaey
ypa0ZaaOaaaeaacaWGgbGaeq4Wdm3aa0baaSqaaiaadMhacaWG5baa
baGaaGOmaaaakiabgUcaRiaadEeacqaHdpWCdaqhaaWcbaGaamiEai
aadIhaaeaacaaIYaaaaOGaey4kaSIaamisamaabmGabaGaeq4Wdm3a
aSbaaSqaaiaadIhacaWG4baabeaakiabgkHiTiabeo8aZnaaBaaale
aacaWG5bGaamyEaaqabaaakiaawIcacaGLPaaadaahaaWcbeqaaiaa
ikdaaaGccqGHRaWkcaaIYaGaamOtaiabeo8aZnaaDaaaleaacaWG4b
GaamyEaaqaaiaaikdaaaaabeaakiabg2da9maakaaabaWaaGbaaeaa
daqadaqaaiaadEeacqGHRaWkcaWGibaacaGLOaGaayzkaaaaleaaaO
Gaayjo+dGaeq4Wdm3aa0baaSqaaiaadIhacaWG4baabaGaaGOmaaaa
kiabgUcaRmaayaaabaWaaeWaaeaacaWGgbGaey4kaSIaamisaaGaay
jkaiaawMcaaaWcbaaakiaawIJ=aiabeo8aZnaaDaaaleaacaWG5bGa
amyEaaqaaiaaikdaaaGccqGHsisldaagaaqaaiaaikdacaWGibaale
aaaOGaayjo+dGaeq4Wdm3aaSbaaSqaaiaadIhacaWG4baabeaakiab
eo8aZnaaBaaaleaacaWG5bGaamyEaaqabaGccqGHRaWkdaagaaqaai
aaikdacaWGobaaleaaaOGaayjo+dGaeq4Wdm3aa0baaSqaaiaadIha
caWG5baabaGaaGOmaaaaaeqaaaaa@863E@
Where,
F
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@
,
G
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@
,
H
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@
,
L
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@
,
M
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@
and
N
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@
are six HILL anisotropic parameters. For
shell elements, only
F
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@
,
G
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@
,
H
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@
and
N
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraaaa@36C1@
are the four HILL parameters
needed.
In LAW78, the HILL criteria is:
(8)
φ
(
A
)
=
1
︸
G
+
H
⋅
A
x
x
2
−
2
r
0
1
+
r
0
︸
2
H
A
x
x
A
y
y
+
r
0
(
1
+
r
90
)
r
90
(
1
+
r
0
)
︸
F
+
H
A
y
y
2
+
r
0
+
r
90
r
90
(
1
+
r
0
)
(
2
r
45
+
1
)
︸
2
N
A
x
y
2
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9
Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHgpGAca
GGOaGaamyqaiaacMcacqGH9aqpdaagaaqaaiaaigdaaSqaaiaadEea
cqGHRaWkcaWGibaakiaawIJ=aiabgwSixlaadgeadaqhaaWcbaGaam
iEaiaadIhaaeaacaaIYaaaaOGaeyOeI0YaaGbaaeaadaWcaaqaaiaa
ikdacaWGYbWaaSbaaSqaaiaaicdaaeqaaaGcbaGaaGymaiabgUcaRi
aadkhadaWgaaWcbaGaaGimaaqabaaaaaqaaiaaikdacaWGibaakiaa
wIJ=aiaadgeadaWgaaWcbaGaamiEaiaadIhaaeqaaOGaamyqamaaBa
aaleaacaWG5bGaamyEaaqabaGccqGHRaWkdaagaaqaamaalaaabaGa
amOCamaaBaaaleaacaaIWaaabeaakmaabmaabaGaaGymaiabgUcaRi
aadkhadaWgaaWcbaGaaGyoaiaaicdaaeqaaaGccaGLOaGaayzkaaaa
baGaamOCamaaBaaaleaacaaI5aGaaGimaaqabaGcdaqadaqaaiaaig
dacqGHRaWkcaWGYbWaaSbaaSqaaiaaicdaaeqaaaGccaGLOaGaayzk
aaaaaaWcbaGaamOraiabgUcaRiaadIeaaOGaayjo+dGaamyqamaaDa
aaleaacaWG5bGaamyEaaqaaiaaikdaaaGccqGHRaWkdaagaaqaamaa
laaabaGaamOCamaaBaaaleaacaaIWaaabeaakiabgUcaRiaadkhada
WgaaWcbaGaaGyoaiaaicdaaeqaaaGcbaGaamOCamaaBaaaleaacaaI
5aGaaGimaaqabaGcdaqadaqaaiaaigdacqGHRaWkcaWGYbWaaSbaaS
qaaiaaicdaaeqaaaGccaGLOaGaayzkaaaaamaabmaabaGaaGOmaiaa
dkhadaWgaaWcbaGaaGinaiaaiwdaaeqaaOGaey4kaSIaaGymaaGaay
jkaiaawMcaaaWcbaGaaGOmaiaad6eaaOGaayjo+dGaamyqamaaDaaa
leaacaWG4bGaamyEaaqaaiaaikdaaaaaaa@8B69@
There are two ways to determine HILL parameters
by using Lankford parameters.
Strain ratio
r
00
,
r
45
,
r
90
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa
aaleaacaaIWaGaaGimaaqabaGccaGGSaGaamOCamaaBaaaleaacaaI
0aGaaGynaaqabaGccaGGSaGaamOCamaaBaaaleaacaaI5aGaaGimaa
qabaaaaa@3F42@
(LAW32, LAW43, LAW72, LAW73)
Yield stress ratio
R
11
,
R
22
,
R
33
,
R
12
,
R
13
,
R
23
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa
aaleaacaaIXaGaaGymaaqabaGccaGGSaGaamOuamaaBaaaleaacaaI
YaGaaGOmaaqabaGccaGGSaGaamOuamaaBaaaleaacaaIZaGaaG4maa
qabaGccaGGSaGaamOuamaaBaaaleaacaaIXaGaaGOmaaqabaGccaGG
SaGaamOuamaaBaaaleaacaaIXaGaaG4maaqabaGccaGGSaGaamOuam
aaBaaaleaacaaIYaGaaG4maaqabaaaaa@487B@
(LAW74, LAW93)
Strain Ratio
The Lankford parameters
r
α
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9
Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadkhapaWaaSbaaSqaa8qacqaHXoqya8aabeaaaaa@396F@
is the ratio of plastic strain in plane and plastic
strain in thickness direction
ε
33
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9
Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
Wdbiabew7aL9aadaWgaaWcbaWdbiaaiodacaaIZaaapaqabaaaaa@39FA@
.
(9)
r
α
=
d
ε
α
+
π
/
2
d
ε
33
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9
Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadkhapaWaaSbaaSqaa8qacqaHXoqya8aabeaak8qacqGH9aqp
daWcaaWdaeaapeGaamizaiabew7aL9aadaWgaaWcbaWdbiabeg7aHj
abgUcaRiabec8aWjaac+cacaaIYaaapaqabaaakeaapeGaamizaiab
ew7aL9aadaWgaaWcbaWdbiaaiodacaaIZaaapaqabaaaaaaa@47D3@
Where,
α
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9
Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaaleaaqaaaaaaaaa
Wdbiabeg7aHbaa@381F@
is the angle to the orthotropic
direction 1.
r
α
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9
Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaqaaaaaaaaa
WdbiaadkhapaWaaSbaaSqaa8qacqaHXoqya8aabeaaaaa@396F@
could be measured with different samples which cut
in different angle with orthotropic direction 1. Like
r
00
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa
aaleaacaaIWaGaaGimaaqabaaaaa@388E@
measured from tensile test in which the loading
direction is along the orthotropic direction 1.
r
90
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa
aaleaacaaI5aGaaGimaaqabaaaaa@3897@
measured from tensile test in which the loading is
perpendicular to orthotropic direction 1.
The strain ratio is the strain in width direction of sample to strain in thickness
direction of sample.
Figure 10.
In this case, the HILL parameters are:
(10)
F
=
r
00
r
90
(
r
00
+
1
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9
Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGgbGaey
ypa0ZaaSaaaeaacaWGYbWaaSbaaSqaaiaaicdacaaIWaaabeaaaOqa
aiaadkhadaWgaaWcbaGaaGyoaiaaicdaaeqaaOGaaiikaiaadkhada
WgaaWcbaGaaGimaiaaicdaaeqaaOGaey4kaSIaaGymaiaacMcaaaaa aa@4322@
(11)
G
=
1
(
r
00
+
1
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9
Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGhbGaey
ypa0ZaaSaaaeaacaaIXaaabaGaaiikaiaadkhadaWgaaWcbaGaaGim
aiaaicdaaeqaaOGaey4kaSIaaGymaiaacMcaaaaaaa@3E93@
(12)
H
=
r
00
(
r
00
+
1
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9
Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibGaey
ypa0ZaaSaaaeaacaWGYbWaaSbaaSqaaiaaicdacaaIWaaabeaaaOqa
aiaacIcacaWGYbWaaSbaaSqaaiaaicdacaaIWaaabeaakiabgUcaRi
aaigdacaGGPaaaaaaa@407A@
(13)
N
=
(
1
+
2
r
45
)
(
r
00
+
r
90
)
2
r
90
(
r
00
+
1
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9
Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGobGaey
ypa0ZaaSaaaeaacaGGOaGaaGymaiabgUcaRiaaikdacaWGYbWaaSba
aSqaaiaaisdacaaI1aaabeaakiaacMcacaGGOaGaamOCamaaBaaale
aacaaIWaGaaGimaaqabaGccqGHRaWkcaWGYbWaaSbaaSqaaiaaiMda
caaIWaaabeaakiaacMcaaeaacaaIYaGaamOCamaaBaaaleaacaaI5a
GaaGimaaqabaGccaGGOaGaamOCamaaBaaaleaacaaIWaGaaGimaaqa
baGccqGHRaWkcaaIXaGaaiykaaaaaaa@4F27@
Here,
G
+
H
=
1
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9
Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGhbGaey
4kaSIaamisaiabg2da9iaaigdaaaa@3A9B@
.
In LAW32, LAW43, and LAW73, the HILL criteria is:
(14)
σ
e
q
=
A
1
σ
1
2
+
A
2
σ
2
2
−
A
3
σ
1
σ
2
+
A
12
σ
12
2
R
=
r
00
+
2
r
45
+
r
90
4
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9
Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGsbGaey
ypa0ZaaSaaaeaacaWGYbWaaSbaaSqaaiaaicdacaaIWaaabeaakiab
gUcaRiaaikdacaWGYbWaaSbaaSqaaiaaisdacaaI1aaabeaakiabgU
caRiaadkhadaWgaaWcbaGaaGyoaiaaicdaaeqaaaGcbaGaaGinaaaa
aaa@437F@
H
=
R
1
+
R
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9
Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibGaey
ypa0ZaaSaaaeaacaWGsbaabaGaaGymaiabgUcaRiaadkfaaaaaaa@3B8D@
A
1
=
H
(
1
+
1
r
00
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9
Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS
baaSqaaiaaigdaaeqaaOGaeyypa0JaamisamaabmaabaGaaGymaiab
gUcaRmaalaaabaGaaGymaaqaaiaadkhadaWgaaWcbaGaaGimaiaaic
daaeqaaaaaaOGaayjkaiaawMcaaaaa@407B@
A
2
=
H
(
1
+
1
r
90
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9
Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS
baaSqaaiaaikdaaeqaaOGaeyypa0JaamisamaabmaabaGaaGymaiab
gUcaRmaalaaabaGaaGymaaqaaiaadkhadaWgaaWcbaGaaGyoaiaaic
daaeqaaaaaaOGaayjkaiaawMcaaaaa@4085@
A
3
=
2
H
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9
Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS
baaSqaaiaaiodaaeqaaOGaeyypa0JaaGOmaiaadIeaaaa@3AA7@
A
12
=
2
H
(
r
45
+
0.5
)
(
1
r
00
+
1
r
90
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9
Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGbbWaaS
baaSqaaiaaigdacaaIYaaabeaakiabg2da9iaaikdacaWGibGaaiik
aiaadkhadaWgaaWcbaGaaGinaiaaiwdaaeqaaOGaey4kaSIaaGimai
aac6cacaaI1aGaaiykamaabmaabaWaaSaaaeaacaaIXaaabaGaamOC
amaaBaaaleaacaaIWaGaaGimaaqabaaaaOGaey4kaSYaaSaaaeaaca
aIXaaabaGaamOCamaaBaaaleaacaaI5aGaaGimaaqabaaaaaGccaGL
OaGaayzkaaaaaa@4BBD@
They all request Lankford parameter (strain ratio)
r
00
,
r
45
,
r
90
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOCamaaBa
aaleaacaaIWaGaaGimaaqabaGccaGGSaGaamOCamaaBaaaleaacaaI
0aGaaGynaaqabaGccaGGSaGaamOCamaaBaaaleaacaaI5aGaaGimaa
qabaaaaa@3F42@
and the HILL parameter
A
i
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyqamaaBa
aaleaacaWGPbaabeaaaaa@37D6@
is automatically computed by Radioss .
Yield Stress Ratio
In LAW93, the yield stress ratio used is:
(15)
R
i
j
=
σ
i
j
σ
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa
aaleaacaWGPbGaamOAaaqabaGccqGH9aqpdaWcaaqaaiabeo8aZnaa
BaaaleaacaWGPbGaamOAaaqabaaakeaacqaHdpWCdaWgaaWcbaGaaG
imaaqabaaaaaaa@4076@
To get yield stress ratio
R
i
j
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa
aaleaacaWGPbGaamOAaaqabaaaaa@38D7@
, yield stress in two loading cases need to be
measured.
Yield stress
σ
11
,
σ
22
,
σ
33
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS
baaSqaaiaaigdacaaIXaaabeaakiaacYcacqaHdpWCdaWgaaWcbaGa
aGOmaiaaikdaaeqaaOGaaiilaiabeo8aZnaaBaaaleaacaaIZaGaaG
4maaqabaaaaa@41A0@
from tensile test
Yield shear stress
σ
12
,
σ
13
,
σ
23
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS
baaSqaaiaaigdacaaIYaaabeaakiaacYcacqaHdpWCdaWgaaWcbaGa
aGymaiaaiodaaeqaaOGaaiilaiabeo8aZnaaBaaaleaacaaIYaGaaG
4maaqabaaaaa@41A0@
from shear test
In LAW93, if parameter input is used, then take initial stress parameter
σ
y
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS
baaSqaaiaadMhaaeqaaaaa@38E4@
as reference yield stress
σ
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS
baaSqaaiaaicdaaeqaaaaa@38A0@
. If curve input is used, then take the yield stress
from curve as reference yield stress
σ
0
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS
baaSqaaiaaicdaaeqaaaaa@38A0@
.
Four HILL parameters for shell are automatically computed by
Radioss .
(16)
F
=
1
2
(
1
R
22
2
+
1
R
33
2
−
1
R
11
2
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOraiabg2
da9maalaaabaGaaGymaaqaaiaaikdaaaGaaiikamaalaaabaGaaGym
aaqaaiaadkfadaqhaaWcbaGaaGOmaiaaikdaaeaacaaIYaaaaaaaki
abgUcaRmaalaaabaGaaGymaaqaaiaadkfadaqhaaWcbaGaaG4maiaa
iodaaeaacaaIYaaaaaaakiabgkHiTmaalaaabaGaaGymaaqaaiaadk
fadaqhaaWcbaGaaGymaiaaigdaaeaacaaIYaaaaaaakiaacMcaaaa@489E@
(17)
G
=
1
2
(
1
R
33
2
+
1
R
11
2
−
1
R
22
2
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiabg2
da9maalaaabaGaaGymaaqaaiaaikdaaaGaaiikamaalaaabaGaaGym
aaqaaiaadkfadaqhaaWcbaGaaG4maiaaiodaaeaacaaIYaaaaaaaki
abgUcaRmaalaaabaGaaGymaaqaaiaadkfadaqhaaWcbaGaaGymaiaa
igdaaeaacaaIYaaaaaaakiabgkHiTmaalaaabaGaaGymaaqaaiaadk
fadaqhaaWcbaGaaGOmaiaaikdaaeaacaaIYaaaaaaakiaacMcaaaa@489F@
(18)
H
=
1
2
(
1
R
22
2
+
1
R
11
2
−
1
R
33
2
)
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamisaiabg2
da9maalaaabaGaaGymaaqaaiaaikdaaaGaaiikamaalaaabaGaaGym
aaqaaiaadkfadaqhaaWcbaGaaGOmaiaaikdaaeaacaaIYaaaaaaaki
abgUcaRmaalaaabaGaaGymaaqaaiaadkfadaqhaaWcbaGaaGymaiaa
igdaaeaacaaIYaaaaaaakiabgkHiTmaalaaabaGaaGymaaqaaiaadk
fadaqhaaWcbaGaaG4maiaaiodaaeaacaaIYaaaaaaakiaacMcaaaa@48A0@
(19)
N
=
3
2
R
12
2
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaiabg2
da9maalaaabaGaaG4maaqaaiaaikdacaWGsbWaa0baaSqaaiaaigda
caaIYaaabaGaaGOmaaaaaaaaaa@3C90@
In LAW74, yield stress ratio
R
i
j
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOuamaaBa
aaleaacaWGPbGaamOAaaqabaaaaa@38D7@
is used with yield stress
σ
11
,
σ
22
,
σ
33
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS
baaSqaaiaaigdacaaIXaaabeaakiaacYcacqaHdpWCdaWgaaWcbaGa
aGOmaiaaikdaaeqaaOGaaiilaiabeo8aZnaaBaaaleaacaaIZaGaaG
4maaqabaaaaa@41A0@
and
σ
12
,
σ
13
,
σ
23
MathType@MTEF@5@5@+=
feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x
fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS
baaSqaaiaaigdacaaIYaaabeaakiaacYcacqaHdpWCdaWgaaWcbaGa
aGymaiaaiodaaeqaaOGaaiilaiabeo8aZnaaBaaaleaacaaIYaGaaG
4maaqabaaaaa@41A0@
input directly, and then six HILL parameters for
solid are automatically computed by
Radioss .
F
=
1
2
(
1
σ
22
2
+
1
σ
33
2
−
1
σ
11
2
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9
Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGgbGaey
ypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaqadaqaamaalaaabaGa
aGymaaqaaiabeo8aZnaaDaaaleaacaaIYaGaaGOmaaqaaiaaikdaaa
aaaOGaey4kaSYaaSaaaeaacaaIXaaabaGaeq4Wdm3aa0baaSqaaiaa
iodacaaIZaaabaGaaGOmaaaaaaGccqGHsisldaWcaaqaaiaaigdaae
aacqaHdpWCdaqhaaWcbaGaaGymaiaaigdaaeaacaaIYaaaaaaaaOGa
ayjkaiaawMcaaaaa@4BF9@
G
=
1
2
(
1
σ
22
2
+
1
σ
33
2
−
1
σ
11
2
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9
Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGhbGaey
ypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaqadaqaamaalaaabaGa
aGymaaqaaiabeo8aZnaaDaaaleaacaaIYaGaaGOmaaqaaiaaikdaaa
aaaOGaey4kaSYaaSaaaeaacaaIXaaabaGaeq4Wdm3aa0baaSqaaiaa
iodacaaIZaaabaGaaGOmaaaaaaGccqGHsisldaWcaaqaaiaaigdaae
aacqaHdpWCdaqhaaWcbaGaaGymaiaaigdaaeaacaaIYaaaaaaaaOGa
ayjkaiaawMcaaaaa@4BFA@
H
=
1
2
(
1
σ
22
2
+
1
σ
33
2
−
1
σ
11
2
)
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9
Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGibGaey
ypa0ZaaSaaaeaacaaIXaaabaGaaGOmaaaadaqadaqaamaalaaabaGa
aGymaaqaaiabeo8aZnaaDaaaleaacaaIYaGaaGOmaaqaaiaaikdaaa
aaaOGaey4kaSYaaSaaaeaacaaIXaaabaGaeq4Wdm3aa0baaSqaaiaa
iodacaaIZaaabaGaaGOmaaaaaaGccqGHsisldaWcaaqaaiaaigdaae
aacqaHdpWCdaqhaaWcbaGaaGymaiaaigdaaeaacaaIYaaaaaaaaOGa
ayjkaiaawMcaaaaa@4BFB@
L
=
1
2
σ
23
2
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9
Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGmbGaey
ypa0ZaaSaaaeaacaaIXaaabaGaaGOmaiabeo8aZnaaDaaaleaacaaI
YaGaaG4maaqaaiaaikdaaaaaaaaa@3DE1@
M
=
1
2
σ
31
2
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9
Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbGaey
ypa0ZaaSaaaeaacaaIXaaabaGaaGOmaiabeo8aZnaaDaaaleaacaaI
ZaGaaGymaaqaaiaaikdaaaaaaaaa@3DE1@
N
=
1
2
σ
12
2
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9
Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGobGaey
ypa0ZaaSaaaeaacaaIXaaabaGaaGOmaiabeo8aZnaaDaaaleaacaaI
XaGaaGOmaaqaaiaaikdaaaaaaaaa@3DE1@
For shell element, take
M
=
N
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9
Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbGaey
ypa0JaamOtaaaa@3909@
and
L
=
N
MathType@MTEF@5@5@+=
feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9
vqaqpepm0xbbG8FasPYRqj0=yi0dXdbba9pGe9xq=JbbG8A8frFve9
Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGnbGaey
ypa0JaamOtaaaa@3909@
.