Jou nal o Applied Ma hema ics and Physics, 2016, 4, 733-748
Published Online Ap il 2016 in SciRes. h p://www.sci p.o g/jou nal/jamp
h p://dx.doi.o g/10.4236//jamp.2016.44084
How o ci e his pape : Němec, I., T cala, M., Še čík, I. and Š ekbaue , H. (2016) New Fo mula o Geome ic S i ness Ma-
ix Calcula ion. Jou nal o Applied Ma hema ics and Physics, 4, 733-748. h p://dx.doi.o g/10.4236//jamp.2016.44084
New Fo mula o Geome ic S i ness
Ma ix Calcula ion
I. Němec1*, M. T cala2, I. Še čík2, H. Š ekbaue 1
1Facul y o Ci il Enginee ing, B no Uni e si y o Technology, B no, Czech Republic
2FEM Consul ing, S.R.O., B no, Czech Republic
Recei ed 4 June 2015; accep ed 24 Ap il 2016; published 27 Ap il 2016
Copy igh © 2016 by au ho s and Scien i ic Resea ch Publishing Inc.
This wo k is licensed unde he C ea i e Commons A ibu ion In e na ional License (CC BY).
h p://c ea i ecommons.o g/licenses/by/4.0/
Abs ac
The s anda d o mula o geome ic s i ness ma ix calcula ion, which is con enien o mos en-
ginee ing applica ions, is seen o be unsa is ac o y o la ge s ains because o poo accu acy, low
con e gence a e, and s abili y. Fo e y la ge comp essions, he angen s i ness in he di ec ion
o he comp ession can e en become nega i e, which can be ega ded as physical nonsense. So in
many cases ubbe ma e ials exposed o g ea comp ession canno be analyzed, o he analysis
could lead o e y poo con e gence. P oblems wi h he s anda d geome ic s i ness ma ix can
e en occu wi h a small s ain in he case o plas ic yielding, which e en ua es e en g ea e p ac-
ical p oblems. The au ho s demons a e ha amo e p ecisional app oach would no lead o such
s ange and heo e ically unjus i ied esul s. An imp o ed o mula ha would elimina e he dis-
ad an ages men ioned abo e and leads o highe con e gence a e and mo e obus compu a ions
is sugges ed in his pape . The new o mula can be de i ed om he p inciple o i ual wo k us-
ing a modi ied G een-Lag ange s ain enso , o om equilib ium condi ions whe e in he choice
o a speci ic s ain measu e is no needed o he geome ic s i ness de i a ion (which can also be
used o de i a ion o geome ic s i ness o a igid uss membe ). The new o mula has been e-
i ied in p ac ice wi h many calcula ions and implemen ed in he RFEM and SCIA Enginee p o-
g ams. The ad an ages o he new o mula in compa ison wi h he s anda d o mula a e shown
using se e al examples.
Keywo ds
Geome ic S i ness, S ess S i ness, Ini ial S ess S i ness, Tangen S i ness Ma ix, Fini e
Elemen Me hod, P inciple o Vi ual Wo k, S ain Measu e
*
Co esponding au ho .
I. Němec e al.
734
1. In oduc ion
S ess s i ening is an impo an sou ce o s i ness and mus be aken in o accoun when analyzing s uc u es.
The s anda d o mula o geome ic s i ness ma ices is in oduced by a numbe o au ho s, such as Zienkie-
wicz, Ba he, Cook, Bely schko, Simo, Hughes, Bone , de Souza Ne o and o he s [1]-[10]. The s anda d o mula
has been shown o be sa is ac o y in a la ge amoun o cases, hough ce ain di icul ies such as low accu acy,
poo con e gence a e and poo solu ion s abili y we e disco e ed when sol ing p oblems ha included he
e alua ion o ex eme s ess and s ain s a es. Some au ho s, e.g. Cook [4], ha e sugges ed an imp o emen o
ba s and some au ho s deal wi h nonlinea models desc ibing la ge ( ini e) de o ma ion (s ain) beha io o ma-
e ials and s uc u es [11]-[21]. Howe e , as a as he au ho s know, no gene al solu ion o he p oblem has
been sugges ed o a 2D o 3D con inuum. Upon his asce ainmen , hough s a ose conce ning he physical es-
sence o geome ic (o s ess) s i ness and he o mula o e alua ing geome ic s i ness ma ices. As a esul , a
new o mula o geome ic s i ness ma ix calcula ion is sugges ed. The p esen a ion o his new o mula, which
should subs an ially imp o e analysis o s uc u es exposed o la ge s ain, is he subjec ma e o his pape . In
Sec ion 2, he s anda d o mula o geome ic s i ness ma ices is p esen ed. Sec ion 3 shows he physical back-
g ound o geome ic s i ness based on equilib ium. In Sec ion 4, he new, imp o ed o mula o geome ic ma-
ices is in oduced. The ad an ages o he new o mula, including a subs an ially imp o ed a e o con e gence
and s abili y, a e demons a ed by examples in Sec ion5. Conclusions a e p esen ed in Sec ion 6.
2. The S anda d Fo mula o S ess-S i ness Ma ices
Le us show he gene al calcula ion algo i hm o he geome ic s i ness ma ix (some imes also called he s ess
s i ness ma ix o ini ial s ess ma ix) o an elemen in an upda ed Lag angian o mula ion.
Le he ollowing hold o each componen
i
u
o displacemen ec o
u
:
1
n
i a ia
a
u Nu
=
=
∑
(1)
whe e
ia
u
is he alue o displacemen
i
u
in node a and n is he numbe o elemen nodes.
Le us de ine ma ix
N
as ollows:
[ ]
12
, ,,n
NN N=N II I
(2)
whe e
I
is he uni diagonal ma ix o he o de 3 × 3, whe e 3 is he dimension o he p oblem. Then, he ol-
lowing ela ion can be w i en o he displacemen ec o :
= ⋅u Nd
(3)
whe e
d
is he ec o o de o ma ion pa ame e s o he elemen con aining all he componen s
ia
u
in such an
a angemen ha o each node a all componen s
i
u
a e lis ed.
Le us de ine ma ix
a
g
con aining he i s de i a i es o base unc ions o node a wi h espec o spa ial
coo dina es
,
,
T
,
ax
a
a ay
az
N
NN
N

∂
= ⊗=

∂

I
g II
xI
(4)
and ma ix
G
, which is o med by sub-ma ices
a
g
[ ]
12
T
,,,,,
an
∂
= =
∂
N
G gg g g
x

(5)
The ope a o
⊗
deno es he enso (K onecke ) ma ix p oduc .
Fu he , le us de ine ma ix
Σ
by mul iplying each componen o he Cauchy s ess enso
σ
by he uni
diagonal ma ix:
sym.
xx xy xz
yy yz
zz
σσσ
σσ
σ


=⊗=



III
I II
I
Σ
σ
(6)
I. Němec e al.
735
I s a e o he s ess is no negligible, he po en ial ene gy o he in e nal o ces should be comple ed by he
ollowing e m:
( )
TT
TT
TT
1 11
dd
2 22
σσ
Ω
Ω
⌠

⌡
∂ ∂ ∂∂
∏ = ⊗ Ω= Ω =
∂∂
∂∂
∫u u NN
I d d dKd
xx
xx
Σ
σ
(7)
Then, he ollowing o mula o he geome ic ma ix o he elemen can be w i en:
T
T
T
dd
σ
Ω
Ω
⌠

⌡
∂∂
= Ω= Ω
∂∂
∫
NN
K GG
xx
ΣΣ
(8)
In eg a ion is ca ied ou on he de o med body
Ω
(in he cu en con igu a ion) and he de i a i es a e pe -
o med wi h espec o he spa ial coo dina es.
The componen o he ma ix
σ
K
ela ing he elemen node a o he elemen node b can also be w i en
simply in ma ix no a ion:
( )
d
ab a b
NN
σ
Ω
= ∇ ⋅∇ Ω
∫
KσI
(9)
o in indicial no a ion:
d , 1, 2, 3
ab
abij kl ij
kl
NN
K ij
xx
σ
σδ
Ω
⌠

⌡
∂∂
= Ω=
∂∂
(10)
Simila o mulae also hold o a o al Lag angian o mula ion, bu he second Piola-Ki chho s ess enso is
hen used ins ead o he Cauchy s ess, and in eg a ion is ca ied ou on he unde o med body
0
Ω
(in he o ig-
inal con igu a ion) while he de i a i es a e pe o med wi h espec o he ma e ial coo dina es.
3. The Sou ce o Geome ic S i ness—The Physical Backg ound
Le us conside he uss membe shown in Figu e 1. Node 2 is loaded by he o ce F pa allel o he x axis and
sliding in he same di ec ion. The equilib ium equa ion in he x di ec ion in node 2 can be w i en as ollows
( ) ( )
0Rx Tx F= −=
(11)
whe e
( )
cosTx N
α
=
is he ho izon al componen o he in e nal o ce a node 2 and
cos xl
α
=
.
()
Rx
is
he esidual o ou -o -balance o ce. The ho izon al s i ness
x
K
a node 2 is de ined simply by he ela ion
2
2
22
22
22
dd d d d d d
ddd d d d d
dd
1 cos sin
dd
x
xM x
R T Nx N N N l N N N x N
K xx
x x x l xl l ll x l l l l
l
Nx N x N N KK
ll l l
ll
σ
αα
     
= = = = += += − +
     
     

= + −= + =+


(12)
This o mula is independen o any s ain measu e o pe inen cons i u i e ela ions. I can be seen ha s i ness
x
K
consis s o wo pa s. The i s pa ,
xM
K
, ep esen s he ma e ial s i ness and depends on he s ain measu e
and cons i u i e ela ions. The second pa ,
x
K
σ
, which does no depend on he ma e ial o he s ain and s ess
measu es chosen, bu only on he geome y and he no mal o ce, ep esen s so-called geome ic s i ness. I can
be seen ha i he angle
α
is ze o, no geome ic s i ness will occu ega dless o he no mal o ce alue.
Le us show a de i a ion o a o mula o geome ic s i ness ma ix o a uss membe (see Figu e 2) in a ini e
elemen o mula ion and le us s a wi h a simple de i a ion based on equilib ium condi ions.
Le
d
be a ec o o he nodal displacemen s o an elemen , and le
be a ec o o esidual o ces; he
s i ness ma ix o he elemen can hen be de ined as ollows:
∂
=∂
Kd
(13)
I. Němec e al.
736
Figu e 1. T uss membe in an a bi a y posi ion in 2D.
Figu e 2. T uss membe : he x axis is he axis o he od in i s o iginal posi ion.
A geome ic (s ess) s i ness ma ix can be ob ained by an equilib ium condi ion when only he ini ial s ess
s a e and pe inen in ini esimal nodal displacemen o each ow o he ma ix is aken in o accoun . Such a de-
ini ion o a geome ic s i ness ma ix is independen o he s ain enso chosen.
To simpli y he ollowing de i a ions le ’s in oduce bo h, he coo dina es
x
wi h he
x
axis aligned wi h he
axis o he od and co esponding displacemen ec o
u
and le ’s es ic he de o ma ion o he
xy
plane.
Le he ec o o he nodal displacemen s o he elemen be
[ ]
T
11 2 2
,, ,u u =d
(14)
whe e
u
and
a e he displacemen componen s in he di ec ion o he
x
and
y
axis, espec i ely. The
well known ma e ial s i ness ma ix o he uss elemen in 2D is hen de ined by he ollowing ela ion:
1 0 10
0000
10 1 0
0000
M
EA
l
−



=
−


K
(15)
No e ha he uss elemen has no la e al ma e ial s i ness.
In gene al, a bi a y e m o a s i ness ma ix
ij
K
is de ined as he de i a i e o an unbalanced o ce
i
wi h espec o he de o ma ion pa ame e
j
d
as is de ined by (13). Based on his de ini ion, he geome ic
s i ness ma ix o he uss elemen subjec ed o ensile o ce N can be easily de i ed. The momen equilib ium
condi ion o he uss membe in he con igu a ion wi h he la e al displacemen
d
in node 1 is su icien o
ob ain he ans e sal diagonal s i ness e m
22
K
:
The momen equilib ium condi ion can be w i en as ollows:
d d cos d 0N Tl
α
−=
(16)
I. Němec e al.
737
Fo he in ini esimal angle
d
α
i can be assumed ha
cos d 1
α
=
, and he ollowing e m o he s i ness
e m
22
K
σ
can be de i ed:
22
d
d
TN
K l
σ
= =
(17)
When in oducing a displacemen
du
in he di ec ion o he axis o he membe , he end o ces a e in equi-
lib ium and no addi ional o ce and he e o e no geome ic s i ness will occu in his di ec ion.
F om equilib ium equa ions and symme y o he s i ness ma ix i is easy o de e mine he o he coe icien s
o he geome ic s i ness ma ix, pa icula ly
24
K
σ
,
42
K
σ
and
44
K
σ
. The emaining coe icien s o he ma-
ix a e ze os. The geome ic s i ness ma ix hen has he ollowing o m:
0000
010 1
0000
0 10 1
N
l
σ


−

=

−

K
(18)
The same o mula co esponds wi h Fo mula (12) and is p esen ed also by Cook in [4], he same as many
o he au ho s. The geome ic s i ness ma ix o a uss membe can also be de i ed om he p inciple o i ual
wo k, which will be desc ibed la e . Then a s ain measu e and cons i u i e law mus be in oduced, which is no
applicable o a igid uss, whe e geome ic s i ness also exis s.
The esul ing angen s i ness ma ix T
K
is de ined as he sum o he ma e ial and geome ic s i ness ma-
ix:
TM
σ
= +KK K
(19)
When applying he gene al s anda d algo i hm o geome ic s i ness ma ices o he uss elemen in ques-
ion, we ob ain:
u

= = ⋅


u Nd
(20)
[]
12 22
,NN=
N II
(21)
whe e
2
I
is he iden i y ma ix o o de 2 and he base unc ions
i
N
a e de ined as ollows:
12
1xx
NN
ll
  
= − , =
  
  
(22)
xx
∂∂
= =
∂∂
uN
d Gd
(23)
whe e
1 0 10
1
0 101
xl
−

∂
= = 
−
∂
N
G
(24)
Subs i u ing in he o mulae
x
N
A
σ
= =Σ
(25)
AlΩ= ⋅
(26)
he o mula o he geome ic s i ness ma ix eads:
TT
1 0 10
010 1
dd
10 1 0
0 10 1
x
l
N
Nl l
σ
σ
Ω
−


−

= Ω= = 
−

−

∫∫
K G G GG
(27)

I. Němec e al.
738
This geome ic s i ness ma ix di e s om ha in Fo mula (18) and in oduces also an axial s i ening. Bu
no eason was ound by he au ho s o concluding ha no mal o ce had led o a change in he axial s i ness o
he elemen . So le us de i e he geome ic s i ness ma ix o a uss elemen in a mo e undispu able way based
on he p inciple o i ual wo k.
Wi h de o ma ion es ic ed o he
xy
plane, he G een-Lag ange s ain enso is de ined
22
1
2
x xx
u u
e
x xx
εη

∂ ∂∂

=+ +=+



∂ ∂∂


(28)
whe e
22
1
,2
xx
u u
ex xx
η

∂ ∂∂

= = +



∂ ∂∂


(29)
Fo uss he p inciple o i ual wo k becomes
d
x x ex
SW
δε
Ω
Ω=
∫
(30)
whe e
x
S
is he 2nd Piola-Ki chho s ess in he
x
axes a he ollowing calcula ed ime s ep + ∆ . Assuming
equali y
x xxxxx
S SS S S
σ
+
≡ = +∆ = +∆

we ob ain he inc emen al exp ession o he (30)
dd d
x x x x ex x x
S We
δε σ δη σ δ
ΩΩ Ω
∆ Ω+ Ω= − Ω
∫∫ ∫
(31)
and he linea ized equa ion o he p inciple o i ual wo k ( i ual displacemen ) simpli ies o:
dd d
x x x x ex x x
Ee e W e
δ σδη σδ
ΩΩ Ω
Ω+ Ω= − Ω
∫∫ ∫
(32)
Assuming (25) we ob ain
xx x x
EAe el N l F u N el
δ δη δ δ
+=−
(33)
whe e
x
u
ex
δ
δ
∂
=∂
and
x
uu
xx xx
δδ
δη
∂∂ ∂∂
= +
∂∂ ∂∂
(34)
[ ]
11010
x
u
exl
∂
= = −
∂d
(35)
[ ]
TT
2
1 1 0 10
0 0000
11 1
1010
1 10 1 0
0 0000
xx
ee ll l
δδ δ
−−
  
  
  
=−=
  
−
  
  
d dd d
(36)
TT T
2
1 0 10
010 1
1
10 1 0
0 10 1
x
uu
xx xx l
δδ
δη δ δ
−


−
∂∂ ∂∂ 
=+= =

−
∂∂ ∂∂ 
−

d G Gd d d
(37)
Then he equa ion o he p inciple o i ual wo k can be w i en as ollows:
T T TT
M ex in
σ
δ δ δδ
+=−
dKd dKd d d
(38)
whe e
1 0 10 1
0000 0
,
10 1 0 1
0000 0
M in
EA N
l
−−
  
  
  
= =
  
−
  
  
K
(39)
I. Němec e al.
739
T
1 0 10
010 1
d10 1 0
0 10 1
N
l
σ
Ω
−


−

= Ω= 
−

−

∫
K GGΣ
(40)
A e ans o ma ion in o global coo dina e sys em
xx
yy
 
=
 
 
R
, whe e
( ) ( )
( ) ( )
cos sin
sin cos
CS
SC
αα
αα
 − −

= =




R
(41)
=d Td
, whe e

=

R
TR
0
0
(42)
and a e elimina ion o he ec o o i ual displacemen s we ge :
T
MM
=K TKT
(43)
22
22
22
22
M
C CS C CS
CS S CS S
EA
lC CS C CS
CS S CS S

−−

−−

=
−−

−−


K
(44)
T
1 0 10
010 1
10 1 0
0 10 1
N
l
σσ
−


−

= = 
−

−

K T KT
(45)
[ ]
T
T
in in NC SCS= =−− T
(46)
( )
M ex in
σ
+=−K Kd
(47)
The geome ic s i ness ma ix (45) is he same as ha ob ained by use he s anda d Fo mula (27) and he i s
ow o he ma ix does no co espond wi h Fo mula (12). Le us y o de i e he geome ic s i ness ma ix o a
uss elemen using a mo e accu a e s ain measu e.
The app oxima e na u e o he linea ela ion be ween he de o ma ion and displacemen can be shown on a
ib e o ini ial leng h
dS
. Wi hou any loss o gene aliza ion, le us in oduce a sys em o coo dina es
x
wi h
he o igin a he s a ing poin o he ib e and wi h he x axis o ien ed in he o iginal di ec ion o he ib e. Le us
deno e by
ds
he leng h o he ib e in he de o med body (Figu e 3).
Le us deno e by he ec o o displacemen o he s a ing poin o he ib e. The end-poin o he ib e will be
displaced by ec o
d+uu
.
Using he o mula o he body-diagonal o a cuboid wi h dimensions
ddSu+
,
d
,
dw
, we can exp ess he
new leng h o he ib e using he ollowing ela ion:
()
222
d dd d d
s Su w= + ++
(48)
In oducing s e ch
ddsS
λ
=
and conside ing
d dd d d
u w
u S Sw S
xx x
δ
∂∂ ∂
= , = , =
∂∂ ∂
(49)
we ob ain he ollowing ela ion o s e ch o he ib e:
22 2 22 2
d
1 1 12
d
x
s u w uu w
S xxx xxxx
λε
∂∂∂ ∂∂∂∂
  
=+==+++ =++++
  
∂∂∂ ∂∂∂∂
  
(50)
I. Němec e al.
740
Figu e 3. Elonga ion o ib e dS.
Le us conside he binomial heo em:
23
11
2 8 16
AA A
A+=+− + +
o
21A<
(51)
and le us ake in o accoun only he i s wo e ms. Then we can w i e:
22 2
1
12
u u w
x xxx
λ

∂ ∂∂∂

=++ + +



∂ ∂∂∂


(52)
and o
x
ε
22 2
1
12
x
u u w
x xxx
ελ

∂ ∂∂∂

= −= + + +



∂ ∂∂∂


(53)
I we wan o be mo e accu a e and ake in o accoun h ee e ms o he binomial expansion, and i we neglec
he hi d and highe powe s o he de i a i es o he displacemen componen s, we ge a mo e accu a e exp ession
o he s e ch:
22
1
12
u w
x xx
λ

∂ ∂∂
 
=++ +

 

∂ ∂∂
 

(54)
and hence
22
1
2
x
u w
x xx
ε

∂ ∂∂
 
=++

 

∂ ∂∂
 

(55)
Fo a 1D p oblem, he e o e, his mo e accu a e exp ession would be iden ical o he o mula o
x
ε
known
om linea mechanics:
x
u
x
ε
∂
=∂
(56)
Using he mo e accu a e s ain measu e we ob ain:
2
1
2
x xx
u
e
xx
εη
∂∂

=+=+

∂∂

(57)
whe e
x
u
ex
∂
=∂
,
2
1
2
x
x
η
∂

=
∂

I. Němec e al.
741
[ ]
TT T T
new new 2
0 0000
1 010 1
11 1
0 101
0 0000
1 0 10 1
x
xx l l l
δ
δη δ δ δ
  
  
−−
∂∂   
== = −=
  
∂∂   
−
  
dGGdd dd d
(58)
whe e is de ined a new ma ix
[]
new
10 101
l
= −
G
ins ead o he s anda d
G
.
The linea ized equa ion o he p inciple o i ual wo k ( i ual displacemen ) modi ies o:
T T TT
10 10 0000 1
0000 010 1 0
1010 0000 1
0000 0 101 0
ex
EA N N
ll
δ δ δδ
−−
 
 
−
 
+=−
 
−
 
−
 
d d d d d d
(59)
A e ans o ma ion in o global coo dina e sys em and elimina ion o he ec o o i ual displacemen s we ge
di e en geome ic s i ness ma ix in he o a ed and hus also in global coo dina e sys em:
T
new new
0000
010 1
d0000
0 10 1
N
l
σ
Ω


−

= Ω= 

−

∫
K GGΣ
(60)
22
22
T
22
22
S SC S SC
SC C SC C
N
lS SC S SC
SC C SC C
σσ

−−

−−

= = 
−−

−−


K T KT
(61)
Resul ing s i ness ma ix
σ
K
de i ed om he p inciple o i ual wo k, using he mo e accu a e s ain
measu e, is he same as ha de i ed om equilib ium condi ions (18) and co esponds wi h Fo mula (12).
I can be seen ha he s anda d o mula has p oduced a di e en geome ic ma ix o he 2D uss elemen
(27) han Fo mulae (18), (12) and (61) de i ed ea lie and heo e ically unjus i ied geome ic axial s i ness was
also p oduced. This o mula would lead o a poo con e gence a e, inaccu acy and e en, in he case o ex eme
comp ession, o singula i y. E.g. o
x
E
σ
= −
, ze o no mal angen s i ness would be ob ained o he uss
elemen , al hough he e is no physical eason o his. Fo
xE
σ
<−
he no mal angen s i ness would e en be
nega i e, which would be absu d. In he case o ension no s abili y p oblem would occu , bu he low con e -
gence p oblem is s ill p esen . E.g. when
xE
σ
=
, he unbalanced nodal o ces o 1/2 o he load inc emen al-
ue would occu in he i s i e a ion o he las inc emen . In he 2nd i e a ion i would be 1/4, and in he i- h i e a-
ion he unbalanced o ce o
12
i
o he load inc emen alue would s ill occu . These p oblems a e known,
and he e o e o he geome ic s i ness o uss elemen s Fo mula (18) is widely used ins ead o Fo mula (27),
which is de i ed om he gene al Fo mula (8) o (9). Then, in many compu e p og ams di e en a es o con-
e gence a e ob ained o a od modeled by a uss elemen han in he case o a uss modeled by solid ele-
men s.
To ob ain he same geome ic s i ness ma ix o he 2D uss elemen (18) as was de i ed abo e om he
equilib ium, he in luence o he membe
ux∂∂
mus be omi ed in he s anda d o mula, i.e. he i s ow o
he
G
ma ix mus be illed in wi h ze os.
4. An Imp o ed Fo mula o a Geome ic S i ness Ma ix
In oducing a ib e o cons an c oss sec ion a ea A in he di ec ion
x
o p incipal s ess in a 2D o 3D con in-
uum ins ead o a od, and assuming only nonze o s ain in he di ec ion o he ib e, and ha all he o he com-
ponen s o he s ain enso a e ze o, we can w i e a simila o mula o (12):
I. Němec e al.
748
h p://dx.doi.o g/10.1016/0045-7825(84)90062-8
[16] Cu nie , A. and Rako omanana, L. (1991) Gene alized S ain and S ess Measu es: C i ical Su ey and New Resul s.
Enginee ing T ansac ions, 39, 461-538.
[17] Chiskis, A. and Pa nes, R. (2000) Linea S ess-S ain Rela ions in Nonlinea Elas ici y. Ac a Mechanica, 146,
109-113. h p://dx.doi.o g/10.1007/BF01178798
[18] Fa ahani, K. and Naghdabadi, R. (2000) Conjuga e S esses o he Se h-Hill S ain Tenso s. In e na ional Jou nal o
Solids and S uc u es, 37, 5247-5255. h p://dx.doi.o g/10.1016/S0020-7683(99)00209-7
[19] Da ijani, H. and Naghdabadi, R. (2010) Cons i u i e Modeling o Solids a Fini ede Fo ma ion Using a Second-O de
S ess-S ain Rela ion. In e na ional Jou nal o Enginee ing Science, 48, 223-236.
h p://dx.doi.o g/10.1016/j.ijengsci.2009.08.006
[20] Hill, R. (1978) Aspec s o In a iance in Solid Mechanics. Ad ances in Applied Mechanics, 18, 1-75.
h p://dx.doi.o g/10.1016/S0065-2156(08)70264-3
[21] Fa ahani, K. and Bahai, H. (2004) Hype -Elas ic Cons i u i e Equa ions o Conjuga e S esses and S ain Tenso s o
he Se h-Hill S ain Measu es. In e na ional Jou nal o Enginee ing Science, 42, 29-41.
h p://dx.doi.o g/10.1016/S0020-7225(03)00241-6
Lis o Va iables
A
C oss sec ion a ea o a beam
C
Ma e ial angen moduli
E
Young modulus
,
M
σ
KK
Ma e ial and geome ic angen s i ness ma ix, espec i ely
,
M
σ
KK
Ma e ial and geome ic angen s i ness ma ix in p incipal axes
i
N
Shape unc ions
N
Ma ix o shape unc ions
R
Ro a ion enso
S
Second Piola-Ki chho s ess
,
in ex
WW
In e nal and ex e nal i ual wo k
d
Vec o o nodal displacemen s
ˆ
e
In ini esimal s ain
e
In ini esimal s ain in p incipalaxes
in
In e nal nodal o ces
ex
Ex e nal nodal o ces
I
Uni diagonal ma ix
l
Membe leng h
Time
u
Displacemen ield
,,u w
Displacemen s in he x, y and z di ec ions espec i ely
,,xyz
Spa ial (Eule ian) coo dina es
,,xyz
Coo dina es in p incipal aces
ε
Modi ied s ain enso in p incipalaxes
,ij
ηη
Quad a ic e ms o he modi ied s ain enso in p incipalaxes
σ
∏
Po en ial ene gy o geome ical s i ness
Σ
= ⊗ IΣ
σ
Σ
= ⊗ IΣ
σ
,ij
σσ
Cauchy s ess enso
σ
Cauchy s ess enso in p incipal axes
0
,
ΩΩ
Domain o cu en (de o med), ini ial (unde o med)