New Formula for Geometric Stiffness Matrix Calculation

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)