RADIOSS THEORY MANUAL 10.0 version January 2009 Large Displacement Finite Element Analysis PART 2

Transcription

1 RADIOSS THEORY MANUAL. verson January 9 Large Dsplaceen Fne Eleen Analyss PART Alar Engneerng, Inc., World Headquarers: 8 E. Bg Beaver Rd., Troy MI USA Phone: Fax: nfo@alar.co

2 RADIOSS THEORY Verson. CONTENTS CONTENTS 6. KINEMATIC CONSTRAINTS 5 6. RIGID BODY RIGID BODY MASS RIGID BODY INERTIA RIGID BODY FORCE AND MOMENT COMPUTATION TIME INTEGRATION RIGID BODY BOUNDARY CONDITIONS 8 6. TIED INTERFACE TYPE SPOTWELD FORMULATION FORMULATION FOR SEARCH OF CLOSEST MASTER SEGMENT RIGID WALL FIXED RIGID WALL MOVING RIGID WALL SLAVE NODE PENETRATION RIGID WALL IMPACT FORCE RIGID LINK SECTION 9 7. LINEAR STABILTY 7. GENERAL THEORY OF LINEAR STABILITY 3 8. INTERFACES 6 8. INTRODUCTION LAGRANGE MULTIPLIER METHOD PENALTY METHOD 8 8. INTERFACE OVERVIEW SURFACE SEGMENT DEFINITION TIED INTERFACE TYPE AUTO CONTACTS INTRODUCTION MODELING OF CONTACTS ALGORITHMS OF SEARCH FOR IMPACT CANDIDATES CONTACT PROCESSING CONTACT DETECTION TYPE 3 - SOLID AND SHELL ELEMENT CONTACT - NO GAP LIMITATIONS COMPUTATION ALGORITHM INTERFACE STIFFNESS INTERFACE FRICTION INTERFACE GAP INTERFACE FAILURE EXAMPLES TYPE 5 - GENERAL PURPOSE CONTACT LIMITATIONS COMPUTATION ALGORITHM INTERFACE STIFFNESS INTERFACE FRICTION INTERFACE GAP INTERFACE ALGORITHM TYPE 6 - RIGID BODY CONTACT LIMITATIONS INTERFACE STIFFNESS INTERFACE FRICTION INTERFACE GAP TIME STEP CALCULATION CONTACT FORCE 53 -jan-9

3 RADIOSS THEORY Verson. CONTENTS 8.8 TYPE 7 - GENERAL PURPOSE CONTACT LIMITATIONS INTERFACE STIFFNESS INTERFACE FRICTION INTERFACE GAP TIME STEP DETECTION AND GAP SIZE VARIABLE GAP GAP CORRECTION FOR NODES WITH INITIAL PENETRATION PENETRATION REACTION FORCE ORIENTATION INTERFACE HINTS TYPE 4 - ELLIPSOIDAL SURFACE TO NODE CONTACT TYPE 4 INTERFACE: HINTS TYPE 5 - ELLIPSOIDAL SURFACE TO SEGMENT CONTACT TYPE 6- NODE TO CURVED SURFACE CONTACT TYPE 7- GENERAL SURFACE TO SURFACE CONTACT SOME COMMON PROBLEMS INCORRECT NEAREST MASTER NODE FOUND INCORRECT NEAREST MASTER SEGMENT FOUND - B INCORRECT NEAREST MASTER SEGMENT FOUND - B INCORRECT NEAREST MASTER SEGMENT FOUND - B INCORRECT IMPACT SIDE - C NO MASTER NODE IMPACT - D MATERIAL LAWS 7 9. ISOTROPIC ELASTIC MATERIAL COMPOSITE AND ANISOTROPIC MATERIALS FABRIC LAW FOR ELASTIC ORTHOTROPIC SHELLS LAWS 9 AND NONLINEAR PSEUDO-PLASTIC ORTHOTROPIC SOLIDS LAWS 8, 5 AND HILL S LAW FOR ORTHOTROPIC PLASTIC SHELLS ELASTIC-PLASTIC ORTHOTROPIC COMPOSITE SHELLS ELASTIC-PLASTIC ORTHOTROPIC COMPOSITE SOLIDS ELASTIC-PLASTIC ANISOTROPIC SHELLS BARLAT S LAW ELASTO-PLASTICITY OF ISOTROPIC MATERIALS JOHNSON-COOK PLASTICITY MODEL LAW ZERILLI-ARMSTRONG PLASTICITY MODEL LAW COWPER-SYMONDS PLASTICITY MODEL LAW ZHAO PLASTICITY MODEL LAW TABULATED PIECEWISE LINEAR AND QUADRATIC ELASTO-PLASTIC LAWS LAWS 36 & DRUCKER-PRAGER CONSTITUTIVE MODEL LAWS & BRITTLE DAMAGE FOR JOHNSON-COOK PLASTICITY MODEL LAW BRITTLE DAMAGE FOR REINFORCED CONCRETE MATERIALS LAW DUCTILE DAMAGE MODEL DUCTILE DAMAGE MODEL FOR POROUS MATERIALS GURSON LAW VISCOUS MATERIALS BOLTZMANN VISCOELASTIC MODEL LAW GENERALIZED KELVIN-VOIGT MODEL LAW TABULATED STRAIN RATE DEPENDENT LAW FOR VISCOELASTIC MATERIALS LAW GENERALIZED MAXWELL-KELVIN MODEL FOR VISCOELASTIC MATERIALS LAW VISCO-ELASTO-PLASTIC MATERIALS FOR FOAMS LAW HYPER VISCO-ELASTIC LAW FOR FOAMS LAW MATERIALS FOR HYDRODYNAMIC ANALYSIS JOHNSON COOK LAW FOR HYDRODYNAMICS LAW HYDRODYNAMIC VISCOUS FLUID LAW LAW ELASTO-PLASTIC HYDRODYNAMIC MATERIAL LAW STEINBERG-GUINAN MATERIAL LAW VOID MATERIAL LAW FAILURE MODEL JOHNSON-COOK FAILURE MODEL WILKINS FAILURE CRITERIA 34 -jan-9

4 RADIOSS THEORY Verson. CONTENTS TULER-BUTCHER FAILURE CRITERIA FORMING LIMIT DIAGRAM FOR FAILURE FLD SPALLING WITH JOHNSON-COOK FAILURE MODEL BAO-XUE-WIERZBICKI FAILURE MODEL STRAIN FAILURE MODEL SPECIFIC ENERGY FAILURE MODEL XFEM CRACK INITIALIZATION FAILURE MODEL 38. MONITORED VOLUME 39. AREA TYPE MONITORED VOLUME 4. PRES TYPE MONITORED VOLUME 4.3 GAS TYPE MONITORED VOLUME 4.3. THERMODYNAMICAL EQUATIONS 4.3. VARIATION OF EXTERNAL WORK VENTING SUPERSONIC OUTLET FLOW GAS TYPE MONITORED VOLUME EXAMPLES 43.4 AIRBAG TYPE MONITORED VOLUME THERMODYNAMICAL EQUATIONS ENERGY VARIATION WITHIN A TIME STEP MASS INJECTION VENTING: OUTGOING MASS DETERMINATION POROSITY INITIAL CONDITIONS JETTING EFFECT REFERENCE METRIC TANK EXPERIMENT 53.5 MONITORED VOLUME TYPE COMMU THERMODYNAMICAL EQUATIONS VARIATION OF EXTERNAL WORK MASS INJECTION VENTING SUPERSONIC OUTLET FLOW JETTING EFFECT REFERENCE METRIC EXAMPLES OF COMMU TYPE MONITORED VOLUME 56. STATIC 6. STATIC SOLUTION BY EXPLICIT TIME-INTEGRATION 6.. ACCELERATION CONVERGENCE 65. STATIC SOLUTION BY IMPLICIT TIME-INTEGRATION 65.. LINEAR STATIC SOLVER 66.. NONLINEAR STATIC SOLVER 66. RADIOSS PARALLELIZATION 73. MEASURE OF PERFORMANCE OF A PARALLEL APPLICATION 73. SMP : SHARED MEMORY PROCESSORS 74.3 SPMD : SINGLE PROGRAM MULTIPLE DATA REFERENCES APPENDICES 83 APPENDIX A - CONVERSION TABLES & CONSTANTS 83 APPENDIX B - MÏE GRÜNEISEN EQUATION-OF-STATE 88 APPENDIX C - BASIC RELATIONS OF ELASTICITY 9 -jan-9 3

5 RADIOSS THEORY Verson. KINEMATIC CONSTRAINTS Chaper KINEMATIC CONSTRAINTS -jan-9 4

6 RADIOSS THEORY Verson. KINEMATIC CONSTRAINTS 6. KINEMATIC CONSTRAINTS Kneac consrans are boundary condons ha are placed on nodal veloces. They are uually exclusve for each degree of freedo DOF, and here can only be one consran per DOF. There are seven dfferen ypes of kneac consrans ha can be appled o a odel n RADIOSS:. Rgd Body. Inal sac equlbru 3. Boundary Condon 4. Ted Inerface Type 5. Rgd Wall 6. Rgd Lnk 7. Cylndrcal Jon Two kneac condons appled o he sae node ay be ncopable. 6. Rgd Body A rgd body s defned by a aser node and s assocaed slave nodes. Mass and nera ay be added o he nal aser node locaon. The aser node s hen oved o he cener of ass, akng no accoun he aser node and all slave node asses. Fgure 6.. shows an dealzed rgd body. Fgure 6.. Idealzed Rgd Body 6.. Rgd body ass The ass of he rgd body s calculaed by: M I EQ I The rgd body's cener of ass s defned by: x G M M I I x x EQ jan-9 5

7 RADIOSS THEORY Verson. KINEMATIC CONSTRAINTS -jan-9 6 y y y I I M M G EQ z z z I I M M G EQ where M s he aser node ass I are he slave node asses G x, G y, G z are he coordnaes of he ass cener. 6.. Rgd body nera The sx coponens of nera of a rgd body are copued by: G G xx G M G M M M xx xx z z y y I z z y y J I EQ G G yy G M G M M M yy yy z z x x I z z x x J I EQ G G zz G M G M M M zz zz y y x x I y y x x J I EQ G G xy G M G M M M xy xy y y x x I y y x x J I EQ G G yz G M G M M M yz yz z z y y I z z y y J I EQ G G xz G M G M M M xz xz z z x x I z z x x J I EQ where j I s he oen of roaonal nera n he j drecon. M j J s he aser node added nera.

8 RADIOSS THEORY Verson. KINEMATIC CONSTRAINTS 6..3 Rgd body force and oen copuaon The forces and oens acng on he rgd body are calculaed by: r r F F M r F EQ r M r M M M SG r r r F EQ where M F r s he force vecor a he aser node F r s he force vecor a he slave nodes M M r s he oen vecor a he aser node M r s he oen vecor a he slave nodes G r s he vecor fro slave node o he cener of ass Resolvng hese no orhogonal coponens, he lnear and roaonal acceleraon ay be copued as: Lnear Acceleraon F γ EQ Roaonal Acceleraon I I3 I ω 3 M EQ α ω I I I3 ω 3 M EQ α ω I I I ω 3α 3 M 3 ω EQ where I are he prncpal oens of nera of he rgd body α are he roaonal acceleraons n he prncpal nera frae reference frae ω s he roaonal velocy n he prncpal nera frae reference frae M are he oens n he prncpal nera frae reference frae -jan-9 7

9 RADIOSS THEORY Verson. KINEMATIC CONSTRAINTS 6..4 Te negraon Te negraon s perfored o fnd veloces of he rgd body a he aser node: r Δ r Δ r ν ν γ Δ EQ r Δ r Δ r ω ω α Δ EQ where v r s he lnear velocy vecor. Roaonal veloces are copued n he local reference frae. The veloces of slave nodes are copued by: r ν r ν r r S Gxω M EQ r ω v M EQ ω 6..5 Rgd body boundary condons The boundary condons gven o slave nodes are gnored. The rgd body has he boundary condons gven o he aser node only. A kneac condon s appled on each slave node, for all drecons. A slave node s no allowed o have any oher kneac condons. No kneac condon s appled on he aser node. However, he roaonal veloces are copued n a local reference frae. Ths reference frae s no copable wh all opons posng roaon such as posed velocy, roaonal, rgd lnk... The only excepon concerns he roaonal boundary condons for whch a specal reaen s carred ou. Connecng shell, bea or sprng wh roaon sffness o he aser node, s no ye allowed eher. 6. Ted Inerface Type Wh a ed nerface s possble o connec rgdly a se of slave nodes o a aser surface. A ed nerface Type can be used o connec a fne esh of Lagrangan eleens o a coarse esh or wo dfferen knds of eshes for exaple sprng o shell conacs. -jan-9 8

10 RADIOSS THEORY Verson. KINEMATIC CONSTRAINTS Fgure 6.. Fne and coarse esh A aser and a slave surface are defned n he nerface npu cards. The conac beween he wo surfaces s ed. No sldng or oveen of he slave nodes s allowed on he aser surface. There are no vods presen eher. I s recoended ha he aser surface has a coarser esh. Acceleraons and veloces of he aser nodes are copued wh forces and asses added fro he slave nodes. Kneac consran s appled on all slave nodes. They rean a he sae poson on her aser segens. Ted nerfaces are useful n rve odelng, where hey are used o connec sprngs o a shell or sold esh. 6.. Spoweld forulaon The slave node s rgdly conneced o he aser surface. Two forulaons are avalable o descrbe hs connecon: - Defaul forulaon - Opzed forulaon 6... Defaul spoweld forulaon SPOTflag When he flag s se o, he spoweld forulaon s a defaul forulaon: - based on eleen shape funcons. - generang hourglass wh under negraed eleens. - provdng a connecon sffness funcon of slave node localzaon. - recoended wh full negraed shells aser. - recoended for connecng brck slave nodes o brck aser segens esh ranson whou roaonal freedo. Forces and oens ransfer fro slave o aser nodes s descrbed n he Fgure 6... : -jan-9 9

11 RADIOSS THEORY Verson. KINEMATIC CONSTRAINTS Fgure 6.. Defaul ed Inerface Type The ass of he slave node s ransferred o he aser nodes usng he poson of he projecon on he segen and lnear nerpolaon funcons: aser Φ p EQ aser slave where p denoes he poson of he slave pon and Φ s he wegh funcon obaned by he nerpolaon equaons. Fgure 6..3 Transfer of slave node effors o he aser nodes SPOTflag S: slave M: slave h Μ P F n S F F n * ΦP M * ΦP F F * ΦP The nera of he slave node s also ransferred o he aser nodes by akng no accoun he dsance d beween he slave node and he aer surface: I aser aser I d Φ p I EQ slave slave The er slave d ay ncrease he oal nera of he odel especally when he slave node s far fro he aser surface. The sably condons are wren on he aser nodes: K K aser roaon aser K K Φ p EQ K aser roaon aser slave roaon K K d Φ p slave slave -jan-9

12 RADIOSS THEORY Verson. KINEMATIC CONSTRAINTS The dynac equlbru of each aser node s hen suded and he nodal acceleraons are copued. Then he veloces a aser nodes can be obaned and updaed o copue he velocy of he projeced pon P by he followng expressons: V ranslaon P ranslaon V Φ p EQ aser V roaon P roaon V Φ p EQ aser The velocy of he slave node s hen obaned: V ranslao n slave n roaon V ranslao V PS EQ P P V V roaon slave roaon P Wh hs forulaon, he added nera ay be very large especally when he slave node s far fro he ean plan of he aser eleen Opzed spoweld forulaon SPOTflag When he flag s se o, he spoweld forulaon s an opzed forulaon: - based on eleen ean rgd oon.e. whou excng deforaon odes - havng no hourglass proble. - havng consan connecon sffness. - recoended wh under negraed shells aser - recoended for connecng bea, sprng and shell slave nodes o brck aser segens. Ths spoweld forulaon s opzed for spowelds or rves. The slave node s joned o he aser segen barycener as shown n Fgure Fgure 6..4 Relaon beween slave node and aser node Forces and oens ransfer fro slave o aser nodes s descrbed n Fgure The force appled a he slave node S s redsrbued unforly o he aser nodes. In hs way, only ranslaonal ode s exced. The oen M CS F s redsrbued o he aser nodes by four forces F such ha: F A r EQ CM -jan-9

13 RADIOSS THEORY Verson. KINEMATIC CONSTRAINTS CM F M CS F where A r s he noral vecor o he segen. Fgure 6..5 Opzed ed Inerface Type S F n F F C P F n /4 F /4 F3 F4 C S M F MF^CS F F In hs forulaon he ass of he slave node s equally dsrbued o he aser nodes. In confory wh effor ranssson as descrbed n 6..., he sphercal nera s copued wh respec o he cener of he aser eleen C: I Slave Slave Slave C I.d EQ where d s dsance fro he slave node o he cener of eleen. In order o nsure he sably condon whou reducon n he e sep, he nera of he slave node s ransferred o he aser nodes by an equvalen nodal ass copued by: -jan-9

14 RADIOSS THEORY Verson. KINEMATIC CONSTRAINTS Y Z X Y X Z Slave Slave I. d Δ, wh I X Y X Z Y Z EQ I,..4 X Z Y Z X Y For hs reason he forulaon causes an ncrease of ass whch ay becoe very poran especally when he node s far fro he ean surface of he aser shell eleen. 6.. Forulaon for search of closes aser segen The aser segen s found va forulaons: - Old forulaon - New proved forulaon 6... Old search of closes aser segen forulaon Isearch When he flag s se o, he search of closes aser segen was based on he old forulaon: A box wh a sde equal o dsearch npu s bul o search he aser node conaned whn hs box. Fgure 6..6 Old search of closes aser segen The dsance beween each aser node n he box and he slave node s copued. The aser node gvng he nu dsance dn s reaned. The segen s chosen wh he seleced node, f he seleced node belongs o segens, one s seleced a rando. -jan-9 3

15 RADIOSS THEORY Verson. KINEMATIC CONSTRAINTS Fgure 6..7 Old search of closes aser segen d < d 3 s chosen d S d New proved search of closes aser segen forulaon Isearch When he flag s se o, he search of closes aser segen s based on he new proved forulaon; a box ncludng he aser surface s bul. The dchooy prncple s appled o hs box as long as he box conans only one aser node and as long as he box sde s equal o dsearch. Fgure 6..8 New proved search of closes aser segen Neares Maser node Neares Maser segen Slave node To copue he nu dsance, dn, we have wo soluons: - The slave node s an nernal node for he aser segen, as shown n Fgure The slave node s projeced orhogonally on he aser segen o gve a dsance ha ay be copared wh oher dsances. Selec he nu dsance: -jan-9 4

16 RADIOSS THEORY Verson. KINEMATIC CONSTRAINTS Fgure 6..9 Orhogonal projecon on he aser segen slave node d 3 4 The segen ha provdes he nu dsance s chosen for he followng copuaon. - The slave node s a node exernal o he aser segen, as shown n Fgure 6... The dsance seleced s ha beween he slave node and he neares aser node. Fgure 6...Neares aser node slave node d 3 4 The segen s chosen usng he seleced node, f he seleced node belongs o segens, one s chosen a rando. -jan-9 5

17 RADIOSS THEORY Verson. KINEMATIC CONSTRAINTS 6.3 Rgd Wall Four ypes of rgd walls are avalable n RADIOSS:. Infne Plane. Infne Cylnder wh Daeer D 3. Sphere wh Daeer D 4. Parallelogra Each wall can be fxed or ovng. A kneac condon s appled on each paced slave node. Therefore, a slave node canno have anoher kneac condon unless hese condons are appled n orhogonal drecons Fxed rgd wall A fxed wall s a pure kneac opon on all paced slave nodes. I s defned usng wo pons, M and M. These defne he noral, as shown n Fgure Fgure 6.3. Fxed Rgd Wall Defnon 6.3. Movng rgd wall A ovng rgd wall s defned by a node nuber, N, and a pon, M. Ths allows a noral o be calculaed, as shown n Fgure The oon of node N can be specfed wh fxed velocy or wh an nal velocy. For splfcaon, an nal velocy and a ass ay be gven a he wall defnon level. -jan-9 6

18 RADIOSS THEORY Verson. KINEMATIC CONSTRAINTS Fgure 6.3..Movng Rgd Wall Defnon A ovng wall s a aser slave opon. Maser node defnes he wall poson a each e sep and poses velocy on paced slave nodes. Ipaced slave node forces are appled o he aser node. The slave node forces are copued wh oenu conservaon. The ass of he slave nodes s no ransed o he aser node, assung a large rgd wall ass copared o he paced slave node ass Slave node peneraon Slave node peneraon us be checked. Fgure shows how peneraon s checked. Fgure Slave Node Peneraon If peneraon occurs, a new velocy us be copued. Ths new velocy s copued usng one of hree possble suaons. These are:. Sldng. Sldng wh Frcon 3. Ted v For a node whch s allowed o slde along he face of he rgd wall, he new velocy V s gven by: v r r r r V V V nn EQ jan-9 7

19 RADIOSS THEORY Verson. KINEMATIC CONSTRAINTS A frcon coeffcen can be appled beween a sldng node and he rgd wall. The frcon odels are developed n secon For a node ha s defned as ed, once he slave node conacs he rgd wall, s velocy s he sae as ha of he wall. The node and he wall are ed. Hence: V v EQ Rgd wall pac force The force exered by nodes pacng ono a rgd wall s found by calculang he pulse by: v I N r F Δ N I I Δ r V W r EQ where N s he nuber of peneraed slave nodes W r s he wall velocy The force can hen be calculaed by he rae of change n he pulse: r r di F EQ d 6.4 Rgd Lnk A rgd lnk poses he sae velocy on all slave nodes n one or ore drecons. The drecons are defned o a skew or global frae. Fgure 6.4. dsplays a rgd lnk. Fgure 6.4..Rgd Lnk Model -jan-9 8

20 RADIOSS THEORY Verson. KINEMATIC CONSTRAINTS The velocy of he group of nodes rgdly lnked ogeher s copued usng oenu conservaon EQ However, no global oen equlbru s respeced. N v V N EQ Angular velocy for he h DOF wh respec o he global or a skew reference frae s: n I jω j j ω EQ N I j j For non-concden nodes, no rgd body oon s possble. A rgd lnk s equvalen o an nfnely sff sprng ype Secon A secon s a cu n he srucure where forces and oens wll be copued and sored n oupu fles. I s defned by: A cung plane A reference pon o copue forces A drecon of he secon. Fgure 6.5. Defnon of a secon for an orened sold M F In RADIOSS he cung plane s defned by a group of eleens and s orenaon by a group of nodes as shown n Fgure jan-9 9

21 RADIOSS THEORY Verson. KINEMATIC CONSTRAINTS Fgure 6.5. Defnon of a secon for a shell esh Then, a pon s defned for he cener of roaon of he secon and a reference frae s aached o hs pon o copue he nernal effors. Fgure Cener of roaon and s assocaed frae for a secon The resulan of all forces appled o he eleens and s applcaon pon are copued by: F EQ f ON f EQ M Fgure Resulan of force and oen for a node I w.r. he roaon pon O -jan-9

22 Chaper LINEAR STABILITY

23 RADIOSS THEORY Verson. LINEAR STABILITY 7. LINEAR STABILTY The sably of soluon concerns he evoluon of a process subjeced o sall perurbaons. A process s consdered o be sable f sall perurbaons of nal daa resul n sall changes n he soluon. The heory of sably can be appled o a varey of copuaonal probles. The nuercal sably of he e negraon schees s wdely dscussed n secon Here, he sably of an equlbru sae for an elasc syse s suded. The aeral sably wll be presened n an upcong verson of hs anual. The sably of an equlbru sae s of consderable neres. I s deerned by exanng wheher perurbaons appled o ha equlbru sae grow. A faous exaple of sable and unsable cases s ofen gven n he leraure. I concerns a ball deposed on hree knds of surfaces as shown n Fgure 7... Fgure 7.. Scheac presenaon of sably a Sable b Unsable c Neural I s clear ha he sae b represens an unsable case snce a sall change n he poson of he ball resuls he rollng eher o he rgh or o he lef. I s worhwhle o enon here ha sably and equlbru noons are que dfferen. A syse n sac equlbru ay be n unsable sae and a syse n evoluon s no necessary unsable. A good undersandng of he sably of equlbru can be obaned by sudyng he load-deflecon curves. A ypcal behavor of a srucure n bucklng s gven n Fgure 7... The load-deflecon curves are slghly dfferen for syses wh and whou perfecon. In he frs case, he srucure s loaded unl he bfurcaon pon B correspondng o he frs crcal load level. Then, wo soluons are aheacally accepable: response whou bucklng BA, response afer bucklng BC. In he case of srucures wh perfecon, no bfurcaon pon s observed. The behavor before bucklng s no lnear and he urnng pon D s he l pon n whch he slope of he curve changes sgn. If he behavor before bucklng s lnear or he nonlneary before he l pon s no hgh, he lnear sably echnque can be used o deerne he crcal load. The ehod s based on he perurbaon of he equlbru sae. As he perurbaons are sall, he lnearzed odel s used. The ehod s dealed n he followng secon. Fgure 7.. Bfurcaon and l pons n load-deflecon curves for syse wh and whou perfecons B: Bfurcaon pon, D: L pon A P Cr P S B D Whou perfecon C Wh perfecon Dsplaceen -jan-9

24 RADIOSS THEORY Verson. LINEAR STABILITY 7. General heory of lnear sably The prncple of vrual power and he nu of oal poenal energy are he varaonal aheacal odels largely used n Fne Eleen Mehod. Under sall-perurbaons assupon hese noons can be appled o he equlbru sae n order o sudy he sably of he syse. Consder he exaple of he ball on he hree knds of surfaces as shown n Fgure 7... If Π s he oal poenal energy, he equlbru s obaned by: δ Π > Sac equlbru EQ.7... Applyng a sall perurbaon o he equlbru sae, he varaon of he oal poenal energy can be wren as: Where Δ δπ δ Π δπ EQ.7... δ Π s he second varaon of he poenal energy. Then, he hree cases can be dsngushed: δ Π > > Sable case a EQ > The energy ncreases around he equlbru sae. δ Π < > Unsable case b EQ > The energy decreases around he equlbru sae. δ Π > Neural sably case c EQ > The energy reans unchanged around he equlbru sae. The las case s used o copue he crcal loads: δ Π δ Π δ Π EQ n ex Where, he ndces n and ex denoe he nerval and exernal pars of he oal poenal energy. Afer he applcaon of he applcaon of fne eleen ehod, he sably equaon n a dscree for can be wren as: e e δ Π δ Π δ Π EQ e ex n Se n n { δf } ds n e ex δ Π δx EQ [ E][ S] [ δe][ δs] dv e δ Πn e EQ Ve Where e desgnae eleen and: { f n }: vecor of he exernal forces δ X : vrual dsplaceen vecor [ E ]: Green-Lagrange sran ensor -jan-9 3

25 RADIOSS THEORY Verson. LINEAR STABILITY [ S ] : Pola-Krchhoff sress ensor The equaon EQ s wren as a funcon of X, he dsplaceen beween he nal confguraon C and he crcal sae C. If X L and S L are he lnear response obaned afer applcaon he load f L n he nal confguraonc, n lnear heory of sably we suppose ha he soluon n C for he crcal load f cr s proporonal o he lnear response: { X } λ{ } cr X L { S } λ{ } EQ.7... cr S L { F } λ{ } cr F L If we ad ha he loadng does no depend on he deforaon sae, he hypohess δ s hen rue. Usng EQ and denong [e] for he lnear par of Green-Lagrange sran ensor and [ η ] for he nonlnear par, we have: { } { e} { η} E EQ.7... Pung hs equaon n EQ.7...9, we oban: Where [] δe [ C]{ δe} λ δη [ C]{ δe} δe [ C][ δη ] S { δ E} dv e δ Π ex L L e EQ.7... e V Or δ Π e ex δx [] k λ [ k X ] [ k ]{ δx } k s he sffness arx, [ k u ] he nal dsplaceen arx, [ σ ] sffness arx and [ C ] he elasc arx. u L σ Π ex EQ k nal sress or geoercal The lnear heory of sably allows esang he crcal loads and her assocaed odes by resolvng an egenvalue proble: [ K ] [ Ku ] [ K ]{ δx } λ σ EQ.7..4 Lnear sably assues he lneary of behavor before bucklng. If a syse s hghly nonlnear n he neghbourhood of he nal sae C, oderae perurbaons ay lead o unsable growh. In addon, f case of pah-dependen aerals he use of ehod s no conclusve fro an engneerng pon of vew. However, he ehod s sple and provdes generally good esaons of l pons. The resoluon procedure consss n wo an seps. Frs, he lnear soluon for he equlbru of he syse under he applcaon of he load { F L } s obaned. Then, EQ.7..4 s resolved o copue he frs desred crcal loads and odes. The ehods o copue he egen values are hose explaned n secon 4.. -jan-9 4

26 Chaper INTERFACES -jan-9 5

27 RADIOSS THEORY Verson. INTERFACES 8. INTERFACES 8. Inroducon Inerfaces solve he conac and pac condons beween wo pars of a odel. Conac-pac probles are aong he os dffcul nonlnear probles o solve as hey nroduce dsconnues n he velocy e hsores. Pror o he conac, he noral veloces of he wo bodes whch coe no conac are no equal, whle afer pac he noral veloces us be conssen wh he penerably condon. In he sae way, he angenal veloces along nerfaces are dsconnuous when sck-slp behavor occurs n frcon odels. These dsconnues n e coplcae he negraon of governng equaons and nfluence perforance of nuercal ehods. Cenral o he conac-pac proble s he condon of penerably. Ths condon saes ha bodes n conac canno overlap or ha her nersecon reans epy. The dffculy wh he penerably condon s ha canno be expressed n ers of dsplaceens as s no possble o ancpae whch pars of he bodes wll coe no conac. For hs reason, s convenen o express he penerably condon n a rae for a each cycle of he process. Ths condon can be wren as: A B γ v v EQ N N N on he conac surface Γ C coon o he wo bodes. A v N and v B N are respecvely he noral veloces n he wo bodes n conac. γ N s he rae of nerpeneraon. EQ sply expresses ha when wo bodes are n conac, hey us eher rean n conac and γ, or hey us separae and γ <. N N On he oher hand, he racons us observe he balance of oenu across he conac nerface. Ths requres ha he su of he racons on he wo bodes vansh: A EQ B N N Noral racons are assued copressve, whch can be saed as: A B < EQ N N N EQ 8... and EQ can be cobned n a sngle equaon sang ha Nγ N. Ths condon sply expresses ha he conac forces do no creae work. If he wo bodes are n conac, he nerpeneraon rae vanshes. On he oher hand, f he wo bodes are separaed γ N < bu he surface racons vansh. As a resul, he produc of he surface racons and he nerpeneraon rae dsappear n all cases. If we noce ha he penerably condon s expressed as an nequaly consran, he condon: γ EQ N N can also be seen as he Kuhn-Tucker condon assocaed wh he opzaon proble conssng n nzng he oal energy EQ....5 subjec o he nequaly consran 8... In pracce, he soluon o a conac proble enals n hree seps: Frs, s necessary o fnd for each pon hose pons n he oppose body whch wll possbly coe no conac. Ths s he geoercal recognon phase. The second phase s o check wheher or no he bodes are n conac and, f he bodes are n conac, f hey are sckng or slppng. Ths sep akes uses of he geoercal nforaon copued n he frs phase. The las sep wll be o copue a sasfacory sae of conac. -jan-9 6

28 RADIOSS THEORY Verson. INTERFACES The geoercal recognon phase s dependen on he ype of nerface. Ths wll be dscussed below n parallel wh he descrpon of nerfaces. On he oher hand, srucural probles wh conac-pac condons lead o consraned opzaon probles n whch he objecve funcon o be nzed s he vrual power subjec o he conac-pac condons. There are convenonally wo approaches o solvng such aheacal prograng probles: he Lagrange ulpler ehod he Penaly ehod. Boh of wo ehods are used n RADIOSS. 8.. Lagrange Mulpler Mehod Lagrange ulplers can be used o fnd he exreu of a ulvarae funcon f x, x,..., x n consran,,..., subjec o he g x x x n, where f and g are funcons wh connuous frs paral dervaves on he open se conanng he consran curve, and g a any pon on he curve where s he graden. To fnd he exreu, we wre: f f f df dx dx... dxn EQ x x x Bu, because g s beng held consan, s also rue ha n g g g dg dx dx... dxn EQ x x x n So ulply EQ by he as ye undeerned paraeer λ and add o EQ. 8..., f x g f g f g λ dx λ dx... λ dxn EQ x x x xn xn Noe ha he dfferenals are all ndependen, so any cobnaon of he can be se equal o and he reander us sll gve zero. Ths requres ha f x k g λ dxk EQ xk g, for all k,..., n, and he consan λ s called he Lagrange ulpler. For ulple consrans, g,..., f λ g λ g... EQ The Lagrange ulpler ehod can be appled o conac-pac probles. In hs case, he ulvarae funcon s he expresson of oal energy subjeced o he conac condons: x, x,..., x Π x, x, & & f & n EQ x x, x,..., x Qx, x, & & x g n EQ where x, x, & & x are he global vecors of d.o.f. The applcaon of Lagrange ulpler ehod o he prevous equaons gves he weak for as: -jan-9 7

29 RADIOSS THEORY Verson. INTERFACES M& x f f Lλ EQ n ex wh Lx b EQ Ths leads o: K L T L x f λ EQ The Lagrange ulplers are physcally nerpreed as surface racons. The equvalence of he odfed vrual power prncple wh he oenu equaon, he racon boundary condons and he conac condons penerably and surface racons can be easly deonsraed [8]. I s ephaszed ha he above weak for s an nequaly. In he dscrezed for, he Lagrange ulpler felds wll be dscrezed and he resrcon of he noral surface racon o be copressve wll resul fro consrans on he ral se of Lagrange ulplers. 8.. Penaly ehod In he soluon of consraned opzaon probles, penaly ehods conss n replacng he consraned opzaon proble wh a sequence of unconsraned opzaon probles. The vrual power connues o be nzed so as o fnd he saonary condon, bu a penaly er s added o EQ....5 so as o pose he penerably condon: where c Γ γ δ Q ρφ dγ EQ φ f γ γ N > φ f γ < γ N N N N ρ s an arbrary paraeer known as he penaly paraeer. The penaly funcon φ s an arbrary funcon of he nerpeneraon and s rae. I s ephaszed ha he weak for ncludng he vrual power and he penaly er EQ s no an nequaly for. The penaly funcon wll be defned n he descrpon of nerfaces. 8. Inerface overvew There are several dfferen nerface ypes avalable n RADIOSS. A bref overvew of he dfferen ypes and her applcaons s shown below. -jan-9 8

30 RADIOSS THEORY Verson. INTERFACES Inerface Descrpons Type Descrpon 3 Ted conac boundary beween an ALE par and a Lagrangan par. Ted conac. Used o sulae pacs and conacs on shell and sold eleens. Surfaces should be sply convex. 5 Used o sulae pacs and conacs beween a aser surface and a ls of slave nodes. Bes sued for bea, russ and sprng pacs on a surface Used o sulae pacs and conacs beween wo rgd surfaces. A general nerface ha reoves he laons of ypes 3 and 5. Drawbead conac for sapng applcaons ALE lagrange wh vod openng and free surface. Ted afer pac wh or whou rebound. Edge o edge or lne o lne pac. Connecs flud eshes wh free, ed or perodc opons ALE or EULER or LAG/ ALE or EULER or LAG. 4 5 Ellpsodal surfaces o nodes conac. Ellpsodal surfaces o segens conac. Each of hese nerfaces was developed for a specfc applcaon feld. However, hs applcaon feld s no he only selecon crera. Soe laons of he dfferen algorhs used n each nerface can also deerne he choce. The algorh laons concern anly he search of he paced segen. Ths search ay be perfored drecly ype 7,, nerfaces, or va a search of he neares node ype 3, 5, 6 nerfaces. Apar fro he laon of he neares node search, soe laons exs for he choce beween he segens conneced o he neares node. These laons are he sae for ype 3, 5 and 6 nerfaces. Type 3, 5 and 6 nerfaces also have soe laons due o he orenaon of he noral segen. Type 7 nerface was wren o eulae ype 3 and 5 nerfaces whou algorh laons. Wh hs nerface, each node can pac one or ore segens, on boh sdes, on he edges or on he corners of he segens. The only laon o hs nerface concerns hgh pac speed and/or sall gap. For hese suaons he nerface wll connue o work properly, bu he e sep can decrease draacally. Types 3, 5 and 7 nerfaces are copable wh all RADIOSS kneac opons. Type nerface s specal opon used wh sold eleens o provde esh ranson. Type nerface s used o connec a Lagrangan and an A.L.E. esh Type nerface s used o connec a fne and a coarse Lagrangan esh. All oher nerface ypes 3, 4, 5, 6 and 7 are used o sulae pacs and conacs. A node ay belong o several nerfaces. -jan-9 9

31 RADIOSS THEORY Verson. INTERFACES 8.. Surface Segen Defnon Surfaces or segens ay be defned n dfferen ways, dependng on he ype of eleen beng used. For a four node hree densonal shell eleen, an eleen s a segen. Fgure 8.. Shell Eleens Segen For a brck eleen, a segen s one face of he brck. Fgure 8.. Brck Eleen Segen For a wo densonal eleen, a segen s one sde. Fgure 8..3 D Eleen Segen 8.3 Ted nerface ype Refer o chaper 6 Kneac Consrans for a dealed descrpon. -jan-9 3

32 RADIOSS THEORY Verson. INTERFACES 8.4 Auo conacs 8.4. Inroducon The physcs of conacs s nvolved n varous phenoena, such as he pac of wo bllard balls, he conac beween wo gears, he pac of a ssle, he crash of a car, ec. Whle he physcs of he conac self s he sae n all hese cases, he an resulng phenoena are no. In he case of bllard balls, s he shock self ha s poran and wll hen be necessary o sulae perfecly he wave propagaon. In he case of gears, s he conac pressure ha has o be evaluaed precsely. The qualy of hese sulaons depends anly on he qualy of he odels spaal and eporal dscrezaon and on he choce of he negraon schee. In srucural crash or vehcle crash sulaons, he ajory of he conacs resul fro he bucklng of ubular srucures and eal shees. Modelng he srucure usng shell and plae fne eleens, he physcs of he conac canno be descrbed n a precse way. The reflecon of he waves n he hckness s no capured and he dsrbuon of conac pressures n he hckness s no aken no accoun. The peculary of he conacs occurrng durng he crash of a srucure les ore n he coplexy of he srucural foldng and he poran nuber of conac zones han n he descrpon of he pac or he conac self. Durng a conac beween wo sold bodes, he surface n conac s usually connuous and only slghly curved. On he oher hand, durng he bucklng of a srucure, he conacs, resulng fro shee foldng, are any and coplex. Globally, he conac s no longer beween wo denfed surfaces, bu n a surface pacng on self. The algorhs able o descrbe hs ype of conac are auo-pacng algorhs. Especally adaped o shell srucures, hey sll can be used o sulae he pac of he exernal surface of a sold 3D eleen on self. The an capables of he auo-conac can be suarzed by he followng funconales: capacy o ake each pon of he surface pac on self capacy o pac on boh sdes of a segen nernal and exernal possbly for a pon of he surface o be wedged beween an upper and a lower par processng of very srong concaves wll coplee foldng reversbly of he conac, hereby auhorzng unfoldng afer foldng or he sulaon of arbag deployen Modelng of conacs The conacs occurrng beween wo surfaces of a fne eleen esh can be odeled n dfferen ways: Conac nodes o nodes The conac s deeced based on he crera of dsance beween he wo nodes. Afer deecon of conac, a kneac condon or penaly forulaon ehod prevens he peneraon aanng he rebound pon "pnball" forulaon. -jan-9 3

33 RADIOSS THEORY Verson. INTERFACES Fgure 8.4. Conac nodes o surface The conac s deeced based on he crera of dsance beween a group of nodes and a eshed surface. The dsance beween a node and he surface of a rangular or quadrangular segen s evaluaed, locally. Syerzed conac nodes o surface The syerzaon of he prevous forulaon akes possble o odel a conac beween wo surfaces, as he group of nodes of he frs surface can pac he second group and vce-versa. Fgure 8.4. Conac edges o edges Ths forulaon akes possble o odel conacs beween wre fraed srucures or beween edges of wodensonal srucures. Conac s deeced based on he crera of dsance beween wo segens. Ths forulaon can also be used o descrbe n an approxae way he surface o surface conac. -jan-9 3

34 RADIOSS THEORY Verson. INTERFACES Surface o surface conac Several approaches can be used o deec he conac beween wo surfaces. If he wo surfaces are quadrangular, he exac calculaon of he conac can be coplex and que expensve. An approxae soluon ay be ade by cobnng he wo prevous forulaons. By evaluang all he conacs of nodes o surface, as well as he edge o edge conacs, he only approxaon s he paral consderaon of he segen curvaure Choce of a forulaon for auo pac In he case of a surface pacng on self, s possble o use one of he prevous forulaons f consderng ceran specfces of he auo-conac. The choce of a forulaon wll depend on wo essenal crera: he qualy of he descrpon of he conac and he robusness of he forulaon. The seleced forulaon has o provde resuls ha are as precse as possble n a noral operaonal suaon, whle sll workng n a sasfyng way n exree suaons. The node-o-node ehod provdes he bes robusness, bu he qualy of he descrpon s no suffcen enough o sulae n a realsc way hose conacs occurrng durng he bucklng of a srucure. The node-o-surface conac s he bes coprose. However, has soe laons, he an one beng ha canno deec conacs occurrng on he edges of a segen. The os crcal suaon occurs when hs leads o he lockng of a par of one surface o anoher. Ths phenoenon, beng rreversble, gh creae rrelevan behavor durng he deployen of a srucure e.g. arbag deployen. An exaple of a lockng suaon s shown n he above ages. To correc hs, s possble o assocae a node-o-surface forulaon o an edgeo-edge forulaon Algorhs of search for pac canddaes Durng he pac of one par on anoher, s possble o predc whch node wll pac on a ceran segen. In he case of he bucklng of a shell srucure, such as a ube, s possble o predc where dfferen conacs wll occur. I s hus necessary o have a farly general and powerful algorh ha s able o search for pac canddaes. The deal of he forulaon of an algorh able o search for pac canddaes wll depend on he choce of he conac forulaon descrbed n he prevous chaper. In a node-o-node conac forulaon, s necessary o fnd for each node he closes node, whose dsance s lower han a ceran value. In he case of he edge-oedge forulaon, he search for neghborng enes concerns he edges and no he nodes. However, we should noe ha n soe algorhs, he search for neghborng edges or segens s obaned by a node proxy calculaon. Moreover, an algorh desgned o search for proxy of nodes can be adaped n order o ransfor no a search for proxy of segens or even for a xed proxy of nodes and segens. I s possble o dsngush four an ypes of search for proxy: drec search opologcally led search algorhs of sorng by boxes bucke sor algorhs of fas sor ocree, quck sor When usng drec search, a each cycle he dsance s calculaed fro each eny node, segen, edge o all ohers. The quadrac cos N*N of hs algorh akes unusable n case of auo-conac. -jan-9 33

35 RADIOSS THEORY Verson. INTERFACES Topologcally led search In a sulaon n fas dynacs, geoercal odfcaons of he srucure are no very poran durng one cycle of calculaon. I s hen possble o consder neghborhood search algorhs usng he nforaon of he prevous cycle of copuaon. If for a node he neares segen s known a he prevous cycle, s hen possble o l he search for hs node o he segens opologcally close o he prevous one he segens havng a leas one coon node. Furherore, f an algorh based on he search of neghborng nodes s used, hen he search ay be led o he nodes of he segens conneced o he prevous closes node. I reans however necessary o do a coplee search before he frs cycle of calculaon. We wll also see ha hs algorh presens sgnfcan resrcons ha l s use o sldng surfaces canno be used for auopacng surfaces. The cos of hs algorh s lnear N, excep a he frs cycle of copuaon, durng whch s quadrac N*N. The cobnaon of hs algorh wh one of he wo followng s also possble Bucke sor Sorng by boxes consss n dvdng space n o seady boxes no necessarly dencal n whch he nodes are placed. The search for closes nodes s led o one box and he weny-sx neghborng boxes. The cos of hs sorng s lnear N for regular eshes. For rregular eshes, an adapaon s possble bu s neres becoes less neresng copared wh he nex soluon. Ths hree-densonal sorng s of he sae knd as one-way sorng wh drec addressng or needle sor In order o l he eory space needed, we frs coun he nuber of nodes n each box, hereby akng possble o l he fllng of hose boxes ha are no epy. In he wo-densonal exaple shown above, nodes are arranged n he boxes as descrbed n he followng able. Wh hs arrangeen, he calculaon of dsances beween nodes of he sae box s no a proble. On he oher hand, akng no accoun he nodes of neghborng boxes s no sraghforward, especally n he horzonal drecon f he arrangeen s frs ade vercally, as n hs case. One soluon s o consder hree coluns of boxes a a e. Anoher soluon, ore powerful bu usng ore eory, would be for each box o conan he nodes already locaed here plus hose belongng o neghborng boxes. Ths s shown n he hrd seres of coluns n he followng able. Once hs -jan-9 34

36 RADIOSS THEORY Verson. INTERFACES sorng has been perfored, he las sep s o calculae he dsances beween he dfferen nodes of a box, followed by he dsance beween hese nodes and hose of he neghborng box. In box,3 for exaple, ffeen dsances us be calculaed. -4, -3, -8, -, -5, -6, -7, -8, 4-3, 4-8, 4-, 4-5, 4-6, 4-7, 4-8. Box Nodes Nodes of he neghborng box, 6, , , , , , , ,5 3 5, , , , , , , , 9 4 4, , In hs exaple we have consdered a search based on nodal proxy. I s possble o adap hs algorh n order o arrange segens or even edges n he boxes. I wll be necessary however o reserve ore eory, for a segen can overlap several boxes Quck sor The ocree s a hree-densonal adapaon of fas sorng. Space s dvded no egh boxes, each one beng subdvded no egh boxes. In hs way a ree wh egh branches per node s obaned. An alernave o he "ocree", closer o he quck sor, consss n successvely dvdng space n wo equal pars, accordng o drecons X, Y, or Z. Ths operaon s renewed for each of he wo resulng pars as long as soe segens or nodes are found n he space concerned. The an advanage o hs algorh, as copared o sorng by boxes, les n he fac ha s perforance s affeced neher by he rregulary of he esh, nor by he rregulary of he odel. The cos of hs algorh s logarhc N*LogN. -jan-9 35

37 RADIOSS THEORY Verson. INTERFACES Fgure Possble Ipacs of Node n In order o llusrae he 3D quck sor, le us consder a search for node-segen proxy. A each sep, space s successvely dvded no wo equal pars, accordng o drecons X, Y or Z. The group of nodes s hus separaed no wo subses. We hereby oban a ree organzaon wh wo branches per node. Afer each dvson, a check s ade o deerne wheher he frs of he wo boxes us be dvded. If so, wll be dvded slar o he prevous one. If no, he nex branch s hen checked. Ths recursve algorh s dencal o he regular fas sorng one. The segens are also sored usng he spaal pvo. The resul of he es can lead o hree possbles: he segen s on he lef sde of he pvo, on he rgh sde of he pvo or asrde he pvo. In he frs wo cases, he segens are reaed slar o he nodes, bu n he hrd suaon, he segen s duplcaed and placed on boh sdes. Aong he dfferen crera ha can be used o sop he dvson are he followng suaons: he box does no conan any nodes he box does no conan any segens he box conans suffcenly few eleens ha he calculaon of dsance of all he couples s ore econocal he denson of he box s saller han a hreshold Conac processng Afer he choce of a good sorng algorh, a forulaon for he handlng of he conac has o be seleced. One can dsngush hree echnques ensurng he condons of connuy durng he conac: kneac forulaon of ype aser/slave. In a conac node o segen, he slave node ranss s ass and force o he aser segen and he segen ranss s speed o he node. Ths forulaon s parcularly adaped o an explc negraon schee, provded ha he nodes do no belong o a aser segen. A node canno be a he sae e slave and aser. Ths approach hen canno be used n he case of auo-conac. he Lagrange ulplers ensure kneac connuy a conac. There s no resrcon as n he prevous forulaon bu he syse of equaons canno be solved n an explc way. The Lagrange ulpler arx has o be reversed a each cycle of copuaon. In he case of auo-conac, he nuber of pons n conac can becoe sgnfcan and hs forulaon hen becoes que expensve. -jan-9 36

38 RADIOSS THEORY Verson. INTERFACES penaly ehods do no ensure kneac conac connuy, bu hey add sprngs a he conac spos. The frs advanage o hs forulaon s s naural negraon n an explc code. Each conac s reaed lke an eleen and negraes self perfecly no he code archecure, even f he prograng s vecoral and parallel. Conrary o he kneac forulaons, he penaly ehod ensures he conservaon of oenu and knec energy durng pac. The forulaon used n RADIOSS s a penaly ype forulaon. The choce of he penaly facor s a ajor aspec of hs ehod. In order o respec kneac connuy, he penaly sprng us be as rgd as possble. If he pedance of he nerface becoes hgher han hose of he srucures n conac, soe nuercal rebounds hgh frequency can occur. To ensure he sably of he negraon dagra, whou havng addonal consrans, hs rgdy us be low. Wh a oo low penaly, he peneraon of he nodes becoes oo srong and he geoercal connuy s no longer ensured. The coprose seleced consss n usng a sffness of he sae order of agnude han he sffness of he eleens n conac. Ths sffness s non lnear and ncreases wh he peneraon, so ha a node s no allowed o cross he surface. These choces provde a clear represenaon of physcs, whou nuercal generaon of nose, bu requre he conac sffness n he calculaon of he crera of sably of he explc schee o be aken no accoun Conac deecon Afer denfyng he canddaes for he pac, s necessary o deerne wheher conac akes place and s precse localzaon. If for a forulaon of node o node conac he deecon of he conac s que easy, becoes ore coplex n he case of a node o segen or edge o edge conac. In he case of edge-o-edge conacs, a drec soluon s possble f he segens are planar. If no, s beer o rangulae one of he segens, whch would hen urn no a node-o-segen conac proble. The search algorh for canddaes s uncoupled fro he res of he processng of he nerfaces. Ths s no he case wh regard o he deecon, localzaon and processng of he conac. These las hree asks sgnfcanly overlap wh each oher so we wll l ourselves o he processng of he conac by penaly for splcy. In he case of conac beween wo sold bodes odeled wh 3D fne eleens, conac can only ake place on he segens of he exernal surface. Ths exernal surface has a ceran orenaon and he pac of a node can coe only fro he ousde. Mos of he node-o-surface conac algorhs use hs orenaon o splfy deecon of conac. In he case of pac of a wo-densonal srucure odeled wh shell or plae fne eleens, conac s possble on boh sdes of he surface. For an orened surface, s necessary o consder conacs of he posve denson sde of he surface on self, conacs of he negave denson sde on self, and he conac of he posve sde on he negave one. Ths las suaon, que rare, can occur n he case of a surface rolled up no self or durng he pac of a wsed surface Node o segen conac The use of a "gap" surroundng he segen s one way of provdng physcal hckness o he surface and akes possble o dsngush he pacs on he op or on he lower par of he segen. The conac s acvaed f he node peneraes whn he gap or f he dsance fro he node o he segen becoes saller han he gap. To calculae he dsance fro he node o he segen, we ake a projecon of hs node on he segen and easure he dsance beween he node and he projeced pon. -jan-9 37

39 RADIOSS THEORY Verson. INTERFACES r r N3 N4, n r Ns N s r -,, s, s r N,- N The projeced pon s calculaed usng soparaerc coordnaes for a quadrangular segen and barycenrc coordnaes for a rangular segen. In he case of any quadrangular segen, he exac calculaon of hese coordnaes leads o a syse of wo quadrac equaons ha can be solved n an erave way. The dvson of he quadrangular segen no four rangular segens akes possble o work wh a barycenrc coordnae syse and gves equaons ha can be solved n an analycal way. Fro he soparaerc coordnaes s, of he projeced pon N, we have all he necessary nforaon for he deecon and he processng of conac. The relaons needed for he deernaon of s and are as follows: he vecor NNs s noral o he segen a he pon N and he noral o he segen s gven by he vecoral produc of he vecors s and. H s / 4 H s / 4 H3 s / 4 H 4 s / 4 ON HON H ON H3ON3 H 4ON 4 r s NN N 4N3 r s NN 4 s N N3 r r r r r n s / s r NNs a n Afer boundng he soparaerc coordnaes beween or -, he dsance fro he node o he segen and he peneraon are calculaed: D NNs P ax, Gap D A penaly force s deduced fro hs. I s appled o he node Ns and dsrbued beween he four nodes N, N, N3, N4 of he segen accordng o he followng shape funcons: Edge o edge conac The forulaon of edge-o-edge conac s slar o ha of node-o-segen conac. The gap surroundng each edge defnes a cylndrcal volue. The conac s deeced f he dsance beween he wo edges s saller han he su of he gaps of he wo edges. The dsance s hen calculaed as follows: N N D N N D / s s N N 3 s N N 4 N N N N / N N N N -jan-9 38

40 RADIOSS THEORY Verson. INTERFACES The force of penaly s calculaed as n node-o-segen conac. I s appled o he nodes N, N, N3, N4 and herefore ensures he equlbru of forces and oens. N3 N N N4 N 8.5 Type 3 - Sold and Shell Eleen Conac - No Gap The an use of hs nerface s wh shell or sold plaes ha are nally n conac. There are no aser and slave surfaces n hs nerface. Each surface s consdered as f were a slave Laons The an laons of ype 3 nerfaces are: The wo surfaces should be sply convex. The surface norals us face each oher. A node ay no exs on he aser and slave sde of an nerface sulaneously. Surfaces us conss of eher shell or brck eleens. I s recoended ha he wo surface eshes be regular wh a good aspec rao. The nerface gap should be kep sall, f no zero. There are soe search probles assocaed wh hs nerface. -jan-9 39

41 RADIOSS THEORY Verson. INTERFACES Fgure 8.5. Surfaces and wh Facng Norals 8.5. Copuaon Algorh The copuaon and search algorhs used for ype 3 nerface are he sae as for ype 5. However, ype 3 nerface does no have a aser surface, so ha he algorhs are appled wce, one for each surface. The surfaces are reaed syercally, wh all nodes allowed o penerae he opposng surface. The nerface sprng sffness apples he opposng peneraon reducon force. Fgure 8.5. Conac Surfaces Treaed Syercally Because he copuaon algorh s perfored wce, accuracy s proved over a ype 5 nerface. However, he copuaonal cos s ncreased. The frs pass soluon solves he peneraon of he nodes on surface wh respec o segens on surface. The second pass solves surface nodes wh respec o surface segens Inerface Sffness When wo surfaces conac, a assless sffness s nroduced o reduce he peneraon's nodes of he oher surface no he surface. The naure of he sffness depends on he ype of nerface and he eleens nvolved. The nroducon of hs sffness ay have consequences on he e sep, dependng on he nerface ype used. The ype 3 nerface sprng sffness K s deerned by boh surfaces. To rean soluon sably, sffness s led by a scalng facor whch s user defned on he npu card. The defaul value and recoended value s.. The overall nerface sprng sffness s deerned by consderng wo sprngs acng n seres. -jan-9 4

42 RADIOSS THEORY Verson. INTERFACES Fgure Inerface Sprngs n Seres K Surface K Surface The equaon for he overall nerface sprng sffness s: K K K K K s EQ where s Sffness Scalng Facor. Defaul s.. K Surface Sffness K Surface Sffness K Overall Inerface Sprng Sffness The scale facor, s, ay have o be ncreased f: K << K or K K << The calculaon of he sprng sffness for each surface s deerned by he ype of eleens. For exaple: K K ples s K K s or K K Shell Eleen If he aser nerface segen s a se of shell eleens, he sffness s calculaed by: K. 5sE EQ where E Modulus of Elascy Shell Thckness The sffness does no depend on he shell sze Brck Eleen If he aser nerface segen s a se of brck eleens, he sffness s calculaed by: sba K.5 EQ V where B Bulk Modulus A Segen Area V Eleen Volue -jan-9 4

43 RADIOSS THEORY Verson. INTERFACES Fgure Brck eleen V A Cobned Eleens If a segen s a shell eleen ha s aached o he face of a brck eleen, he shell sffness s used Inerface Frcon Type 3 nerface allows sldng beween conac surfaces. Coulob frcon beween he surfaces s odelled. The npu card requres a frcon coeffcen. No value defaul defnes zero frcon beween he surfaces. The frcon on a surface s calculaed by: K ΔF r CC EQ where K Inerface Sprng Sffness C C conac node dsplaceen vecor Fgure Coulob Frcon C Conac pon a e C Conac pon a e Δ -jan-9 4

44 RADIOSS THEORY Verson. INTERFACES Fgure Frcon on nerface ype 3 Frs pac Maer Ks p Gap Slave node K V D The noral force copuaon s gven by: Fn K sp EQ Gap where K s K Gap P K Inal nerface sprng sffness as n ype 5 The angenal force copuaon s gven by: F KD EQ Kn where K If he frcon force s greaer han he lng suaon, F > μ Fn, he frconal force s reduced o equal he l, F μ Fn, and sldng wll occur. If he frcon s less han he lng condon, F μfn, he force s unchanged and sckng wll occur. Te negraon of he frconal forces s perfored by: r r r new old F F ΔF EQ where Δ F r resul fro equaon Fgure Frcon on ype 3 nerface F μf n X -jan-9 43

45 RADIOSS THEORY Verson. INTERFACES Inerface Gap Type 3 nerfaces have a gap ha deernes when conac beween wo segens occurs. Ths gap s user defned, bu soe nerfaces wll calculae an auoac defaul gap. The gap deernes he dsance for whch he segen neracs wh he hree nodes. If a node oves whn he gap dsance, such as nodes and, reacon forces ac on he nodes. Type 3 nerface have a gap: - only noral o he segen, as shown n Fgure on he conac sde of he segens, whch s defned by he surface noral. The sze of he gap defned for ceran nerface ypes s crcal. If he gap s oo sall, he soluon e sep ay be draacally reduced or a node ay ove across he enre gap n one e sep. However, f he gap s oo large, nodes no assocaed wh he drec conac ay becoe nvolved. Fgure Inerface Gap Inerface Falure Exaples There are a nuber of suaons n whch ype 3 eleens ay fal. A couple of hese are shown below. Care us be aken when defnng conac surfaces wh large deforaon sulaons. If he noral defnons of he conac surfaces are ncorrec, node peneraon wll occur whou any reacon fro eher surface. Fgure Iproper Noral Drecon -jan-9 44

46 RADIOSS THEORY Verson. INTERFACES Referrng o Fgure 8.5.9, he frs suaon shows he esh deforng n a way ha allows he norals o be facng each oher. However, n he second case, he deforaon oves wo surfaces wh norals all facng he sae drecon, where conac wll no be deeced. Large roaons can have a slar effec, as shown n Fgure 8.5. Fgure 8.5. Inal and Defored Mesh Before Ipac Kneac oon ay reposon he esh so ha norals do no correspond. I s recoended ha possble pac suaons be undersood before a sulaon s aeped. 8.6 Type 5 - General Purpose Conac Ths nerface s used o sulae he pac beween a aser surface and a ls of slave nodes, as shown n Fgure The peneraon s reduced by he penaly ehod. Anoher ehod s possble: Lagrange ulplers. Bu spaal dsrbuon forces s no sooh and nduces hourglass deforaon. Ths nerface s anly used for: Sulaon of pac beween bea, russ and sprng nodes on a surface. Sulaon of pac beween a coplex fne esh and a sple convex surface. A replaceen for a rgd wall Laons The an laons of ype 5 nerface are: The aser segen norals us be orened fro he aser surface owards he slave nodes. The aser sde segens us be conneced o sold or shell eleens. The sae node s no allowed o exs on boh he aser and slave surfaces. Soe search probles refer o he probles A, B, B, B3, C, D n chaper I s recoended ha he aser surface esh be regular, wh a good aspec rao and ha a sall or zero gap be used o deec peneraon. -jan-9 45

47 RADIOSS THEORY Verson. INTERFACES Fgure 8.6. Surface nodes and surface segens 8.6. Copuaon Algorh The copuaon and search algorhs used for ype 5 are he sae as for ype 3. Refer o chaper Inerface Sffness When wo surfaces conac, a assless sffness s nroduced o reduce he peneraon's nodes of he oher surface no he surface. The naure of he sffness depends on he ype of nerface and he eleens nvolved. The nroducon of hs sffness ay have consequences on he e sep, dependng on he nerface ype used. For a ype 5 nerface, he sprng sffness K s deerned by he aser sde only. The sffness scalng facor defaul value and recoended value s.. For a sof aser surface aeral and sff slave surface, he sffness scalng facor should be ncreased by he elasc odulus rao of he wo aerals. The calculaon of he sprng sffnesses s he sae as n a ype 3 nerface. If a segen s a shell as well as he face of brck eleen, he shell sffness s used. The overall nerface sprng sffness s deerned by consderng wo sprngs acng n seres. Fgure 8.6. Inerface Sprngs n Seres K Surface K Surface The equaon for he overall nerface sprng sffness s: K K K K K s EQ jan-9 46

48 RADIOSS THEORY Verson. INTERFACES where s Sffness Scalng Facor. Defaul s.. K Surface Sffness K Surface Sffness K Overall Inerface Sprng Sffness The scale facor, s, ay have o be ncreased f: K << K or K K << The calculaon of he sprng sffness for each surface s deerned by he ype of eleens. For exaple: K K ples s K K s or K K Shell Eleen Refer o chaper Brck Eleen Refer o chaper Cobned Eleens Refer o chaper Inerface Frcon Type 5 nerface allows sldng beween conac surfaces. Coulob frcon beween he surfaces s odelled. The npu card requres a frcon coeffcen. No value defaul defnes zero frcon beween he surfaces. The frcon copuaon on a surface s he sae as for ype 3 nerface. Refer o chaper Darsad and Renard odels for frcon are also avalable: Darsad law: Renard law: where, C C V C 4V C V μ EQ C e p C e p C e 3 5 V V C μ C C 3 C f [, C5] C V C C6 C V C C C 5 5 μ C C C 3 f V [ ] μ C f V C6 V C6 C C s, C μ μ d V EQ C 5,C 6 -jan-9 47

49 RADIOSS THEORY Verson. INTERFACES C 3 μ ax, C4 μn C V C V 5 cr, 6 cr Possbly of soohng he angen forces va a fler: 4F Τ Τ α F α F EQ where he coeffcen a depends on he Iflr flag value. Τ Inerface Gap Refer o chaper for ype 3 nerface Inerface Algorh The algorh used o calculae nerface neracon for each slave node s:. Deerne he closes aser node.. Deerne he closes aser segen. 3. Check f he slave node has peneraed he aser segen. 4. Calculae he conac pon. 5. Copue he peneraon. 6. Apply forces o reduce peneraon. For ore nforaon, refer o Chaper Deecon of Closes Maser Node The RADIOSS Sarer and Engne use dfferen ehods o deerne he closes aser node o a parcular slave node. The Sarer searches for he aser node wh he nu dsance o he parcular slave node. The Engne carres ou he followng algorh, referrng o Fgure Fgure Search Mehod -jan-9 48

50 RADIOSS THEORY Verson. INTERFACES. Ge he prevous closes segen, S, o node N a e.. Deerne he closes node, M, o node N whch belongs o segen S. 3. Deerne he segens conneced o node M S, S, S and S Deerne he new closes aser node, M, o node N a e. The new aser node us belong o one of he segens S, S, S or S Deerne he new closes aser segen S, S 4, S 5 and S 6. The Sarer CPU Cos s calculaed wh he followng equaon: CPUsarer a x Nuber of Slave Nodes x Nuber of Maser Nodes The Engne CPU Cos s calculaed wh he followng equaon: CPUengne b x Nuber of Slave Nodes The algorh used n he engne s less expensve bu does no work n soe specal cases. In Fgure 8.6.4, f node N s ovng fro N o N and hen o N, he closes aser nodes are M and M. When he fnal node oveen o N s aken, he pac on segen S wll no be deeced snce none of he nodes on hs segen are consdered as he closes aser node. Fgure Undeeced Ipac Deecon of Closes Maser Segen The closes aser segen o a slave node N, shown n Fgure 8.6.5, s found by deernng a reference quany, A. Fgure Closes Maser Segen Deernaon Mehod -jan-9 49

51 RADIOSS THEORY Verson. INTERFACES The A value for segen S s gven by: r r r r A MB MP MB MP < where P Projecon of N on plane M B B B and B Tangenal Surface Vecors along segen S edges. The A value for segen S s gven by: r r r r A MB MP MB MP 3 > where P Projecon of N on plane M B B 3 The sae procedure s carred ou for all aser segens ha node M s conneced o. The closes segen s he segen for whch A s a nu. In soe specal cases curved surfaces, s possble ha:. All values of A are posve.. More han one value of A s negave. EQ EQ Deecon of Peneraon Peneraon s deeced by calculang he volue of he erahedron ade by slave node N and he aser nodes of he correspondng aser segen, as shown n Fgure For a gven noral n, he sgn of he volue ells f peneraon has occurred. Fgure Terahedron used for Peneraon Deecon -jan-9 5

52 RADIOSS THEORY Verson. INTERFACES Fgure Negave Volue Terahedron - Peneraed Node Reducon of Peneraon The peneraon, p, s reduced by he nroducon of a assless sprng beween he node, N, and he conac pon, C. Fgure Forces Assocaed wh Peneraon The force appled on node N n drecon NC s: F Kp EQ where K Inerface Sprng Sffness Reacon forces F, F, F 3 and F 4 are appled on each aser node as shown n Fgure n he oppose drecon o he peneraon force, such ha: F F F3 F4 F EQ Forces F,, 3, 4 are funcons of he poson of he conac pon, C. They are evaluaed by: -jan-9 5

53 RADIOSS THEORY Verson. INTERFACES F where d 4 d j 4 d j j N F S, N j c S, c c c EQ or ore sply: F N S F EQ c, c where N Sandard lnear quadralaeral nerpolaon funcons S c and c Isoparaerc coordnaes conac pon The penaly ehod s used o reduce he peneraon. Ths provdes: Accuracy Generaly Effcency Copably 8.7 Type 6 - Rgd Body Conac Ths nerface s used o sulae pacs beween wo rgd bodes. I works lke ype 3 nerface excep ha he oal nerface force s a user defned funcon of he axu peneraon. The npu and copuaonal algorhs are he sae as for ype 3 nerfaces. Ths nerface s used exensvely n vehcle occupan sulaons, eg. knee bolsers Laons Soe of he an laons for hs nerface ype are: Surface us be par of one and only one rgd body. Surface us be par of one and only one rgd body. The nerface sffness user defned funcon can reduce he e sep. Oher laons are he sae as for ype 3 nerfaces. Fgure 8.7. Surfaces and wh Facng Norals -jan-9 5

54 RADIOSS THEORY Verson. INTERFACES 8.7. Inerface Sffness When wo surfaces conac, a assless sffness s nroduced o reduce he peneraon's nodes of he oher surface no he surface. The naure of he sffness depends on he ype of nerface and he eleens nvolved. If a segen s a shell as well as he face of brck eleen, he shell sffness s used Inerface Frcon Type 6 nerface allows sldng beween conac surfaces. Coulob frcon beween he surfaces s odelled. The npu card requres a frcon coeffcen. No value defaul defnes zero frcon beween he surfaces. The frcon copuaon on a surface s he sae as for ype 3 nerface refer o chaper Inerface Gap Refer o chaper for ype 3 nerface Te Sep Calculaon The sable e sep used for e negraon equaons s copued by: M Δ. EQ K where M nm rgd body, M rgd body K Tangen of user force funcon Te sep Δ s affeced by he acual sffness derved fro funcon f: M Δ EQ f p The funcon f refers o a funcon nuber gven n npu and has o be provded by user Conac force Fgure 8.7. Specally sued for rgd bodes -jan-9 53

55 RADIOSS THEORY Verson. INTERFACES FN r r p q f r r MAX p, q N r r k j EQ p j qk q r N s he conrbuon o node N of vecor q r dsrbued on he segen peneraed by node Q. 8.8 Type 7 - General Purpose Conac Ths nerface sulaes he os general ype of conacs and pacs. Type 7 nerface has he followng properes:. Ipacs occur beween a aser surface and a se of slave nodes, slar o ype 5 nerface.. A node can pac on one or ore aser segens. 3. A node can pac on eher sde of a aser surface. 4. Each slave node can pac each aser segen excep f s conneced o hs segen. 5. A node can belong o a aser surface and a se of slave nodes, as shown n Fgure A node can pac on he edge and corners of a aser segen. None of he prevous nerfaces allow hs. 7. Edge o edge conacs beween aser and slave segens are no solved by hs nerface Laons All laons encounered wh nerface ypes 3, 4 and 5 are solved wh hs nerface. I s a fas search algorh whou laons. Fgure 8.8. Slave and Maser Node Ipac There are no search laons wh hs nerface concernng node o surface conacs. All possble conacs are found. -jan-9 54

56 RADIOSS THEORY Verson. INTERFACES There s no laon on he use of large and sall segens on he sae nerface. Ths s recoended o have a good aspec rao eleens or a regular esh o oban reasonable resuls; however, s no an oblgaon. There s no laon o he sze of he sprng sffness facor. The sprng sffness s uch greaer han nerfaces 3 and 5, wh he defaul sffness facor se o.. Ths s o avod node peneraons larger han he gap sze, reovng probles ha were assocaed wh he oher nerfaces Inerface Sffness When wo surfaces conac, a assless sffness s nroduced o reduce he peneraon of one surface nodes o he oher surface. The naure of he sffness depends on he ype of nerface and he eleens nvolved. The nroducon of hs sffness ay have consequences on he e sep, dependng on he nerface ype used. The nerface sprng sffness calculaon s no as sple as for ypes 3, 4 and 5. The nal sffness s calculaed usng he ehods for ype 3 nerfaces. However, afer nal peneraon, he sffness s gven as a funcon of he peneraon dsance and he rae of peneraon. A crcal vscous dapng coeffcen gven on he npu card vsc allows dapng o be appled o he nerface sffness. dp F f p vsc KM EQ d The sffness s uch larger han he oher nerfaces o accoodae hgh speed pacs wh nal crossng of surfaces. The consequence of hs s ha a sable e sep s calculaed o anan soluon sably Inerface Frcon Type 7 nerface allows sldng beween conac surfaces. Coulob frcon beween he surfaces s odelled. The npu card requres a frcon coeffcen. No value defaul defnes zero frcon beween he surfaces. In ype 7 nerface a crcal vscous dapng coeffcen s defned, allowng vscous daped sldng. The frcon on a surface ay be calculaed by wo ehods. The frs ehod suable for conac angenal velocy greaer ha /s conss n copung a vscous angenal growh by: r r Δ F C V EQ In he second ehod an arfcal sffness K S s npu. The change of angen force F s obaned he followng equaon: Δ F δ EQ K S where δ s he angen dsplaceen. The noral force copuaon s gven by: dp Fn KsP C EQ d Gap where K s K Gap P C VIS s K M s K Inal nerface sprng sffness as n ype 5 VIS s Crcal dapng coeffcen on nerface sffness npu -jan-9 55

57 RADIOSS THEORY Verson. INTERFACES Fgure 8.8. Coulob Frcon C Conac Pon a e C Conac Pon a e Δ Fgure Frcon on ype 7 nerface - schee The angenal force copuaon s gven by: FRIC * F F F n, EQ where F C V ad C VIS F K M F ad adheson force s n ad VIS F Crcal dapng coeffcen on nerface frcon npu -jan-9 56

58 RADIOSS THEORY Verson. INTERFACES If he frcon force s greaer han he lng suaon, F > μ Fn, he frconal force s reduced o equal he l, F μ Fn, and sldng wll occur. If he frcon s less han he lng condon, F μfn, he force s unchanged and sckng wll occur. Noe ha he frcon coeffcen μ ay be obaned by Coulob, Darsad and Renard odels as descrbed n secon Te negraon of he frconal forces s perfored by: r r r new old F F ΔF EQ where ΔF r s obaned fro EQ or EQ Fgure Frcon on ype 7 nerface F μf n V Inerface Gap Type 7 nerfaces have a gap ha deernes when conac beween wo segens occurs. Ths gap s user defned, bu soe nerfaces wll calculae an auoac defaul gap. Shown n Fgure 8.8. s a segen ype 7 nerface wh hree nodes n close proxy. The gap, as shown, deernes he dsance for whch he segen neracs wh he hree nodes. If a node oves whn he gap dsance, such as nodes and, reacon forces ac on he nodes. Fgure Type 7 Inerface Gap Type 7 nerface has a gap ha covers boh edges of he segens, as shown n Fgure jan-9 57

59 RADIOSS THEORY Verson. INTERFACES Te Sep A e sep s calculaed o anan sably when a ype 7 nerface s used. dp The kneac or nerface e sep s calculaed f > by: d Gap p Δn. 5 dp EQ d The sable e sep or nodal e sep s gven by: M Δ nod EQ K where M Nodal ass K Kn er Kel Nodal sffness The e sep used for he nerface s he saller of he wo. If he nerface sprng sffness s oo grea, he e sep can be reduced draacally. If he wo aerals nvolved n he conac are he sae, he defaul nerface sffness facor can be reaned. Ths s he case when odellng shee eal. However, he sffness facor ay need adjusen f he wo aerals sffnesses vary oo uch; for exaple, seel and foa Mehods o Increase Te Sep The e sep can be alered by wo dfferen ehods, by alerng he sze of he gap and by ncreasng he nal sffness. Fgure shows hree force-peneraon curves for a ype 7 nerface. Boh ehods change he naure of he sffness whch affecs he e sep. Force Fgure Force - Peneraon Curves F K K 3 4 K K K Peneraon Gap Gap Usng a larger gap sze, curves and keep he sae nal sffness, hence he nal e sep reans he sae. Snce he pac slowng force s appled over a greaer dsance, he sffness s no changed as uch, bu ncreases. Increasng he nal nerface sffness, alhough decreasng he e sep nally, wll ncrease he e sep f peneraon s large. -jan-9 58

60 RADIOSS THEORY Verson. INTERFACES Deecon and Gap Sze A slave node can be deeced near a aser segen fro all drecons, as shown n Fgure The sze of he gap can be user defned, bu RADIOSS auoacally calculaes a defaul gap sze, based on he sze of he nerface eleens. For shell eleens, he copued gap s he average hckness. For brck eleens s equal o one enh of he nu sde lengh Varable gap By defaul he gap s consan on all aser segens. If he varable gap opon s acvaed, a dfferen gap s used for each conac akng no accoun he physcal hckness on he aser and slave sdes. Fgure Varable gap For shell eleens, he aser gap s equal o one half of he shell hckness. The slave gap s equal o one half of he larges hckness of all conneced shell eleens. For sold eleens, he aser gap s zero. If he slave node s only conneced o sold eleens, he slave gap s zero. For bea or russ eleens conneced o he slave node, he slave gap s one half of he square roo of he secon area. If a slave node s conneced o dfferen eleens shell, brck, bea, russ he larges gap value s used. The oal gap s he su of he slave and aser gaps. The oal gap canno be saller han a nu gap user npu gap Gap correcon for nodes wh nal peneraon Type 7 nerface s very sensve o nal peneraons. One ehod for solvng he resulng probles s o use an auoac gap correcon INACTI 5. Wh auoac gap correcon he effecve gap s correced o ake no accoun he nal peneraon. The correcon s only appled o he nally peneraed nodes. If he node peneraon decreases, he correcon s reduced. The copued peneraon s llusraed n Fgure jan-9 59

61 RADIOSS THEORY Verson. INTERFACES Fgure Correced gap Peneraon Reacon Lke he oher nerface ypes, ype 7 has a sprng sffness as a slave node peneraes he nerface gap prevous secon. However, here are soe fundaenal dfferences n he deernaon of he reacon force. Fgure shows a graph of force versus peneraon of a node on a aser segen. Ths fgure also shows a pcoral dagra of node peneraon and he assocaed forces. Fgure Peneraon Reacon Force The reacon force s no a lnear relaon lke he prevous nerfaces. There s a vscous dapng whch acs on he rae of peneraon. The force copuaon s gven by: dp F K s P C EQ d -jan-9 6

62 RADIOSS THEORY Verson. INTERFACES where Gap K s K Gap P C VIS s K M s K Inal nerface sprng sffness as n ype 5 VIS s Crcal dapng coeffcen on nerface sffness npu The nsananeous sffness s gven by: Gap K K Gap P EQ Fgure 8.8. Force - Peneraon Graph A crcal vscous dapng s requred o be defned on he ype 7 npu card for boh dapng on he sprng sffness and for nerface frcon dapng Force Orenaon Due o he gap on a ype 7 nerface exendng around he edges of a segen, he reacon forces over a surface wll be sooh. Fgure 8.8. shows he reacon forces on a node a varous posons around wo adjonng segens. -jan-9 6

63 RADIOSS THEORY Verson. INTERFACES Fgure 8.8. Force Orenaon Poson n Fgure 8.8. shows he force acng radally fro he edge of he segen. The sze of he force depends on he aoun of peneraon. A poson he force s noral o he segen surface. In poson 3 wo segens nersec and her gaps overlap. The resul s ha each segen apples a force o he node, noral o he respecve segen; hs can double he force for he dsance of gap overlap Inerface hns Man proble One an proble reans naely: deep peneraons are no easly oleraed They lead o hgh penaly forces and sffness, and consequenly o a drop n e sep. When such a proble occurs, you ay see: a very sall e sep an nfne loop essage os lkely due o dvergence a large conac force vecor n anaon Deep peneraons.e. close o gap value canno be avoded n os car crash sulaons. They occur n he followng cases: nal peneraons of adjacen plaes edge pacs: wrong sde conacs full collapse of one srucural regon rgd body pac on anoher rgd body or on fxed nodes or on very sff srucures pac beween heavy sff srucures hgh pac speed sall gap -jan-9 6

64 RADIOSS THEORY Verson. INTERFACES The elasc conac force s calculaed wh he forulaon: gp F k EQ g p Wh k.5 STFAC*E* The elasc conac energy s calculaed wh he forulaon: g p CE kg p gln EQ g When node o eleen d-plane dsance s saller hen - gap, he node s deacvaed. The axu poenal conac energy s: elasc conac energy CE 3 k g Drasc e sep droppng s osly due o cases where node s forced no he gap regon Reedes o he an proble There are several ways o resolve hs proble: - Increase Gap Increasng he gap s he bes reedy, bu check ha no nal peneraons resul fro hs. - Increase Sffness Increase STFAC densonless sffness facor or provde an approprae effecve global sffness value v3 and up. 3- DT/INTER/DEL Engne opon Soe node wll be allowed o cross he paced surface freely before peneraon reaches - gap. 4- /DT/INTER/CST Engne opon Nodal ass wll be odfed o anan e sep consan. Ths opon should be avoded when rgd body slave nodes are slaves of a ype 7 nerface. The nal peneraons are osly due o dscrezaon and odelzaon probles. They resul n hgh nal forces ha should be avoded. 5- Modfy geoery New coordnaes are proposed n he lsng fle for all nally peneraed nodes. These are he coordnaes used n he auoac coordnae change opon. However, hs gh no suffce. Several eraons are soees necessary. RADIOSS wll creae a fle RooDA conanng he odfed geoery. 6- Reduce gap When here are only sall peneraons wh a. gap, hs should be reduced, oherwse care should be aken as hs wll reduce poenal conac energy. 7- Deacvae node sffness Ths soluon s he sples. I wll generally no unduly affec your resuls. For sake of precson, use hs opon only for nal peneraons reanng afer geoercal adjusens. Edge conac proble A specal algorh s developed for hs purpose. Modelzaon should evenually be adaped o preven suaons where nodes of an eleen ove o oppose sdes of a surface. For sold o sold conacs, he exernal closed surfaces ay be used. -jan-9 63

65 RADIOSS THEORY Verson. INTERFACES 8.9 Type 4 - Ellpsodal surface o node conac Ths nerface sulaes pacs beween an hyper-ellpsodal rgd aser surface and a ls of slave nodes. Ths nerface s used for MADYMO o RADIOSS couplng. The hyper-ellpsodal surface s reaed as an analycal surface hyper-ellpsodal surfaces are dscrezed only for pos-processng. The nerface allows user defned behavor. - User defnes oal elasc force versus axu peneraon of nodes. - A local frcon coeffcen s copued a each paced node, dependng upon elasc force copued a s locaon by scalng he oal elasc force by he followng facor: peneraon of he node dvded by su of node peneraons. - A local vscosy coeffcen n he noral drecon o he surface s copued a each paced node, dependng upon hs node's velocy or he copued elasc force a s locaon. I s also possble o only defne a consan sffness facor, a consan frcon coeffcen or a consan vscosy coeffcen. A e sep s copued o ensure sably Type 4 nerface: Hns As he nerface s defned as nodes pacng upon a surface, pac canno be deeced f he esh s oo coarse. In general, use a esh whch sze s fner han he lowes se-axs of he aser surface. Fgure 8.9. No pac s deeced The nerface s desgned o allow peneraon of slave nodes. However, he conac algorh does no ensure ha a node wll no cross he ellpsod when sldng; nodes ay slde along he ellpsod unl hey fully cross he ellpsod, resulng n ha he srucure self fully crossng he surface and conac force s no longer appled o as shown n fgure 8.9., where perfec sldng s consdered. Increase nerface sffness or frcon o avod hs proble. -jan-9 64

66 RADIOSS THEORY Verson. INTERFACES Fgure 8.9. Sldng unl srucure fully crosses surface 8. Type 5 - Ellpsodal surface o segen conac Type 5 nerface beween surfaces ade up of 4-node or 3-node segens and hyper-ellpsods s a penaly conac nerface whou dapng. I apples o ype 4 nerface, especally when he esh s coarser han he ellpsod sze. Reeber ha n such a case, ype 4 nerface s able o copue low qualy conac forces even f fals o fnd conac, as shown below. Fgure 8.. No conac s deeced Inerface Type 4 : No conac s deeced beween heel and coarse esh of floor Inerface sffness s a non-lnear ncreasng funcon of peneraon, copued n order o avod peneraons up o half he ellpsod: Deph K K where K s an npu sffness facor. Deph Peneraon -jan-9 65

67 RADIOSS THEORY Verson. INTERFACES Fgure 8.. Peneraon s deeced Deph P realzes axu peneraon upon he eleen a e Peneraon F n noral force K * Peneraon A Kneac Te Sep s copued so ha he eleen does no cross he lne L whn one e sep. A frcon coeffcen s npu. Inerface akes no accoun sldng/rollng effecs. Coulob Frcon condon s expressed as: F F n for each peneraed eleen. 8. Type 6- Node o Curved Surface Conac Inerface ype 6 wll allow o defne conac condons beween a group of nodes slaves and a curve surface of quadrac eleens aser par as shown n Fgure 8.. for a syerc conac. The aser par ay be ade of 6-node hck shells or node-brcks. The Lagrange Mulpler Mehod LMM s used o apply he conac condons. By he way ha he LMM s used, no gap s necessary o be appled. Soe applcaons of hs nerface are sldng conacs whou gaps as n gear box odellng. Fgure 8.. Node o curved surface conac n nerface ype 6 8. Type 7- General Surface o Surface Conac The nerface s used n he odellng of surface-o-surface conac. I s a generalzed for of ype 6 nerface n whch he conac on he wo quadrac surfaces are drecly resolved whou needs of gap as he Lagrange Mulpler Mehod s used Fgure 8... The conac s supposed o be sldng or ed. -jan-9 66

68 RADIOSS THEORY Verson. INTERFACES Fgure 8.. Quadrac surface o quadrac surface conac 8.3 Soe Coon Probles The followng secons conan exaples of soe coon probles n he conac nerfaces and soluons o overcoe he Incorrec Neares Maser Node Found If he nerface surface s no sply convex, he splfed aser node search ay fnd an ncorrec neares aser node. Ths proble occurs wh nerface ypes 3, 6 and 5 aser sde only. The soluon o hs proble s o use ype 7 nerface. Fgure 8.3. Incorrec Maser Node Found 8.3. Incorrec Neares Maser Segen Found - B In soe cases he neares aser node s no conneced o he neares segen. Ths proble can occur wh nerface ypes 3, 6 and 5 aser sde only. The soluon s o eher use ype 7 nerface or o refne he esh. -jan-9 67

69 RADIOSS THEORY Verson. INTERFACES Fgure 8.3. Wrong Neares Maser Segen Incorrec Neares Maser Segen Found - B In soe cases he neares aser node s no conneced o he neares segen. Ths proble can occur wh nerface ypes 3, 5 and 6 aser sde only. The soluon s o eher use ype 7 nerface or change he esh for nal esh proble. Fgure Incorrec Neares Maser Segen Incorrec Neares Maser Segen Found - B3 If he angle beween segens s less han 9 degrees, he ncorrec neares segen ay soees be found, as n Fgure Ths proble can occur wh nerface ypes 3, 5 and 6. The soluon s o use a ype 7 nerface or o refne he esh. Wh a fner esh, he shape s sooher. -jan-9 68

70 RADIOSS THEORY Verson. INTERFACES Fgure Maser Segen Angle o Acue Incorrec Ipac Sde - C A node can only pac on he posve sde of a segen for nerface ypes 3, 6, and 5 aser sde. The soluon s o use a ype 7 nerface. Fgure Wrong Noral Drecon No Maser Node Ipac - D Wh ype 5 nerface, only slave nodes pac aser segens; aser nodes canno pac slave segens. Ths can be solved by eher nverng he slave and aser sdes, or by changng he ype of nerface. Inerface ypes 3 and 7 wll solve hs proble adequaely. -jan-9 69

71 RADIOSS THEORY Verson. INTERFACES Fgure Maser Node Peneraon -jan-9 7

72 Chaper MATERIALS -jan-9 7

73 9. MATERIAL LAWS A large varey of aerals s used n he srucural coponens and us be odeled n sress analyss probles. For any knd of hese aerals a range of consuve laws s avalable o descrbe by a aheacal approach he behavor of he aeral. The choce of a consuve law for a gven aeral depends a frs o desred qualy of he odel. For exaple, for sandard seel, he consuve law ay ake no accoun he plascy, ansoropc hardenng, he sran rae, and eperaure dependence. However, for a roune desgn aybe a sple lnear elasc law whou sran rae and eperaure dependence s suffcen o oban he needed qualy of he odel. Ths s he analys desgn choce. On he oher hand, he sofware us provde a large consuve lbrary o provde odels for he ore coonly encounered aerals n praccal applcaons. RADIOSS aeral lbrary conans several dsnc aeral laws. The consuve laws ay be used by he analys for general applcaons or a parcular ype of analyss. The user can also progra a new aeral law n RADIOSS. Ths s a powerful resource for he analys o code a coplex aeral odel. Theorecal aspecs of he aeral odels ha are provded n RADIOSS are descrbed n hs chaper. The avalable aeral laws are classfed n Table 9... Ths classfcaon s n copleenary wh hose of RADIOSS npu anual. The reader s nved o consul ha one for all echncal nforaon relaed o he defnon of npu daa.

74 RADIOSS THEORY Verson. MATERIALS Table 9.. Maeral law descrpon Type Descrpon Model Law nuber n RADIOSS MID Isoropc Elascy Lnear elasc Hooke Hyper elasc Ogden-Mooney-Rvln 4 Lnear elasc for orhoropc shells Fabrc 9 Nonlnear elasc for ansoropc shells Fabrc 58 Nonlnear pseudo-plasc orhoropc Honeycob 8 Copose and Ansoropc Maerals Elaso-plascy of Isoropc Maerals Vscous Maerals solds whou sran rae effec Cossera Medu 68 Nonlnear pseudo-plasc orhoropc Crushable foa 5 solds wh sran rae effec Elaso-plasc ansoropc shells Elaso-plasc orhoropc coposes von Mses hardenng whou daage von Mses hardenng wh brle daage Hll 3 Hll abulaed 43 3-Paraeer Barla 57 Copose Shell 5 Copose Shell Chang-Chang 5 Copose Sold 4 Tsa-Wu forula for sold Foa odel 53 Johnson-Cook Zerll-Arsrong Zhao 48 Cowper-Syonds 44 Tabulaed pecewse lnear 36 Tabulaed quadrac n sran rae 6 Rgd aeral 3 Drucker-Prager for rock or concree Druger-Prager for rock or concree by polynonal Hänsel odel 63 Ugne and ALZ approach 64 Elasoer 65 Alunu, glass, ec. 7 Pred rves 54 Renforced concree 4 von Mses hardenng wh ducle Ducle daage for solds and shells daage Ducle daage for solds 3 von Mses wh vsco-plasc flow Ducle daage for porous aerals, 5 Gurson Bolzan 34 Generalzed Kelvn-Vog 35 Vsco-elasc Tabulaed law 38 Generalzed Maxwell-Kelvn 4 Hyper vsco-elasc 6 Tabulaed law - hyper vsco-elasc 7 Vsco-plasc Closed cell, elaso-plasc foa 33 Sran rae and eperaure Johnson-Cook 4 dependence on yeld sress Turbulen vscous flow Hydrodynac vscous 6 Hydrodynac Elaso-plasc hydrodynac von Mses soropc hardenng wh 3 polynoal pressure Hydrodynac aeral Lee-Tarver aeral 4 Elaso-plasc hydrodynac wh Senberg-Gunan 49 heral sofenng Vod Vod aeral Fcous -jan-9 73

75 RADIOSS THEORY Verson. MATERIALS 9. Isoropc Elasc Maeral Two knds of soropc elasc aerals are consdered: Hooke s law for lnear elasc aerals, Ogden and Mooney-Rvln laws for nonlnear elasc aerals. These aeral laws are used o odel purely elasc aerals, or aerals ha rean n he elasc range. The Hooke s law requres only wo values o be defned; he Young's or elasc odulus E, and Posson's rao, υ. The law represens a lnear relaon beween sress and sran. The Ogden s law s appled o slghly copressble aerals as rubber or elasoer foas undergong large deforaon wh an elasc behavor [34]. The sran energy W s expressed n a general for as a funcon of W,, : λ λ λ3 α p α p α p λ λ λ μ p W 3 3 α p p where λ, h prncpal srech λ ε, ε beng he h prncpal engneerng sran and consans. Ths law s very general due o he choce of coeffcen α p and μ p. α p and EQ For a copressble aeral, he coponens of Cauchy sress ensor are gven n s prncpal reference: σ λ W μ p aeral EQ J λ λ λ λ wh, J 3, relave volue: dlaaon: λ λ λ λ 3 ρ J. For an ncopressble aeral, we have J. For unfor ρ EQ Consderng he general case of copressble aeral and defnng new varables: 3 λ * J λ EQ * * * * The new sran energy funcon s expressed as λ, λ, λ, J W λ, λ, λ W. We can wre: 3 * * * λ W λ W λ j W J σ EQ * J λ J j λ j λ J λ 3 3 * J j 3 As λ J and J λ 3 λ λ * λ j for j and J λ 3 3 λ λ j for j, EQ s splfed o: * 3 * * * W W W σ λ J EQ * * J λ 3 j λ j J -jan-9 74

76 RADIOSS THEORY Verson. MATERIALS The pressure s: 3 * W p σ j EQ J j Ths can be rewren as: where p K fbulk J J EQ fbulk a user defned funcon relaed o he bulk odulus K : μ 3 υ υ K EQ μ p α p μ p EQ μ beng he ground shear odulus, and ν he Posson's rao. W * s obaned usng EQ and EQ : W * μ, μ Ln λ λ λ α * * * p * α * α p * α p λ λ, λ3 λ λ λ3 3 p p p * * * p 3 p EQ * * W * α p λ * μ p λ EQ λ p * W whch nsure he condons: λ λ λ3. I should be noed ha n pure dlaaon σ * * * * λ λ λ3 λ and λ λ λ, we have also W σ 3 bu, he pressure s copued whou * usng W. Noe: For an ncopressble aeral we have υ. 5. However, υ. 495 s a good coprose o avod oo sall e seps n explc codes. Mooney-Rvln aeral law ads wo basc assupons: The rubber s ncopressble and soropc n unsraned sae, The sran energy expresson depends on he nvarans of Cauchy ensor. The hree nvarans of he Cauchy-Green ensor are: λ λ λ3 I λ λ λ λ3 λ3 λ I EQ I λ λ λ for ncopressble aeral The Mooney-Rvln law gves he closed expresson of sran energy as: -jan-9 75

77 RADIOSS THEORY Verson. MATERIALS I C 3 W C I EQ wh: μ C μ C EQ α α The odel can be generalzed for a copressble aeral. 9. Copose and Ansoropc Maerals The orhoropc aerals can be classfed no followng cases: Lnear elasc orhoropc shells as fabrc, Nonlnear orhoropc pseudo-plasc solds as honeycob aerals, Elasc-plasc orhoropc shells, Elasc-plasc orhoropc coposes. The purpose of hs secon s o descrbe he aheacal odels relaed o copose and orhoropc aerals. 9.. Fabrc law for elasc orhoropc shells laws 9 and 58 Two elasc lnear and a nonlnear odels exs n RADIOSS Fabrc lnear law for elasc orhoropc shells law 9 A aeral s orhoropc f s behavor s syercal wh respec o wo orhogonal plans. The fabrc law allows o odel hs knd of behavor. Ths law s only avalable for shell eleens and can be used o odel an arbag fabrc. Many of he conceps for hs law are he sae as for law 4 whch s approprae for copose solds. If axes and represen he orhoropy drecons, he consuve arx C s defned n ers of aeral properes: C E υ E υ E E G G 3 G 3 EQ where he subscrps denoe he orhoropy axes. As he arx C s syerc: -jan-9 76

78 RADIOSS THEORY Verson. MATERIALS υ υ EQ E E Therefore, sx ndependen aeral properes are he npu of he aeral: E Young's odulus n drecon E Young's odulus n drecon υ Posson's rao G, G 3, G 3 Shear odul for each drecon The coordnaes of a global vecor V r s used o defne drecon of he local coordnae syse of orhoropy. The angle Φ s he angle beween he local drecon fber drecon and he projecon of he global vecor V r as shown n Fgure 9... Fgure 9.. Fber Drecon Orenaon The shell noral defnes he posve drecon for Φ. Snce fabrcs have dfferen copresson and enson behavor, an elasc odulus reducon facor, R E, s defned ha changes he elasc properes of copresson. The forulaon for he fabrc law has a σreducon f σ < as shown n Fgure 9... Fgure 9.. Elasc Copresson Modulus Reducon -jan-9 77

79 RADIOSS THEORY Verson. MATERIALS 9... Fabrc nonlnear law for elasc ansoropc shells law 58 Ths law s used wh RADIOSS sandard shell eleens and ansoropc layered propery ype 6. The fber drecons warp and wef defne he local axes of ansoropy. Maeral characerscs are deerned ndependenly n hese axes. Fbers are nonlnear elasc and follow he equaon: σ dσ dε Eε Bε, wh > EQ The shear n fabrc aeral s only supposed o be funcon of he angle beween curren fber drecons axes of ansoropy: and τ G α τ f α αt EQ τ an G an α G τ f α > αt A G A G G an α, T G T G wh τ G an α an α T where αt s a shear lock angle, G T s a angen shear odulus a G, he defaul value s calculaed o avod shear odulus dsconnuy a α T, and G s a shear odulus a α. If α T : G G. Fgure 9..4 Elasc Copresson Modulus Reducon wef α φ warp α s an nal angle beween fbers defned n he shell propery ype 6. The warp and wef fber are coupled n enson and uncoupled n copresson. Bu here s no dsconnuy beween enson and copresson. In copresson only fber bendng generaes global sresses. Fgure 9..5 llusraes he echancal behavor of he srucure. -jan-9 78

80 RADIOSS THEORY Verson. MATERIALS Fgure 9..5 Local frae defnon Warp racon free n wef drecon Warp copresson no racon n wef A local cro odel descrbes he aeral behavor Fgure Ths odel represens jus ¼ of a warp fber wave lengh and ¼ of he wef one. Each fber s descrbed as a non lnear bea and he wo fbers are conneced wh a conacng sprng. These local non lnear equaons are solved wh Newon eraons a ebrane negraon pon. Fgure 9..6 Local frae defnon Wef Warp Tracon fber couplng Copresson no couplng 9.. Nonlnear pseudo-plasc orhoropc solds laws 8, 5 and Convenonal nonlnear pseudo-plasc orhoropc solds laws 8 and 5 These laws are generally used o odel honeycob aeral srucures as crushable foas. The acroscopc behavor of hs knd of aerals can be consdered as a syse of hree ndependen orhogonal sprngs. The nonlnear behavor n orhogonal drecons can be hen deerned by experenal ess. The behavor curves are njeced drecly n he defnon of law. Therefore, he physcal behavor of he aeral can be obaned by a sple law. However, he croscopc elaso-plasc behavor of a aeral pon canno be represened by decoupled undreconal curves. Ths s he ajor drawback of he consuve laws based on hs approach. The cell drecon s defned for each eleen by a local frae n he orhoropc sold propery. If no propery se s gven, he global frae s used. -jan-9 79

81 RADIOSS THEORY Verson. MATERIALS Fgure 9..7 Local frae defnon The Hooke arx defnng he relaon beween he sress and sran ensors s dagonal, as here s no Posson's effec: σ E σ σ 33 σ σ 3 σ 3 E E 33 G An soropc aeral ay be obaned f: G 3 ε ε ε 33 ε ε 3 G3 ε 3 EQ E E E E33 and G G3 G3 EQ Plascy ay be defned by a voluc sran or sran dependen yeld curve Fgure The npu yeld sress funcon s always posve. If he aeral undergoes plasc deforaon, s behavor s always orhoropc, as all curves are ndependen o each oher. Fgure 9..8 Honeycob ypcal consuve curve The falure plasc sran ay be npu for each drecon. If he falure plasc sran s reached n one drecon, he eleen s deleed. The aeral law ay nclude sran rae effecs law 5 or no law 8. -jan-9 8

82 RADIOSS THEORY Verson. MATERIALS 9... Cossera edu for nonlnear pseudo-plasc orhoropc solds law 68 Convenonal connuu echancs approaches canno ncorporae any aeral coponen lengh scale. However, a nuber of poran lengh scales as grans, parcles, fbers, and cellular srucures us be aken no accoun n a realsc odel of soe knds of aerals. To hs end, he sudy of a crosrucure aeral havng ranslaonal and roaonal degrees-of-freedo s underlyng. The dea of nroducng couple sresses n he connuu odellng of solds s known as Cossera heory whch reurns back o he works of brohers Cossera n he begnnng of h cenury []. A recen renewal of Cossera echancs s presened n several works of Fores e al [], [], [3], and [4]. A shor suary of hese publcaons s presened n hs secon. Cossera effecs can arse only f he aeral s subjeced o non-hoogeneous sranng condons. A Cossera edu s a connuous collecon of parcles ha behave lke rgd bodes. I s assued ha he ransfer of he neracon beween wo volue eleens hrough surface eleen ds occurs no only by eans of a racon and shear forces, bu also by oen vecor as shown n Fgure Fgure 9..9 Equlbru of Cossera volue eleen μ 3 σ σ μ 3 σ σ L σ μ3 Surface forces and couples are hen represened by he generally non-syercal force-sress and couple-sress ensors σ and μ uns MPA and MPa-: j j σ jn j ; μjn j EQ The force and couple sress ensors us sasfy he equlbru of oenus: σ f ρ& u& μ j, j ε σ c I & φ j, j kl kl EQ where f are he volue forces, c volue couples, ρ ass densy, I he soropc roaonal nera and he sgnaure of he perurbaon,k,l. In he ofen used couple-sress, he Cossera cro-roaon s consraned o follow he aeral roaon gven by he skew-syerc par of he deforaon graden: ε kl φ ε jku j, k EQ jan-9 8

83 RADIOSS THEORY Verson. MATERIALS The assocaed orson-curvaure and couple sress ensors are hen raceless. If a Toshenko bea s regarded as a one-densonal Cossera edu, consran EQ s hen he counerpar of he Euler-Bernoull condons. The resoluon of he prevous boundary value proble requres consuve relaons lnkng he deforaon and orson-curvaure ensors o he force- and couple-sresses. In he case of lnear soropc elascy, we have: σ λe δ μ e j j kk μ α k δ β κ kk j j Sy. j Sy. j μ e γκ Skew Sy. c j Skew Sy. j EQ Sy. SkewSy. where e j and ej are respecvely he syerc and skew-syerc par of he Cossera deforaon ensor. Four addonal elascy odul appear n addon o he classcal Laé consans. Cossera elasoplascy heory s also well-esablshed. Von Mses classcal plascy can be exended o cropolar connua n a sraghforward anner. The yeld creron depends on boh force- and couple-sresses: f 3, j j EQ μ a s s a s s b μ μ b μ μ R σ j j j j j j where s denoes he sress devaor and a, b are he aeral consans. Cossera connuu heory can be appled o several classes of aerals wh crosrucures as honeycobs, lqud crysals, rocks and granular eda, cellular solds and dslocaed crysals Hll s Law for Orhoropc Plasc Shells Hll s law odels an ansoropc yeld behavor. I can be consdered as a generalzaon of von Mses yeld crera for ansoropc yeld behavor. The yeld surface defned by Hll can be wren n a general for: σ σ G σ σ H σ σ Lσ Mσ Nσ F EQ where he coeffcens F, G, H, L, M and N are he consans obaned by he aeral ess n dfferen orenaons. The sress coponens σ are expressed n he Caresan reference parallel o he hree planes of j ansoropy. EQ s equvalen o von Mses yeld crera f he aeral s soropc. In a general case, he loadng drecon s no he orhoropc drecon. In addon, we are concerned wh he plane sress assupon for shell srucures. In planar ansoropy, he ansoropy s characerzed by dfferen srenghs n dfferen drecons n he plane of he shee. The plane sress assupon wll allow o splfy EQ , and wre he expresson of equvalen sress σ eq as: σ eq A σ Aσ A3σ σ Aσ EQ The coeffcens A are deerned usng Lankford s ansoropy paraeer r α : 9 r r45 r R ; 4 H R ; R A H EQ r -jan-9 8

84 RADIOSS THEORY Verson. MATERIALS A H ; A H r 9 A H r 3 ; r r where he Lankford s ansoropy paraeers r α are deerned by perforng a sple enson es a angle α o orhoropc drecon : dε N F G H π α H 4 α dε 33 r Sn α Cos α F Sn α G Cos α EQ The equvalen sress he sran rae ε& law 3: σ σ eq s copared o he yeld sress σ y whch vares n funcon of plasc sran ε ε n. ax & ε & ε EQ y a p, ε p and Therefore, he elasc l s obaned by: ε n & σ a. ε EQ The yeld sress varaon s shown n Fgure 9... Fgure 9.. Yeld sress varaon The sran raes are defned a negraon pons. The axu value s aken no accoun: dε d dε dε x y d ax,, ε EQ d d d xy In RADIOSS, s also possble o nroduce he yeld sress varaon by a user defned funcon law 43. Then, several curves are defned o ake no accoun he sran rae effec. I should be noed ha as Hll s law s an orhoropc law, us be used for eleens wh orhoropy properes as Type 9 and Type n RADIOSS Elasc-Plasc Orhoropc Copose Shells Two knds of copose shells ay be consdered n he odelng: Copose shells wh soropc layers, Copose shells wh a leas one orhoropc layer. -jan-9 83

85 RADIOSS THEORY Verson. MATERIALS The frs case can be odeled by an soropc aeral where he copose propery s defned n eleen propery defnon as explaned n chaper 5. However, n he case of copose shell wh orhoropc layers he defnon of a convenen aeral law s needed. Two dedcaed aeral laws for copose orhoropc shells exs n RADIOSS: Maeral law COMPSH 5 wh orhoropc elascy, wo plascy odels and brle ensle falure, Maeral law CHANG 5 wh orhoropc elascy, fully coupled plascy and falure odels. These laws are descrbed here. The descrpon of elasc-plasc orhoropc copose laws for solds s presened n he nex secon Tensle behavor The ensle behavor s shown n Fgure 9... The behavor sars wh an elasc phase. Then, reached o he yeld sae, he aeral ay undergo an elasc-plasc work hardenng wh ansoropc Tsa-Wu yeld crera. I s possble o ake no accoun he aeral daage. The falure can occur n he elasc sage or afer plasfcaon. I s sared by a daage phase hen conduced by he foraon of a crack. The axu daage facor wll allow hese wo phases o separae. The unloadng can happen durng he elasc, elasc-plasc or daage phase. The daage facor d vares durng deforaon as n he case of soropc aeral laws law 7. However, hree daage facors are copued; wo daage facors d and d for orhoropy drecons and he oher d 3 for delanaon: σ σ σ d E ε d E ε EQ d d γ G where d and d are he ensle daages facors. The daage and falure behavor s defned by nroducon of he followng npu paraeers: ε Tensle falure sran n drecon ε Maxu sran n drecon ε Tensle falure sran n drecon ε Maxu sran n drecon d ax Maxu daage resdual sffness afer falure -jan-9 84

86 RADIOSS THEORY Verson. MATERIALS Fgure 9.. Tensle behavor of copose shells -d ax E Delanaon The delanaon equaons are: σ σ d 3 3 d G 3 γ EQ G 3 γ EQ where d3 s he delanaon daage facor. The daage evoluon law s lnear wh respec o he shear sran. Le γ hen: γ 3 γ 3 d γ γ for 3 d γ for 3 γ EQ jan-9 85

87 RADIOSS THEORY Verson. MATERIALS Fgure 9.. Delanaon wh shear sran Plasc behavor The plascy odel s based on he Tsa-Wu creron, whch allows o odel he yeld and falure phases. The crera s gven by [57]: Where σ F σ Fσ Fσ Fσ F44σ Fσ σ F EQ F F F σ σ ; F c σ σ 44 c y c y y y σ σ ; c y y F y σ σ c y y y σ σ ; α F FF α s he reducon facor. The sx oher paraeers are he yeld sresses n enson and copresson for he orhoropy drecons whch can be obaned unaxal loadng ess: σ y Tenson n drecon of orhoropy σ y Tenson n drecon of orhoropy c σ y Copresson n drecon of orhoropy c σ y Copresson n drecon of orhoropy c σ y Copresson n drecon of orhoropy σ y Tenson n drecon of orhoropy The Tsa-Wu crera s used o deerne he aeral behavor: σ < F : elasc sae σ F : plasc adssble sae EQ σ > F : plascally nadssble sresses -jan-9 86

88 RADIOSS THEORY Verson. MATERIALS For F σ he cross-secons of Tsa-Wu funcon wh he planes of sresses n orhoropc drecons s shown n Fgure Fgure 9..3 Cross-secons of Tsa-Wu yeld surface for F σ σ σ y σ y c σ y σ σ y c σ y c σ y σ If F σ >, he sresses us be projeced on he yeld surface o sasfy he flow rule. σ axu value F W p varyng n funcon of he plasc work W p durng work hardenng phase: n F s copared o a F σ F W p bw p EQ wh: b Hardenng paraeer n Hardenng exponen Therefore, he plascy hardenng s soropc as llusraed n Fgure jan-9 87

89 RADIOSS THEORY Verson. MATERIALS Fgure 9..4 Isoropc plascy hardenng Falure behavor The Tsa-Wu flow surface s also used o esae he aeral rupure by eans of wo varables: ax plasc work l W p, axu value of yeld funcon F ax. If one of he wo condons s sasfed, he aeral s rupured. The evoluon of yeld surface durng work hardenng of he aeral s shown n Fgure Fgure 9..5 Evoluon of Tsa-Wu yeld surface The odel wll allow he sulaon of he brle falure by foraon of cracks. The cracks can eher be orened parallel or perpendcular o he orhoropc reference frae or fber drecon, as shown n Fgure -jan-9 88

90 RADIOSS THEORY Verson. MATERIALS For plasc falure, f he plasc work W P s larger han he axu value W for a gven eleen, hen he eleen s consdered o be rupured. However, for a ul-layer shell, several crera ay be consdered o odel a oal falure. The falure ay happen: ax P W P ax P W P ax P W P ax P W P ax P W P ax P W P ax P W P If W > for one layer, If W > for all layers, If W > or ensle falure n drecon for each layer, If W > or ensle falure n drecon for each layer, If W > or ensle falure n drecons and for each layer, If W > or ensle falure n drecon for all layer, If W > or ensle falure n drecon for all layer, If ax W P > W P or ensle falure n drecon and for each layer. The las wo cases are he os physcal behavors; bu he use of falure crera depends, a frs, o he analys s choce. In RADIOSS he flag I OFF defnes he used falure crera n he copuaon. Fgure 9..6 Crack orenaon ax P In pracce, he use of brle falure odel allows o esae correcly he physcal behavor of a large rang of coposes. Bu on he oher hand, soe nuercal oscllaons ay be generaed due o he hgh sensbly of he odel. In hs case, he nroducon of an arfcal aeral vscosy s recoended o sablze resuls. In addon, n brle falure odel, only enson sresses are consdered n crackng procedure. The ducle falure odel allows plascy o absorb energy durng a large deforaon phase. Therefore, he odel s nuercally ore sable. Ths s represened by CRASURV odel n RADIOSS. The odel akes also possble o ake no accoun he falure n enson, copresson and shear drecons as descrbed n he followng Sran rae effec The sran rae s aken no accoun whn he odfcaon of EQ whch acs hrough a scale facor: n & ε p bwp cln & ε o F W EQ jan-9 89

91 RADIOSS THEORY Verson. MATERIALS wh: W p W & p : plasc work, : axu value of he plasc work raes of he sran ensor, W & po : reference plasc work rae for sac loadng, c : sran rae coeffcen equal o zero for sac loadng. The las equaon ples he growng of he Tsa-Wu yeld surface when he dynac effecs are ncreasng. The effecs of sran rae are llusraed n Fgure Fgure 9..7 Sran rae effec n work hardenng CRASURV odel The CRASURV odel s an proved verson of he forer law based on he sandard Tsa-Wu crera. The an changes concern he expresson of he yeld surface before plasfcaon and durng work hardenng. Frs, n CRASURV odel he coeffcen F 44 n EQ depends only on one npu paraeer: F 44 EQ σ y Anoher odfcaon concerns he paraeers F j n EQ whch are expressed now n funcon of plasc work and plasc work rae as n EQ : Wp F Wp σ F Wp σ F Wp σ F Wp σ F44 Wp σ F Wp σσ c c c c n c & ε y σ b Wp c Ln & ε o c c c c n c & ε y σ b Wp cln & ε o F σ σ σ σ σ n & ε σ b Wp c Ln EQ & ε o n & ε σ b Wp cln & ε o n & ε σ b Wp cln & ε o y y y where he fve ses of coeffcens b, n and c should be obaned by experence. The work hardenng s shown n Fgure jan-9 9

92 RADIOSS THEORY Verson. MATERIALS Fgure 9..8 CRASURV plascy hardenng The CRASURV odel wll allow he sulaon of he ducle falure of orhoropc shells. The plasc and falure behavors are dfferen n enson and n copresson. The sress sofenng ay also be nroduced n he odel o ake no accoun he resdual Tsa-Wu sresses. The evoluon of CRASURV crera wh hardenng and sofenng works s llusraed n Fgure jan-9 9

93 RADIOSS THEORY Verson. MATERIALS Fgure 9..9 Flow surface n CRASURV odel Chang-Chang odel Chang-Chang law [58], [59] ncorporaed n RADIOSS s a cobnaon of he sandard Tsa-Wu elasc-plasc law and a odfed Chang-Chang falure crera [6]. The effecs of daage are aken no accoun by decreasng sress coponens usng a relaxaon echnque o avod nuercal nsables. Sx aeral paraeers are used n he falure crera: S Longudnal ensle srengh S Transverse ensle srengh S Shear srengh C Longudnal copressve srengh C Transverse copressve srengh β Shear scalng facor. where s he fber drecon. The falure creron for fber breakage s wren as: Tensle fber ode : σ > e σ σ β. f S S < faled elasc plasc EQ jan-9 9

94 RADIOSS THEORY Verson. MATERIALS Copressve fber ode : σ < σ e c. C < For arx crackng, he falure creron s: Tensle arx ode : σ > e σ σ β. S S < Copressve arx ode : σ < σ C σ σ e d. S < S C S faled elasc plasc faled elasc plasc faled elasc plasc EQ EQ EQ If he daage paraeer s equal o or greaer han., he sresses are decreased by usng an exponenal funcon o avod nuercal nsables. We use a relaxaon echnque by decreasng gradually he sress: [ σ ] f [ σ ] EQ d r f exp and r where: T he e, r he sar e of relaxaon when he daage crera are assued, T he e of dynac relaxaon. r Wh: σ σ [ σ d r ] he sress coponens a he begnnng of daage for arx crackng [ ] d d r σ d 9..5 Elasc-Plasc Orhoropc Copose Solds The aeral law COMPSO 4 n RADIOSS allows o sulae orhoropc elascy, Tsa-Wu plascy wh daage, brle rupure and sran rae effecs. The consuve law apples o only one layer of lana. Therefore, each layer needs o be odeled by a sold esh. A layer s characerzed by one drecon of he fber or aeral. The overall behavor s assued o be elaso-plasc orhoropc. v v v Drecon s he fber drecon, defned wh respec o he local reference frae r, s, as shown n Fgure jan-9 93

95 RADIOSS THEORY Verson. MATERIALS Fgure 9.. Local reference frae For he case of undreconal orhoropy.e. E 33 E and G 3 G he aeral law 53 n RADIOSS allows o sulae an orhoropc elasc-plasc behavor by usng a odfed Tsa-Wu crera Lnear elascy When he lana has a purely lnear elasc behavor, he sress calculaon algorh s as follows:. Transfor he lana sress, σ, and sran rae, d j, fro global reference frae o fber reference frae. j. Copue lana sress a e Δ by explc e negraon: j Δ σ j DjkldklΔ 3. Transfor he lana sress, σ Δ σ EQ j, back o global reference frae. The elasc consuve arx C of he lana relaes he non-null coponens of he sress ensor o hose of sran ensor: { σ } [ D]{} ε EQ The nverse relaon s generally developed n er of he local aeral axes and nne ndependen elasc consans: E ε ε ε 33 γ γ 3 γ 3 ν E E Sy. ν E ν E E G G 3 σ σ σ 33 σ σ 3 σ 3 G3 EQ where E j are he Young s odulus, due o he dsoron. Gj shear odulus and ν j Posson s raos. γ j are he sran coponens -jan-9 94

96 RADIOSS THEORY Verson. MATERIALS Fgure 9.. Sran coponens and dsoron Orhoropc plascy Lana yeld surface defned by Tsa-Wu yeld crera s used for each layer: W F σ Fσ F3σ 3 F f p F σ Fσ F33σ 3 F44σ F55σ 3 F66σ 3 wh: F F σ σ F3σ σ 3 F3σ σ 3 σ σ c EQ ; F c σ σ F FF ; F 3 F where represens he yeld envelope evoluon durng work hardenng wh respec o sran rae effecs: where σ s he yeld sress n drecon. c and denoe respecvely for copresson and enson. f f p EQ n & ε W B W.ln p c & ε W p s he plasc work, B he hardenng paraeer, n he hardenng exponen and c sran rae coeffcen. f W p s led by a axu value f ax : W p f W p f ax σ σ y ax EQ If he axu value s reached he aeral s faled. In EQ , he sran rae affecs on he evoluon of yeld envelope. However, s also possble o ake no accoun he sran rae ε& effecs on he axu sress σ as shown n Fgure 9... ax -jan-9 95

97 RADIOSS THEORY Verson. MATERIALS a Sran rae effec on Fgure 9.. Sran rae dependency σ ax b No sran rae effec on σ ax & ε σ σ y c.ln & ε & ε σ ax σ ax c. ln & ε f ax σ σ y ax & ε σ σ y c.ln & ε σ ax σ ax Undreconal Orhoropy Law 53 n RADIOSS provdes a sple odel for undreconal orhoropc solds wh plascy. The undreconal orhoropy condon ples: E 33 E EQ G 3 G The orhoropc plascy behavor s odeled by a odfed Tsa-Wu creron EQ n whch: F σ 45c F F F F F44 EQ c σ y 45c σ y where y s yeld sress n 45 undreconal es. The yeld sresses n drecon,,, 3 and 45, are defned by ndependen curves obaned by undreconal ess Fgure The curves gve he sress varaon n funcon of a so-called sranε : [] v ε exp Trace ε EQ v -jan-9 96

98 RADIOSS THEORY Verson. MATERIALS Fgure 9..3 Yeld sress curve for a undreconal orhoropc aeral σ j User defned Yeld curve j Tracon Copresson [] ε exp Trace ε v 9..6 Elasc-plasc ansoropc shells Barla s law Barla s 3- paraeer plascy odel s developed n [] for odellng of shee under plane sress assupon wh an ansoropc plascy odel. The ansoropc yeld sress creron for plane sress s defned as: K a K K c K σ F a K EQ where σ e s he yeld sress, a and c are ansoropc aeral consans, Barla s exponen and K and K are defned by : σ xx hσ yy K EQ e K σ xx hσ yy p σ xy where h and p are addonal ansoropc aeral consans. All ansoropc aeral consans, excep for p whch s obaned plcly, are deerned fro Barla wdh o hckness sran rao R fro: R R9 a EQ R R9 c a h R R R R9 9 -jan-9 97

99 RADIOSS THEORY Verson. MATERIALS The wdh o hckness rao for any angle ϕ can be calculaed accordng o [] by: σ R ϕ EQ e F F σ xx σ yy σ ϕ σ ϕ s he unaxal enson n he ϕ drecon. Le ϕ 45, EQ gves an equaon fro whch he where ansoropy paraeer p can be copued plcly by usng an erave procedure: σ e F F σ xx σ yy σ 45 R 45 EQ I s worhwhle o noe ha Barla s law reduces o Hll s law when usng. -jan-9 98

100 RADIOSS THEORY Verson. MATERIALS 9.3 Elaso-Plascy of Isoropc Maerals The sran hardenng behavor of aerals s a ajor facor n srucural response as eal workng processes or plasc nsably probles. A proper descrpon of sran hardenng a large plasc srans s generally perave. For any plascy probles, he hardenng behavor of he aeral s sply characerzed by he sran-sress curve of he aeral. For he proporonal loadng hs s generally rue. However, f he loadng pah s cobned, he characerzaon by a sple sran-sress curve s no longer adequae. The ncreenal plascy heory s generally used n copuaonal ehods. Plascy odels are wren as rae-dependen or ndependen. A rae-dependen odel s a one n whch he sran rae does affec he consuve law. Ths s rue for a large rang of eals a low eperaure relave o her elng eperaure. Mos soropc elasc-plasc aeral laws n RADIOSS use von Mses yeld crera as gven n secon.7.. Several knds of odels are negraed. The odels nvolve daage for ducle or brle falures wh or whou dslocaon. The cuulave daage law can be used o access falure. The nex few paragraphs descrbe heorecal bases of he negraed odels Johnson-Cook plascy odel law In hs law he aeral behaves as lnear elasc when he equvalen sress s lower han he yeld sress. For hgher value of sress, he aeral behavor s plasc. Ths law s vald for brck, shell, russ and bea eleens. The relaon beween descrbng sress durng plasc deforaon s gven n a closed for: where: n Τ & ε σ a bε p cln & ε σ Flow sress Elasc Plasc Coponens ε p Plasc Sran True sran a Yeld Sress b Hardenng Modulus n Hardenng Exponen c Sran Rae Coeffcen ε& Sran Rae ε& Reference Sran Rae Teperaure exponen Τ 98 Τ Τ 98 el EQ Τel s he elng eperaure n Kelvn degrees. The adabac condons are assued for eperaure copuaon: En Τ Τ EQ ρc ρ Volue Where: ρ C p he specfc hea per un of volue Τ nal eperaure n degrees Kelvn -jan-9 99

101 RADIOSS THEORY Verson. MATERIALS E n nernal energy Two oponal addonal npus are: σ ax Maxu flow sress ε ax Plasc sran a rupure Fgure 9.3. shows a ypcal sress-sran curve n he plasc regon. When he axu sress s reached durng copuaon, he sress reans consan and aeral undergoes deforaon unl he axu plasc sran. Eleen rupure occurs f he plasc sran s larger han ε ax. If he eleen s a shell, he rupured eleen s deleed. If he eleen s a sold eleen, he rupured eleen has s devaorc sress ensor peranenly se o zero, bu he eleen s no deleed. Therefore, he aeral rupure s odeled whou any daage effec. Fgure 9.3. Sress - Plasc Sran Curve Sran rae defnon Regardng o he plasfcaon ehod used, he sran rae expresson s dfferen. If he progressve plasfcaon ehod s used.e. negraon pons hrough he hckness for hn-walled srucured, he sran rae s: dε dε dε x y d ax,, ε xy EQ d d d d ε xy γ xy EQ Wh global plasfcaon ehod, we have: dε de / d EQ d σ VM where E s he nernal energy. For sold eleens, he axu value of he sran rae coponens s used: & ε & ε & ε & ε & ε & ε & ε ax EQ x y z xy yz xz -jan-9

102 RADIOSS THEORY Verson. MATERIALS Sran rae flerng The sran raes exhb very hgh frequency vbraons whch are no physcal. The sran rae flerng opon wll allow o dap hose oscllaons and herefore oban ore physcal sran rae values. If here s no sran rae flerng, he equvalen sran rae s he axu value of he sran rae coponens: eq & ε & ε & ε & ε & ε & ε & ε ax EQ x y z xy yz xz For hn-walled srucures, he equvalen sran s copued by he followng approach. If ε s he an coponen of sran ensor, he kneac assupons of hn-walled srucures allows o decopose he nplane sran no ebrane and flexural deforaons: ε κz ε EQ Then, he expresson of nernal energy can by wren as: E Therefore: κz ε σε dz Eε dz E dz EQ κ z ε κεz dz E κ z ε z z EQ E E κε The expresson can be splfed o: E 3 E κ ε Eε eq EQ ε eq ε κ EQ The expresson of he sran rae s derved fro EQ : & ε κz & & ε EQ Adng he assupon ha he sran rae s proporonal o he sran.e.: Therefore: & ε α ε EQ & κ α κ EQ & ε αε EQ jan-9

103 RADIOSS THEORY Verson. MATERIALS Referrng o EQ , can be seen ha an equvalen sran rae can be defned usng a slar expresson o he equvalen sran: & ε eq αε eq EQ & ε eq k& & ε EQ For sold eleens, he sran rae s copued usng he axu eleen srech: & ε & eq λ ax EQ The sran rae a negraon pon, n /ANIM/TENS/EPSDOT/ <<n s calculaed by he followng expresson: & ε & ε & ε b n EQ Where: ε& s he ebrane sran rae /ANIM/TENS/EPSDOT/MEMB ε& s he bendng sran rae /ANIM/TENS/EPSDOT/BEND b The sran rae n upper and lower layers are copued by: & ε u & ε l & ε ε& b & ε ε& b /ANIM/TENS/EPSDOT/UPPER EQ /ANIM/TENS/EPSDOT/LOWER EQ The sran rae s flered by usng he followng equaon: ε f a & ε a & ε & EQ f where: a π d F cu d e nerval F cu cung frequency ε& flered sran rae f Exaple: Sran rae flerng An exaple of aeral characerzaon for a sple ensle es s gven n RADIOSS Exaple Manual. For he sae exaple a sran rae flerng allows o reove hgh frequency vbraons and oban soohed he resuls. Ths s shown n Fgures 9.3. and where he cu frequency F cu KHz s used. -jan-9

104 RADIOSS THEORY Verson. MATERIALS Fgure 9.3. Force coparson n exaple Fgure Frs prncpal sran rae coparson ax % 9.3. Zerll-Arsrong plascy odel law Ths law s slar o Johnson-Cook plascy odel. The sae paraeers are used o defne he work hardenng curve. However, he equaon ha descrbes sress durng plasc deforaon s: ε C C exp & C3 C4 ln C ε Τ Τ ε & σ 5 where: σ Sress Elasc Plasc Coponens n p EQ ε p Plasc Sran Τ Teperaure copued as n Johnson Cook plascy -jan-9 3

105 RADIOSS THEORY Verson. MATERIALS C Yeld Sress n Hardenng Exponen ε& Sran Rae, us be s - convered no user's e un ε& Reference Sran Rae Addonal npus are: σ ax Maxu flow sress Plasc sran a rupure ε ax The ε ax allows o defne eleen rupure as n he forer law. The heorecal aspecs relaed o sran rae copuaon and flerng are also he sae Cowper-Syonds plascy odel law 44 Ths law odels an elaso-plasc aeral wh: soropc and kneac hardenng ensle rupure crera The daage s negleced n he odel. The work hardenng odel s slar o he Johnson Cook odel law whou eperaure effec where he only dfference s n he sran rae dependen forulaon. The equaon ha descrbes he sress durng plasc deforaon s: p σ & EQ n a bε p ε c where, σ Flow sress Elasc Plasc Coponens ε p Plasc Sran True sran a Yeld Sress b Hardenng Modulus n Hardenng Exponen c Sran Rae Coeffcen ε& Sran Rae /p sran rae exponen The planed odel n RADIOSS allows he cyclc hardenng wh a cobned soropc-kneac approach. The coeffcen C hard varyng beween zero and uny s nroduced o regulae he wegh beween soropc and kneac hardenng odels. In soropc hardenng odel, he yeld surface nflaes whou ovng n he space of prncple sresses. The evoluon of he equvalen sress defnes he sze of he yeld surface, as a funcon of he equvalen plasc sran. The odel can be represened n one densonal case as shown n Fgure When he loadng drecon s changed, he aeral s unloaded and he sran reduces. A new hardenng sars when he absolue value of he sress reaches he las axu value Fgure 9.3.4a. -jan-9 4

106 RADIOSS THEORY Verson. MATERIALS Fgure Isoropc and Kneac hardenng odels for deforaon decrease a Isoropc hardenng b Prager-Zegler kneac hardenng σ y σ y Ths law s avalable for solds and shells. Refer o he RADIOSS Inpu Manual for ore nforaon abou eleen/aeral copables Zhao plascy odel law 48 The elaso-plasc behavor of aeral wh sran rae dependence s gven by Zhao forula [6], [6]: σ A Bε C Dε & ε E & ε n k p p.ln EQ & ε where: ε p plasc sran, ε& sran rae A Yeld sress B hardenng paraeer n hardenng exponen C relave sran rae coeffcen D sran rae plascy facor Relave sran rae exponen E sran rae coeffcen k sran rae exponen In he case of aeral whou sran rae effec, he hardenng curve gven by EQ s dencal o hose of Johnson-Cook. However, Zhao law allows a beer approxaon of sran rae dependen aerals by nroducng a nonlnear dependency. As descrbed for Johnson-Cook law, a sran rae flerng can be nroduced o sooh he resuls. The plasc flow wh soropc or kneac hardenng can be odeled as descrbed for n secon The aeral falure s happened when he plasc sran reaches a axu value as n Johnson-Cook odel. However, wo ensle sran ls are defned o reduce sress when rupure sars: ε ε σ n σ n EQ ε ε where: ε larges prncpal sran ε and ε rupure sran ls -jan-9 5

107 RADIOSS THEORY Verson. MATERIALS ε > ε, he sress s reduced by EQ When ε > ε he sress s reduced o zero. If Tabulaed pecewse lnear and quadrac elaso-plasc laws laws 36 and 6 The elasc-plasc behavor of soropc aeral s odeled wh user defned funcons for work hardenng curve. The elasc poron of he aeral sress-sran curve s odeled usng he elasc odulus, E, and Posson's rao, ν. The hardenng behavor of he aeral s defned n funcon of plasc sran for a gven sran rae Fgure An arbrary nuber of aeral plascy curves can be defned for dfferen sran raes. For a gven sran rae, a lnear nerpolaon of sress for plasc sran change, can be used. Ths s he case of law 36 n RADIOSS. However, n law 6 a quadrac nerpolaon of he funcons allows o beer sulae he sran rae effecs on he behavor of aeral as s developed n law 6. For a gven plasc sran, a lnear nerpolaon of sress for sran rae change s used. Copared o Johnson-Cook odel law, here s no axu value for he sress. The curves are exrapolaed f he plasc deforaon s larger han he axu plasc sran. The hardenng odel ay be soropc, kneac or a cobnaon of he wo odels as descrbed n secon The aeral falure odel s he sae as n Zhao law. For soe knds of seels he yeld sress dependence o pressure has o be ncorporaed especally for assve srucures. The yeld sress varaon s hen gven by: σ f p y σ y ε p EQ where p s he pressure defned by EQ Drucker-Prager odel descrbed n secon gves a nonlnear funcon for f p. However, for seel ype aerals where he dependence o pressure s low, a sple lnear funcon ay be consdered: y y ε C p ε σ σ p p where C s user defned consan and p he copued pressure for a gven defored confguraon. Fgure Pecewse lnear sress-sran curves EQ σ & ε & & ε ε & ε & ε & ε ε p The prncpal sran rae s used for he sran rae defnon: dε dε dε x y d d d dε dε x y d d For sran rae flerng, refer o secon dγ d xy EQ jan-9 6

108 RADIOSS THEORY Verson. MATERIALS Drucker-Prager consuve odel laws & For aerals lke sols and rocks he frconal and dlaaonal effecs are sgnfcan. In hese aerals, he plasc behavor depends on he pressure as he nernal frcon s proporonal o he noral force. Furherore, for frconal aerals, assocave plascy laws, n whch he plasc flow s noral o he yeld surface, are ofen napproprae. Drucker-Prager [63] yeld creron uses a odfed von Mses yeld crera o ncorporae he effecs of pressure for assve srucures: A A P A F J EQ P where: J second nvaran of devaorc sress J P pressure A, A, A aeral coeffcens s j sj Fgure shows EQ n he plane of J and P. The creron expressed n he space of prncpal sresses represens a revoluonary surface wh an axs parallel o he rsecng of he space as shown n Fgure Ths represenaon s n conras wh he von Mses crera where yeld creron has a cylndrcal shape. Drucker-Prager creron s a sple approach o odel he aerals wh nernal frcon because of he syery of he revoluon surface and he connuy n varaon of noral o he yeld surface. The pressure n he aeral s deerned n funcon of voluerc sran for loadng phase: μ P f for loadng d μ > EQ where f s a user defned law or a cubc polynoal funcon law. For unloadng phase, f he voluerc sran has a negave value, a lnear relaon s defned as: P C μ for unloadng d μ < and μ < EQ For unloadng wh a posve voluerc sran, anoher lnear funcon ay be used: P Bμ for unloadng d μ < and > μ EQ In RADIOSS Drucker-Prager odel s used n laws and. None of hese laws can reproduce he onodensonal behavor. In addon, no vscous effec s aken no accoun. Fgure Yeld Crera n he plane of J and P. -jan-9 7

109 RADIOSS THEORY Verson. MATERIALS Fgure Drucker-Prager yeld crera n space of prncpal sresses σ S σ 3 σ Fgure Maeral pressure varaon n funcon of voluerc sran -jan-9 8

110 RADIOSS THEORY Verson. MATERIALS Brle daage for Johnson-Cook plascy odel law 7 Johnson-Cook plascy odel s presened n secon For shell applcaons, a sple daage odel can be assocaed o hs law o ake no accoun he brle falure. The crack propagaon occurs n he plan of shell n he case of ono-layer propery and hrough he hckness f a ul-layer propery s defned Fgure Fgure Daage Affeced Maeral Crack orenaon Layer crackng The elasc-plasc behavor of he aeral s defned by Johnson-Cook odel. However, he sress-sran curve for he aeral ncorporaes a las par relaed o daage phase as shown n Fgure The daage paraeers are: ε Tensle rupure sran n drecon ε Maxu sran n drecon d ax Maxu daage n drecon ε f Maxu sran for eleen deleon n drecon The eleen s reoved f one layer of eleen reaches he falure ensle sran, ε f. The nonal and effecve sresses developed n an eleen are relaed by: n eff d σ σ EQ where < d < s he daage facor. The srans and he sresses n each drecon are gven by: σ vσ ε EQ d E E σ vσ ε EQ E E σ γ EQ G d d d v E σ EQ [ ] ε vε -jan-9 9

111 RADIOSS THEORY Verson. MATERIALS E σ EQ ε d vε [ d v ] The condons for hese equaons are: < d < ε ε ; d ε ε ; d A lnear daage odel s used o copue he daage facor n funcon of aeral sran. d ε ε EQ ε ε The sress-sran curve s hen odfed o ake no accoun he daage by EQ Therefore: σ ε ε E p ε ε EQ ε ε The sofenng condon s gven by: ε ε ε ε EQ p The aheacal approach descrbed here can be appled o he odelng of rves. Pred law n RADIOSS allows acheveen of hs end by a sple odel where for he elasc-plasc behavor a Johnson-Cook odel or a abulaed law 36 ay be used. Fgure 9.3. Sress-sran curve for daage affeced aeral Brle daage for renforced concree aerals law 4 The odel s a connuu, plascy-based, daage odel for concree. I assues ha he an wo falure echanss are ensle crackng and copressve crushng of he concree aeral. The aeral law wll allow o forulae he brle elasc plasc behavor of he renforced concree. -jan-9

112 RADIOSS THEORY Verson. MATERIALS The npu daa for concree are: E c Young's odulus e.g.: 3 MPa ν c Posson's rao e.g.:. fc Unaxal copressve srengh ex : 3 MPa f /f c Tensle srengh rao defaul. f b /f c Baxal srengh rao defaul. f /f c confned srengh rao defaul 4. s /f c confnng sress rao defaul.5 Experenal resuls allow o deerne he aeral paraeers. Ths can be done by n-plane undreconal and b-axal ess as shown n Fgure The expresson of he falure surface s n a general for as: where:, J, f, f, θ c f σ EQ c b J second nvaran of sress I σ ean sress J3 θ lode angle wh cos3θ 3 / J A scheac represenaon of he falure surface n he prncpal sress space s gven n Fgure The yeld surface s derved fro he falure envelope by nroducng a scale facor k presened n Fgure σ,θ. The erdan planes are The seel drecons are defned dencally o aeral law 4 by a ype 6 propery se. If no propery se s gven n he eleen npu daa, r,s, Ψ are aken respecvely as drecon,, 3. For quad eleens, drecon 3 s aken as he Ψ drecon. Seel daa properes are: E Young's odulus σ y Yeld srengh E Tangen odulus α Rao of renforceen n drecon α Rao of renforceen n drecon α 3 Rao of renforceen n drecon 3 -jan-9

113 RADIOSS THEORY Verson. MATERIALS Fgure 9.3. Falure surface n plane sress σ Unaxal enson Unaxal copresson σ Baxal enson Baxal copresson Fgure 9.3. Falure surface n prncpal sress space Falure Surface σ 3 σ σ -jan-9

114 RADIOSS THEORY Verson. MATERIALS Fgure Merdans of falure and yeld surfaces σ f von c Falure n copresson Falure n racon Yeld n copresson.5 Yeld n racon..5 P f c Ducle daage odel In secon 9.3.7, a daage odel for brle aerals s presened. I s used n RADIOSS law 7 vald for shell eshes. The daage s generaed when he shell works n racon only. A generalzed daage odel for ducle aerals s ncorporaed n RADIOSS laws, and 3. The daage s no only generaed n racon bu also n copresson and shear. I s vald for solds and shells. The elasc-plasc behavor s forulaed by Johnson- Cook odel. The daage s nroduced by he use of daage paraeer δ. The daage appears n he aeral when he sran s larger han a axu value ε : δ o If ε < ε da δ Law s dencal o law. o If ε ε da E da δ E and ν da δ δ ν da Ths ples an soropc daage wh he sae effecs n enson and copresson. The npus of he odel are he sarng daage sran ε da and he slope of he sofenng curve E as shown n Fgure For brck eleens he daage law can be only appled o he devaorc par of sress ensor s j and Eda Gda. Ths s he case of law n RADIOSS. However, f he applcaon of daage law o ν da sress ensor σ s expeced, RADIOSS law 3 ay be used. j -jan-9 3

115 RADIOSS THEORY Verson. MATERIALS Fgure Ducle daage odel The sran rae defnon and flerng for hese laws are as explaned n secon The sran rae ε& ay affec or no he axu sress value σ accordng o he user's choce as shown n Fgure ax a Sran rae effec on Fgure Sran rae dependency σ ax b No sran rae effec on σ ax & ε σ σ y c.ln & ε & ε σ ax σ ax c. ln & ε & ε σ σ y c.ln & ε σ ax σ ax 9.3. Ducle daage odel for porous aerals Gurson law 5 The Gurson consuve law [64] odels progressve crorupure hrough vod nucleaon and growh. I s dedcaed o hgh sran rae elaso-vscoplasc porous eals. A coupled daage echancal odel for sran rae dependen voded aeral s used. The aeral undergoes several phases n he daage process as descrbed n Fgure jan-9 4

116 RADIOSS THEORY Verson. MATERIALS Fgure Daage process for vsco-elasc-plasc voded aerals nal sae of he aeral Growh of he exsng crovods Nucleaon by decoheson of he arx-parcules nerface or by fracure of he ncluson Coalescence of rcovods and ducle fracure of he arx : Incluson or parcle second phase : Vod The consuve law akes no accoun he vod growh, nucleaon and coalescence under dynac loadng. The evoluon of he daage s represened by he vod volue fracon, defned by: f V V V a EQ a Where V, a V are respecvely he eleenary apparen volue of he aeral and he correspondng eleenary volue of he arx. The rae of ncrease of he vod volue fracon s gven by: f & f& g f& n EQ The growh rae of vods s calculaed by: g p [ D ] f & f Trace EQ Where Trace[D p ] s he race of he acroscopc plasc sran rae ensor. The nucleaon rae of vods s gven by he followng expresson: f& n ε M ε N f N S N e & ε M EQ S N π Where f N s he nucleaed vod volue fracon, S N s he Gaussan sandard devaon, ε N s he nucleaed effecve plasc sran and ε M s he adssble plasc sran. The vscoplasc flow of he porous aeral s descrbed by: σ eq 3 σ Ω evp q f cosh q σ M σ M σ eq Ωevp q f 3 σ M q f q f 3 f f σ > σ EQ jan-9 5

117 RADIOSS THEORY Verson. MATERIALS Where σ eq s he von Mses effecve sress, σ M s he adssble elaso-vscoplasc sress and σ s he f s he specfc coalescence funcon whch can be wren as: hydrosac sress and f f f f c f f u F f f c c f f c f f f f f > f c c EQ Where: f c crcal vod volue fracon a coalescence, f F crcal vod volue fracon a ducle fracure, f u he correspondng value of he coalescence funcon q * f u, u f f F f. The varaon of he specfc coalescence funcon s shown n Fgure f * Fgure Varaon of specfc coalescence funcon * f f N * f f C * f f F Nucluaon Growh Coalescence ε N f * ε M The adssble plasc sran rae s copued as follows: p σ : D & ε EQ M f σ M Where σ s he Cauchy sress ensor, σ M s he adssble plasc sress and D p s he acroscopc plasc sran rae ensor whch can be wren n he case of he assocaed plascy as: wh p Ωevp D & λ EQ σ Ω evp he yeld surface envelope. The vscoplasc ulpler s deduced fro he conssency condon: Ω evp Ω& EQ evp Fro hs las expresson we deduce ha: -jan-9 6

118 RADIOSS THEORY Verson. MATERIALS & Ωevp λ EQ Ωevp e Ωevp Ωevp σ Ω Ω M evp evp : C : A f : I A A σ σ σ f M ε M σ where: A Ω σ : evp ε M ε N σ f N S N ; A e f σ M SN π EQ jan-9 7

119 RADIOSS THEORY Verson. MATERIALS 9.4 Vscous aerals General case of vscous aerals represens a e-dependen nelasc behavor. However, specal aenon s pad o he vscoelasc aerals such as polyers exhbng a rae- and e-dependen behavor. The vscoelascy can be represened by a recoverable nsananeous elasc deforaon and a non-recoverable vscous par occurrng over he e. The characersc feaure of vscoelasc aeral s s fadng eory. In a perfecly elasc aeral, he deforaon s proporonal o he appled load. In a perfecly vscous aeral, he rae of change of he deforaon over e s proporonal o he load. When an nsananeous consan ensle sress σ s appled o a vscoelasc aeral, a slow connuous deforaon of he aeral s observed. When he resulng e dependen sran ε, s easured, he ensle creep coplance s defned as : ε D EQ σ The creep behavor s anly coposed of hree phases: prary creep wh fas decrease n creep sran rae, secondary creep wh slow decrease n creep sran rae and erary creep wh fas ncrease n creep sran rae. The creep sran rae s he slope of creep sran o e curve. Anoher knd of loadng concerns vscoelasc aerals subjeced o a consan ensle sran, ε. In hs case, he sress, σ whch s called sress relaxaon, gradually decreases. The ensle relaxaon odulus s hen defned as: σ E EQ ε Because vscoelasc response s a cobnaon of elasc and vscous responses, he creep coplance and he relaxaon odulus are ofen odeled by cobnaons of sprngs and dashpos. A sple scheac odel of vscoelasc aeral s gven by he Maxwell odel shown n Fgure The odel s coposed of an elasc sprng wh he sffness E and a dashpo assgned a vscosy μ. I s assued ha he oal sran s he su of he elasc and vscous srans: ε e v ε ε EQ Fgure 9.4. Maxwell odel The e dervaon of he las expresson gves he expresson of he oal sran rae: & ε & & e v ε ε EQ As he dashpo and he sprng are n seres, he sress s he sae n he wo pars: σ σ v σ e EQ jan-9 8

120 RADIOSS THEORY Verson. MATERIALS The consuve relaons for lnear sprng and dashpo are wren as: e σ Eε hen Eε& e & EQ σ σ με& v EQ Cobnng EQ , EQ and EQ , an ordnary dfferenal equaon for sress s obaned: σ & σ E & ε or μ σ & σ E & ε EQ τ where μ τ s he relaxaon e. A soluon o he dfferenal equaon s gven by he convoluon negral: E σ dε ' dε ' d' EQ d' [ '/ τ ] E e d' R ' d' where R s he relaxaon odulus. The las equaon s vald for he specal case of Maxwell one-densonal odel. I can be exended o he ul-axal case by: σ dε ' C jkl EQ d' ' d' where C jkl are he relaxaon odul. The Maxwell odel represens reasonably he aeral relaxaon. Bu s only accurae for secondary creep as he vscous srans afer unloadng are no aken no accoun. Anoher sple scheac odel for vscoelasc aerals s gven by Kelvn-Vog sold. The odel s represened by a sple sprng-dashpo syse workng n parallel as shown n Fgure Fgure 9.4. Kelvn-Vog odel The aheacal relaon of Kelvn-Vog sold s wren as: σ E ε ηε& EQ When η no dashpo, he syse s a lnearly elasc syse. When E no sprng, he aeral behavor s expressed by Newon's equaon for vscous fluds. In above relaon, we have consdered a one-densonal odel. For ulaxal suaons, he equaons can be generalzed and rewren n ensor for. The Maxwell and Kelvn-Vog odels are approprae for deal sress relaxaon and creep behavors. They are no adequae for os of physcal aerals. A generalzaon of hese laws can be obaned by addng oher sprngs o he nal odels as shown n Fgures and The equaons relaed o he generalzed Maxwell odel are gven as: -jan-9 9

121 RADIOSS THEORY Verson. MATERIALS σ σ σ EQ σ E ε EQ & σ σ ε E η & EQ The aheacal relaons, whch hold he generalzed Kelvn-Vog odel are: ε ε ε e k EQ e σ σ k σ e σ ε ; E σ E V k ε ; & ε k V σ η The cobnaon of hese equaons allows o oban he expresson of sress and sran raes: & σ & ε & ε & ε & E ε e k k EQ σ ηε E ε k k & EQ E E σ ε σ ε EE E & η η & EQ Fgure Generalzed Maxwell odel Fgure Generalzed Kelvn-Vog odel -jan-9

122 RADIOSS THEORY Verson. MATERIALS The odels descrbed above concern he vscoelasc aerals. The plascy can be nroduced n he odels by usng a plasc sprng. The plasc eleen s nacve when he sress s less han he yeld value. The odfed odel s able o reproduce creep and plascy behavors. The vscoplascy law 33 n RADIOSS wll allow o pleen very general consuve laws useful for a large rang of applcaons as low densy closed cells polyurehane foa, honeycob, pacors and pac lers. The behavor of vscoelasc aerals can be generalzed o hree densons by separang he sress and sran ensors no devaorc and pressure coponens: s j ej Ψ τ dτ EQ τ ε kk σ kk 3K τ dτ EQ τ where s j and e j are he sress and sran devaors. shear and bulk relaxaon odul. ε kk, Ψ and K are respecvely he dlaaon and he 9.4. Bolzann Vscoelasc odel law 34 Ths law vald for sold eleens can be used for vscoelasc aerals lke polyers, elasoers, glass and fluds. Elasc bulk behavor s assued. Ar pressure ay be aken no accoun for closed cell foas: wh: and: P Kε EQ kk P ar P γ V P ar ; γ γ γ Φ V EQ V ε kk ln EQ V γ s he voluerc sran, Φ, he porosy, P he nal ar pressure, γ he nal voluerc sran and K he bulk odulus. For devaorc behavor, he generalzed Maxwell odel s used. The shear relaxaon odul n EQ s hen defned as: β G G e Ψ EQ s G l l s G G EQ where G s he shor e shear odulus, G l he long e shear odulus and β s he decay consan, defned as he nverse of relaxaon e τ s : ηs β ; wh τ s EQ G τ s The coeffcens η, s Gs and s Gl are defned for he generalzed Maxwell odel as shown n Fgure jan-9

123 RADIOSS THEORY Verson. MATERIALS Fgure Generalzed Maxwell Model for Bolzann law Fro EQ , he value of β governs he ranson fro he nal odulus G o he fnal odulus Ψ and when G G G For, we oban G hen l Ψ. For a lnear response, we pu I. G l Generalzed Kelvn-Vog odel law 35 Ths law uses a generalzed vscoelasc Kelvn-Vog odel whereas he vscosy s based on he Naver equaons. The effec of he enclosed ar s aken no accoun va a separae pressure versus copresson funcon. For open cell foa, hs funcon ay be replaced by an equvalen "reoved ar pressure" funcon. The odel akes no accoun he relaxaon zero sran rae, creep zero sress rae, and unloadng. I ay be used for open cell foas, polyers, elasoers, sea cushons, duy paddngs, ec. In RADIOSS he law s copable wh shell and sold eshes. The sple scheac odel n Fgure descrbes he generalzed Kelvn-Vog aeral odel where a e-dependen sprng workng n parallel wh a Naver dashpo s pu n seres wh a nonlnear rae-dependen I sprng. If σ s he ean sress, he devaorc sresses s j a seps n and n are copued by he 3 expressons: n sj σ δ σ δ j else δ n j j n for EQ s n j s n j s& d j EQ wh: G G G. G s& j Ge& j sj ej η j η G G G. G s& j Ge& s e η j η for EQ for EQ jan-9

124 RADIOSS THEORY Verson. MATERIALS where G and E G Mn G are defned as: v Ae& B, v v EQ E G EQ In EQ he coeffcens A and B are defned for Young's odulus updaes E E ε& E. Fgure Generalzed Kelvn-Vog odel for RADIOSS law 35 The expressons used by defaul o copue he pressure s: where: dp d K K KK C K & ε kk C σ kk C3 ε EQ λ η 3λ η kk E K 3 EQ K v E 3 v EQ P σ kk EQ V ε kk ln EQ V λ and η are he Naver Sokes vscosy coeffcens whch can be copared o Lae consans n elascy. η V λ s called he voluerc coeffcen of vscosy. For ncopressble odel, ε kk and λ 3 μ and μ. In EQ , C, C and C 3 are Boolean ulplers used o defne dfferen responses. For 3 exaple, C, C C 3 refers o a lnear bulk odel. Slarly, C C C 3 corresponds o a vsco-elasc bulk odel. For polyurehane foas wh closed cells, he skeleal sphercal sresses ay be ncreased by: P γ Par EQ γ φ -jan-9 3

125 RADIOSS THEORY Verson. MATERIALS where γ s he voluerc sran, Φ, he porosy, P he nal ar pressure. In RADIOSS, he pressure ay ρ also be copued wh he P versus μ, by a user defned funcon. Ar pressure ay be assued as ρ an "equvalen ar pressure" vs μ. The user can defne hs funcon used for open cell foas or for closed cell by defnng a odel dencal o aeral law FOAM_PLAS 33 see followng secons Tabulaed sran rae dependen law for vscoelasc aerals law 38 The law ncorporaed n RADIOSS can only be used wh sold eleens. I can be used o odel: polyers, elasoers, foa sea cushons, duy paddngs, hyperfoas, hypoelasc aerals. In copresson, he nonal sress-sran curves for dfferen sran raes are defned by user Fgure Up o 5 curves ay be npu. The curves represen nonal sresses versus engneerng srans. Fgure Nonal sress-sran curves defned by user npu funcons The frs curve s consdered o represen he sac loadng. All values of he sran rae lower han he assued sac curve are replaced by he sran rae of he sac curve. I s reasonable o se he sran rae correspondng o he frs curve equal o zero. For sran raes hgher han he las curve, values of he las curve are used. For a gven value of ε&, wo values of funcon a ε for he wo edaely lower ε& and hgher ε& sran raes are read. The relaed sress s hen copued as: a & ε & ε σ σ σ σ & ε & ε b EQ Paraeers a and b defne he shape of he nerpolaon funcons. If a b, hen he nerpolaon s lnear. The fgure shows he nfluence of a and b paraeers. -jan-9 4

126 RADIOSS THEORY Verson. MATERIALS Fgure Influence of a and b paraeers The couplng beween he prncpal nonal sresses n enson s copued usng ansoropc Posson's Rao: v j c v v R ε v exp EQ c v j ν s he axu Posson's rao n enson, ν c beng he axu Posson's rao n copresson, and R ν, he exponen for he Posson's rao copuaon n copresson, Posson's rao s always equal o ν c. In copresson, aeral behavor s gven by nonal sress vs nonal sran curves as defned by he user for dfferen sran raes. Up o 5 curves ay be npu. The algorh of he forulaon follows several seps:. Copue prncpal nonal srans and sran raes.. Fnd correspondng sress value fro he curve nework for each prncpal drecon. 3. Copue prncpal Cauchy sress. 4. Copue global Cauchy sress. 5. Copue nsananeous odulus, vscosy and sable e sep. Sress, sran and sran raes us be posve n copresson. Unloadng ay be eher defned wh an unloadng curve, or else copued usng he "sac" curve, correspondng o he lowes sran rae Fgures and. -jan-9 5

127 RADIOSS THEORY Verson. MATERIALS Fgure unloadng behavor no unloadng curve defned Fgure 9.4. unloadng behavor unloadng curve defned I should be noed ha for sably reasons, dapng s appled o sran raes wh a dapng facor: n n & ε & ε & ε R EQ n D The sress recovery ay be appled o he odel n order o ensure ha he sress ensor s equal o zero, n an undefored sae. A hyseress decay ay be appled when loadng, unloadng or n boh phases by: βε σ σ H n, e EQ Where, H s he hyseress coeffcen, and β he relaxaon paraeer. Confned ar conen ay be aken no accoun, eher by usng a user defned funcon, or usng he followng relaon: V V P ar P EQ V Φ V -jan-9 6

128 RADIOSS THEORY Verson. MATERIALS The relaxaon ay be appled o ar pressure: P Mn P P exp R ar ar, ax. p Generalzed Maxwell-Kelvn odel for vscoelasc aerals law 4 Ths law ay only be appled o sold eleens. Bulk behavor s assued o be lnear: dp Kε& kk EQ d Shear behavor s copued usng a shear facor as follows: G G e G 3 β EQ Fgure 9.4. Maxwell-Kelvn Model Vsco-elaso-plasc aerals for foas law 33 Ths aeral law can be used o odel low densy closed cell polyurehane foas, pacors, pac lers. I can only be used wh sold eleens. The an assupons n hs law are he followng: The coponens of he sress ensor are uncoupled unl full voluerc copacon s acheved Posson's raon.. The aeral has no dreconaly. The effec of enclosed ar s consdered va a separae Pressure vs Voluerc Sran relaon: P γ P ar EQ γ Φ V wh: γ γ V EQ jan-9 7

129 RADIOSS THEORY Verson. MATERIALS γ beng he voluerc sran, Φ, he porosy, P he nal ar pressure, γ he nal voluerc sran. s The skeleal sresses σ Fgure 9.4. : before yeld follow he Maxwell-Kelvn-Vogh vscoelasc odel see Fgure 9.4. Maxwell-Kelvn-Vogh Model s s E E s EE σ j Δ σ j E j j & ε σ ε j Δ EQ η η The Young's Modulus used n he calculaon s: E ax E, E ε& E Plascy s defned by a user defned curve vs voluerc sran, γ, or σ A B Cγ Plascy s appled o he prncpal skeleal sresses. The full sress ensor s obaned by addng ar pressure o he skeleal sresses: σ j s σ P δ j ar j y EQ Hyper vsco-elasc law for foas law 6 Experenal ess on foa specens workng n copresson llusrae ha he aeral behavor s hghly nonlnear. The general behavor can be subdvded no hree pars relaed o parcular deforaon odes of aeral cells. When he sran s sall, he cells workng n copresson defor n ebrane whou causng bucklng n s laeral hn-walls. In he second sep, he laeral hn-walls of he cells buckle whle he aeral undergoes large deforaon. Fnally, he las sep he cells are copleely collapsed and he conac beween he laeral hn-walled cells ncreases he global sffness of he aeral. As he vscous behavor of foas s deonsraed by varous ess, s worhwhle o elaborae a aeral law ncludng he vscous and hyper elascy effecs. Ths s developed n [] where a decouplng beween vscous and elasc pars s nroduced by usng fne ransforaons. Only he devaorc par of sress ensor s concerned by vscous effecs. The soropc hyper-elasc law appled o he sphercal par of sress ensor s slar o ha presened n secon 9.. The vscosy odel s based on a generalzaon of Maxwell s odel for hree-densonal case usng fne ransforaons n hree seps []: Splng he ensor of deforaon graden F on wo pars; srech sphercal, dsoronal devaorc, Decouplng assupon beween sphercal and dvaorc pars, Generalsaon of forulaon by convoluon negral n hree-densonal case. F J 3 F EQ dev F sp def I EQ jan-9 8

130 RADIOSS THEORY Verson. MATERIALS wh J def whch gves he decoposon of he graden of deforaon as: F sp Fdev 3 F EQ The hypohess of decoposon appled o he elasc poenal allows separang he densy of oal nal energy no wo pars: Ψ Ψ EQ sp Ψ dev where ndces «sp» and «dev» denoe respecvely sphercal and devaorc pars. Usng Hll s crera o express he densy of hyperelasc poenal for crushable foas, can be wren as a funcon of prncpal sreches λ, λ, λ3 and deforaon graden: Ψ n μ λ, 3 3 EQ α β α α α β λ, λ λ λ λ 3 J α where n s he law order, μ and α he aeral consans, for,,, n. The reader s nved o consul references [] and [] for ore deals. β ν wh ν ν he Posson coeffcens 9.5 Maerals for Hydrodynac Analyss The followng aeral laws are coonly used for flud sulaons: Johnson-Cook odel for sran rae and eperaure dependence on yeld sress law 4, Hydrodynac vscous aeral for Newonan or urbulen fluds law 6, Elaso-plasc hydrodynac aerals wh von Mses soropc hardenng and polynoal pressure law 3, Senberg-Gunan elaso-plasc hydrodynac law wh heral sofenng law 49, Boundary eleen aerals law : Purely heral aerals law 8 RADIOSS provdes a aeral daabase ncorporaed n he nsallaon. Many paraeers are already defned by defaul and gve accurae resuls. Soe of he are descrbed n he followng secons: energy and pressure equaons are solved sulaneously Johnson Cook Law for Hydrodynacs law 4 Ths aeral law cobnes boh laws and 3, as well as addng heral characerscs such as elng eperaure. The equaons descrbng he sae of sress and pressure are: σ n & ε A Bε C ln Τ p & ε C4 C μ E n EQ p C Cμ Cμ C3μ 5 EQ jan-9 9

131 RADIOSS THEORY Verson. MATERIALS where Τ Τ Τel Τ Τ Maeral paraeers are he sae as n law 3. The paraeers are: C Sran rae coeffcen ε& Reference sran rae. Teperaure exponen Τ el Melng eperaure Τ ax Maxu Teperaure. For T > Tax : s used. p Cp specfc hea per un volue For an explanaon abou sran rae flerng, refer o chaper Hydrodynac Vscous Flud Law law 6 Ths law s specfcally desgned o odel lquds and gases. The equaons used o descrbe he aeral are: S j ρvej & EQ C4 C μ E n 3 p C Cμ Cμ C3μ 5 EQ where S j Devaorc sress ensor ν Kneac vscosy e& j Devaorc sran rae ensor The kneac vscosy ν s relaed o he dynac vscosy, η by: η ρ v EQ Modellng a perfec gas To odel a perfec gas, all coeffcens C, C, C, C 3 us be se o equal zero. Also: C C γ EQ P E n EQ γ A perfec gas allows copressbly and expanson and conracon wh a rse n eperaure. However, for any suaons, especally very slow subsonc flows, an ncopressble gas gves accurae and relable resuls wh less copuaon. -jan-9 3

132 RADIOSS THEORY Verson. MATERIALS Modellng an ncopressble gas To odel an ncopressble gas, he coeffcens should be se o: C C C C C E EQ C ρ EQ c where, c speed of sound Incopressbly s acheved va a penaly ehod. The sound speed s se o a leas es he axu velocy. Ths classcal assupon s no vald when flud and srucures are coupled. In hs case, se he sound speed n he flud so ha he frs egen frequency s a leas es hgher n he flud han n he srucure Elaso-plasc Hydrodynac Maeral law 3 Ths law s only used wh sold brck and quadrlaeral eleens. I odels he elasc and plasc regons, slar o law, wh a non lnear behavor of pressure and whou sran rae effec. The law s desgned o sulae aerals n copresson. The sress - sran relaonshp for he aeral under enson s: n A Bε σ EQ p The copresson relaonshp s gven n ers of pressure, p: 3 p C C C μ C μ C C μ E EQ where μ ρ / ρ 4 n μ 3 5 C, C, C, C 3, C 4, C 5 Hydrodynac consans E n Energy per un of nal volue The pressure and energy values are obaned by solvng wo lnear equaons: E Δ E ρ EQ p p Δ Δ ρ G μ p E Δ p Δ F Δ Δ μ EQ F μ C Cμ Cμ C3μ 3 where C 4 C μ μ 5 G Inpu requres Young's or he elasc odulus, E, and Posson's rao, υ. These quanes are used only for he devaorc par. They do no need o be conssen wh he bulk odulus, C. The plascy aeral paraeers are: A Yeld Sress B Hardenng Modulus n Hardenng Exponen σ ax Maxu flow sress ε ax Plasc sran a rupure -jan-9 3

133 RADIOSS THEORY Verson. MATERIALS A pressure cu off, p n, can be gven o l he pressure n enson. The pressure cu off us be less han zero. Fgure 9.5. shows a ypcal curve of he hydrodynac pressure. Fgure 9.5. Hydrodynac Pressure Relaonshp The coeffcens C 4 and C 5 are used o conrol eperaure effecs. The nal energy per un volue, E n, s also requred. Sulaons nvolvng eperaure effecs can ncrease copuaonal e subsanally snce energy and pressure equaons are solved sulaneously Senberg-Gunan aeral law 49 Ths law defnes as elasc-plasc aeral wh heral sofenng. When aeral approaches eng pon, he yeld srengh and shear odulus reduces o zero. The elng energy s defned as: E E c T EQ cold v E cold s cold copresson energy and T elng eperaure supposed o be consan. If he nernal Where energy E s less han G E, he shear odulus and he yeld srengh are defned by he followng equaons: 3 c G b pv h 3 cv fe EE E E e EQ fe E E E Ec 3 σ y σ b pv h 3 e c EQ v Where b, b, h and f are he aeral paraeers. σ s gven by a hardenng rule: [ βε ] n σ σ EQ p The value of σ s led by σ ax. The aeral pressure s defned by a polynoal equaon of sae as n law 3 EQ jan-9 3

134 RADIOSS THEORY Verson. MATERIALS 9.6 Vod Maeral law Ths aeral can be used o defne eleens o ac as a vod, or epy space. 9.7 Falure odel In addon o he possbly o defne user s aeral falure odels, RADIOSS V negraes several falure odels. These odels use generally a global noon of cuulave daage o copue falure. They are osly ndependen o consuve law and he hardenng odel and can be lnked o several avalable aeral laws. A copably able s gven n he RADIOSS Reference Gude. The followng able gves a bref descrpon of avalable odels. Table 9.. FAILURE MODEL DESCRIPTION Falure Model Type Descrpon CHANG Chang-Chang odel Falure crera for coposes ENERGY Energy sorop HASHIN Copose odel Hashn odel JOHNSON Ducle falure odel Cuulave daage law based on he plasc sran accuulaon LAD_DAMA Copose delanaon Ladeveze delanaon odel PUCK Copose odel Puck odel TBUTCHER Falure due o fague Fracure appears when e negraon of a sress expresson becoes rue WILKINS Ducle Falure odel Cuulave daage law FLD Forng l dagra Inroducon of he experenal falure daa n he sulaon SPALLING Ducle Spallng Johnson Cook falure odel wh Spallng effec TENSSTRAIN Tracon WIERZBICKI Ducle aeral 3-D falure odel for sold and shells XFEM Ducle brle falure odel Modfed Tuler-Bucher odel 9.7. Johnson-Cook falure odel Hgh-rae ess n boh copresson and enson usng he Hopknson bar generally deonsrae he sress-sran response s hghly soropc for a large scale of eallc aerals. The Johnson-Cook odel s very popular as ncludes a sple for of he consuve equaons. In addon, also has a cuulave daage law ha can be accesses falure: d Δ ε ε f * & ε * wh: ε [ D D exp D σ ] D ln [ D T ] f 3 4 & ε 5 EQ EQ where Δε s he ncreen of plasc sran durng a loadng ncreen, and he paraeers * σ σ σ D he aeral consans. Falure s assued o occur when d. VM he noralzed ean sress -jan-9 33

135 RADIOSS THEORY Verson. MATERIALS 9.7. Wlkns falure crera An early connuu odel for vod nucleaon s presened n [98]. The odel proposes ha he decoheson falure sress σ s a crcal cobnaon of he hydrosac sress σ and he equvalen von Mses sress σ VM : σ σ σ c c EQ VM In a slar approach, a falure crera based on a cuulave equvalen plasc sran was proposed by Wlkns. Two wegh funcons are nroduced o conrol he cobnaon of daage due o he hydrosac and devaorc loadng coponens. The falure s assued when he cuulave reaches a crcal value d c. The cuulave daage s obaned by: d c n WW d p WW Δε p ε EQ wh: W, ap 3 j where: α P σ, β jj W A, s s A Max,, s s s3 s3 s Δ ε p s an ncreen of he equvalen plasc sran, W s he hydrosac pressure weghng facor, W s he devaorc weghng facor, s are he devaorc prncpal sresses, a, α and β are he aeral consans Tuler-Bucher falure crera A sold ay break ownng o fague due o Tuler-Bucher crera [99]: σ σ d d EQ where consan called daage negral. r λ σ r s he fracure sress, λ aeral consan, s he e when sold cracks and d anoher aeral Forng L Dagra for falure FLD In hs ehod he falure zone s defned n he plane of prncpal srans Fg The ehod usable for shell eleens allows nroducng he experenal resuls n he sulaon. -jan-9 34

136 RADIOSS THEORY Verson. MATERIALS Fgure 9.7. Generc forng l dagra FLD ε ajor % Falure zone ε n % Spallng wh Johnson-Cook Falure odel In hs odel, he Johnson-Cook falure odel s cobned o a Spallng odel where we ake no accoun he spall of he aeral when he pressure acheves a nu value p n. The devaorc sresses are se o zero for copressve pressure. If he hydrosac enson s copued, hen he pressure s se o zero. The falure equaons are he sae as n Johnson-Cook odel Bao-Xue-Werzbck Falure odel Bao-Xue-Werzbck odel [5] represens a 3-D fracure creron whch can be expressed by he followng equaons: n f n ax n n / [ ε ε ] ξ ε ε EQ Cη ε ax Ce C4η ε n C3e ax Where C, C, C 3, 4 defned as followng: wh: for solds: for shells: n σ : hydrosac sress C, γ and are he aeral consans, n he hardenng paraeer and η and ξ are σ 7 3 η ; ξ 3 σvm σvm σ σ η ; ξ η η VM J 7 3 -jan-9 35

137 RADIOSS THEORY Verson. MATERIALS σ VM 3 sss3 : von Mses sress J : Thrd nvaran of prncpal devaorc sresses Fgure 9.7. Graphcal represenaon of Bao-Xue-Werzbck falure crera ε f ε f ηcons ε ax ε n ε f - ξ Sran Falure Model Ths falure odel can be copared o he daage odel n law 7. When he prncpal enson sran ε reaches ε a daage facor D s appled o reduce he sress as shown n Fg The eleen s rupured when D. In addon, he axu srans ε and ε ay depend on he sran rae by defnng a scale funcon. Fgure Sran falure odel -jan-9 36

138 RADIOSS THEORY Verson. MATERIALS Specfc Energy Falure Model When he energy per un volue acheves he value E, hen he daage facor D s nroduced o reduce he sress. For he l value E, he eleen s rupured. In addon, he energy values E and E ay depend on he sran rae by defnng a scale funcon. Fgure Sran falure odel XFEM Crack Inalzaon Falure Model Ths falure odel s avalable for Shell only. The falure ode crera are wren as: For ducle aerals, he cuulave daage paraeer s: where, σ r s he fracure sress σ s he axu prncpal sress λ s he aeral consan s he e when shell cracks for naon of a new crack whn he srucure D s anoher aeral consan called daage negral For brle aerals, he daage paraeer s: σ r σ - D b -jan-9 37

139 Chaper MONITORED VOLUME -jan-9 38

140 RADIOSS THEORY Verson. MONITORED VOLUME. MONITORED VOLUME An arbag s defned as a onored volue. A onored volue s defned as havng one or ore 3 or 4 node shell propery ses. The defned surface us be closed he noral o shell eleens us be all orened ouward as shown n Fgure... The shell noral us be orened ousde he volue. I s possble o reverse he shell norals for a gven propery se by enerng a negave propery nuber. Duy properes propery ype and aerals aeral ype can be used. Fgure.. Tre odel: volue closed There are fve ypes of MONITORED VOLUME: AREA Type: volue and surface oupu pos processng opon, no pressure PRES Type: user funcon defnng pressure versus relave volue GAS Type: adabac pressure volue relaon. P V Vnc Cs AIRBAG Type: Sngle arbag PV nrt rt E c c T γ ρc T V P ρ rt ρ γ p v v γ wh γ COMMU Type: Chabered, councang, folded arbag arbag wh councaons Sae basc equaons: Typcal use of onored volue s for re, fuel ank, arbag. For re use PRES or GAS ype onored volue. For fuel ank use PRES or GAS ype onored volue. For sple unfolded arbag use AIRBAG ype onored volue. For chabered arbag use wo ore ore COMMU ype onored volues. For folded arbag use a se of COMMU ype onored volues. c c p v -jan-9 39