1 exp( c) 1 ( )

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1 Maerial Model LS-DYNA Theory Manual For exponenial relaionhip: 0 ε 0 cε h( ε) exp ε > 0 c 0 exp( c) Lmax ε Lmax ε > 0 c 0 where Lmax SSM ; and c CER. Sre of Damping Elemen i: σ D ε ε (9.56.5) 3 max Toal Sre i: σ σ σ σ3 (9.56.6) Maerial Model 58: Rae Seniive Compoie Fabric See maerial ype 58, Laminaed Compoie Fabric, for he reamen of he compoie maerial. Rae effec are aken ino accoun hrough a Maxwell model uing linear vicoelaiciy by a convoluion inegral of he form: where g ( τ ) kl ε kl σ gkl ( τ) dτ (9.58.) 0 τ i he relaxaion funcion for differen re meaure. Thi re i added o he re enor deermined from he rain energy funcional. Since we wih o include only imple rae effec, he relaxaion funcion i repreened by ix erm from he Prony erie: () g N m Gme β (9.58.) m We characerize hi in he inpu by he hear moduli, G i, and he decay conan, β i. An arbirary number of erm, no exceeding 6, may be ued when applying he vicoelaic model. The compoie failure i no direcly affeced by he preence of he vicou re enor. Maerial Model 59: Coninuou Surface Cap Model Thi i a cap model wih a mooh inerecion beween he hear yield urface and hardening cap, a hown in Figure The iniial damage urface coincide wih he yield urface. Rae effec are modeled wih vicoplaiciy. 9.86

2 LS-DYNA Theory Manual Maerial Model Figure General Shape of he concree model yield urface in wo-dimenion. Sre Invarian. The yield urface i formulaed in erm of hree re invarian: i he fir invarian of he re enor, i he econd invarian of he deviaoric re enor, and 3 i he hird invarian of he deviaoric re enor. The invarian are defined in erm of he deviaoric re enor, S and preure, P, a follow: 3 3P S S 3 S S jk S ki (9.59.) Plaiciy Surface. The hree invarian yield funcion i baed on hee hree invarian, and he cap hardening parameer, κ, a follow: (,,, κ ) f R F F (9.59.) 3 f c Here F f i he hear failure urface, F c i he hardening cap, and R i he Rubin hreeinvarian reducion facor. The cap hardening parameer κ i he value of he preure invarian a he inerecion of he cap and hear urface. Trial elaic re invarian are emporarily updaed via he rial elaic re enor, σ T. Thee are denoed T, T, and 3 T. Elaic re ae are modeled when f ( T, T, 3 T,κ Τ ) 0. Elaic-plaic re ae are modeled when f ( T, T, 3 T,κ Τ ) > 0. In hi cae, he plaiciy algorihm reurn he re ae o he yield urface uch ha f ( P, P, 3 P,κ P ) 0. Thi i accomplihed by enforcing he plaic coniency condiion wih aociaed flow. Shear Failure Surface. The rengh of concree i modeled by he hear urface in he enile and low confining preure regime: F f ( ) α λ exp θ β (9.59.3) 9.87

3 Maerial Model LS-DYNA Theory Manual Here he value of α, β, λ, andθ are eleced by fiing he model urface o rengh meauremen from riaxial compreion (TXC) e conduced on plain concree cylinder. Rubin Scaling Funcion. Concree fail a lower value of 3 (principal re difference) for riaxial exenion (TXE) and orion (TOR) e han i doe for TXC e conduced a he ame preure. The Rubin caling funcion R deermine he rengh of concree for any ae of re relaive o he rengh for TXC, via RF f. Srengh in orion i modeled a Q F f. Srengh in TXE i modeled a Q F f, where: ' Q Q α λ exp α λ exp β β θ θ (9.59.4) Cap Hardening Surface. The rengh of concree i modeled by a combinaion of he cap and hear urface in he low o high confining preure regime. The cap i ued o model plaic volume change relaed o pore collape (alhough he pore are no explicily modeled). The ioropic hardening cap i a wo-par funcion ha i eiher uniy or an ellipe: ( F c where L( i defined a: [ L( ] [ L( L( ], κ ) (9.59.5) [ X( L( ] κ if κ > κ0 L( κ ) (9.59.6) κ 0 oherwie The equaion for F c i equal o uniy for L(. I decribe he ellipe for > L(. The inerecion of he hear urface and he cap i a κ. κ 0 i he value of a he iniial inerecion of he cap and hear urface before hardening i engaged (before he cap move). The equaion for L( rerain he cap from reracing pa i iniial locaion a κ 0. The inerecion of he cap wih he axi i a X(. Thi inerecion depend upon he cap ellipiciy raio R, where R i he raio of i major o minor axe: X ( L( κ ) RF f ( L( κ )) (9.59.7) The cap move o imulae plaic volume change. The cap expand (X( and κ increae) o imulae plaic volume compacion. The cap conrac (X( and κ decreae) o imulae plaic volume expanion, called dilaion. The moion (expanion and conracion) of he cap i baed upon he hardening rule: D (XX 0) D (XX 0) ( ) (9.59.8) p ε v W exp 9.88

4 LS-DYNA Theory Manual Maerial Model p Here ε v he plaic volume rain, W i he maximum plaic volume rain, and D and D are model inpu parameer. X 0 i he iniial locaion of he cap when κκ 0. The five inpu parameer (X 0, W, D, D, and R) are obained from fi o he preurevolumeric rain curve in ioropic compreion and uniaxial rain. X 0 deermine he preure a which compacion iniiae in ioropic compreion. R, combined wih X 0, deermine he preure a which compacion iniiae in uniaxial rain. D, and D deermine he hape of he preure-volumeric rain curve. W deermine he maximum plaic volume compacion. Shear Hardening Surface. In unconfined compreion, he re-rain behavior of concree exhibi nonlineariy and dilaion prior o he peak. Such behavior i be modeled wih an iniial hear yield urface, N H F f, which harden unil i coincide wih he ulimae hear yield urface, F f. Two inpu parameer are required. One parameer, N H, iniiae hardening by eing he locaion of he iniial yield urface. A econd parameer, C H, deermine he rae of hardening (amoun of nonlineariy). Damage. Concree exhibi ofening in he enile and low o moderae compreive regime. d vp ( d σ σ ) (9.59.9) A calar damage parameer, d, ranform he vicoplaic re enor wihou damage, denoed σ vp, ino he re enor wih damage, denoed σ d. Damage accumulaion i baed upon wo diinc formulaion, which we call brile damage and ducile damage. The iniial damage hrehold i coinciden wih he hear plaiciy urface, o he hrehold doe no have o be pecified by he uer. Ducile Damage. Ducile damage accumulae when he preure (P) i compreive and an energy-ype erm, τ c, exceed he damage hrehold, τ 0c. Ducile damage accumulaion depend upon he oal rain componen, ε i j, a follow: τc σε (9.59.0) The re componen σ are he elao-plaic ree (wih kinemaic hardening) calculaed before applicaion of damage and rae effec. Brile Damage. Brile damage accumulae when he preure i enile and an energy-ype erm, τ, exceed he damage hrehold, τ 0. Brile damage accumulaion depend upon he maximum principal rain, ε max, a follow: τ E ε (9.59.) max Sofening Funcion. A damage accumulae, he damage parameer d increae from an iniial value of zero, oward a maximum value of one, via he following formulaion: 9.89

5 Maerial Model LS-DYNA Theory Manual Brile Damage Ducile Damage D d( τ ) D C( τ ) D exp τ 0 max B d( τ ) d c B A( τ ) Bexp c τ 0c The damage parameer ha i applied o he ix ree i equal o he curren maximum of he brile or ducile damage parameer. The parameer A and B or C and D e he hape of he ofening curve ploed a re-diplacemen or re-rain. The parameer dmax i he maximum damage level ha can be aained. I i inernally calculaed and i le han one a moderae confining preure. The compreive ofening parameer, A, may alo be reduced wih confinemen, uing he inpu parameer pmod, a follow: pmod d max 0.00) A A( (9.59.) Regulaing Meh Size Seniiviy. The concree model mainain conan fracure energy, regardle of elemen ize. The fracure energy i defined here a he area under he rediplacemen curve from peak rengh o zero rengh. Thi i done by inernally formulaing he ofening parameer A and C in erm of he elemen lengh, l (cube roo of he elemen volume), he fracure energy, G f, he iniial damage hrehold, τ 0 or τ 0c, and he ofening hape parameer, D or B. The fracure energy i calculaed from up o five uer-pecified inpu parameer (G fc, G f, G f, pwrc, pwrc). The uer pecifie hree diinc fracure energy value. Thee are he fracure energy in uniaxial enile re, G f, pure hear re, G f, and uniaxial compreive re, G fc. The model inernally elec he fracure energy from equaion which inerpolae beween he hree fracure energy value a a funcion of he re ae (expreed via wo re invarian). The inerpolaion equaion depend upon he uer-pecified inpu power pwrc and pwr, a follow. pwr - f f f f 3 pwrc f f fc f 3 if he preure i enile G G ran(g G ) where ran if he preure i compreive G G ran(g G ) where ran The inernal parameer ran i limied o range beween 0 and. Elemen Eroion. An elemen loe all rengh and iffne a d. To preven compuaional difficulie wih very low iffne, elemen eroion i available a a uer opion. An elemen erode when d > 0.99 and he maximum principal rain i greaer han a uer upplied inpu value, -ERODE. 9.90

6 LS-DYNA Theory Manual Maerial Model Vicoplaic Rae Effec. A each ime ep, he vicoplaic algorihm inerpolae beween he elaic rial re, σ i T j, and he invicid re (wihou rae effec), σ p, o e he vp vicoplaic re (wih rae effec), σ : vp T p ( γ γ wih ) Δ / η γ Δ / η Thi inerpolaion depend upon he effecive fluidiy coefficien, η, and he ime ep, Δ. The effecive fluidiy coefficien i inernally calculaed from five uer-upplied inpu parameer and inerpolaion equaion: if he preure i enile η η ran( η η ) ran 3 if he preure i compreive η η ran( η η ) ran c 3 η0 η0c η ηc η Sraeη N Nc ε ε - pwr pwrc The inpu parameer are η 0 and N for fiing uniaxial enile re daa, η 0c and N c for fiing he uniaxial compreive re daa, and Srae for fiing hear re daa. The effecive rain rae i ε. Thi vicoplaic model may predic ubanial rae effec a high rain rae (ε >00). To limi rae effec a high rain rae, he uer may inpu overre limi in enion (over) and compreion (overc). Thee inpu parameer limi calculaion of he fluidiy parameer, a follow: if E ε η > over hen η over ε where over over when he preure i enile, and over overc when he preure i compreive. The uer ha he opion of increaing he fracure energy a a funcion of effecive rain rae via he repow inpu parameer, a follow: rae G f G f repow E f (9.59.3) rae Here G f i he fracure energy enhanced by rae effec, and f i he yield rengh before applicaion of rae effec (which i calculaed inernally by he model). The erm in bracke i greaer han, or equal o one, and i he approximae raio of he dynamic o aic rengh. 9.9