LS-DYNA Manual Vol2 971R700

VOLUME II Material Models

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;EOS
LS#345 ha& hi&torically reference( e:-ation& of &tate 9y type i(entifier&. Below the&e i(entifier& are given with the corre&pon(ing eywor( name in the or(er that they appear in the man-al. The e:-ation& of &tate can 9e -&e( with a &-9&et of the material& that are availa9le for &oli( element&. Type 1" i& line( to the type 2 thic &hell element an( can 9e -&e( to mo(el engine ga&et&.
/Y3E %( ;EOS=LINEAR=3OLYNOMIAL
/Y3E )( ;EOS=LINEAR=3OLYNOMIAL=WI/1=ENER2Y=LEAK
/Y3E -( ;EOS=I2NI/ION=AND=2ROW/1=O.=REA0/ION=IN=1E
/Y3E ?( ;EOS=/A<ULA/ED=0OM3A0/ION
/Y3E ,( ;EOS=/A<ULA/ED
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/Y3E %$( ;EOS=IDEAL=2AS
/Y3E %)( ;EOS=MIE=2RUNEISEN
/Y3E $%!"( ;EOS=USER=DE.INED

DE.INI/IONS @ NO/ES ON SOME 0OMMONLY USED 3ARAME/ERS(
An or(er to pre&cri9e the 9o-n(ary an(Eor initial thermo(ynamic con(ition, man-al comp-tation& are often nece&&ary. Convention& or (efinition& m-&t 9e e&ta9li&he( to &implify thi& proce&&. Some 9a&ic varia9le& are (efine( in the following. Since many of the&e varia9le& have alrea(y 9een (enot# e( 9y (ifferent &ym9ol&, the notation& -&e( here are -ni:-e in thi& &ection only They are pre&ente( to only clarify their -&age. corre&pon(ing SA -nit &et i& al&o pre&ente( a& an example.
+ir&t con&i(er a few vol-metric parameter& &ince they are a mea&-re of compre&&ion or expan&ion.
ol-me ≈ () a&& ≈ () C-rrent &pecific vol-me per ma&& = = ≈
'eference &pecific vol-me = = ≈
'elative vol-me = = ( ⁄ )( ⁄ ) = = C-rrent normaliHe( vol-me increment = = 1 − = 1 − fre:-ently -&e( vol-metric parameter i& = − 1 = = − = − 1 Sometime& another vol-metric parameter i& -&e() ! = = Th-& the relation 9etween an( ! i& = = ! − 1 The following ta9le &-mmariHe& the&e vol-metric parameter&.
'ABL<S COP'<SSAO5 5O LO3 <MP5SAO5
= = "" > 1 1 ? 1
V0 – INITIAL RELATIVE VOLUME

LSDYNA 1#11 <OS
" = "#$ 3en&ity at nominalEreference &tate, -&-ally non#&tre&& or non#(eforme( &tate.
"%&' 3en&ity at t $
" C-rrent (en&ity
'ecalling the c-rrent relative vol-me, = = , an( the relative vol-me at time$ i& then
= %&' = % = % . =enerally, the $ inp-t parameter in an I<OSJcar(, refer& to thi& ." i& generally the (en&ity (efine( in the ITJcar(. ence, if a material i& mechanically com# pre&&e( at t$, $, or , the initial relative vol-me, may 9e comp-te( an( inp-t accor(ingly * +. The UreferenceV &tate i& a -ni:-e &tate with re&pect to which the material &tre&& ten&or i& comp-te(. Therefore i& very critical in comp-ting the pre&&-re level in a material. Ancorrect choice of wo-l( lea( to incorrect pre&&-re comp-te(. An general, i& cho&en &-ch that at Hero compre&&ion or expan&ion, the material &ho-l( 9e in e:-ili9ri-m with it& am9ient &-rro-n(ing. An many of the e:-a# tion& &hown in the <OS &ection, i& fre:-ently -&e( a& a mea&-re of compre&&ion or expan&ion. owever, the -&er& m-&t clearly (i&ting-i&h 9etween an( .
E0 INTERNAL ENERGY
Anternal energy repre&ent& the thermal energy &tate temperat-re (epen(ent component of a &y&tem. One (efinition for internal energy i&
, = -./ ≈ (0234) 5ote that the capital U,V here i& the a9&ol-te internal energy. At i& not the &ame a& that -&e( in the &-9&e:-ent I<OS eywor( inp-t, or &ome e:-ation& &hown for each I<OSJcar(. Thi& internal ener# gy i& often (efine( with re&pect to a ma&& or vol-me -nit.
Anternal energy per -nit ma&& al&o calle( &pecific internal energy)
4 = , = -/ ≈ 5 0234 6 Anternal energy per -nit c-rrent vol-me)
4 = -/ = "-/ = -/ ≈ 5 0234 = 786 Anternal energy per -nit reference vol-me)
4 = -./ = "-./ = -./ ≈ 5 0234 = 786 .

1#12 <OS LSDYNA
4 typically refer& to the capital U<V &hown in &ome e:-ation& -n(er thi& U<OSV &ection. ence the initial Uinternal energy per unit reference volume V, <$, a eywor( inp-t parameter in the I<OS &ec# tion can 9e comp-te( from
4%&' = "-/%&' To convert from 4 to 4, &imply (ivi(e 4 9y
4 = "-/ = 9"-/: "" = 4 EQUATION OF STATE EOS )
thermo(ynamic &tate of a homogeneo-& material, not -n(ergoing any chemical reaction& or pha&e change&, may 9e (efine( 9y two &tate varia9le&. Thi& relation i& generally calle( an e:-ation of &tate. +or example, a few po&&i9le form& relating pre&&-re to two other &tate varia9le& are
; = ;("</) = ;(<4) = ;( < 4) = ;(< 4) The la&t e:-ation form i& fre:-ently -&e( to comp-te pre&&-re. The <OS for &oli( pha&e material& i& &ometime& partitione( into 2 term&, a col( pre&&-re an( a thermal pre&&-re
; = ;() > ;?(<4) ;() i& the col( pre&&-re hypothetically eval-ate( along a $#(egree#@elvin i&otherm. Thi& i& &ome# time& calle( a $#@ pre&&-re#vol-me relation or col( compre&&ion c-rve. ;?(<4) i& the thermal pre&&-re component that (epen(& on 9oth vol-metric compre&&ion an( thermal &tate of the material.
3ifferent form& of the <OS (e&cri9e (ifferent type& of material& an( how their vol-metric compre&# &ion or expan&ion 9ehavior&. The coefficient& for each <OS mo(el come from (ata#fitting, phe# nomenological (e&cription&, or (erivation& 9a&e( on cla&&ical thermo(ynamic&, etc.
!INEA" #O$P"ESSION
An low pre&&-re proce&&e&, pre&&-re i& not &ignificantly affecte( 9y temperat-re. hen vol-metric compre&&ion i& within an ela&tic linear (eformation range, a linear 9-l mo(-l-& may 9e -&e( to re# late vol-me change& to pre&&-re change&. 'ecalling the (efinition of an i&otropic 9-l mo(-l-& i& W+-ng 10"X @ = − A. Thi& may 9e rewritten a& ; = B−@ C = . The 9-l mo(-l-&,, th-& i& e:-ivalent to - in I<OSJLA5<'JPOL45OAL when all other coefficient& are Hero. Thi& i& a &imple&t form of an <OS. To initialiHe a pre&&-re for &-ch a material, only m-&t 9e (efine(.
INITIAL CONDITION SETTING

LSDYNA 1#1% <OS
+or many of the <OS availa9le, pre&&-re i& &pecifie( given, an( the -&er m-&t mae an a&&-mption on either 4%&' or . Con&i(er two po&&i9ilitie& a /%&' i& (efine( or a&&-me( from which4%&' may 9e comp-te(, or 2 "%&' i& (efine( or a&&-me( from which may 9e o9taine(.
WHEN TO USE THE EOS
+or &mall &train& con&i(eration&, a total &tre&& ten&or may 9e partitione( into a (eviatoric &tre&& com# ponent an( a mechanical pre&&-re.
DEF = DGEF > DHHI JEF = DGEF − ;JEF ; = − DHHI K DHHI = −;
The pre&&-re component may 9e written from the (iagonal &tre&& component&.
5ote that LMM = 9LNNOLPPOL: i& po&itive in ten&ion while P i& po&itive in compre&&ion.
Similarly the total &train ten&or may 9e partitione( into a (eviatoric &train component vol-me# pre&erving (eformation an( a vol-metric (eformation.
QEF = QGEF > QHHI JEF where RMM i& calle( the mean normal &train, an( QHH i& calle( the (ilatation or vol-me &train change in vol-me per -nit initial vol-me
QHH = −

;EOS=LINEAR=3OLYNOMIAL
P-rpo&e) 3efine coefficient& for linear polynomial <OS an( initialiHe the initial thermo(ynamic &tate of the material. Thi& i& (one 9y (efining <$ an( $ 9elow.
Car( 1 1 2 % ! " 0 7 /
aria9le <OSA3 C$ C1 C2 C% C ! C" C0
Type / + + + + + + +
aria9le <$ $
Type + +
VARIA<LE DES0RI3/ION
<OSA3 <:-ation of &tate A3, a -ni:-e n-m9er or la9el not excee(ing / character& m-&t 9e &pecifie(.
C$ The $th polynomial e:-ation coefficient.
... ...
C0 The 0th polynomial e:-ation coefficient.
<$ Anitial internal energy per -nit reference vol-me &ee the 9eginning of the I<OS &ection.
$ Anitial relative vol-me &ee the 9eginning of the I<OS &ection.
Re5ars(
1. The linear polynomial e:-ation of &tate i& linear in internal energy. The pre&&-re i& given 9y)
; = - > - > -8 8 > - > (-W > -X > -Y 8) , Z

-[ \ are &et to Hero if ] + < = −1 < ^_` i& the ra#
tio of c-rrent (en&ity to reference (en&ity. " i& a nominal or reference (en&ity (efine( in the ITJ58LL car(.
The linear polynomial e:-ation of &tate may 9e -&e( to mo(el ga& with the gamma law e:-a# tion of &tate. Thi& may 9e achieve( 9y &etting)
- = - = -8 = - = -Y = + an( -W = -X = a − 1 where a = bcbd i& the ratio of &pecific heat&. Pre&&-re for a perfect ga& i& then given 9y)
e = (a − 1) , < ha& the -nit of pre&&-re where " an( "
28 hen - = - = -8 = - = -Y = +, it (oe& not nece&&arily mean that the initial pre&# &-re i& Hero, ;+ * -+ The initial pre&&-re (epen(& the val-e& of all the coefficient& an( on%f=+ an( ,%f=+. The pre&&-re in a material i& comp-te( from the whole e:-ation a9ove,; = ;(g< ,). At i& alway& prefera9le to initialiHe the initial con(ition 9a&e( on %f=+ an(,%f=+. The -&e of - = - = -8 = - = -Y = + m-&t 9e (one with ca-tion a& it may change the form an( 9ehavior of the material. The &afe&t way i& to -&e the whole <OS e:-a# tion to man-ally chec for the pre&&-re val-e. +or example, for i(eal ga&, it i& wrong to (e# fine -W = -X = a − 1 an( - = - = -8 = - = -Y = + at the &ame time.

;EOS=JWL
Car( 1 2 % ! " 0 7 /
aria9le <OSA3 B '1 '2 O<= <$ O
Type / + + + + + + +
See e:-ation in 'emar&.
$ Anitial relative vol-me.
Re5ars( The KL e:-ation of &tate (efine& the pre&&-re a&
e = h 1 − ij N 4j N >k 1 − ij P 4j P > il < an( i& -&-ally -&e( for (etonation pro(-ct& of high explo&ive&.

;EOS=SA0K=/UESDAY
Car( 1 2 % ! " 0 7 /
aria9le <OSA3 1 2 % B1 B2 <$ $
Type / + + + + + + +
1
2
The Sac e:-ation of &tate (efine& pre&&-re a&

;EOS=2RUNEISEN
Car( 1 2 % ! " 0 7 /
aria9le <OSA3 C S1 S2 S% =O <$
Type / + + + + + + +
aria9le $
Type +
C
S1
S2
S%
=O
$ Anitial relative vol-me Re5ars(
The =r-nei&en e:-ation of &tate with c-9ic &hoc velocity#particle velocity v&#v p (efine& pre&&-re for compre&&e( material& a&
e = b P p BOqP prP pP C s(t N) pt P uPuvN t u(uvN)Pw P > (a > x ) ,Z
an( for expan(e( material& a&

LSDYNA 1#1 <OS

;EOS=RA/IO=O.=3OLYNOMIALS
Car( 1 1 2 % !
Type + + + +
Type + + + +
Car( 0 1 2 % !
Type + + + +
Type + + + +
1$
11
Re5ars(
The ratio of polynomial& e:-ation of &tate (efine& the pre&&-re a&
e = yNOyPlOylPOyzl y{Oy|lOy}lP (1 > ~)
where
•E = € F' hEF F ‚ = ƒ „… † ] I = − 1 ‚ = I „… † ‡ I

1#2! <OS LSDYNA
;EOS=LINEAR=3OLYNOMIAL=WI/1=ENER2Y=LEAK
Car( 1 1 2 % ! " 0 7 /
aria9le <OSA3 C$ C1 C2 C% C! C" C0
Type / + + + + + + +
aria9le <$ $ LCA3
Type + + A
C$
C1
C2
C%
C!
C"
C0
LCA3 Loa( c-rve A3 (efining the energy (epo&ition rate
Re5ars(
Thi& polynomial e:-ation of &tate, linear in the internal energy per initial vol-me, ,, i& given 9y
e = - > - > -88 > - > (-W > -X > -Y8),

LSDYNA 1#2" <OS
in which -< -< -8< -< -W< -X< an( -Y are -&er (efine( con&tant& an(
= − 1 Z where i& the relative vol-me. An expan(e( element&, we &et the coefficient& of 8 to Hero, i.e.,

1#20 <OS LSDYNA
;EOS=I2NI/ION=AND=2ROW/1=O.=REA0/ION=IN=1E
Car( 1 1 2 % ! " 0 7 /
aria9le <OSA3 B MP1 MP2 +'<' = '1
Type / + + + + + + +
aria9le '2 '% '" '0 +MA= +'<R ='O1 <
Type + + + + + + + +
Car( % 1 2 % ! " 0 7 /
aria9le '1 <S1 CP C' <<TL CC'AT <5R TP$
Type + + + + + + + +
aria9le ='O2 '2 <S2 <5 +M=' +5='
Type + + + + + +
Pro(-ct KL con&tant &ee &econ( e:-ation in 'emar&

LSDYNA 1#27 <OS
= ωCv of pro(-ct
'1 8nreacte( KL con&tant &ee fir&t e:-ation in 'emar&
'% ωCv of -nreacte( explo&ive
'" 8nreacte( KL con&tant &ee fir&t e:-ation in 'emar&
+MA= axim-m + for ignition term
+'<R Con&tant in ignition term of reaction e:-ation
='O1 Con&tant in growth term of reaction e:-ation
< Con&tant in growth term of reaction e:-ation
'1 Con&tant in growth term of reaction e:-ation
<S1 Con&tant in growth term of reaction e:-ation
CP eat capacity of reaction pro(-ct&
C' eat capacity of -nreacte( <
<<TL Con&tant in ignition term of reaction e:-ation
CC'AT Con&tant in ignition term of reaction e:-ation
<5R eat of reaction
='O2 Con&tant in completion term of reaction e:-ation
'2 Con&tant in completion term of reaction e:-ation
<S2 Con&tant in completion term of reaction e:-ation
<5 Con&tant in completion term of reaction e:-ation
+M=' axim-m + for growth term

1#2/ <OS LSDYNA
Re5ars(
<:-ation of State +orm 7 i& -&e( to calc-late the &hoc initiation or fail-re to initiate an( (etona# tion wave propagation of &oli( high explo&ive&. At &ho-l( 9e -&e( in&tea( of the i(eal < 9-rn op# tion& whenever there i& a :-e&tion whether the < will react, there i& a finite time re:-ire( for a &hoc wave to 9-il( -p to (etonation, an(Eor there i& a finite thicne&& of the chemical reaction Hone in a (etonation wave. t relatively low initial pre&&-re& >2#% =Pa, thi& e:-ation of &tate &ho-l( 9e -&e( with material type 1$ for acc-rate calc-lation& of the -nreacte( < 9ehavior. t higher initial pre&&-re&, material type can 9e -&e(. KL e:-ation of &tate (efine& the pre&&-re in the -nreacte( explo&ive a&
;# = ‰4X# > ‰84Y# > ‰ ?## (‰ = Š#‹Œ) where e an( Te are the relative vol-me an( temperat-re, re&pectively, of the -nreacte( explo&ive. nother KL e:-ation of &tate (efine& the pre&&-re in the reaction pro(-ct& a&
; = x 4 > 4 8 > ? ‘ = Š‹Œ’“ where p an( Tp are the relative vol-me an( temperat-re, re&pectively, of the reaction pro(-ct&. & the chemical reaction convert& -nreacte( explo&ive to reaction pro(-ct&, the&e KL e:-ation& of &tate are -&e( to calc-late the mixt-re of -nreacte( explo&ive an( reaction pro(-ct& (efine( 9y the fraction reacte( ++O implie& no reaction, +1 implie& complete reaction. The temperat-re& an( pre&&-re& are a&&-me( to 9e e:-al /# = /,e# = e an( the relative vol-me& are a((itive, i.e.,
= (1 − •)# > The chemical reaction rate for conver&ion of -nreacte( explo&ive to reaction pro(-ct& con&i&t& of three phy&ically reali&tic term&) an ignition term in which a &mall amo-nt of explo&ive react& &oon after the &hoc wave compre&&e& itD a &low growth of reaction a& thi& initial reaction &prea(&D an( a rapi( completion of reaction at high pre&&-re an( temperat-re. The form of the reaction rate e:-a# tion i&
”y”& = S‰4(1 − •)$#(4 − 1 − ––‰†f)##&—˜ Agnition
>™š›1(1 − •)#œ•—e# =rowth
>™š›\(1 − •)#œ8 S—8e# Completion

LSDYNA 1#2 <OS
3etail& of the comp-tational metho(& an( many example& of one an( two (imen&ional &hoc initia# tion an( (etonation wave calc-lation can 9e fo-n( in the reference& Cochran an( Chan W17X, Lee an( Tarver W1/$X. 8nfort-nately, &-fficient experimental (ata ha& 9een o9taine( for only two &oli( explo&ive& to (evelop very relia9le &hoc initiation mo(el&) PBM#"$! an( the relate( M#9a&e( explo&ive& LM#1!,LM#1$,LM#$!, etc. an( LM#17 the in&en&itive TTB#9a&e( explo&ive. 'eactive flow mo(el& have 9een (evelope( for other explo&ive& T5T, P<T5, Compo&ition B, propellant&, etc. 9-t are 9a&e( on very limite( experimental (ata.

1#%$ <OS LSDYNA
;EOS=/A<ULA/ED=0OM3A0/ION
aria9le <OSA3 = <$ $
Type / + + +
Re4eat 0ards $ and ! Bor 0i# /i# and K i 8 A total oB , 7ards 5ust 9e deBined8
Car( 2 1 2 % ! "
ε1, ε2,.. ε5 ol-metric &train, ln
@1,@2,..@5 B-l -nloa(ing mo(-l-&
Re5ars(

;EOS=/A<ULA/ED=0OM3A0/ION ;EOS
LSDYNA 1#%1 <OS

+ig-re 1#1. Pre&&-re ver&-& vol-metric &train c-rve for <:-ation of &tate +orm / with

;EOS=/A<ULA/ED
aria9le <OSA3 = <$ $ LCC LCT
Type / + + + A A
DeBine ) additional 7ards iB and onl iB L00 and L0/ eCual ero8 Re4eat 0ards $ and ! to deBine 0i and /i8
Car( 2 1 2 % ! "
= a <$ Anitial internal energy
$ Anitial relative vol-me
LCC Loa( c-rve (efining ta9-late( f-nction C. See e:-ation in 'emar&. The a9&ci&&a val-e& of LCC an( LCT m-&t increa%e monotonically. The (efini# tion can exten( into the ten&ile regime.

<1,<2, ...<5 ol-metric &train, ln, where i& the relative vol-me. The fir&t a9&ci&&a point, <1, m-&t 9e $.$ or po&itive if the c-rve exten(& into the ten&ile re# gime with &-9&e:-ent point& &ecrea%ing monotonically.
C1,C2,..C5 Ta9-late( point& for f-nction C.
T1,T2,..T5 Ta9-late( point& for f-nction T.
Re5ars(
The ta9-late( e:-ation of &tate mo(el i& linear in internal energy. Pre&&-re i& (efine( 9y

;EOS=3RO3ELLAN/=DE.LA2RA/ION
Thi& <:-ation of &tate 1$ ha& 9een a((e( to mo(el air9ag propellant&.
Car( 1 1 2 % ! " 0 7 /
aria9le <OSA3 B MP1 MP2 +'<'
Type / + + + + +
aria9le g '1 '2 '% '"
aria9le '0 +MA= +'<R ='O1 <
Type + + + + +
Car( ! 1 2 % ! " 0 7 /
aria9le '1 <S1 CP C' <<TL CC'AT <5R TP$
Type + + + + +
aria9le ='O2 '2 <S2 <5 +M=' +5='
Type + + + + + +

+'<' 8nreacte( Co#vol-me
'% 8nreacte( ωCv
='O1 +ir&t 9-rn rate coefficient
< Pre&&-re <xponent 1&t term
'1 <xponent on + 1&t term
<S1 <xponent on 1#+ 1&t term
CP eat capacity Cv for pro(-ct&
C' eat capacity Cv for -nreacte( material
<<TL <xtra, not pre&ently -&e(
CC'AT Pro(-ct co#vol-me
<5R eat of 'eaction

<S2 <xponent on 1#+ 2n( term
<5 Pre&&-re <xponent 2n( term
+M=' axim-m + for 1&t term
+5=' inim-m + for 2n( term
Re5ars(
(eflagration 9-rn rate reactive flow mo(el re:-ire& an -nreacte( &oli( e:-ation of &tate, a reac# tion pro(-ct e:-ation of &tate, a reaction rate law an( a mixt-re r-le for the two or more &pecie&. The mixt-re r-le for the &tan(ar( ignition an( growth mo(el WLee an( Tarver 1/$X a&&-me& that 9oth pre&&-re& an( temperat-re& are completely e:-ili9rate( a& the reaction procee(&. owever, the mixt-re r-le can 9e mo(ifie( to allow no thermal con(-ction or partial heating of the &oli( 9y the reaction pro(-ct ga&e&. +or thi& relatively &low proce&& of air9ag propellant 9-rn, the thermal an( pre&&-re e:-ili9ri-m a&&-mption& are vali(. The e:-ation& of &tate c-rrently -&e( in the 9-rn mo(el are the KL, =r-nei&en, the van (er aal& co#vol-me, an( the perfect ga& law, 9-t other e:-ation& of &tate can 9e ea&ily implemente(. An thi& propellant 9-rn, the ga&eo-& nitrogen pro(-ce( 9y the 9-rning &o(i-m aHi(e o9ey& the perfect ga& law a& it fill& the air9ag 9-t may have to 9e mo(ele( a& a van (er aal]& ga& at the high pre&&-re& an( temperat-re& pro(-ce( in the propellant cham9er. The chemical reaction rate law i& pre&&-re, particle geometry an( &-rface area (epen(ant, a& are mo&t high#pre&&-re 9-rn proce&&e&. hen the temperat-re profile of the reacting &y&tem i& well nown, temperat-re (epen(ent rrheni-& chemical inetic& can 9e -&e(.
Since the air9ag propellant compo&ition an( performance (ata are company private information, it i& very (iffic-lt to o9tain the re:-ire( information for 9-rn rate mo(eling. owever, Amperial Chemi# cal An(-&trie& ACA Corporation &-pplie( pre&&-re exponent, particle geometry, pacing (en&ity, heat of reaction, an( atmo&pheric pre&&-re 9-rn rate (ata which allowe( -& to (evelop the n-merical mo(el pre&ente( here for their 5a5% ^ +e2O% (river air9ag propellant. The (eflagration mo(el, it& implementation, an( the re&-lt& for the ACA propellant are pre&ente( in Wall:-i&t, et.al., 1$X.
The -nreacte( propellant an( the reaction pro(-ct e:-ation& of &tate are 9oth of the form)

LSDYNA 1#%7 <OS
ω an( # v pl-& the correct (i&tri9-tion 9etween Ucol(V compre&&ion an( internal energie& are -&e(, the calc-late( temperat-re& are very rea&ona9le an( th-& can 9e -&e( to chec propellant perfor# mance.
The reaction rate -&e( for the propellant (eflagration proce&& i& of the form)
”y”& = (1 − •) • eŸ ¡¢ £ y £ y¤¥¦¥N
> (1 − •)§ •‰eœ ¡¢ y¤¥¦¥P £ y£

where F i& the fraction reacte( F $ implie& no reaction, F 1 i& complete reaction, t i& time, an( p i& pre&&-re in 9ar&, r , %,u,,, -, y( F limit 1 an( F limit 2 are con&tant& -&e( to (e&cri9e the pre&&-re (epen(ence an( &-rface area (epen(ence of the reaction rate&. Two or more pre&&-re (epen(ant reaction rate& are incl-(e( in ca&e the propellant i& a mixt-re or exhi9ite( a &harp change in reaction rate at &ome pre&&-re or temperat-re. B-rning &-rface area (epen(encie& can 9e approximate( -&ing the *.+y +x term&. Other form& of the reaction rate law, &-ch a& rrheni-& temperat-re (epen(ent e.E/"T type rate&, can 9e -&e(, 9-t the&e re:-ire very acc-rate temperat-re& calc-lation&. ltho-gh the theoretical G-&tification of pre&&-re (epen(ent 9-rn rate& at ilo9ar type pre&&-re& i& not com# plete, a va&t amo-nt of experimental 9-rn rate ver&-& pre&&-re (ata (oe& (emon&trate thi& effect an( hy(ro(ynamic calc-lation& -&ing pre&&-re (epen(ent 9-rn acc-rately &im-late &-ch experiment&.
The (eflagration reactive flow mo(el i& activate( 9y any pre&&-re or particle velocity increa&e on one or more Hone 9o-n(arie& in the reactive material. S-ch an increa&e create& pre&&-re in tho&e Hone& an( the (ecompo&ition 9egin&. Af the pre&&-re i& relieve(, the reaction rate (ecrea&e& an( can go to Hero. Thi& feat-re i& important for &hort (-ration, partial (ecompo&ition reaction&. Af the pre&# &-re i& maintaine(, the fraction reacte( event-ally reache& one an( the material i& completely con# verte( to pro(-ct molec-le&. The (eflagration front rate& of a(vance thro-gh the propellant calc-lat# e( 9y thi& mo(el for &everal propellant& are :-ite clo&e to the experimentally o9&erve( 9-rn rate ver# &-& pre&&-re c-rve&.
To o9tain goo( agreement with experimental (eflagration (ata, the mo(el re:-ire& an acc-rate (e# &cription of the -nreacte( propellant e:-ation of &tate, either an analytical fit to experimental com# pre&&ion (ata or an e&timate( fit 9a&e( on previo-& experience with &imilar material&. Thi& i& al&o tr-e for the reaction pro(-ct& e:-ation of &tate. The more experimental 9-rn rate, pre&&-re pro(-c# tion an( energy (elivery (ata availa9le, the 9etter the form an( con&tant& in the reaction rate e:-a# tion can 9e (etermine(.
Therefore, the e:-ation& -&e( in the 9-rn &-9ro-tine for the pre&&-re in the -nreacte( propellant
;§ = ¨1© 4 jX ©ª >¨\ © 4 jY© ª > j ©?ªª yjlj where - an( T- are the relative vol-me an( temperat-re re&pectively of the -nreacte( propellant. The relative (en&ity i& o9vio-&ly the inver&e of the relative vol-me. The pre&&-re Pp in the reaction pro(-ct& i& given 9y)

;EOS ;EOS=3RO3ELLAN/=DE.LA2RA/ION
1#%/ <OS LSDYNA
& the reaction procee(&, the -nreacte( an( pro(-ct pre&&-re& an( temperat-re& are a&&-me( to 9e e:-ili9rate( T u T p T , p P u P p an( the relative vol-me& are a((itive)
= (1 − •) © § > • © where ' i& the total relative vol-me. Other mixt-re a&&-mption& can an( have 9een -&e( in (ifferent ver&ion& of 34523E%3. The reaction rate law ha& the form)
”y”& ='O1P ^ +'<R< + ^ +MA=' 1 1 # + ^ +MA=<S1
^ ='O2P ^ +'<R<5 + ^ +MA=' 2 1 # + ^ +MA=<S2
Af F excee(& +M=' , the ='O1 term i& &et e:-al to Hero, an(, if F i& le&& than +5=' , the ='O2 term i& Hero. Th-&, two &eparate or overlapping 9-rn rate& can 9e -&e( to (e&cri9e the rate at which the propellant (ecompo&e&.

LSDYNA 1#% <OS
;EOS=/ENSOR=3ORE=0OLLA3SE
Car( 1 1 2 % ! " 0 7 /
aria9le <OSA3 5L3 5C' 81 82 A<$ <C$
Type / + + + + + +
5L3 irgin loa(ing loa( c-rve A3
5C' Completely cr-&he( loa( c-rve A3
81 <xce&& Compre&&ion re:-ire( 9efore any pore& can collap&e
82 <xce&& Compre&&ion point where the irgin Loa(ing C-rve an( the Com# pletely Cr-&he( C-rve inter&ect
A<$ Anitial Anternal <nergy
Re5ars(

1#!$ <OS LSDYNA
8nloa(ing occ-r& along the virgin loa(ing c-rve -ntil the exce&& compre&&ion &-rpa&&e& g. fter that, the -nloa(ing follow& a path 9etween the completely cr-&he( c-rve an( the virgin loa(ing c-rve. 'eloa(ing will follow thi& c-rve 9ac -p to the virgin loa(ing c-rve. Once the exce&& com# pre&&ion excee(& g8, then all -nloa(ing will follow the completely cr-&he( c-rve.
+or -nloa(ing 9etween g an( g8 a partially cr-&he( c-rve i& (etermine( 9y the relation&hip)
e () = e (Op®) (Op )Op¯°± − 1 Z where
o = ; (; ²³´) an( the &-9&cript& pc an( cc refer to the partially cr-&he( an( completely cr-&he( &tate&, re&pective# ly. Thi& i& more rea(ily -n(er&too( in term& of the relative vol-me ' .
+ig-re 1#2. Pre&&-re ver&-& compaction c-rve

= O p ; () = ; ®¯µ¶
Thi& repre&entation &-gge&t& that for a fixe( ²·¸ = p¯°±O the partially cr-&he( c-rve will &epa#
rate linearly from the completely cr-&he( c-rve a& ' increa&e& to acco-nt for pore recovery in the material.
The 9-l mo(-l-& 0 i& (etermine( to 9e the &lope of the c-rrent c-rve time& one pl-& the exce&& compre&&ion
= ”A”p (1 > ) The &lope ”A”p for the partially cr-&he( c-rve i& o9taine( 9y (ifferentiation a&)
”A”p = ”A¹¹ 5‘N v u®“(N v u)(Nu¯°±) 6 ( O p ®) ”p ( O p¯°± )
Simplifying,
= ”A¹¹ (pr )”p (1 > —) where
— = ( O p®) ( O p)( O p ¯°±) − 1 Z The 9-l &o-n( &pee( i& (etermine( from the &lope of the completely cr-&he( c-rve at the c-rrent pre&&-re to avoi( in&ta9ilitie& in the time &tep.

;EOS=IDEAL=2AS
P-rpo&e) Thi& i& e:-ation of &tate form 12 for mo(eling i(eal ga&. At i& an alternate approach to -&# ing I<OSJLA5<'JPOL45OAL with C! C" a#1 to mo(el i(eal ga&. Thi& ha& a &lightly improve( energy acco-nting algorithm.
Car( 1 1 2 % ! " 0 7 /
aria9le <OSA3 C$ CP$ CL CR T$ $
Type / + + + + + +
C$ 5ominal con&tant#vol-me &pecific heat coefficient at STP
CP$ 5ominal con&tant#pre&&-re &pecific heat coefficient at STP
CL Linear coefficient for the variation& of Cv an( Cp ver&-& T
CR R-a(ratic coefficient for the variation& of Cv an( Cp ver&-& T
T$ Anitial temperat-re
$ Anitial relative vol-me &ee the 9eginning of the I<OS &ection
Re5ars(
1. The pre&&-re in the i(eal ga& law i& (efine( a&
e = " ‘- − -.“/- = - > -º/ > -»/8-. = -. > -º/ > -»/8
= = ( ⁄ )( ⁄ ) = = ""

The initial relative vol-me, %&' $, initial temperat-re, /%&'T$, an( heat capacity in# formation are (efine( in the I<OSJA3<LJ=S inp-t. 5ote that the UreferenceV (en&ity i& typically a (en&ity at a non#&tre&&e( or nominal &tre&& &tate. The initial pre&&-re &ho-l( al# way& 9e chece( man-ally again&t &im-lation re&-lt.
2. hen (ealing with <-lerianEL< mo(el&, the i(eal ga& mo(el i& implemente( to pre&erve the a(ia9atic &tate (-ring a(vection. The a(ia9atic &tate i& con&erve( on the expen&e of a per# fect internal energy con&ervation.
%8 The i(eal ga& mo(el i& goo( for low (en&ity ga& only. 3eviation from the i(eal ga& 9ehavior may 9e in(icate( 9y the compre&&i9ility factor (efine( a&
= ;¨/ hen Q (eviate& from 1, the ga& 9ehavior (eviate& from i(eal.

;EOS=JWL<
Thi& i& <:-ation of &tate +orm 1!. The KLB Kone&#ilen&#Lee#Baer e:-ation of &tate, (evel# ope( 9y Baer W11X an( f-rther (e&cri9e( 9y Baer an( Oro&H W11X, (e&cri9e& the high pre&&-re regime pro(-ce( 9y over(riven (etonation& while retaining the low pre&&-re expan&ion 9ehavior re# :-ire( for &tan(ar( acceleration mo(eling. The (erive( form of the e:-ation of &tate i& 9a&e( on the KL form (-e to it& comp-tational ro9-&tne&& an( a&ymptotic approach to an i(eal ga& at high ex# pan&ion&. ((itional exponential term& an( a varia9le =r-nei&en parameter have 9een a((e( to a(e# :-ately (e&cri9e the high#pre&&-re region a9ove the Chapman#Ko-g-et &tate.
Car( 1 1 2 % ! " 0 7 /
aria9le <OSA3 1 2 % ! "
aria9le '1 '2 '% '! '"
aria9le L1 L2 L% L! L"
Type + + + + +
aria9le BL1 BL2 BL% BL! BL"
Type + + + + +
aria9le 'L1 'L2 'L% 'L! 'L"
Type + + + + +
aria9le C O<= < $
1 <:-ation of &tate coefficient, &ee 9elow.
2 <:-ation of &tate coefficient, &ee 9elow.
% <:-ation of &tate coefficient, &ee 9elow.
! <:-ation of &tate coefficient, &ee 9elow.
" <:-ation of &tate coefficient, &ee 9elow.
'1 <:-ation of &tate coefficient, &ee 9elow.
'2 <:-ation of &tate coefficient, &ee 9elow.
'% <:-ation of &tate coefficient, &ee 9elow.
'! <:-ation of &tate coefficient, &ee 9elow.
'" <:-ation of &tate coefficient, &ee 9elow.
L1 Aλ1, e:-ation of &tate coefficient, &ee 9elow.
L2 Aλ2, e:-ation of &tate coefficient, &ee 9elow.
L% Aλ%, e:-ation of &tate coefficient, &ee 9elow.
L! Aλ!, e:-ation of &tate coefficient, &ee 9elow.

C <:-ation of &tate coefficient, &ee 9elow.
O<= <:-ation of &tate coefficient, &ee 9elow.
< <nergy (en&ity per -nit initial vol-me
Re5ars(
The KLB e:-ation#of#&tate (efine& the pre&&-re a&
e = € hEXE' 1 − ¾j¥ 4j¥ > ¾l > - 1− ¾i(iO)
¿ = € ( h¾E > k¾E)4jÀ¥ > ŠXE'
where i& the relative vol-me, < i& the energy per -nit initial vol-me, an( hE, ¨E, h¾E, k¾E, ¨¾E, -, an( Š are inp-t con&tant& (efine( a9ove.

LSDYNA 1#!7 <OS

;EOS=2ASKE/
Thi& mo(el wor& with &oli( element& an( the thic &hell -&ing &elective re(-ce( 2 x 2 integration <L+O'2 on S<CTAO5JTS<LL to mo(el the re&pon&e of ga&et&. +or the thic &hell only, it i& completely (eco-ple( from the &hell material, i.e., in the local coor(inate &y&tem of the &hell, thi& mo(el (efine& the normal &tre&&, σHH, an( (oe&n]t change any of the other &tre&& component&. The mo(el i& a re(-ction of the ITJ=<5<'LJ5O5LA5<'J03O+J3ASC'<T<JB<.
Car( 1 1 2 % ! " 0 7 /
aria9le <OSA3 LCA31 LCA32 LCA3% LCA3!
Type / A A A A
Car( 2 1 2 % ! " 0 7 /
aria9le 85LO3 @ 3P+ T+S C+S LO++S<T AS
Type + + + + + + +
LCA31 Loa( c-rve for loa(ing.
LCA32 Loa( c-rve for -nloa(ing.
LCA3% Loa( c-rve for (amping a& a f-nction of vol-metric &train rate.

<R.$.$) Loa(ing an( -nloa(ing follow loa(ing c-rve
<R.1.$) Loa(ing follow& loa(ing c-rve, -nloa(ing follow& -nloa(ing c-rve. The -nloa(ing c-rve A3 if -n(efine( i& taen a& the loa(# ing c-rve.
<R.2.$) Loa(ing follow& loa(ing c-rve, -nloa(ing follow& -nloa(ing &tiffne&&, @, to the -nloa(ing c-rve. The loa(ing an( -nloa(ing c-rve& may only inter&ect at the origin of the axe&.
<R.%.$) R-a(ratic -nloa(ing from pea (i&placement val-e to a perma# nent off&et.
@ 8nloa(ing &tiffne&&, for 85LO32 only.
3P+ 3amping factor for &ta9ility. al-e& in the neigh9orhoo( of -nity are rec# ommen(e(. The (amping factor i& properly &cale( to eliminate time &tep &iHe (epen(ency.
T+S Ten&ile fail-re &train.
C+S Compre&&ive fail-re &train.
O++S<T Off&et factor 9etween $ an( 1.$ to (etermine permanent &et -pon -nloa(ing if the 85LO3%.$. The permanent &et& in compre&&ion an( ten&ion are e:-al to the pro(-ct of thi& off&et val-e an( the maxim-m compre&&ive an( ten&ile (i&placement&, re&pectively.
AS Anitial vol-me &train.

;EOS=MIE=2RUNEISEN
Thi& i& <:-ation of &tate +orm 10, a ie#=r-nei&en form with a p#` compaction mo(el.
Car( 1 1 2 % ! " 0 7 /
aria9le <OSA3 = 1 2 % P<L PCO 5
Type / + + + + + + +
Car( 2 1 2 % ! " 0 7 /
aria9le LP$ <$ $
<OSA3 <:-ation of &tate i(entification. -ni:-e n-m9er or la9el not excee(ing / character& m-&t 9e &pecifie(.
= =r-nei&en gamma.
PCO Compaction pre&&-re
5 Poro&ity exponent
LP$ Anitial poro&ity
Re5ars(
The e:-ation of &tate i& a ie#=r-nei&en form with a polynomial -goniot c-rve an( a p#` compac# tion mo(el. +ir&t, we (efine a hi&tory varia9le repre&enting the poro&ity ~ that i& initiali&e( to ~ Á1. The evol-tion of thi& varia9le i& given a&
~(f) = Â^ÃÄ1< Â„_ Å~<Â„_œÆ& Ç 1 > ( ~ − 1) s¹È¦c(œ)¹È¦cÉ¤ wÊËÌÍ where e(f) in(icate& the pre&&-re at time t . +or later -&e, we (efine the cap pre&&-re a&
e = eÎ − (eÎ − e#˜) B ÏÏCÐÊ
The remain(er of the <OS mo(el i& given 9y the e:-ation&
e("< 4) = Ñ~"4 > eÒ(!)s1− 1\ Ñ!w eÒ(!) = h! > h8!8 > h!
together with
;EOS=USER=DE.INED
The&e are e:-ation& of &tate 21#%$. The -&er can &-pply hi& own &-9ro-tine&. See al&o ppen(ix B. The eywor( inp-t ha& to 9e -&e( for the -&er interface with (ata.
Car( 1 1 2 % ! " 0 7 /
aria9le <OSA3 <OST LC 5 A<CT <O O B8L@
Type / A A A A + + +
DeBine LM0 5aterial 4ara5eters usin ? 4ara5eters 4er 7ard8
Car( 2 1 2 % ! " 0 7 /
aria9le P1 P2 P% P! P" P0 P7 P/
Type + + + + + + + +
<OST 8&er e:-ation of &tate type 21#%$ incl-&ive. n-m9er 9etween 21 an( %$ ha& to 9e cho&en.
LC Length of material con&tant array which i& e:-al to the n-m9er of material con&tant& to 9e inp-t. LC \ !/
5 5-m9er of hi&tory varia9le& to 9e &tore(, &ee ppen(ix 3.
A<CT ectoriHation flag on1. vectoriHe( -&er &-9ro-tine m-&t 9e &-pplie(.
<O Anitial internal energy.
B8L@ B-l mo(-l-&. Thi& val-e i& -&e( in the calc-lation of the contact &-rface &tiffne&&.
P1 +ir&t material parameter.
P2 Secon( material parameter.

LSDYNA 2#1 T
;MA/ LS#345 ha& hi&torically reference( each material mo(el 9y a n-m9er. & &hown 9elow, a three (igit n-merical (e&ignation can &till 9e -&e(, e.g., ITJ$$1, an( i& e:-ivalent to a corre&pon(ing (e&criptive (e&ignation, e.g., ITJ<LSTAC. The two e:-ivalent comman(& for each material mo(el, one n-merical an( the other (e&criptive, are li&te( 9elow. The n-m9er& in &:-are 9racet& &ee ey 9elow i(entify the element form-lation& for which the material mo(el i& implemente(. The n-m9er in the c-rly 9racet&, FnG, in(icate& the (efa-lt n-m9er of hi&tory varia9le& per element inte# gration point that are &tore( in a((ition to the 7 hi&tory varia9le& which are &tore( 9y (efa-lt. +or the type 10 f-lly integrate( &hell element& with 2 integration point& thro-gh the thicne&&, the total n-m# 9er of hi&tory varia9le& i& / x n^7. +or the Belyt&cho#T&ay type 2 element the n-m9er i& 2 x n^7.
n a((itional optionJ /I/LE may 9e appen(e( to a ;MA/ eywor( in which ca&e an a((itional line i& rea( in /$a format which can 9e -&e( to (e&cri9e the material. t pre&ent, LS#345 (oe& not mae -&e of the title. Ancl-&ion of title& &imply give& greater clarity to inp-t (ec&.
Ke to nu59ers in sCuare 9ra7ets
" Solids %1 1uHesLiu 9ea5 %< <elts7Ho resultant 9ea5 %I <elts7Ho interated solid and tu9ular 9ea5s %/ /russ %D Dis7rete 9ea5 %SW S4ot:eld 9ea5 $ SHells !a /Hi7 sHell Bor5ulation % !9 /Hi7 sHell Bor5ulation $ !7 /Hi7 sHell Bor5ulation ! !d /Hi7 sHell Bor5ulation > * S4e7ial air9a ele5ent > S31 ele5ent ) A7ousti7 solid - 0oHesie solid ?A Multi5aterial ALE solid &alidated+
?< Multi5aterial ALE solid &i54le5ented 9ut not alidated%+ , Me59rane ele5ent
ITJ33J<'OSAO52 ITJ33JP<'<BALT4 ITJ33JPO'<JA'
1 <rror a&&ociate( with a(vection inherently lea(& to &tate varia9le& that may 9e incon&i&tent with nonlinear con&tit-tive ro-tine& an( th-& may lea( to nonphy&ical re&-lt&, noncon&ervation of energy, an( even n-merical in&ta9ility in &ome ca&e&. Ca-tion i& a(vi&e(, partic-larly when -&ing the 2 n( tier of material mo(el& implemente( for L< m-lti#material &oli(& (e&ignate( 9y W/BX which are largely -nte&te( a& L< material&.

ITJ33JT<'LJ<MP5SAO52 ITJ5O5LOCL2

LSDYNA 2#% T

2#! T LSDYNA

LSDYNA 2#" T

2#0 T LSDYNA
ITJ2!") ITJPLJNOPTAO5T'OPACJ<LSTAC W$X N%$ ITJ2!0) ITJPLJ58LL W$X N27 ITJ2"1) ITJTALO'<3JP'OP<'TA<S W2X N0 ITJ2"") ITJPA<C<AS<JLA5<'JPLSTACJT<'L W$,2,%a9c(X ITJ2"0) ITJO'PO8SJSOLA3SJ+A5AT<JST'A5 W$X ITJ200) ITJTASS8<J3ASP<'S<3 W$X ITJ207) ITJ<A=TJCA5J'8BB<' W$,"X ITJ20) ITJB<'=ST'OJBO4C<J'8BB<' W$,"X ITJ27$) ITJC W$,"X ITJ271) ITJPO3<' W$,"X ITJ272) ITJ'T W$,"X ITJ270) ITJC'O5OLO=ACLJASCO<LSTAC W2,%a9c(X
+or the (i&crete type 0 9eam element&, which are -&e( to mo(el complicate( (amper& an( m-lti# (imen&ional &pring#(amper com9ination&, the following material type& are availa9le)
ITJ$00) ITJLA5<'J<LSTACJ3ASC'<T<JB< W13X ITJ$07) ITJ5O5LA5<'J<LSTACJ3ASC'<T<JB< W13X ITJ$0/) ITJ5O5LA5<'JPLSTACJ3ASC'<T<JB< W13X ITJ$0) ITJSA3J3P<'J3ASC'<T<JB< W13X ITJ$7$) ITJ43'8LACJ=SJ3P<'J3ASC'<T<JB< W13X ITJ$71) ITJCBL<J3ASC'<T<JB< W13X ITJ$7!) ITJ<LSTACJSP'A5=J3ASC'<T<JB< W13X ITJ$%) ITJ<LSTACJ03O+JSP'A5=J3ASC'<T<JB< W13X ITJ$!) ITJA5<LSTACJSP'A5=J3ASC'<T<JB< W13X ITJ$") ITJA5<LSTACJ03O+JSP'A5=J3ASC'<T<JB< W13X ITJ11) ITJ=<5<'LJ5O5LA5<'J03O+J3ASC'<T<JB< W13X ITJ121) ITJ=<5<'LJ5O5LA5<'J13O+J3ASC'<T<JB< W13X ITJ1!0) ITJ13O+J=<5<'LAQ<3JSP'A5= W13X ITJ10) ITJ=<5<'LJSP'A5=J3ASC'<T<JB< W13X ITJ17) ITJS<ASACJASOLTO

LS-DYNA Manual Vol2 971R700

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Transcript of LS-DYNA Manual Vol2 971R700