Comparison of Nonlinear Models for Prediction of Continuum Damage in Aluminum under Different Loading

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1 ;< +:=>3 9): / +,( )*! + (##$ %&' () *+, -,/ "! + (##$ %&' () *+, -,/ *+,!! C"$ : "#$ % = + # ; *+, - 9: " ) ; 3 ) 583 % GH3 L %I J&K GH3 D 3 E?F >!C 3 &' ) 3?@ A >, ) ' ) > + 3 M 3 Q 3 3 D 3?@ A,> H F 3 = N H 3 E3 M & Q %I (USDFLD); 3 & 3 3 P+ % ;: + ) O,> ;,! # 3 S, 3 K, 3 I M) ) Q GH $R #,-) + 3M 3!3 F 3 > D$ -KH + ) > 3@?@ MK3 V U3 E?F -, T$ 3 M,> ;, L 3 > 3 > 583% $ P4 N = " :+H+,E:F Comparison of Nonlinear Models for Prediction of Continuum Damage in Aluminum under Different Loading A M A Farsi Mechanical Engineering, Astronautic Research Institute, Iranian Space Research Center, Tehran, Iran A R Sehat Mechanical Engineering, Astronautic Research Institute, Iranian Space Research Center, Tehran, Iran Abstract In this paper, the continuum damage mechanics principles based on onora model and limiter model as a non-linear model are studied Nucleation, development and propagation of damage in Al 583 are investigated Different loading such as cyclic loading, triaxial stress and temperature influence on damage mechanism is investingated The FE simulation of material behavior during failures is carried out by AAQUS software package and USDFLD subroutine The results are validated with experimental tests These comparison show that the simulation results are verified by experimental tests, and the onora model accuracy to behavior modeling is better Also, capability of model to prediction of fracture is not acceptable when the temperature is high Keywords: Continuum Damage Mechanics, FEM, Aluminum 583, onora Model, Limiter Model 3 3 V 3 >' %-, (&' 775 % F ;##$ 4 % ): ; [4] % L! (E?F,) 3 + H ), + ;3 ) = = "N % 9&$ ) &R,N [5] IJ KL8/ F ) (##$,) ^V, j, Z! 3 %I 4 4 &' 3 _?# k? M $ & " 3 ) 9R: Q 3,-) Q #3? [6] OR lp >a ) ) R ) -KH W %' -KH NP@ L 3 M %' + no,$ ; S! "C,M ) _) Continuum Damage Mechanic(CDM) * - 9: 3 F P3 = " W " [P3 " \,9 X Y, &Z ]CC W ; P4' ( L 9&$ ) ) + ) 3 3 " W &@, X % -KH = I % ^ ) _F ` *+, ER 3 9?@ A = I, O:? 3 3 " " \, 9 X 3 ;W? &' 3 : a,n 3 C H b S&' E 3 Z E?F ) KH S, ) L 3,) E ) ) 3 &R 9' ), + -C M, 4! " ; G []3 *+,! ): :,) Rf@,) 3 %I ) > ): 3 (##$ [] M, "3 >` ' 3 [3] * farsi@ariacir :

2 -H&H > ##: [4]! Z D E > -" U [P3,9 X ' 9&$ 3 x 3 ' W 3 " P A " \?@ A = " ]CC 9@ " ; 9Z?@ 3 $ P4 N [5] F 3 L, X O 9&$ 4 $ " = M 3 & > 4 %' M ) $," ) H O 3 > 3 3 GH3 $ ) + 3?3 = ) O 3 % (P&+ 3 & > " [6] Z []3 =?@ A > [7] \& 4 : S& M 3 M { 93# ( L > ; + H T o ) KH 3, " P A + M 3,> 3 ;##$ 4 P ; % [8] Z % ; + 3, X ;E 3 3 I " ) V K z\ = > "! O U+ A533- pr E 3 P+ % " \ " U 9 E?F 3 % -KH+ ) 3 > ; $ E?F ) 9R: =, X + H 3 3 &E p',! + H F O + M) ) 9R: # 3 $ o ^ 3 $3 [9] # " P = ( X 3 > " > ; Z " &@, X ]C!C GH3 L 3 [] >E M L M >R 3 &@ O: L - [] 3 = N H & ) O 3 ) 33 >' %-, % 3 [] 3 $OR M 3M 3 E4 > ), %I (##$ 4 3 D J &K 3 Z = ) 3, /3 E3,M _) ;Y, J&K GH3 %)U ; 3 3 ( O,> ; Z F M P Z,> }?,> E ): (##$ % # ; 3 J&K GH3 D 3 / 3 *+, ) : +, 3 a@ H F 3*+ (E?F ; S!D 3 ) O %?@ 3D Z,> A) 3! ), ): F 3 3 ]K &' 3 3 = >, ) = ) E3,M 3,> O ' ) 3 I,) { 3583b % p4)e?f ; 3 M (ASTM E8) Z 4 > [7]jC, j, > 3 ;K 3 H NP@ % ) = " >R Z G j?e ;+ ) = ) % -KH = " ' 3 o + 3, O: [5], O: p ]$,) " U " 4 qk 3?3 [8]! O: : " \ S 93F A ١ 3 3! E P-& # N pr -" U (R 3 + t +, O: 3 SI: ( X ;Y, (K 3)9 " \ T + 9 X,F O: ", M MP+ 3! + H M, % 3 " " P A > 989> L $ ($OR,o $3 qk [9] E x C D " \ ),9 X yp4 D3 3, $3,o N H 3 > ; # 3 $3,3 ;Y,! + E $ #,-), X ]CC = " P > D " W &@ 3 >E M >R L M 3 = > ; [] M 3 4 3?@ A ( X > ;! Q ) O 3 > ; I $R, " \ ) HY > ; > %I % b,-) N 3 3 I! $, ) O Z H D! ;E 3 fk E 3 I (, 3?@ A 3 9 X 3,?@ A = I, O:,9 X %-,, > ; [] " \ E E) _3 " \ 3 > ; " \ L M ^I =: 3?@ A ; ;Y, > 3@ 3 % -KH (O &: 3 M 3 $ M L k 3 ze3 > [] E A533 U+ % 4 P -@ 3 = " -KH I &R M# " \ 9 X,GH3 3 =?@ A > 3 SE 93F 3 D Z > &R P [3] SE & M) Q) O 3! >, $ & GH3 M, O: 9f $ # ) 3 4 > " " \ U - 3 "C &@ 3M?@ A,> # = Rayleigh-Ritz Nonisotropic

3 $R V + &'$ N O 8: N/* P/-M " (583 %) 3 ; 3 4 3, O:,o _) t + ;-, " P!4 3 3 (D) = I: D = - / = : I: _?# k?, O: : -3 A _?# k? 3 ' F $ GH3 : (٢) M J E ;+ H T 3 >' _4 I: " P t + 3 = () G-4 = J E(3)?3 (R 3 L M?3 3 =!"#! = : + L M J E 3 3 : (UE - (' 3 L M L $ L M (O ; 3 ' : (UE 9` = (3) (UE M ;P-4 ]CC qk " & (UE 3 3 O :, X ) " G H 3, + E & : " " JR 3 ;Y, % _3 (Ψ) P&, ) b ) zue : F O 3 3 &@, X y i &@ 3, X %&= '(()) '*& X L x i &@, X ;3-3 (4?3) -C(,M! = : 3 ( L 4 ; (4)?3 3 [], A : 9 " \ ( L 3 " U (4) ((5)?3)3 - ) 9# - (' H FM -KHT$ %I P(E?F & GH 3 ) " \ $ M?@ AM L 3 3 M P ) 3{ " +3 + ' ) > 3;Y, H FO,> ) 3O 3 P+% ) : 3 + ( O ~@ J E!4 3 (USDFLD) ; 3 ) P+ % 4 )9 R: Q,> ; 9&$ 3 3 O # ',9&$ Q 3 3 I,M) 4 56 (, { % & 3 (P&+ -3 { P ( : 4 (P& ;, {, ^ ; ) E?F E3 GH3 : " \ % + X 3, % # 9&$," ) H F 3 [3] 4@ >' L = ) ;+ ) ' 3 X " = 9 X = 3 D 3 9$ 9` G H 3 ( X I ^ " \ 3 I 4 3, O:,o ) 4 -KH _?# k? M, (R 3 = P+ H ) b, I: f' ;, X % _4 I: J E + H T,o ) 4 a@ 3 _?# k? M, ) 3 4 3,=,-, O: [P3,*+ 3 #, "C I: _) " \ G H 3 M 3, =, # _4 I: # # M D P,,o, O: zu: & : $? -C & 3 P+ (R 3 = X 4 ) _a# $OR 3,o o 9f+,o 3 3 $OR " 3 (9) J E _4 I: D ( ) = () () [5] Z = X J E 3 ()?3> () I: _?# k? A > 3?3 ; ] K 3 3 = X L k? O?3 ) JR 3 3 A () _4 > 3 3 ) = ) e(, -,T, D)+ ) P( T, r, x) K &@ X r T?3 ; (5) 4 K &@ X Crack Nucleation Representative Volume Element

4 ()?3 (R 3 D Z 9 _) [5] ` F = (ab!)(!") (c` ) (ab!) () K n ~fk S 3L t + 3 (3?3) - [ K F S!?3 3 OR 3 3 : = # ; f =gh!/ [3] $ = N H (3) K F G-4 (),?3 ) H } 3, 3 = _3 # a3 ( $ %I 9 _3 3 > 3 = 3! I l K > : ) (R = : (UE?@ " U ]CC X 3 (7) (6),?3 G 3 Y=( 67 6# Y=- > 9 :;(!"#) <=( > 9) E M b ) ) { ", Mσ A )P M >E ) e 9 ) 3 = 3 Y (6) (7) D3 ; ( X 3 3 'F F 3 (UE 9 3 " U GH 3 : ) &@, X 3 3 j\ ; " % F )U :G 3 ` 3 3M?@ A,> # = D-D =(D z{ D )( -[ (L ^ L rs ) ( t pq^ trs ) ]a ) D D z{ = ( c`) : (P-P z{ ) 3 lm # k j 3 M [5] 3 (6)?3 ) O lm = n (6) o t pq L 3 pq $ : _F ` 3 %)U t rs L rs f( v w " 3 3 ) $ M : [] 3 3 v xy +? +Q K,P+ % E! M % I! " E? 3 O,N ),P+ yp 4N " 3 ' ) z ' ) 4 #9 : 3' ; %I $ ; -,?3 9:P ZP4 9 O(UE (U E 3! ) 9 O(UE 9jG: N 9 : &9 ` ',N 3 E 9 O [4] 6 p 3 T O&K,P+% 3M,&, + ) ) 3 P +% 34 3 }#$ ; + H 3 % I ) > 3 )?@ A9Z 9&$ 3 9 ) O C3D8 O 9&$ ) (E?F 3 M 3 (4) (5) X) H, 3?@ " ( ] K H 3 E E3[5] ),% + E MP + 3 I M ) 9# Q %$?# E?F I `: M,,C : G F G * Ġ - Y - { x Ṁ = ( X { D M { εl?3 ; (8) K ( X, X ( X E " F 3 ) 3 (F)j\ 9 ) > &@ " \ 4 I,-) Q, ) O []3 4 - ) _) 9 (R 3 9 _) 3 3 9Y +- / 9 " % F V 3 &@, X ) E3 = N (,O,P)+ # (F) [], 3 K K R?3 ; J E ()?3 (R 3 = X 3 ( X?3 = - Q 6R 6S = 9 _)F D (9) () " \ M { λ?3 ; M { -KH { ;3 9-3?3 ; (-λ )" \ = V& 3 ^I " \ 9 _) ;E 3 ) & (UE _) " % F 3 (F )= ; 3 ]$ _3 " 3 = 9 ;Y, 3 `, O: ) 4 9: a@ 3 % 3 _) _) _3,> []?@ A ( 3! = (E? 3 # ; Z O&K 9 I?@ A > %I [3] ]K -KH ()?3 (R 3 Z 3 & _) # = [(! : )("S W ) : ( W!"# )]# YZ [\] ^ [ C <_^ [] () 3 Ramberg-Osgood hardening law 4 Finite Element Method(FEM) 5 AAQUS 6 ANSYS Isotropic hardening kinematic hardening 4

5 $R V + &'$ ) 9 R: 59 %I 3 M M ) ;, MM) O ',) qk M -M > 4( R 3M M) )9R: # 6 9 )E3> M) E?F 9;Y, ::-JH-)6L/R-4M (F:-)6 L/R L3:-(F ) (mm) 3 5/8 9/3 9/3 9/6 7/ A C D G R T 3 PC 3 ( X qk # " 3! &45 z # : ; ) 3 3 I M) %-, GH3 $ ) D % " T ; 3 >'3 P+% 3 (R > 9 > f ( R 3 ( C! k? ) E?F : 3 ) I D#+ (! k? ) - 3, ) 4 M, ; a FV SDV PH ' 3 = X, P +% ; >E+P+ % T ST MK3 + ) D= X>' $ EF D (USDFLD) ; 3 "3; 33 ) -KH + 3, H, XJ E 3(USDFLD) ; 3, X I fk, ) )E3 ' 33 9:+ E 3;Y, Z ; 3 ; [5] O -L/R-M σ(mpa),> 3S+,-)6 }F 3,) )9R: Q GH$R -5 M ) % I ) ), l K ;Y, 3 %I M (R 3,M) ; 3 I GT-> ", > M - )T ;, O -,/ b -) # 7-LC (3 9) 5 ( )I (F:-)6 L/R )3-- S K:US-5M C : L3 (F: -)6 L/R-6M O > M) E?F 3 z\ E?F ) % M) M $ > M) 3 M) ; E?F 3 & GH3 " \ #? + H F :?@ A,> M) ; " 3 GH3,: ) ) 3 & GH3 & ] (R 3 >' 3 D >', ) E?F G L -3 ) ; ;Y, 8F, (44) -JH CS T-3M GT-7-LC (49 ) 583% p 4) E?F>M ) > 4 E3 3(ASTM E8) 3 Solution dependent state variable(sdv) Predefined field variable 5

6 3 5 σ(mpa) :X:4 L/R L3:-3(F (mm)) 8 A D T 3 = 9&$ &F GH3 Z 79 M) I3 O -JH -)6 : 6 C3 V:4-(F # 36MPa 75 MPa 53 MPa 64 S& M! %$ P M M 3M?@ A,> # = 3 σ(mpa) ( )I )6 )3 Y:3 - S K:US-9M 5 5 ( )I )E3 % M) E?F- M 7F -)6 L/R )3-- S K:US-7M + 7D -)6L/R)3-- SK:US-M :T 5 Z) 3-6 ( R 3,M ) + %I,M) 343 (USDFLD) ; 3 "3 3 P+% & 3' # 3,) )9R: Q ) 9&$ #,M) Q M ) ) 9R: Q 3 4 3,) s 3 I 3 []_4 D! # D$ + H T 4>4 (R 3 ), >' %-, + 3 M,> }3? { E?F ) % A M 99 % M) I O 3 >4 3 E3 3 ) E3 % M) E?F f ;Y, Z 9 'F 3 =?@ A,> ; ; 3 4 3! F E }#$ ;! 3 3 P U3 z = N H + 3 M G + H F 4 O S! ^V ; 3M = N H 3 L 3 3 %!C M) ) T ; 3 %I %-,?@ A > F $ %I H %)U 9R: ) Q + H F M %$ S& M 3 =, :X:4 L/R-8M # 3 ' ) ƒ\r 4! 6

7 $R V + &'$ "P OR 3 3 I ) %-, 33 + # I34 # )?@ OR -Y, ; ; 3 P ' 9&$ 3 Q L - 3 P+ % ) O 3 %I ) D H ) + ; GH3 9: ) O?, 6>4 39 9R: Q F P : ;?@ A,>, ; 3 4 3, & GH3 ) > 3 3@ 3, " U $ M -M =) " U > ; 3K3, > ; X GH3 3, (GH3 % > M) Q # 3 >' X :OH " \ $ ] GH3 : " M, ( M ) 3) ) ; ; 3 G?@ A, > X C 3 F 93F F ) T$ + 3M > ) σ(mpa)!3 P : ; 3 > F ;, 3 > M) > ε ' 9&$ ) 9R: Q 3 > M) ) 3 Q 9) # ˆ Š Œ 9 Q 3 > 3 ( 5>4 + Q [ )*-4(F ƒ ƒ 64 ~ 36 } 75 E (GPa) 7 ʋ 3 s 5 3>, 5>49 a, T + 3M 3 > 3 3, Z M 8 F :4 3 ( +,/ $ Q"-5(F?@ R 3 > 3 # /9 /95 / /5 75E 75E 73E 68E R >?@ 3 # / /44 / /5 /75 / 5!3 ) %-, š œ ž Ÿž 78E 78E 76E 7E 73E 7E 7E 65E ;-?@ 7 8 F :4 3 ( + Q3 M -F K:US- M 7F -)6 Z) 3 )* 3 CQ 8 F :4 3 +,( +,/ $Q"-6(F >?@ R # 3 % 3 Steps 7F -)6+,: R >?@ # % 3 š œ ž Ÿž /6 54E / 7 55E 3E+8 /3 57E / 4 58E 64E /6 58E / 3 6E 38E /8 64E / 9 64E+8 356E+8 /3 65E / 64E 4E /4 / 5?@ ; σ(mpa) F 8 ( :4 3( + Q3M -F K:US- (F: -)6 Z) M 3) 3 = > 3),N ) 3 & ] GH3 %-, T$ 3M F & GH3 $ T ; 3 (E?F X $ 3 3 GH3 + ) %-, + H, { E?F! (E+ 3 " \ 9 ε

8 -K H ^ F T$; E " U $ + T$ ; E,> F > F 9 3F F ; M _ ) $ " : 3 # : ; ;! ) : E 3 (f ( > ) 9)= t + j\@ ; 9U E?F -, ) + H T = X D3?3 ; %)U F 3 t + ; { ; # H }#$ $ }F t + a K3 ; ; ;Y,,@ EF : ) 3 P " P + ; -3 Q ; ;Y, " P 9a J &K,> E % I 3 (##$ ( ##$ ) E3 M : σ(mpa) 74 -)6 L/R )3-- S K:US-4 M -K H + GH L S!, 9' ) =?@ A,> E ;Y, ; 3 G + H!3 " ) ^ V ; V D $ ) M3,> T ; 3 limiter bonora TEST Elongation 5 5 _F 4 ; (##$ -, b ; ER 3 ; " M H ak3 ) *+, -,/ 3 3 ) }#$ ; [6] H 4 3 ) %I H R: Q O )3?@ A > F 3 4 Z 69 = # H) 9&$ ) [5] )?3 ) O 3! ^F ) 9F ' N ) & 3 9: = # (7?3) $ D = E E "! > E i- GH3 3, " U > O 3 = # D i E i (7) &F GH3 " U 7>4 (R 3 = Q ; ) +:33 F +:=>3 )6@ 56: US-7(F ) ' Q >F 93F K, -3 >4 ; Q GH3 3 /36 - /7 /7, = P, ;Y, &' 3 MP+ " \ % + X $ 33 %-, = 3 M,> &F,) EF, (E?F 3 $ " M >' ~fk3 $ " A (R 3 z' ' M 3 ( X E3 M L 3 3 >' $ O { E?F " ) ) + 3 M ), 9$ M _) $ ;K S,3 = { 4 D#+ 3 I M _) = _) C ' 9&$ L 9 I34 ' ) &R+ 3 = J&K # M M ] ; 33 /58 /6 M) ) X M # { M # ; 3 O H;- N ) = ; O! ;- $ 4 9 }3??# ) &R+) D 3 D$ 3 # ]K (E?F { 3M?@ A,> # = 74 -)6 L/R M"S +:3 \<: +,8: -3 M 3M 3 # ; M / 3 F = # M Pattern 8

9 $R V + &'$ D 3) O U 3 /3 L )] GH3 %I,> ;!3 9 3 = K C4-8 K #, X J E 3 USDFLD ; 3 I fk, ) ) E3 ' 3 9 : + E 3 ; Y, qk X O ~@ 3 3 (USDFLD) ; 3 ) I SUROUTINE USDFLD(FIELD,STATEV,PNEWDT,DIRECT,T,CELE NT, TIME,DTIME,CMNAME,ORNAME,NFIELD,NSTATV, NOEL, LAYER, KSPT,KSTEP,KINC,NDI,NSHR,COORD,JMAC,JMAT YP,MATLAYO) C INCLUDE 'AA_PARAMINC' C CHARACTER*8 CMNAME,ORNAME CHARACTER*3 FLGRAY(5) DIMENSION FIELD(NFIELD),STATEV(NSTATV),DIRECT(3,3), T(3,3),TIME() DIMENSION ARRAY(5),JARRAY(5),JMAC(*),JMATYP(*),COOR D(*) user coding to define FIELD and, if necessary, STATEV and PNEWDT RETURN END : (USDFLD) ; 3, X J E FIELD(NFIELD) T?#, X : " STATEV(NSTATV), X : 3 3 9: : " PNEWDT O 3 ) MP ) MP+ DIRECT(3,3) CELENT TIME() CMNAME NFIELD NSTATV NOEL NPT,, : " > 4 t ' >a ) " 9:, )A &:, ) # fk % K #, X E 9: 3 3, X E, E 3)* :4 3 ( )3-- SK:US-6 M :> 5+,TJ)6Z) + 3 M > F C,, Q ) A T$ 3 K3 z # >F 93F 3 = N H zu : U ; 3 M = T K F x r (&' ( X } F C(- [ K F) > + H F 3 % + X ) > ; 33 y 93F G 3M! T$ = / 4 q# " ' 3 ^V ; 3 ;##$ %)U 3!3 P : ; > F N\ +> % -K H S! 9' ) = N H 9 > ), %I 3 (##$ SA &' -7,) # ; Z = E4 +)!4 = j E > 3) % I ' 3 I (R 3 -KH + ) O b ; S!) 583% H 3 3 I,) ; (,9:,*+ A,> )' ) >!C 3 3 I,M) O P +% + H F E? ) 3?@ 3> # ; H%I E?F Q H> '(USDFLD) ; 3 & 3 3 M M ) " 3, 3 I,M), l K ' ) > 3%)U -K H + ) > 3@?@ > $ " GH3 %-, # A,> ; -, + 3M 3!3 F 3,> ; F ) E3 E?F 3 ' M _) L 3 3 %I }#$ ;Y, = 3 M,> ;, = N H,!3 -)3 ) FU3 P A 3E? H! (##$ Tempering 9

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