Improvement on the Modeling of Rate-Dependent Plasticity and Cyclic [PDF]

V ol. 18

No. 1

CH INES E JO U RN A L O F AER ON A U TICS

February 2005

Improvement on the Modeling of Rate Dependent Plasticity and Cyclic Hardening by Bodner Partom Model SH I Duo qi, YANG Xiao guang , WANG Yan rong ( School of Jet Prop ulsion, Beij ing Uni versity of A eronaut ics and A stronautics, Beij ing Abstract:

100083 , China )

An additional isotr opic internal variable is utilized to extend the Bodner P ar to m unified v is

coplastic constitutiv e model ( orig inal B P ) to improve the modeling of rate dependent plasticity and cyclic hardening behaviors o f metals. T he extended model ( new B P) contains two isotr opic internal variables: one plays t he role of repr esenting t he fast hardening in smaller inelastic strain range, while the other evo lutes at slower speed accompanied by lar ger accumulated inelastic deformation, such as cyclic har dening. T o ex amine the validity of the ex tended co nstitutive model, the rate dependent plasticity of a N i base su peralloy U dimet 720L i at 650! and 700 ! are char acterized using bo th mo dels. N ot only numer ical sim ulations ar e co nducted for various loading condit ions by implementing both models into ABA QU S using a user mater ial subroutine, also a systematic comparison between tw o models is completed. Numer ical re sults show that the ex tended material constants in the new model pro vide mor e flexible capability in mod eling the inelastic behavior of the material w ith sound accuracy. Key words:

space pro pulsion; superalloy; constitutive relation; viscoplasticity; internal state v ar iable

对 B P 本 构模拟材料率相关塑性变形和循环硬化的改进. 石多奇, 杨晓光, 王延荣. 中国航空学报 ( 英文版) , 2005, 18( 1) : 83- 89. 摘

要: 对 Bo dner P ar to m 统一黏塑性本构理论( B P 模 型) 进行了 修正, 改善了 其对材料 加载速 率

相关的黏塑性和循环硬化的模拟精度。改进模型包含两个各向同 性硬化内 变量: 一 个用于描 述小 非弹性应变范围内快速的各向同性硬化 , 另 一个以 缓慢的 演化速 率来表述 大的累 积非弹 性变形, 如循环硬化过程中非弹性应变累积。利用 ABA QU S 的用户材 料子程序 将改进模型 编成有限 元计 算代码, 通过对镍基合金 U dimet 720Li 率相 关黏塑 性特征 的本 构模拟 对其 进行了 验证, 并 与 B P 模型作了对比。结果表明改进模型能够以更高精度、 更加灵活 地对材料 的非弹性力 学行为进 行建 模。 关键词: 航空航天推进系统; 高温合金; 本构关系; 黏塑性; 内变量 文章编号: 1000 9361( 2005) 01 0083 07

中图分类号: V252; T G 132. 3 + 2

In the past t hree decades, considerable prog ress has been made in the area of the so called

文献标识码: A

jis and Zolochevsky ( 2000) ( 2000)

[ 12]

[ 11]

, J rvstr t ( 2002)

, Rosakisa, et al

[ 13]

, est ablished dif

unif ied const it ut ive theories and several new or

f erent const it ut ive equations based on thermody

improved thermo viscoplast ic formulae have been

namics and dislocat ion dynamics. T he kernel of t he

proposed and evaluat ed in hig h temperature appli

viscoplastic equations is t hat all observed effects,

cat ions, mainly in components subject ed to t her

such as rat e dependent plast icity, cyclic hardening

mal mechanical loading condit ions. For example,

( sof tening ) , creep and st ress relax at ion, are a re

M roz( 1969) [ 1] , Philips and Wu( 1973) [ 2] , Bodner

sult of t he same int ernal physical processes[ 14] .

and Partom ( 1975) ( 1980)

[ 5]

[ 3]

, Kriege( 1975)

[ 4]

, Drucker and Palgen( 1981)

berg and Yen ( 1981 ) Chaboche ( 1989)

[ 9]

[ 7]

, Dafalias

[ 6]

, Eisen

, Walker ( 1981 )

, Yoshida ( 2000)

[ 10]

[ 8]

,

, Voyiad

T hat is, single or mult iple internal variables could be used t o represent time dependent creep and time independent plast icit y. T he evolut ionary e quat ions of the internal state variables are based on

R eceived dat e: 2004 05 31; R evision received dat e: 2004 11 05 Foundation it em: EU FP5 E3E Disc Lif ing © 1994-2010 China Academic Journal Electronic Publishing House. Open access under CC BY-NC-ND license. http://www.cnki.net

% 84 %

SHI Duo qi, Y ANG Xiao guang, WAN G Yan rong

CJA

their microstructure evolut ion, which are regarded

some modification to t he original Bodner Part om

as being a bet ter approach of t he governing physical

theory t hat can accurately charact erize the mechan

processes than the classical plastic and creep t heo

ical behaviors. A systemat ic comparison of t he model and their propert ies are made in the vis

ries for those complex loading condit ions. T here f ore, the unif ied met hod is ex pected to have superi or capabilit ies for prediction of nonlinear responses of material, especially at elevat ed temperature. Bodner Partom model uses plast ic w ork to de

coplastic framework and for t he monotonic tension and t ension compression cases. It is found t hat t he ext ended mat erial const ants in more t han one isot ropic hardening rule provides more f lexible ca

scribe t he materials resist ance to plast ic flow rat her

pabilit y in modeling the inelast ic behavior of t he

than inelast ic strain, w it hout an explicit yield crite

mat erial w ith sound accuracy .

rion. Since 1970∀ s, the theory was w idely used in modeling the const it ut ive characterist ics of various coppers and superalloys and w as made great prog ress, especially in NASA HOST pro g ram [ 15 17] . T here are many applicat ions using ba

1

Numerical Simulat ion by B P M odel T he Bodner Part om t heory is consist ed of vis

coplastic flow rule and the evolut ionary equat ions of

sic Bodner Part om t heory in const it ut ive modeling

internal state variables. An ex plicit y ield criterion is not required. T he basic const it ut ive equations are

of v arious engineering materials, such as RMI

as f ollow ing [ 15, 20, 23]

[ 3]

[ 18, 19]

40 , annealed OFHC copper , Ni based su peralloy B1900 + H f[ 17] , H ast elloy X and alu [ 20]

m inum alloy 8009

[ 21]

, INCONEL 690

, OFE

[ 22]

copper and 70#30 brass . Most of t hese works w ere mainly focused on the w ork hardening charac

ij ij , in

=

ij , e +

= D 0 ex p -

( 1)

ij , in

1 ( Z I + Z D) 2 3J 2

2

n

Sij ( 2) J2

Z I = m 1 Z 1 - Z I ( t ) W p( t ) A 1 Z1

t erist ic of B P model for monotonic loading . F ew

r1

ZI( t ) - Z2 Z1

( 3)

paid att ention t o the mechanical responses for cyclic

Z D ( t ) = !ij ( t ) uij ( t )

loading and creep loading. Generally, t he def ormat ion of superalloy used

!ij ( t ) = m 2 Z 3 uij ( t ) - !ij ( t ) W p A 2 Z1

in turbomachine is very small, viz. < 1 0% , and it is dif ficult t o distinguish elast ic and plast ic com ponents. Hence, from t he view point of lif e predic t ion of t urbine components, the st ress st rain be haviors under cyclic and t ensile deformation condi

uij ( t ) =

( !kl !kl ) Z1

1/ 2

r

( 4)

2

vij

( 5)

∀ij ( t ) !ij ( t ) ; v ij ( t ) = ( 6) ( ∀kl ∀kl ) 1/ 2 ( !kl !kl ) 1/ 2

Here, W p = ∀ij ij , in is t he inelast ic work. T he ini t ial condit ion is Z I ( 0) = Z 0 and !ij ( 0) = 0. Physi

t ion has to be described precisely by advanced con st it ut ive equat ion, because t he subsequent creep de

cal meaning of mat erial constant s D 0 , Z 0 , Z 1 ,

f orm at ion of t he component s is evaluated based on

discussed and could be obtained throug h an opt i mum program in Ref. [ 23] . For simplicity, t hese

the calculated st at e of both stress and strain. How ever, numerical simulat ion of the m echanical be havior of a ∃ st reng thened Ni base polycrystalline superalloy U dimet 720L i revealed t hat in the origi nal Bodner Part om unified const itutive t heory, on ly one isot ropic hardening variable w as limit ed in

Z 2 , Z 3 , m 1 , m 2 , n, A 1 , A 2 , r 1 and r 2 , w ere

basic set of material paramet ers is reffered to as old B P const ants. T he simulat ed results for Udimet 720L i are show n in F ig . 1 and F ig . 2. It is observed t hat acut ely t ransient nonlinear

describing t he diff erent mechanisms of hardening

change occurs at t he knee of the stress strain curves, and t he original B P model overpredicts t he

under monotonic loading and cyclic loading [ 23, 24] .

hardening st ress range for some initial cycles, but

T herefore, an object ive of t he present st udy is t o consider improving possibilit ies that are offered by

gradually approaches to complete shakedow n of cyclic hardening, w hich is in cont rast to the exper

© 1994-2010 China Academic Journal Electronic Publishing House. Open access under CC BY-NC-ND license. http://www.cnki.net

February 2005

Improvement on t he M odeling of Rate Dependent Plast icity and Cycl ic Hardening by Bodner Partom M odel

iment al t rend, F ig . 2.

% 85 %

t onic loading case, the w ork hardening rate versus stress curves const ruct ed from tensile t est curves reveal t hat t he material hardening consists of tw o kinds of mechanisms: isot ropic and kinem at ic hard enings[ 25] . If only this t ype of loading is const itu t ively modeled, the ex cellent agreement bet ween calculated and ex perimental dat a can be yielded by the original B P equat ions w it h one isotropic vari able and one kinemat ic variable. H ow ever, It is not suff icient t o service structures w orking under com plex t hermal mechanical loading condit ions. T he

( a) T = 650 !

old B P const ants give not only sharp t ransit ion of t he t ensile st ress st rain figures, but also unsat is f ied result s of t he cyclic hardening w hen bot h tw o loading cases are considered. T he basic reason is that isot ropic hardening under sym met ric cyclic loading evolutes slow ly cycle by cy cle, which is complet ely dif ferent f rom that fast one under monotonic tensile loading.

2 2. 1

Improvement of the B P M odel

Additional isotropic hardening rule

F rom above st udy, one can f ind that only one isot ropic variable plus one kinematic variable is not ( b) T = 700 !

Fig. 1

Simulations of rate dependent tension curves by old B P constants

suff icient t o characterize accurat ely t he responses of monotonic and cyclic loadings simult aneously. It is necessary that some improvements of the const itu t ive model are needed to elim inat e the def iciency. One t ype of t he modifications of B P model was [ 22]

proposed by Khen and Rubin ( 1992) and veri [ 26] f ied by Bodner and Lindenf eld ( 1995) , Kroupa [ 27]

and Bartsch ( 1998) in w hich t he param eters m 1 and m 2 are considered to be t he funct ions of isot ropic variable and kinemat ic variable, respec t ively. How ever, t he physical meaning is not so clear

Fig . 2

Simulation of cyclic hardening curves by old B P constants

T he authors think t hat the shakedow n is

[ 22]

. In t his paper, it is considered t hat t he f eature

of hardening in cyclic loading is dist inct ive from that of monotonic loading and should be represent ed by diff erent variables. T hat is, the isot ropic state variable of B P model in Eq. ( 2) could be

mainly caused by the incorporat ion of the flow rule w it h only one isot ropic internal variable describing

const it ut ed by tw o components, ZI = ZI, a + ZI, b

both t he f ast and the slow hardenings. In mono

both Z I, a and Z I, b obey t he same evolut ionary rule

© 1994-2010 China Academic Journal Electronic Publishing House. Open access under CC BY-NC-ND license. http://www.cnki.net

( 7)

% 86 %

SHI Duo qi, Y ANG Xiao guang, WAN G Yan rong

including hardening, dynamic recovery and st at ic Z I, a = m 1a Z 1a - Z I, a ( t ) A 1aZ 1a

Z I, a( t ) - Z 2 a Z1 a

Z I, b = m 1b Z 1b - Z I, b ( t ) A 1bZ 1b

Z I, b ( t ) - Z 2b Z 1b

one Z I, a for larger deformation. Table 1

thermal recovery term r 1a

( 8) W p( t ) r

1b

( 9)

T he equat ions: Z I, a ( 0) = Z 0a and Z I, b ( 0) = Z 0b , are initial conditions. T he direct ional variables Z D and !ij ( t ) keep the same f orms as Eqs. ( 4) and

The new B P constants f or Udimet 720Li m 1a /

m 1b /

650

M Pa- 1 0. 0166

M Pa- 1 6. 626

717. 7

982. 8

841. 1

700

0. 0392

3. 334

879. 1

1193. 9

943. 4

T/ !

Z 1b /

Z3 /

m 2/

Z2a /

Z 2b /

M Pa

M Pa

M Pa- 1

M Pa

M Pa

650

1170. 0

372. 3

0. 140

700. 0

800. 0

700

1496. 5

387. 3

0. 152

507. 0

1260. 0

T/ !

n

r

A/ 10- 5 s- 1

E/ M Pa

650

2. 52

0. 81

4. 2

188 717

700

1. 87

0. 65

8. 8

182 345

T/ !

W p( t ) -

CJA

Z 0a / M Pa

Z 0b / M Pa

Z 1a / M Pa

( 5) . T he first state variable Z I, a can be used t o charact erize the slow er hardening behavior under

2. 3

cyclic loading, and the second one Z I, b can be used in monot onic loading, see Fig. 3.

T o ex am ine t he validity of t he ex tended con st itut ive model and to apply it t o complex struc

Simulations

t ures, numerical sim ulat ions are conduct ed for vari ous loading condit ions by implement ed it int o ABAQU S. T he result ing f it s for t he calculat ed and experimental monot onic t ensile stress strain curves from t he slowest to the f ast est st rain rates by new const ants are show n in F ig . 4. It is not ed that t he inclusion of new isot ropic variable at bot h t emperat ures in the analysis is im port ant since otherw ise t he calculat ed curves w ould be sharper aft er t he init ial yield reg ion compared t o F ig. 3 Evolut ion of isotr opic variable for the mo dified rule

In summ ary, tw elve material constant s in t he

Fig. 1, which is the int erest st rain rang e f or high temperature applicat ions. It is also indicat ed t hat:

original B P model are t ransformed int o Z 0 a , Z 0b ,

& in t he reg ion of smaller inelast ic st rain, kine

Z 1a , Z 1b , Z 2a , Z 2b , m 1a , m 1b , Z 3 , m 2 , n,

mat ic component plays t he main role of hardening and saturates f aster, w hereas as t he inelast ic strain

A

1a ,

r 1a , A 1b and r 1b in modified B P equat ions.

It is convenient to set A 1a = A 1b = A and r 1a = r 1b = r . T hese paramet ers are referred to as new

ascends, isotropic component begins to be domi nant; ∋ t he new B P constant s yield bet ter pre

B P constants. T he increased number of int ernal v ariables should allow much more flex ibility in rep resent ing the viscoplastic, t ime and t emperat ure dependent responses, w hich w ill be discussed later. 2. 2

Material constants

Applying the optimum algorithm proposed in Ref . [ 23] , t he new B P const ants f or U dimet 720L i are listed in T able 1. It can be seen that m 1b is much larger in quantit y than m 1a at bot h temperatures, w hich means that Z I is composed of a f ast er component Z I, b for smaller inelast ic def orm at ion and a slower

( a) T = 650 !

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February 2005

Improvement on t he M odeling of Rate Dependent Plast icity and Cycl ic Hardening by Bodner Partom M odel

% 87 %

F rom Fig. 1 t o Fig. 5, it implies that

new

B P const ants have more flex ibility t han t hose of old B P const ants in modeling monot onic tension and cyclic hardening behaviors simultaneously. At these tw o temperatures, t he new B P constant s produce less cyclic hardening. T he reason is that the first isot ropic component Z Ia m ainly t akes ef f ect on the cyclic hardening at slower evolutionary rate w ith smaller m 1a . On t he other hand, anot her component Z Ib goes great ly fast er during tensile loading w ith a larger speed of m 1b . Anyw ay, t he ( b) T = 700 !

modif ied forms of the equat ions yields bett er agree

F ig. 4 Simulations of rate dependent tension cur ves

ment than t he basic form to calculat e monotonic tension and cyclic hardening for Udimet 720Li at

by new B P constants

dict ion in t he transformed region of t he curves be

650 ! and 700 ! .

cause of t he addition of t he independent com po

T he predict ed and ex perimental cyclic st able

nents of isot ropic hardening variable. F ig . 5 shows the properties of cyclic hardening

hyst eresis loops of U diment 720Li subject ed t o

of Udimet 720L i f or st rain cont rolled sy mmet ric

symmet ric cyclic loading at 650 ! and 700 ! are compared in Fig. 6 and F ig . 7. Both old and new

test s at 650 ! and 700 ! . T he model curves exhib it t he basic trends of the experimental data and

of t he cyclic loops in w hich isotropic sof tening phe

demonst rat e good predict ive capabilit ies at both

nomena of t he material are observed. It mig ht be

temperatures. It is indicated that the cyclic harden

feasible t o int roduce some other improvements int o the direct ional hardening rules.

ing develops cycle by cycle, t hat is, stress range #∀ goes up as cycle number increases. T he evolu

const ant s are unable t o generat e sat isf ied outcome

t ion w ill stop at a cycle that hardening is sat urat ed. It is also show n that t he st able cycle number de creases at elevat ed temperature maybe because t he mat erial resist ance t o plast ic flow has been furt her w eakened at higher temperature.

F ig. 6 T he stable cyclic loo p of U dimet 720 L i at 650 ! ( = 10 - 3s- 1 , # = ( 0. 8% )

For creep simulat ion, t he

new

B P con

st ants y ield t he same prediction as t he old ones, see Fig. 3 in Ref. [ 23] . It indicat es t hat the t heory cannot model t he t ert iary creep t hat is charact erized Fig . 5

Simulation of cyclic hardening curves by new B P constants

by rapid increase of t he creep st rain rat e and mate rial damag e accumulation. T he reason is t hat t he damage is not t aken int o account.

© 1994-2010 China Academic Journal Electronic Publishing House. Open access under CC BY-NC-ND license. http://www.cnki.net

% 88 %

SHI Duo qi, Y ANG Xiao guang, WAN G Yan rong

CJA

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[ 2]

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[ 3]

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[ 4]

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( = 10 s

, # = ( 0. 8% )

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[ 7]

3

viscoplasticit y [ J ] . Journal of Applied M echanics, 1981, 48

Conclusions

F irst ly, the modeling capabilit y for monot onic

( 6) : 276- 284. [ 8]

pendent constit utive relat ionships [ R ] . N ASA CR 165553,

discussed using original B P model. Secondly, a

1981. [ 9]

1989, 5: 247- 302. [ 10]

v arious loading conditions. T he results of t he pre sent study can be summarized as follow s:

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[ J] . Journal of t he M echanics and Physics of Solids, 2000,

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( 3) T o examine the validity of the ext ended

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duct ed for various loading condit ions and bet ter re sult s are achieved.

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[ 14]

loadings at elevated tem perature. const it ut ive model, numerical simulations are con

R osakisa P. A thermodynamic internal variable model f or the part it ion of plast ic w ork int o heat and stored energy in m etals

( 2) T o improve the descript ion of rat e depen

hardening mechanisms f or monotonic and cyclic

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pressing t he dif ferent mechanism hardenings under

t ional isot ropic variable is int roduced to ex tend t he

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[ 11]

( 1) T he original B P model is limit ed in ex monotonic loading and cyclic loading.

Chaboche J L. Const itut ive equat ions f or cyclic plasticit y and cycl ic viscoplasticit y [ J] . Int ernat ional Journal of Plast icit y,

lidity is ex am ined through numerical simulations of the viscoplast ic def orm at ion of Udimet 720L i under

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and st rain cont rolled symmetric cyclic loading is modif ied form of t he model is proposed, and its va

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1985. [ 16]

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Acknow ledgments

1986. [ 17]

T he aut hors thank Dr. Joachim . Villw ock

t hermal recovery in met als [ J ] . Journal of Engineering M at e rials and Technology, 1988, 110: 1- 8.

and Dr. Vladislav. L evkovit ch of Rolls Royce Deut schland Lt d. f or t heir kind helps in the discus sion about t he experiment s and the modeling.

Chan K S. Phenom enological model ing of hardening and

[ 18]

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© 1994-2010 China Academic Journal Electronic Publishing House. Open access under CC BY-NC-ND license. http://www.cnki.net

February 2005

Improvement on t he M odeling of Rate Dependent Plast icity and Cycl ic Hardening by Bodner Partom M odel

100: 388- 394. [ 19]

[ 26]

Bodner S R, Partom Y, Partom I. U niaxial cyclic loading of

[ 20]

[ 21]

residual st ress and st rengt h of a tit anium matrix composit e af t er thermomechanical fatigue[ J ] . Composit es Part B, 1998,

[ J] .

29B: 633- 642.

Journal of Engineering M at erials and Technology,

1996, 118: 19- 27.

Biographies:

V enkatesh V , Rack H J. Elevated t emperature hardening of

SHI Duo qi Born in 1975, he received his do ctoral degree fro m the School of Jet Pro pulsion, Beijing U niv . of Aero. & Astro ( BUA A) in M arch, 2004 and began his postdoctor al research in the institute of solid mechanics ( BU AA) . His re search interests include mechanical behavior of mater ials, heterogeneous solid mechanics, numer ical optimizatio n, life evaluation of eng ineering structures. T el: ( 010) 82316362, E mail: shdq@ buaa. edu. cn YANG Xiao guang Born in 1962, he is the professor of School of Jet Propulsio n in BU AA . T he r esearch inter ests in volve constitutive theor y, fatigue and fr actur e, ther mal bar r ier coating ( T BC) , CAD/ CAE. T el: ( 010) 82316362, 82317787, E mail: yx g@ buaa. edu. cn WANG Yan rong Born in 1965, he is the professor of School of Jet Pr opulsion in BU AA . His major r esearch inter ests include flow induced vibr atio n in turbomachinery, fa tig ue and fracture of components at elevated temperature, CAE and reliability eng ineering. T el: ( 010) 82316362, E mail: yr wang @ buaa. edu. cn

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