Fatigue Damage Calculated by the Ratio-Method to Materials and Its Machine Part

V ol. 16　 N o . 3 　　　　　　　　　　　　 CHIN ESE JO U RN A L OF A ERO N A U TICS　　　　　　　　　　　　　 A ugust 2003

Fatigue Damage Cal culated by the Ratio-Method to Material s and Its Machine Part YU Yan-gui ( Inf ormation Science & E ngineering College, W enz hou Universit y , W enz hou　325027, China) Abstract: 　 Several new calculating equatio ns on the dam ag e-evolving r ate ar e sug gested for describing the elastic-plastic behavior of some materials under un -sy mmetric cy clic loading . A nd the estimating form ulas are giv en of the life relativ e to v aried damag e value D oi at each loading history . T he method is to adopt the r atio of plastic str ain range to elastic str ain range as t he stress-strain parameter, using the staple material constants as the material par amet er s in damage calculating ex pression . A nd it g ives out a new concept of the compositive material constant , that has a functional r elation with the staple material constants, averag e str ess, av er age strain and cr itical loading time. In addition, it calculat es fatigue damage as ex ample for a par t o f car, its calculating r esults are accor dant with the L andg raf' s equation , and calculating pr ecision is more r igo rous, so could avoid unnecessar y fatig ue tests and will be o f practical sig nificance to st int times , manpow er and capitals , and to prov ide conv enience for engineering applications. Key words: 　unsymm etr ical cycle; r atio ; staple material co nstant; damage evolving rate; life 用比值法计算材料及其机件的疲劳损伤. 虞岩贵. 中国航 空学报( 英文版) , 2003, 16( 3) : 157- 161. 摘　要: 提出了在非对称循环加载下描述材 料弹塑性材料行为 的损伤演变速率方 程式与各个历程 与不同损伤量 D oi 相对应的寿命 N o i 估算式。 其方法是采 用以塑性应变幅同弹 性应变幅之比值 e

p

/

作为应力应变参量, 以常用的材料常数作为 材料参数。而且, 还提出 了与常用材料常数、平均应

力与平均应变、加载临界历程长短有着 函数关系的综合性 材料常数的新概念。此外, 以汽车的某一 零件为例, 计算了它的 疲劳损伤。其计算结果与 L andg raf 方 程式计算结果 一致, 且 计算精度 较高。 这对避免过多而重复的疲劳试验, 对方便工程应用, 对节 约人力、 时间和资金有着实际意义。 关键词: 非对称循环; 比值法; 常用材料常数; 损伤演变速率; 寿命 文章编号: 1000-9361( 2003) 03-0157-05　　　中图分类号: Q 346. 5　　　文献标识码: A

　 　 Numerous scientists have sugg est ed various 2

=

e

2E

+

2k

p ′

1/ n′

( 2)

kinds of calculat ing ex pressions on fatigue damag e of st ruct ures and m at erials, w hich include t he

T he special f eat ure of t hese equations is that t hey

Dow ling 's equat ion, the energ y equation, t he

had all used the fatigue st rengt h coef ficient

′ f

, t he

′ 1

L andgraf' s equation, t he equation of local st ress and strain and so on . T hese works have done valuable contribut ions t o ex periment al research and en-

fat igue st rength exponent b , the fatigue duct ilit y

≠ 0 t he

ing exponent n . And t hese material constant s had

gineering application. L andgraf' s equat ion 1 D= N = 2

[ 1]

And f or

m

is as follow s ′ f

′ f

E

e

(

′ p f ′ f

-

′

′

cyclic st rengt h coeff icient K and t he strain harden′

been always accepted and applied ex tensively in ′ 1

′ 1

1/ ( b - c ) m

′

coef ficient f , the fat igue ductilit y exponent c1 , t he

)

( 1)

And t he equat ion of local stress and strain is in t he follow ing form ,

Received dat e: 2002-09-16; R evision received dat e: 2003-05-05 A rt icle U RL : ht t p: / / ww w . hkx b. n et . cn/ cja / 2003/ 03/ 0156/

each eng ineering domain. But these equations do not include any concret e physical param et ers show ing struct ure damages ( for inst ance, t he damag e param eter D , t he micro -crack size a) . On t he ot her hand, many other scient if ic w orkers have sug-

Y U Y an-gui

・ 158・

CJA

gest ed t he damage evolving equat ions in connect ion

In order t o use curves ex plaining the new

w it h the damage parameter D and a in m odern f atigue damage discipline. Murakami[ 2] and . .

damage evolving rat e and relat ing life ex pression sugg est ed by the ratio-met hod, here it g ives t he

!∀#∃%&∋ had

[ 3]

proposed t he small crack initiat ion

t wo directions double logarit hmic coordinate sys-

and propagation problem s corresponding t o t he small crack size a.

tem and t wo direct ions curv e ( Fig. 1) . It is well

da/ dN = B( da/ dN = C(

m

( 3)

n

( 4)

) a　( f rom Ref. [ 2] ) p

) a　( f rom Ref . [ 3] )

Good qualities of t hese equations are t he presence

know n t hat for m at erials w it h elast ic st rain component larg er t han plast ic strain component ( e > p ) , t he relat ion bet w een st rain and life ( e / ( 2 2N oi ) ) is present ed by t he negat ive direct ion coor-

of damage paramet ers having specific physical meanings. How ev er t hese equat ions include new

dinate w it h curve 1 ( A 1B A , f or local

material constant s B , C , m , n, and so on, w hich are not used by general m at erial handbooks in t he

betw een damage evolving rate and damage st ress fact or range ( dD / dN - H ) is presented by posit ive

near future. T heref ore t he new equat ions have not been applied ex t ensively in engineering now , because it m ust st ill prosecute more t ests, unt il t hese

direct ion coordinat e with curves 1 and 2. Its equation is that[ 4, 5]

material const ants are quit e reliable. In t his w ay, it has t o giv e yet a great volume of t he manpower ,

w here H is def ined as the damag e stress f actor

materials and bankrolls. On t he ot her hand, in these equations of a good many, t hey have been selected as the loading parameter only using a single stress paramet er e or strain parameter p to

curve 2( A 3B 1A 2, f or local

range,

H=

staple const ant s

′ f

′

, b1 ,

′ f

′

′

′

, c1, K and n , and f inds there are funct ion relat ionships. T heref ore, t he author has suggest ed new damag e evolving equations for describing elastic-plast ic behavior of m at erial, w here bot h the above m at erial const ants used by engineering f ar -rang ing applicat ion, and t he damage parameter D ( or a et c. ) are adopt ed. And it is used as t he st ress -st rain paramet er w it h t he p to elastic strain rat io of plastic strain range range e . It is conceivable t hat works of t he au-

thor maybe av oid unnecessary f at igue test s and w ill be of pract ical significance t o stint times, manpow er and capit als.

= 0) and

≠0) , and t he relation

m ′ dD/ dN = A ′ 1 H 1 = A 1(

) m1 D ′

1/ m

( 5) ′

, m 1 = - 1/ b1 , and A 1 is a composite const ant of a material. But w hen t he influence of average st ress ( m ≠0) must be t hought and should be correct ed as follow s( curve 2) , it beD

1

comes

calculat ing f at ig ue damag e, so it is not comprehensive, neither is t here nicet y. T he aut hor of t he present paper st udies, again and again , the new material paramet ers B , C, m and n wit h t hat six

m

m

′ A′ 1 = 2[ 2 f ( 1 -

( lnD

m ac

m

- m 1 / ′ f) ]

- lnD 0 )

( 6)

m ac

w here D is a damage value corresponding to mac macro crack form ing size a f or a material of m ac m ac specim en , as a = 0. 7-1. 0 mm , D = 0. 7-1. 0 ( f or

m

≠0, at point A 2 or lg ( f -

m

) / E ) . T he

D 0 is a baseline damage value corresponding t o t he micro-crack forming size. And each history life N oi relat ed t o varied dam age value D oi can be f ound by int egral of Eq. ( 5) ) - m 1/

N oi = ( lnD 0i - lnD 0 ) ( ′

2[ 2 f ( 1 -

′

m

/ f) ]

- m

1

( lnD

mac

- D0)

( 7)

While t he curve 2 ( A 2B 1 A 3 ) presented by t he negat ive direct ion coordinate syst em ( / 2-2N oi ) in Fig. 1 just is corresponding to Eq. ( 7) . On t he other hand, curves 3 and 6 ( F ig . 1, 6BC 1 ) show t he mat erial behavior f or plast ic strain t o be t he

1　Dam age Calculations for U n-symm etric Cyclic L oading 1. 1　Two directions coordinate and two

main com ponent ( p >

e

) ; its equat ion of t he dam-

ag e evolving rate for describing the posit ive direc[ 4, 5]

tion curve is

′

directions curve

m1 dD/ dN = B ′ 1 I = B′ 1

′

m1 p

D

( 8)

Fat igue D amage Calculated by the R at io-M et hod t o M aterials and It s M ach ine Part

A ugust 2003

・ 159・

T hus, the est im at ing expression of varied hist ory lif e N oi can be derived f rom Eq. ( 11) under unsymm et ric cyclic loading N oi = ( lnD 0i - lnD 0) ( ′ 1

2[ (

-

m

)/(

′ f

-

p

′

/

e

′

m

′

) - m1 m1 / ( m 1 - m 1 ) /

′

) ] m1 m1 / ( m 1 - m1 ) ( lnD mac - lnD 0) ( 12)

And the curve 5 ( C 1B C) present ed by t he negat ive direct ion coordinat e syst em ( / 2-2N oi ) in Fig. 1 just is corresponding to Eq. ( 12) . Assuming that ( f or

m

≠0)

C * = 2[ (

′ f

-

m

( lnD

m ac

′

′ f

)/(

-

m

′

) ] m1 m1 / ( m 1- m 1 )

- lnD 0 )

( 13)

And it should be point ed out that t he influence of averag e strain is lesser , so it also may be assumed t hat m = 0, C * = 2[ ( Fig . 1　 T w o direction cur ves

rect ed as f ollows ′ 1

w here C

*

B = 2[ 2 ( 1 -

m

′ f

/ )]

- m′ 1

( lnD

m ac

- lnD 0)

′

′

2[ 2 f ( 1 -

m

/ f) ]

- m

1

( lnD

′

p mac

) - m1 / - D 0)

( 10)

1. 2 　 New damage evolving rate equations and l ife estimation expression by the ratiomethod From Eq. ( 7) it can derive the , and f rom Eq. ( 10) it can also derive the p. As studied and proved again and again , t he rat io p / e can be used as calculating damage. So one can obt ain ′ ′ p f( 1 m/ f ) p/ = = ′ ′ eE f( 1 m / f) m - m′/ ( m m ′ ) 1

1

1

1

( 2N oi )

- ( m - m ′) / ( m m′) 1

1

m ac

′

) / f′] m1 m 1 / ( m1 - m1 )

- lnD 0 )

( 14)

w hich is t he f unct ional relation wit h f , f , m , m , ′ mac m 1, m 1 , and ( D - D 0) . T herefore , t he damag e evolving rat e equat ion t o synthet ically consider elastic-plastic behavior of mat erials is dD/ dN = C (

( 9) mac ′ Here t he D is a damag e value of corresponds t o f ( at the point c1 ) . According t o the same method ment ioned above, t he est imat ing ex pression of each hist ory life N oi relat ing t o t he varied damag e value D oi should be as follow s N oi = ( lnD 0i - lnD 0) (

m

is t he compositive m at erial constant ,

*

′ f

lnD oi - lnD 0 m ac lnD - lnD 0

-

( lnD

w here the I is def ined t o be t he damag e strain ′ 1/ m 1 ′ ′ fact or, I = pD , m′ 1 = - 1/ c1 , B 1 is also a composite const ant of material, f or t he p > e ; considering t he influence of average st rain( m ≠0) , the composit e const ant of a mat erial should be cor-

′

′ f

p

/

)

m m ′/ (m - m′) 1

1

1

1

D

( 15)

Here has been propounded the new equat ion ( 15) of damage evolving rat e f or describing elast ic -plastic behavior of some metallic mat erials under unsymm et rical cyclic loading , just corresponding to t he curve 5 ( C B C 1 ) shown by the positive direction coordinate syst em ( dD / dN / - / 2) in Fig . 1. It w ill be seen f rom this t hat if t he varying course of t he damage values D oi is not t hought upon varied loading hist ory , and the liv elong process ( D oi = mac

D ) is only considered from the baseline damag e value D 0 t o m acro-crack f orming for a component , t hen it can be assumed in Eqs. ( 12) and ( 15) lnD oi - lnD 0 = lnD m ac - lnD 0 ( 16) ′

′

′

And according to m 1 = - 1/ b1 and m 1 = - 1/ c1 , such, t he equat ions ( 15) and ( 1) w ill be in f ull accord.

2　An Example ,…

A part in a car is made of rolled st eel. It s

( 11)

curves betw een nominal stress and time and be-

1 1

Y U Y an-gui

・ 160・

CJA

tw een local st ress and st rain are as F ig . 2. When it is loaded, one can calculat e the local st ress and strain. Here it adopt s t he Neuber' s equat ion ( 17) and the equat ion ( 2) of cy clic st ress-st rain t o calculat e . = K 2 ( s) 2 / E

( 17)

w here t he s is a rang e of t he nom inal stress; K is a ef fect ive coeff icint of stress concent rat ion . E is an elastic modulus. On t he other hand one can also calculat e f or it s damage to Fig. 2 w hich is by t he L andgraf' s equat ion ( 1) and t he ratio-met hod-

equat ion ( 15) and ( 12) . T hen t he calculat ing results of variedmethods put in are T able 1, so that it can compare expediently.

Table 1　The contrasts of varied methods to damage calcul ation

Calculative dat a

Calculation of st ress and st rain

C alculat io n o f damage C um ulate damage

Life, B , N E quation ( 12)

M aterial Rolled steel Range of nominal stress

s/

!

MPa

320

0. 67

so 1 / MP a 395. 5

E / M Pa K / M Pa 192000

n

1125. 9

s12 / MP a 699

0. 193

f / MPa

935. 9

b1

K 2. 6

s56/ MPa

521

791

434

240

s67 / MPa 656. 7

Calculating results

( 2)

/ MPa 780 520

0. 0122 0. 0038

m / MPa - 21. 7 3. 8

1-4-7

( 17)

910

0. 024

3. 3

1-4-7

m1 2. 22

s45 / MPa

2-3-2 5-6-5

2-3-2 5-6-5

c1 - 0. 07

s14 / MPa

Num ber of equations

m

Ca lculat ing results

D 1 = 1/ N = 3. 03×10-4 D 2 = 1/ N = 3. 565×10- 6 ( 1) D 3 = 1/ N = 1. 835×10- 3 D = D 1 + D 2 + D 3 = 2. 142* 10- 3

p

- 0. 0017 - 0. 0047

By the Landg raf s equation number of equations

f

0. 26

s23 / MPa

Cy clic kind

Cy clic kind

m1

- 0. 095 10. 526

0. 0041 0. 0027

0 0. 0183 By the ratio-method -equation

0. 0047

N umber of equatio ns

Calculating results

( 15) ( 15)

( dD / dN) 1 = 3. 43×10 - 4 ( dD / dN ) 2 = 3. 38×10-6

( 1) ( 1)

e

0. 0082 0. 0010

( 15) ( dD / dN ) 3 = 1. 801×10-3 dD / dN = ( dD / dN ) 1 + ( dD / dN ) 2+ ( dD / dN ) 3 = 2. 147×10-

B= 1/ D = 466. 98( Number of cyclic loa ding seg ment)

3

B = 1/ D= 465. 7( Num ber of cyclic loading seg ment)

N = h B . T he h is a t im e of per cyclic loading seg ment.

3　Discussions T he peculiarities of these equat ions suggested in t he present document consist in: ( 1) It gives definit ely the calculating damag e evolving rat e Eq. ( 15) and t he life ex pression N oi Eq. ( 12) by corresponding varied damag e values D oi at each hist ory. ( 2) it has sug gest ed a new concept of t he compositive mat erial constant C * that has a f unctional relat ion with t he staple material constant s f , ′ b′ 1 , f , c1 and averag e st ress

m

, averag e strain m ,

m ac and critical loading hist ory ( lnD - D 0) . ( 3) T he damage paramet ers D in each equa-

tion f or calculat ing damage rat e Eq. ( 15) and varied hist ory lif e N oi Eq. ( 12) may be y et converted int o other physical paramet ers besides t he calculat-

ing w ay ment ioned above. It should and could concret ely describe the dam age about a mat erial ( for ex am ple, using a micro -crack size a, a cont raction of area, etc. ) , and its converting w ays and m eans are sim pler. For inst ance, when the D is converted by a sm all crack a, Eqs. ( 5) , ( 8) can be such as Eqs. ( 3) and ( 4) . Here ordaining t hat D 0 < D ≤ mac

, D

mac

= 0. 7～1. 0, it s unit is a value of no-dim ac m ac mension . And a0 < a≤ a , a = 0. 7～1. 0 m m , it s unit is a millim et er. So D and a can be t hought D

equivalent . ( 4) It suggest s in Ref. [ 6] t hat t here is a consanguineous relation bet ween the dam ag e parameter D and the contract ion of area of a specimen, w here the D is defined as D = 1-

Af A fd

( 18)

w here t he A f and A fd are the sectional areas of stat i-

Fat igue D amage Calculated by the R at io-M et hod t o M aterials and It s M ach ine Part

A ugust 2003

cally tensional f ract ure for un-damaged or damaged specim en respect ively . In all appearance, Eq . ( 18) is only subst it ut ed into the equat ions of damag e rat e or life, so it is not dif ficult to derive out t he various formal convert ed ex pressions.

ram eter ( or a or ∀, or A fd , etc. ) . T heref ore it w ill be of practical significance to st int ex periment al t imes, m anpow er and capit als. References [ 1] 　 王德俊. 疲劳强度设 计理论 与方 法[ M ] . 沈阳: 东北 工学 院, 1991. 147- 153.

( 5) T he calculat ing results f rom T able 1 can also be seen that it is alm ost coincident al by the ratio-met hod-equat ions and t he Landgraf' s equat ion. But calculating precision by t he ratio -met hod -e-

Wang D J. Th e t heory and met hod of design on f at igue st rengt h [ M ] . Shenyang: t he Publishing Company of N ort heast U niversity, 1991. 147- 153. ( in Chinese) [ 2] 　 M urak ami Y , Harada S. Correlat ions am ong grow t h law of small cracks , low -cycle f at igue law and applicability of

quations is m ore rigorous, because it has considered t he influence f or

m

M iner' s R ule[ J ] . Engineerin g Fract ure M echanics, 1983,

≠0 and calculated by Eq. *

!∀#∃%&∋

, ( )∗+∃)! , −. −∋!%&. .!∀∃/0∗∋&1 & ∀!.)&2&1

34 2!%∃4 2∋∃#∃∀!.∀35∗∋&1 ) 6∗∀%&2∋78 4 2!%18[ J] . 9∀∃:%∗;7 9∀∃<∋∃4 2&, 1990, ( 4) : 12- 21.

4　Conclusions ( 1) F or un-symmetric cyclic loading, the equation ( 15) of dam age evolving rat e and it s lif eest im at ion -ex pression ( 12) calculat ed by t he rat io method are full coincident al w it h t he L angraf ' s equation ( 1) under lnD oi - lnD 0 = lnD

18, ( 5) : 909-924. [ 3] 　

( 13) f or com posit e mat erial const ant C .

・ 161・

m ac

- lnD 0 . But t he significance of the present Eq. ( 15) and ( 12) consist s in : it not only suggests def init ely t he lif e-est im at ion expression relative to v aried damag e value D oi but also gives out a new concept of t he *

compositive material const ant C having functional ′ ′ m ac relat ion w ith t he f , f , m , m , b1 , c 1 and ( lnD - lnD 0) .

M argolin B Z, Svechova B A . A nalysis of crack germination and grow th on fatigue and f racture to th e pearlit e-st eel[ J] . Journal of St rengt h Problem , 1990, ( 4) : 12- 21. ( in Russian) [ 4] 　 Y u Y G, Zh ao E J. Calculat ions t o damage evolving rat e under symmet ric cyclic loading [ A ] . In: W u X R and W ang Z R. Proceedings of t he Sevent h Int ern at ional Fat igue Congress[ C] . Beijing: 1999. 1137- 1142. ( in Russian) [ 5] 　 Y U Y G . T he Calculat ions to it s damage evolving rate and life f or a component under un-sym met rical cyclic load [ A ] . Proceedings of A sian -Pacific Conf erence on Fract ure and St rengt h of St ruct ure' 93 [ C] . Sponsored by M at erials and M ech anics Division En gineers, 1993. 615- 618. [ 6] 　 李细广, 琚定一. 疲劳损伤临界 值和疲劳 寿命[ A ] . 疲劳损 伤理论研讨会论文集[ C] . 洛阳: 中国航空学会, 中国力学 学会, 1992. 104.

( 2) In the damag e calculat ing , it considers

Li X G, Ju D Y . Th e fatigue damage crit ical value and t he fat igue life [ A ] . Proceeding of t he Fatigue Damage T heoret i-

bot h t he com ponent of t he e and t he component of t he p in Eqs. ( 15) , ( 12) and ( 1) , so it is

cal Proseminar [ C ] . Luoyang : Th e A eronaut ic A cademy of China an d t he M echanical Academy of China , China , 1992.

more comprehensive, and is ex act . But f or

m

≠0,

calculat ing precision is m ore rig orous by t he rat iomethod -equations than by Eq . ( 1) . ( 3) T he equat ions given by the ratio-met hod, could both avoid t he shortcomings of unnecessary fat igue t ests and use-dif ficult y for new mat erial constant s, and assimilat e t he advant ag e using t he six st aple material const ant s ( K , n ,

f

, b1 ,

f

,

c1) and t he modernistic damag e paramet er D. ( 4) T he damage paramet ers D in each equation for calculat ing damage rate and varied hist ory lif e N oi can be converted int o another physical pa-

104- 106. ( in Chinese)

Biography: YU Yan-gui 　 M ale, bo rn in M arch 1936 from L eqing Zhejiang , he is a pr ofessor of W enzhou U niversity , Plur alistic pro fessor of Zhejiang Science & T echnology U niv er sity, studying the eng ineer ing applications on fatigue-damage-fr acture subjects, the technology of assistant desig n calculatio ns and auto control with co mputer on co mpr essor s and pressur e vessel. E-mail: yuyangui @ sina . co m. cn, Fax : + 86-577-88373335, T el: + 86-57788373359