Non-linear creep damage under one-dimensional variable tensile stress

lnt J ~on Linear ~4e~ham~s Vol 16 No 1 pp 27-38 1981 Printed m Great Britain

002~7462/81/010027 1250200/0 ~) 1981 Pergamon Press Ltd

NON-LINEAR CREEP DAMAGE UNDER ONE-DIMENSIONAL VARIABLE TENSILE STRESS F A COZZARELLI Department of Engineering Soence, State Umverslty of New York at Buffalo, Buffalo. New York, U S A and

G BERNASCONI lstltuto dl Meccamca e Costruzmne delle Macchlne, Pohtecmco dl Mllano, Milan, Italy

(Recewed 17 December 1979) ~.bstract--A non-hnear damage relatxon, containing the axml strata history a n d a time integral over the stress h~story, ~s proposed for the case of one-d~mensmnal time dependent tensde stress Non-hnear steady and transtent creep terms are included m the axml strata relatxon, and elastic and creep Polsson's ratms are introduced into the lateral strata relation Using these relat,ons, complete damage solutmns are obtained for the constant stress rate, step stress, relaxat,on and constant load tests Observations are made concerning the associated rupture times

1 INTRODUCTION

The problem of creep damage and rupture is currently of great pracUcal importance, especially m the field of nuclear safety This subject is very complex and much remains to be accomplished in the development of a complete and useful creep mechanics model for the estlmatmn of rupture time The publication based on the Nov 1978 Ispra course on creep [1] contains detailed discussion on every aspect of this subject as well as an exhaustwe b~bllography, also see Hult [2] and the books by Hult [3] and OdqvIst [4] In a recent contribution to the subject of creep rupture, Bellonl, Bernasconl and Plattl [-5] have employed material density variation as a measure of damage m steel under constant onedimensional tensde load By extenswe data analysis they found that the damage depends on the level of the initial stress, the time expllotly and also on the axial creep strain as a function of time By employing a damage relation expressed in terms of power functions In these three variables and an axial creep strain relation consisting simply of a stress power law for steady creep, they were able to estimate the creep rupture time and thereby assign a physical interpretation (i e, in terms of density varmtlon) to the Kachanov rupture parameter (see [6]) Further results on this approach to creep damage were presented at two recent congresses [7, 8] In this paper we discuss m detml an extension of the damage relation presented in [5, 7, 8] to the case of time dependent one-dlmensmnal tensde stress We first modify th~s relatmn by stipulating that ~t be employed for constant stress rather than constant load, and then postulate that for variable stress the damage rate be a function of the axial creep strata rate, axml creep strata, damage and stress Also, we employ a fmrly general axial strata relation containing linear elastic, nonhnear transient creep and non-linear steady creep terms The temperature is assumed to be constant in both the damage and strata constltutwe relations Using these relatmns, damage solution are then obtained for several prescr|bed oned~mensmnal stress and strain histories as well as for the case of constant load accompanied by lateral contraction Such solutmns are useful m the design and analys~s of tensile test programs, and also for the estimation of rupture times Sectmn 2 contains the one-dlmensmnal model for damage and strain (both axial and *This research was supported m part by the Office of Naval Research under Contract No N00014-75-C-0302 27

28

F A COZZARELLIand G BERNAS(ONI

lateral), first for constant stress and then as extended to varmble stress In Section 3 we obtain complete damage soluUons for the constant stress rate test, the step stress test and the relaxation test with transient creep excluded m this last case SoluUons are then obtained in Section 4 for the constant load test with lateral contraction, first for large deformation w~th transmnt creep excluded (or included approximately m the elastic t e r m / a n d second for small deformation m d u d m g transient creep as a separate term A short d~scuss~on containing some suggestions for possible testing programs then concludes the paper 2 CREEP M O D E L

2 1 Damage relatmn For constant one-d~mens~onal tensile stress at constant temperature we employ the damage relation

D(t) -- Cec(t)~a~ot~

(1)

Here, D(t) and edt) are the damage and axial creep strata respectwely at time t, a o is the constant stress, and C, ~t, ?, 6 are posmve material damage constants at a parUcular temperature In [5] a 0 was the m m a l stress for constant load and the damage D(t) was gwen as -Ap/po where Ap is the densRy change p(t)-po (negatwe) due to void formauon and growth, here D(t) is not necessarily restricted to th~s particular measure of damage EquaUon (1) may also be written as

d [

D(t)11/6_o.w6

dt Lc~dt)'J

I2)

o

Note that e~(t) does not include the elastic strata, which ~s consistent with the v~ew that damage (such as due to void formation and growth) is not assocmted with recoverable deformation In generahzmg equation (1) to the case of variable stress a(t), we postulate that the damage rate be a function of the axml creep strata rate, axml creep strata, damage and stress, ~e,

D(t)=J [e~(t), ec(t), O(t), a(t)] above a variable indicates d/dt An expression consistent

(3)

where the dot wRh this function is obtained directly if we simply replace a o in equatmn (2) by tr(t), which upon integration yields the desired result

D(t)=C~(t)~[~I a(t')'/~dt'l ~

(4)

F o r a(t)= a o equation (4) dearly sJmphfies to equation (1), and furthermore It possesses the physically necessary characteristic that D(t) be continuous even ff the stress history is discontinuous The necessRy of considering general damage accumulation laws, such as equatmns (3) and (4), was recently discussed m [9]

2 2 Strain relattons For the axml creep strata at constant tensile stress and temperature appearing m equation (1), we employ m this paper the sum of axial creep strata components

~c(t)=~s(t) +~,(t)

(5)

where es(t) =

t,

et(t)

\It/

(1 - e - " )

(6)

Here, edO and et(t)represent the axial stratusdue to steady creep and translent creep respectively,and 2, n,/~, q, z are posmve material creep constants Expressions of this type have been widely used in the study of creep m various metals and polymers, including the practically Important case of stainless steel (see Garofalo [10]l The power form in time is also

Non-hnear creep damage under one-d~mens~onal varmble tensile stress

29

widely used, but for the sake of brevity we restrict our attention here to strain m the form of equations (5) and (6) In generalizing equations (6) to variable stress a(t) we follow the strain-hardening approach of Cozzarelh and Shaw [11], whereby each component of creep strain rate Is expressed as a function of the same component of creep strata and the stress, i e,

~,{t)=g[~[tl,

~,(t)=h[et(t),

(7(t)],

(7(t)]

(7)

After approprmte d~fferenttatlon and mampulatlon, one then obtains the desired generahzatlons of equations [6) m integral form as e~(t)=

fo ( ~ ) "

cIt',

e,(t)=~1 e ''~ fo

(~(t'}y \Y /

e ~'~ dt'

(8)

Thus, the axial creep strata for use m equation (4) ts given by equation (5) with equations (8) In order to obtain the solutions m the next two sections, we shall also require expressions for the total axial strata and the total lateral strain For the total axial strain ~(t) we s~mply add a linear elastic term to the axial creep strain, i e, (y

e(t) = ~

+e~(t)

(9)

where E is the elastic modulus And for the total lateral strain e~(t)we follow the approach o f C o u r t m e , Cozzarelh and Shaw [12], and describe lateral contraction by means of

E~(t)= - v ~ +v~(tl +v,~,(tt

I°

(10)

1

where v IS the usual linear elastic Polsson's ratio, while v~ and v, are steady creep and transient creep Polsson's ratios respectively For metals, v~ is close to 1/2 (incompressible) whereas v, Is between v and v~ Note that we have not employed above the simplified approach of combining elastic and transient creep strain into a single non-hnear instantaneous term (e g, see [4])

2 3 Rupture ttme The presence of the creep strain m the damage relation (4) was motivated by the experimental observations in [5, 7, 8], and results in a couphng of the damage and creep strain relations which is necessary for a physically reasonable estimation of rupture time We have not followed Rabotnov's approach [ 13] of Including the damage in the creep strain relation, since it was felt that this additional couphng would comphcate the formulation unnecessarily Therefore the present formulation does not account for tertiary creep at constant stress, but this is not necessarily a restriction If for example one defines the rupture nine as that t~me at which rap~d tertiary creep and rupture commence Furthermore, the s~gnlficance of tertiary creep up to the point of actual rupture can be mlnsmlzed if care ~s taken to base the creep model on constant stress data (as done here) rather than on constant load data Specifically, let t, be the rupture time corresponding to a critical value of damage for rupture D, For the case of constant stress equations (1), (5)and (6) then yield

D,=C

t, + 6° q(1--e -t''*) \Y/

a~t~

(11)

where t, g~ves the rupture t~me m ~mphctt form If we ~gnore transient creep ( l ~ oc )we may obtain t, explicitly as

which agrees with the result given m [5], except that there a o was the initial stress for constant load It was also pointed out m [5] that the posttive quantity (n~ +)')/(~ +6) is analagous to the exponent of stress (generally designated as v) in the Kachanov damage law [6] In the next two sections ~ e examine several more complicated tests

30

F A COZZARELLIand G BERNASCONI

3 DAMAGE

SOLUTIONS

FOR

STRESSES

VARIOLS

PRESCRIBED

AND STRAINS

Some insight into the physical nature of the proposed creep model may be gained by obtaining explicit expressmns for the damage D(t) with various prescribed stress and strain histories Thus, m this sectmn we obtain solutions to equatmn (4) for the constant stress rate, step stress and relaxatmn tests, and also make some observations on the assocmted rupture times

3 1 Constant stress rate test As a first example, consider the case of a stress increasing hnearly with time from a zero value at t = 0, ~e,

~(t)=ktH(t),

k>0

(13)

where k ss the constant stress rate and H(t) ts the umt step functmn Restricting q to odd positive integers, equations (5) and (8) yield for this case the axml creep strata e~(t)=

[ ( k ) " t "+1 + ( k y ( ~ 2

(-1)'z'q' )] n-+--1 \U./ \,=0 ~-----~ tq-l+zqq)e-t/" H(t)

(14)

Equation (4) with equation (13) then yields the damage m terms of ec(t) as

D(t)

F( +a} ~ l"k't'+"lH(t

)

(15)

If we ignore transient creep, equatmns (14) and (15) yield the simple expressmn for damage

D(t) = (n +C l y (~_~)~ 2_.~k~+.~t~+~(.+ 1)+~H(t)

(16)

Note that in equatmns (15) and (16) D(t)~oe as t ~ o e , and as in the constant stress test there will always exist a finite rupture t~me t~ at which the damage reaches a critical value D.

3 2 Step stress test Now consider the case of a constant stress apphed at t = 0 and then suddenly changed to K % at t=tl, l e ,

a(t)=ao{[H(t)-H(t-t,)]

+ K H ( t - t l ) ~, K>O

(17)

In this case, equations (5) and (8) gwe ec(t)=

[tH(t) + ( K " - 1)(t-tOH(t-tl) ] t7

q

+(~)[(l-e-'/')H(t)+(Kq-1)(l-e-(t-tl)/')H(t--tl)]

(18)

Next, equation (14) gives

D(t) -~[t"H(t) Ce~(t) + (K~ - 1)(t~- tf)H(t - t 1)]

(19)

Again, if we ignore transient creep a slmphfied expressmn for damage is obtained as

D(t) = C,~-~"~+"~{ta+~H(t) + [ ( K ~ - 1)t~(ta - t ~ ) +(K ~ " - 1)ta(t~ - t~) +(K ~- 1XK~"- 1)(t~ - t~)(t~ - t~)]H(t - tl)}

(20)

As m the constant stress rate test, D(t)--* oo as t--* oo m the present case and a fimte rupture Ume t. again exists Also note that although in accordance with equation (17) a(t) is discontinuous at t = q, the damage D(t) In accordance with equations (19) and (18) ~s continuous at this point while D(tO is discontinuous

Non-hnear creep damage under one-d~mensmnalvariable tensile stress

31

3 3 Relaxatmn test As a third example, consider the case of a constant strata suddenly apphed at t =0, i e,

e(t)=eoH(t )

(21)

When e(t) is prescribed, equation (9) (with equaUons (5) and (8)) is a nonhnear differential equation m a(t) which may tn general be difficult to solve However, ff we Ignore transmnt creep, this dffferentml equatmn with condmon (21) slmphfies to

1 da [a'~"

-E ~-t-+~) =O,

t>O

(22)

where a(0 +)-- Ee 0 For n > 1, the solutmn to equatton (22) is readily obtained as

a(t) = E[e o "+ 1 +(n - 1)E"2-"t] 1/~-. + l~H(t)

(23)

The relaxatmn test with transient creep neglected is thus eqmvalent to a test where stress is prescribed m accordance with equatmn (23) Substituting equatmns (21) and (23) into equatmn (9) we then obtain the creep strain as

~c(t) = {Co-- [Co "+1 +(n-- 1)E").-"t-] 1/~-"+ 1)}H(t)

(24)

Finally, equatmn (4) with equatmn (23) yields for o~=y--6(n--1) the damage m terms of ec(t) as

D(t) / E - l)."b \~ ( Cedt)-[.b(-~Z_l-~_y ) ~[(Eeo) -"+1 + ( n - 1 ) E 2 - " t , ] t~'"- 1,-']/6'"- 1, _

[(Eeo)-. + 1]ta¢.- 1)- ~1/0~.-1)}~H(t)'

(E-12"~ a

=!~--n-ZT--1] {ln [1 +(n-1)E"e~)-x2-"t]}~H(t),

¢o4:0,

(25a)

o2=0

(25b)

In the hmlt as t--* oo we obtain from equatmns (24) and (25)

(t-o.+~-a~.- 1)E~-~.{2.6/D,_b(n_l)]}a,

hm D(t) =~.--~o r.oo

oO

o~>0, ,

~<0.

(26)

We thus obtain the interesting result that whereas for the material power combination co > 0 the damage may not necessarily ever obtain a crmcal value D . the existence of a fimte rupture t~me t. is guaranteed when to < 0 A similar result ~s obtained when one studies damage durmg stress relaxatmn using the Kachanov damage formulatmn [14,] We treat the constant load test separately tn the next sectmn, since that case ~s somewhat more comphcated than the three cases considered m th~s sectmn

4 DAMAGE S O L U T I O N S FOR CONSTANT LOAD Consader now a bar of instantaneous cross-sectional area A(t) under a constant load Po suddenly apphed at t = 0 The bar experiences a continuously mcreasmg stress Po a(t)=h-~)

(27)

due to the effect of lateral contractmn, and the ex~stance of a fimte rupture t~me Is assured Employing the logarithmic definmon for large strata, we write the lateral strata rate as

,, R(t) 1 A(t) el(t) = R(t)-- 2 A(t)

(28)

where R(t) Is the instantaneous cross-sectional radms In th~s section we determine the solutions for A(t), a(t), ec(t) and D(t), first for the case of large deformaUon with transient creep excluded (or included approximately m the elastic term) and then for the case of small deformation with transient creep included

32

F ht COZZARELLIand G BERNASCON1

4 1 Large deformatzon--trans~ent creep excluded (or included approximately) For this case we set e t = 0 in equation (10) and also assume that the steady creep is Incompressible (v~= 1/2), whereupon with the use of equations (8) and (27) we obtain -v

-2\A(t)2]'

t>0

(29)

This result In combination with equation (28) leads to the differential equation in ACt) (It = - - ~ - A"- 2 dA - A"- x dA,

t> 0

(30)

One may obtain a rough correction for the exclusion of transient creep by replacing E with g*, a smaller fictitious modulus of elasticity The mmal condition for equation (30) is prescribed at t = 0 ÷ (1 e, immediately after the load is applied), i e, A(0+)=Ao

(311

Since the strain at t = 0 + is small we may write el(0 +) ~I/2[A(O+)-A(O-)]/A(O +) and thereby obtain

2vP o A(0 +) ~A(0- ) - ~ -

(32)

from which we see that A(0 +) differs from the known A(0- ) by a very small amount Differential equation (30) with initial condition (31) may be integrated to yield Ao.]k

(n-1)E

=1

(n-1)E

n

t,

t>0

(33)

where a o = P o / A o Equation (33) gives the function Ao/A(t)=F(t ) m implicit form, but explicit expressions may be obtained for approximate cases For example, if we neglect the elastic term ( E ~ oo) equation (33) gives the simple relation A° [-1 A~k-n

"t

,

t

>0

(34)

On the other hand, we can retain the elastic term, but approximate Ao/A within the brackets by umty, and accordingly obtain

A(t)

(n - 1)E - 2nvtr o

Since equation (35) contains equation (34) as a specml case, we shall use equation (35) in the remainder of this subsection, with E replaced by E* when transient creep IS approximated The stress now follows Immediately from equation (35) as

[

Ao n(n-1)E tr(t)=tr o ~ ) ~tr o 1 - ( n _ l ) E _ 2 n V ~ o

"t

1-"

,

t>0

(36)

Next, the creep strata as obtained by substituting equation (36) into the first of equations (8) IS given by

[

In 1-- ( n _ l ) E _ 2 n v t r °

ec(t) .~ --

n ( n - 1)E (n -- 1)E - 2nvtr o

t ,

t >0

(37)

Note that ec(t)~ oo as t--*tmax, where

tmax=

(n-1)E-2nVtro(~_~)-" n(n -- 1)E

which IS analagous to the ductile rupture criterion due to Hoff [15] Finally, the damage is obtained in terms of ec(t) from equations (4) and (36) as

(38)

Non-hnear creep damage under one-dimensional varmble tenstle stress

°("

~Csc(t) z a~ [ \ 7 _ fin) tmax

[( ' T -''''" 1}', 1 - tmax

-- 1

33

t>0

(39)

N o t e that as expected a finite rupture time t, extsts corresponding to a gwen crtUcal damage Dr, but now m contrast with the constant stress test [equations (11) and (12)] D(t)--* oe as t--*tma x Also note that m general ? ~ 6 n in equation (39), and in fact for the very special case ? = fin equation (39) slmphfies to

[

(,---'ll' tmax]/

(40)

~Cec(t) ~tr~ tmax In \

In order to illustrate the character of the above strata and damage soluUons, we employ the data gwen m [8] for the following two steels materml a AISI 310 stamless steel at 600°C, materml b 2 25Cr 1Mo ferrmc steel at 550°C The vartous material parameters (elasttc--E*, v, creep---n, 2-% damage - ~ , ~, 6, C) for these two materials are summarized m Table i, where E* m&cates we are using the rough correcUon for the excluston of transient creep Usmg these values w~th an m m a l stress ao equal to 30 k g r a m - 2 for material a and 25 kg m m 2 for material b, we obtained the constant load curves ejt)eo [equation (37)-] and D(tleo [equation (39)] shown plotted m Figs 1 and 2 Table 1 Material constants Material

E*, kg m m - 2

a

b

1000

1000

v

03

03

n 2- ", mm2"/kg"-hr

7 0 74 x 10 - t 3

56 0 45 x 10- to

c~

07

065

6

006 0012 4 3 x 10- 3

25 025 5 6 x 10- 7

C, mm2VkgLhr a

K._eey to Subscrlpt._ss a -AISI 3t0 (600oc)

O60

/

b -2 25Cr 4Mo(550*C) PO-ConstantLoadTest

/ /

O 50 - "° - C°nstant Stress Test

040

/

_ (%)a = 50 kg/mmz

(0"0)b=25kg/mm

030-

/

/

/

/

(ec)bP°'-~ /

0 20 -

/ / /

//~....

(Ec)b%

O10 -

40

,'o 20

'o 30

&50 12o ;o 60 70

40

(tmax)b

do 80 i2o 90 (tmax)a

t(hr) Ftg 1 Strams at constant load and stress Large deformauon Transient creep included approxtmately in elasttc term NLM

16

1 - C

34

F A COZZARELLIand G BERNASCONI

Key to Subscripts a -AISI 3t0 (600"C) b - 2 25Cr 4Mo (550"C) Po -Constont Lood Test 50 -- °'° -Constont Stress Test 60-

/ /

(O-o)a = :50 kg/mrn ~' (O'o)b =25 kcJ/mm z (Dr) o = 25 x t0 -4

40 -

":

/

/ / ].~ / ~Dbp '

/

3o-

/

I ]

/]

o

....

°

t:~ (Dr)a

.

.

.

(D r )b

t0 0

.

.

.

.

.

i ~ ]

t0

~

o ao.°

I

20

I

30

I I

I

I I

I

40

50

(t r I bp0

I

60

11r ) boo

I II

]

70

80

I

90

(t r lapo

t(hr)

Fig 2 Damage at constant load and stress Large deformahon Transient creep included approximately m elasUc term

respectively vs t i m e in h o u r s F o r p u r p o s e s of c o m p a r i s o n we h a v e also s h o w n in F i g s 1 a n d 2 the c o n s t a n t stress curves ec(t)~o a n d D(t),o [ e q u a t i o n s (1), (5) a n d (6) with e t = 0 ] for these s a m e values of the m a t e r i a l c o n s t a n t s a n d a o It is n o t e w o r t h y t h a t the curves at c o n s t a n t l o a d differ very significantly from t h o s e at c o n s t a n t stress I n F i g 1 we have i n d i c a t e d the ductile r u p t u r e times (tmax) c o r r e s p o n d i n g to ec--, c~ Also, m F i g 2 we h a v e m a r k e d the r u p t u r e times at c o n s t a n t l o a d a n d stress [(tr)Po, (trLo] corr e s p o n d i n g to c r m c a l values o f d a m a g e Dr e q u a l to 25 x 10 - 4 for m a t e r i a l a a n d 15 x 10 - 4 for m a t e r i a l b (from [8]) These v a r i o u s r u p t u r e times are s u m m a r i z e d in T a b l e 2, as expected the r u p t u r e times at c o n s t a n t l o a d a r e less t h a n b o t h the ductile r u p t u r e times a n d the r u p t u r e times at c o n s t a n t stress Table 2 Rupture times Constant load and stress--large deformatmn--trans~ent creep included approximately m E* Material

tmax, hr (L)po,hr (trLo,hr

a

b

86 4 77 6 196 1

57 8 33 6 55 9

4 2 Small deformatmn--mcludmg transzent creep as a separate term C o n s i d e r a g a i n the d e r i v a t i o n of a differential e q u a t i o n in A(t), except t h a t n o w we r e t a i n the t r a n s i e n t c r e e p t e r m et(t) a n d also d o n o t restrict v s to 1/2 T o this e n d we use e q u a t i o n (28) to form the c o m b i n a t i o n

8,(t) +l ez(t) 1 ( ~ ) z

=2

A + - -1 -2z A

(41)

If we n o w subsUtute e q u a t i o n (10) with e q u a t i o n s (8) a n d (27) into the left-hand side of e q u a t i o n (41), we get the n o n - h n e a r differential e q u a t i o n

Non-hnear creep damage under one-&menstonal varmble tensile stress

E

#~-B+v~

(~)

(~)"

35

( P7 °- ) q m = 21 ( ~ ) + 1 (2-c B ) , t>0,

(42)

where B(t) = 1/A(t) Equatmn (42) cannot m general be solved m closed form, and thus we shall mmphfy the problem by mtroducmg the assumptmn of small deformatmn Accordingly, we write

A(t)=A o-A(t), /i(t) <~Ao,

(43)

where fi~(t)is a small area increment and A o has been defined by equation (31) with equatmn (32) Substituting equatmn (43) into equatmn (42) and neglecting higher order terms, we now obtain a hnear &fferentlal equatmn m A(t) as t>O,

(44) where, as before, a o = Po/Ao For conventence we introduce the non&menmonal variables

/i

3. =Aoo'

t

7=-,

t45)

Equation (44) then becomes a5A" +a2A--a3A=aa/n, "' ~

~>0,

(46)

where the prime mdmates d/dL and the non&menslonal coefficmnts are defined by

1 al=2 a2

=2

vao

(47a)

E

E

nv,z

(47b)

--qv t

a3=rlVs'C(~) n

(47c)

As lnmal condmons for equation (46) we require 4(0 +) and .~'(0 +) The former follows &rectly from equations (31), (43) and (45) as

,~(0+) =0

(48)

In obtaining the latter we first note that ~(0 ÷) =e,(0+)=0, and use equations (8), (10) and (28) to obtain A(0 +) vP o + v, a o q ao " e~(0+):~ Aoo :(A~oE) A(0 ) - T ( - - f ) - v ~ ( ~ - )

(49,

Introducing equations (43), (45) and (47) into this result and again assuming small deformatmn we obtain the desired condmon as A'(0+)= as 4 1

aln q

as

atq

a2

(50)

alq

The solution to differential equation (46) with mmal condmons (48) and (50) is easily obtained For example, m the specml case o f q = n we get

1 [--a2+2al +A) (-a2+A~-[+(_a2-2al+A) ~(i)=- n+~, 2nA exp \- 2aal ] 2hA

(-a2-A'~ ?, ~>0, 2a, ]

exp k

(51) where A =(a 2 +4ala3) t/2

(52)

36

F A COZZARELL1and G BERNASCON1

The stress then follows directly from equaUon (51) m accordance with a(t)--%[1 +/](t)], 7>0 (53) Substituting equation (53) into equation (5) with equation (8) and neglecting higher order terms, we obtain the creep strata as 2al )(O'o)"] ec(~) = - a-21z (_~)" + t - a 2 +2a 1 +A'~[ z ( 2a, ~(ao'~" +{ aa 2A Ik~q-A,]\2-,/ \2al ~aa2 +z~ 7 {-a 2 +k~)+(a=--2a, x exp ~- -~aI \ ~

// 2a, "~l'ao'~" )L/: ~ ~ ; ~ - ~ - = - ) -+-(2al--a22al_ A ) ( ~ ) n l

+A'~r

1,4> Although the creep model at constant stress [l e, equat]ons (5) and (6)] does not contain a ternary creep stage, equation (54) does ymld a region of increasing creep rate after a crmcal time tc Setting e;(tc) =0 we obtain this crmcal time as - a 1 {((-a2-A)[(2al -a2-A),(ao/).)" +(-a2-A)(ao/#)"]] tc=~ln - a z + A)[(2al - a2 + A)v(ao/2)" + ( -a2 + A)(ao/#)"] J Finally, the damage for small deformatmns ts obtained from equatmn (4) as

(55)

D(t' {( y)/:~-__ y'cal y/: ('-a2-k-2al-k-A_']( 2al "~ {/--a2-k-A"~ C~-~)=- = a~) 1 - ~nn 6-n a-~ + 6- \ 2~A -,]\ - a2 ~-k ] ex p !k 2a7 -,] 7 yr(a2--2al+A']( 2a, )exp(--a2--A'~'~a +-6 2nk /(-a:-~-A \- ~a 7 / j

(56)

where we again find that tr IS fimte In order to dlustrate the above solutmns the creep strata was calculated using the parameters assembled m [16] from the data of Garafalo et al [10, 17] for the following steel material d AISI 316 stainless steel at 816°C These elastic, steady creep and transmnt creep parameters (E, n, 2-", q,/,-q, z) are given m Table 3 along with the addmonal assumed Pmsson's ratms v=O 3, v,=0 4 and Vs=0 5 Key to Subscr=pts 025-

d-AISI .3t6 (846=C) Po- Constant Load Test a"o- Constant Stress Test

0 20-

(~c)dPo-~ (°'°)d=703kg/mrnZ

~'/~"

i

0t5 -

I

O

kU oto o-O

o o5

o

40

20

50

40

50

60

70

]

I

80 tc 90

I

400

I

440

t(hr) Fig

3 Strainsat constant load and stre~s Smalldeformatmn Transientcreep includedas separate term

Non-hnear creep damage under one-dimensional varmble tensile stress

37

Table 3 Materml constants Materml d E kg m m - 2 v, ~,

n=q / -" mm2"/kg"-hr It q, mm2q/kgq z, hr

10,500 03 04 05 35 0 795 × 10 6 0 802 x 10 '* 21 7

Figure 3 shows the ec curves for constant load [equation (54)] and constant stress [equations (5) and (6)] for a o equal to 7 03 kg m m - 2, and we again see that these two solutions differ by a considerable a m o u n t U p o n comparing Figs 1 and 3 we see that the ductile rupture time tma x IS no longer fimte when deformations are small However, we now have the crltmal time t~ [equation (55)] corresponding to the point of reflection, and m the present example this occurred at 84 hours (see F~g 3) 5

DISCUSSION

Starting with a damage relation for constant one-dimensional tensile load given in [5], an extended damage relation was proposed for time dependent one-dimensional tensile stress histories This relation contains a power of the strain history and also a power of a time integral over a power of the stress history, thereby ensuring a continuous damage history even for a discontinuous stress history Employing an axtal strain relation containing both steady and transient creep terms with a lateral strain relation containing three Polsson's ratios, damage solutions were obtained first for the constant stress rate, step stress and relaxation tests and then for the constant load test In all tests considered, except the relaxation test, there existed a finite rupture time corresponding to a critical value of damage, the existence of a finite rupture time was guaranteed in the case of the relaxation test only when a material power combination was negative Constant load test solutions were obtained both for the case of large deformation with transient creep excluded (or included approximately in the elastic terms) and for the case of small deformation with transient creep included as a separate term, m both cases the creep strain and damage solutions are considerably greater than the corresponding solutions at constant stress In the former case a ductile rupture t~me was obtained correspondlng to infinite strain In the latter case a critical time was observed marking the begmnmg of a region of Increasing creep rate A testing program for the determination of the material parameters would most logically begin with the one-dimensional constant tensile stress test, w~th the axial creep strain and damage measured as a function of t~me Density variation ~s one useful index of damage, but it may be necessary to also use other methods such as the actual counting of voids The most reliable procedure would probably be to first fit the axial creep strain data to determine the material parameters m the model for ec(t), and then to smooth the damage data with the calculated ~c(t) before finally fitting this damage data to determine the material parameters in the model for D(t) Then the vahdlty of the proposed model for one-dimensional time dependent tensile stress could be checked by performing one or more of the tests discussed in this paper, w~th the step stress and constant load tests being possibly the most useful Acknowledgement The authors wish to thank Chm-Sheng Lee, Research Assistant State L m~erslty of New York at Buffalo for his assistance m the preparation of the curves

REFERENCES G PlattL and G Bernascom (eds I, Creep oj Engineering Materials and Structures Apphed Science P u b l , London i 19801

38

F A COZZARELLIand G ~ERNASCONI

2 J Hult, Creep In continua and structures, in Topics in Apphed Continuum Mechanics (Edited by J L Zeman and F Zlegler) Sprmger-Verlag, Vienna, pp 137-155 (1974) 3 J Hult, Creep in Engmeerln 9 Structures Blaisdell Publ Co, Waltham, Mass (1966) 4 K G Odqvlst, Mathematical Theory of Creep and Creep Rupture Oxford Clarendon Press Oxford (1974) 5 G Bellonl, G Bernasconl and G Pxattl, Creep damage and rupture In AISI 310 austemtlC steel Meccanlca 12(2), 84-96 (1977) 6 L M Kachanov, Time of the rupture process under creep conditions lzv Akad Nauk, SSSR, Otd Tekh Nauk, Nr 8 (1958) 7 R Matera, G Pratt1, G Belloni and S Matteazzl, Dannegglamento a scorrimento VlSCOSOdell'acclalo 2 25 Cr 1 Mo Proc Fourth AIMETA Congress, Florence (Oct 1978) 8 G Plattl, G Bernascom and F A Cozzarelh, Damage equations for creep rupture in steels Paper L 11/4, 5th Int Conf on Structural Mechamcs m Reactor Technology, Berhn (Aug 1979) 9 W J Ostergren and E Krempl, A umaxlal damage accumulation law for time-varying loading including creep-fatigue interactmn J Pressure Vessel Tech 101, 118-124 (1979) 10 F Garofalo, Fundamentals of Creep and Creep-Rupture m Metals MacMillan Co, London (1965) 11 F A Cozzarelli and R P Shaw, A combined strain and time-hardening non-linear creep law lnt J NonLinear Mechanics 7, 221-234 (1972) 12 D Courtme, F A Cozzarelh and R P Shaw, Effect of time dependent compressibility on non-linear viscoelastic wave propagation lnt J Non-Linear Me~hamcs 11, 365 383 (1976) 13 Yu N Rabotnov, Creep Problems in Structural Members (English translation by F A Leckle), Ch 6, Sec 88 North-Holland Publ Co, Amsterdam (1969) 14 J Hult, Creep strength of structures Lectures dehvered at Centre International des Sciences Mecanlques, Udlne, Italy (1973) 15 N J Hoff, The necking and rupture of rods subjected to constant tension loads J Appl Mech Z0, 105-108 (1953) 16 F A Cozzarelll and G Tlttemore, Free lateral vibration of an axially creeping beam-column under initial axml compression Meccam¢a 12(3), 151-158 (1977) 17 F Garofalo, C Richmond, W F Domlsand F von Gemmlngen, Straln-tlme, rate-stressandrate-temperature relations during large deformations m creep, Proc Joint Int Conf Creep Inst Mech Eng (1963)

R~sura~ On propose dans le cas d'une contralnte de traction monodlmenslonnelle fonctlon du temps une relatlon de d~faut non lln~alre contenant l ' h l s t o r l q u e de la d~format~on axlale et une int~grale sur le temps de l ' h l s t o m q u e de contralnte On i n c l u t dans la relatlon de d~formatlon axlale des termes non l~n~alres de fluage permanent et trans~tolre et on Introd u l t dans la relatlon de d~formatlon lat6rale des rapports de Polsson en ~ l a s t l c l t ~ et en fluage En ut111sant ces relatlons on obtlent les solutlons compl~tes de d~faut pour les tests de taux de contra~nte constant, contralnte en escaller, relaxatlon et charge constante. On f a l t des observatlons au sujet des temps de rupture assocle~

Zusammenfassung E1ne n l c h t l l n e a r e Schadensbezlehung, dle das vorhergehende Dehnungsgeschehen In a c h s l a l e r Rlchtung und eln Z e l t l n t e g ral des vorhergehenden Spannungsgeschehens e n t h a l t , wlrd f u r den Fall e l n e r elndlmenslonalen, Z e l t abhanglgen Zugspannung vorgeschlagen N1chtllneare, s t a t l o n a r e und Ubergangsglleder f u r das Krlechen slnd In der Bezlehung f u r dle Achslaldehnung enthalten Polssonsche Zahlen f u r die e l a s t l s c h e und dle Krlechverformung werden in dle Bezlehung f u r dle Querdehunung e l n g e f u h r t Unter Verwendung d l e s e r Bezlehungen werden v o l l s t ~ n d l g e Schadenslosungen f u r Experlmente m l t konstanter Spannungsanderung, sprunghafter Spannungs~nderung, Spannungsrelaxat~on und m l t konstanter Last a u f g e s t e l l t Welterhln werden d~e zugehorlgen Bruchzelten erw~hnt