High temperature creep damage under biaxial loading: INCO 718 and 316 (17-12 SPH) steels

Nuclear Engineering and Design 114 (1989) 365-377 North-Holland, Amsterdam

365

HIGH TEMPERATURE CREEP DAMAGE I N C O 718 A N D 316 (17-12 S P H ) S T E E L S P. D E L O B E L L E ,

F. T R I V A U D E Y

UNDER

a n d C. O Y T A N A

BIAXIAL LOADING:

*

Laboratoire de M~canique Appliqu~e (A U 04), Facult~ des Sciences et des Techniques, route de Gray, 25030 Besangon, France

Received August 1988

The respective influence of the Von-Mises equivalent stress and of the maximum principal stress on high temperature creep damage of two industrial alloys (INCO 718 and 17-12 SPH stainless steel) are pointed out in a quantitative way through tensile-torsion biaxial tests. Through inversions of the shear component, the important part taken by the principal direction corresponding to the maximum principal stress is also shown. The results are observed to be opposite according to whether the alloy suffers cyclic hardening as 17-12 SPH does or cyclic softening which is the case of Inco 718. These results are supported by metallographic observations. They demand an anisotropic form for the damage variable D, while besides a time dependence, the kinetic equation must include the part taken by the strain.

1. Introduction

2. Experimental method

The aim of the present paper is to point out the part played by the stress state on damage and fracture through creep tests in biaxial loadings. Two industrial alloys - Inconel 718 at 6 5 0 ° C and a 316 austenitic stainless steel (French specificity 17-12 SPH) at 6 0 0 ° C - were considered. Indeed, the simultaneous but different roles of the maximal principal stress and the equivalent (in the Von-Mises meaning) one are well known [1-5]. The material science point of view generally claims that crack nucleation and growth at the origin of damage are respectively controlled by equivalent stress and the maximum principal stress [2,6-8]. The increasing damage in the material may be accounted for, in flow rules, by introducing a damage internal variable having its own kinetics and leading to the concept of an effective stress 8 = o / ( 1 - D); 8 is interpreted through the consideration of an effective area of a sample containing cavities and debonding. According to various theories about the nature of D and its intrinsic definition, this variable is considered either as scalar (isotropic damage) [2,9], tensorial of the order two (anisotropic damage) [3,10] or of the order four [4,15]. The reported tensile-torsion tests will be analysed on the basis of these theories.

The tests were performed on a servo-controlled tensile-torsion creep machine. Large stiffnesses together with an accurate extensometry led to reliable and precise results [12]. The samples were tubes with internal and external diameters respectively equal to 8 and 10 ram, the gage length was 50 m m between the threaded heads. The Inconel samples were obtained from bars heat treated, quenched and then annealed in such a way to get a structural precipitation hardening ('t and "t' phases). The composition of this superalloy is given on table 1. The stainless steel samples were machined from plates with 30 m m thickness, quenched from 1200°C. N o further heat treatment was done. The composition of this stainless steel is given on table 2. To separate the effects of the different stress components, tests were performed either at constant equivalent stress ~ or at constant principal stress opl, the two constants being chosen in such a way that in a pure tensile test they correspond to the same tensile stress

* #_.quipe appartenant au GRECO 'Grandes D6formations et Endommagement'. 0029-5493/89/$03.50

Table 1 Composition of the INCO alloy (wt%) Ni, 52.26; Cr, 18.26; Fe, 19; Nb, 5.3; Ti, 1.02; A1, 0.54; Mo, 3.06; C, 0.035

© E l s e v i e r S c i e n c e P u b l i s h e r s B.V.

366

P. Delobelle et aL / High temperature creep damage

5OO

~uze

~.,(MPa) ~-... --. INCO

300 ~-...

.

200

~

)

20(

-loo

SPH

T.650°C,T.600~ ~

tn(h)

t~h)

1 330 170

IO O l . - / I / l ~ \

0 -200

INCO

!

2 406

381

3 260 4 495

9"/0

,00

,o o

615

5 213? 1%8 6 780

;F ; ' V / 7

9 720 IC 935 11 820

-

(11 -501

Fig. 1. Loading points loci in the stress space.

%z. Therefore in the plane o~=, oze (the two components of the applied stress), the representative points of the stress states used in the various tests are on the two following loci:

3. Experimental results

3.1. Relationships between :~n~ and ozz, 5 and %1 In fig. 2 the equivalent strain:

5

=

(oz 2 + 30z2) '/2

=

c

(1)

and

o~, = ( o~ + ( o~ + 40~ )t'~)/2 = ~,

(2)

with c = 640 MPa for I N C O 718 and c = 320 MPa for the stainless steel. The stress states used in experiments are reported in fig. 1 together with the loci of eqs. (1) and (2). It must be noticed that the tests were performed at constant stress components, i.e. the applied force and torque varied according to:

F = Fo(lo/l )

and

C = Co(to~l) 3/2

= (Ez2-,l 2 / 1/2 ~- ~(z0)

(4)

is reported versus time for various tests on the two alloys. Thus one observes for a given 5, that primary creep gives a unique curve,_ in agreement with L e v y - M i s e s flow rules. Then the curves separate leading to decreasing rupture times when the stress state gets closer to the tensile axis. This points out the influence of the maximum principal stress %1 on the damage evolution of these two alloys. The exponents of minimum creep rate sensitivities to 5 and to opt, i.e. respectively:

(3)

1 and 10 are respectively the actual and initial lengths of the sample, eq. (3) is based on the assumption of a constant volume.

n = (2 in ~ / o

in 5)T.op,

and

n, = (2 In i _ _ / ~ In o~)r,~ Table 2 Composition of the 17-12 SPH alloy (wt%)

C, 0.028; i n , 1.88; Si, 0.3; S, 0.001; P, 0.028; Ni, 12.46; Cr, 17.31; Mo, 2.44; N2, 0.077; B = 1 2 ppm, Co, 0.135; Cu, 0.175; Fe, balance

(5)

can be determined. One obtains: n < 16 and n 1 = 2 for 600 < 5 < 750 M P a in the case of INCO. 8 < n N 14 and n 1 = 2.2 for 250 < 5 < 400 MPa in the case of the 17-12 SPH. F o r n, the values so obtained agree with results reported elsewhere and determined for uniaxial tensile creep tests [13,14]. But, at a given 5, ~ i , depends on

367

P. Delobelle et a L / High temperature creep damage

:l c o O-p,\

/

t11

...J

#

Fig. 2. Some examples of creep curves for INCO and 17-12 SPH showing the part taken by Opl.

Opl for the two alloys considered. This observation correlates with the rupture times evolution (see below) and shows that the usual assumption of a quasi-nul damage at i , m is not valid. Therefore the minimum creep rate is not physically meaningful; it only arises from the opposite effects of hardening and damaging.

%1; this shows that %1 plays a significant part in damage. To a first approximation the results can be written as:

3.2. Evolution of the rupture time t R with ozz, ~ and Op]

v-14;

P1=12

v-9;

3
In fig. 3, for both alloys, rupture time has been reported versus ozz, ff and O'pl. As mentioned previously, it is observed that for a given if, t R depends on

Ln(tR(h))

3~

3.8

[] INCO

4

4.2

tROt(O,~)-" ;

tRa(~)-" ;

tRa(Opl) -'2

with and

v2 - - 2 f o r I N C O 7 1 8 ,

and

(6)

2.5
The value v2 = 2 obtained for the I N C O alloy agrees with the value v2 = 1.8 reported by Dyson and McLean

Ln(tl~h)),-~.~ r ~]

~ Lni i~ 3

3.2

ff-125PH

3.4

Lrl 0~i~

Fig. 3. Evolution of t R with oz: , o, Opl and (a,, :t: o,o) for the two alloys considered.

P. Delobelle et al. / High temperature creep damage

368

[6] for a N I M O N I C which is also a nickel based superalloy. The m e a n value v2 = 4 (17-12 SPH) is the same as that o b t a i n e d o n n o t c h e d axisymmetric samples tested in tensile creep by Beziat et al. [15]. As will be detailed further a more accurate analysis shows that in the case of the stainless steel, 1,2 is a n increasing function of opt, with 2 < v2 < 4 w h e n 190 < Opl < 320 M P a (fig. 3). This p o i n t appears to be s u p p o r t e d b y the quantitative study of the ratio of the n u m b e r of cracks over the n u m b e r of total defects which also increases with increasing %1 (see later). It m a y be noticed that the empirical relationship, v = 0.7n, o b t a i n e d o n material is correctly fitted here: one finds p / n = 0.87 (for I N C O alloy), p / n = 0.64 w h e n the largest m e a s u r e d n values are used (for 17-12 SPH).

tory a n d it is thus s h o w n that the familiar relation t~ 1 is, in fact, a particular value of n l / v 2.

( m i n (3(

3.4. Shear component inversion tests A i m e d at pointing out the possible influence of the principal direction associated with the m a x i m u m principal stress (see for example [16,17]) some biaxial creep tests with a n inversion every 48 h of the shear stress c o m p o n e n t o20 were p e r f o r m e d (fig. 1). In fig. 4, the ratio ( t r ~ / t g ) of the r u p t u r e time in tests with inversions (tRi) to the r u p t u r e time for m o n o tonic creep tests at the same stress state ( t g ) is plotted versus the r o t a t i o n of the principal direction due to inversion: Aq~. The results o b t a i n e d with the same m e t h o d on copper b y M u r a k a m i , S a n o m u r a a n d Saito [17] are also plotted. T h e significant p a r t played by the principal direction clearly appears, ( t R s / t R ) being generally greater t h a n one. However, unlike c o p p e r which exhibits a ratio always between o n e a n d two, the 17-12 S P H stainless steel m a y have ratios u p to three (which shows a strong benefit due to the inversions) while the I N C O alloy exhibits values less t h a n o n e ( h a r m f u l effect of inversions). O n e can notice that in the case of the ductile 17-12 SPH material, the samples tested with inversion sequencies do n o t exhibit any significant buckling at rupture, c o n t r a r y to the p r o p o r t i o n a l loadings close to pure torsion. This last b e h a v i o u r induces a n inaccuracy in the d e t e r m i n a t i o n of r u p t u r e times a n d

3.3. Relationship between imi~ and t R As imm a n d t R b o t h d e p e n d o n 6 a n d %1, some relationship should exist between these two parameters. Indeed, one can write:

(-'/"]) ]~~ . T rain a[~, t R

and

i rain a [[ 'dR- ' l / ~ 2 ) ]~o p , , T ,

(7)

which should constitute a n a-posteriori verification for the e x p o n e n t values in eqns. (5) a n d (6). The calculated values of the n l / / , 2 ratio are: 1 in the I N C O case, a n d 0.54 in the 17-12 S P H one, while the values o b t a i n e d directly from the representative curves of imi. versus t R are respectively 1 a n d 0.51. T h e agreement is satisfac-

17_12SPH INCO 718 I Cu [Murakam;] -El.exp. --A.exp. I-e-exp. ,-.(~)- 'lP,.4, ~2-4 -(]]-V.12, ~2 _ . " 2 )calculuswith.

,.~.

V:8,

~8

-~v.~,

v2.~

)

,, . ' . .

. .

^

experiment n~

"

'r,0,

iCe)

[t, il h~] 3 2.5

/t l

i

2 . . . .

v

Fig. 4. Relationship between t g i / t g and zig,: experimental results and predictions obtained from eqs. (18 to 20) with: ~ = 4, ~2 = 4 and i, = 1'2 = 8 in the case of the 17-12 SPH alloy. P2 = 2, p = 12, i, = P2 = 14 and P2 = 0, p = 12 in the case of the INCO alloy.

P. Delobelle et aL / High temperature creep damage therefore in the ratio (tRi//tR) (fig. 4). This artefact does not occur with the INCO alloy. As will be shown later the buckling initiation may be related to the important dissymmetry of the cracks orientation spectrum observed in proportional loadings (while during tests with inversions this spectrum keeps an axial symmetry). However, as far as a quantitative analysis is to be considered, the close connection between damage and cyclic behaviour must be kept in mind. Therefore a global model (i.e. a model in which the flow and damage rules are coupled) is necessary. Indeed, INCO suffers cyclic softening [13], while the 17-12 SPH stainless steel strongly hardens in cyclic tests [18]. Moreover the ratio (Vz/r) is 0.16 in the first material compared to 0.5 obtained in the second; this clearly shows that the INCO alloy is much less sensitive to the effects of the principal stress than 17-12 SPH stainless steel.

369

+ INCO • 17_12SPH ~ p, Arctg(~zz) .

.

.

.

.

.

.

• .

.

.

.

.

.

.

exp6rimentn' .

.

~--~4J

431

3.5. Study of the flow directions 3. 5.1. Proportional loadings In fig. 5 the flow angle

~oxp = tan-' ( ,,0/,=, )

(8)

is plotted versus ( t / t a ) for various loadings. It is seen that the flow direction is stable over most of the life, a rotation towards the o=o axis occurs only in the last step during which a necking zone appears. During this necking part, oz0 and o,= are no longer constant and o~0 increases more rapidly than oz, which explains the observed rotation. However, in the case of INCO, some scatter occurs for t / t R < 0.3. This scatter also appears on the shear modulus and could be induced by some structural changes occurring at the beginning of the plastic strain. Moreover a good agreement is found between fl,xp and flth determined from the Levy-Mises flow rules i.e. t a n - l ( % 0 / % ~ ) =/3~,p = flth = t a n - a ( 3 o ~ e / 2 ° ~ ) -

(9)

This could show that the damage variable D acts in the flow rules as a scalar, which means that either damage is a scalar or is a tensor D taking a part in these rules through one of its invariants (the second one for instance).

In fact, even if a completely anisotropic form is used for D (as is proposed in § 4 where D is a second rank tensor and acts in an anisotropic way on the state equation), the solution df the tensile-torsion problem (dotted line in fig. 5) shows that the strain directions rotate very little (of te order of a few degrees in the most favourable case i.e. pure torsion) towards the

0

QS

, I f ( t i t R)

Fig. 5. Evolution of the flow direction for proportional loadings: experimental results (fie,p) and predictive curve in the anisotropic case (full line) Von Mises case and anisotropic case (broken line).

principal direction. Moreover, it is difficult to detect this, as the inverse rotation due to the necking pointed out in the beginning of this section occurs. Therefore, the fact that the flow direction is constant during a proportional test until rupture is not sufficient to derive definite conclusions about the nature of D (scalar or tensorial).

3.5.2. Loading with periodic torque inversions In fig. 6 an example of the evolution of the flow direction fl~p during torque inversions in the 17-12 SPH case is given. The flow directions rotate in agreement with the predictions of the normality rule applied to Von-Mises equipotential surfaces shifted from the origin by a kinematical hardening [14,19]. It can be remarked that if the inversion period were to be a bit longer eq. (9) would be obeyed again at long time after the last torque change i.e. (%o/%,) = 3/2(a,e/a~=). This comes from the rule of proportionality between the kinematical tensorial variable a and the applied stress one u, which is always true during proportional load-

370

II

P. Delobelle et al. / High temperature creep damage

° Arctg(~l~'zr) INC0.~[6_)

0/6:

creep test at 275 MPa are shown for the 17-12 SPH ahoy in fig. 7. While the rupture strains do not suffer significant modifications, the rupture time becomes an increasing function of the pre-load. This is true for the stainless steel which exhibits cycling hardening; the opposite effects seem to take place in the case of the INCO alloy (which, on the contrary, softens during cyclic testing). These observations support the conclusion of § 3.4: it is necessary to include strain in the damage rules. Indeed, rules using only time dependence, with no strain effects taken into account, lead to rupture times with no dependence on pre-deformation. 3. 7. Study of the cracks orientation

Fig. 6. Examples of the evolution of flc,,p during consecutive %~ inversions (full line) experimental results and Von Mises case (broken line). ings and towards which a tends during non-proportional ones [14,19]. If hardening were to be purely isotropic, relationship (9) (see dotted line in fig. 6) would apply over the whole tests (ii), immediately after an inversion; this is not supported by experiments. It can be noticed that damage does not seem to modify the rotation magnitude but only the strain amplitudes and this at every period. 3.6. Influence of a hardening pre-loading test on rupture time

The changes induced by loading samples respectively at 320 and 373 MPa during 2 rain on a consecutive

t £ (10-21

oO

200

17-12SPH;CF~.275MPo

400

600

800

,~

1000 t[hl

Fig. 7. Creep at 275 MPa: effect of a predeformation on the consecutive rupture time and on the equivalent rupture strain: (1) direct loading at 275 MPa; (2) preloading at 320 MPa; and (3) preloading at 373 MPa.

The fracture features are clearly of a ductile type in the 17-12 SPH case. On the contrary, though the nature of the rupture zone is essentially ductile, some twinned zones appear close to the external surface in the I N C O samples. A quantitative study of the orientation and of the number of intergranular cracks was performed after polishing along some generating lines of the broken tubes. On each face thus obtained, an area about 2 nun × 2 mm was examined at distances from the fracture surface ranging between 0.5 and 3 ram. The magnification was × 220. Those defects verifying the criterion l / e > 2 and l_>_10 # m (l and e being respectively the length and opening of the defect) were considered as cracks. The number of cracks, together with the total number of defects were measured. Let Dc be the surface density of cracks (according to the above criterion) and Dt be the surface density of defects of any type (including cracks). For each sample three faces were observed (i.e. 6 mm 2 in total). 3.7.1. Case of l N C O 718 The surface of the samples showed a pronounced intergranular debonding; however the polished faces exhibit only a very small number of microcracks and a lot of quasi-spherical microdefects very often localised at grain boundaries. For instance, test 2 gives 5% cracks and 95% voids, with Dt = 87 defects/mm 2. Moreover, the defects of a crack-type do not have very pronounced crack characteristics and a statistical analysis of crack orientation is difficult to perform. From this point of view, damage appears as quasi-isotropic at a microscopic scale; this is consistent with the analysis of rupture times which pointed out that the contributions of the maximum principal stress and of the corresponding principal direction (v 1 >> v2 and

P. Delobelle et al. / High temperature creep damage

371

zz

~N5

11"(8) J-ZZ

~o~1 ~o~ b)~ ~o~o~o J'npl

J 'Pl

O

Fig. 8. (a,b,c) 17-12 SPH histograms for the orientation of cracks in the cases of proportional loadings (nos. 2 and 3) and of non-proportional loadings (nos. 6, 7, 8, fig. 1).

372

P. Delobelle et al. / High temperature creep damage

Photograph I. Example of cracks at grain boundaries.

( t m / t a ) = 1, see fig. 4) are much smaller than that of the equivalent stress. 3. 7.2. Case o f the stainless steel 17-12 S P H For this alloy the results are quite different: the observed defects are intergranular and often verify the criterion proposed to distinguish cracks. An example of intergranular cracks is given in photograph 1. (a) Cracks orientation histograms. They are given in fig. 8 (a, b, c); the number of cracks N having an orientation 0 is reported versus 0, fig. 8(a), for an area 4 to 6 turn 2. For the proportional tests (no. 2) and (no. 3) a single crack family occurs while two of them are definitely seen for shear component inversion tests. The maximum of each population approximately corresponds to directions perpendicular to the two positions occupied by the principal stress direction during cycling. In fig. 8 this is marked as _L npl , J_ np~, making an angle with the sample axis:

~ = ½tan-'(

2°~e t.

of cracks belonging to the family around that maximum in the histograms which correspond to the np~ value obtained at the end of the even half-cycle (left peaks in fig. 8) is smaller than that induced during odd halfcycles. This is consistent with the fact that rupture always occurs during an odd ha/f-cycle. (b) Evolution o f De and D t with the distance d. In fig. 9 the evolutions of the densities D c and Dt have been reported versus the distance d of the observed area to the fracture surface. Of course, these two parameters are decreasing functions of d and it has been considered that taking a value d of I mm when comparing the various tests was an appropriate choice. Fig. 10(a) shows the evolution of the ratio D i D t with %1 for d = 1 mm. This ratio which characterizes the crack shape of defects increases with op~. This result is consistent with that obtained by plotting u2/p versus °pl (P2 the sensitivity coefficient of rupture time to %1 was obtained by linearizing the curves In t a = f ( l n Opl ) of fig. 3, step by step). Moreover, this supports the observation of ~'2 depending of op~. In fig. 10(b), P2

7c-

2C

DclD~ (ram -2} .... Dc e; ~ ~ D,-/ , - - - - ~ '

~"

0 70

r~-(2)-j (6)--r (5]....

L I

i

|u

1

2

3

'

n9(3)

':

1(7)---

(10)

It is thus shown that crack opening occurs perpendicularly to the principal direction associated with the maxim u m principal stress Opl. This explains the results about rupture time, i.e. the large part taken by %1 and nr, ~ (I' 1 -- I'2, t R i / t R much larger than 1, see figs. 3 and 4). The damage anisotropy clearly appears during microscopic observations and is quite consistent with macroscopic measurements; moreover the unique result obtained on the same alloy in a tensile-torsion test at 5 5 0 ° C by Levaillant et al. [20] agrees well with the present results. It must also be noticed that the number

-(I)--

....

I I

. ~

1 I

2

d

Id[-mmJ 3

-

Fig. 9. Surface densities De and D t versus the distance d to the rupture area.

P. Delobelle et aL / High temperature creep damage versus op~ is also plotted. T h e results o b t a i n e d at E M P (Ecole des M i n e s de Paris) t h r o u g h the a d j u s t m e n t of d a m a g e profiles o n axisymmetric n o t c h e d samples using a finite element m e t h o d [21] are also reported in fig. 10(b). In this last work values of v2 = 4 at the highest stresses a n d P2 = 2 at the smallest ones were obtained. This result is consistent with that of the present work• A n o t h e r c o n f i r m a t i o n of the increase of v 2 with Opl comes from the c o m p a r i s o n of the rupture times of s m o o t h samples with those of n o t c h e d specimens. F o u r tests were p e r f o r m e d o n n o t c h e d specimens the shape of w h i c h being the same as the one used in the work of E M P (i.e. n o t c h e d radius R = 5 ram, m i n i m u m sample radius in the n o t c h e d part a = 3.8 mm, therefore a / R = 0.76). T h e experiments were tensile tests with a tensile stress far from the n o t c h Oa~z. F o r a given applied stress %,~, such that %~, _< 250 MPa, the linear relationship between In t R a n d In % ~ c a n b e approximately deduced from that of the unn o t c h e d samples t h r o u g h a simple shift along the stress axis (fig. 11); this is j u s t due to the fact that the actual stress in the m e d i u m n o t c h plane, is given b y the

5(MPa 50 ~-~'%~'-:.<(rR 1-450 . ^

:~ ~

-R.UU

373

~ R.Smm. ID R=Smm. ~_ R=l rnm. tfr°m E.M'P" =.

" ~

,l'~mj~oo ~

-3 0

ZOO

10 i

10-2

10~

I

0.5

1~3. f

200 i

I

2p t

i

• I/2

i

i

1

i

i

300 I

I

i

l, Pal

• E.M.P. [2'1]

8

-5

I

I\

t ,~L~

rt,u

Fig. 11. Comparisons of rupture times for smooth and notched samples. Determination of o a and of the shift ratio %~,/Oe~,.

B r i d g m a n formula (which takes into account the stress triaxiality) i.e.: --

[I)c/Dt] - + 0.7

"~04I

I

a

-I

,11,

T h e observed translation in fig. 11 between the curves of n o t c h e d a n d u r m o t c h e d specimens is j u s t equal to ln(%zz/%~z ). Eq. (11) is b a s e d o n a purely isotropic plastic model of the V o n Mises type; the agreement between eq. (11) a n d the observed shift shows that d a m a g e essentially depends o n ~ leading to a weak g2 value w h e n %zz a n d therefore %1 is small. F o r %~z > 420 M p a (fig. 11) the r u p t u r e times of n o t c h e d a n d u n n o t c h e d specimens b e c o m e equal for equal loadings, d a m a g e is therefore essentially governed by oa~~ ~ op]; this supports the large values of v2 obtained at higher applied stresses. T h e two above conclusions h o l d true w h e n other n o t c h dimensions are used a n d particularly for R = 1 m m [23] for which Oaz~/Oe~, = 1.58 (see fig. 11). T h e d e p e n d e n c y of v2 o n a N b e i n g n o w established the following evolution e q u a t i o n m a y b e proposed:

7 •

@

i

~

I

"

v2 = V ( ° p i / ° R )

I

I

Fig. 10. (a) Similarities in the evolutions of v 2 / v and D c / D t with %1- (b) Relationship v2 = f ( % l )-

where o R = 580 M P a .

(12)

It should b e noticed that o R is close to the ductile rupture stress at a n imposed strain rate ( i t ~- 10 - a s - l ) ; this value appears to c o r r e s p o n d to the intersect of the various In o R versus In t R experimental curves plotted in fig. 11 for the various types of sample geometries• v 2 = v w h e n %1 = °azz OR• =

P. Delobelleet al. / High temperature creep damage

374

The same behaviour, but much less pronounced, is exhibited by I N C O 718. It can be approximated as: p P2 = 3 ( O P l f l O R )

where

o R = 1 2 0 0 MPa,

(13)

The microscopic point of view may be used as a conclusion: it seems from the study of these two materials that the nucleation and the first steps in the isotropic growth of voids are governed by the equivalent stress while during the transition of the defects from the void to the intergranular crack shape, their opening essentially depends on the maximum principal stress.

being performed. Therefore, relationship (16) will be retained in the present state of the study; however, it might appear later that a contribution from a compressive state of stress to damage should be taken into account; this could be obtained through an additive combination of %1 and a. Taking into account the influence of the principal direction npl can be obtained through the introduction of the second rank tensor (npl ®npl), i.e. the tensor product of npl. Thus, the tensor D is defined as:

together with: / ) / / ) o = [(o;1~ 6 ( " - " 2 ) / o ~ ) ( 1 / ( 1 - D ) ) n ]

4. Interpretation and modelling

(,

with v2 = V(Opl/Og),

4.1. Damage evolution rules

or in an other contracted form:

The one-dimensional analysis of Rabotnov-Kachanov [9,4,5] leads to a damage variable obeying the equation: D//) o

=

( O / O o ) ~ ( l / ( 1 - D ) ) n,

(14)

w h e r e / ) o and oo are dimensional constants. Its solution predicts a rupture time ( D = 1) given by: t R = (1/(~

"~- 1 ) ) D 0 ( o o / O ) ' .

(15)

The generalisation of eqs. (14) and (15) to the case of complex stress states can be obtained using a function A(oo/oo) of the stress tensor invariants. In such a way, Hayhurst [1] introduces a linear function of the maxim u m principal stress %1, the first stress invariant J1 and the second invariant of the deviatory stress j~/2 (x ~. Contrary to the ductile plastic rupture case, Jx is generally negligible during creep. Analyses based on material science use instead of a product of the previous invariants [2,6,8]. In the present case, as in eq. (6),

bo=t~°0J~ °

t

~

(17)

[

1 ]

J

'

v = 9 for the 17-12 SPH and v = 14 for the I N C O alloy. The eqs. (17) constitute a simple approach to damage in agreement with the experimental results about the rupture times for monotonic loadings and which can be solved in an analytical way. A more complete theoretical modelling was proposed by Murakami et al. [3,24]. It consists in extending the uniaxial R a b o t n o v - K a c h a n o v basic concept of an effective stress 6 = ( o / ( 1 - D)) to the stress tensor. For this a second rank tensor • is defined as: = (!- D)-'

(1 unit tensor)

(18)

and the effective stress tensor, symmetrical, of the order two 8, is given by: 6= ½(o~+ ~o).

(19)

By analogy to the processes leading to eqs. (14) and (15) one can write:

" = "1 -}- " 2 ' A ( o i j / O o )

may be written as: J A (Oij/.O)

= ( "plJ'2v--0'-')'O

/ O 0" ~'/") .

(16)

\ i'~

Thus corroborating the Cane results [8]. Clearly, relationship (16) predicts D = 0 in a purely compressive test for which the maximum principal stress is zero; this might disagree with experiments. To verify this and assess the damaging capability of compressive states of stress, several cyclic creep tests with rectangular shaped cycles (i.e. tension, torsion with inversion, compression) including holding times in the compressive state are

P

l

P

J'l

(2o)

where: ~

"tl/2

= [~l~ij~ij )

~

13xl/2/

, a = (~)

~l ~t " l l / 2 (OijOij) ,

[

]

o~ being a component of the deviatory effective stress. It must be noticed that no ~ invariant leads to (1/(1 D)) when the uniaxial case is considered; however the term ( ~ ( ' - ~ ) ) is kept in (20), in the same way as ~ 4= v is introduced in (14). That is to say, ~/4= v has no clear physical meaning but it provides a supplementary

P. Delobelleet al. / High temperature creep damage parameter aimed at improving the fitting of the experimental creep curves by the model as does the factor (~o~-,)). It must be noticed that when the same rupture time in uniaxial tensile tests is taken for the two models, (17) and (18) to (20), their respective numerical predictions about proportional tensile-torsion curves are close to each other. The damage evolution rules (18) to (20) introduced above make damage depend only on time in creep tests, therefore tro/tR determination does not depend on the flow rules, t m / t R was calculated with 71= v in eqs. (18) to (20). The two extreme cases for INCO: v = v2 or /)2 = 0 are reported in fig. 4. i n the completely anisotropic case (first one, curve 2 in fig. 4) tpd//t R increases from 1 to 2 when At/, if changed from 0 to 90 °. In contrast, using the average experimental values for v and v2, leads to a predicted ratio tr~/t g always less than 2 (curve 1 in fig. 4). Moreover, fig. 4 also shows that damage anisotropy is larger for stainless steel 17-12 SPH than for INCO alloy, which agrees with experiments. However the model predicts that tRi/tR can never exceed two or be less than one and this is not in agreement with the experiments. In fact, as previously mentioned, it is likely that damage does not depend only on time but also on strain or strain rate; therefore damage and flow rules can no longer be separated. Adding a strain parameter aimed at accounting for the cyclic properties of the material should give more freedom for the predicted evolution of tav/tR and make it possible to get tr~/tR< 1 for cyclic softening and t m / t R > 2 for materials exhibiting cyclic hardening. The analysis of such model is presently proceeding. It is worth noting at this point that, contrary to the current eqs. (17) to (20), Pineau et at. proposed a model in which only strain takes a part [18,20].

4.2. Flow rules It has already been mentioned that cyclic properties and the flow direction rotations (fig. 6) can be accounted for through two internal variables respectively isotropic and kinematic; the recovery of which is not negligible because of t~ae high temperatures. A model developed by the authors [25,26] for the undamaged states of the two alloys under consideration gave very satisfactory predictions. In a simplified form, damage is introduced through the relations (18) to (20) and the model can be expressed as follows. A state equation: •

r

1- I

-

1

iU =-32,0 s i n h [ - - - ~ ] [ ----8----~],

(21a)

375

with (21b)

= ½(a~ + ~a).

A criterion to distinguish micro- and macro-viscoplastic flows: & - &2= Y,

(22)

with a consistency equation (which will relate &ij, &2q, Y and their time derivatives): -t

~t

,LI

a,7= a2, 7 + ( Y / ~ " ) ( ~xij"- a2,j) .

(23)

The evolution rules of the internal variables: (2) The kinematical one:

°2,, p2(Y ,j- .,j,) -.2 2 .t

~

t

-

=m

~t

(24)

%

(initial condition ~2u(0) = 0). (b) The isotropic one: I ? = p 3 ( y sat- y ) ~ - R 3 Ym,

(25)

(initial condition Y(0) = Y0)- One gets Y0 < ysat in the case of 17-12 SPH, (leading to cyclic hardening) and Y0 > ysat, for the INCO alloy (cyclic softening). (c) Damage: see (18) to (20); in these equations, (2//3)l/2(iijiij) 1/2, P2' RE; ysat, P3, R3 and m are constants depending on temperature. =

[3Xl/2Z =

t

t

,j

t

,,1/2

,p

;

~2 being given by an equation such as (21b). The model is constituted of eqs. (18) to (25)• A more detailed description can be found in [25] and [26]. It must be pointed out that the straightforward introduction of eq. (19) in the (21) to (24) system does not a priori fulfil the principle of the symmetry of the relationship between the stress components and the strain rate components. Following Murakami et al. [3,24], this difficulty can be solved in two ways• On the one hand a fourth rank tensor ¥ ( 0 ) allowing another definition of the effective stress can be introduced: g = l ( ~ t ( ~ ) : o + ( ' ~ ( ~ ) : o)T);

(26)

on the other hand, a simpler isotropic theory of the D tensor, for instance = (1 + C trace(D))e,

(27)

can be used. The same holds true for the effective internal variable & definition. In the present case, it can be shown through numerical tests that the misfit to the symmetry principle is significant only when rupture is

P. Delobelle et a L / High temperature creep damage

376

gO0-3)

-~-- ' • l • I % ]

1

|

.°=:r~z}ex p.

3

--r/_..- 2 4 , r/ -..V = 1 0 ~calculus -'-V .14 , ~ -,. u ,

o

o

o o o__._

0.5

. -

100

t 4

INCO[n..61

.,,

ZZ

"

__ _ . . _ ~ . ~ - ~ f

_ . ....4~ s e o e e e e eeee

soee eeooeoe

1

" "~-'-

300

2

°zelexp INCOIn'81 ,~=~ ] c a k d u s ~ . . . oo"

so

,50-'--.....]200

500

:

/ o oO 3';

o o ° ° °I

,[hi

2 |

. ."

""

~

z ° o

Fig. 12. (a,b) Examples of predictions obtained from eqs. (21)-(25). While the beginning of the strain curves and the rupture times are correctly predicted, rupture strains are strongly underestimated whatever the value of the (~1 - v) parameter: ~/- p = 0; ~ = 14 and (~/- v) = 10, ~/= 24. -Cases of a proportional loading and of a test with torque inversions.

approached ( D sufficiently high) and therefore that it only modifies slightly the model responses and consequently the conclusions drawn from the numerical study. Moreover, as the symmetry is ignored only in the last parts of the test, in future, it should be worth considering the effects of the anisotropic damage on the constitutive laws in an isotropic way. The first complete predictions of eqs. (17) to (25) on I N C O 718 (examples in fig. 12(a,b)) and 17-12 SPH alloys show that: - Primary and steady creep curves are correctly described (monotonic loading and loading with torque inversions, fig. 12(a,b); this is not surprising as they correspond to undamaged states for which the model is valid, - The rupture times are fairly well predicted for proportional loadings, (fig. 12(a,b); here too this had to be expected as eq. (20) was derived from the analysis of rupture times. As mentioned above, in the stainless steel case, the measured values of. t R are too small for pure torsional test because of buckling. - On the contrary, for both alloys, theoretical tertiary creep is too weak when * 1 - v = 0 and viscoplastic strain at rupture is clearly underestimated (fig. 12(a,b), the same holds true when ~1- ;' ~ 0 as rupture strain is a decreasing function of r l - v. This artefact is illustrated in fig. 12(a) for two 7 / - P values. One can notice that apart any physical considerations, it is possible to approximate in an analytical way rup-

ture strain iR using the experimental parameter X = = (17 + 1 ) / ( I 7 - - /1 + 1). The value , / = 8,75 is obtained ( i R = 8.8%). When the definitions of damage and effective stresses (eq. (18) and (19)) are used this leads to (71 - v) < 0 in eq. (20), which is not consistent from a physical point of view. This is why the lowest possible ~/ value was taken equal to v (i.e. 14) (fig. 12(a,b) in the case of INCO). Therefore, once more, introducing strain in the damage rule becomes necessary to improve the solution of the rupture strain problem. It should be noticed that this difficulty arises in all the performed simulations as all of them underestimated rupture strain. ~R//tR~min

5. Conclusions

The part played in damage by the equivalent stress 8, the maximum principal stress %1 and the corresponding principal direction n pl has been pointed out through torsion-tensile creep experiments on two alloys with different mechanical behaviours. Damage rules using a damage tensorial variable in a similar way to the model of Murakami et al. [3,24] are proposed; it is aimed at completing a model including kinematical and isotropic hardening variables. The inadequacies of a model where D depends only on time, (RabotnovKachanov extensions) i.e. the evolution of the t m / t R ratio ( t ~ / t R > 2 a n d / o r ~ 1) not predicted, the underestimation of rupture strains, the non-prediction of the effects of preloading the samples are pointed out. It is

P. Delobelle et al. / High temperature creep damage therefore proposed to introduce strain or strain rate in the d a m a g e rules; this proposal is currently b e i n g investigated.

Acknowledgement This study is p a r t of a p r o g r a m c o n d u c t e d b y the G R E C O ' G r a n d e s D 6 f o r m a t i o n s et E n d o m m a g e m e n t ' . It was s u p p o r t e d t h r o u g h a c o n t r a c t in the case of the 17-12 S P H stainless steel ( E D F - K0665) a n d helped b y S N E C M A in the case of I N C O .

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377

[14] P. Delobell¢ and C. Oytana, Experimental study of the flow rules of a 316 stainless steel at high and low stresses, Nucl. Eurg. & Des. 83 (1984) 333-348. [15] J. Bcziat, A. Diboine, C. Levaillant and A. Pineau, Creep damage in a 316 stainless steel under triaxial stresses, IV Int. Sem. on Inelastic Analysis and life Prediction in high Temperature environnement, Chicago-Illinois, USA (1983). [16] W.A. Trampczynski, D.R. Hayhurst and F.A. Leckie, Creep rupture of copper and aluminium under non-proportional loading, J. Mech. Phys. Solids 29, 5 / 6 (1981) 353-374. [17] S. Murakami, Y. Sanomura and K. Saitoh, Formulation of cross hardening in creep and its effect on the creep damage process of copper, J. Eng. Mat. Techn. 108 (1986) 167-173. [18] J.L. Chaboche, K. Dang-Van and G. Cordier, Modification of the strain memory effect on the cyclic hardening of 316 stainless steel Proc. of SMIRT 5, Division L, Berlin (1979). [19] P. Delobelle, D. Varchon and C. Oytana, A biaxial tension-torsion constant stress creep testing machine in: D.J. Gooch and I.M. How, eds., Techniques for Multiaxial creep testing, Chap. V, (Elsevier Applied Science, New York, 1986) 93-101. [20] C. Levaillant, A. Pineau, M. Yoshida and R. Piques, Creep tests on axisymmetric notched bars; global displacement measurements and metallographic determination of local strain and damage in: D.J. Gooch and I.M. How, eds., Techniques for Multi-axial Creep Testing, Chap. XI (Elsevier Applied Science, New York, 1986) 199-208. [21] H. Carrerot, Diplbme d'Etudes Approfondies Report, Ecole Nationale Sup~rieure des Mines de Paris, 1986. [22] P.W. Bridgman, Study in Large Plastic Flow and Fracture (McGraw-Hill, New York, 1952) 1-37. [23] E. Contesti, G. Cailletaud and C. Levaillant, Creep damage in 17-12 SPH stainless steel notched specimens, metallographical study and numerical modeling, J. Press. Vessel Techn. 109 (1987) 228-235. [24] S. Murakami and N. Ohno, A continuum theory and creep and creep damage, I.U.T.A.M. Symp., Leicester, in: A.R.S. Ponter and D.R. Hayhurst, eds., Creep in Structures (Springer-Verlag, Amsterdam, 1980) 422-443. [25] P. Delobelle and C. Oytana, Etude des lois de comportement h haute temp6rature .en plasticit6-fluage d'un acier inoxydable aust6nitique 17-12 SPH, J. Nucl. Mater. 139 (-1986) 204-227. [26] P. Delobelie and C. Oytana, Modeling of 316 stainless steel (17-12 SPH) mechanical properties using biaxial experiments; Part I: Experiments and basis of the model; Part II: Model and simulation, J. Press. Vessel Techn. 109 (1987) 449-454; 455-459.