Some elements on multiaxial behaviour of 316 L stainless steel at room temperature

Mechanics of Materials 3 (1984) 333-347 North-Holland

333

SOME ELEMENTS ON MULTIAXIAL BEHAVIOUR OF 316 L STAINLESS STEEL AT ROOM TEMPERATURE G. CAILLETAUD Centre des Matbriaux, Ecole Nationale Sup&ieure des Mines de Paris, France *

H. KACZMAREK and H. POLICELLA Office National d'Etudes et de Recherches Abrospatiales Chhtillon, France Received 29 October 1984; revised version received 3 December 1984

This paper presents some experimental results on 316 L stainless steel at room temperature. Tubular specimens are subjected to axial and torsional strain controlled loadings. The main investigation deals with the additional hardening due to multiaxiality: it is present every time that the strain (or stress) path is non-proportional (two-level tests in tension and torsion, circular or square paths in the strain plane). In order to characterize the amount of hardening, two-dimensional simulations on the volume element are performed with a classical constitutive model with isotropic and non-linear kinematic hardening.

1. Introduction The non-uniqueness of the cyclic stress-strain curve is a well-known phenomenon for materials like stainless steel: the so called 'memory effect' has been studied in the past and several plasticity theories are able to describe the additional hardening due to high-low loading under uniaxial cyclic loading (Chaboche, Dang Van and Cordier, 1979; Dafalias and Seyed Ranjbari, 1982; Ohno, 1982); some multiaxial proportional prior loadings can also be represented and the resulting state of hardening after industrial thermochemical treatment (drawing, forging, rolling...) is finally characterized by the maximum equivalent plastic strain range during the forging process (Nouailhas et al., 1983). In stainless steel, this type of 'memory effect' clearly relates to the presence of a cell microstructure, the size of which is strain dependent (Chaboche et al., 1982). This paper is devoted to the description of * Tests and calculations were done as the author was with Office National d'Etudes et de Recherches A6rospatiales

another type of path dependence of cyclic hardening: the additional hardening due to multiaxiality. This effect has already been pointed out in the literature by means of tests involving out of phase axial-torsional loadings in the strain plane on copper (Lamba and Sidebottom, 1978) and 1-CrMo-V Steel (Kanazawa, Miller and Brown, 1979). In the present study, the same type of tests is performed on 316 L stainless steel, but we also try to quantify the amount of hardening by investigating other strain paths (e.g., two level tests in tension and torsion, square paths, small angle out of phase loading). The hardening will be characterized by reference to uniaxial experimental data and multiaxial calculations.

2. Material, specimens, test procedure The material employed in Z2 CND 17-13 (316 L type stainless steel), taken from 37 mm diameter bars and subjected to a solution heat treatment before machining. Some 25 mm diameter bars with the same treatment have been tested from the same heat (n ° T0089 supplied by Creusot Loire)

0167-6636/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)

334

G. Cailletaud et al. / Multiaxial behaviour o/' 310 L S.S.

and served for the uniaxial characterization of the material using classical smooth fatigue specimens (6 mm diameter, 12 mm length of the cylindrical part). These uniaxial test results will be shown in the following as 'uniaxial reference'. This type of alloy has been studied extensively under uniaxial loading. Incremental tests, two level tests under stress or strain control have already been performed (Chaboche, Dang Van and Cordier, 1979, Chaboche, Kaczmarek and Raine, 1980; Nouailhas et al., 1983, Chaboche et al., 1979). The specimens used for tension-torsion tests are thin-walled tubes, with inside and outside diameter of 10 and 12 mm respectively and a gauge length of 18 mm. The tests are made at room temperature and the specimens are instrumented with electrical resistance gauges. Testing is performed under angle and displacement control so that the destruction of the gauges (after about

5 A

1000 10000 cycles at low strain levels but after only 10 cycles for a strain level of more than 3%) does not affect the test. In fact the readings of the gauge allow for checking the strain level in the cylindrical part. In the first cycles, an important strengthening appears in the plastically deformed cylindrical part and the strain amplitude decreases while the total angle amplitude is kept constant. After a few cycles equilibrium is reached, and the plastic strain profile along the specimen is then unchanged with cycles; under these conditions the local axial and shear strains remain constant and it is possible to characterize equivalent lengths for plastic behaviour in tension and in torsion. Different experiments show that these equivalent lengths are practically independent of the type of loading, and hence they are taken constant for all the tests. The principle of the analysis when strain gauge measurements are no longer possible are given elsewhere (Cailletaud, 1982). A computer allows to apply any type of loading to the servohydraulic machine, the signals being numerically generated. The data acquisition and the graphic output are made with the same computer. The following notations are used: expression of the total displacement in tension -

a. 8

~~

D = D e + Dp = ~

% -5 -0.5

5.

!

-0.025 Fig. 1.

+ L~epO.

(1)

expression of the total angle in torsion B = fie + fly = C / R

7 0.5

0.025

b.

F

J 0.025

+ Lp% 0 ,

(2)

( F is the axial loading, C the torque, R* and k are experimental stiffnesses, Lp and Lp are equivalent lengths, ep0 and C~p0 are the plastic strain and the plastic rotation per unit length, the plastic engineering shear strain yp being obtained as: yp = ~p x Maximum radius). The shear stress r is evaluated on the outside surface of the specimen using a viscoplasticity correction (Delobelle, Oytana and Mermet, 1979), so that finally the chosen value lies between the elastic and perfectly plastic calculations. Finally the total axial strain e and the total engineering shear strain y are calculated from: = %o + o / E Y = YpO + r / t z

( E = Young's modulus),

(3a)

(~ = shear modulus).

(3b)

G. Cailletaud et al. / Multiaxial behaviour of 316 L S.S.

The method has been tested on a number of loading cases. Fig. 1 and Fig. 2 show the results obtained for a complex test (no. 20 in Table 1). The imposed straining path follows " X " branches in the angle-displacement plane (Fig. l(a)) The analysis by the equivalent length method predicts a different path in the strain plane for the cylindrical part of the specimen. As a matter of fact, the resulting loop (shown in Fig. l(b)), is very similar to the results coming from strain gauges recording. The accuracy of the analysis for each strain component is finally established by the curves in Fig. 2(a) and 2(b), for which the calculated axial and shear strains are plotted vs. the corresponding strain gauge signal for cycle 10. Note that in Fig. 2 and in the following, equiv0.025

335

Table 1

,i, ]215I,.

Pure tension Pure torsion

In phase, small phase difference

3

10

00o

90° phase difference

15

21

Two level tests

I

17

I

e (calc)

,V

Complex paths

×i

alent stress and plastic strain are used:

2,

/

A similar quantity is introduced for the total strain e(gauge}

- 0.025 - 0.025

0.025

v

Note also that there is not a simple relation between ~, ~p and 6.

0.025

3.

/ - 0.025 - 0.025

Fig. 2.

+ T02/3.

'

"X/'X/3"(gauge) 0.025

Experimental

results

All the tests are performed using symmetric loading, under constant angle and displacement amplitude control. As shown in Table 1 five categories of tests were made: - reference tests in either pure tension or pure torsion, test with 0 °, 8 ° or 20 ° phase difference, tests with 90 ° phase difference, two level tests in tension or torsion, - complex paths.

336

G. Cailletaud et al.

/

Multtaxial behamour

In Fig. 3 the results obtained in pure tension or pure torsion are plotted. They show that von Mises criterion correlates the torsion and tension data. It is clear also that changing the type of loading produces a strengthening. This effect is very important for high-low type loadings; so it is confirmed that the 'strain memory effect' already observed in one-dimensional tests is still present here. After a first level in tension or torsion with an equivalent strain range of about 3%, the behaviour in torsion or tension (test 8 or 9) for an equivalent strain range of about 1% is abnormal: the stresses are 150 MPa higher than those of the 1% reference data points. But a little strengthening is also present if the same equivalent strains are used for both levels (tests 6 and 7): this 'crosshardening' effect will be discussed in Section 6. Representative points of the other tests (out of phase tests, complex paths) are plotted in Fig. 4. This diagram clearly shows the strengthening due to multiaxiality: the increase of maximum stress for a given plastic strain amplitude is very important (more than 250 MPa) for 90 ° phase difference, which presents the highest 'degree of multiaxiality' i.e. the direction of the strain increment is always perpendicular to the strain vector deft-

~^,V~/2 & /

(Mr'a) OH

600

~"fo") ,~00-

9

~ ~ Umaxial

÷~

/

~

/

gefeFence

+ [ens~on o l()rStOt)

/

/

o

200 --

~ 316 L

S.S.

800

0

UDtaxlal ~~ge[efet)co

1

,X

"

I

t

o

/

+ tens.ion ~ .tor s!on • in phase tension-torsion %68°pha.se difference in tension torsion (tests n ° 3, 10, 15) 9. test n ° 21

/

ZOO

/ / Cl/

200

1

)

cptq(%)

Fig. 4.

ning the present plastic state. Intermediate values are obtained for the other loading cases (tests no. 18-20-21). In the following we give a more detailed account of each type of test. For a better understanding of what is and what is not ' n o r m a l ' in the stress-strain behaviour, a numerical simulation with a classical viscoplasticity model is superimposed on the experimental loops. This model (Chaboche and Rousselier, 1981) introduces one or several non-linear kinematic variables, the sum of which represents the center of the elastic domain, the radius of this domain being kept constant as long as only the stabilized cycle is modelled. The expression is: = 3•

0 .~

-

Xr

(5a)

EpH 0 Fig. 3.

T--

±, = c i ( ~ a i ~ p - x i p ) ,

(58)

I

(superimposed dot denotes the time derivative; ep,

G. Cailletaud et al. / Multiaxial behaviour of 316 L S.S.

1000

xi and o are second order tensors o' and x' are their deviatoric part), with

337

(7 (MPa)

n

J(o-x)=

(1.5(o'-x'):

(o'-x'))

(5d)

1/2

a.

0

and x = E x,

(5e)

i

where n, K, R, c~, a~ are material coefficients chosen as below (units, MPa, s).

(%) -

1000 -

n

K

R

al

cl

a2

c2

25

151

180

150

400

270

12.5

Using such a type of model is justified by the viscosity of 316 stainless steel even at room temperature. In fact all the tests are performed with similar strain rates (10 s for crossing the twisting angle or displacement range), and equivalent results would be found with a plasticity model without time effect.

0

2.5

500

"rx/3- (MPaf~/j

b. 0 4. Reference tests, small angle phase difference As shown in Fig. 5, pure tension or pure torsion are consistent for a given strain level. Let us note that the presence of a memory of the maximum strain in the alloy studied makes simultaneous prediction of the shape of the stress-strain loop and of the cyclic hardening curve difficult, even with an elaborate model. For this reason the coefficients are fitted to give good results for equivalent strains of about 3%, which covers most of the tests. The same coefficients will be used for all the simulations. The basic results concerning these tests are reported in Table 2. For the test where the twisting angle and the axial displacement are in phase (Table 3), the axial and shear strains are out of phase. For producing in-phase local strains in the cylindrical part, one has to impose a phase difference at the extremities of the specimen (see (1), (2), (3)). In the following, the delay of twisting angle relative to the displacement is denoted by ,~, the delay of the shear strain

t r

P

- 500

25

-2.5

C_.-J

y

0

2.5

Fig. 5.

relative to the axial strain ~p. This delay will be characterized as a percentage of the period for triangular waveform loading (for instance tests 4 and 11) and by a value in degrees for sinuso~dal wave form. This relationship is first illustrated by results of test no. 23 (Fig. 6) for which the twisting angle and the axial displacement vary sinusoidally, the angle signal being delayed by 8 ° relative to the displacement. In these conditions the e-y diagram (Fig. 6(d)) does not exhibit any phase difference (4 = 0) and the numerical simulation remains in good agreement with the experimental stress-strain

o'1

0

Ig

I o

o

oZ_..

0

I

~-

a-

I

g

p

o

o

Cl

P~

ol

0

O 0

I

l

q

--'1 ;,"

~

n

4>

o

I

o

l

i,~

o

.

!

A

r~

g

c~

x

L~

g,

g,

o

p

.~

o

1+

I+

0

~.1

p

t+

L.D

~

o

I+

"e

~

o~

X

~h

~'

m-

b~

B

E

339

G. Cailletaud et al. / Multiaxial behauiour of 316 L S.S.

loops in axial and shear plane (Fig. 6(a) and Fig. 6(b)). Nevertheless it should be noted that the shape of the calculated loops is identical in Fig. 6(a) and Fig. 6(b) but that a difference is found for the experimental loops. A stress saturation appears in the shear diagram while the loop in the axial plane remains sharp. The shape of the stress-strain loops is no longer classical if a small phase difference is" applied. For the test no. 11, the local delay of e is 0,022 (triangular waveform), that corresponds to a value of - 8 ° for ~b: as shown on Fig. 7(a), the maximum shear stress is not obtained for the maximum shear strain. This trend has to be correlated with the normality rule in plasticity and can be observed also in the numerical simulation of the test. Nevertheless, the value of this maximum shear stress is not correctly represented; the difference between the experimental and numerical maximum shear stress is about 80 Mpa. Note that at the same time

the difference between experiment and simulation remains very small for the axial component (about 10 MPa for the maximum stress) and that the maximum equivalent stress is obtained at the maximum axial strain. After test no. 11, Fig. 7(b) shows the result obtained using the opposite phase difference. For the test no. 22, the local phase difference is ~ = 8.5 ° and the aspects of the stress-strain loop are inversed compared to the previous case. Now the pointed loop is obtained for the shear components, and the prediction is good in this case. The additional strengthening appears on the axial stress and the calculated maximum axial stress is 80 MPa too small. For these last two sets the disagreement between simulation and experiment appears clearly in Fig. 8 which shows the loading paths in the stress plane.

500

5°° I

0

Y/ ',,_..J -

500

-

- 0.025

0.025

E

500 0.025

0.025

a.

500

500 r

-

5ooi

~/'J3 -

-

Fig. 7.

0.025

0.025

500 0.025 -

0.025

340

G. Cailletaud et al. / Multiaxial beha~iour of 310 L S.S.

rx/3~

1000 U---=

I

-

/ // /

-

I// ~//

500

500

_ l o o o L_ . . . . . -

5oo!~/~

i

.

o ) 1000

. . . . . . . . 7 ..................

/

/

,

o

500 t

500

-

i

1000

o.o25

/ -

:

500

_0.025[__ . . . .

Fig. 8,

-

Fig.

5. 90 ° phase difference Two types of tests are considered here. For specimens number 3, 10, 15, the maximum shear and axial strains are equivalent; for test 21, the maximum shear strain has been reduced to the third of e (see Table 4). In fact, as shown in Fig. 9(a),the true phase difference on axial and shear strains at the center of the specimen is about q~= 68 ° for test 3; the same value is valid for the

[. . . . . . . . . .

ej

0.025

0.025

9.

other tests. During the initial cycles, the maximum strain decreases and reaches a stabilized value after about 10 cycles; nevertheless at the same time, an important strengthening can be noted on the axial and shear stresses, as shown on Fig. 9(b). Figure 10 summarizes the shape of the resulting curves for stable conditions in o-r~ diagram for the four tests.

Table 4

Test

Displacement (ram)

Angle

A£

A¥/~5l

%

%

v~ (degree)

(degree)

A~;p %

~Lyp %

]

M'Pxa~/2

''lMP3/Tf/2 a

(degree)

+0.55

+6,77

-90

3.21

3.09

-68

2.50

2.56

813

797

I0

+0.428

+4.88

-90

2.28

2.07

-68

1 .53

I .36

690

739

15

+0.282

+2.42

-90

1.00

0.83

-68

0.45

0.35

5t0

[

505

21

+0.282

+1.20

-90

1.04

0.32

-68

0.60

0. I0

425

[

360

3

G. Cailletaud et aL / Multiaxial behauiour o/316 L S.S. I

(~(MPa)

10~ \

:~\

{ ....

W

l

,v'5 CM%~ )

Fig. 10.

On each axis there are plotted two pairs of points, corresponding to uniaxial references (tension or torsion). Triangles are related to an axial

1000[o-

[

341

straining of 1% (o axis) or to an equivalent shear straining of 0.3% (~-v~ axis) (i.e. the same strains as in test 21). Squares are related to 3% axial or 3% equivalent shear strain (i.e. the same strains as in test 3). So the respective differences between triangles and loop 21 and between squares and loop 3 provide a measure of hardening due to multiaxiality in each case. The amount of hardening is about 140 MPa in the first case and 280 MPa for the second. This hardening seems to be constant over the whole cycle, even for test 21 in which the shear component is very small. It becomes surprisingly high, as the stresses are increased by factors 1.4 to 1.8. In addition characteristic points are noted on the loop of test 3 (twisting angle or displacement rain, max, or zero), showing the phase difference between stress and strain. This effect can be directly marked on the stress-strain loops which have no apparent elastic domain. This is obviously due to the fact that the loading path does not include any unloading period since the equivalent stress 6 remains constant. This is shown in Fig. 11: the only linear part is the first ~hear loading, when axial load is not applied.

6. Two level tests

' -

.......

1000

- 0.025

1000

-

1000

0.025

T

- - -

- 0 . 0 2 5

Fig. 11.

[. . . . . . . . . . . . .

-

,i 0.025

Table 5 summarizes the maximum stress and strain values in this type of test. The loading change starts just before the half life for the first loading level; the reported values correspond to the last cycles (stabilized conditions) for the first level and to the 10th cycle for the second level. As previously explained, a 'cross hardening' is present at the second level, even for the equivalent strain range, this hardening being (at least partially) evanescent, see Fig. 12 for the stress evolution vs. number of cycles for test 14. Let us note that for test n ° 14 a phase difference equivalent to ~b= - 8 ° can be seen on the strain signal (as for test no. 4) and that the phase difference is magnified in the stress plane: Fig. 13 shows the resulting loop in this plane just before and just after the loading change.

342

G. Cailletaud et aL / Multiaxial behat)iour o[ 316 l, S.S.

600-

500

kC

'\

,~00-

/

k /~oI ////k~X

200-

301 ~' 350

-200-

,

/

-400-

-60(

-

-

0

500

500 - 500

500

Fig. 13.

Fig. 12,

Table 5

Test

Test

Imposed loading level i, level 2

Ayp

.'~ :,v/,/y%

MP a

+ 6.77 °

2.56

491

3.03

+

0.55

2.15

580

2.78

+

0.55

2.45

510

3.00

+ 6.77 °

2.49

526

3.00

+ 0.55 mm

2.40

510

2,95

+ 2.13 °

0.40

421

0.80

+ 6.77 °

2.45

480

291

+ 0,201mm

0,15

390

0.57

~

~

ASp

gyp

/<,

OM

TM¢~

~

~

0

-0.022

1.43

1.89

400

420

1.86

2.2]

-0,5

-0.522

1.37

1.83

420

420

1.82

2.21

14

Imposed

loading

for test

]4 = J 0.40 mm, J 4.92 °

343

G. Cailletaud et al. / Multiaxial behaviour of 316 L S.S.

7.Complexpaths

neous cycling, (2) during the ten subsequent cycles, even if the loading increment is perpendicular to the first direction, the normal to the surface still presents a component in this direction and plastic flow exists. So this effect can be modelled using the viscoplastic equation previously described. Another type of effect can be noted in test 18. 'Cross hardening' is present in this case every ten cycles, and Fig. 15 indicates that hardening is not saturated at all after the first loading change. It can be observed during the whole life, so that the final stress levels are very high. In this test, it must be noted that the increase of hardening due to multiaxiality is (at least partially) evanescent when going back to radial loading. The hardening mech-

Figures 14, 15 and 16 give an overview of what happens in tests 18, 19 and 20. Figure 14(d) shows the consecutive ten cycles in e and y direction, Figure 14(a) and 14(b) exhibits the consecutive stress-strain loops in axial and shear plane: during the cycling in shear plane, there is a drop of axial stress (from A to B in the axial plane, see also the Fig. 14(c)) and the opposite is true (drop of shear stress from C to D, Fig. 14(b) and 14(c)) during the axial cycling. This phenomenon has to be related to the presence of kinematic hardening in the material, then (1) the center of the yield surface has moved in the direction of the instanta-

500

5 0 0 ,.

_

r~/3

3~.590 ~ It

600

Cycle 5 8 0 cycle 5 9 0 a.

- 500 - 0.025

0 025

5OO

- 500

0.025

II

500

~/,,/~

Cycle 5 9 0 cycle 6 0 0

b.

- 500 - 0.025

Fig. 14.

j

:,/,/5 0.025

- 0.02

- 0.025

0.025

344

G. Cailletaud et al. / Multiaxial behaeiour of 316 L S.S.

I (M,o,~)

anisms are reactived at each change in direction of strain. That confirms the observations made on the two-level tests (Fig. 12), which cannot be explained by a classical model: for a good representation the radius of the yield surface must be isotropically increased by using an integral variable keeping memory of the loading changes (Mac Dowell, 1983), or this yield surface must be considered as anisotropic (for instance elliptical, the larger diameter being perpendicular to the loading direction (Baltov and Sawczuck. 1961).

,/~"~\

/

....

30O 200 ~

N

T

tO

IO0

lO00

Fig. 15. Angle

3 Displacement

a,

'

4 4

/

~

2

Time .---

3

0.025 ~/.,,/~

T. t ! i

,ooo or~- ......... F. . . . . . . . . . . . .

b.

I

-,,

--

3e - ~ _ J

4

I

I

1i - 0.025 - - - 0.025

.£_

.....

i

0.025

i

\ L-V//

- l oool

J_

- 0.025

I-

looo !~,/~

12S f ! \ , ,

< 0.025

\

c

\

N

\ 3 _

1000

L

-

1000

Fig. 16.

i

",-4 /

/

/ 4

i

z/

!

!

E

:

t

[

o 1000

- 0.025

0.025

G. Cailletaud et al. / Multiaxial behat, iour of 316 L S.S.

As for the square path, the problem is not so complex: - Figure 16(a) shows the imposed twisting angle and displacement and Fig. 16(b) the resulting axial and shear strains at the center of the specimen,

Angle

Time

Displacement

Time

0.025

r-

-

]- .

1~/ff3

i

3

i

. . . . . . . . . . . .

I

1

4

[

I i

5

I i

7

-o.o25 _ ............... J__ -

0.025

1000

~ _

,,/3

0.025

..........

-r- . . . . . . . . . . . . . .

!

i

I

i

7

i

! i

-

1000' -

Fig. 17.

0.025

oj 0.025

345

with the usual phase difference if compared to displacement and angle. - In Figure 16(d) and 16(e) are plotted the resulting stress-strain loops in the corresponding diagrams. - Figure 16(c) indicates that the loading path in the stress plane is almost circular. This explains the very high 'measured' hardening in this test (Fig. 4). It is found to be practically equivalent to out-of-phase tests. Of course this hardening cannot be represented by a classical model. For test no. 20, the imposed loading is quite complex (see Fig. 17(a) and (b)) and it is not surprising to find also a complex response (see Fig. 17(c)). But it must be noted that the qualitative aspect of the stress-strain loops (large-little loops, angles, location of maximum stress ...) is given by the classical model. Nevertheless the quantitative value of the maximum stresses is not obtained (compare Fig. 18(a) and 18(b), Fig. 18(c) and 18(d)). Finally, the additional hardening due to multiaxiality is qualitatively characterized in Fig. 19. On the y-axis is reported the 'stress default', that is the difference on axial or shear stress component between experiment and the numerical prediction and not the difference on equivalent stress which would be less sensitive. It is obvious that the hardening increases when the 'degree of multiaxiality' increases. The amount of hardening ( - 80 MPa) is found to be small for tests with 8.5 ° phase difference, but very high for square path (350 MPa) or 68 ° phase difference. It is influenced by the phase difference, and not by the maximum strain range. In tests 3 and 10, for a maximum equivalent strain of 1,6% and 1,08% the stress default is 300 MPa and 280 MPa and in test 15, for a maximum equivalent strain of only 0,5% it is still equal to 225 MPa. Numerous studies are in progress to give a right expression of the 'degree of multiaxiality'. It must be noted from Fig. 19 that this variable deals with the plastic strain and not the total strain. For this reason, a sudden drop of stress default is found for test 21, in which the phase difference is still 68 °, but where plastic deformations are very small.

G. Cailletaud et al. / Multiaxial behaviour of 316 L S.S.

346

1000

5

b.

4

J -

t

2

c (Exp.)

000 - 0.025

e (Calc.)

0.025

1000

~

13

C,

d.

4 -

"7/~/3- (Exp.)

1000

- 0.025

0.025

Fig. 18.

Conclusion Stress default (MPa) 000) 0[3)

oo~ 200

Xr~ °(20

tOO¸

Degree of multiaxibilife (to be defined)

8~o Phase difference Fig. 19.

Phase difference

Various strain paths were performed in tension and torsion on 316 L type stainless steel at room temperature. The hardening is normal as long as the strain path is proportional. Any change in direction of straining increases the hardening level. This hardening is (at least partially) evanescent when the strain path becomes radial, but the hardening mechanisms are continuously activated in out of phase loading so that the additional hardening is very high. In this case the equivalent stress can reach 800 MPa for strain ranges around 3%, that is 300 MPa more than the uniaxial cyclic hardening curves. This paper does not propose any modelling of these phenomena, but only shows the experimental data for future interpretation. The work to be done can be evaluated from the text by comparing experiments and numerical simulations made with classical viscoplastic model. It clearly shows (see

G. Cailletaud et al. / Multiaxial behaviour of 316 L S.S.

Figs. 7, 16, 18) that w i t h a n o r m a l i t y rule a n d v o n M i s e s c r i t e r i o n , the q u a l i t a t i v e a p p e a r a n c e of the stress strain c u r v e is o b t a i n e d , b u t the a g r e e m e n t is n o t g o o d f r o m a q u a n t i t a t i v e p o i n t of view.

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