The residual strain of polycrystalline graphite (II)

Journal of Nuclear Materials 57 (1975) 205-211 © North-Holland Publishing Company

T H E R E S I D U A L S T R A I N O F P O L Y C R Y S T A L L I N E G R A P H I T E (II)

Relationship between the strain and the energy dissipated during compression tests M o t o k u n i ETO

Japan Atomic Energy Research Institute, Tokai-mura, Ibaraki.ken 319-11, Japan Received 17 January 1975

Studies on the relationships between the residual strain and the energy dissipated during the compressive loading and unloading cycles are carried out for nuclear and other commercial graphites. It is also investigated whether there are correlations between the fracture energy and mechanical properties such as the compressive strength of these materials. The results show that for needle coke and fine-grained isotropic graphites the residual strain e 0 = K1/:~-, while for Gilsocarbon graphites e o = K2 ,E~. Here, E is the dissipated energy and K~ and K 2 are constants dependent on the kind of graphite and the specimen orientation. These relationships can be derived on some assumptions concerning the mechanisms for deformation of needle coke or Gilsocarbon graphite. It is also found that both the compressive strength and the fracture strain, are proportional to the square root of the fracture energy for various polygraphites. Des ~tudes sur les relations entre la d&ormation r~siduelle et l'~nergie dissip~e durant des cycles de charge par compression et de d~charge ont 6t~ entreprises sur des graphites nucl6aires et sur d'autres graphites commerciaux. On a ~galement ~tudi~ s'il existe des relations entre l'~nergie de rupture et les propri~t~s m~caniques telles que la r6sistance ~ la compression de ces mat~riaux. Les r~sultats montrent que pour des cokes en aiguille et pour des graphites isotropes z~grain fin e0 = KIE~, tandis que pour les cokes de Gilsonite e o =K2E~. E repr6sente l%nergie dissip~e et K 1 et K 2 sont des constantes dependant du type de graphite et de l'orientation de l'echantillon. Ces relations peuvent 6tre ddduites de quelques hypotheses concernant les m~canismes de ddformation du coke en aiguilles et du graphite de Gilsonite. On a aussi trouv6 qu'h la lois la r~sistance ~ la compression et la d~formation h la rupture sont proportionnelles ~ la racine carrie de l'6nergie de rupture de diff~rents polygraphites. An Graphit t-fir die Kerntechnik und andere kommerzielle Zwecke wurden Untersuchungen zur Beziehung zwischen der Restdehnung und der abgegebenen Energie wahrend einer zyklischen Druckbeaufschlagung durchgefiihrt. Ferner wurde ein etwaiger Zusammenhang zwischen der Bruchenergie und den mechanischen eigenschaften, wie Druckfestigkeit dieser Materialien, untersucht. Nach den Ergebnissen betr/igt die Restdehnung f'tir Nadelkoks und isotropen Feinkorngraphit co = K1E~ wogegen fflr Gilsonite-Graphit e 0 = K2E'~ ist. Dabei sind E die abgegebene Energie und K l und K 2 Konstanten, die yon der Art des Graphits und der Proben-orientierung abh~ingen. Diese Beziehungen k6nnen unter einigen Abnahmen, die sich auf den Verformungsmechanismus yon Nadelkoks oder Gilsonite-Graphit beziehen, abgeleitet werden. Es stellt sich ferner heraus, class die Druckfestigkeit und die Bruchdehnung der Quadratwurzel der Bruchenergie verschiedener polykristalliner Graphite proportional sind.

1. Introduction T h e r e have b e e n a n u m b e r o f investigations i n t o t h e s t r e s s - s t r a i n r e l a t i o n s h i p a n d t h e residual strain o f p o l y c r y s t a l l i n e g r a p h i t e [ 1 - 1 1 ]. H o w e v e r , t h o u g h a few investigators h a v e related the i n t e r n a l - f r i c t i o n d a t a to t h e hysteresis l o o p o f s t r e s s - s t r a i n curve [12, 13], r a t h e r little a t t e n t i o n has b e e n paid to the relation-

ship b e t w e e n the residual strain and t h e e n e r g y dissip a t e d or stored d u r i n g loading and u n l o a d i n g cycles, i.e. t h e q u a n t i t y r e p r e s e n t e d as the area o f the h y steresis l o o p OPQ in fig. 1, w h e r e t h e s t r e s s - s t r a i n curve is s h o w n s c h e m a t i c a l l y for p o l y g r a p h i t e . T h e p u r p o s e o f this p a p e r is to investigate the s t r e s s - s t r a i n p r o p e r t i e s o f the m a t e r i a l in view o f the energy described above in t h e case o f c o m p r e s s i v e tests, i.e. to

M. l£to /Residual strain (II)

206

b • Dissipoted

O3 CO t.~

".. ......

¢/)

Energy

R

p/./'~l ,. . . . . .

Looding "

~

Frecfu re i

,

i I I

%3 x

I I

z

ading

/ 0 _

/ /

/ E'o

- o

[

I I T

6"m

I

I I

S ~'f STRAIN, 8

£1

Fig. 1. Schematic graph of the stress strain relationship of polycrystalline graphite.

report the relationships between the residual strain and the energy and to derive them on some assumptions concerning the mechanism for the generation of the residual strain. It is also investigated whether there are correlations between the fracture energy, which is believed to correspond to the area ORS in fig. 1, and the mechanical properties such as the fracture stress of nuclear and other commercial graphites.

2. Experimental Materials used, specimen size and the manner in which the compressive tests were carried out are the same as those in our previous paper [14]. 'Dissipated' energy and fracture energy were calculated from the area of the hysteresis loop OPQ or the area ORS shown in the schenratic graph in fig. 1, respectively. The area was measured using a planimeter.

'

260

'

",,~ 800

660

~6o-

SQUARE ROOT OF THE ENERGY [ (J/m 3) ]

Fig. 2. Some examples o f the results on the residual strain of the needle coke graphite H327 and fine-grained isotropic graphite 7477/PT as a function of the square root of the dissipated energy.

where K 1 and K 2 are constants depending on the kind o f graphite and the specimen orientation. Some examples of these relationships are shown in figs. 2 and 3. Table 1 summarizes the mean values o f K 1 or K 2 for the graphites investigated in the present experiments. The number of specimens examined was 2 to 5 for each direction, except in the case o f Angloform graphite for which only one specimen was used for with-grain and across-grain directions, respec tively. Plots of the fracture stress of and the fracture strain e l- versus the fracture energy Ef, which is the 5 IM2- 24

~th- g

across g. -

~ •

•

acro~-g. 1 " ,/o

/

3. R e s u l t s <

The results obtained show that for the needle coke and floe-grained isotropic graphites the residual strain e 0 is expressed in terms of the energy E described above e as 1

e0 = K 1 E ~

,

~w2~ t~

(])

e0 = K 2 E~3 ,

2'o

2o

~o

CUBIC ROOT OF THE ENERGY

and for the Gilsocarbon graphites (2)

8'o

[ (.T/m3) ~3 ]

,oo

Fig. 3. Some examples of the results on the residual strain o f the Gilsocarbon graphites as a function of the cubic root o f the dissipated energy.

M. Eto/Residual strain (11)

207

Table 1 t 1 List of the values of K 1 or K 2 in the equation eo = K 1 E 2 or eo = K 2 E~. ~Brand Directi~

11327

SMG

With-grain Across-grain K 1 or K 2

6.4 10.1

6.2 5.8 8.3 8.1 K 1 × 106(J/m 3) ~

1001 ~" /

T .~-4

Z

G163A

7477PT

IM2-24

Angloform

5.9 6.9

3.6 3.4

3.4 3.3 K 2 X 105(J/m3) --~

3.8 5.1

differences in the values for the various kinds of graphite were observed in the Charpy tests.

/" Slope=l/2

-~-

op~=~t2

z

10 ILl ~-

SEG-5H

4. Discussion

10-2t~ r ~ : With-grain

- - : Acr'a6s-.grain

0~-

o

H327

•

IM2"2&

LL

• ~"

SMG G163AS

[] •

Angloform 71.771PT

v

SEG-5H

o

SEG-RM-H

lb6 1( FRACTURE ENERGY [Jim 3]

I.t/ a:

4.1. Model for deriving the relationships between e 0 and E

LL

In order to derive the relationships e 0 = K1E~ and e 0 = K2E~, some assumptions were made concerning

=0-3

Fig. 4. Plots of the fracture stress and strain versus the fracture energy for various graphites exanfined.

• &A

A

•

•

c~o

> {1-

0

I 5 FRACTURE

L 10 ENERGY

115

20

( 1 O S 3 / m 3)

Fig. 5. Plots of the Charpy impact value versus the fracture energy for various graphites examined. The designation of the specimens is the same as that in fig. 4. total energy dissipated up to fracture, i.e. the area ORS in fig. 1, are shown in fig. 4. It would be expected that the fracture energy might correlate strongly with the Charpy impact value. However, fig. 5 indicates that almost no correlation between these values was observed in the present experiments. Less pronounced

the mechanism by which the residual strain is caused. These are as follows. (1) Some parts o f a specimen are 'plastically' deformed to cause a residual strain. The energy is dissipated or stored in these deformed regions and the amount of the energy is proportional to the deformed volume. (2) The deformed volume is increased with increasing stress. (3) For needle coke and fine-grained isotropic graphites, the shapes o f the volume are disk-like and/or plate-like with thickness b the value o f which is unaltered regardless o f stressing. In the case o f Gilsocarbon graphites the shape of the volume is spherical. (4) The residual strain is proportional to the mean value o f the radii o f the disks and/or the widths and lengths o f plates for the needle coke and fine-grained graphites, and the radii o f the spheres for the Gilsocarbon graphites. The assumption (3) was made in view o f the fact that the Gilsonite coke is spherical whereas the needle coke is of rather lamellar structure. From the assumption ( 1 ) w e obtain E = KV

(3)

where E, K and V are the energy, a constant and the total deformed volume, respectively. Suppose that in a given specimen there are n deformable regions. Then, assumption (3) gives V = mr b F 2

(4)

M. Eto/Residual stra& {1I)

208

and/or

(4')

V=n~ W T

for the needle coke or fine-grained isotropic graphites. Here, b, e-, T, ~ , respectively, denote the mean value of the thicknesses of the deformed r@ons, the mean of the radii of the disks, the means of the widths and lengths of the plates. Since w/- can be written as /- = (F') 2, eqs. (4) and (4') are expressed by the equation,

v = K ~-2,

(5)

where • is a constant. Also for the Gilsocarbon graphites

g=n~r~3=X~

3,

(6)

where ?- is the mean of the radii of the deformed spheres and X, a constant. The assumption (4) gives e0 = L r ,

(7)

where e 0, L are the residual strain and a constant, respectively. From eqs. (3), (5) and (7), 1

1

1

e 0 = [L/(xK)~ l E~ = K1 E~

(8)

for the needle coke or fine-grained isotropic graphites, and from eqs. (3), (6) and (7), e0

=

[L/(XK) 5' ] E 5' = K 2

E-~

graphites), or a sphere (for the Gilsocarbon graphites) with increasing stress. The idea concerning the deformed volume is somewhat similar to that propounded by Woolley [5]. The effect of stressing on electrical resistivity of graphite is one of the interesting aspects of the structural change which causes the residual strain. Resistivity change of graphite during compression tests has been already reported in previous papers [15, 16]. Here, in order to investigate the relation between the residual strain and 'residual' resistivity, the measurements of the resistivity change caused by increasing cyclic stress were carried out for some graphites. A few examples o f the results are shown in figs. 6 and 7. These figures show that the resistivity after stressing is a bit lower than that before stressing, at least when the applied stress is lower than about a half of the fracture stress. The fact suggests that at lower stresses the structural changes causing the residual strain contribute to resistivity decrease, i.e. the 'residual' resistivity.is negative, or even if they would have a resistivity-increasing effect~ the amount of resistivity increase caused by them is not large enough to cancel the resistivity-decreasing effects. It may be believed that the closure of pores and cracks present before stressing is caused by stressing to contribute to the resistivity decrease. The results on the effect of stressing on ther-

(9) 1.!

for the Gilsocarbon ~raphites. Here, K 1 and K 2 are L/(KK) ~ and L/(XK) ~ , respectively. Comparing eqs. (8) and (9) with eqs. (1) and (2) we realize that the relationships obtained from the experimental results can be derived from the assumptions described above.

SMG(//)

1u

4.2. Origin o f the residual strain ~1.1

Mechanisms for the residual strain have been proposed in our previous paper [14]. It is noted that no definite mechanism is proposed in the derivation of the relationships above, i.e. eqs. (8) and (9). It can be considered that the strain results from the deformation within grists and/or at grain boundaries. Among the assumptions made above, one which is in relation to the structure of graphite is that the deformed volume would increase in the form of a disk and/or a plate (for the needle coke or fine-grained isotropic

Q: /

Unstressed

,.c

o',2

o14

6.6

o'.~

NORMALIZED STRESS (o~/o~f}

1.o

Fig. 6. Resistivity change due to cyclic increasing stress for a needle coke graphite. Applied stress is normalized to the compressive strength of, and the specimen direction is parallel to the extrusion axis.

M. Eto/R esidual strain {I1)

209

! 1.10 Stressed Unstresse(

2 Q-,

m1.05 0 Z -'to

m6C

~to~ w

0.~

02 o'.6 o18 1.o NORMALIZED STRESS (c,-'Ic~) Fig. 7. Resistivity change due to cyclically increasing stress for Gilsocarbon or fine-grained isotropic graphite. The specimen direction is parallel to the press axis. o12

real expansion o f graphite are consistent with this consideration, i.e. decrease in accommodation [17, 18]. On the other hand, the defects such as point defects and dislocations produced by deformation of grists would have a resistivity-increasing effect. Anyway, as a result, the resisitivity o f graphites after stressing is rather lower than that before stressing. As the stress goes up to about 0.5 of the 'residual' resistivity is increased. This corresponds to the fact that the optically resolvable cracks are produced at stresses higher than about 0.5 of, as was shown in previous investigations [9, 15, 16, 19].

4.3. Relationship between the fracture energy and mechanical properties The results presented in fig. 4 show that the compressive strength of and the fracture strain ef are approximately expressed in terms of the fracture energy Ef as l

of =: c~E~

(10)

and J

et. = 3 E~. ,

(1l)

where c~ and/3 are constants independent o f specimen brand and orientation. From fig. 4, c~ and 3 are calculated asc~ = 6.08 X 104 "v'~/m,3 = 2.40 X 10 -5 m/x/~. Eqs. (10) and (11-) give

of(N/m 2) = 2.53 X 109 e f ,

(12)

I.L

2C

o,

i

~

FRACTURE STRAIN (%)

5

Fig. 8. Plots of the fracture stress versus fracture strain for various graphites examined. The designation of the specimens is the same as that in fig. 4. which indicates that for various graphites the fracture stress is approximately proportional to the fracture strain, That is, the value 2.5 X 109 N/m 2 is an apparent modulus for the relationship between of and ef, common to various polygraphites. This is shown in fig. 8. We also obtain from eqs. (10) and (I1)

(l/tq3) efof = E l .

(13)

If the material is completely elastic a/3 = 2. In the present case, however, a/3 = 1.46. This implies that for polycrystalline graphites the fracture energy should be calculated using eq. (13), i.e. Ef = 0.685 of ef. The difference in the value of the constant factor, i.e. 0.5 and 0.685, would be attributed to the 'plastic' nature of graphite. For the strength o f neutron-irradiated graphite the constant-energy [2] or the constant. strain [11] criterion was reported. In this connection the dependence of fracture stress or strain of irradiated specimens on the fracture energy should be investi. gated in the future. The constant-energy criterion is 02 = k E * ,

(14)

and the constant-strain criterion, of = k E * ,

(15)

where k is a constant and E * the Young's modulus.

210

M. Eto /R esidual strain {II)

In view of the relationship represented in eq. (10), eq. (14) predicts that Ef

=

(K/~2)E * ,

I

E3 A v

o~o

)

7477,PTt , I IM2-24

(//)j

(16)

whereas 0.4

(17)

Ef = ( k / ~ ) 2 E .2

can be derived from eq. (15). The results obtained in the present experiments, however, show that neither eq. (16) nor eq. (17) is obeyed if we try to apply them to the non-irradiated specimens of various kinds of graphites. For example, the Young's rood ulus of withgrain H327 is 1.25 to 1.5 times larger than that of IM2-24 or 7477/PT. On the other hand, the fracture energy is 5 to 10 times larger for IM2-24 or 7477/PT than for H327. By modifying eqs. (14) and (15): (Of/O0) 2 = k E * / E ? ,

(14')

and of E* --=k o0 E~

(15')

the constant-energy and constant-strain criteria were applied to an analysis of the data on the strength o f irradiated graphites; the results will be published elseP where. In eqs. (14') and ( 1 5 ) , E 0 and o 0 are the Young's modulus and the fracture stress before irradiation. It is to be noted that eqs. (10) and (l 1) can apply for various polygraphites. This is interesting in view

("sl

0.8

"'~ •

0.6 ',O

o~

.

E

•

&

0.4

0.2 0

i 0.2

i 0,4 NORMALIZED

t 0.6 STRESS

i 0.8

O

0.2

[3

• •

V A

•

0'2

o

j]

0 & o

1

00

•

0'.4 NORMALIZED

0.6 STRESS

i

0.8 ( °-/of )

1.0

Fig. 10. Ratio of the dissipated energy to the energy necessary for deformation as a function of applied stress for Gilsocarbon and fine-grained isotropic graphites parallel to the press axis. of the fact that the fracture energy Ef consists of both elastic and 'plastic' terms, i.e. the energy which would be released by unloading and that dissipated in the compression process. Figs. 9 and i0 show some examples o f the plots of the ratio of the dissipated energy E, the area OPQ in fig. 1 to the total energy necessary for deforming Era, the area OPT in fig. 1 versus the normalized applied stress. Extrapolation of the ratio to a = of makes us expect that the dissipated energy at o = of might be 0.4 to 0.6 of the fracture energy, common to the graphites examined. This is believed to be the reason why eqs. (10) and (i 1) are obeyed by various brands o f polygraphites. That is, even if the dissipated or stored energy should be considered as a fracture criterion, relationships similar to eqs. (10) and (1 1) might hold in the way that af = ~ a ~ and ef = v#015 /3 v~t'. Attention should also be paid to the fact that the behaviour o f the change in the ratio of/:" to E m is somewhat similar to that of the resistivity change shown in figs. 6 and 7. This suggests that the amount of the dissipated energy correlates with the 'residual' resistivity, and pronounced changes in the structure of the graphite must be caused by stresses above about a half of the compressive strength [9, 15, 16, 19].

1.0

(cr/crf)

Fig. 9, Ratio of the dissipated energy to the energy necessary for deformation as a function of applied stress for needle coke graphites parallel to the extrusion axis.

5. Conclusions (1) The residual strain e 0 of graphite is represented

M Eto/Residual strain {II) in terms o f the energy E dissipated during the loading and unloading cycles as e 0 = K 1 I:'-~ for the needle coke or fine-grained isotropic graphites, and e0 = K 2 E } for the Gilsocarbon graphites. These relationships can be derived on some assumptions concerning the manner in which the energy is dissipated and the residual strain is caused. (2) The constant K 1 for the needle coke graphites is smaller in the case o f the with-grain specimens than the across-grain ones, whereas for the fine-grained isotropic and the Gilsocarbon graphites the difference in K 1 or K 2 between with-grain and across-grain directions is less pronounced. (3) Both the fracture stress of and the fracture strain et. are in proportion to the square root o f the fracture energy El. The fracture energy is Ef = 0.685 o f e f , which shows that the fracture energy o f polygraphite is about 1.4 times larger than that calculated on the assumption that the material is completely elastic. The relationship between crf and ef is of(N/m 2) = 2.53 X 109 e f , which indicates that for of and ef o f the various graphites examined, an apparent elastic modulus E*' is obtained a s E * ' = 2.53 × 109 N/m 2.

2l 1

Acknowledgements The author wishes to thank Mr. Shinzo Nomura for very helpful suggestions. He is also indebted to Mr. Katsuo Fujisaki for technical assistance.

References [ 1] Ph. P. Arragon and R.M. Berthier, Industrial Carbon and Graphite (Soc. Chem. Ind., London, 1958) p. 565. [2] H.H.W. Losty and J.S. Orchard, Proc. 5th Carbon Conf., vol. 1 (Pergamon Press, New York, 1962) p. 519. [3] G.M. Jenkins, Brit. J. Appl. Phys. 13 (1962) 30. [4] R.V. Hesketh, J.Appl. Phys. 35 (1964) 3604. [5] R.L. Woolley, Phil. Mag. 11 (1965) 799. [6] G.M. Jenkins, Carbon 3 (1965) 93. [7] J. Rappeneau, G. Jouquet and M. Guimard, Carbon 3 (1966)407. [8] E.J. Seldin, Carbon 4 (1966) 177. [9] O.D. Slagle, J.Am. Cer. Soc. 50 (1967) 495. [10] G.M. Jenkins, J. Nucl. Mater. 29 (1969) 322. [11] M.R. Everett and F. Rindealgh, tligh Temp.-High Pressures 4 (1972) 329. [ 12] T. Tsuzuku and M.H. Saito, Chemistry and Physics of Carbon, ed. by P.L. Walker, Jr. (Marcel Dekkcr, New York, 1968) p. 185. [13] J. Merlin, P. Gobin, G. Jouquct and J. Rappeneau, J. Nucl. Mater. 24 (1967) 200. [14] M. Eto andT. Oku, J. Nucl. Mater. 57 (1975) 198. [15] M. Eto, T. Usui and T. Oku, J. Nucl. Mater. 45 (1972/ 73) 347. [16] M. Eto and T. Oku, J. Nucl. Mater. 54 (1974) 245. [17] I.W. Gazda, Carbon 8 (1970) 511. [18] P.E. Hart, Carbon 10 (1972) 233. [19] G.M. Jenkins, J. Nucl. Mater. 5 (1962) 280.