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1 This article was downloaded by:[battelle Energy Alliance] On: 9 July 2008 Access Details: [subscription number ] Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK Philosophical Magazine Publication details, including instructions for authors and subscription information: The yield curve and the compressive strength of polycrystalline graphite R. L. Woolley a a C.E.G.B., Berkeley Nuclear Laboratories, Berkeley, Gloucestershire Online Publication Date: 01 April 1965 To cite this Article: Woolley, R. L. (1965) 'The yield curve and the compressive strength of polycrystalline graphite', Philosophical Magazine, 11:112, To link to this article: DOI: / URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article maybe used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 r The Yield Curve and the Compressive Strength of Polycrystalline Graphite By R. L. WOOLLEY C.E.G.B., Berkeley Nuclear Laboratories, Berkeley, Gloucestershire [Received 10 November ABSTRACT The form of the yield curve of polycrystalline graphite, and the value of its compressive strength, are explained quantitatively in terms of dislocations associated with the sub-grain boundaries. Q 1. INTRODUCTION THE stress-strain curve of polycrystalline reactor-grade graphite, and the conditions at fracture, have been measured by several workers (Woods et al. 1955, Losty 1962, Jenkins 1962). Jenkins (1962) pointed out that the stress-strain curve is parabolic at low strains. He proposed a model based on an array of springs and friction elements, which led to a parabolic law but which did not make any quantitative prediction of the value of the quadratic coefficient. The theory was explicitly limited to small stresses producing plastic deformation only in a few isolated parts of the material. In this paper we examine a somewhat different model which is not restricted to small deformations. It suggests that the stress-strain curve should be an exponential, whose parameters can be related quantitatively to a dislocation structure previously used to account for the anomalous elasticity of graphite (\\Toolley 1905). The model also predicts an acceptable value for the compressive strength. fj 2. MODEL OF A PLASTICALLY DEFORMED MATERIAL We consider a material containing a distribution of ATo dislocations per unit volume, each dislocation separately being characterized by a yield stress uy, such that when the local tensile stress is less than uy, the dislocation is locked, but when it exceeds uy the dislocation can move. Each dislocation is able to move a distance g when it has yielded, without increase of stress. This motion produces a local stress-relief in a volume v surrounding the dislocation, and in this volume the stress will remain near to uy even when the external stress is raised further. We assume that the boundaries of the volume v suffer the same strain as the overall strain of the specimen. Let Young s modulus of the bulk material be Y. Each dislocation can then

3 800 R. L. Woolley O ~ the L also be characterized by its tensile yield strain, cy = uy/ Y, where ry is the strain at which the given dislocation would yield if no other dislocation yielded first. The distribution of dislocations with respect to yield strain may thus be written: =I@(%) dey,..... * - (1) where.f is the distribution function. Fig. 1 0 Strain 0. 5 % Stress-strain curve of graphite; comparison of theory and experiment. Suppose that at a given strain, there are n dislocations per unit volume which have yielded. The effective plastic volume is then nv per unit volume, and the fraction of the volume which is still elastic is 1 -nv. The number of dislocations per unit volume which can become plastic in a tensile strain-increment dr is then : We will consider here the special case where Integrating (2) we obtain : dn=(l-nv)f(c)dc (a) f (c) = constant = a (3) nv= 1 -exp (- vac) (4) The mean tensile stress is then the average of the stresses in the yielded elements and in the elastic regions, and is : u=(l-nv)yc+ ff Ycvdn J o = Y(l-exp(-vac))/va (5) Writing co for llva we thus have : u= Yc,(l-exp(-c/EO)) (6)

4 Yield Curve and Compressive Strength of Polycrystdline Graphite 801 In fig. 1 we have plotted the tension and compression data of Losty (1962), which all fall on one curve. To this we have fitted the curve : which is a very good fit at all points. a parabolic curve : o= 3440(1 -exp (- 395r))p.s.i.,..... (7) For comparison we have also fitted ~=6*0~10-~~+2*6~10~u~..... (8) to the low-strain data. This however, is not such a good fit above a strain of 0.25%. The important difference between eqns. (7) and (8) is that in the parabolic form (eqn. (8)) the stress increases without limit, whereas in the exponential form (eqn. (7)) it has an upper bound, i.e. a finite compressive strength, at 3440 p.s.i. or 2.38 x 1O8dyn/cm2 ; this is typical of compressive strengths observed experimentally (Losty 1962) DISLOCATIONS STRUCTURE IN POLYCRYSTALLINE GRAPHITE The principal dislocation structure in polycrystalline graphite is that which surrounds the lamellar sub -grains (Bacon 1958, Dawson and Follett 1959). These sub-grains are of diameter G parallel to the basal plane and height H perpendicular to the basal plane. The base of each sub-grain is a twist boundary consisting of a network of three sets of partial screw dislocations (Williamson 1960), and the vertical sides are tilt boundaries of edge dislocations. Fig. 2 Q G _ 0 Network of screw dislocations on the base of a sub-grain. Dislocations such as csbc are repelled out of the net by the stress field of the other dislocations. Figure 2 shows a net of dislocations forming the base of a typical subgrain. Application of a suitably oriented and suitably large shear stress will ultimately cause one of the longer peripheral dislocations, such as abc, P.M. 3F

5 802 R. L. Woolley on the to expand outwards similarly to a Frank-Read sufficient to take it through the semi-circle ab'c, expand indefinitely into an increasing loop, ab"c. that the repulsion of the other dislocations in source; if the stress is it will then continue to In Appendix I we show the net, un, is usually sufficiently large to cause this to happen to some of the longer peripherel dislocations. Only one or two dislocations per net can be expanded in this way because the outward repulsion diminishes as the number in the net diminishes and as the number outside increases. Each expanded dislocation moves away as a result of the repulsive stress urn (Appendix I), which decreases the further it goes. Finally it meets an obstacle, namely (a) another basal net or (b) another dislocation in the same plane or (c) another dislocation in a neighbouring plane. In case (a) the dislocation will either combine with the net or will be strongly repelled by it, according to sign; neither of these makes this dislocation of further interest in discussing yield. Case (b) is similar to case (a). In case (c), the second dislocation could be one in another basal net, or could be a dislocation expanded out of another basal net by the mutual repulsion of the net dislocations. If the coordinates of the one dislocation relative to the other are (2, y) then they attract or repel each other by a force of bud parallel to the basal plane, where... (9) where Ks= 3-12 x 1011dyn/crna and ' (Chou 1962). The stress Ud has a maximum value of Udm= Ksb/4nr)y when x= +r)y, so the dislocation-pair corresponds to a plastic element whose yield stress for further movement of the dislocation is a shear stress of: Uy' = Udm - Un (10) For dislocations separated by one basal plane, y= 3.35 A.U ; and the corresponding value of Udm is 7.8 x 1Osdyn/cmz. For a separation of two basal planes the maximum stress is half this. Since H corresponds to six layer planes, values of y greater than two plane-spacings are not likely to be frequent, and a mean value of Udm will be about 6 x 108dyn/cm2. As these pairs of mutually trapped dislocations are derived from the peripheries of the nets of screw dislocations associated with nearby lamellar sub-grains, it follows that in every sub-grain we shall expect to find a small number, a, of trapped pairs of dislocations of this type, where a is not large compared with unity. The number of sub-grains per unit volume is +G3H so the number of trapped dislocations is : No = 8a/nGaH Each of these dislocations is regarded as being of length G. (11)

6 Yield Curve and Compressive Strength of Polycrystalline Graphite 803 The stresses Udm and u, of eqn. (10) are of the same order of magnitude, and a, has a large variance. In consequence, all values of their difference, uy), up to a maximum of the order of Udm, are equally probable. In consequence we can show (Appendix 11) that the number of these trapped dislocations which have yield strains below ey is : Comparison with eqns. (1) and (3) then gives : N/No= Y y/trudm (12) a = NoY/rudm. - (13) As H N 6 layer planes, the mean separation of basal nets in any given layer plane is of the order of 46G. When a trapped dislocation yields under the influence of a suitable external stress, it will therefore be able to go a distance g = PG before meeting another obstacle, where,9 is of the order of unity. If graphite were isotropic, the volume v would then be Y(/~G)~ where y- 1. For screw dislocations in anisotropic material, the stresses drop off in the c-direction more rapidly than in the a-direction, by a factor of q. We therefore take: From eqns. (ll),(13) and (14) we have then: v = Y(PG)3/7? (14) YeO= Y/va=r2qudmH/8GaP3y (15) Substituting 2.38 x lo* and 6 x 10*dyn/cm2 for YE, and Udm, and 500 and ~OA.U. for G and H (cf. Woolley 1965) we obtain : ap3y= (16) If,9-1 and y N 1, then a N 1 as earlier supposed. We note in consequence that the number of trapped dislocations in a volume v is N,v - 5. Of these, only one, the weakest, actually yields. $ 4. DISCUSSION The agreement between theory and experiment (fig. 1 and eqn. (16)) indicates that the theory is probably correct in general terms. A number of approximations, however, have been made. The model of $2 is closely related to one previously discussed (Woolley 1963) and is based on uniform strain of the boundaries of the plastic elements. This, though better than uniform stress (Taylor 1938), is not exact ; it does allow for the average increase of stress in the still-elastic regions, but takes no account of the fact that this stress-increase is non-uniform and is greatest in any elastic region lying adjacent to a plastic region. It is difficult to predict reliable theoretical values of a, P, y and Udm. AS the elastic moduli and stress-strain curve of graphite depend on its method of preparation and degree of preferred orientation, and there is a wide spr6ad in the experimental value of G/H, it is not felt that it would be profitable to refine the theory further at present,. 3F2

7 804 R. L. Woolley on the There is evidence that polycrystalline graphite contains other dislocation structures (Jenkins et al. 1962, Thrower and Reynolds 1963) but there are no quantitative estimates of their concentration, and their possible contribution to yield is not readily estimated. It is of interest to calculate the mean displacement, 6, of the yielded dislocations when the plastic strain is say 1%, approaching the point of fracture. The number of dislocations which then have yielded is l/v, and each is of length G, so : 6 = Ev/Gb N 130 A.U (17) This confirms our supposition that g N G. If the plastic elements of the model of 9 2 have symmetric yield properties, then the stress-strain curve during unloading should be identical with the lower part of the stress-strain curve of a virgin specimen, but with doubled scale and reversed sign (Woolley 1953, Jenkins 1962). At low strains this is experimentally observed (Jenkins 1962) but at strains above 0.2% the amount of plastic recovery during unloading increases to about twothirds of the original plastic strain (Losty 1962). Considering first the low-strain case, we see from $3 that the individual dislocations probably have very asymmetric yield properties. The plastic elements of volume D each contain about five trapped dislocations, and it is the weakest of thepe in each direction which determines the corresponding yield strength, which will again therefore be asymmetric. However, the distribution of forward and reverse yield strengths is presumably symmetric, so that on the average the aggregate will behave as if the individual elements were symmetric. Considering now the high-strain case, the yielded dislocation-pairs are now widely separated and each dislocation is being forced towards a basal net in its own plane. Consider the Burgers vectors of the approaching dislocation and of the net-dislocation which is nearest to it. These are either anti-parallel or at f 60". In the first case the twodislocations will combine, forming perfect lattice over their common length, and the yielded dislocation will remain there on unloading. In the second two cases, the two dislocations repel, and, on unloading, the yielded dislocation will be forced to return towards its original position. Thus about two-thirds of the plastic strain will be recovered on unloading CONCLUSIONS The tensile and compressive yield curves of polycrystalline graphite, and its compressive yield strength, can be adequately explained in terms of the experimentally observed dislocation structure. ACKNOWLEDGMENT This paper is published by permission of the Central Electricity Generating Board.

8 Yield Curve and Compressive Strength of Polycrystalline Graphite 805 APPENDIX I THE EQUILIBRIUM OF A SET OF DISLOCATIONS Consider the v dislocations forming one of the three sets of parallel screw dislocations of fig. 2. If these are regarded as infinite in length, their positions are found by the method of Eshelby et al. (1961). Hence we calculate the stress at the position of the vth dislocation, due to the combined stress field of the 1,2,... (v- 1)th dislocations, and we obtain : Ksbv(v- 1)/4~G = vkgb/4rrl (18) The other two sets of dislocations together make a further contribution of the half of this, making : u, = 3vK,b/8?rL (19) The dislocations abc which close the periphery of the basal net have lengths from zero to 2L/2/3. The stress to activate a Frank-Read source of length S is 2T/Sb, where T( N K,b2/2?r) is the tension in the dislocation, and for S = 2L/43 this becomes : u= 42/3Ksb/8?rL (20) But v - 5, so that 3v > 443 and the dislocation will expand past the position ab'c. Taking L= 120~.u., the value of u, becomes 2.2 x logdyn/cma. If the v dislocations were infinite in length, the stress which they produce at a large distance, r, would diminish as G/r. As they are only of length G, however, the stress will diminish as G2/r2. Thus at a distance r, u, x lod G2/r2 dyn/cm2. APPENDIX I1 THE DISTRIBUTION OF YIELD STRAINS We here evaluate f( ey) of eqn. (1). In 5 2 we assumed that Y was uniform. This is not true in polycrystalline graphite ; but the grains whose orientation gives a large Y are also oriented to give a high uy. These contribute little tof(ey) so for our present purposes it is a reasonable approximation to treat Y as uniform. Figure 3 represents a volume containing a slip-plane Oxy with slipdirection Ox, subjected to a tensile stress u= YE. Shear stresses u1 and u2 and a tensile stress u3 act on Oxy in the directions Ox, Oy end Ox respectively. Then : u1 = u sin 0 cos 0 cos +, (21 a) a2=usinbcos8sin+, (21 b) = cos2 e (21c)

9 806 R. L. Woolley on the Fig. 3 4" Resolved components of applied stress. The baa1 planes of unirradiated graphite contain cracks of mean length X and mean spacing 2 (fig. 4). The micrographs of Thrower and Reynolds (1963) suggost that X/Z, here denoted by 2f, has values between 2 and 6, i.e. N 2. The shear stress u1 produces elastic stress concentrations at the ends of the cracks, but the high ratio of the crystal elastic moduli cll/c44 prevents these stress concentrations from penetrating far in the 2 direction ; in consequence the material between the cracked planes experiences merely u1. The tensilestress a, produces transverse forces P - Xu3/2 on the ends of the element abcd (fig. 4) ; the high values of cll/c4 and C,,/C~~ produce effectively a simple shear of the element (cf. Baker and Kelly 1964) with Fig. 4 Lenticular cracks in graphite; effect of applied stress.

10 Yield Curve and Compressive Strength of Polyc ystalline Graphite 807 shear stress PlZ = [u3 parallel to the basal plane. This becomes 5u3 cos t,h when resolved parallel to an arbitrary direction in this plane. In a given slip system (fig. 3), u therefore produces a total shear stress of: ay =asinbcosbcos$+[ucos2bcos*..... (22) Without loss of generality we may take t,h = $. The probability of finding a dislocation oriented in db and d$ is sin BdB. 2d$/~, where B and $ extend from 0 to Hence the fraction of the trapped dislocations with yield stress uy and oriented in db and d$ is : d2n 2 4 uy - =sinbdb (23) NO 71 udm Substituting uy from eqn. (22) and integrating, the fraction of the trapped dislocations which have yield strains < ey is : N - 2Yey 1+[ To- F m. 3 * We divide this by 2, because each dislocation has easy yield (Udm-un) in only one direction, and not in the opposite direction (Udm +an). Taking 2 we obtain : t= NINo = Y y/tudm * (25) REFERENCES BACON, G. E., 1958, U.K.A.E.A., A.E.R.E., Rep M-R2702. BAKER, C., and KELLY, A., 1964, Phil. Mag., 9,927. CHOU, Y. T., 1962, J. appl. Phya., 33,2747. DAWSON, I. M., and FOLLETT, E. A. C., 1959, Proc. roy. Soc. A, 253, 390. ESHELBY, J. D., FRANK, F. C., and NABARRO, F. R. N., 1951, Phil. Mag., 42, 351. JENKINS, G. M., 1962, Brit. J. appl. Phys., 13, 30. JENKINS, G. M., TURNBULL, J. A., and WILLIAMSON, G. K., 1962, J. nucl. Mater., 7, 216. LOSTY, H. H. W., 1962, cited in Nightingale, R. E., Nuclear araphite (Academic Press). TAYLOR, G. I., 1938, J. Inst. Met., 62, 307. THROWER, P. A., and REYNOLDS, W. N., 1963, J. nuc2. Mater., 8,221. WILLIAMSON, G. K., 1960, Proc. roy. SOC. A, 257, 457. WOODS, W. K., BUPP, L. P., and FLETCHER, J. F., 1955, Proc. Conf. Peaceful Uses of Atomic Energy (U.N.O.) 7, 455. WOOLLEY, R. L., 1953, Phil. Mag., 44,597; 1965, Phil. Mag., 11, 475.