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The principal facet stress as a parameter for predicting creep rupture under multiaxial stresses
Acfa metall. Vol. 37, No. 4, pp. 1067-1077,
1989
OOIII-6160/89 $3.00+ 0.00 Copyright 0 1989Pergamon Press plc
Printed in Great Britain. All rights reserved
THE PRINCIPAL FACET STRESS AS A PARAMETER FOR PREDICTING CREEP RUPTURE UNDER MULTIAXIAL STRESSES W. D. NIX,? J. C. EARTHMAN, G. EGGELER and B. ILSCHNER Laboratoire de Mttallurgie Mi?canique, Ecole Polytechnique Fed&ale de Lausanne, 34, ch. de Bellerive, CH-1007 Lausanne, Switzerland (Received
10 August 1987; in revised form 18 August 1988)
Abstract-A multiaxial stress parameter for predicting creep rupture of metals and alloys under different loading conditions is proposed. The parameter is based on the stress redistribution associated with grain boundary sliding and is expected to be valid for alloys that exhibit grain boundary sliding and fail by cavitation of grain boundaries perpendicular to the maximum principal stress. An analysis given by Anderson and Rice is used to show that, prior to cavitation, the average tensile stress on grain boundary facets perpendicular to the maximum principal stress is approximately (or = 2.24 (r, - 0.62 (u2 + u3) where 0, > c2 > or are the principal stressses. We show that this quantity, called the principal facet stress, can be used to predict multiaxial creep rupture from uniaxial rupture data for several materials. Cases for which this parameter does not accurately predict multiaxial creep rupture are also discussed. R&nun&-Nous proposons un parametre de contrainte multiaxiale pour predire la rupture en fluage des metaux et des alliages soumis a differentes conditions de charge. Ce parametre est base sur la redistribution des contraintes qui est associee au glissement intergranulaire, et I’on s’attend a ce qu’il soit valable pour les alliages qui presentent du glissement intergranulaire et cassent par cavitation des joints de grains perpendiculaires a la contrainte principale maximale. Selon une analyse due a Anderson et Rice, la contrainte de traction moyenne sur les facettes intergranulaires perpendiculaires a la contrainte principale maximale est approximativement &gale, avant la cavitation a or = 2,24 (r, - 0,62(0, + or) ou (r, > e2 > CT)sont les contraintes principales. Cette quantite, dbnommee contrainte sur la facette principale, peut servir a predire la rupture en fluage multiaxial a partir des rtsultats de la rupture uniaxiale pour plusieurs materiaux. Nous discutons tgalement les cas od ce paramttre ne predit pas avec precision la rupture en fluage multiaxial. Zusammenfas-sung-Es wird ein Parameter fiir mehrachsige Spannungzustande vorgeschlagen, der zur Voraussage des Kriechbruches in Metallen und Legierungen unter unterschiedlichen Belastungsbedingungen benutzt werden kann. Dieser Parameter beruht auf der mit der Komgrenzgleitung zusammenhiingenden Spannungsverteilung; er wird als giiltig fiir Legierungen angesehen, die Komgrenzgleitung aufweisen und durch Porenbildung an den Komgrenzen senkrecht zur maximalen Hauptspannung brechen. Mit einer von Anderson und Rice angegebenen Analyse wird gezeigt, dat3 die mittlere Zugspannung auf die Komgrenzfacetten senkrecht zur griit3ten Hauptspannung vor der Porenbildung etwa ur = 2,24 u, - 0,62(u, + u,) betrlgt, wobie u, > u2 > u3 die Hauptspannungen angeben. Wir zeigen, da8 diese GrBBe, genannt Facetten-Hauptspannung, benutzt werden kann, aus Ergebnissen zum einachsigen Bruch fiir mehrere Materialien den mehrachsigen Kriechbruch vorherzusagen. Fllle, in denen dieser Parameter zu ungenauen Voraussagen fiihrt, werden diskutiert.
INTRODUCTION Creep rupture by cavity growth and coalescence has long been known as an important failure mode for high temperature components in power generation equipment and facilities. This failure mode has heen
TPermanent address: Department of Materials Science and Engineering, Stanford University, Stanford, CA 94305, U.S.A.
studied extensively, most commonly under uniaxial stress conditions. Although uniaxial stress experiments have led to a good understanding of the physical processes involved, they do not provide sufficient information to predict failure under multiaxial stress conditions. Bending and torsion are examples of loading that can cause multiaxial stresses to develop in smooth components. Multiaxial stress states are also produced at notches and other geometric irregularities, even when the remote loading is
1067
1068
NIX er al.:
PREDICTING
CREEP RUPTURE
purely uniaxial. Thus, the prediction of creep rupture in engineering structures requires that the multiaxial stresses be taken into account.
UNDER MULTIAXIAL
STRESSES
and J2 is the second stress invariant defined by
Continuum mechanics approach
Hayhurst and Leckie [l-3] and their colleagues [4] have studied the effects of multiaxiality on creep rupture and have used the principles of damage mechanics [5,6] to describe these effects. Here we give a brief review of their results. They show that for a smooth round bar subjected to uniaxial tension, the rupture lifetime at a given tem~rature can be expressed as tf=Mo-n
(1)
where e is the uniaxial stress and A4 and r are parameters that characterize the evolution of damage at the temperature in question, The need to consider multiaxial stresses is shown by considering the creep rupture properties of notched round bars. Hayhurst and Leckie [2] pointed out that equation (1) does not correctly predict creep rupture of notched bars even if the nominal stress in the notch is used in the expression. Instead equation (1) must be generalized to allow other components of the multiaxial stress state to enter the equation. Following Hayhurst et al. [4] and Hayhurst et al. [7], we define the notch rupture stress, 6,, as the average stress acting on the minimum cross-section of the notch and the uniaxial rupture stress, brr as the stress acting in a smooth sided specimen which has the same lifetime as the notched bar. When cr, > o,, the notch has a strengthening effect whereas for a, < (I~, it has a weakening effect. Hayhurst et al. [7j have called X:,= rr,/tr, the normalized representative rupture stress and used it as a measure of the effect of the notch on creep rupture. Perhaps a more intuitive quantity is the reciprocal of Z,, which we call the notch effect factor, NEF = c,/u,. The rupture stress of the notched bar is found by multiplying the rupture stress of the smooth bar, (T*,by the notch effect factor. It is evident from this description that when 0, #a,, the stress in a notched bar does not produce damage at the same rate as for a smooth bar. Thus equation (1) cannot be used for notched bars. Hayhurst and Leckie [3] have argued that equation (1) must be generalized to allow different components of the multiaxial stress state to influence the rupture life. They retain the form of equation (1) and write, for a general state of multiaxial stress t, = M(aa, + BJ, + rJ*)-*
(2)
for the rupture time. In this expression and in others in this paper we describe the multiaxial stress state in terms of the principal stresses, 6, > ur > 03. Here ui is the ma~mum principal stress, J, is the first stress invariant defined by Ji = a, + o2 + a3 = 3an
(3)
+(aZ-cr3)2+(a3-41)2]112=0~.
(4)
The first stress invariant is related to the hydrostatic tension stress through J, = 3 un and the second invariant is the von Mises equivalent shear stress or the effective stress, J2 = a,. The coefficients a, /3 and y describe the relative cont~butions of the different multiaxial stress components to the rupture life. The terms M and x again characterize the evolution of damage; their values are assumed to be independent of stress state. In order for equation (2) to be consistent with equation (I) for the case of uniaxial tension the coefficients ~1,/I and y must be chosen such that c( + fl + y = 1. Then for the case of uniaxial tension (J1 = G,, J2 = a,), equation (2) reduces to equation (1) (a, = a). It is of interest to consider the physical basis for including the various multiaxial stress terms in equation (2). It is well known that the diffusive growth of integranular cavities is driven by the tensile stresses acting on the grain boundaries. Since intergranular fracture usually occurs first on those grain boundaries perpendicular to the maximum principal stress, it is reasonable to assume that this component of the stress state plays a central role in creep rupture. Cavity growth at high temperatures can be driven also by the hydrostatic tension component of the multi~ial stress state, in the same way as it is in ductile rupture. Thus J, = 3 en should, in principle, appear in the generalized creep rupture expression. The nucleation of cavities is also required for creep rupture. This process requires very high stress concentrations that are produced only by inhomogeneous plastic deformation such as grain boundary sliding and slip band formation [8]. Because these defo~ation processes are driven by shear stresses, it follows that some form of shear stress should also enter the equation for creep rupture. Also, diffusive cavity growth on a grain boundary facet is often constrained by creep of the surrounding matrix. Even though the constrained diffusive growth process is driven by the normal stress acting on the grain boundaries, the overall rate of the process is governed by the creep rate of the su~oundings and this includes the e%ct of shear stresses. This too, suggests that a shear stress should be included in the creep rupture equation. These arguments provide a rationale for including the effective stress, o,, in equation (2). Hayhurst and Leckie [3] have reviewed the multiaxial creep rupture data for several metals and have found that the maximum principal stress, cri, and the effective stress o, are much more important than the hydrostatic stress cr,, in determining creep rupture. Thus they let /? = 0 in equation (2) as an
NIX et al.:
approximation
PREDICTING
CREEP RUPTURE UNDER MULTIAXIAL
and rewrite the equation as tr =
M[aa,
+
(1 -
a)aJx
(5)
where a is a single parameter that describes the relative importance of u, and a, in determining rupture. It is evident that the parameter a cannot be determined from uniaxial creep rupture tests alone. Multiaxial tests in which 0, and u, can be varied independently are needed to determine a. Table 1 illustrates some of the sample geometries that have been used to study creep rupture under multiaxial stress conditions. Uniaxial tension (a,, cr2 = oj = 0) is included in most experiments and is included here for completeness. The effective stress for uniaxial tension is equal to the maximum principal stress, oe = 0,. For torsion, a state of pure shear exists in the sample (a,, gz = 0, c3 = -a,). For this case the effective stress is J3 rri . Thus by comparing creep rupture results for uniaxial tension with those for torsion it is possible to vary the values of 0, and a, independently and in this way determine the value of a in equation (5). It is also possible to determine a by comparing creep rupture results for uniaxial tension with those for notched bars. The relations needed for this are shown in Table 1 for the different kinds of notches considered in this paper. It is, of course, necessary to know the stress state in the notch. Generally this is found by making a finite element analysis of creep deformation in the notch Table
I. Sample geometries
Uniaxial
ur=
and stress states for multiaxial
tension
Torsion
S-l ~ S
( > tests
stresses
0, = maximum (T*= bl = 0
principal
stress
6, = maximun Q* = 0 (r3 = -fJ
principal
stress
6, = maximum
principal
stress
u 1
Notched
bar
S-l
w (r
1
1069
and determining the stress state when steady state flow has been established. Although the stresses in the notch plane cannot be completely uniform, the stresses in the notch plane near the axis of the bar do not depend strongly on the distance from the axis. Hayhurst and Henderson [9] have obtained steady state solutions for a circular notch and for the British Standard “V” notch. The circular notch treated by them was introduced originally by Bridgman but is here called the Hayhurst circular notch to distinguish it from the Cane circular notch below. The finite element calculations of the maximum principal stress, ul, and the effective stress, cre, have been reported by Hayhurst et al. [4]. We have used their results to determine the stress states in the notches under consideration. As shown in Table 1, we define the severity of the notch, S, as the ratio of the maximum principal stress to the effective stress (evaluated at the center of the notch plane): S = a,/~,. The values of S for the different notches considered are shown in the table. For the axisymmetric stress state in the notch, we may express the effective stress as u,=u,-a,, where uT = u2 = u) is the transverse stress acting in the notch (at r = 0). Using the definition of the notch severity we may write
Principal
Stress State
STRESSES
British standard S=1.25 cr,=o,=O.Za, notch Hayhurst circular s= 1.49 o,=o,=0.33a, notch Cane circular s=1.9 u,=o,=0.470, notch S = severitv of the notch = 0, io.
u1
(6)
1070
NIX et al.:
PREDICTING
CREEP RUPTURE UNDER MULTIAXIAL
STRESSES
more severe than the Hayhurst circular notch. To distinguish between the two we refer to the notch geometry used by Cane as the Cane circular notch. The data taken from Cane’s paper include the rupture time, the maximum principal stress and the effective stress and are shown in Table 2, along with the other principal stresses that have been calculated for each test. Also shown in the table are values of tl that have been calculated using equation (5). A value of CLcan be obtained for each torsion or notched bar test with the cl, and a, for that test with the corresponding M and x values determined from uniaxial test data. We note that for this set of data the value of OL is not constant, but varies from 0.39 for one of the shortest multiaxial tests to 0.79 for the longest test. The parameter a might be expected to vary in this way if the fracture mode changes significantly from the shortest test to the longest, but this was not reported by Cane and is not expected for this material. In the next section of the paper we offer an alternative approach to creep rupture under multiaxial stresses. This approach, which is based on the principal facet stress, is capable of rationalizing not only the data of Cane [13] and Hayhurst et al. [7], but also the multiaxial creep fracture data of other investigators [lo, 141.
for the transverse stress in the notch. Equivalently, we may express the notch severity as
s-f!-. @I-uT
We note that S = 1 for uniaxial tension (cr = 0) and that S becomes indefinitely large in the limit of hydrostatic tension (uT = or,). Using the notch parameters shown in Table 1, Hayhurst et al. [7] have determined the values of o!for several materials. Their results are: a = 0 for an alu~num alloy, d = 0.7 for copper and CI= 0.75 for 316 stainless steel. The effective stress appears to be much more important than the maximum principal stress in creep rupture of the aluminum alloy. The results for copper and 316 stainless steel suggests that both the maximum principal stress and the effective stress influence the rupture process. It should be noted that slightly different values of cz were reported for copper in earlier papers. Revisions in the value of CIare directly attributable to refined estimates of the stress state in the notch. Multiaxial (tension-torsion) creep rupture experiments have also been conducted on pre-cavitated copper by Stanzl et al. (lo]. Although they did not express their results in terms of equation (5), a value of CI= 1 can be determined from their finding that the rupture time for this material is uniquely related to the maximum principal stress. This result is consistent with ex~tations; since the pre-cavitation process results in cavities of all of the grain boundaries, it might be expected that the effective stress would play no role in the fracture process. Apparently, the maximum principal, stress which is the driving force for cavity growth and coalescence, completely determines the rupture lifetime for this material. Other data sets for copper also indicate that the rupture time is well characterized by the maximum principal stress [ll, 121. The damage mechanics approach can also be tested using the multiaxial creep rupture data for a 2: Cr-1 MO steel published by Cane [13]. This set of data includes uniaxial tension, torsion and notched bar results. The shape of the notch in the Cane experiments is circular but the dimensions make it
The principal facet stress
It is well known that cavitation is concentrated on those grain bounda~es that are ~~ndi~ul~ to the maximum principal stressf7, 10, 152 11.In some cases this concentration is extreme, with inclined grain boundaries showing very little cavitation. One way to account for this strong cavitational anisotropy is to consider the stress redistribution that occurs when the grain boundaries slide. The shear stresses on grain boundaries are relieved when they slide and this causes a redistribution of the normal stresses to occur. Cocks and Ashby [22] and Nix [23] have studied the stress redistribution for a two dimensional hexagonal grain structure with remote uniaxial tension, rr, applied ~~ndicular to one set of grain facets. They showed that after grain boundary sliding the average stress on the transverse facets is 1.5 cr, while that on the inclined facets is zero. This helps to
Table 2. Creep rupture data for 2iCr-1 MO steel [data from Cane (1982)] Stress state Uniaxial tension
Rupture time (s)
$i%,
(M&j
1.12 x 4.09 x 9.69 x 1.52 x 2.34 x 4.00 x
106 IO* IO6 10’ IO’ IO’
178 147 I24 108 93 77
I 0 0
Torsion
5.45 x 1.49 x 3.06 x 5.91 x
106 IO’ IO’ IO’
103 85 ::
0 0 s
Cane circular notch
1.55 x 10s 1.90 x IO6
338 236
159 111
Cane circular notch parameters S = s,/uq =
0 0
0 0 0 0 0 0 -103 -85 -72 -62 159 111
(?&a)
(&ii&)
178 147 124 108 93 77
399 329 278 242 208 172
178 147 124 108
29.5 243 206 177
0.52 0.60 0.68 0.79
178 124
559 391
0.41 0.39
1.9cz = cJ = bT = PI (S - 1)/S = 0.47 b,.
a -
NIX et al.:
PREDICTING
CREEP RUPTURE UNDER MULTIAXIAL
explain why cavitation is concentrated on the transverse grain boundaries. Because grain boundary sliding and the associated stress re~st~bution are expected to occur under creep conditions, it is reasonable to suppose that the stress directly influencing the cavitation which leads failure is not the applied stress, but the concentrated normal stress acting on the transverse or nearly transverse grain boundaries. We call this the principal facet stress and use it as a parameter for describing creep rupture under multiaxial stress conditions. The simple estimate given above of the principal facet stress for a two dimensional hexagonal grain structure subjected to uniaxial tension is not adequate for our purposes. It does not include multiaxial stresses and it is not based on three dimensional grain structure. An estimate of the principal facet stress for a three dimensional grain structure can be made using the results of an analysis given by Anderson and Rice [24]. They have recently analyzed grain boundary sliding and cavitation in a polycrystalline solid composed of a periodic array of identical grains shaped like Wigner-Seitz cells. Each grain has 6 square facets and 8 hexagonal facets and all of the edges have the same length. Two orientations were considered: one with the tensile axis perpendicular to one set of square facets and one with the tensile axis perpendicular to one set of hexagonal facets. Anderson and Rice analyzed the average normal stress acting on the transverse grain boundaries for each of these two orientations. Before cavitation has started to occur but after grain boundary sliding has relaxed the shear stresses on the boundaries, they show that the principal facet stress is a? = 3a, - 25 for the tensile axis perpendicular facets and
(8) to one set of square
cr? = 1.67a, - 0.67crr
(9)
for the tensile axis perpendicular to one set of hexagonal facets. Here CF]is the maximum principal stress and ur = err = g3 is the axisymmetric transverse stress. To obtain the average principal facet stress for a polycrystalline structure we take a weighted average of the facet stresses given by equations (8) and (9). Since there are 14 facets, of which 6 are square and 8 are hexagonal, we define the average principal facet stress as d _.!%@l+& F-14
F
hex 14OF
*
(101
Using equations (8) and (9) this becomes oF = 2.24a, - 1.24~~
(11)
Although the analysis of Anderson and Rice [21] was for an axisymmetric stress state (8* = crj = er), we generalize equation (11) by replacing the transverse stress with the mean value of the two transverse stresses a, and cr,: Q~= (a* + a,)/2. With this
modification, given by
STRESSES
the average principal
1071
facet stress is
bF = 2.24a, - 0.62@, + 03).
(12)
We note that for a hydrostatic stress state (a, = a2 = u3 = uu) the principal facet stress is simply equal to crH.This corresponds to the limit of no grain boundary sliding and no stress redistribution. It should be emphasized that our estimate of the principal facet stress is expected to be valid after grain boundary sliding has caused a redistribution of the stresses to occur but before the transverse grain boundaries have started to cavitate, During the course of creep, cavitation on the transverse boundaries will lead to additional changes in the facet stresses. Eventually the transverse facets will fail completely and the stresses will then be supported entirely by the other boundary facets. Presumably the inclined facets would then begin to cavitate rapidly leading to final rupture. These complexities, although important to a prediction of the absolute rupture time, need not be included in the multiaxial creep rupture parameter.
RESULTS AND DISCUSSION In this section of the paper we show that the principal facet stress can be used to describe creep fracture under multiaxial stress conditions for several materials. Six different sets of multiaxial creep rupture data are used in this study: (I) the Cane data for a 2$ Cr-I MO steel, described in the previous section, (2) uniaxial tension and notched bar data for a 316 stainless steel, reported by Hayhurst et al. [7], (3) tension and torsion data for uncavitated copper, reported by Stanzl et al. [lo], (4) tension and torsion data for Nimonic 80A, reported by Dyson and McLean [14], (5) creep rupture data for commercially pure copper subjected to combined tension and torsion, reported by Henderson 1121 and (6) creep rupture data for pure copper with superimposed hydrostatic pressure, reported by Needham and Greenwood [ll]. The data taken from these studies are presented in Tables 2-7, respectively. Figure I shows the creep rupture data of Cane [ 131 for a 2: Cr-1 MO steel. In this figure, and the ones to follow, three different stress parameters: the effective stress, a,; the maximum principal stress, 0, and the principal facet stress, cF are plotted against the logarithm of the rupture time. In this way the correlation between the different stress parameters and the rupture time can be assessed visually. All of the data used in the figures can be found in the tables. In Fig. 1 the data for uniaxial tension are distinguished from those for the torsion and notched bar experiments by different symbols. Evidently, neither the effective stress nor the maximum principal stress correlate well with the time to rupture for 2; Cr-1 MO steel. We consider especially the failure of the max-
NIX et al.:
PREDICTING
CREEP RUPTURE
UNDER
MULTIAXIAL
STRESSES
Table 3. Creep rupture data for 316 stainless steel [data from Hayhurst et uf. (1984)]
Stress state
Rupture time (s)
(&a)
$a)
x rd x Id X 106 x IO6 x 10’
348 312 275 245 209
0 0 0 0 0
British standard notch
2.54 X lo6 4.74 x 106 6.23 x 10’
306 282 203
61 56
Hayhurst circular notch
5.80 x 106 9.80 x lo6
305 284
Uniaxial tension
5.06 8.78 2.15 6.18 2.83
(h&L)
(Gk)
(h&
348 312 275 245 209
780 699 616 549 468
41
56 41
245 226 162
610 562 404
101 94
101 94
205 191
558 520
a” 0 0 0 61
British standard notch parameters S = o, /a, = 1.25 a, = oj = a, = 6, (S - 1)/S = 0.20 G,. Hayhurst circular notch parameters S = q /a, = 1.49 u2 = + = a, = o, (S - 1)/S = 0.33 bl, Table 4. Crew ruuture data for initially uncavitated cout~r [data from Stanzi er al. (1983N Stress state Uniaxial tension
Rupture time (s) 2.10 4.32 7.02 3.60 4.60
x x x x x
lo4 IO4 IO4 105 105”
6.80 x id 1.50 X I@ 8.60 x 10d 1.75 X 105 9.27 x IO5
Torsion
(&a)
(&a)
30.0 27.6 22. i 19.3 16.6
0 0 0 0 0
32.4 26.0 18.9 16.2 13.5
0 0 0 0
0
@Isa) 0 0
(?&a)
: 0
30.0 27.6 22.t 19.3 16.6
67.2 61.8 49.5 43.2 37.2
- 32.4 -26.0 - 18.9 - 16.2 -13.5
56.1 45.0 32.7 28.0 23.4
92.7 74.4 54.1 46.3 38.6
‘This rupture time was obtained from Fig. 8 of Stanzi et at.; the time reported in their Tabb 2 appears to be in error. Table 5. Creep rttpture data for Nimonic SOA [data from Dyson and McLean (1977)] Stress state Uniaxial tension
Torsion
Rupture time Is)
(M?a,
(&a,
6.41 x 106 5.62 X 106 2.62 x IO6 2.45 x IO6 1.40 X 106 1.73 x 103 3.90 X 104 1.65 X to’ 4.69 X Id
153 154 187 203 216 326 411 494 567
0 0 0 0 0 0 0 0 0
4.10 X 10’ 1.57 X 10’ 7.08 x 10” 3.42 x IO6 9.39 x 10s 3.59 X 10s 1.73 x 10s 5.63 X 104
57.9 84.9 111 138 184 211 243 281
0 0 0 0 0 0 0 0
(MGa) 0
0 0 0 0 0 0 0
0 - 57.9 -84.9 -111 -138 -184 -211 - 243 -281
153 154 187 203 216 326 411 494 567
342 345 418 45.5 483 731 920 1106 1270
100 147 192 238 319 366 420 486
166 243 318 394 527 605 694 803
Table 6. Creep rupture data copper [data from Henderson (1979)] Stress state
Rupture time
&a,
Uniaxial tension
2.69 x 10’ 3.87 X lo6 4.54 X lb 1.77 X 10s
30.6 46.2 46,2 53.9
a/z = 3
1.02 X 106 2.47 X 106
a/z -2
Ihzaf
(&a) 0
(&a)
&?a)
0 0 0
30.6 46.2 46.2 53.9
68.6 104 104 121
61.5 50.6
-5.6 -4.6
64.5 53.0
141 116
1.65 x Id 1.10 X 107
55.9 37.2
-9.6 -6.4
61.2 40.7
131 87.2
1.10x 106 4.04 X lo*
57.0 49.9
0
0
-21.8 -19.1
70.5 61.7
141 124
a/r = 0.4
5.17 X 106
41.9
0
-28.2
61.1
111
Torsion
1.77 X 10’
30.3
0
-30.3
52.5
a/t =
1
u = tensile stress; r = shear stress.
86.7
NIX ef al.:
PREDICTING
Table 7. Creep rupture stress state Uniaxial tension with hydrostatic pressure
Uniaxial tension
CREEP
data for OFHC Rupture time (s)
RUPTIJRE
UNDER
MULTIAXIAL
copper [data from Needham (A)
STRESSES
and Greenwood
(197_5)]
&a)
(hza)
7.20 1.35 1.00 5.50
x x x x
10” 106 106 105
10.1 14.4 17.0 20.2
- 10.6 -6.3 -3.7 -0.5
-10.6 -6.3 - 3.7 -0.5
20.7 20.7 20.7 20.7
35.8 40.1 42.7 45.9
8.70 1.95 1.10 6.50 5.30 3.50
x x x x x x
IO6 106 106 IO’ 10s 10s
10.1 13.6 17.9 20.2 24.0 24.0
-14.1 - 10.6 -6.3 -4.0 -0.2 -0.2
-14.1 - 10.6 -6.3 -4.0 -0.2 -0.2
24.2 24.2 24.2 24.2 24.2 24.2
40.1 43.6 47.9 50.2 54.0 54.0
2.50 1.15 8.50 3.00 1.50
x x x x x
IO6 106 IO5 IO’ 10s
13.7 17.0 21.4 24.2 27.2
- 13.9 -10.6 -6.2 - 3.4 -0.4
- 13.9 - 10.6 -6.2 -3.4 -0.4
27.6 27.6 27.6 27.6 27.6
47.9 51.2 55.6 58.4 61.4
2.00 x 10s
27.6
27.6
61.8
imum principal stress to describe the rupture data. At the same maximum principal stress, the samples tested in torsion fail sooner than those tested in tension while the notched samples fail later. By contrast, the principal facet stress, as defined by equation [12], brings all of the creep rupture data onto a single curve. The success of the principal facet stress in bringing the data together in Fig. 1 suggests that the data points for torsion and the notched bar experiments are separated from those for uniaxial tension (at a fixed maximum principal stress) because of different degrees of grain boundary sliding and stress redistribution. High principal facet stresses are produced in torsion and they cause the torsion samples to fail sooner than samples subjected to uniaxial tension. Conversely, small principal facet stresses are developed in the notch, with the consequence that the notched samples fail later than those subjected to uniaxial tension. Cane has studied grain boundary sliding and cavitation in 2aCr-1 MO steel under uniaxial creep conditions at 838 K [17]. Using longitudinal scratches as markers, grain boundary displacements were measured on uniaxial creep specimens. Stresses can be redistributed to transverse facets when the displacements associated with grain boundary sliding are of the order of the relative elastic displacements of the grains, 6, = (a/E)& where cr is the applied stress, E is Young’s modulus and L is the grain diameter. Cane’s measurements indicate that grain boundary sliding displacements satisfy this condition for 2f Cr-1 MO steel. For example, for a stress of 108 MPa at 1.5% strain, the mean displacement associated with grain boundary sliding is approximately 0.1 pm, which is of the order of 6, (approximately equal to 0.08 pm taking L = 150 pm [ 171and E = 2 x lo5 MPa). It follows that the grain boundary deformation behavior of 2iCr-1 MO steel is consistent with the assumption of grain boundary sliding made in the derivation of the principal facet stress.
0
0
(A)
1073
(&i)
The creep rupture data of Hayhurst et al. [7] for 316 stainless steel are shown in Fig. 3. Again the different stress parameters are plotted against the logarithm of the rupture time. Here the data for uniaxial tension tests are distinguished from those for the notched bar tests. The data for two different notch geometries are shown: The British Standard notch and the Hayhurst circular notch. The rupture times for the notched samples are greater than those for uniaxial tension at the same maximum principal stress. This represents a notch strengthening effect. 210 190170 -
.
0
. 0
150 130 110 -
90
.
o
Y .
70 -
2 l/4 Cr tMo Steel Temperature = 838KCane (19R2)
n
.
0
. 0
Torsion Circular
.
0
Tension
0
Notch
2”
400 2 114 Cr 1Mo Steel Temperature = 838K Cane (1982)
. 300-
100 105
“““‘I
“““I 106 Time to Rupture
““‘. 107
10x
(s)
Fig. 1. Creep rupture data for a 2aCr-I MO steel plotted as the effective stress (top), the maximum principal stress (middle) and the principal facet stress (bottom) vs the logarithm of the rupture time.
1074
NIX et uf.: PREDICTING
CREEP RUPTURE
We note that the Hayhurst circular notch is more severe than the British Standard notch and it causes a greater notch strengthening effect. Here again we find that the principal facet stress is a correlating parameter that brings all of the data onto a single curve. Evidently the transverse stresses acting in the notch makes the principal facet stress smaller than for uniaxial tension. This causes the rupture life of the notched bar to be greater than that for uniaxial tension under the same maximum principal stress. Stanzl et al. [lo] have measured the creep rupture life of uncavitated copper at 500°C in both tension and torsion. Their results are shown in Fig. 5. We observe that samples tested in torsion fail sooner than those tested in tension at the same maximum principal stress just, as for the 2iCr-1 MO steel data discussed above. We find again that the principal facet stress brings the tension and torsion data together. Figure 4 shows the creep rupture data of Dyson and McLean [14] for Nimonic SOA. We find that neither the effective stress nor the maxims principal stress alone can characterize the creep rupture data. Again we see that the principal facet stress correlates well with the time to rupture. In the four examples cited above the principal facet stress beings the multiaxial creep rupture data into coincidence with the uniaxial data. This correlation should prove to be quite useful because it allows a prediction of the rupture time under multiaxial stresses from uniaxial creep rupture data. Of course, this prediction requires a knowledge of the multiaxial stress state imposed. Although the principal facet stress is expected to describe multiaxial creep fracture for many materials, there are some materials for which it fails to bring the uniaxial and multiaxial data into coincidence. One example is found in the work of Hayhurst et al. [7l who showed that creep fracture of an alu~num alloy is well characterized by the effective stress (a = 0). Although we do not fully understand the failure of the principal facet stress in this case, we believe it may be related to the failure of this material to exhibit grain boundary sliding and cavitation of transverse boundaries, as envisioned by the present approach. It is known that pure aluminum does not cavitate easily. If the aluminum alloy cited here behaves like pure aluminum, then the rupture time would not be expected to correlate with the principal facet stress. Stanzl et al. [lo] have also performed tension and torsion experiments for precavitated copper specimens. They have found that, for a given maximum principal stress, the time to rupture for precavitated copper under tension and torsion is the same. It follows that the principal facet stress theory is not valid for this case. Precavitated copper is so brittle that rupture may occur before the stresses are adequately redistributed by grain boundary sliding. if so we would not expect the principal facet stress to be applicable.
UNDER MULTIAXIAL
STRESSES
Three separate studies of creep fracture of copper under multiaxial stresses have indicated that failure of this materia1 is characterized completely by the maximum principal stress f7, 11, 121. The results of one study are shown in Table 8. The data were given by Hayhurst, Dimmer and Morrison [7] in terms of the normalized maximum principal stress, C, = a,/~,, where cr, is the maximum principal stress in a smooth bar and 0, is the average stress on the minimum section of a notched bar for a given time to rupture. Taking rr, to be equal to the average maximum principal stress, cri, the notch effect factor, equal to the reciprocal of E,, is given by NEF = Q!/a,. Values of er and 6) normalized by a, can be determined for each specimen geometry using relations derived from finite element results [4,9]. A normalized value of the principal facet stress, ur/tr,, is calculated by inserting the normalized principal stresses into equation (12). These normalized stress values are shown in Table 8 with corresponding values of the notch severity, S, [4]. The normalized principal facet stress is not independent of specimen geometry. Thus, it is not a valid parameter for predicting creep rupture for this material. The principal facet stress theory is based on the assumption that grain boundary sliding and stress redistribution occurs and that failure occurs primarily by cavitation on transverse boundaries. It is, at first, surprizing that err is not valid for notched copper specimens since it appears to be valid for the copper data of Stanzl et al. [lo]. An explanation of this discrepancy is found by comparing the micrographs of notched bar cross-sections of copper with the micrographs of notched bar cross-sections of 316 stainless steel reported by Hayhurst et al. [7]. This comparison shows that the cavitation damage in notched bars of copper is much more localized on the transverse plane at the center of the notch than it is for 316 stainless steel. The failure of notched specimens of copper appears to be more representative of crack propagation than of uniform creep rupture. The validity of cr for the copper data of Stanzl et al. is explained by noting that their data were obtained for unnotched specimens (uniaxial tension and torsion). Henderson [ 1I] has reported creep rupture data for samples of commercially pure copper subjected to varying amounts of loading in tension and torsion. We have calculated the various stress parameters from that data; they are plotted vs the time to rupture in Fig. 5. It is clear that the effective stress fails to describe the rupture data for this material. A much better correlation is found using the maximum principal stress. The correlation of the principal facet stress with the time to rupture is also shown in the figure. Although the amount of scatter in the plot of principal facet stress vs time to rupture is greater than that found in the plot of maximum principal stress vs the time to rupture, the differences are not large. One concludes that the maximum principal stress is a better correlating parameter for this material, but it
NIX et al.:
PREDICTING
CREEP RUPTURE
? 316 Stainless Steel Temperature = 823K Hayhunt. Dimmer & Morrison (1984)
400 Cl o 3M)-
600
o
Tension
.
British Standard Notch
.
Circular Notch
“““” ‘.““‘,
100
.
.““.‘-
.““..I
0
.
1075 ““-
.“‘.“-
Nimonic 80A Temperature = 1023K Dyson&McLean (1977)
.
-
l
0
.
0 0
.
.
“““‘I .‘...‘I
““‘-
316 Stainless Steel Temperature = 823K Hayhurst. Dimmer & Morrison (1984).
0
STRESSES
0
400.
400.
.“‘.“-
x0-
0
200 -
UNDER MULTIAXIAL
Nimonic 80A Temperature = 1023K Dyson & McLean (1977)
0
500.
0
400300 ‘200 _
100-
. o
Tension
.
Torsion
_
0 .
.
.O
0
l =J ‘*.
0
“.‘..‘n ...““-
““‘.‘m .. “‘I
““.‘I ““.‘I
..’ “‘I “““I
”
--y
Nimonic 80A Temperah!% = 1023K Dyson & McLean (1977)
500 -
.
British Standard Notch
l
Circular Notch
t 0103
I
Time to Rupture (s)
I
o
20
l
o
Tension Torsion
10’
.
0
.
I
106
.
. . .
.
.
10’
..“_j 108
cl
,“..I
4. Creep rupture data for Nimonic 80A plotted as the effective stress (top), the maximum principal stress (middle) and the principal facet stress (bottom) vs the logarithm of the rupture time.
40 -
0
“.“‘I
I
105
Time to Rupture (s)
Fig. 2. Creep rupture data for 316 stainless steel plotted as the effective stress (top), the maximum principal stress (middle) and the principle facet stress (bottom) vs the logarithm of the rupture time.
30
.
104
’
3. _ 0
‘.....I
•1
copper Temperature = 523K Henderson (1979)
0
20 ’
0 Tension
. Oh=3 0 o/7=2 . o/7=1 _ . o/7=0.4 A Torsion
Copper Temperature = 523K Henderson (1979)
30
loo-
.
..“‘I
‘..“I
90
“‘.”
t
Temperature = 773K Stanzl. Argon & Tschegg (1983)
80
120
70
0
0 Tension . d7=3 0 G/7=2 . d7=1 . o/7=0.4 A Torsion
140
60
100
50
Copper 80 - Temperature = 523K Henderson (1979)
40 30’ IO’
A
‘..‘.‘I
104
“”
* 16
....-I
106
Time to Rupture (s)
Fig. 3. Creep rupture data for pure copper plotted as the effective stress (top), the maximum principal stress (middle) and the principal facet stress (bottom) vs the logarithm of the rupture time.
: 60
106
107 Time to Ruptu~ (s)
Fig. 5. Creep rupture data for commercially pure copper plotted as the effective stress (top), the maximum principal stress (middle) and the principal facet stress (bottom) vs the logarithm of the rupture time.
1076
NIX et al.: PREDICTING CREEP RUPTURE UNDER MULTIAXIAL STRESSES Table
8. Creep
Uniaxial tension British standard notch Circular notch
rupture data for copper-stress states for equal rupture times [data from Hayhurst et al. (1984)]
1.00
1.oo
0
0
2.24
1.25
0.899
0.180
0.180
1.790
1.49
1.053
0.346
0.346
1.930
NEF = Notch effect factor. or is the rupture stress for a smooth bar loaded in uniaxial tension with the same rupture time. a,=a,=u,=a,(S-1)/S.
is only marginally better. We do not know why the principal facet stress fails to give a better correlation in this case. Perhaps the oxides that are present in commercially pure copper inhibit grain boundary sliding and prevent the necessary stress redistribution from occurring. Needham and Greenwood [12] have studied the creep and creep rupture properties of OFHC copper subjected to uniaxial with superimposed hydrostatic pressure. Their study is an important one because the stress states in their experiments were better known than in the notched bar experiments. Their results are shown in Fig. 6. Again we see that for copper the effective stress fails to describe the time to rupture data. As Needham and Greenwood concluded, the maximum principal stress appears to give the best correlation with the data. Although the correlation based on the maximum principal stress is not perfect, it is better than that based on the principal facet stress. This data set raises the most important question about the validity of using the principal facet stress as a correlating parameter for creep fracture. Pure copper is known to exhibit both grain boundary sliding and intergranular cavitation at the temperature of these tests 773 K (SOP), so the principal facet stress should be expected to correlate well with the time to rupture. Since the principal facet stress works well for many other materials, it is of interest to determine why it does not work well in this case. We believe the failure of the principal facet stress to describe these data may be related to the particular multiaxial stress states that are produced by subjecting the samples to hydrostatic pressure. In particular, we suggest that the principal facet stress underestimates the time to rupture when large hydrostatic pressures are present. Consider the tension-torsion data of Stanzl et al. [lo] (Fig. 3) which were obtained for the same material (nominally) and at the same temperature as the NeedhamGreenwood data [12]. The tension-torsion data give the best correlation with the principal facet stress whereas the data obtained using hydrostatic pressure gives a better correlation with the maximum principal stress. Thus, the two techniques for producing multiaxial stresses appear to give different results. We note that the hydrostatic pressure is either negative or zero in the tension-torsion experiments whereas it is positive in most of the tests involving the use of hydrostatic
pressure. A close inspection of the data shown in Table 7 and Fig. 6 indicates that the principal facet stress underestimates the time of rupture of samples subjected to hydrostatic pressure relative to the one subjected to uniaxial tension only. This seems to suggest that the constant in the second term on the right hand side of equation (12) is too large. This term predicts that the principal facet stress for a round bar subjected to radial pressure P, (a, = 0, CT~ = u3 = -P,) is 1.24P,. According to the present model, such a radial pressure is expected to cause grain boundary sliding to occur and to develop tensile stresses on those grain boundary facets perpendicular to the axis of the sample. For soft metals such as pure copper this may not be possible. The deformation associated with such a stress state may be more homogeneous than that envisioned in the present
OoO...o: .O
g*5
1
20
2
35
B j$
25-
.
.
.
. .
/
i
O0
0.
Pure copper Temperature= 773K
.
Needham &Greenwood
(1975) m i .I 4
,5 _ o 0,=27.6MPa 5
. .
O
.
.O
0. Oe= 20.7MPa MPa 0, = 24.2
. . .
.
.
CI,= 20.7 MPa
Time to Rupture (s)
Fig. 6. Creep rupture data for pure copper plotted as the effective stress (top), the maximum principal stress (middle) and the principal facet stress (bottom) vs the logarithm of the rupture time.
NIX et al.:
PREDICTING
CREEP RUPTURE UNDER MULTIAXIAL
treatment. This would cause the principal facet stress to underestimate the time to rupture of samples subjected to hydrostatic pressure relative those subjetted to uniaxial tension, as we have seen. Although such an underestimation is not desirable, it is preferable to an overestimate of the time to rupture. By comparison, using the maximum principal stress as the correlating parameter can lead to overestimates of the time to rupture. This is shown most clearly in Figs 1, 3 and 4 (middle) where the samples subjected to torsion fail sooner than would be expected on the basis of uniaxial tension experiments. SUMMARY We have shown that creep rupture under multiaxial stresses can be predicted from uniaxial creep rupture data using the principal facet stress as a creep rupture parameter. Grain boundary sliding causes a stress redistribution to occur and this leads to preferential cavitation on grain boundaries perpendicular to the maximum principal stress. The stresses on the cavitating boundaries depend on stress state and this provides a basis for using the principal facet stress as a multiaxial creep rupture parameter. For three different materials: (1) 2: Cr-1 MO steel, (2) 316 stainless steel, (3) Nimonic 80A, the time to rupture at a given temperature depends uniquely on the principal facet stress. The correlation with the principal facet stress has also been found to be valid for pure copper tested in tension and torsion. Thus, the principal facet stress can be used to predict the time to rupture under multiaxial stresses from measurements of the time to rupture under uniaxial stresses. However, the principal facet stress fails to give the best correlation with time to rupture for an aluminum alloy, which is governed by the effective stress, for precavitated copper, which is governed by the maximum principal stress and for three other data sets for uncavitated copper, which appear to be governed by the maximum principal stress. We discuss the probable causes of the failure of the principal facet stress in these cases. Acknowledgements-Financial support by the Swiss Federal Government within the framework of the European cooper-
STRESSES
1077
ation COST 505 is gratefully acknowledged. Additional support from Stiftung Volkswagenwerk is also appreciated. Professor W. D. Nix was on sabbatical leave from Stanford at the Cole Polytechnique Fbderale de Lausanne by invitation of the president of the EPFL, Professor B. Vittoz. REFERENCES 1. D. R. Hayhurst, J. Mech. Phys. Solids 20, 381 (1972). 2. D. R. Hayhurst and F. A. Leckie, J. Mech. Phys. Solids 21, 431 (1973).
3. D. R. Hayhurst and F. A. Leckie, in J. Carlsson and N. G. Ohlson (editors) Mechanical Behavior of Materials, Proc. ICM4, Vol. 2, p. 1195. Pergamon Press, Oxford (1984). 4. D. R. Hayhurst and F. A. Leckie and C. J. Morrison, Proc. R. Sot. A360, 243 (1978). 5. L. M. Kachanov, IZV Akad. Nauk. SSSR Otd. Teckh. Nuuk. 8, 26 (1958). 6. Y. N. Robotnov, Creep Problems in Structural Members
(English translation edited by F. A. Leckie). North Holland, Amsterdam (1969). 7. D. R. Hayhurst, P. R. Dimmer and C. J. Morrison, Phil. Trans. R. Sot. A 311, 103 (1984).
8. H. Riedel, Fracture at High Temperatures, p. 85. Springer, Berlin (1987). 9. D. R. Hayhurst and J. T. Henderson, Int. J. mech. Sci. 19, 133 (1977). 10. S. E. Stanzl, A. S. Argon and E. K. Tschegg, Acta metall. 31, 833 (1983). Il. J. Henderson, J. Engng Mater. Tech., Trans. A.S.M. E. 101, 356 (1979).
12. N. G. Needham and G. W. Greenwood,
Metal Sci.
9, 258 (1975).
13. B. J. Cane, in Advances in Fracture Research (Fracture 8f), Vol. 3, p. 1285. Pergamon Press, Oxford (1982). 14. B. F. Dyson and D. McLean, Metal Sci. 11, 37 (1977). 15. P. W. Davies, K. R. Williams and B. Wilshire, Phil. Mag. 18, 197 (1968). 16. T. Johannesson and A. Tholen, J. Inst. Metals 97, 243 (1969).
17. B. J. Cane, Metals Sci. 13, 287 (1978). 18. I.-W. ,.^^.. Chen and A. S. Argon, Acla metall. 29, 132 (IYVI).
19. R. T. Chen and J. R. Weertman, Mater. Sci. Engng 64, 15 (1984). 20. K. S. Yu and W. D. Nix, Scripta mefall. 18, 173 (1984). 21. G. Eggeler, J. C. Earthman, N. Nilsvang and B. Ilschner. To be published. 22. A. C. F. Cocks and M. F. Ashby, Prog. Mater. Sci. 27, 265 (1982). 23. W. D. Nix. To be published. 24. P. M. Anderson and J. R. Rice, Acta metall. 33, 409 (1985).
1989
OOIII-6160/89 $3.00+ 0.00 Copyright 0 1989Pergamon Press plc
Printed in Great Britain. All rights reserved
THE PRINCIPAL FACET STRESS AS A PARAMETER FOR PREDICTING CREEP RUPTURE UNDER MULTIAXIAL STRESSES W. D. NIX,? J. C. EARTHMAN, G. EGGELER and B. ILSCHNER Laboratoire de Mttallurgie Mi?canique, Ecole Polytechnique Fed&ale de Lausanne, 34, ch. de Bellerive, CH-1007 Lausanne, Switzerland (Received
10 August 1987; in revised form 18 August 1988)
Abstract-A multiaxial stress parameter for predicting creep rupture of metals and alloys under different loading conditions is proposed. The parameter is based on the stress redistribution associated with grain boundary sliding and is expected to be valid for alloys that exhibit grain boundary sliding and fail by cavitation of grain boundaries perpendicular to the maximum principal stress. An analysis given by Anderson and Rice is used to show that, prior to cavitation, the average tensile stress on grain boundary facets perpendicular to the maximum principal stress is approximately (or = 2.24 (r, - 0.62 (u2 + u3) where 0, > c2 > or are the principal stressses. We show that this quantity, called the principal facet stress, can be used to predict multiaxial creep rupture from uniaxial rupture data for several materials. Cases for which this parameter does not accurately predict multiaxial creep rupture are also discussed. R&nun&-Nous proposons un parametre de contrainte multiaxiale pour predire la rupture en fluage des metaux et des alliages soumis a differentes conditions de charge. Ce parametre est base sur la redistribution des contraintes qui est associee au glissement intergranulaire, et I’on s’attend a ce qu’il soit valable pour les alliages qui presentent du glissement intergranulaire et cassent par cavitation des joints de grains perpendiculaires a la contrainte principale maximale. Selon une analyse due a Anderson et Rice, la contrainte de traction moyenne sur les facettes intergranulaires perpendiculaires a la contrainte principale maximale est approximativement &gale, avant la cavitation a or = 2,24 (r, - 0,62(0, + or) ou (r, > e2 > CT)sont les contraintes principales. Cette quantite, dbnommee contrainte sur la facette principale, peut servir a predire la rupture en fluage multiaxial a partir des rtsultats de la rupture uniaxiale pour plusieurs materiaux. Nous discutons tgalement les cas od ce paramttre ne predit pas avec precision la rupture en fluage multiaxial. Zusammenfas-sung-Es wird ein Parameter fiir mehrachsige Spannungzustande vorgeschlagen, der zur Voraussage des Kriechbruches in Metallen und Legierungen unter unterschiedlichen Belastungsbedingungen benutzt werden kann. Dieser Parameter beruht auf der mit der Komgrenzgleitung zusammenhiingenden Spannungsverteilung; er wird als giiltig fiir Legierungen angesehen, die Komgrenzgleitung aufweisen und durch Porenbildung an den Komgrenzen senkrecht zur maximalen Hauptspannung brechen. Mit einer von Anderson und Rice angegebenen Analyse wird gezeigt, dat3 die mittlere Zugspannung auf die Komgrenzfacetten senkrecht zur griit3ten Hauptspannung vor der Porenbildung etwa ur = 2,24 u, - 0,62(u, + u,) betrlgt, wobie u, > u2 > u3 die Hauptspannungen angeben. Wir zeigen, da8 diese GrBBe, genannt Facetten-Hauptspannung, benutzt werden kann, aus Ergebnissen zum einachsigen Bruch fiir mehrere Materialien den mehrachsigen Kriechbruch vorherzusagen. Fllle, in denen dieser Parameter zu ungenauen Voraussagen fiihrt, werden diskutiert.
INTRODUCTION Creep rupture by cavity growth and coalescence has long been known as an important failure mode for high temperature components in power generation equipment and facilities. This failure mode has heen
TPermanent address: Department of Materials Science and Engineering, Stanford University, Stanford, CA 94305, U.S.A.
studied extensively, most commonly under uniaxial stress conditions. Although uniaxial stress experiments have led to a good understanding of the physical processes involved, they do not provide sufficient information to predict failure under multiaxial stress conditions. Bending and torsion are examples of loading that can cause multiaxial stresses to develop in smooth components. Multiaxial stress states are also produced at notches and other geometric irregularities, even when the remote loading is
1067
1068
NIX er al.:
PREDICTING
CREEP RUPTURE
purely uniaxial. Thus, the prediction of creep rupture in engineering structures requires that the multiaxial stresses be taken into account.
UNDER MULTIAXIAL
STRESSES
and J2 is the second stress invariant defined by
Continuum mechanics approach
Hayhurst and Leckie [l-3] and their colleagues [4] have studied the effects of multiaxiality on creep rupture and have used the principles of damage mechanics [5,6] to describe these effects. Here we give a brief review of their results. They show that for a smooth round bar subjected to uniaxial tension, the rupture lifetime at a given tem~rature can be expressed as tf=Mo-n
(1)
where e is the uniaxial stress and A4 and r are parameters that characterize the evolution of damage at the temperature in question, The need to consider multiaxial stresses is shown by considering the creep rupture properties of notched round bars. Hayhurst and Leckie [2] pointed out that equation (1) does not correctly predict creep rupture of notched bars even if the nominal stress in the notch is used in the expression. Instead equation (1) must be generalized to allow other components of the multiaxial stress state to enter the equation. Following Hayhurst et al. [4] and Hayhurst et al. [7], we define the notch rupture stress, 6,, as the average stress acting on the minimum cross-section of the notch and the uniaxial rupture stress, brr as the stress acting in a smooth sided specimen which has the same lifetime as the notched bar. When cr, > o,, the notch has a strengthening effect whereas for a, < (I~, it has a weakening effect. Hayhurst et al. [7j have called X:,= rr,/tr, the normalized representative rupture stress and used it as a measure of the effect of the notch on creep rupture. Perhaps a more intuitive quantity is the reciprocal of Z,, which we call the notch effect factor, NEF = c,/u,. The rupture stress of the notched bar is found by multiplying the rupture stress of the smooth bar, (T*,by the notch effect factor. It is evident from this description that when 0, #a,, the stress in a notched bar does not produce damage at the same rate as for a smooth bar. Thus equation (1) cannot be used for notched bars. Hayhurst and Leckie [3] have argued that equation (1) must be generalized to allow different components of the multiaxial stress state to influence the rupture life. They retain the form of equation (1) and write, for a general state of multiaxial stress t, = M(aa, + BJ, + rJ*)-*
(2)
for the rupture time. In this expression and in others in this paper we describe the multiaxial stress state in terms of the principal stresses, 6, > ur > 03. Here ui is the ma~mum principal stress, J, is the first stress invariant defined by Ji = a, + o2 + a3 = 3an
(3)
+(aZ-cr3)2+(a3-41)2]112=0~.
(4)
The first stress invariant is related to the hydrostatic tension stress through J, = 3 un and the second invariant is the von Mises equivalent shear stress or the effective stress, J2 = a,. The coefficients a, /3 and y describe the relative cont~butions of the different multiaxial stress components to the rupture life. The terms M and x again characterize the evolution of damage; their values are assumed to be independent of stress state. In order for equation (2) to be consistent with equation (I) for the case of uniaxial tension the coefficients ~1,/I and y must be chosen such that c( + fl + y = 1. Then for the case of uniaxial tension (J1 = G,, J2 = a,), equation (2) reduces to equation (1) (a, = a). It is of interest to consider the physical basis for including the various multiaxial stress terms in equation (2). It is well known that the diffusive growth of integranular cavities is driven by the tensile stresses acting on the grain boundaries. Since intergranular fracture usually occurs first on those grain boundaries perpendicular to the maximum principal stress, it is reasonable to assume that this component of the stress state plays a central role in creep rupture. Cavity growth at high temperatures can be driven also by the hydrostatic tension component of the multi~ial stress state, in the same way as it is in ductile rupture. Thus J, = 3 en should, in principle, appear in the generalized creep rupture expression. The nucleation of cavities is also required for creep rupture. This process requires very high stress concentrations that are produced only by inhomogeneous plastic deformation such as grain boundary sliding and slip band formation [8]. Because these defo~ation processes are driven by shear stresses, it follows that some form of shear stress should also enter the equation for creep rupture. Also, diffusive cavity growth on a grain boundary facet is often constrained by creep of the surrounding matrix. Even though the constrained diffusive growth process is driven by the normal stress acting on the grain boundaries, the overall rate of the process is governed by the creep rate of the su~oundings and this includes the e%ct of shear stresses. This too, suggests that a shear stress should be included in the creep rupture equation. These arguments provide a rationale for including the effective stress, o,, in equation (2). Hayhurst and Leckie [3] have reviewed the multiaxial creep rupture data for several metals and have found that the maximum principal stress, cri, and the effective stress o, are much more important than the hydrostatic stress cr,, in determining creep rupture. Thus they let /? = 0 in equation (2) as an
NIX et al.:
approximation
PREDICTING
CREEP RUPTURE UNDER MULTIAXIAL
and rewrite the equation as tr =
M[aa,
+
(1 -
a)aJx
(5)
where a is a single parameter that describes the relative importance of u, and a, in determining rupture. It is evident that the parameter a cannot be determined from uniaxial creep rupture tests alone. Multiaxial tests in which 0, and u, can be varied independently are needed to determine a. Table 1 illustrates some of the sample geometries that have been used to study creep rupture under multiaxial stress conditions. Uniaxial tension (a,, cr2 = oj = 0) is included in most experiments and is included here for completeness. The effective stress for uniaxial tension is equal to the maximum principal stress, oe = 0,. For torsion, a state of pure shear exists in the sample (a,, gz = 0, c3 = -a,). For this case the effective stress is J3 rri . Thus by comparing creep rupture results for uniaxial tension with those for torsion it is possible to vary the values of 0, and a, independently and in this way determine the value of a in equation (5). It is also possible to determine a by comparing creep rupture results for uniaxial tension with those for notched bars. The relations needed for this are shown in Table 1 for the different kinds of notches considered in this paper. It is, of course, necessary to know the stress state in the notch. Generally this is found by making a finite element analysis of creep deformation in the notch Table
I. Sample geometries
Uniaxial
ur=
and stress states for multiaxial
tension
Torsion
S-l ~ S
( > tests
stresses
0, = maximum (T*= bl = 0
principal
stress
6, = maximun Q* = 0 (r3 = -fJ
principal
stress
6, = maximum
principal
stress
u 1
Notched
bar
S-l
w (r
1
1069
and determining the stress state when steady state flow has been established. Although the stresses in the notch plane cannot be completely uniform, the stresses in the notch plane near the axis of the bar do not depend strongly on the distance from the axis. Hayhurst and Henderson [9] have obtained steady state solutions for a circular notch and for the British Standard “V” notch. The circular notch treated by them was introduced originally by Bridgman but is here called the Hayhurst circular notch to distinguish it from the Cane circular notch below. The finite element calculations of the maximum principal stress, ul, and the effective stress, cre, have been reported by Hayhurst et al. [4]. We have used their results to determine the stress states in the notches under consideration. As shown in Table 1, we define the severity of the notch, S, as the ratio of the maximum principal stress to the effective stress (evaluated at the center of the notch plane): S = a,/~,. The values of S for the different notches considered are shown in the table. For the axisymmetric stress state in the notch, we may express the effective stress as u,=u,-a,, where uT = u2 = u) is the transverse stress acting in the notch (at r = 0). Using the definition of the notch severity we may write
Principal
Stress State
STRESSES
British standard S=1.25 cr,=o,=O.Za, notch Hayhurst circular s= 1.49 o,=o,=0.33a, notch Cane circular s=1.9 u,=o,=0.470, notch S = severitv of the notch = 0, io.
u1
(6)
1070
NIX et al.:
PREDICTING
CREEP RUPTURE UNDER MULTIAXIAL
STRESSES
more severe than the Hayhurst circular notch. To distinguish between the two we refer to the notch geometry used by Cane as the Cane circular notch. The data taken from Cane’s paper include the rupture time, the maximum principal stress and the effective stress and are shown in Table 2, along with the other principal stresses that have been calculated for each test. Also shown in the table are values of tl that have been calculated using equation (5). A value of CLcan be obtained for each torsion or notched bar test with the cl, and a, for that test with the corresponding M and x values determined from uniaxial test data. We note that for this set of data the value of OL is not constant, but varies from 0.39 for one of the shortest multiaxial tests to 0.79 for the longest test. The parameter a might be expected to vary in this way if the fracture mode changes significantly from the shortest test to the longest, but this was not reported by Cane and is not expected for this material. In the next section of the paper we offer an alternative approach to creep rupture under multiaxial stresses. This approach, which is based on the principal facet stress, is capable of rationalizing not only the data of Cane [13] and Hayhurst et al. [7], but also the multiaxial creep fracture data of other investigators [lo, 141.
for the transverse stress in the notch. Equivalently, we may express the notch severity as
s-f!-. @I-uT
We note that S = 1 for uniaxial tension (cr = 0) and that S becomes indefinitely large in the limit of hydrostatic tension (uT = or,). Using the notch parameters shown in Table 1, Hayhurst et al. [7] have determined the values of o!for several materials. Their results are: a = 0 for an alu~num alloy, d = 0.7 for copper and CI= 0.75 for 316 stainless steel. The effective stress appears to be much more important than the maximum principal stress in creep rupture of the aluminum alloy. The results for copper and 316 stainless steel suggests that both the maximum principal stress and the effective stress influence the rupture process. It should be noted that slightly different values of cz were reported for copper in earlier papers. Revisions in the value of CIare directly attributable to refined estimates of the stress state in the notch. Multiaxial (tension-torsion) creep rupture experiments have also been conducted on pre-cavitated copper by Stanzl et al. (lo]. Although they did not express their results in terms of equation (5), a value of CI= 1 can be determined from their finding that the rupture time for this material is uniquely related to the maximum principal stress. This result is consistent with ex~tations; since the pre-cavitation process results in cavities of all of the grain boundaries, it might be expected that the effective stress would play no role in the fracture process. Apparently, the maximum principal, stress which is the driving force for cavity growth and coalescence, completely determines the rupture lifetime for this material. Other data sets for copper also indicate that the rupture time is well characterized by the maximum principal stress [ll, 121. The damage mechanics approach can also be tested using the multiaxial creep rupture data for a 2: Cr-1 MO steel published by Cane [13]. This set of data includes uniaxial tension, torsion and notched bar results. The shape of the notch in the Cane experiments is circular but the dimensions make it
The principal facet stress
It is well known that cavitation is concentrated on those grain bounda~es that are ~~ndi~ul~ to the maximum principal stressf7, 10, 152 11.In some cases this concentration is extreme, with inclined grain boundaries showing very little cavitation. One way to account for this strong cavitational anisotropy is to consider the stress redistribution that occurs when the grain boundaries slide. The shear stresses on grain boundaries are relieved when they slide and this causes a redistribution of the normal stresses to occur. Cocks and Ashby [22] and Nix [23] have studied the stress redistribution for a two dimensional hexagonal grain structure with remote uniaxial tension, rr, applied ~~ndicular to one set of grain facets. They showed that after grain boundary sliding the average stress on the transverse facets is 1.5 cr, while that on the inclined facets is zero. This helps to
Table 2. Creep rupture data for 2iCr-1 MO steel [data from Cane (1982)] Stress state Uniaxial tension
Rupture time (s)
$i%,
(M&j
1.12 x 4.09 x 9.69 x 1.52 x 2.34 x 4.00 x
106 IO* IO6 10’ IO’ IO’
178 147 I24 108 93 77
I 0 0
Torsion
5.45 x 1.49 x 3.06 x 5.91 x
106 IO’ IO’ IO’
103 85 ::
0 0 s
Cane circular notch
1.55 x 10s 1.90 x IO6
338 236
159 111
Cane circular notch parameters S = s,/uq =
0 0
0 0 0 0 0 0 -103 -85 -72 -62 159 111
(?&a)
(&ii&)
178 147 124 108 93 77
399 329 278 242 208 172
178 147 124 108
29.5 243 206 177
0.52 0.60 0.68 0.79
178 124
559 391
0.41 0.39
1.9cz = cJ = bT = PI (S - 1)/S = 0.47 b,.
a -
NIX et al.:
PREDICTING
CREEP RUPTURE UNDER MULTIAXIAL
explain why cavitation is concentrated on the transverse grain boundaries. Because grain boundary sliding and the associated stress re~st~bution are expected to occur under creep conditions, it is reasonable to suppose that the stress directly influencing the cavitation which leads failure is not the applied stress, but the concentrated normal stress acting on the transverse or nearly transverse grain boundaries. We call this the principal facet stress and use it as a parameter for describing creep rupture under multiaxial stress conditions. The simple estimate given above of the principal facet stress for a two dimensional hexagonal grain structure subjected to uniaxial tension is not adequate for our purposes. It does not include multiaxial stresses and it is not based on three dimensional grain structure. An estimate of the principal facet stress for a three dimensional grain structure can be made using the results of an analysis given by Anderson and Rice [24]. They have recently analyzed grain boundary sliding and cavitation in a polycrystalline solid composed of a periodic array of identical grains shaped like Wigner-Seitz cells. Each grain has 6 square facets and 8 hexagonal facets and all of the edges have the same length. Two orientations were considered: one with the tensile axis perpendicular to one set of square facets and one with the tensile axis perpendicular to one set of hexagonal facets. Anderson and Rice analyzed the average normal stress acting on the transverse grain boundaries for each of these two orientations. Before cavitation has started to occur but after grain boundary sliding has relaxed the shear stresses on the boundaries, they show that the principal facet stress is a? = 3a, - 25 for the tensile axis perpendicular facets and
(8) to one set of square
cr? = 1.67a, - 0.67crr
(9)
for the tensile axis perpendicular to one set of hexagonal facets. Here CF]is the maximum principal stress and ur = err = g3 is the axisymmetric transverse stress. To obtain the average principal facet stress for a polycrystalline structure we take a weighted average of the facet stresses given by equations (8) and (9). Since there are 14 facets, of which 6 are square and 8 are hexagonal, we define the average principal facet stress as d _.!%@l+& F-14
F
hex 14OF
*
(101
Using equations (8) and (9) this becomes oF = 2.24a, - 1.24~~
(11)
Although the analysis of Anderson and Rice [21] was for an axisymmetric stress state (8* = crj = er), we generalize equation (11) by replacing the transverse stress with the mean value of the two transverse stresses a, and cr,: Q~= (a* + a,)/2. With this
modification, given by
STRESSES
the average principal
1071
facet stress is
bF = 2.24a, - 0.62@, + 03).
(12)
We note that for a hydrostatic stress state (a, = a2 = u3 = uu) the principal facet stress is simply equal to crH.This corresponds to the limit of no grain boundary sliding and no stress redistribution. It should be emphasized that our estimate of the principal facet stress is expected to be valid after grain boundary sliding has caused a redistribution of the stresses to occur but before the transverse grain boundaries have started to cavitate, During the course of creep, cavitation on the transverse boundaries will lead to additional changes in the facet stresses. Eventually the transverse facets will fail completely and the stresses will then be supported entirely by the other boundary facets. Presumably the inclined facets would then begin to cavitate rapidly leading to final rupture. These complexities, although important to a prediction of the absolute rupture time, need not be included in the multiaxial creep rupture parameter.
RESULTS AND DISCUSSION In this section of the paper we show that the principal facet stress can be used to describe creep fracture under multiaxial stress conditions for several materials. Six different sets of multiaxial creep rupture data are used in this study: (I) the Cane data for a 2$ Cr-I MO steel, described in the previous section, (2) uniaxial tension and notched bar data for a 316 stainless steel, reported by Hayhurst et al. [7], (3) tension and torsion data for uncavitated copper, reported by Stanzl et al. [lo], (4) tension and torsion data for Nimonic 80A, reported by Dyson and McLean [14], (5) creep rupture data for commercially pure copper subjected to combined tension and torsion, reported by Henderson 1121 and (6) creep rupture data for pure copper with superimposed hydrostatic pressure, reported by Needham and Greenwood [ll]. The data taken from these studies are presented in Tables 2-7, respectively. Figure I shows the creep rupture data of Cane [ 131 for a 2: Cr-1 MO steel. In this figure, and the ones to follow, three different stress parameters: the effective stress, a,; the maximum principal stress, 0, and the principal facet stress, cF are plotted against the logarithm of the rupture time. In this way the correlation between the different stress parameters and the rupture time can be assessed visually. All of the data used in the figures can be found in the tables. In Fig. 1 the data for uniaxial tension are distinguished from those for the torsion and notched bar experiments by different symbols. Evidently, neither the effective stress nor the maximum principal stress correlate well with the time to rupture for 2; Cr-1 MO steel. We consider especially the failure of the max-
NIX et al.:
PREDICTING
CREEP RUPTURE
UNDER
MULTIAXIAL
STRESSES
Table 3. Creep rupture data for 316 stainless steel [data from Hayhurst et uf. (1984)]
Stress state
Rupture time (s)
(&a)
$a)
x rd x Id X 106 x IO6 x 10’
348 312 275 245 209
0 0 0 0 0
British standard notch
2.54 X lo6 4.74 x 106 6.23 x 10’
306 282 203
61 56
Hayhurst circular notch
5.80 x 106 9.80 x lo6
305 284
Uniaxial tension
5.06 8.78 2.15 6.18 2.83
(h&L)
(Gk)
(h&
348 312 275 245 209
780 699 616 549 468
41
56 41
245 226 162
610 562 404
101 94
101 94
205 191
558 520
a” 0 0 0 61
British standard notch parameters S = o, /a, = 1.25 a, = oj = a, = 6, (S - 1)/S = 0.20 G,. Hayhurst circular notch parameters S = q /a, = 1.49 u2 = + = a, = o, (S - 1)/S = 0.33 bl, Table 4. Crew ruuture data for initially uncavitated cout~r [data from Stanzi er al. (1983N Stress state Uniaxial tension
Rupture time (s) 2.10 4.32 7.02 3.60 4.60
x x x x x
lo4 IO4 IO4 105 105”
6.80 x id 1.50 X I@ 8.60 x 10d 1.75 X 105 9.27 x IO5
Torsion
(&a)
(&a)
30.0 27.6 22. i 19.3 16.6
0 0 0 0 0
32.4 26.0 18.9 16.2 13.5
0 0 0 0
0
@Isa) 0 0
(?&a)
: 0
30.0 27.6 22.t 19.3 16.6
67.2 61.8 49.5 43.2 37.2
- 32.4 -26.0 - 18.9 - 16.2 -13.5
56.1 45.0 32.7 28.0 23.4
92.7 74.4 54.1 46.3 38.6
‘This rupture time was obtained from Fig. 8 of Stanzi et at.; the time reported in their Tabb 2 appears to be in error. Table 5. Creep rttpture data for Nimonic SOA [data from Dyson and McLean (1977)] Stress state Uniaxial tension
Torsion
Rupture time Is)
(M?a,
(&a,
6.41 x 106 5.62 X 106 2.62 x IO6 2.45 x IO6 1.40 X 106 1.73 x 103 3.90 X 104 1.65 X to’ 4.69 X Id
153 154 187 203 216 326 411 494 567
0 0 0 0 0 0 0 0 0
4.10 X 10’ 1.57 X 10’ 7.08 x 10” 3.42 x IO6 9.39 x 10s 3.59 X 10s 1.73 x 10s 5.63 X 104
57.9 84.9 111 138 184 211 243 281
0 0 0 0 0 0 0 0
(MGa) 0
0 0 0 0 0 0 0
0 - 57.9 -84.9 -111 -138 -184 -211 - 243 -281
153 154 187 203 216 326 411 494 567
342 345 418 45.5 483 731 920 1106 1270
100 147 192 238 319 366 420 486
166 243 318 394 527 605 694 803
Table 6. Creep rupture data copper [data from Henderson (1979)] Stress state
Rupture time
&a,
Uniaxial tension
2.69 x 10’ 3.87 X lo6 4.54 X lb 1.77 X 10s
30.6 46.2 46,2 53.9
a/z = 3
1.02 X 106 2.47 X 106
a/z -2
Ihzaf
(&a) 0
(&a)
&?a)
0 0 0
30.6 46.2 46.2 53.9
68.6 104 104 121
61.5 50.6
-5.6 -4.6
64.5 53.0
141 116
1.65 x Id 1.10 X 107
55.9 37.2
-9.6 -6.4
61.2 40.7
131 87.2
1.10x 106 4.04 X lo*
57.0 49.9
0
0
-21.8 -19.1
70.5 61.7
141 124
a/r = 0.4
5.17 X 106
41.9
0
-28.2
61.1
111
Torsion
1.77 X 10’
30.3
0
-30.3
52.5
a/t =
1
u = tensile stress; r = shear stress.
86.7
NIX ef al.:
PREDICTING
Table 7. Creep rupture stress state Uniaxial tension with hydrostatic pressure
Uniaxial tension
CREEP
data for OFHC Rupture time (s)
RUPTIJRE
UNDER
MULTIAXIAL
copper [data from Needham (A)
STRESSES
and Greenwood
(197_5)]
&a)
(hza)
7.20 1.35 1.00 5.50
x x x x
10” 106 106 105
10.1 14.4 17.0 20.2
- 10.6 -6.3 -3.7 -0.5
-10.6 -6.3 - 3.7 -0.5
20.7 20.7 20.7 20.7
35.8 40.1 42.7 45.9
8.70 1.95 1.10 6.50 5.30 3.50
x x x x x x
IO6 106 106 IO’ 10s 10s
10.1 13.6 17.9 20.2 24.0 24.0
-14.1 - 10.6 -6.3 -4.0 -0.2 -0.2
-14.1 - 10.6 -6.3 -4.0 -0.2 -0.2
24.2 24.2 24.2 24.2 24.2 24.2
40.1 43.6 47.9 50.2 54.0 54.0
2.50 1.15 8.50 3.00 1.50
x x x x x
IO6 106 IO5 IO’ 10s
13.7 17.0 21.4 24.2 27.2
- 13.9 -10.6 -6.2 - 3.4 -0.4
- 13.9 - 10.6 -6.2 -3.4 -0.4
27.6 27.6 27.6 27.6 27.6
47.9 51.2 55.6 58.4 61.4
2.00 x 10s
27.6
27.6
61.8
imum principal stress to describe the rupture data. At the same maximum principal stress, the samples tested in torsion fail sooner than those tested in tension while the notched samples fail later. By contrast, the principal facet stress, as defined by equation [12], brings all of the creep rupture data onto a single curve. The success of the principal facet stress in bringing the data together in Fig. 1 suggests that the data points for torsion and the notched bar experiments are separated from those for uniaxial tension (at a fixed maximum principal stress) because of different degrees of grain boundary sliding and stress redistribution. High principal facet stresses are produced in torsion and they cause the torsion samples to fail sooner than samples subjected to uniaxial tension. Conversely, small principal facet stresses are developed in the notch, with the consequence that the notched samples fail later than those subjected to uniaxial tension. Cane has studied grain boundary sliding and cavitation in 2aCr-1 MO steel under uniaxial creep conditions at 838 K [17]. Using longitudinal scratches as markers, grain boundary displacements were measured on uniaxial creep specimens. Stresses can be redistributed to transverse facets when the displacements associated with grain boundary sliding are of the order of the relative elastic displacements of the grains, 6, = (a/E)& where cr is the applied stress, E is Young’s modulus and L is the grain diameter. Cane’s measurements indicate that grain boundary sliding displacements satisfy this condition for 2f Cr-1 MO steel. For example, for a stress of 108 MPa at 1.5% strain, the mean displacement associated with grain boundary sliding is approximately 0.1 pm, which is of the order of 6, (approximately equal to 0.08 pm taking L = 150 pm [ 171and E = 2 x lo5 MPa). It follows that the grain boundary deformation behavior of 2iCr-1 MO steel is consistent with the assumption of grain boundary sliding made in the derivation of the principal facet stress.
0
0
(A)
1073
(&i)
The creep rupture data of Hayhurst et al. [7] for 316 stainless steel are shown in Fig. 3. Again the different stress parameters are plotted against the logarithm of the rupture time. Here the data for uniaxial tension tests are distinguished from those for the notched bar tests. The data for two different notch geometries are shown: The British Standard notch and the Hayhurst circular notch. The rupture times for the notched samples are greater than those for uniaxial tension at the same maximum principal stress. This represents a notch strengthening effect. 210 190170 -
.
0
. 0
150 130 110 -
90
.
o
Y .
70 -
2 l/4 Cr tMo Steel Temperature = 838KCane (19R2)
n
.
0
. 0
Torsion Circular
.
0
Tension
0
Notch
2”
400 2 114 Cr 1Mo Steel Temperature = 838K Cane (1982)
. 300-
100 105
“““‘I
“““I 106 Time to Rupture
““‘. 107
10x
(s)
Fig. 1. Creep rupture data for a 2aCr-I MO steel plotted as the effective stress (top), the maximum principal stress (middle) and the principal facet stress (bottom) vs the logarithm of the rupture time.
1074
NIX et uf.: PREDICTING
CREEP RUPTURE
We note that the Hayhurst circular notch is more severe than the British Standard notch and it causes a greater notch strengthening effect. Here again we find that the principal facet stress is a correlating parameter that brings all of the data onto a single curve. Evidently the transverse stresses acting in the notch makes the principal facet stress smaller than for uniaxial tension. This causes the rupture life of the notched bar to be greater than that for uniaxial tension under the same maximum principal stress. Stanzl et al. [lo] have measured the creep rupture life of uncavitated copper at 500°C in both tension and torsion. Their results are shown in Fig. 5. We observe that samples tested in torsion fail sooner than those tested in tension at the same maximum principal stress just, as for the 2iCr-1 MO steel data discussed above. We find again that the principal facet stress brings the tension and torsion data together. Figure 4 shows the creep rupture data of Dyson and McLean [14] for Nimonic SOA. We find that neither the effective stress nor the maxims principal stress alone can characterize the creep rupture data. Again we see that the principal facet stress correlates well with the time to rupture. In the four examples cited above the principal facet stress beings the multiaxial creep rupture data into coincidence with the uniaxial data. This correlation should prove to be quite useful because it allows a prediction of the rupture time under multiaxial stresses from uniaxial creep rupture data. Of course, this prediction requires a knowledge of the multiaxial stress state imposed. Although the principal facet stress is expected to describe multiaxial creep fracture for many materials, there are some materials for which it fails to bring the uniaxial and multiaxial data into coincidence. One example is found in the work of Hayhurst et al. [7l who showed that creep fracture of an alu~num alloy is well characterized by the effective stress (a = 0). Although we do not fully understand the failure of the principal facet stress in this case, we believe it may be related to the failure of this material to exhibit grain boundary sliding and cavitation of transverse boundaries, as envisioned by the present approach. It is known that pure aluminum does not cavitate easily. If the aluminum alloy cited here behaves like pure aluminum, then the rupture time would not be expected to correlate with the principal facet stress. Stanzl et al. [lo] have also performed tension and torsion experiments for precavitated copper specimens. They have found that, for a given maximum principal stress, the time to rupture for precavitated copper under tension and torsion is the same. It follows that the principal facet stress theory is not valid for this case. Precavitated copper is so brittle that rupture may occur before the stresses are adequately redistributed by grain boundary sliding. if so we would not expect the principal facet stress to be applicable.
UNDER MULTIAXIAL
STRESSES
Three separate studies of creep fracture of copper under multiaxial stresses have indicated that failure of this materia1 is characterized completely by the maximum principal stress f7, 11, 121. The results of one study are shown in Table 8. The data were given by Hayhurst, Dimmer and Morrison [7] in terms of the normalized maximum principal stress, C, = a,/~,, where cr, is the maximum principal stress in a smooth bar and 0, is the average stress on the minimum section of a notched bar for a given time to rupture. Taking rr, to be equal to the average maximum principal stress, cri, the notch effect factor, equal to the reciprocal of E,, is given by NEF = Q!/a,. Values of er and 6) normalized by a, can be determined for each specimen geometry using relations derived from finite element results [4,9]. A normalized value of the principal facet stress, ur/tr,, is calculated by inserting the normalized principal stresses into equation (12). These normalized stress values are shown in Table 8 with corresponding values of the notch severity, S, [4]. The normalized principal facet stress is not independent of specimen geometry. Thus, it is not a valid parameter for predicting creep rupture for this material. The principal facet stress theory is based on the assumption that grain boundary sliding and stress redistribution occurs and that failure occurs primarily by cavitation on transverse boundaries. It is, at first, surprizing that err is not valid for notched copper specimens since it appears to be valid for the copper data of Stanzl et al. [lo]. An explanation of this discrepancy is found by comparing the micrographs of notched bar cross-sections of copper with the micrographs of notched bar cross-sections of 316 stainless steel reported by Hayhurst et al. [7]. This comparison shows that the cavitation damage in notched bars of copper is much more localized on the transverse plane at the center of the notch than it is for 316 stainless steel. The failure of notched specimens of copper appears to be more representative of crack propagation than of uniform creep rupture. The validity of cr for the copper data of Stanzl et al. is explained by noting that their data were obtained for unnotched specimens (uniaxial tension and torsion). Henderson [ 1I] has reported creep rupture data for samples of commercially pure copper subjected to varying amounts of loading in tension and torsion. We have calculated the various stress parameters from that data; they are plotted vs the time to rupture in Fig. 5. It is clear that the effective stress fails to describe the rupture data for this material. A much better correlation is found using the maximum principal stress. The correlation of the principal facet stress with the time to rupture is also shown in the figure. Although the amount of scatter in the plot of principal facet stress vs time to rupture is greater than that found in the plot of maximum principal stress vs the time to rupture, the differences are not large. One concludes that the maximum principal stress is a better correlating parameter for this material, but it
NIX et al.:
PREDICTING
CREEP RUPTURE
? 316 Stainless Steel Temperature = 823K Hayhunt. Dimmer & Morrison (1984)
400 Cl o 3M)-
600
o
Tension
.
British Standard Notch
.
Circular Notch
“““” ‘.““‘,
100
.
.““.‘-
.““..I
0
.
1075 ““-
.“‘.“-
Nimonic 80A Temperature = 1023K Dyson&McLean (1977)
.
-
l
0
.
0 0
.
.
“““‘I .‘...‘I
““‘-
316 Stainless Steel Temperature = 823K Hayhurst. Dimmer & Morrison (1984).
0
STRESSES
0
400.
400.
.“‘.“-
x0-
0
200 -
UNDER MULTIAXIAL
Nimonic 80A Temperature = 1023K Dyson & McLean (1977)
0
500.
0
400300 ‘200 _
100-
. o
Tension
.
Torsion
_
0 .
.
.O
0
l =J ‘*.
0
“.‘..‘n ...““-
““‘.‘m .. “‘I
““.‘I ““.‘I
..’ “‘I “““I
”
--y
Nimonic 80A Temperah!% = 1023K Dyson & McLean (1977)
500 -
.
British Standard Notch
l
Circular Notch
t 0103
I
Time to Rupture (s)
I
o
20
l
o
Tension Torsion
10’
.
0
.
I
106
.
. . .
.
.
10’
..“_j 108
cl
,“..I
4. Creep rupture data for Nimonic 80A plotted as the effective stress (top), the maximum principal stress (middle) and the principal facet stress (bottom) vs the logarithm of the rupture time.
40 -
0
“.“‘I
I
105
Time to Rupture (s)
Fig. 2. Creep rupture data for 316 stainless steel plotted as the effective stress (top), the maximum principal stress (middle) and the principle facet stress (bottom) vs the logarithm of the rupture time.
30
.
104
’
3. _ 0
‘.....I
•1
copper Temperature = 523K Henderson (1979)
0
20 ’
0 Tension
. Oh=3 0 o/7=2 . o/7=1 _ . o/7=0.4 A Torsion
Copper Temperature = 523K Henderson (1979)
30
loo-
.
..“‘I
‘..“I
90
“‘.”
t
Temperature = 773K Stanzl. Argon & Tschegg (1983)
80
120
70
0
0 Tension . d7=3 0 G/7=2 . d7=1 . o/7=0.4 A Torsion
140
60
100
50
Copper 80 - Temperature = 523K Henderson (1979)
40 30’ IO’
A
‘..‘.‘I
104
“”
* 16
....-I
106
Time to Rupture (s)
Fig. 3. Creep rupture data for pure copper plotted as the effective stress (top), the maximum principal stress (middle) and the principal facet stress (bottom) vs the logarithm of the rupture time.
: 60
106
107 Time to Ruptu~ (s)
Fig. 5. Creep rupture data for commercially pure copper plotted as the effective stress (top), the maximum principal stress (middle) and the principal facet stress (bottom) vs the logarithm of the rupture time.
1076
NIX et al.: PREDICTING CREEP RUPTURE UNDER MULTIAXIAL STRESSES Table
8. Creep
Uniaxial tension British standard notch Circular notch
rupture data for copper-stress states for equal rupture times [data from Hayhurst et al. (1984)]
1.00
1.oo
0
0
2.24
1.25
0.899
0.180
0.180
1.790
1.49
1.053
0.346
0.346
1.930
NEF = Notch effect factor. or is the rupture stress for a smooth bar loaded in uniaxial tension with the same rupture time. a,=a,=u,=a,(S-1)/S.
is only marginally better. We do not know why the principal facet stress fails to give a better correlation in this case. Perhaps the oxides that are present in commercially pure copper inhibit grain boundary sliding and prevent the necessary stress redistribution from occurring. Needham and Greenwood [12] have studied the creep and creep rupture properties of OFHC copper subjected to uniaxial with superimposed hydrostatic pressure. Their study is an important one because the stress states in their experiments were better known than in the notched bar experiments. Their results are shown in Fig. 6. Again we see that for copper the effective stress fails to describe the time to rupture data. As Needham and Greenwood concluded, the maximum principal stress appears to give the best correlation with the data. Although the correlation based on the maximum principal stress is not perfect, it is better than that based on the principal facet stress. This data set raises the most important question about the validity of using the principal facet stress as a correlating parameter for creep fracture. Pure copper is known to exhibit both grain boundary sliding and intergranular cavitation at the temperature of these tests 773 K (SOP), so the principal facet stress should be expected to correlate well with the time to rupture. Since the principal facet stress works well for many other materials, it is of interest to determine why it does not work well in this case. We believe the failure of the principal facet stress to describe these data may be related to the particular multiaxial stress states that are produced by subjecting the samples to hydrostatic pressure. In particular, we suggest that the principal facet stress underestimates the time to rupture when large hydrostatic pressures are present. Consider the tension-torsion data of Stanzl et al. [lo] (Fig. 3) which were obtained for the same material (nominally) and at the same temperature as the NeedhamGreenwood data [12]. The tension-torsion data give the best correlation with the principal facet stress whereas the data obtained using hydrostatic pressure gives a better correlation with the maximum principal stress. Thus, the two techniques for producing multiaxial stresses appear to give different results. We note that the hydrostatic pressure is either negative or zero in the tension-torsion experiments whereas it is positive in most of the tests involving the use of hydrostatic
pressure. A close inspection of the data shown in Table 7 and Fig. 6 indicates that the principal facet stress underestimates the time of rupture of samples subjected to hydrostatic pressure relative to the one subjected to uniaxial tension only. This seems to suggest that the constant in the second term on the right hand side of equation (12) is too large. This term predicts that the principal facet stress for a round bar subjected to radial pressure P, (a, = 0, CT~ = u3 = -P,) is 1.24P,. According to the present model, such a radial pressure is expected to cause grain boundary sliding to occur and to develop tensile stresses on those grain boundary facets perpendicular to the axis of the sample. For soft metals such as pure copper this may not be possible. The deformation associated with such a stress state may be more homogeneous than that envisioned in the present
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.
.
.
. .
/
i
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.
Needham &Greenwood
(1975) m i .I 4
,5 _ o 0,=27.6MPa 5
. .
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.
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.
.
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Time to Rupture (s)
Fig. 6. Creep rupture data for pure copper plotted as the effective stress (top), the maximum principal stress (middle) and the principal facet stress (bottom) vs the logarithm of the rupture time.
NIX et al.:
PREDICTING
CREEP RUPTURE UNDER MULTIAXIAL
treatment. This would cause the principal facet stress to underestimate the time to rupture of samples subjected to hydrostatic pressure relative those subjetted to uniaxial tension, as we have seen. Although such an underestimation is not desirable, it is preferable to an overestimate of the time to rupture. By comparison, using the maximum principal stress as the correlating parameter can lead to overestimates of the time to rupture. This is shown most clearly in Figs 1, 3 and 4 (middle) where the samples subjected to torsion fail sooner than would be expected on the basis of uniaxial tension experiments. SUMMARY We have shown that creep rupture under multiaxial stresses can be predicted from uniaxial creep rupture data using the principal facet stress as a creep rupture parameter. Grain boundary sliding causes a stress redistribution to occur and this leads to preferential cavitation on grain boundaries perpendicular to the maximum principal stress. The stresses on the cavitating boundaries depend on stress state and this provides a basis for using the principal facet stress as a multiaxial creep rupture parameter. For three different materials: (1) 2: Cr-1 MO steel, (2) 316 stainless steel, (3) Nimonic 80A, the time to rupture at a given temperature depends uniquely on the principal facet stress. The correlation with the principal facet stress has also been found to be valid for pure copper tested in tension and torsion. Thus, the principal facet stress can be used to predict the time to rupture under multiaxial stresses from measurements of the time to rupture under uniaxial stresses. However, the principal facet stress fails to give the best correlation with time to rupture for an aluminum alloy, which is governed by the effective stress, for precavitated copper, which is governed by the maximum principal stress and for three other data sets for uncavitated copper, which appear to be governed by the maximum principal stress. We discuss the probable causes of the failure of the principal facet stress in these cases. Acknowledgements-Financial support by the Swiss Federal Government within the framework of the European cooper-
STRESSES
1077
ation COST 505 is gratefully acknowledged. Additional support from Stiftung Volkswagenwerk is also appreciated. Professor W. D. Nix was on sabbatical leave from Stanford at the Cole Polytechnique Fbderale de Lausanne by invitation of the president of the EPFL, Professor B. Vittoz. REFERENCES 1. D. R. Hayhurst, J. Mech. Phys. Solids 20, 381 (1972). 2. D. R. Hayhurst and F. A. Leckie, J. Mech. Phys. Solids 21, 431 (1973).
3. D. R. Hayhurst and F. A. Leckie, in J. Carlsson and N. G. Ohlson (editors) Mechanical Behavior of Materials, Proc. ICM4, Vol. 2, p. 1195. Pergamon Press, Oxford (1984). 4. D. R. Hayhurst and F. A. Leckie and C. J. Morrison, Proc. R. Sot. A360, 243 (1978). 5. L. M. Kachanov, IZV Akad. Nauk. SSSR Otd. Teckh. Nuuk. 8, 26 (1958). 6. Y. N. Robotnov, Creep Problems in Structural Members
(English translation edited by F. A. Leckie). North Holland, Amsterdam (1969). 7. D. R. Hayhurst, P. R. Dimmer and C. J. Morrison, Phil. Trans. R. Sot. A 311, 103 (1984).
8. H. Riedel, Fracture at High Temperatures, p. 85. Springer, Berlin (1987). 9. D. R. Hayhurst and J. T. Henderson, Int. J. mech. Sci. 19, 133 (1977). 10. S. E. Stanzl, A. S. Argon and E. K. Tschegg, Acta metall. 31, 833 (1983). Il. J. Henderson, J. Engng Mater. Tech., Trans. A.S.M. E. 101, 356 (1979).
12. N. G. Needham and G. W. Greenwood,
Metal Sci.
9, 258 (1975).
13. B. J. Cane, in Advances in Fracture Research (Fracture 8f), Vol. 3, p. 1285. Pergamon Press, Oxford (1982). 14. B. F. Dyson and D. McLean, Metal Sci. 11, 37 (1977). 15. P. W. Davies, K. R. Williams and B. Wilshire, Phil. Mag. 18, 197 (1968). 16. T. Johannesson and A. Tholen, J. Inst. Metals 97, 243 (1969).
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19. R. T. Chen and J. R. Weertman, Mater. Sci. Engng 64, 15 (1984). 20. K. S. Yu and W. D. Nix, Scripta mefall. 18, 173 (1984). 21. G. Eggeler, J. C. Earthman, N. Nilsvang and B. Ilschner. To be published. 22. A. C. F. Cocks and M. F. Ashby, Prog. Mater. Sci. 27, 265 (1982). 23. W. D. Nix. To be published. 24. P. M. Anderson and J. R. Rice, Acta metall. 33, 409 (1985).