Comparison of creep rupture lifetimes of single and double notched tensile bars

Acta metall, mater. Vol. 41, No. 4, pp. 1215-1222, 1993

0956-7151/93 $6.00+ 0.00 PergamonPress Ltd

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COMPARISON OF CREEP RUPTURE LIFETIMES OF SINGLE A N D DOUBLE NOTCHED TENSILE BARS A. M. OTHMAN 1, J. LINLt, D. R. HAYHURSTI't and B. F. DYSON z ~Department of Mechanical and Process Engineering, University of Sheffield, Mappin Street, P.O. Box 600, Shefffield S1 4DU and 2Divisionof Materials Metrology, National Physical Laboratory, Teddington, Middlesex TW11 0LW, England (Received 3 July 1992)

Abstract---Creep rupture behaviour and lifetimes have been studied for two notched bars--one having a single notch and the other a double notch--by use of the creep finite element solver, Damage XX, based on Continuum Damage Mechanics (C.D.M.). A physically-basedconstitutive equation which utilises two damage parameters was used to model tertiary creep behaviour. Its versatility allows a wide spectrum of material parameters to be used which simulated a range of material behaviour. All computations showed that lifetimesfor the double-notch testpiece were within 95% of those for the equivalent single notch, and therefore validated the double notch geometry being studied. A consequence of the calculations has been the verification of a previous suggestion that the lifetimes of materials which exhibit tertiary creep due to dislocation strain softening (such as nickel-basedsuperalloys)could be less affected by stress-state effects on ductility, than materials which do not exhibit such softening (such as pure metals and simple solid solution alloys).

1. INTRODUCTION Notched bars loaded in uni-axial tension have been used as a convenient method to produce in the laboratory those tel-axial states of stress which are encountered in service in real components. Stress and strain histories in semi-circular and British Standard (B.S.) notched bars which undergo creep deformation due to steady loading were numerically studied by Hayhurst and Henderson [1]. Notch strengthening and weakening were explained in terms of the multiaxial stress rupture criterion satisfied by the material [2]. It was shown how the circular notch may be used as a materials test and that the B.S. notch is a good means of assessing the sensitivity of structural behaviour to the multi-axial stress rupture criterion of the material. The development of continuum damage in the creep rupture of the circular and B.S. notched bars was investigated by Hayhurst et aL [3] both experimentally and by using a time-iterative numerical procedure which describes the formation and growth of creep damage as a field quantity. Three other notched bars with different geometries had been previously studied by Hayhurst et al. [4] in an attempt to pin-point the design features which enable a reasonably large volume of material to be subjected to a uniform state of tri-axial tension stress and strain. Stationary-state stresses and strains were determined using finite element procedures for the

tPresent address: Department of Mechanical Engineering, UMIST, Manchester, England.

three geometries. One of the notch geometries is given in Fig. l(c, d); and a double notched bar having the same geometry as the single one is presented in Fig. l(a, b). The merit of the double notched bar is that it allows a micro and macro examination to be made of the state of damage in the testpiece just prior to fracture. High temperature testing of materials is both longterm and expensive and there is a need to collect as much macro- and micro-scopic information from a single test as possible. Two notches in one testpieee provide data both at failure, from the separated notch, and data just before failure, in the ostensibly unfailed notch. In this way the mechanisms-based information advocated by Dyson [5] can be collected. The provision of such data is essential to the establishment of materials databases necessary for use in the Computer Aided Design analysis techniques which have been developed [6] for use in component life assessment. The objective of this research was to determine by finite element analysis, the errors in fracture lifetime incurred by the use of two notches. A notch geometry has been chosen which has been used in practice [7] and which is advocated in a new Code of Practice for creep testing of notched bars [8]. To assess the robustness of the chosen design to the material constitutive law, a physically-based equation has been used which not only accounts for tertiary creep due to two mechanisms, but is constructed in a computerfriendly way which allows a wide spectrum of material variables to be input and so simulate a range of material behaviour.

1215

1216

OTHMAN et al.: CREEP RUPTURE IN NOTCHED TENSILE BARS cro

¢ro

(c)

(a)

b

b

|

i

(b)

for most cases of interest. The first damage state variable, w~, is defined from the physics of dislocation softening [I0] to lie within the range 0-1 for mathematical convenience and C quantifies its rate of evolution; and the second damage variable, w2, is defined from the physics of nucleation-controlled creep-constrained cavitation to range from zero to the failure point at w2 = 1/3; D = 1/3el, where ef is the failure strain. The use of the hyperbolic sine relationship in constitutive equations, although little used now, has a long history having been suggested by Nadai (1938) [I1] as being appropriate for many engineering alloys. Recently, a hyperbolic sine relationship has been shown to have a physical basis in nickel-based superalloys [12]. The strain history can be calculated by numerical integration of equation (1) as a coupled set for any arbitrary stress or temperature history, within the limits of applicability of the physical model. Othman et al. [9] demonstrated how equation set (1) could be modified to describe multi-axial creep with the stress function being either a hyperbolic sine or a power-law relationship.

(d) 2 2 Multi-axial creep with the sinh law and twodamage variables

I

b1~=1.9, n/~=0.5,

bib=a,

~/a=4

Fig. 1. Geometry of the notched bars. Isometric presentation and finite element presentation (a) and (b) for double notched bar; (c) and (d) for single notched bar. 2. CONSTITUTIVE EQUATIONS AND THEIR NORMALISATION 2.1. Uniaxial creep with the sinh law and two damage variables

Continuum Damage Mechanics (C.D.M.) with two damage state variables has been justified and then used by Othman et al. [9] to model tertiary creep softening in nickel-based superalloys caused by: (i) grain boundary cavity nucleation and growth, and (ii) the multiplication of mobile dislocations. The equation set in uniaxial form is de

dt

1 = A (1 - wl)(l - w2)~ sinh(Ba) CA

_

sinh(Bcr)

dw 2 1 dt = D A O - w l ) ( 1 - w 2 ) "

sinh(B~r)

(1)

where A, B, C and D are material constants and n is given by (Ba)coth(Ba), which approximates to Bo

First consider the strain rate equation (1) without the damage parameters wmand w2: it then reduces to de/dt = A sinh(Bo). This equation can be generalised for multi-axial conditions by assuming an energy dissipation rate potential ~ = (A/B)cosh(Boc) where tre=(gu~q~j/2) 1/2 and ~q~j are the stress deviators ( = % - o~6~//3). Assuming normality and the associated flow rule the multi-axial relation is given by =~

=

sinh(Bo~).

On reintroduction of the damage variables wl and w2 the multi-axial equation set discussed by Othman et al. [9] is recovered. de U = 3__AA(~qU~ dt 2 \ix2/(1

sinh(___._Bo',_)_ -

wl)(1

-

w2)n

dWl = CA (1 - Wl) sinh(Boe) dt (l - w2),n dw2 (~el) ' H sinh(B°e) dt = DA (l - wl)(1 -- w2)n

(2)

where n = Baocoth(Boc). The additional material constant v describes the stress state sensitivity of grain boundary cavity nucleation: (0~/0,) lacing unity under uni-axial conditions, equation set (1); H is a parameter to indicate the state of loading; for cr~ tensile, H = 1; and for o~ compressive, H = 0. The creep strain rates, iq, are dependent upon both the effective stress oe and the deviatoric stresses ~q~jwhile the first damage rate is related to the effective stress tre; and the second damage rate to the maximum principal tension stress 0 z and the effective stress oe.

OTHMAN et al.: CREEP RUPTURE IN NOTCHED TENSILE BARS Table

of the material constants for the constitutive equations with sinh stress dependence

1. Values

A (h -~)

B (MPa)

C

D

v

E (MPa)

2 x 10 -6

0.016

300

2.0

2

2 x l0 s

When C = D = 0, equation set (2) reduces to gij=(3A/2)(~o/tr,)sinh(Bae) which will be used to provide base-line information in order to determine t h e effect of damage on the creep behaviour of the testpiece.

2.3. Normalisation The constitutive equation sets (1) and (2) can be normalised by introduction of the following parameters -- (~ij

Eij

Si j = ~ij

Equation set (2) then becomes d2ij = 3 Sij sinh(ctZo) dz 2 r e (1 - wl)(l - w2)"

(3)

dw~ d T = Ceo (1 - Wl) sinh(uEe)

(4)

(1 -

w2)"

H sinh(ctZe ) dw2 = Deo ( ~ ) v dz (1 - wi)(l - w2)~ __

(5)

where the normalised time scale is given by z = At/eo, ~t = Btr0, e0 = ao/E, and n = Ze coth(~tZe).

2.4. Material data The investigation of physically based constitutive equations with two-damage-state-variables for the hyperbolic sine function have been reported by Othman et al. [9]. The material constants for the constitutive equations which make use of the sinh functions used in the initial numerical calculations are summarised and listed in Table I. They are derived from the work by Dyson and Loveday [7]. The tertiary material parameters C, D and v were systematically varied in subsequent calculations in order to study their influence on the robustness of the double notch design. 3. CONTINUUM DAMAGE MECHANICS FINITE ELEMENT SOLVER Creep damage describes the material degradation which gives rise to the acceleration of creep rate known as tertiary creep. The material is treated as a continuum under load and the constitutive equations are used to describe how the stress, strain and damage (or strain rate and damage rate) are interrelated at a point (or at an element) in the material. Different points in the body suffer different stress levels and are at different stages of material deterioration. The constitutive equations remain valid until the specified rupture criterion is reached. At this time, the point under consideration can no longer sustain load and,

1217

at this point (or element), the material is deemed to have failed. Ultimately a region of failed material will progress through the body until the applied load can no longer be sustained and the body fails by a global collapse mechanism. The numerical procedure used to solve the boundary value problem for creep damage deformation is that used by Hayhurst et al. [13]. It is based on the finite element method and employs constant strain triangular elements, which have been extensively used to model the behaviour of notched and cracked creep components [3]. The solution method is based on the theory of Continuum Damage Mechanics. The procedure takes the elastic solution as its starting point and integrates normalised creep strains, 2ij, and creep damage state variables w t , w2, with respect to the normalised time. The integration is carried out over a series of discrete normalised time steps using a fourth order Runge-Kutta technique; this procedure involves the repeated solution of the boundary value problem to determine the field quantities required for the numerical solution. Creep damage, as represented by the two damage state variables, develops monotonically with time throughout the structure, and failure of an element is deemed to have occurred when the second damage state variable, w2, attains the prescribed value of 0.3. The material element is then unable to transmit or sustain load and it is removed from the model. The boundary value problem is then redefined to allow either a crack, or damage zone, to develop and spread. Once the boundary value problem is redefined, the time integration is continued by taking the field variables just before the local failure occurred as the new starting point. The procedure is then repeated until complete failure of the cracked member occurs. 4. THE GEOMETRY, MESH AND NUMERICAL PROCEDURES The geometry of the single and double notched bars due to Dyson and Loveday [7] is shown in Fig. 1 and presented in Table 2. A single quadrant of each axisymmetric tension bar has been selected for investigation. The quadrant has been subdivided into triangular elements by the finite element mesh generation package, Femgen. The size of these elements had been redefined in the notched region of the bars where a high level stress concentration occurs. A coarse mesh is selected in the parallel portion of the testpiece where the stresses are uniform and change little. The meshes generated for the numerical investigation contain 642 constant strain triangular elements and 363 nodes for the single notched bar; 1150 elements and 631 nodes for the

b/a 1.9

Table 2. Notch geometry Rib h/b 0.5

3

s/a 4

1218

OTHMAN et al.: 1 I

2 I

3 I

CREEP RUPTURE IN NOTCHED TENSILE BARS 4

14

I

"-= 8

-

12

-

10

-

6

-

4

-."~ "o -

0

1

2

3

2

0

4

Fig. 2. Variation of ratio of predicted lifetimesfor notched, and parallel bar, to, testpieees for single ( ) and for double ( - - - ) notched bars with stress-state index v for a range of values of D with C = 0. The curves for D = 2 and C ---0 are plotted on the upper displaced axes. double notched bar, respectively. The half band width of the stiffness matrix has been optimised, using special purpose software, to the minimum possible by automatically renumbering the element nodes. The procedure of optimizing the band width of the stiffness matrix is important for the efficient usage of computer resources, and also to achieve accuracy of the numerical computations. The optimised mesh was then input to the Continuum Damage Mechanics solver Damage XX. The loaded boundary of the mesh was subjected to a constant normalised stress 27rr = 1; and the boundaries of symmetry were subjected to the conditions of 27xr = 0 and of zero normal displacement. The computations were carried out on the IBM 3090-600E computer at the Science and Engineering Research Council, Rutherford Appleton Laboratory. The computed results of stresses, strains, displacements, damage state variables, were stored for subsequent examination by the postprocessor Femview. The computer modelling results of the creep damage distribution and lifetimes for the single and double notched bars are presented below.

Computations were carried out with either C = 0 or C = 300 with a range of values of D and v to simulate materials of different uniaxial ductility and stress state dependencies. 5.1. L i f e t i m e s

(a) C = 0. The material represented by equation set (3)-(5) is damaged only by intergranular cavitation and simulates a pure metal or simple solid solution alloy. The parameter D was set initially to equal 2 (equivalent to a uniaxial strain at fracture of approximately 16%) and calculations performed with the stress state index v varied in unit increments between 0 and 4. The resultant computational lifetimes were normalised by the lifetime of a parallel sided bar having the same net section stress of 425 MPa and plotted in Fig. 2 as open symbols for the single notch and solid symbols for the double notch. It is evident that fracture of the double notched testpiece always occurs in a slightly shorter time--roughly 5%----even though the lifetimes of the notched testpiece are systematically reduced as v increases. The calculations were repeated using a value of D = 8 (equivalent to a uniaxial ductility of approximately 4%) and also D = 32 (equivalent to a uniaxial ductility of approximately 1%). Again, open symbols refer to the single notch and solid symbols to the double notch. It is seen in Fig. 2 that lifetimes of the double notch testpiece are, with one exception, approximately 5% less than those with a single notch: when D = 32 and v = 4, the lifetimes coincide. (b) C = 300. The material now simulates a nickelbased superalloy and its behaviour is described by equation set (3)-(5). The calculations reported in section (a) above were repeated and the results 14

C = 300 n

12 _ ' X - \xx~ 10 : ,

8

°x

\X.\

--

•

X'\\ \\

\

6 -

D = 2

OO D=8 0 * D=32

\\

N\ x

\

",, 2 -

5. RESULTS OF COMPUTATIONS Finite element computations were carried out on both notch geometries by imposing a constant stress o0 of approximately 118 MPa at the outer boundary of each testpiece to produce an average axial stress at the notch throat of a = 425 MPa.

~x

-

0

"o. " I'" " 1

I' ~ 2

V

3

4

Fig. 3. Variation of ratio of predicted lifetimesfor notched, to, and parallel bar, tp, testpieces for single ( ) and for double ( - - - ) notched bars with stress-state index v for a range of values of D with C = 300.

Ca)

(b)

i8,ges ,0",8

!! .791 :

.677 .62

.563 II , s o s

,4.48

II .3gl .334 .277 ,22 I , L53

(c)

(d)

Fig. 4. Comparison of distributions of the damage variables wt, w2, for double notched bar with those for single notched bar immediately before failure, for the material constants C = 300, D = 2 and v = 2. Fields plots (a) and (b) are for w2; and field plots (c) and (d) are for wI.

1219

(a)

(b)

.277 .254 ,231

!!, , 2e8 J.o5 . I.(,2 I . 136

z..*ss | ,g23P'-t ,692 Ir-I

•

~ ,4g2E-I ,~'3 I E - t

Y

Y

L,

,513 ,~7 .427 :~ . 3 8 5 ,342 .299 .257 .214 .172 .129

.56 .513 :~ ,,11.67 ,4~'

1

.374 .327 ,28 .234 ,187 ,1.41 ,941(-[

.863E-1 .436E-1

.4,76E- l

(c)

(a)

Fig. 5. Comparison of distributions of the damage variables wI , w2, for double notched bar with those for single notched bar immediately before failure, for the material constants C = 300, D = 32 and v = 4. Fields plots (a) and (b) are for w2; and field plots (c) and (d) are for wl.

1220

OTHMAN et al.: CREEP RUPTURE IN NOTCHED TENSILE BARS presented in Fig. 3. The general trend is similar to that found in Fig. 2 with the double notch testpiece again exhibiting lifetimes that are approximately 5% less than the single notch under identical conditions. gain the exception being the test where D = 32 and v = 4, when the lifetimes are coincident. 5.2. Damage fields

Computed zones of damage close to failure are shown in Fig. 4 for both types of notch with C = 300, D = 2 and v = 2. For this situation it may be seen from Fig. 3 that the notched bar lifetime is approximately 8 times the plain bar lifetime. Figure 4(a) and (b) show how the second damage state variable, w2, spreads out from the notched regions for the double and single notched bars respectively. The zone of highest material damage does not initiate at the minimum section of the notched bar as may be expected, but at a point on the notch radius close to the bar shank. The zone of damage grows and spreads out to the centre of the notch minimum section. Comparison of the distributions of the failed regions shown in Fig. 4(a. b) illustrates the similarity of the zone of cavitation fracture damage in both types of notched bar. However there is not perfect symmetry, which might be expected and the relevance of this is discussed in a later section. Figures 4(c, d) show the distribution of the first variable, Wl, close to failure for both bars. The material damage is confined to the minimum section of the notch for both bars. The distribution of the first damage variable, w~, for the double notched bar shown in Fig. 4(c) is similar to that for the single one shown in Fig. 4(d). To a first approximation the degree of symmetry achieved is slightly better than for the field plots of w2. It may be seen from Fig. 3 that for C = 3 0 0 , D = 32 and v = 4 the notched bar has a lifetime equal to approximately one half of that for the plain bar: in engineering parlance, the material has notch weakened. For this extreme condition it is worth investigating the character of the damage growth in the notches; and, the degree of similarity of the two damage fields, wt and w2, in the single and double notched testpieces. Figure 5(a) and (b) show the field variations of the second damage state variable, w2, for this material with the double and single notched bars respectively. For both testpieces the damage fields are more distinct and closely confined to the notch throat, as compared with those given in Fig. 4(a,b) for C = 3 0 0 , D = 2 and v = 2 . From comparison of Fig. 4(a, b) and of Fig. 5(a, b) it is clear that the increased stress-state sensitivity v = 4, (,r~/Z+)~, used for the conditions of Fig. 5 results in steeper gradients of damage and hence to more well defined, and distinct damage trajectories. In addition, the effect of material ductility will play a role in achieving the more diffuse damage fields of Fig. 4: for C = 300, D = 2 and v = 2 the uniaxial ductility is approximately 1 6 ° , and for C = 300, D = 32, v = 4

1221

the uniaxial ductility is approximately 1%. The directions in which the damage zones have moved across the minimum sections of the notches are different for v = 2 and v = 4. In the case of v = 4 it can be seen from Fig. 5(a, b) that the damage field initiates at the edge of the notch on the plane of symmetry, it then propagates for a short distance either on, or almost parallel to, the plane of symmetry and subsequently deviates at an angle of approximately 30 ° to it. To a first approximation, the behaviour of the single and double notches is the same. The detailed differences between Fig. 4(a, b) and Fig. 5(a) and (b) can only be attributed to the detailed histories of stress redistribution which are characterised by the different basic material properties. Figure 5(c, d) show the spatial variation of the damage variable w~ which, in comparison with those shown in Fig. 4(c, d), is more highly confined to the notch throat region. Damage levels are also consistently lower, being typically w~ = 0.55-0.6, for Fig. 5(c, d), compared with w~ = 0.96 for Fig. 4(a, d). This is a reflection of the dependence of w~ on the effective strain, E+, which is much lower in Fig. 5. Comparison of the detailed distributions of w~ for the double and single notched testpieces in Fig. 5(c, d) respectively indicates the same general shape of the damage regions, but slightly different levels of intensity. The observation that the damage fields of Fig. 5 are well-confined to the notch throats of both double and single notched testpieces, and are clearly noninteracting, is consistent with the two testpieces having the same lifetimes as given in Fig. 3. For each data set used in Figs 4 and 5 the mechanisms of evolution, and the shapes of the damage zones, for both wL and w2, are very similar for both double and single notched testpieces. This confirms that the notches in the double notched testpiece are sufficiently well-separated to cause minimal interaction for the wide range of materials which have been modelled. 6. DISCUSSION The results presented in Figs 2 and 3 demonstrate that the perturbation of the stress fields by the second notch--although clearly present--affects lifetimes by less than 5%, even with the wide range of material input parameters studied. A difference of 5% is small compared to errors introduced by material scatter and testing uncertainties and it is concluded that the present double notch geometry, used by Dyson and Loveday [7] to study cavitation and crack damage fields, is a vaild test method. A Code and Practice for notched-bar creep testing has recently been published [8] in which St. Venants principle was used to select the minimum spacing of two notches covering a range of geometries. The present work validates the procedure used in the Code of Practice for the range of materials studies here.

1222

OTHMAN et al.: CREEP RUPTURE IN NOTCHED TENSILE BARS

An interesting side issue of this work is the effect that the parameter C (the intrinsic dislocation strain softening coefficient) has on the magnitude of the reduction in lifetimes found in notched testpieces as the stress state parameter v increases from 0 to 4. Examination of Figs 2 and 3 reveals that lifetimes are reduced less when C = 300 compared with C = 0: marginally so when the uniaxial ductility is low (D = 32), but when D = 2 and the uniaxial ductility is relatively high (for nickel-base superalloys) the difference becomes considerable, being only 1/3 of the reduction found when C = 0. This effect of intrinsic strain softening on counteracting the reductions in lifetime caused by multiaxial effects an ductility (represented here by the parameter v) has previously been discussed by Dyson and Gibbons [14] for staticallydeterminate components. These ideas were used to suggest that the reason why certain low uniaxial ductility materials exhibit a von Mises rupture criterion is because they have an unusually high value of C, which makes lifetimes relatively independent of multiaxial ductility. Figure 3 demonstrates that a similar effect occurs in the statically-indeterminate notch testpiece. This is consistent with the recent finding of Othman et al. [9] that the notch bar testpiece exhibits a skeletal point stress even in the presence of tertiary damage, which is tantamount to saying that the notch bar can be treated as a statically-determinate testpiece. 7. CONCLUSIONS 1. Finite element computations have shown that fracture lifetimes of the double notch geometry given in Fig. 1 were within 5% of those of the equivalent single notch geometry, even though the material input parameters were varied over a wide spectrum. This validates the double notch geometries in a recent Code of Practice for a wide range of materials. 2. Studies of the damage fields in the testpieces have shown very similar distributions of w~ and w2 for double and single notched bars for a wide range of material behaviour. For materials with C = 300, where the uniaxial ductility is approximately 16% (D = 2 and v = 2), spatial damage gradients are less severe than when the uniaxial ductility is approxi-

mately 1% (D -- 32 and v = 4). With C = 300, D -- 2 and v = 2, the regions over which damage evolves is larger than with C = 300, D = 32 and v = 4. However, in both cases damage is confined to the notch throat region. 3. The observation that a high value of the intrinsic strain softening parameter, C, attenuates the reductions in lifetime caused by the effect of the stress state parameter, v, is suggested as being consistent with the recent finding that these notches exhibit a skeletal point stress, even in the presence of tertiary creep. Acknowledgements--The research reported in this paper was carded out as part of the "Material Measurement Programme", a programme of underpinning research financed by the United Kingdom Department of Trade and Industry. The authors gratefully acknowledge the provision of computing facilities and support by SERC and by IBM.

REFERENCES 1. D. R. Hayhurst and J. T. Henderson, Int. J. Mech. Sci. 19, 243 (1977). 2. D. R. Hayhurst, F. A. Leckie and C. J. Morrison, Proc. R. Soc. Lond. A 360, 243 (1978). 3. D. R. Hayhurst, P. R. Dimmer and C. J. Morrison, Phil. Trans. R. Soc. Lond. A 311, 103 (1984). 4. D. R. Hayhurst, F. A. Leckie and J. T. Henderson, Int. J. Mech. Sci. 19, 147 (1977). 5. B. F. Dyson, Proc. Int. Conf. on High-Temp. Structural Design (edited by L. H. Larsson), pp. 335-354. M.E.P. Publications, London (1992). 6. D. R. Hayhurst, Proc. Int. Conf. on High-Temp. Structural Design (edited by L. H. Larsson), pp. 317-334. M.E.P. Publications, London (1992). 7. B. F. Dyson and M. S. Loveday, I U T A M Syrup. on Creep in Structures (edited by A. R. S. Ponter and D. R. Hayhurst), pp. 406-421. Springer, Berlin (1980). 8. G. A. Webster, B. J. Cane, B. F. Dyson and M. S. Loveday, High-Temperature Testing Committee Report. National Physical Laboratory, Teddington, ISBN-0946754-13-6 (1991). 9. A. M. Othman, D. R. Hayhurst and B. F. Dyson, Proe. R. Soc. A, in press. 10. B. F. Dyson and M. McLean, ISIJ Int. 30, 802 (1990). 11. A. Nadai, Stephen Timoshenko Anniversary Volume. Macmillan, New York (1938). 12. B. F. Dyson, Rev. Phys. Appl. 23, 605 (1988). 13. D. R. Hayhurst, P. R. Brown and C. J. Morrison, Phil. Trans. R. Soc. Lond. A 311, 131 (1984). 14. B. F. Dyson and T. B. Gibbons, Acta metall. 35, 2355 (1987).