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CRACK PROPAGATION IN METALS DURING THERMAL SHOCK
Volume 2, ICM 3, Cambridge,
England,
August 1979
CRACK PROPAGATION IN METALS DURING THERMAL SHOCK R. P. Skelton Materials Division, Central Electricity Research Laboratories, Kelvin Avenue, Leatherhead, Surrey KT22 7SE, U.K.
INTRODUCTION Multiple surface cracking can arise from repeated thermal shocks (Northcott and Baron, 1956) but only recently have growth rates below the surface been measured. These may occur uniformly (Howes, 1973) but usually, after an initial acceleration, there is a deceleration with increasing penetration (Mowbray, Woodford and Brandt, 1973; Newton, 1976). Thermal shock may arise in the fast reactor where sudden changes in sodium temperature can rapidly chill AISI 316 component surfaces. This can be simulated by water quenching, as in conventional tests (Newton, 1976). Thus it is necessary to know the cyclic propagation rate and whether a crack can reach the opposite face. Thermal gradients are set up across a specimen or component following surface temperature transients. If no external stresses are acting the corresponding stress profile changes sign such that r - o
a dx = —^
r -r. o I
2 /
° r dr
=
(1)
°
r. I
for a slab or tube of thickness £ or (rQ-r-[) respectively. For a linear gradient Booth and Blomfield (1974) have shown that the stress intensity is a maximum at the crossover point at the centre. For a parabolic gradient this maximum is dis placed towards the shocked face (Chell and Ewing, 1977; Hellen, Price and Harrison, 1975). Crack arrest will occur at a threshold value of stress intensity. Should external stresses a^pp act in addition to transient stresses (e.g. by prevention of free contraction during a quench) then equation (1) is put equal to a^pp and the crack will be driven further forward. Maximum growth rate is assumed to occur at the peak thermal transient. This work derives expressions for cyclic crack growth during thermal shock which take into account (i) end constraints, (ii) component shape and (iii) the effect of yielding.
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DETERMINING THE ELASTIC STRESS PROFILE Temperature Transients in Tube The volume mean temperature T a v decides where the stress profile changes sign. Biot (1957) has shown that the exact solution for temperature (and hence elastic stress) profile (Carslaw and Jaeger, 1959; Elizarov, 1971) may be simplified. In the following we will assume r
(l-x/jD2 + n] /(1+n) (2) o ^ where oQ is the quenched surface stress at any time. The value of n depends on geometry and end loading. In the tube of Fig. 1, by putting x = r-r-j;, I = ^o"ri i-n equation (2) and substituting for a in equation (1) we find upon integration and after some reduction o
= o
n
= (3u 4 -8u 3 +6u 2 -l)/6(l-u 2 )(l-u) 2
where r^ = u r Q , u < 1. Table 1 summarizes n values for different wall thickness ratios and also gives the corresponding distance of the crossover point from the shocked face. TABLE 1 Position of Volume Mean Temperature for Zero Applied Load and No Yielding. Tube Quenched from Inside u
n
0 0.1 0.2 0.3 0.348 0.4
•0.167 ■0.197 ■0.222 ■0.244 ■0.253 ■0.262
x/1 0.592 0.556 0.529 0.506 0.497 0.488
u 0.5 0.6 0.7 0.8 0.9
0.9.
n •0.278 ■0.292 0.304 ■0.315 0.325 0.329
x/1 0.473 0.460 0.449 0.439 0.430 0.426
By suitable insertion of thermocouples, temperature profiles were determined in a specimen of solution treated AISI 316 steel (Fig. 1) which was quenched from 625°C. After ^ Is the transient gradient became linear but before that time was parabolic. The rise and decay of T-T av at several points through the thickness is indicated in Fig. 2; the peak occurred between 0.4 and 0.5s near the bore. Assuming perfect elasticity, an equibiaxial thermal stress is generated (Timoshenko and Goodier, 1970), given by o = Ea(T-Tav)/(1-v) where E at 625°C is 1.5xl0^MPa, a is the corresponding coefficient of expansion (1.9xlO~5) atl(i v =0.3. The product Ea is approximately independent of temperature. The peak pseudoelastic stress profile calculated from this relation and the observed temperature profile were found to be in good agreement with equation (2), taking, for the present specimen, n = -0.25, see Table 1. Effect of Restraint during Quenching The expressions for longitudinal thermal stress in a cylinder (Timoshenko and Goodier, 1970) may be simplified to czz
= aET(l-T
/T)/(l-v)
(3a)
I High a' zz
=
aET(l-v T
av
15
Temperature
/T)/(l-v)
(3b)
for zero and full restraint respectively i.e. ozz > a z z . As T a v departs from an initial temperature T Q we suppose that the stress profile, which may be considered initially coincident with the temperature profile in an unrestrained cylinder, is progressively displaced from it. The crossover conditions are given respectively by T = T a v and T = v T a v , equations (3a,b). This is shown schematically in Fig. 3 where the momentary zero temperature is taken at the unshocked face. The peak stress level is raised everywhere by OB or n(l-v)a0/(l+n) for full axial restraint (i.e. when n = -0.25, Fig. 1 and Table 1, a 23% increase is produced). It may be shown that the raised profile is given by a
=
OQ
[(l-x/£)Z + njj
/(1+n)
(4)
where n f = vn (i.e. in our example, the stress now changes sign at x/l = 0.73 cf. Table 1 ) .
-i
TENSION DURING SHOCK COMPRESSION DURING SHOCK
T/C POSITIONS
/ * \ « 1 \
0.3 mm FROM INNER SURFACE
7 [) Fig. 1. Thermal shock specimen
Fig. 2. Thermal transients
Elastic follow up may be allowed for by replacing v with A where l>A>v. In this case n f =An and the funrestrained1 profile (Fig. 3) is displaced to the lesser extent n(l-A)o0/(l+n) during the transient. It may be shown that E(l-A)/(l-v) is an apparent modulus E* depending on the system stiffness. In our apparatus this was E/14.5 which gives A=0.95 so that only a 1.6% increase in stress was produced at the peak transient (0.5s). Quenching time was thus increased in order to raise stresses generally across the section. Putting equation (1) equal to QAPP> and substituting for o from equation (4):
APP (1+n)
r 2 3 4 j l-6u +8u -3u
(5)
L6(l-u 2 )(l-u) 2 J
Vfoen n f >l, equations (4,5), the crossover point disappears. important consequences in describing crack penetration.
This clearly has
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EFFECT OF YIELDING ON STRESS PROFILE An elastic (parabolic) stress gradient is almost linear approaching the quenched face. Yield thus occurs over a depth d as shown in Fig. 4. The stress oQ has dropped to o\ and in the direction O'X the material describes an equibiaxial stress strain curve: a
=
B eTy
(6)
B and y(^0.3) are constants. Figure 5 presents biaxial cyclic stress-strain curves (in terms of stress and strain range) for 316 steel calculated (Skelton, 1979a) from uniaxial elevated temperature data (Dawson and others, 1967). The position of 0', Fig. 4, depends on end restraint and quench severity. The readjusted curve after yielding is thus imagined to be in two parts, passing through 0 f and is calculated for a single quench as follows: (i)
(ii)
The penetration of the yielded region is given by equation (4) with o=o : q = d/£ = 1- [(l+n)o /a -n 1 j (7) The profile (measured from 0 1 ) to the back face is elastic and given by o = o
(iii)
(l-x'/£') 2 + n " J
(8)
/U+n") 1
The relative distribution of equations (6,8) about 0 is based on the equilibrium condition, equation (1). It may be shown (Skelton, 1979a) that for a tube quenched from the inside
(9)
n" = -(A+B)/(l-v +A) where
A
- '(v 2 -u 2 )(p+a 1 /o y )/(l+p)-(l-u 2 )a App /a ]
B =
(l-6v 2 +8v 3 -3v 4 )/6(l-v) 2
v
[q(l-u)+uj
=
UNRESTRAINED RESTRAINED
1 0'
^
Fig. 3. Displacement of profile
x,'
1/
i, r
Fig. 4. The yielded profile
I High
Temperature
17
In equations (7,9) true values of o\loy are obtained from the notional ratios o0/oy using the stress strain curve, Fig. 5. The complete profile for a single quench is then known. Residual stresses in the yield zone after each quench can be assessed from the difference between actual and elastic profiles, Fig. 4. With repeated quenching it is necessary to allow the stresses to shake down. The method is discussed elsewhere (Skelton, 1979a). It is found that the yield zone depth d at peak tension successively decreases while that in residual compression correspondingly increases so the saturation peak tensile stress at the surface is obtained after very few cycles. The shakedowi value of d is obtained immediately by putting Oy = 2o y in equation (7) where 2oy is the sum of tensile and compressive yield stress, see Fig. 5. Similarly, in equation (9) the value of o\lov is obtained from the ratio a 0 /2a y , as shown in Fig. 5. The final stress profiles calculated for several end loads in repeated quenching from 625°C are shown in Fig. 6 for the experimental system of Fig. 1. At a certain value of end constraint the crossover point disappears.
0.1
0.2
0.3
0.4 i
05 i
06 i
0.7
0.8 1
0.9 1
1.1 1
^ ^ N S \
a
|-
^
i
1
L 0.002
0.004
0.006 0.008 0.010 0012 TOTAL STRAIN RANGE
0014
1
i \ , 2
, \ .
^
APP 0.20
^ v , 3
^^^^^J.084
DEPTH,mm
**"""-—-£—
0 016
Fig. 5. Cyclic stress strain curves
Fig. 6. Profiles at shakedown
APPLICATION TO CRACK GROWTH Attenuation Factors In push pull (Skelton 1978a,b) or reverse bend (Thomas, 1977) fatigue, cyclic crack growth rates accelerate as the crack extends. If these results were used directly in thermal shock, estimates of lifetime would be pessimistic. We thus expect crack growth rates to be attenuated values of those under uniform loading conditions. Buchalet and Bamford (1976) show that the stress intensity under arbitrary loading is given by: K=a(Tra)
2 3 4-* A F 1 + 0 . 6 3 7 A 1 F o | - + 0 . 5 A_F_ - 0 + 0 . 4 2 4 A . F . - . + 0 . 3 7 5 A . F _ -]- \ o 1 1 21 2 3 2 3 4 3 o j
(10)
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where A 0 , Ai etc. are coefficients in the polynomial giving the peak stress transient in the uncracked body (e.g. equations (2,4)) and F - ^ ^ etc. are com pliances. It may be shown (Skelton, 1979a) that for our specimen, F^,F2 etc. =1 both for circumferential and longitudinal cracks. Thus taking n = -0.25 in equation (2) for zero end load (Table 1) we have an attenuation factor F at each crack depth (i.e. ratio of K value to that under uniform loading). For our fully restrained tube, O A P P / O 0 ^ 23% giving the displaced profile of equation (4) with n = -0.25, n f = -0.076 (equation (5)) and so attenuation factors may be similarly calculated for this and other end loads. To account for yielding, the two-part curves of Fig. 6, for example, can be fitted by quartic regression analysis. Fuller details are given elsewhere (Skelton, 1979a). The normalized variation in stress intensity across the specimen section, derived from equation (10), is shown in Fig. 7. Whilst it appears that conditions are more onerous for the yielded distribution, in fact the lower surface stress o-y results in a reduced maximum in stress intensity. The elastic solution agrees well with an independent finite element calculation for this specimen by D.C. Martin. The elastic results apply for any quenching temperature, but not the plastic, where the yield depth will vary.
Prediction from Stress Intensity Relations for cyclic crack growth in 316 steel in air between 500 and 625°C are given (Marshall, 1978; Skelton 1978b) by AK 3 mm/cycle
Upper bound, high AK: da/dN = 1.4xlO~ Best fit, low AK:
da/dN = 3.2xlo"
6
AK
2
(11)
mm/cycle.
(12)
For zero end load, the variation of AK through the section is given by Fig. 7 and corresponding growth rates (from equation (11)) are shown in Fig. 8. A maximum rate at 1mm crack depth is predicted but elastic values are an order of magnitude greater than propagation rates in a yielded distribution. This is not surprising since, in readjusting the stresses, no account has been taken of accompanying changes in strain. a/l
0.1 "I
0.2 0.3 04 1 1 1
0.5 06 0.7 0.8 0.9 1.0 1 1 I 1 I 1
* DEPTH,**
Fig.
3
«
7. Normalised stress intensities
0.1
1.0 CRACK DEPTH, mm
Fig. 8. Predicted growth rates, no constraint
J High
Temperature
19
The difference can be lessened by calculating an equivalent stress intensity from a strain intensity AK^ (Haigh and Skelton, 1978). Equivalent equibiaxial stresses in the yield zone are obtained from the expression (ep+0.5ee)E/(1-v) and are shown in Fig. 6. In calculating attenuation factors, it is not necessary to readjust the remainder of each profile. Equivalent stress intensities across the section are given in Fig. 7. The resulting crack growth rates, Fig. 8, from equations (11, 12) lie between the two previous extremes.
Prediction from High Strain Fatigue Data When crack growth rates from conventional high strain fatigue tests are available, corresponding rates in thermal shock may be generated alternatively, having identified the surface conditions. Isothermal crack growth data on square-sided specimens (Wareing, 1975) in air at 625°C may be taken as an upper limit. Cyclic stress strain data on solution treated 316 steel, as in Fig. 5, are also reasonably independent of temperature (see Berling and Slot, 1969; Kanazawa and Yoshida, 1975). By successively applying the Neuber (1961) relation to calculate the increase in strain concentration at the crack tip during each increment of advance, crack growth rates in high strain fatigue are given (Skelton, 1979b) by: da/dN
=
(e T K t 2 ^ + 6 ) / k ' ) 1 / a '
(13)
a* , kf describe the corresponding endurance curve. y and $ are obtained from uniaxial cyclic stress strain data e.g. as in Fig. 5. K t is the elastic stress concentration factor. If it is assumed that this is attenuated, following the actual stress profiles of Fig. 6 (because plastic strains are allowed for in equation (13)), then conventionalfcrack growth rates at each depth, Fig. 9, are modified by the factor F*YM a + p)a, Taking the equibiaxial elastic stress a 0 = 730MPa for zero end load, the equivalent uniaxial plastic strain range obtained from Fig. 5 is 0.002. High strain fatigue growth rates at this strain range are given in Fig. 9. (Similarly, with an end load given by cj^pp/aQ = 0.125, the equivalent plastic strain range is 0.0023). Taking a 1 = 0.38 at 500°C (Marshall,
Fig. 9. Circumferential crack growth rates
Fig. 10. Alternative form of Fig. 9
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1978) and 3 = 0.28, y = 0.45 (Skelton, 1979b) predicted growth rates in thermal shock are shown in Fig. 9 for several end loads. EXPERIMENTAL COMPARISON The hollow specimen, Fig. 1 was tested in an Instron machine alongside a temper ature ramp and R.F. heating unit. Displacement-controlled thermo mechanical cycles were produced as before (Skelton, 1978c) except that water quenching replaced the steady cool. Approximately 30-40 quenching cycles, each of 2s duration, were possible in lh. Cracking initiated from the bore and breakthrough at the outer surface was detected by a loss of vacuum in the surrounding chamber. Average growth rates were determined by the d.c. potential drop method at the out side surface ('back face1 calibration). Circumferential crack progress from the bore is indicated in Fig. 10. With zero end load, crack arrest had occurred by 18,000 cycles at 0.46 of the wall thickness. In contrast a peak end stress of 92MPa (o^pp/^o = 0.125, a^pp/ay=0.41) caused one crack to break through by about 14,000 cycles (Fig. ll):all other cracks had pene trated correspondingly deeper. An intermediate end stress of 60MPa (c^pp/a0 = 0.084, CAPp/ay = 0.27) caused breakthrough in 32,000 cycles. Metallography confirmed that longitudinal cracks (Fig. 11) arrested at ^0.46 of the depth, in dependently of end stress. The potential drop method could not of course detect them. All cracking was transgranular. The results of Fig. 10 are compared with the predictions from high strain fatigue data in Fig. 9. These curves overestimate crack growth rates: by summing the cycles for each growth increment, breakthrough is predicted to occur at roughly half the experimental values. Similar estimates are furnished by the strain intensity method used with equation (11).
CIRCUMFERENTIAL CRACK, FULLY RESTRAINED BODY
i
Fig. 11. Circumferential, longitudinal cracks, X8.5
i i i i i i i i i
Fig. 12. As for Fig. 7, large body
CONCLUDING REMARKS 1.
For a multiple array of cracks the stress intensity at the tip of each is relieved (Baratta, 1978; Bowie, 1973). It may be shown that the 'relief
I High
Temperature
21
factor* in AK for our longitudinal and circumferential cracks is ^ 0.5. However, from equation (11), for example, this would predict an eight fold reduction every where in da/dN. Since one crack can break away with imposed end loads, this mod ification should be considered dangerous in assessments. 2. Figure 12 is similar to Fig. 7 (elastic case), but for a much larger body. It is seen (i) that the increased wall thickness leads to an increase in stress in tensity for the same surface conditions, implying that cracks penetrate(relatively) deeper and (ii) that longitudinal cracks are more dangerous in the (relatively) thin-walled geometry of Fig. 12. This arises from Buchalet and Bamford's (1976) analysis where the coefficients F^etc. in equation (10) become > 1. 3. If a threshold AK 0 is known, then Figs. 7,11 provide a rapid estimation of the maximum depth of penetration. Thus taking AK0=8MPa ml (Skelton, 1978b) and an equivalent surface stress at zero end load of 417MPa, Fig. 6, then arrest is pre dicted, from Fig. 7 at a=2.7mm, safely overestimating the experimental value of 2.1mm. 4. From this work we thus have a thermal fatigue analysis for simple components in terms of crack growth. If the predicted curves e.g. Figs. 8,9 intersect the back face of the component within the experimental range of growth rates, breakthrough is expected. Alternatively, by integrating the curves, predicted crack depth is given directly against number of cycles. 5. The present tests have exaggerated quench severity in order to characterize the effects of end load. Current tests are examining the effect of quenching (i) on the ferritic 9CrlMo steel (which has a greater thermal conductivity and lower ex pansion coefficient than 316 steel i.e. a reduced thermal stress) and on (ii) much larger sections of 0.5Cr-Mo-V steel as used in turbine chests and casings. ACKNOWLEDGEMENTS I would like to thank Dr. B. Tomkins for providing some calibration data and Drs. D.C. Martin, G.G. Chell, D.J.F. Ewing and D. Raynor for helpful discussions. The work was carried out at the Central Electricity Research Laboratories and the paper is published by permission of the Central Electricity Generating Board. REFERENCES Baratta, F.I. (1978). Eng. Fract. Mech., 10, 691-697. Berling, J.T. and Slot, T. (1969). Fatigue at High Temperature, Spec.Tech.Publ. 459, Amer. Soc. Testing Mater., Philadelphia, pp. 3-30. Biot, M.A. (1957). J. Aeronaut Sci., 2A_9 857-873. Booth, S.J. and Blomfield, J.A. (1974). Unpublished work. Bowie, O.L. (1973). In G.C. Sih(Ed.), Mechanics of Fracture l,Noordhoff, Leyden, pp. 1-55. Buchalet, C.B. and Bamford, W.H. (1976). Mechanics of Crack Growth, Spec.Tech. Publ.590,Amer.Soc.Testing Mater., Philadelphia, pp. 385-402. Carslaw, H.S. and Jaeger, J.C. (1959). Conduction of Heat in Solids, 2nd ed., Clarendon Press, Oxford, p. 92. Chell, G.G. and Ewing, D.J.F. (1977). Intemat. J. Fract., 13, 467-479. Dawson, R.A.T., Elder, W.J., Hill, G.J. and Price, A.T. (1967). Thermal and High Strain Fatigue, The Metals and Metallurgy Trust, London, pp. 239-269. Elizarov, D.P. (1971). Teploenergetica, 18, 78-82 (117-123 English version). Haigh, J.R. and Skelton, R.P. (1978). Mater. Sci. Eng., .36, 133-137. Hellen, T.K., Price, R.H. and Harrison, R.P. (1975). 3rd Internat.Conf.on Struc tural Mechanisms in Reactor Technology, Imperial College, London,paper L7/4. Howes, M.A.H. (1973). Fatigue at Elevated Temperatures, Spec.Tech.Publ.520, Amer. Soc. Testing Mater., Philadelphia, pp.242-254. Kanazawa, K. and Yoshida, S. (1975). Conf. on Creep and Fatigue in Elevated Temperature Applications, Inst. Mech. Engrs., London, pp. 226.1-226.10.
ICM 3,
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Marshall, P. (1978). Unpublished work. Mowbray, D.F., Woodford, D.A. and Brandt, D.E. (1973). Fatigue at Elevated Temp eratures, Spec. Tech.Publ.520, Amer.Soc. Testing Mater., Philadelphia, pp. 416-426. Neuber, H. (1961). Trans. A.S.M.E., Series E, ^ 8 , 544-550. Northcott, L. and Baron, H.G. (1956). J. Iron Steel Inst., 184, 385-408. Newton, J.D. (1976). J. Austral. Inst. Met., n_, 94-102. Skelton, R.P. (1978a). Mater. Sci. Eng., 312, 211-219. Skelton, R.P.(1978b). Time Dependent Degradation of Pressure Boundary Materials, I.A.E.A. meeting, Innsbruck, 20-21 November. Skelton, R.P. (1978c). Mater. Sci. Eng. 35^ 287-298. Skelton, R.P. (1979a). C.E.G.B. Report RD/L/R. Skelton, R.P. (1979b). Fatigue Eng. Mater. Struct., to be published. Thomas, G. (1977). Unpublished work. Timoshenko, S.P. and Goodier, J.N. (1970). Theory of Elasticity, 3rd ed., McGraw Hill, New York, pp. 433-484. Wareing, J. (1975). Met.Trans., 6A, 1367-1377. APPENDIX Biot (1957) has defined the transit time t^ to be when the temperature of the outer face just begins to fall. It is given by t^=0.088 l^/D where D is the diffusivity, in turn given by k/pc where k is the thermal conductivity, p is the den sity (7.8xl03kg m"3) and c the specific heat (5.1xl02 J kg" 1 °C""1) . For thick section turbine components, transit times occur many minutes after the onset of cooling, see Table 2 for a comparison of measured and predicted values. In ex periments we have found that transit time slightly anticipates the arrival of peak stress. TABLE 2 Comparison of Predicted and Experimental Transit Times Average Temp. C
Material
550 450 500 530 300
AISI 316 AISI 316 0.5CrMoV lCrMoV 12Cr
24 22 35 35 29
4.5 10 35 356 550
21 min 61 min
100
Low Alloy Steel
42
603
51 min
*
Thermal Conductivity k,
W m-1 °Cl
Transit Time tx Thickness — mm Predicted Measured 0.3s 1.6s
12s
0.4s 2.0s
12s
19 min* >40min* <80min >33min <60min
Ref.
Present Work Tomkins, 1977 Skelton, 1978 Copp, 1972 Briner and Beglinger, 1975 Rathbone, 1929
Computer calculation from known heat transfer coefficients. REFERENCES
Briner, M. and Beglinger, V. (1975). Conf. on Creep and Fatigue in Elevated Temperature Applications, Inst. Mech. Engrs., London, pp. 220.1-220.8. Copp, L.H. (1972). Unpublished work. Rathbone, T.C. (1929). Quoted in Nadai, A. (1963). Theory of Flow and Fracture of Solids, Vol. II. McGraw Hill, New York, p. 399. Skelton, R.P. (1978). Unpublished work. Tomkins, B. (1977). Unpublished work.
England,
August 1979
CRACK PROPAGATION IN METALS DURING THERMAL SHOCK R. P. Skelton Materials Division, Central Electricity Research Laboratories, Kelvin Avenue, Leatherhead, Surrey KT22 7SE, U.K.
INTRODUCTION Multiple surface cracking can arise from repeated thermal shocks (Northcott and Baron, 1956) but only recently have growth rates below the surface been measured. These may occur uniformly (Howes, 1973) but usually, after an initial acceleration, there is a deceleration with increasing penetration (Mowbray, Woodford and Brandt, 1973; Newton, 1976). Thermal shock may arise in the fast reactor where sudden changes in sodium temperature can rapidly chill AISI 316 component surfaces. This can be simulated by water quenching, as in conventional tests (Newton, 1976). Thus it is necessary to know the cyclic propagation rate and whether a crack can reach the opposite face. Thermal gradients are set up across a specimen or component following surface temperature transients. If no external stresses are acting the corresponding stress profile changes sign such that r - o
a dx = —^
r -r. o I
2 /
° r dr
=
(1)
°
r. I
for a slab or tube of thickness £ or (rQ-r-[) respectively. For a linear gradient Booth and Blomfield (1974) have shown that the stress intensity is a maximum at the crossover point at the centre. For a parabolic gradient this maximum is dis placed towards the shocked face (Chell and Ewing, 1977; Hellen, Price and Harrison, 1975). Crack arrest will occur at a threshold value of stress intensity. Should external stresses a^pp act in addition to transient stresses (e.g. by prevention of free contraction during a quench) then equation (1) is put equal to a^pp and the crack will be driven further forward. Maximum growth rate is assumed to occur at the peak thermal transient. This work derives expressions for cyclic crack growth during thermal shock which take into account (i) end constraints, (ii) component shape and (iii) the effect of yielding.
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DETERMINING THE ELASTIC STRESS PROFILE Temperature Transients in Tube The volume mean temperature T a v decides where the stress profile changes sign. Biot (1957) has shown that the exact solution for temperature (and hence elastic stress) profile (Carslaw and Jaeger, 1959; Elizarov, 1971) may be simplified. In the following we will assume r
(l-x/jD2 + n] /(1+n) (2) o ^ where oQ is the quenched surface stress at any time. The value of n depends on geometry and end loading. In the tube of Fig. 1, by putting x = r-r-j;, I = ^o"ri i-n equation (2) and substituting for a in equation (1) we find upon integration and after some reduction o
= o
n
= (3u 4 -8u 3 +6u 2 -l)/6(l-u 2 )(l-u) 2
where r^ = u r Q , u < 1. Table 1 summarizes n values for different wall thickness ratios and also gives the corresponding distance of the crossover point from the shocked face. TABLE 1 Position of Volume Mean Temperature for Zero Applied Load and No Yielding. Tube Quenched from Inside u
n
0 0.1 0.2 0.3 0.348 0.4
•0.167 ■0.197 ■0.222 ■0.244 ■0.253 ■0.262
x/1 0.592 0.556 0.529 0.506 0.497 0.488
u 0.5 0.6 0.7 0.8 0.9
0.9.
n •0.278 ■0.292 0.304 ■0.315 0.325 0.329
x/1 0.473 0.460 0.449 0.439 0.430 0.426
By suitable insertion of thermocouples, temperature profiles were determined in a specimen of solution treated AISI 316 steel (Fig. 1) which was quenched from 625°C. After ^ Is the transient gradient became linear but before that time was parabolic. The rise and decay of T-T av at several points through the thickness is indicated in Fig. 2; the peak occurred between 0.4 and 0.5s near the bore. Assuming perfect elasticity, an equibiaxial thermal stress is generated (Timoshenko and Goodier, 1970), given by o = Ea(T-Tav)/(1-v) where E at 625°C is 1.5xl0^MPa, a is the corresponding coefficient of expansion (1.9xlO~5) atl(i v =0.3. The product Ea is approximately independent of temperature. The peak pseudoelastic stress profile calculated from this relation and the observed temperature profile were found to be in good agreement with equation (2), taking, for the present specimen, n = -0.25, see Table 1. Effect of Restraint during Quenching The expressions for longitudinal thermal stress in a cylinder (Timoshenko and Goodier, 1970) may be simplified to czz
= aET(l-T
/T)/(l-v)
(3a)
I High a' zz
=
aET(l-v T
av
15
Temperature
/T)/(l-v)
(3b)
for zero and full restraint respectively i.e. ozz > a z z . As T a v departs from an initial temperature T Q we suppose that the stress profile, which may be considered initially coincident with the temperature profile in an unrestrained cylinder, is progressively displaced from it. The crossover conditions are given respectively by T = T a v and T = v T a v , equations (3a,b). This is shown schematically in Fig. 3 where the momentary zero temperature is taken at the unshocked face. The peak stress level is raised everywhere by OB or n(l-v)a0/(l+n) for full axial restraint (i.e. when n = -0.25, Fig. 1 and Table 1, a 23% increase is produced). It may be shown that the raised profile is given by a
=
OQ
[(l-x/£)Z + njj
/(1+n)
(4)
where n f = vn (i.e. in our example, the stress now changes sign at x/l = 0.73 cf. Table 1 ) .
-i
TENSION DURING SHOCK COMPRESSION DURING SHOCK
T/C POSITIONS
/ * \ « 1 \
0.3 mm FROM INNER SURFACE
7 [) Fig. 1. Thermal shock specimen
Fig. 2. Thermal transients
Elastic follow up may be allowed for by replacing v with A where l>A>v. In this case n f =An and the funrestrained1 profile (Fig. 3) is displaced to the lesser extent n(l-A)o0/(l+n) during the transient. It may be shown that E(l-A)/(l-v) is an apparent modulus E* depending on the system stiffness. In our apparatus this was E/14.5 which gives A=0.95 so that only a 1.6% increase in stress was produced at the peak transient (0.5s). Quenching time was thus increased in order to raise stresses generally across the section. Putting equation (1) equal to QAPP> and substituting for o from equation (4):
APP (1+n)
r 2 3 4 j l-6u +8u -3u
(5)
L6(l-u 2 )(l-u) 2 J
Vfoen n f >l, equations (4,5), the crossover point disappears. important consequences in describing crack penetration.
This clearly has
16
ICM 3,
Volume
2
EFFECT OF YIELDING ON STRESS PROFILE An elastic (parabolic) stress gradient is almost linear approaching the quenched face. Yield thus occurs over a depth d as shown in Fig. 4. The stress oQ has dropped to o\ and in the direction O'X the material describes an equibiaxial stress strain curve: a
=
B eTy
(6)
B and y(^0.3) are constants. Figure 5 presents biaxial cyclic stress-strain curves (in terms of stress and strain range) for 316 steel calculated (Skelton, 1979a) from uniaxial elevated temperature data (Dawson and others, 1967). The position of 0', Fig. 4, depends on end restraint and quench severity. The readjusted curve after yielding is thus imagined to be in two parts, passing through 0 f and is calculated for a single quench as follows: (i)
(ii)
The penetration of the yielded region is given by equation (4) with o=o : q = d/£ = 1- [(l+n)o /a -n 1 j (7) The profile (measured from 0 1 ) to the back face is elastic and given by o = o
(iii)
(l-x'/£') 2 + n " J
(8)
/U+n") 1
The relative distribution of equations (6,8) about 0 is based on the equilibrium condition, equation (1). It may be shown (Skelton, 1979a) that for a tube quenched from the inside
(9)
n" = -(A+B)/(l-v +A) where
A
- '(v 2 -u 2 )(p+a 1 /o y )/(l+p)-(l-u 2 )a App /a ]
B =
(l-6v 2 +8v 3 -3v 4 )/6(l-v) 2
v
[q(l-u)+uj
=
UNRESTRAINED RESTRAINED
1 0'
^
Fig. 3. Displacement of profile
x,'
1/
i, r
Fig. 4. The yielded profile
I High
Temperature
17
In equations (7,9) true values of o\loy are obtained from the notional ratios o0/oy using the stress strain curve, Fig. 5. The complete profile for a single quench is then known. Residual stresses in the yield zone after each quench can be assessed from the difference between actual and elastic profiles, Fig. 4. With repeated quenching it is necessary to allow the stresses to shake down. The method is discussed elsewhere (Skelton, 1979a). It is found that the yield zone depth d at peak tension successively decreases while that in residual compression correspondingly increases so the saturation peak tensile stress at the surface is obtained after very few cycles. The shakedowi value of d is obtained immediately by putting Oy = 2o y in equation (7) where 2oy is the sum of tensile and compressive yield stress, see Fig. 5. Similarly, in equation (9) the value of o\lov is obtained from the ratio a 0 /2a y , as shown in Fig. 5. The final stress profiles calculated for several end loads in repeated quenching from 625°C are shown in Fig. 6 for the experimental system of Fig. 1. At a certain value of end constraint the crossover point disappears.
0.1
0.2
0.3
0.4 i
05 i
06 i
0.7
0.8 1
0.9 1
1.1 1
^ ^ N S \
a
|-
^
i
1
L 0.002
0.004
0.006 0.008 0.010 0012 TOTAL STRAIN RANGE
0014
1
i \ , 2
, \ .
^
APP 0.20
^ v , 3
^^^^^J.084
DEPTH,mm
**"""-—-£—
0 016
Fig. 5. Cyclic stress strain curves
Fig. 6. Profiles at shakedown
APPLICATION TO CRACK GROWTH Attenuation Factors In push pull (Skelton 1978a,b) or reverse bend (Thomas, 1977) fatigue, cyclic crack growth rates accelerate as the crack extends. If these results were used directly in thermal shock, estimates of lifetime would be pessimistic. We thus expect crack growth rates to be attenuated values of those under uniform loading conditions. Buchalet and Bamford (1976) show that the stress intensity under arbitrary loading is given by: K=a(Tra)
2 3 4-* A F 1 + 0 . 6 3 7 A 1 F o | - + 0 . 5 A_F_ - 0 + 0 . 4 2 4 A . F . - . + 0 . 3 7 5 A . F _ -]- \ o 1 1 21 2 3 2 3 4 3 o j
(10)
ICM 3, Volume
18
2
where A 0 , Ai etc. are coefficients in the polynomial giving the peak stress transient in the uncracked body (e.g. equations (2,4)) and F - ^ ^ etc. are com pliances. It may be shown (Skelton, 1979a) that for our specimen, F^,F2 etc. =1 both for circumferential and longitudinal cracks. Thus taking n = -0.25 in equation (2) for zero end load (Table 1) we have an attenuation factor F at each crack depth (i.e. ratio of K value to that under uniform loading). For our fully restrained tube, O A P P / O 0 ^ 23% giving the displaced profile of equation (4) with n = -0.25, n f = -0.076 (equation (5)) and so attenuation factors may be similarly calculated for this and other end loads. To account for yielding, the two-part curves of Fig. 6, for example, can be fitted by quartic regression analysis. Fuller details are given elsewhere (Skelton, 1979a). The normalized variation in stress intensity across the specimen section, derived from equation (10), is shown in Fig. 7. Whilst it appears that conditions are more onerous for the yielded distribution, in fact the lower surface stress o-y results in a reduced maximum in stress intensity. The elastic solution agrees well with an independent finite element calculation for this specimen by D.C. Martin. The elastic results apply for any quenching temperature, but not the plastic, where the yield depth will vary.
Prediction from Stress Intensity Relations for cyclic crack growth in 316 steel in air between 500 and 625°C are given (Marshall, 1978; Skelton 1978b) by AK 3 mm/cycle
Upper bound, high AK: da/dN = 1.4xlO~ Best fit, low AK:
da/dN = 3.2xlo"
6
AK
2
(11)
mm/cycle.
(12)
For zero end load, the variation of AK through the section is given by Fig. 7 and corresponding growth rates (from equation (11)) are shown in Fig. 8. A maximum rate at 1mm crack depth is predicted but elastic values are an order of magnitude greater than propagation rates in a yielded distribution. This is not surprising since, in readjusting the stresses, no account has been taken of accompanying changes in strain. a/l
0.1 "I
0.2 0.3 04 1 1 1
0.5 06 0.7 0.8 0.9 1.0 1 1 I 1 I 1
* DEPTH,**
Fig.
3
«
7. Normalised stress intensities
0.1
1.0 CRACK DEPTH, mm
Fig. 8. Predicted growth rates, no constraint
J High
Temperature
19
The difference can be lessened by calculating an equivalent stress intensity from a strain intensity AK^ (Haigh and Skelton, 1978). Equivalent equibiaxial stresses in the yield zone are obtained from the expression (ep+0.5ee)E/(1-v) and are shown in Fig. 6. In calculating attenuation factors, it is not necessary to readjust the remainder of each profile. Equivalent stress intensities across the section are given in Fig. 7. The resulting crack growth rates, Fig. 8, from equations (11, 12) lie between the two previous extremes.
Prediction from High Strain Fatigue Data When crack growth rates from conventional high strain fatigue tests are available, corresponding rates in thermal shock may be generated alternatively, having identified the surface conditions. Isothermal crack growth data on square-sided specimens (Wareing, 1975) in air at 625°C may be taken as an upper limit. Cyclic stress strain data on solution treated 316 steel, as in Fig. 5, are also reasonably independent of temperature (see Berling and Slot, 1969; Kanazawa and Yoshida, 1975). By successively applying the Neuber (1961) relation to calculate the increase in strain concentration at the crack tip during each increment of advance, crack growth rates in high strain fatigue are given (Skelton, 1979b) by: da/dN
=
(e T K t 2 ^ + 6 ) / k ' ) 1 / a '
(13)
a* , kf describe the corresponding endurance curve. y and $ are obtained from uniaxial cyclic stress strain data e.g. as in Fig. 5. K t is the elastic stress concentration factor. If it is assumed that this is attenuated, following the actual stress profiles of Fig. 6 (because plastic strains are allowed for in equation (13)), then conventionalfcrack growth rates at each depth, Fig. 9, are modified by the factor F*YM a + p)a, Taking the equibiaxial elastic stress a 0 = 730MPa for zero end load, the equivalent uniaxial plastic strain range obtained from Fig. 5 is 0.002. High strain fatigue growth rates at this strain range are given in Fig. 9. (Similarly, with an end load given by cj^pp/aQ = 0.125, the equivalent plastic strain range is 0.0023). Taking a 1 = 0.38 at 500°C (Marshall,
Fig. 9. Circumferential crack growth rates
Fig. 10. Alternative form of Fig. 9
ICM 3,
20
Volume
2
1978) and 3 = 0.28, y = 0.45 (Skelton, 1979b) predicted growth rates in thermal shock are shown in Fig. 9 for several end loads. EXPERIMENTAL COMPARISON The hollow specimen, Fig. 1 was tested in an Instron machine alongside a temper ature ramp and R.F. heating unit. Displacement-controlled thermo mechanical cycles were produced as before (Skelton, 1978c) except that water quenching replaced the steady cool. Approximately 30-40 quenching cycles, each of 2s duration, were possible in lh. Cracking initiated from the bore and breakthrough at the outer surface was detected by a loss of vacuum in the surrounding chamber. Average growth rates were determined by the d.c. potential drop method at the out side surface ('back face1 calibration). Circumferential crack progress from the bore is indicated in Fig. 10. With zero end load, crack arrest had occurred by 18,000 cycles at 0.46 of the wall thickness. In contrast a peak end stress of 92MPa (o^pp/^o = 0.125, a^pp/ay=0.41) caused one crack to break through by about 14,000 cycles (Fig. ll):all other cracks had pene trated correspondingly deeper. An intermediate end stress of 60MPa (c^pp/a0 = 0.084, CAPp/ay = 0.27) caused breakthrough in 32,000 cycles. Metallography confirmed that longitudinal cracks (Fig. 11) arrested at ^0.46 of the depth, in dependently of end stress. The potential drop method could not of course detect them. All cracking was transgranular. The results of Fig. 10 are compared with the predictions from high strain fatigue data in Fig. 9. These curves overestimate crack growth rates: by summing the cycles for each growth increment, breakthrough is predicted to occur at roughly half the experimental values. Similar estimates are furnished by the strain intensity method used with equation (11).
CIRCUMFERENTIAL CRACK, FULLY RESTRAINED BODY
i
Fig. 11. Circumferential, longitudinal cracks, X8.5
i i i i i i i i i
Fig. 12. As for Fig. 7, large body
CONCLUDING REMARKS 1.
For a multiple array of cracks the stress intensity at the tip of each is relieved (Baratta, 1978; Bowie, 1973). It may be shown that the 'relief
I High
Temperature
21
factor* in AK for our longitudinal and circumferential cracks is ^ 0.5. However, from equation (11), for example, this would predict an eight fold reduction every where in da/dN. Since one crack can break away with imposed end loads, this mod ification should be considered dangerous in assessments. 2. Figure 12 is similar to Fig. 7 (elastic case), but for a much larger body. It is seen (i) that the increased wall thickness leads to an increase in stress in tensity for the same surface conditions, implying that cracks penetrate(relatively) deeper and (ii) that longitudinal cracks are more dangerous in the (relatively) thin-walled geometry of Fig. 12. This arises from Buchalet and Bamford's (1976) analysis where the coefficients F^etc. in equation (10) become > 1. 3. If a threshold AK 0 is known, then Figs. 7,11 provide a rapid estimation of the maximum depth of penetration. Thus taking AK0=8MPa ml (Skelton, 1978b) and an equivalent surface stress at zero end load of 417MPa, Fig. 6, then arrest is pre dicted, from Fig. 7 at a=2.7mm, safely overestimating the experimental value of 2.1mm. 4. From this work we thus have a thermal fatigue analysis for simple components in terms of crack growth. If the predicted curves e.g. Figs. 8,9 intersect the back face of the component within the experimental range of growth rates, breakthrough is expected. Alternatively, by integrating the curves, predicted crack depth is given directly against number of cycles. 5. The present tests have exaggerated quench severity in order to characterize the effects of end load. Current tests are examining the effect of quenching (i) on the ferritic 9CrlMo steel (which has a greater thermal conductivity and lower ex pansion coefficient than 316 steel i.e. a reduced thermal stress) and on (ii) much larger sections of 0.5Cr-Mo-V steel as used in turbine chests and casings. ACKNOWLEDGEMENTS I would like to thank Dr. B. Tomkins for providing some calibration data and Drs. D.C. Martin, G.G. Chell, D.J.F. Ewing and D. Raynor for helpful discussions. The work was carried out at the Central Electricity Research Laboratories and the paper is published by permission of the Central Electricity Generating Board. REFERENCES Baratta, F.I. (1978). Eng. Fract. Mech., 10, 691-697. Berling, J.T. and Slot, T. (1969). Fatigue at High Temperature, Spec.Tech.Publ. 459, Amer. Soc. Testing Mater., Philadelphia, pp. 3-30. Biot, M.A. (1957). J. Aeronaut Sci., 2A_9 857-873. Booth, S.J. and Blomfield, J.A. (1974). Unpublished work. Bowie, O.L. (1973). In G.C. Sih(Ed.), Mechanics of Fracture l,Noordhoff, Leyden, pp. 1-55. Buchalet, C.B. and Bamford, W.H. (1976). Mechanics of Crack Growth, Spec.Tech. Publ.590,Amer.Soc.Testing Mater., Philadelphia, pp. 385-402. Carslaw, H.S. and Jaeger, J.C. (1959). Conduction of Heat in Solids, 2nd ed., Clarendon Press, Oxford, p. 92. Chell, G.G. and Ewing, D.J.F. (1977). Intemat. J. Fract., 13, 467-479. Dawson, R.A.T., Elder, W.J., Hill, G.J. and Price, A.T. (1967). Thermal and High Strain Fatigue, The Metals and Metallurgy Trust, London, pp. 239-269. Elizarov, D.P. (1971). Teploenergetica, 18, 78-82 (117-123 English version). Haigh, J.R. and Skelton, R.P. (1978). Mater. Sci. Eng., .36, 133-137. Hellen, T.K., Price, R.H. and Harrison, R.P. (1975). 3rd Internat.Conf.on Struc tural Mechanisms in Reactor Technology, Imperial College, London,paper L7/4. Howes, M.A.H. (1973). Fatigue at Elevated Temperatures, Spec.Tech.Publ.520, Amer. Soc. Testing Mater., Philadelphia, pp.242-254. Kanazawa, K. and Yoshida, S. (1975). Conf. on Creep and Fatigue in Elevated Temperature Applications, Inst. Mech. Engrs., London, pp. 226.1-226.10.
ICM 3,
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Marshall, P. (1978). Unpublished work. Mowbray, D.F., Woodford, D.A. and Brandt, D.E. (1973). Fatigue at Elevated Temp eratures, Spec. Tech.Publ.520, Amer.Soc. Testing Mater., Philadelphia, pp. 416-426. Neuber, H. (1961). Trans. A.S.M.E., Series E, ^ 8 , 544-550. Northcott, L. and Baron, H.G. (1956). J. Iron Steel Inst., 184, 385-408. Newton, J.D. (1976). J. Austral. Inst. Met., n_, 94-102. Skelton, R.P. (1978a). Mater. Sci. Eng., 312, 211-219. Skelton, R.P.(1978b). Time Dependent Degradation of Pressure Boundary Materials, I.A.E.A. meeting, Innsbruck, 20-21 November. Skelton, R.P. (1978c). Mater. Sci. Eng. 35^ 287-298. Skelton, R.P. (1979a). C.E.G.B. Report RD/L/R. Skelton, R.P. (1979b). Fatigue Eng. Mater. Struct., to be published. Thomas, G. (1977). Unpublished work. Timoshenko, S.P. and Goodier, J.N. (1970). Theory of Elasticity, 3rd ed., McGraw Hill, New York, pp. 433-484. Wareing, J. (1975). Met.Trans., 6A, 1367-1377. APPENDIX Biot (1957) has defined the transit time t^ to be when the temperature of the outer face just begins to fall. It is given by t^=0.088 l^/D where D is the diffusivity, in turn given by k/pc where k is the thermal conductivity, p is the den sity (7.8xl03kg m"3) and c the specific heat (5.1xl02 J kg" 1 °C""1) . For thick section turbine components, transit times occur many minutes after the onset of cooling, see Table 2 for a comparison of measured and predicted values. In ex periments we have found that transit time slightly anticipates the arrival of peak stress. TABLE 2 Comparison of Predicted and Experimental Transit Times Average Temp. C
Material
550 450 500 530 300
AISI 316 AISI 316 0.5CrMoV lCrMoV 12Cr
24 22 35 35 29
4.5 10 35 356 550
21 min 61 min
100
Low Alloy Steel
42
603
51 min
*
Thermal Conductivity k,
W m-1 °Cl
Transit Time tx Thickness — mm Predicted Measured 0.3s 1.6s
12s
0.4s 2.0s
12s
19 min* >40min* <80min >33min <60min
Ref.
Present Work Tomkins, 1977 Skelton, 1978 Copp, 1972 Briner and Beglinger, 1975 Rathbone, 1929
Computer calculation from known heat transfer coefficients. REFERENCES
Briner, M. and Beglinger, V. (1975). Conf. on Creep and Fatigue in Elevated Temperature Applications, Inst. Mech. Engrs., London, pp. 220.1-220.8. Copp, L.H. (1972). Unpublished work. Rathbone, T.C. (1929). Quoted in Nadai, A. (1963). Theory of Flow and Fracture of Solids, Vol. II. McGraw Hill, New York, p. 399. Skelton, R.P. (1978). Unpublished work. Tomkins, B. (1977). Unpublished work.