An analytical model for the ductile failure of biaxially loaded type 316 stainless steel subjected to thermal transients

Journal of Nuclear Materials North-Holland, Amsterdam

171

144 (1987) 171-180

AN ANALYTICAL MODEL FOR THE DUCTILE FAILURE OF BIAXIALLY 316 STAINLESS STEEL SUBJECTED TO THERMAL TRANSIENTS

LOADED

TYPE

R.J. DIMELFI 3444 Ullnwn St.. Sun Dqo, Received

16 December

CA 92106, USA

1985: accepted

15 July 1986

Failure properties are calculated for the case of biaxially loaded type 316 stainless steel tubes that are heated from 300 K to near melting at various constant rates. The procedure involves combining a steady state plastic-deformation rate law with a strain hardening equation. Integrating under the condition of plastic instability gives the time and plastic strain at which ductile failure occurs for a given load. The result is presented as an analytical expression for equivalent plastic strain as a function of equivalent stress, temperature, heating rate and material constants. At large initial load, ductile fracture is calculated to occur early, at low temperature, after very little deformation. At very small loads deformation continues for a long time to high temperatures where creep rupture mechanisms limit ductility. In the case of intermediate loads. the plastic strain accumulated before the occurrence of unstable ductile fracture is calculated. Comparison of calculated results is made with existing experimental data from pressurized tubes heated at 5.6 K/s and 111 K/s. When the effect of grain growth on creep ductility is taken into account from recrystallization data. agreement between measured and calculated uniform ductility is excellent. The general reduction in ductility and failure time that is observed at higher heating rate is explained via the model. The model provides an analytical expression for the ductility and failure time during transients for biaxially loaded type 316 stainless steel as a function of the initial temperature and load. as well as the material creep and strain hardening parameters.

1. Introduction Increasingly sophisticated technologies are requiring that structural materials be taken to their limit of mechanical endurance. Complex loading conditions and severe thermal cycling can produce tensile stresses that cause failure of critical components. To avoid failure at the design stage or predict its occurrence, one must understand deformation and fracture mechanisms in engineering materials and their relation to thermal history and loading geometry. There are a number of advanced technologies in which components in service are subjected to a constant multiaxial load while experiencing substantial increase in temperature. For example, such conditions can exist within important components of breeder reactors, space nuclear reactors, space weaponry, and fusion reactors. One problem that has received considerable attention because of its relevance in nuclear technology is the response of austenic stainless steel to constant biaxial loading during heating. Predicting failure is an important aspect of these analyses. Unfortunately, in many such studies, ductile failure is treated as a phenomenon separate from other

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mechanical properties of the material. Often, a cumulative damage law and life fraction rule are invoked. Their use involves a failure criterion like the Larson-Miller parameter [l]. Such parameters are used indiscriminately in these analyses even in temperature regimes where they are neither applicable nor reliably accurate. In other work [2] in this area a rate equation of “state” and a strain-hardening law are properly defined. However, low temperature failure ductility is treated separately as though independent of hardening rate, by substitution into the analysis of experimentally determined ductility values as a function of temperature. Further, at high temperature where creep rupture (intergranular fracture) mechanisms are observed, failure is incorrectly attributed to flow stress reductions due to thermal annealing. It is our intention in this analysis to calculate the circumstances of failure in this complex loading situation by keeping track of the fundamental mechanisms of deformation and failure of the material. We then apply this understanding within the context of the thermo-mechanical loading history. These fundamental mechanisms are better extracted from simple tests. B.V.

172

R.J. DiMelfi / Ductile failure

Laboratory fracture experiments help us learn about fracture mechanisms under various controlled conditions that are considerably simpler than those experienced in service. Two such experiments that have provided considerable insight are tensile creep tests and uniaxial tensile tests. In the creep test, samples are deformed at constant stress and temperature until rupture occurs. Most times, creep fracture is understood in terms of intergr~ul~ separation mechanisms 131. The failure ductility is roughly the plastic strain accumulated at a constant rate until fracture occurs. Intergranular creep crack propagation often obeys the same kinetic law as creep deformation. Therefore it is logical to expect a rupture ductility independent of temperature and stress and this is often observed [4]. The creep test is a high temperature test, but applied stresses and resulting plastic strain rates are small. The unia.xial tensile test is typically performed at lower temperatures than creep experiments. Ideally, a constant strain rate is imposed on the sample and the tensile stress increases until fracture occurs. At these rates and temperatures, the grain boundaries can usually sustain plastic deformation as well as the grain matrix me+:rial, so that ductile failure in tension is not controlled by intergranular fracture. It is another attribute that leads to ductile failure under these conditions. Unlike creep deformation, the plastic strain rate in a tensile test at low temperature is extremely sensitive to flow stress. Any local flaw that increases the stress somewhat will increase the local strain rate substantially. If the material cannot harden sufficiently to inhibit this process, the flaw will propagate rapidly and failure will occur. This catastrophic process is termed plastic instability. These two limiting cases: high-temperature creep rupture by intergranular fracture and low-temperature ductile failure by plastic instability can be analyzed quite readily because of the simplicity of the tests. The resulting analysis can even be applied to engineering cases where conditions are not quite as simple as the laboratory experiments. For example, we have analyzed deformation and failure of pressurized stainless steel tubes subjected to thermal ramps [S]. This situation is particularly relevant to conditions experienced by fuelcladding tubes in nuclear reactors. We have found that at very small applied loads (low pressure), failure does not occur until temperatures are well into the creep regime (at or above half absolute melting). Tube failure in this case can be well-predicted by assuming that creep-rupture mechanisms are dominant. At large loads, substantial deformation begins early during heating, and the prediction of failure as a low-temperature phe-

ofbiaxially loaded rype 316 SS nomenon dominated by plastic instability is quite successful. So, our understanding of the fracture processes in the two simple limiting cases can be applied to an engineering situation where temperature and stress are increasing with time, and the stress state is multiaxial. This approach is applicable to other engineering conditions that involve similar thermomechanical loading (i.e. multiaxial loading and thermal transients). Of course, the approach discussed above has its limitations. There is certainly an intermediate loading range where one cannot assert that fracture is controlled by one or the other mode. It is more difficult to calculate failure ductility in this intermediate range than it is at very large or very small loads. At very large loads, plastic instabi~ty begins when plastic flow starts and the useful ductility is zero. At very small loads, the creep ductility of the material will be the failure ductility. In the intermediate case, substantial plastic flow can occur before the onset of instability and ductility calculations become more complicated and have not yet been done. Also, it has been observed in the above experiments [6] that the failure ductility is lower in the intermediate loading range for higher heating rate tests. Inasmuch as the failure time depends on the ductility, it is important to calculate the ductility in the intermediate range. This calculation was not made in our earlier analysis [S]. In that work, the effect of heating rate on failure time was taken into account by substitution of experimentally reported ductilities measured at different heating rates into the formula for failure time expressed as a function of applied stress. In this work we actually calculate the ductility at intermediate loads.

2. Analysis In this note, we analyze plastic deformation and failure of biaxially stressed type 316 stainless steel tubes subjected to thermal transients. We include treatment of the case where loads are in the intermediate range discussed earlier. This is important because during thermal transients, when components do fail, fracture in the intermediate mixed-mode range can be the most natural occurrence. Temperature “ramping” will take the material through a broad spectrum of deformation modes even when the load remains constant. Consequently, a component will accumulate strain and creep damage at rates that increase with temperature. At higher strain rates, the material becomes more sensitive to unstable flow and this process eventually leads to failure. In this analysis, we try in a straightforward way to keep track of these processes and arrive at an analyti-

113

R.J. DiMelfi / Ductrle failure of biaxially loaded type 316 SS

cal expression for the failure ductility. The effect of heating rate on ductility is an inherent part of the calculation. This allows true prediction of its effect on failure time without resorting to substitution of experimentally measured strains. In order to perform this calculation, one needs a deformation rate law that describes the entire range of rate behavior of the material. As we have noted, at low temperature the plastic strain rate of most engineering materials is highly sensitive to changes in the stress, and hence more sensitive to unstable flow. At high temperature this is not the case. Also, one needs an equation to describe the way in which the material hardens with strain. If a material can harden at a rate higher than that at which local stresses increase, unstable flow can be delayed. Increasing temperature and plastic strain usually serve to decrease the hardening rate. Finally, one needs to keep track of the operative fracture mechanisms. We have chosen to perform this analysis on type 316 stainless steel first because of its use in advanced technologies. Also, there are considerable data on its mechanical properties. Finally, as we mentioned above we performed earlier calculations based on the behavior of this alloy. In order to compare calculated results with experiment we will focus our attention on the behavior of thin-wall tubes subjected to constant internal pressure and heated at a constant rate. Philosophically, the approach used here is a simple one. In the intermediate range of loading pressures, failure can still be prescribed to occur at the onset of plastic instability, but substantial strains can accumulate before this point is reached. Also, since failure temperatures are somewhat higher in this loading range than they are at very high stresses, the creep failure mechanism (intergranular cracking) can supercede plastic instability as the failure mode. To describe the plastic strain rate we use a rate law we have recently developed [7] for type 316 stainless steel based on the classical chemical rate theory. The advantage of this equation is its broad applicability, and the flexibility of its mathematical form for the manipulations required here. Further, we have shown that its form is consistent with the useful Larson-Miller rupture parameter. In this context, the model was shown to be quantitatively consistent ‘with a broad range of rupture data plotted in accordance with the Larson-Miller parameter. This equation for the plastic strain rate is

i,=*i,

exp(-$1

sir&(g),

which

for stresses

and temperatures

of interest

in this

note reduces

to

In these equations (I is the equivalent stress (in SI units) for plastic strain, AC,, is the activation free energy at zero stress given by AG,,/k = 38533 K, s2 is the activation volume given by m3

6.57 x 1O-28 (1.66 - 5.75 x 10m4 T)’

’

where E is Young’s Modulus given by Ea(1.66 - 5.75 X 10m4 T), E,= 130.5 GPa, s2, is the value of D at E = E,,, T is absolute temperature in Kelvins, i, = 1.95 x lOs/s, and k is Boltzmann’s constant in SI units. We have found that eq. (2) describes the deformation-rate behavior of type 316 stainless steel reasonably well quantitatively over wide ranges of stress and temperature. The reader should seek the previous work [7] for details.The true-stress, true-strain behavior of type 316 stainless steel exhibits a “saturation” stress, i.e., a steady-state flow stress at large values of strain. For this reason, we employ a Vote-type stress-strain law [8,9] which has the form for the flow stress u=u,-(us-ul)exp(-t//r,).

(3)

The quantity i in this equation is a hardness parameter that is referenced to the strength of fully annealed material. In general, it represents positive and negative contributions to the strength such as prior cold work, irradiation hardening, annealing recovery, and of course accumulated equivalent plastic strain cr. The saturation stress a,, the yield strength ui, and eC are functions of strain rate and temperature (see ref. [9] for details). The saturation stress is the steady state flow stress in eq. (1) so that

We have obtained satisfactory agreement flow stress-strain curves if we assume u, = 0.20,.

(5)

In eq. (3), cc = (us - u,)/B, hardening rate given by 8, = 3.66 x lO_‘G, G = (92 - 4.02 X 10e2T)

with observed

where

0i

is the initial

(5’) GPa.

In this analysis we focus our attention on internally pressurized thin-wall tubes heated at constant rate. As we said earlier, high pressure results in failure at low temperature during thermal transient testing. Failure

R.J. DIMelf

174

/ Ductile failure

occurs in this range by plastic instability, when the material’s inherent ability to harden per increment of plastic strain cannot keep up with increases in stress caused by geometric changes. Also, this ability to harden can be reduced by increases in temperature during transients. This failure mode is, of course, a function of loading geometry. Also, the fact that at low temperature the deformation rate is highly stress-sensitive [see eq. (2)] means that any local thinning of the sample resulting in higher local stresses will propagate ever faster without accumulating much overall equivalent plastic strain. Under these conditions, failure can be prescribed to occur essentially just when the condition for plastic instability is satisfied. Pressurized tubes incur a two-to-one biaxial stress rate. It can be shown that for this geometry [lo] the condition for instability is

of biaxially loaded type 3/6 SS 20% cold-worked type 316 stainless steel; i.e., i,, = 0.2. The quantity u, is the true equivalent applied stress. In the constant pressure experiments modeled here, the true stresses is not held constant but increases as the tube becomes thinner. From kinematics we know that, in the constant-pressure loaded tube geometry, = ui exp(JSe,),

where 0,” is the initial applied equivalent stress at initial tube radius n, and initial thickness ho, and a and h are the instantaneous radius and thickness, respectively. By substitution from eq. (9) the argument of the second exponential function in eq. (8) may be expressed as Ai

-

-$[i m(1+

$=I/Ti0. P

e,

exp( -i/e,)

&/b)

(6)

However, we will show later that an instability condition having a right hand side equal to 2&o gives excellent agreement with data [ll]. This implies that loading during those experiments does not produce a strictly two-to-one biaxial tensile stress state locally near the failure site. Using eq. (3) we get for our failure criterion = 60,

(7)

which we solve simultaneously with eqs. (3) and (4) which embody the rate and temperature dependence extracted from eq. (2). At small loading pressures, tube failure would occur at high transient temperatures. Once in this regime, the stress-sensitivity of the deformation rate is low, local geometric flaws are therefore not as critical and considerable amounts of plastic strain can accumulate before failure. However at very high temperature, grain boundaries become weak and creep failure mechanisms can dominate. As we said earlier, at high temperatures the deformation and rupture processes are closely allied [4] resulting in an essentially constant creep ductility. This would place a ceiling on transient failure strain in this regime, caused by intergranular fracture. To address the subject of instability within the framework we have described we combine eqs. (3), (4) and (7), substitute for z, and solve for i,, obtaining

(8) In eq. (S), t^= i,, + ep, where i,, is the initial hardness

of

(9)

-Order(g)2].

=

(10) where we have defined A = fl,Q/O.S

and

For small strains, the second term on the right side of eq. (10) is small and we will ignore it to facilitate integration of eq. (8). This is tantamount to assuming a constant true stress experiment which is not the case for the constant load tests modeled here. This will delay somewhat the calculated transient time at which the onset of plastic instability and ductile failure occurs. This is because only two of the three causes for unstable flow are taken into account: a decreased hardening rate caused by plastic straining and by increasing temperature during the transient. However, the third factor contributing to instability, increasing true stress, is not. However, our limited goal of calculating failure ductility in the intermediate loading range and illustrating heating-rate effects is more readily achieved with this approximation. For a constant heating rate ?, eq. (8) may be expressed as di=[$exp(-$)exp(-$$)Idl,

(11)

where i = e p + Pa, and i, is the initial hardness of 20% cold-worked material. If we designate T* as the failure temperature, note that [6] ep - 0 for T < T*, and multiply both sides of eq. (11) by exp( Ai/bkT*) and integrate. we get (note: on the right hand side, i = i,, for

R.J. DiMelfi / Ductile failure of biaxially loaded type 316 SS T-CT*)

- z(

f

- A)]

(12) where i* is the hardness at failure, and Ti is the initial temperature. Using the series approximation for the exponential integral on the right hand side we obtain for the plastic strain at failure (see Appendix) 1,A bi‘T*(AG,

’

175

First, we address the subject of failure at very high temperature where creep mechanisms dominate. Intergranular fracture limits ductility in this regime and therefore cuts off the strain predicted by eq. (13). Also, since both creep cracking and creep deformation occur for the most part at a constant rate and obey essentially the same kinetic law one observes a constant creep ductility [4]. However, Yamada [ll] has observed substantial grain growth to occur at the very highest temperatures during his experiments. We have found [3,5] that the creep crack propagation rate in type 316 stainless steel obeys the empirical relation

+ i,A,,‘b)

(13) where A,, is defined in the Appendix, and T, = 300 K [ll]. The failure temperature is defined during the transient as the temperature at the onset of plastic instability or when E: + co. The value E: attained just before T* (at T* - 1) is the uniform failure ductility calculated for the specified heating rate and applied stress. Both this ductility and the failure temperature increase with decreasing applied stress. However, at high temperatures and low stresses the deformation becomes more creep like and the grain boundaries become sufficiently weakened so that the ductility is limited by creep failure mechanisms as discussed earlier. Under these conditions, the failure temperature is determined by equating the above calculated ductility with the steady state creep ductility, which is essentially constant ( = 40%) [12]. The observed [6] reduced ductility in the intermediate stress range at high heating rate is calculated automatically and not imposed artificially as before.

3. Results and discussion

We will compare results calculated from eq. (13) with the experimental data of Yamada [ll] and Hunter et al. [6]. We modify this equation slightly in accordance with our earlier comments regarding the instability condition (eq. (6)). This is accbmplished by replacing 0, (defined in eq. (5’)) by 8,/2. Agreement with test results implies that in the failure region a local stress state exists for which plastic instability is defined by do/d< = 260 instead of 60.

where K, is the appropriate stress intensity factor. Also, if grains grow during creep at constant stress, fewer and fewer grain boundaries are sustaining the propagating cracks. It has been found [5,13] that K,

a Cl/=,

where G is the normalized grain size. That is, as grains grow the stress driving creep crack propagation becomes more intensified. We have followed this logic in our earlier work obtaining an expression for creep ductility E,, = E~~G-5 35’2,

(14)

where czr = 40% [12] is the creep ductility (converted to diametral strain) of this material without grain growth. In fig. 1 we show a plot of eq. (14) for the two heating rates relevant to the experiments of Hunter et al. [6] and Yamada [ll]. The kinetics of grain growth used in eq. (14) are extracted from real recrystallization data [14] on this material as discussed in refs. [2] and [9]. We will see next that creep fracture as reflected in fig. 1 limits our calculated (eq. (13)) failure ductility only at the very highest temperature. We now compare in fig. 2 the predicted failure temperature as a function of initially applied hoop stress with the data of Yamada [ll] and Hunter et al. [6] for tests performed at T = 5.6 K/s. Considering the scatter in experimental results, the agreement is excellent. In fig. 3, the calculated failure ductility (uniform diametral strain) as a function of failure temperature is shown along with Yamada’s results. The data of Hunter et al. are not shown because they did not measure uniform diametral strain at the failure site as did Yamada. It is also interesting to note in fig. 4 that an apparently unique relationship predicted between failure ductility and applied hoop stress is borne out by the Yamada data. Hunter et al. observed that at higher heating rate,

176

R. J. DiMelf

/ Ductile failure of braxially loaded type 316 SS

0.45 0.40

\

0.35 $ E

\

\ \

0.30

\ \

‘\ III

0.25

g

\

2

0.20

E I 5 n

0.15

\

\

\

\

0.10 0.05 0.00

I

I

I

I

1200

I

I

1310

I

FAILURE

I

I

1420

I

I

1530

1

I

1640

1750

TEMPERATURE , K

Fig. 1. The calculated creep ductility (diametral strain) as a function of failure temperature for the two heating rates shown.

failure at a given applied stress is delayed to higher temperatures. This shift in failure temperature is adequately predicted by the model for the two heating rates tested, 5.6 K/s and 111 K/s, shown in fig. 5. Another rate effect that was observed is an apparent reduction in failure ductility at a given value of failure temperature. Fig. 6 illustrates the calculated reduction in comparison with observed results. Both sets of results are each normalized with respect to the maximum ductility (measured or calculated) for the case where T= 5.6 K/s. It 1650 y

I

I

I

should be noted that the absolute values of measured ductility are much less than those calculated, because the Hunter group reported strains measured away from the failure site where temperatures were considerably lower because of thermal gradients. Hence, both sets of data are shifted by 100 K along the abscissa for the purpose of comparison with calculated results. Fig. 6 is therefore presented only as a crude qualitative illustration of the calculated reductions in ductility apparent at higher heat rates. I

I

I

I

I

I

I

I

I

I

I

r

I

I

1550

E 1450 2 2 1350 aw f

1250

g 1 1 2

II50

+ REFERENCE 6 o REFERENCE I I

1050 950

I

I

6.8

I

I

I

7.6

7.2

Loglot Fig. 2. The failure temperature

HOOP

8.0 STRESS,

8.4

PA)

as a function of initial applied hoop stress for transient tests conducted the calculated result shown in comparison with experiment [6,11].

at 5.6 K/s.

The solid curve is

177

R.J. DiMerf / Ductile failure of biaxially loaded type 316 SS

0.35 0.30 f 0.25 (L Li 0.20 2 E 0.15 w 2 0

0.10 0.05 I

0.00

I

I

ii00

I

I

t

1200

I

1300

I

I

I

I

1400

1500

FAILURE TEMPERATURE Fig. 3. The uniform

diametral

strain at failure as a function of failure temperature. comparison with experiment Ill].

modes ranging from low temperature “ time-independent” plasticity to high temperature creep. Under these circumstances, it is important to be able to predict the time and ductility at failure for a given applied load. In this note, we have derived an analytical expression that can be used to determine these failure parameters over the entire useful loading range. This has been accom-

In this note, we have analyzed the deformation and failure behavior of an important commercial alloy subjetted to complex thermomechanical loading. Components in many advanced technology systems are exposed to constant multiaxial loading while experiencing some form of thermal cycling. Such temperature changes take the material through a spectrum of deformation

0.35

I

I

I

I

I

The solid curve is the calculated result shown in

I

I

I

II

I

I

I

0.30 $ & 2 g w Gf

0.25 0.20 0.15 0.10 0.05 0.00

L

0

40

80

HOOP

120

160

200

240

STRESS, MPA

Fig. 4. The relationship between diametral failure strain and initial hoop stress. The experimental at 5.6 K/s. The curves are calculated results at two heating rates. Note that they superimpose processes dominate failure at low stress.

data [ll] (symbols) were obtained and agree with data until creep

178 300 2250 t - 200 2 E & 150 % 0 100 I 50 0

1200

1100

1300

1400

FAILURE Fig. 5. The failure temperature

calculated

TEMPERATURE,

at two heating

plished by combining a wide-ranging deformation rate law with a strain hardening law. Both laws are physitally-based and have been fit to fund~ent~ mechanical property data on 316 stainless steel. We then integrate the combined equations under the imposed condition of plastic instability. The mathematical form of the rate equations together with the simplifying assumptions that we have made (see Appendix) allow an essen-

0.9 2

1500

1600

1700

K

rates shown in comparison with data from ref. 161.

tially analytical result (eq. (13)). Analytical solutions such as this are very useful because they help one to keep track of the physics of the problem. The material is characterized by such quantities as AC,, the creep activation energy (often the activation energy for self diffusion), 9, the stress dependence of the strain rate, 0,, the hardening rate at zero plastic strain, the saturation stress and the yield stress. Our analytical result

-I-5.6 K/s

0.8

I I 1 K/s

0

z(y 0.7

0.1

1’

I

1100

fi

I

I200

I

I

1300 FAILURE

Fig. 6. The calculated strains are normalized

I

I

I

1400 TEMPERATURE,

I

\

1500

I

\I

1600

K

failure ductility at two heating rates shown in comparison with data [6] as a function of failure temperature. All with respect to the maximum strain (measured or calculated) at 5.6 K/s. All data are shifted along the abscissa by 100 K.

179

R.J. DiMeifi / Ductile failure of biaxraNy loaded rype 316 SS

allows one to easily determine the sensitivity of predicted results to changes in these physical parameters as well as to changes in the independent variables such as applied load, initial temperature or heating rate. This ease of sensitivity analysis can be particularly useful when comparing alloys, characterized as above, for service under the loading conditions described here. With regard to the circumstances where creep fracture cuts off the calculated ductility, it should be noted that this occurs only at the highest temperatures, approaching melting, where grain growth is substantial. In real engineering situations, this event should be avoided at the design stage. Nonetheless, creep ductilities and grain growth kinetics have been measured for many materials, and when accounted for (fig. 1) here, provide good agreement with data at high temperature. In view of the complexity of the experiments analyzed, the analytical result that we have obtained shows excellent agreement with experimental data.

Appendix

data ref. [6]). This is the same as asserting that T = T, (constant) until cp = E;F. This allows straightforward integration of the left-hand side of eq. (A.3) giving for that integral

The integral (A.3) is

exp(Ai/bkT*)dP=

+exp[-g(i

on the right-hand AC, + A&,/b kT

side (RHS)

of eq.

(A.5)

dT.

We ignore the temperature dependence of the slowly and focus on A = 0,52/0.8. varying b = ln[fie~“/(Ol)] By definition, A may be expressed as A = 1.25 x 3.66 x lo-‘Gx

6.57 x 1O-2”( E,/E)‘. (A.6)

Since both E and G have the form OL- PT, and E = 2(1 + v)G, and assuming v, Poisson’s ratio constant (about 0.35 for stainless steel) one can show that A=

To obtain eq. (12), we have multiplied eq. (11) by exp(AP/bkT*). This gives

(A.41

I Lns=y[exp(g)-l].

both sides of

8.75 x lo-l9 1 - 3.46 x 10m4T

= 8.75 x lo-l9

+ 3.03 x IO-“T (A.7)

Letting A, = 8.75 X lo-l9 and A, = 3.03 substituting into eq. (A.5) we get

- A)]

(A.11

I,,,

= exp(-Z)/:‘exp(

- AGo +zo’b)

x

10e2’

and

dT.

Recall that i=Ela + zp and therefore di = de,. Also, note that since ep = 0 for T < T*, the argument of the first exponential on the right-hand side (RHS) of eq. (A.l) is either zero (when T= T*) or (-AZ,/&) [l/T - l/T*] (when T < T*). Substituting, we get

Changing variables by letting x = (AC, + A,i,/b)/kT, and using the exponential integral Ei( - X) we get

exp( &) exp( &&)de,

IRHs =

(A.81

AG, -A&/b kT

T* 7-I

(A.91 =%exp(&)exp(-$$)exp(-z)dT.

(A.21 The first exponential factors on each side cancel another. Integrating both sides of eq. (A.2) we get

Substituting eq. (A.9) into (A.5) and back into the right side of eq. (A.3) [the integral on the left side of eq. (12) being straightforward] one obtains eq. (13).

one Acknowledgements

AC, + Ac^,/b kT

dT.

(A.3)

This is simply another version of the integrals expressed in eq. (12). We have assumed ep = 0 until T = T* (see

This work was supported by Sandia National Laboratory under contract number 95-2929-9 (UNM # 281-547-9). The author is grateful to his colleague Dr. John. M. Kramer for many helpful discussions. He also wishes to thank Ms. Gayla Angel for preparing the manuscript.

180

R. J. DiMelfi

/ Ductile failure

References [l] J.L. Straalsund, R.L. Fish and G.D. Johnson, Nucl. Tech. 25 (1975) 531-540. [2] G.L. Wire, N.S. Cannon and G.D. Johnson, J. Nucl. Mater. 82 (1979) 317-328. [3] R.J. DiMelfi and W.D. Nix, Intern, J. Fracture 13 (1977) 341-348. [4] F.C. Monkman and N.J. Grant, Proc. ASTM 56 (1956) 593-605. [S] J.M. Kramer and R.J. DiMelfi, J. Engrg. Mater. Techn. 101 (1979) 293-298. [6] C.W. Hunter, R.L. Fish and J.J. Holmes, Nucl. Techn. 27 (1975) 3766388. [7] R.J. DiMelfi and J.M. Kramer, Res. Mech. 11 (1984) 245-294.

of biaxially loaded type 316 SS [8] E. Vote. J. Inst. Metals 74 (1948) 553-562. [9] R.J. DiMelfi and J.M. Kramer, J. Nucl. Mater. 89 (1980) 338-346. [lo] W. Johnson and P.B. Mellor, Engineering Plasticity (Ellis Horwood, Chichester, 1983) p. 254. [ll] H. Yamada. J. Nucl. Mater. 78 (197X) 24432. [12] F. Garofalo, 0. Richmond, W.F. Domis and F. van Gemmingen, in: Proc. Joint. Intern. Conf. Creep 1963, Vol. I (Int. Mech. Eng., London) pp. 1-31-1-39. [13] W.D. Nix, D.K. Matlock and R.J. DiMelfi, Acta Metall. 25 (1977) 495-503. [14] M.M. Paxton and J.J. Holmes, Hanford Engineering Development Laboratory, Report #HEDL-TME-71-126 (1971).