Constant creep data accumulation for design stress analysis on cladding material thickness around cylindrical reactor fuel elemen

Whitmarsh, C. L., Jr.

English

NASA Technical Reports Server

NASA 
N 
rr) 
m 
P 
n 
z + 
4 m 
4 z 
TECHNICAL NOTE D-4352 
METHOD FOR CALCULATING 
ALLOWABLE CREEP STRESS 
IN LINEARLY INCREASING 
STRESS ENVIRONMENT 
I 
1 '  
by Charles L. Whitmarsh, Jr. 
Lewis Research Center 
Cleueland, Ohio 
NATIONAL AERONAUTICS A N D  SPACE A D M I N I S T R A T I O N  WASHINGTON,  D. C. J A N U A R Y  1968 
https://ntrs.nasa.gov/search.jsp?R=19680017292 2020-03-23T22:59:51+00:00Z
u-dcs' ERRATA 
NASA Technical Note D-4352 
- .' Y 
METHOD FOR CALCULATING ALLOWABLE CREEP STRESS 
IN LINEARLY INCREASING STRESS ENVIRONMENT 
By Charles L. Whitmarsh, Jr. 
January 1968 
Page 5: Equation (IO) should read 
Page 6, figure 1: The values of the slope of the Larson-Miller correlation 
should be -1.9X10- , -1.8, -1. 3, - 1 . 2 ,  and -1.  I. 4 
.- 
NASA-Langley, 1968 Issued 5-21-68 
TECH LIBRARY KAFB, NM 
METHOD FOR CALCULATING ALLOWABLE C R E E P  STRESS 
IN LINEARLY INCREASING STRESS ENVIRONMENT 
By C h a r l e s  L. Whi tmarsh ,  Jr. 
Lewis  R e s e a r c h  C e n t e r  
Cleveland, Ohio 
NATIONAL AERONAUT ICs AND SPACE ADM I N I STRATI ON 
For sale by the Clearinghouse for Federal Scientific and Technical Information 
Springfield, Virginia 22151 - CFSTI price $3.00 
METHOD FOR CALCULATING ALLOWABLE CREEP STRESS 
IN LINEARLY INCREASING STRESS ENVIRONMENT 
by Charles L. Whitmarsh, Jr. 
Lewis Research Center 
SUMMARY 
An analytical technique was  developed to calculate the allowable design stress for a 
specified creep limit of a material under conditions of linear s t ress  buildup and constant 
temperature. The analysis utilizes constant-stress creep data in the integration of a 
s t ress  -dependent creep rate over the material history. Allowable design s t ress  limits 
for 1'-percent creep of refractory metals in the temperature range 800' to 1600' K a r e  
increased by 25  to 57 percent when compared with calculations based on the constant end- 
of -life s t ress  condition. 
INTRODUCTION 
An accurate assessment of the allowable s t resses  that cause high-temperature creep 
in materials is important for space power systems currently under study. A great body 
of experimental creep data has been obtained for materials of interest under conditions of 
constant stress and temperature. The Larson-Miller (ref. 1) correlation, for example, 
which plots allowable constant stress against a combined time and temperature parameter 
for a fixed value of creep, is widely used for interpolation among the data. However, con- 
ditions of variable stress and/or temperature may be encountered in practice, and few 
creep data for such conditions exist. The situation that prompted the present work was  
the calculation of cladding thickness for a cylindrical nuclear -reactor fuel element de- 
signed to contain fission gases being generated at a constant rate. Internal pressure and, 
therefore, s t resses  are not constant (although temperature can be reasonably assumed to 
be), but increase linearly from some initial low value. Assuming that the end-of-life 
stress is constant over the entire reactor lifetime is unduly conservative, and the calcu- 
lational results of cladding thicknesses and core volumes are greater than necessary. 
This assumption may severely penalize high -temperature -reactor concepts intended for 
space power. 
If the creep rate of a material under stress is assumed to be independent of prior 
creep sustained at other conditions, then total creep can be accumulated. This idea is 
not new (ref. 2) and has been experimentally verified for several materials (refs. 3 to 5). 
Recently, this technique has been used for the design of radioisotope capsules where 
helium, a decay product, builds up internal pressure (ref. 6). 
for  variable-stress situations and to use as the basis for the analysis experimental data 
obtained under constant -stress conditions. 
The purpose of this analysis is to apply the concept of accumulation of total creep 
ANALY S I S 
General Case 
The basic assumption used in the following analysis is that the creep rate of a mate- 
rial under s t ress  is independent of prior creep sustained at other conditions and that 
total creep, therefore, can be accumulated, according to the relation 
lEf de = JTf ;(U, T)dT 
where E is the creep of a sample; i ( o ,  T) is the creep rate of a sample and is in general 
a function of the stress u and the temperature T to which the sample is subjected; and 
cf and T~ represent the final fractional elongation and final time, respectively. 
If the creep rate is now assumed to be constant for the time period of an experiment 
at constant stress and temperature (second-stage creep only), then the creep rate 
be written as the ratio of some fractional elongation to the time tO(u, T) over which 
this elongation occurred. Thus 
can 
To facilitate the translation of these data to other values of E and t, the time tf(u, T) 
required for the design creep Ef to occur can be defined as 
2 
I 
where the subscript f is used to differentiate between design conditions and the experi- 
mental test conditions in equation (2). Substitution of this relation into equation (1) and 
integrating the left side yield 
To carry out the integration, tf(u7 T) is written as an explicit function of stress and tem- 
perature. A useful relation to  accomplish this is the Larson-Miller (ref. 1) correlation. 
Under those conditions where the Larson-Miller correlation can be represented by a 
straight line on a semilog graph, it can be written as 
log u = m P  + log a = mT(K + log to) + log a (5) 
where 
m slope of equation 
P Larson-Miller parameter 
a ordinate u intercept 
T absolute temperature 
K constant dependent on material 
to time of exposure 
Rearranging equation (5) into an explicit relation of time to yields 
and combining equations (2), (3), (4), and (6) yields 
3 
Implicit in the derivation of equation (7) is the use of constant-stress - constant- 
temperature experimental data to describe the creep occurring under conditions of vari- 
able stress and temperature. However, this procedure is proper only if the basic 
assumption holds that creep rate is independent of prior creep. It should be noted also 
that, in the derivation of equation (7), no assumptions a r e  required on the variation of 
stress and temperature over the operating lifetime of the sample. These variables often 
can be described as functions of the operating history of the sample; for example, 
(T = U(T) and T = T(T), which reduce the integrand to  a function of time T. Depending on 
the circumstances, equation (7) can then be integrated, either numerically or in closed 
form, to determine the allowable end-of-life stress for a fixed core life or  vice versa. 
Linearly Varying Stress and Constant Temperature 
A closed-form integration of equation (7) is obtainable for the relatively simple and 
interesting case for which the operating temperature remains constant over the sample 
history, while the stress varies linearly with time. This case is applied to the calcula- 
tion of creep in the cladding of a pin-type nuclear-reactor fuel element operating at con- 
stant power and constant temperature. Since fission-product gases a re  generated at a 
nearly constant rate, the pressure and, hence, s t resses  within the fuel element can be 
considered to build up linearly with time. Under this condition, the rate of increase of 
stress is constant and can be set equal to the stress at the end of the operating life di- 
vided by the time period over which this rate occurred. Thus, 
du - 'f - - -  
where the implicit assumption has been made that the stress is initially zero. 
tion (7) from operating time T to s t ress  0. Equation (7) then becomes 
Equation (8) can now be used to effect a change of variable of integration in equa- 
€ 0  K l/mT Uf = - 10 7 fa 
€f 
4 
Integration yields 
m T  
At this point, examination of equation (6) reveals that the quantity a ( l O K ~ f )  is equal 
to the stress value predicted by the Larson-Miller correlation for  the condition 
This point is denoted as aLM and a stress modifying factor 
= cf. 
is defined as 
f -4-1/mT 
Thus, equation (9) becomes 
and states that, when the stress is linearly increasing and the temperature is constant 
over the operating history, the allowable end-of-life stress of to achieve a desired 
creep value is equal to the product of the s t ress  oLM (obtained from the Larson-Miller 
correlation for an equivalent value of creep) and a factor + that depends on the material 
and the operating temperature. 
terest and is shown plotted as a function of temperature in figure 1. Data for these ma- 
terials were taken from references 7 and 8 and were  assumed to have a constant Larson- 
Miller slope over the temperature range plotted. For the temperature range 800' to 
1600' K, the least sensitivity is exhibited by NAS-36 with + values of 1.25 to 1.4, re- 
spectively, and the greatest sensitivity is exhibited by FS-85 with + values of 1.37 to 
1.57, respectively. From figure 1 it can be seen that + increases with operating tem- 
perature and is nearly linear over a broad range. 
To illustrate the use of the + factor, imagine a hypothetical application requiring 
10 000 hours' service at a constant temperature of 2200' F (1480' K) for  which the stress 
on the sample is linearly increasing. If 1-percent creep and the material T-111 a r e  
specified, for example,. use of the Larson-Miller data given in reference 7 indicates a 
design stress limit aLM of about 5000 psi (0.18 MN/m2). From figure 1, + for  T-111 
at 2200' F (1480' K) is about 1.42. Hence, a stress of of about 7100 psi (0.26 MN/m2) 
at the end of life could be allowed. 
The + factor was  evaluated for  several high-temperature refractory alloys of in- 
5 
li. 
I 
1.55 
1. 50 
3 
s 
~ 1.45 
f-l 
of w
m 
1.40 2 
1.35 
1. 30 
1 7.54 
1.60} 
- 
- 
- 
- 
I 
- 
.- 
e 
Material Composition Slope of 
Larson- Miller 
correlation, 
OK- 1 
/ T-lll / / 
-LL, - l  -.. 1 . -2- __ I _. -- 
Cb-28Ta-10.5w-O.9Zr 
W-25Re 
Ta-W-2Hf 
Cb-1Zr 
Ta-5.7W-1.56Re- 
0.7Mo-0.25Hf-0.13Zr- 
0.015C-0.015N 
- 5 . 9 x 1 0 - ~  
-5. 8 
-3.9 
-3. 8 
-3.4 
Temperature, OF 
I . I  1 I I .-I -~ I I I I 
800 900 1000 1100 1200 1300 1400 1500 1600 1700 
Temperature, OK 
Figure 1. - Calculational factor to increase allowable design s t r e s s  for 1-percent c reep  in several  refractory-metal 
alloys. 
Application to Reactor-Fuel-C lad Design 
In a study of a small, fast, gas-cooled, high-temperature reactor employing pin- 
type fuel elements, the cladding thickness was  calculated for the constant stress aLM 
as well as fo r  the linearly varying stress. The allowable design stress and the pressure 
on the fuel cladding were related by means of the tangential stress equation for thick wal l  
cylinders (ref. 9) evaluated at the inside surface of the tube wall  
(ri + r;)Pi - 2rOpo 2 
a =  
(rz - r:) 
6 
0 
I 
I 
where r is the tube radius, P is the pressure, and the subscripts i and o refer to 
inside and outside conditions, respectively. Equation (11) was used to evaluate the 
allowable stress from which the cladding thickness was determined by equation (12). 
The cladding material was tungsten - 25 percent rhenium, and the operating conditions 
were 25 000 hours at 1800’ K with 1 percent creep. 
conditions, with a 60-percent increase allowed in design stress over the constant-stress 
assumption. A saving in cladding thickness was effected, which allowed an increased 
fuel loading that resulted in a core volume reduction of 18 percent with the use of the 
technique reported herein as compared with that of the constant-stress assumption. 
tions. 
a constant pressure on the outside of the element, the cladding is initially in compression. 
As  the fission gases build up throughout the reactor life, this compression decreases and 
changes to tension when the fission-gas pressure exceeds the coolant pressure. 
end of life, the net pressure on the cladding equals the fission-gas pressure minus the 
coolant pressure. 
dashed line shows the fission-gas pressure alone, and the lower dashed line shows the 
net pressure (fission gas minus coolant) on the cladding. The solid line starting at zero 
EE r 
The factor + was 1.6 under these 
? 
The foregoing analysis utilized the following approximation to actual reactor condi - 
Because the initial fission-gas pressure was  zero and the gaseous coolant exerts ? 
A t  the 
These pressure histories a re  presented in figure 2. The upper 
I 
I 
0 
Fission gas pressure at end of life, P 7 
fg ‘\I 
Tf 
Time 
Figure 2. - Pressure history of a nuclear reactor fuel 
element with respect to containment of fission gases. 
7 
and ending at the end-of-life net pressure Pf was the assumed pressure history used for  
the stress calculations. This assumption is considered conservative, since the assumed 
tensile s t ress  was always equal to or greater than the actual. 
DlSCU S S ION 
The analytical method presented herein appears mathematically suitable for any 
total creep limit within the constant-creep-rate assumption and is not restricted to the c 
creep limit of the experimental test data. 
should be limited to low creep limits. Also, uncertainties exist in the use of the Larson- 
Miller creep rate correlation where longtime test data for materials of interest a r e  
usually nonexistent, and thus extended extrapolations a r e  required. Fortunately, the 
stress-increase factor I$ plotted in figure 1 is relatively insensitive to changes in both 
the value of the slope of the Larson-Miller correlation and the temperature. For exam- 
ple, a 100-percent increase in either m o r  T causes the factor to increase about 15 
per cent. 
be used if  a closed analytical solution were not required. Thus, equation (4) could be 
solved numerically by use of a digital computer if sufficient test data and/or problem 
history data were available. 
which the fuel cladding thickness is determined by creep as opposed to those situations 
in which limitations are related to material elastic properties or strength. In general, 
this application will include high-power-density - high-temperature systems that are 
required to retain fission gases in the core. Of course, fission gas pressure is only one 
of the factors upon which cladding thickness depends, and in a detailed reactor design, 
additional items, such as thermal cycling, heat-transfer effects, and fabrication, must 
be considered. 
To assure such application, however, its use 
I 
In theory, any stress-temperature- creep correlation and/or any stress history could 
Method application to nuclear-reactor design is restricted to those situations in 
S U M M A R Y  OF RESULTS 
An analytical method was developed to calculate allowable end-of-life stress for a 
design creep limit of a material under conditions of linearly increasing stress and con- 
stant temperature. For a given material and creep limit, a temperature dependent fac- 
tor  was calculated which represents the ratio of the allowable end-of-life stress to the 
constant stress obtained from Larson-Miller data. 
Calculations were performed to show that the allowable end-of-life s t ress  for 
8 
- . . . . . . ... .. , 
i 
i 
i 
refractory metals, 
and a temperature 
1 
i 
, 
t 
! 
for the conditions of l-percent creep, and linearly increasing stress, 
range of 800' to 1600' K, was increased by 25 to 57 percent when 
compared with constant-stress calculations. 
Also, the core volume of a small, fast-spectrum, high-temperature, gas- cooled 
reactor was reduced by 18 percent when the method presented herein was used for  the 
determination of the required fuel cladding thickness. 
Lewis Research Center, 
National A e r onauti cs and Space Administration, 
Cleveland, Ohio, November 7, 1967, 
120-27-06-05-22. 
1. I 
I 
4 
P 
2. 
3. 
4. 
5. 
6. 
I 
I 
7. 
8. 
9. 
REFERENCES 
Larson, F. R. ; and Miller, James: A Time-Temperature Relationship for Rupture 
and Creep Stresses. Trans. ASME, vol. 74, no. 5, July 1952, pp. 765-771; Disc., 
pp. 771-775. 
Robinson, Ernest L. : Effect of Temperature Variation on the Long-Time Rupture 
Strength of Steels. Trans. ASME, vol. 74, no. 5, July 1952, pp. 777-781. 
Serensen, S. V. ; and Strelaiev, V. S. : Creep Resistance and Low-Cycle Fatigue of 
Fiberglass Plastics. Joint International Conference on Creep. Institute of Mechan- 
ical Engineers, 1963, pp. 3-93 to 3-98. 
McCoy, H. E. : Creep-Rupture Properties Under Conditions of Constant and Varying 
Stresses. Nucl. Applications, vol. 2 ,  no. 6, Dec. 1966, pp. 481-485. 
Kennedy, Alfred J. : Processes of Creep and Fatigue in Metals. John Wiley and Sons, 
Inc., 1963, pp. 401-428. 
Nichols, J. P. ; and Winkler, D. R. : A Program for Calculating Optimum Dimensions 
of Alpha Radioisotope Capsules Exposed to Varying Stress and Temperature. Rep. 
No. ORNL -TM-1735 (NASA CR-72172), Oak Ridge National Lab., Apr. 1967. 
Moss, Thomas A. : Materials Technology Presently Available for Advanced Rankine 
Systems. Nucl. Applications, vol. 3, no. 2,  Feb. 1967, pp. 71-81. 
Sawyer, J. C. ; and Steigerwald, E. A. : Generation of Long Time Creep Data on 
Refractory Alloys at Elevated Temperatures. TRW Equipment Labs. (NASA 
CR-72185), Jan. 17, 1967. 
Singer, Ferdinand L. : Strength of Materials. Harper and Brothers, 1951. 
NASA-Langley, 1968 - 22 E-4188 9 
"The aeronautical and space activities of the United States shall be 
conducted so as to contribute . . . to  the expansion of human knowl- 
edge of phenomena in the atmosphere and space. The Admillistration 
shall provide for the widest practicable and appropriate dissemination 
of information concerning its activities and the results thereof." 
-NATIONAL AERONAUTICS AND SPACE ACT OF 1958 
NASA SCIENTIFIC AND TECHNICAL PUBLICATIONS 
TECHNICAL REPORTS: Scientific and technical information considered 
important, complete, and a lasting contribution to existing knowledge. 
TECHNICAL NOTES: Information less broad in scope but nevertheless of 
importance as a contribution to existing knowledge. 
TECHNICAL MEMORANDUMS: Information receiving limited distribu- 
tion because of preliminary data, security classification, or other reasons. 
CONTRACTOR REPORTS: Scientific and technical information generated 
under a NASA contract or grant and considered an important contribution to 
existing knowledge. 
TECHNICAL TRANSLATIONS: Information published in a foreign 
language considered to merit NASA distribution in English. 
SPECIAL PUBLICATIONS: Information derived from or of value to NASA 
activities. Publications include conference proceedings, monographs, data 
compilations, handbooks, sourcebooks, and special bibliographies. 
TECHNOLOGY UTILIZATION PUBLICATIONS: Information on tech- 
nology used by NASA that may be of particular interest in commercial and other 
non-aerospace applications. Publications include Tech Briefs, Technology 
Utilization Reports and Notes, and Technology Surveys. 
1 
I 
Details on the availability of these publications may be obtained from: 
SCIENTIFIC AND TECHNICAL INFORMATION DIVISION 
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 
Washington, D.C. PO546 


Method for calculating allowable creep stress in linearly increasing stress environment

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server