NASA Contractor Report 175094

Effects of State Recovery on Creep

Buckling Under Variable Loading

(NASA-CR-175094) £PFECTS;01 STATf RECOVERY N86-2531Q

ON CHEEP BUCKLING UNDEfi VfiBIABLE LOADING

Pinal Beport (Akrcn Duiv. , Ohio,), 45 p

flC A03/MF A01 I CSCLf03B Onclas

• G3/90 -- 42904

D.N. Robinson and S.M. Arnold

/'

University of Akron

Akron, Ohio

April 1986

Prepared for

Lewis Research Center

Under Grant NAG 3-379

NASA

National Aeronautics and

Space Administration
EFFECTS OF STATE RECOVERY ON CREEP BUCKLING UNDER VARIABLE LOADING

D. N. Robinson*

S. M. Arnold**

ABSTRACT

Structural alloys embody internal mechanisms that allow recovery of

state with varying stress and elevated temperature, i.e., they can return to

a softer state following periods of hardening. Such material behavior is

known to strongly influence structural response under some important thermomechanical loadings, for example that involving thermal ratchetting.

Here, we investigate the influence of dynamic and thermal recovery on the

creep buckling of a column under variable loading. The column is taken as

the idealized (Shanley) sandwich column. The constitutive model,, unlike the

commonly employed Norton creep model, incorporates a representation of both

dynamic and thermal (state) recovery. The material parameters of the con-

"stitutive model are chosen to characterize Narloy Z, a representative copper

alloy used in thrust nozzle liners of reusable rocket engines. Variable

loading histories include rapid cyclic unloading/reloading sequences and intermittent reductions of load for extended periods of time; these are superimposed on a constant load. The calculated results show that state recovery

significantly affects creep buckling under variable loading. Failure to

account for state recovery in the constitutive relations can lead to nonconservative predictions of the critical creep buckling time.

* Professor, Department of Civil Engineering, University of Akron, Akron,

Ohio, 44325 and NASA Lewis Research Center.

**" Graduate Student, Department of Civil Engineering University,

. University of Akron, Akron, Ohio 44325.
INTRODUCTION

The influence of dynamic and thermal recovery on the high temperature

behavior of structural alloys is well.recognized [1,2] (see Figs. 1 and 2).

For example, recovery is believed to play a major role in one of the central

structural problems relating to liquid metal breeder reactor design, the

probl&irrqf thermal ratchetting. Failure to account for recovery effects in

structural analyses involving repeated thermal transients has been shown to

give cjualitatively incorrect and rioncbnservative predictions of ratchetting

'in sortie instances [3]. Special provisions for taking;recovery into account

(e.g., the so called d- reset procedure) [4] are now finding their way into

documents for guiding structural analysis ,.in the U.S. nuclear industry. The

effects of thermal ratchetting have also tieen observed in thrust nozzle

liners of reusable rocket engines.

A second area in which recovery effects are thought to play a primary

role, and one that impacts essentially all high temperature system design,

is that of creep crack growth under creep/fatigue (variable stress) conditions. Creep (state) recovery is believed to be one of the main causes of

acceleration of crack growth in the creep regime and under variable loads.

A study by Kubo [5], using the same constitutive relations [2,6] employed in

the present work, has shown that the experimentally observed acceleration of

creep cracks under variable stress can be predicted through the inclusion of

appropriate mechanisms of recovery in the constitutive relations.

In this paper we discuss a third area of mechanics in which recovery is

expected to have significant influence, that is the problem of creep buckling under variable loading. The presence of mechanisms of recovery
can allow creep strain rates to increase following periods of hardening and

thus cause acceleration of creep buckling. Load reversals may be accompanied

by dynamic recovery as in Fig. 1 and load reductions by thermal recovery as

in Fig.' 2; in either case, creep rates are increased upon reloading, tending

to reduce the critical time to buckling.

In the vast majority of creep buckling studies the constitutive model

used has been of the Norton type [7-9] in which the (steady state) creep

rate is taken as a function of stress and temperature alone. Some investigations have made use of hardening theories [10-12] that include a primary

creep phase. However, these theories do not generally allow for inelastic

state recovery and, consequently, for rejuvenation of primary creep. These

constitutive theories are adequate in the case of constant loading but may

significantly over predict the time to buckling under certain types of variable loading.

Here, we examine the elevated temperature creep buckling problem under

variable loading using the simple Shanley model [13] of a column but with a

constitutive model [2,6] that embodies a representation of both dynamic and

thermal recovery. The constitutive model is that developed principally by

Robinson at Oak Ridge National Laboratory and NASA Lewis Research Center

and, as indicated earlier, was that used in the studies concerning creep

crack growth by Kubo.

The present study was motivated partly by the occurence of what appears

to be local ratchetting-buckling failures in the throat liner of the main

thrust nozzles of reusable rocket engines, notably the NASA Space Shuttle

main engine. For that reason the material parameters used in the constitutive equations are chosen to represent a copper alloy, Narloy - Z, which is

typical of materials used in rocket engine thrust nozzle liners.
We shall first state the constitutive model and, secondly, specify the

geometry of the Shanley column model. We then investigate creep buck!ing

behavior under constant loading; cyclic loading which includes relatively

rapid load reductions and reversals (dynamic recovery); and cyclic loading

which includes intermittent reductions of load for extended periods of time

(thermal recovery). Some of the calculations are repeated fqr comparison

using a roughly equivalent Norton type creep model. A discussion is given

of some physical aspects associated with the constitutive model and the results and finally, we state the conclusions drawn from the study.

THE CONSTITUTIVE THEORY

As indicated earlier we make use of the constitutive law reported in

[2,6] where a complete multiaxial statement of the model is given. Here,

we state the model in uniaxial terms together with the corresponding parameter values appropriate for the copper alloy Narloy-Z.

PToW Taw:

AF sgn(o-a) ; F > 0 and o(o-a) > 0

,p I . (1)

e =

; F < 0 or F > 0 and o(a-a)<0

Evolutionary law:

a , =•

I ,B cP - R|a| Sgna; a>a and oo>0

~— -n . m-0

0 e - R a Sgna; a<a or oa<0
.2

in which F = A^i - i . . - (3)

k'

and A = 1.60 x ID'S

n = 4

m = 8.73

S = 1.07 x ID'6 (T2) + 1.60

7k

2

 = 209.6 - 0.20 (T)

H = 1.46 x 10

 7

R = 1.06 x 10'7

a0 = 0.2

•p ,

e indicates the inelastic strain rate in h"1 and o the applied uniaxi al stress in MPa. In the general form of the theory, < (MPa) is a Bingham^

Prager threshold stress playing the role of a scalar state variable, here it

is taken to have the constant value given above. For present purposes,

a (MPa) is the single inelastic state variable. It represents the uniaxial

component of a tensorial state variable (internal stress) that appears in

the multiaxial formulation of the theory. a0 is the minimum attainable

value of |a| and plays a primary role in the representation of dynamic recovery (see Discussion) . The remaining parameters, in some cases functions

of temperature (T), are consistent with the units MPa, h and degrees Kelvin.

The elastic response is characterized by a linearly temperature dependent

Young's modulus E (MPa) given by

E = 1.47 x 105 - 70.5 (T) (5)
Equations (1-4) incorporate both dynamic and thermal recovery and therefore

are capable of predicting the type of behavior illustrated in Figs. (1) and

(2).

The parameter values for Narloy - Z were determined from uniaxial tensile and stress relaxation test data. As the complete data base normally

required to characterize a particular alloy using the present constitutive

theory was not available the tensile and relaxation data had to be considered sufficient.

Tensile data were found [14] over a wide temperature range (~30-811K)

but at only a single strain rate 0.002/s. Limited stress relaxation data

[15] were found at the temperature 811K (1000F); these were used in the

absence of creep data. The most serious data deficiency in the present

' '.-'•:<•' , \ : I '."' ' ' ' ' '

context concerns the lack of tests giving a direct measurement of recovery

effects, i.e., tests such as strain or stress transient dip tests or open

loop cyclic tests involving partial stress reversals. Nevertheless, it is

believed that the material characterization is adequate for a reasonably

quantitative study of creep buckling behavior under variable stress.

Figure 3 shows a comparison of the tensile data with predictions based

on equations (1) - (4). Figure 4 gives a similar comparison of typical relaxation responses with a prediction. Neither figure is intended as a

demonstration of the predictive capability of the constitutive model but

simply as an assessment of the correlation with existing data. The predictive capability of the constitutive theory has been adequately demonstrated

relative to other alloys in earlier publications [2,6].
THE SHANLEY COLUMN

The column model adopted here is the sandwich idealization introduced

by Shanley [13] and used by Kachanov [16] and others. The column geometry

is indicated in Fig. 5. All of the deformation is presumed to occur in the

slender bar elements 1 and 2 of length h and cross sectional area A/2. The

remainder of the column of length L (L»h) remains rigid. The width of the

column, i.e., the distance separating bars 1 and 2, is also taken as h.

The column is loaded by a time dependent load P(t) (shown positive);

the lateral displacement of the load point at any time is denoted by u(t).

The stress and total strain in bars 1 and 2 are denoted by oj, 02 and

GI = 6j/h and 62 = ^2^» respectively.

Equilibrium requires that

°

2

 + °i = 2o o

and

4o u

•2 - "1 • -H-

 (7>

where b0 = P/A.

Differentiation with respect to time provides the rate form of the

equilibrium equations,

o + o = 2o0 (8)

and

* In.' A *L (9)

From geometrical considerations (Fig. 5) the compatibility condition

is obtained as
6 -6 u - u

-^r-1 = -r-^ (10)

or-.; , .,, ,-,'••. • - . • • • '

• . u .- u

;in which, u0 is the initial displacement (imperfection). The rate form of

the compatibility relation, is .

•••' ^Decomposing the total strain rates in each bar into elastic and inelastic contributions gives

where E is the Young's modulus as specified in equation (5). The inelastic

*P "P strain rates t\ and $2

 are» °^ course, obtained by applying the inelastic

constitutive equations (1) T (4) to each bar.

Combining the equilibrium equations (8) and (9), the compatibility

equation (12), and the constitutive relationships (13) and (14) leads to
Under constant compressive stress (i.e., o0 <0 and o0 - 0), and in

* P "P the absence of inelasticity (e = e =0), equation (15) gives the Euler

critical stress for the idealized column as

°c

 = 4L

Further, calling

P = ~ (17)

and

n-.^j- (18)

where n is termed the nondimensional displacement, we have from equation (15)

Now, from equation (6)

• • •

°2

 = 2oo - °1

and from equations (12), (13), (14) and (18)

The coupled system of equations (1), (2), (3), (19), (20) and (21) together with the appropriate initial conditions, including the initial imperfection n0 = n(0), allow the (nondimensional) displacement n(t) to be calculated for a specified temperature T, Euler critical stress oc and history of

loading o0(t). The results of several such calculations for various loading

histories are presented in the following section.
RESULTS

. ,r..Al 1 of the calculated results are isothermal with the temperature taken

to be :811K (1000F). The Euler critical stress ,oc is 200 MPa and the loading/

•

unloading ramp rate |o0| is 96 MPa/sec. The governing system of equations

was integrated using a self-adapting Adams-Bashforth predictor-corrector

method with a 4th order Runge-Kutta method as a starter. The calculations

were performed in double precision on a Prime 850 computer with an upper er1 I ror bound of 10~4.

In all cases, the criterion defining the critical time to buckling tc is

taken as

n- %-l (22)

Th'is is/equivalent, under constant load, to the criterion

r °1

 =

 ° (23)

i.e., the stress in bar element 1 becoming zero.

Behavior Under Constant Load

In all calculations the compressive load is first applied from zero,

where the deforming elements are considered in a virgin state (i.e.,

a]_ = a2 - 0), to a nominal value of the applied stress o0 = P/A of -35 MPa.

Thus,. with the Euler stress oc = 200 MPa the nominal value of the ratio P

(equation 17) is . ...

r

=

10
In this section we present the calculated results for a constant load,

i.e., with p held constant at the above value. Figure 6 shows the response

H(T) under these conditions for four different values of initial (nondimensional) displacement n0 = 0.02, 0.05, 0.10, and 0.2. The time T = t/t0 in

Fig. 6 is nondimensional, being normalized with respect to the critical time

corresponding to n0 = 0.02, i.e., t0 - 6h.

It is seen in Fig. 6 that the critical time to buckling is reduced by a

factor of almost 8 with an order of magnitude increase in the initia l displacement.

For the sake of comparison, all subsequent calculations are taken to

have the initia l imperfection n0 = 0.02 and are presented in terms of the

nondimensional time T = t/t0.

i: Behavior Under Variable Load

We first consider the effect of rapid load reductions and reversals

superimposed on the constant load o0 = -35 MPa (p = -0.175). As shown in the

"•••••' inserts of Fig. 7, load interuptions occur at time intervals of TO = 0.15

with varying amplitude. These include reductions (in the tensile direction)

of o0 to -20 MPa and to 0 MPa (insert a), to +20 MPa (insert b), and a complete reversal to +35 MPa (insert c). The calculated effects of these histories on the creep buckling response n(f) are shown in the respective curves

a, b, and c of Fig. 7.

Curve a corresponds to the loading histories indicated in insert a, i.e.

with reductions to -20 MPa and 0 MPa. The response curve for each is identical

to that of Fig. 6 for the same initial displacement 0.02, indicating that no

change in the buckling response has occurred. Response curve b corresponds

to insert b in which there is a load reversal to OQ = +20 MPa. Here, we begin

to see a measurable change in the calculated creep buckling response, the

11
critical time having diminished from the constant load case by about 10%.

Finally, in curve c, we observe a reduction of more than 30% in the time to

buckling. This corresponds to the history of insert c where a complete load

reversal from o0 = -35 MPa to OQ = +35 MPa occurs at each load interruption.

Immediately following each reversal, we see evidence of the reappearance of

primary creep. This is attributed to the presence of dynamic recovery (Fig.

1) where creep (or relaxation) is observed to be accelerated with stress reversals - even in the absence of significant reversed inelastic strain. Although, to the knowledge of the authors, these effects have not been observed

directly in creep buckling phenomena, it is expected that such effects can

occur, on the basis of the experimental observations illustrated schematically .in Fig. 1.

The influence of dynamic recovery is best understood by considering the

•*

state space (o,a) of Fig. 8. Note that in Fig. 8 and all subsequent representations of the state space, compressive o and a are shown as positive and

are plotted upward and to the right, respectively. Hereafter, the relevant

quadrants of the state space will be referred to as the first•(oo>o) and the

fourth (oa<0). In Fig. 8 the trajectory of the state point (03, 02) ^or

bar element 2 is shown corresponding to the constant load response curve a

in Fig. 6. The segment OA traces the path of the state point during initial

load-up to o0 = -35 MPa. Some inelasticity is indicated over path OA by the

.increase in the inelastic state variable 02- As °o ^s ^e^ constant, bar 2

•creeps under nearly constant stress and the state point moves toward B. As

the .geometric nonlinearity becomes prevalent the stress in bar 2 increases

.(as: that in bar 1 decreases) and the state point moves toward C. Point C

corresponds to the buckled condition n = 1 in Fig. 6 (curve a).

12
Similarly, Fig. 9 shows the trajectory of the state point for the loading; histories illustrated in insert a of Fig. 7, i.e., involving load reductions to o0 = -20 MPa and to o0 = 0. Here, we see the effect of the abrupt

load changes as vertical (elastic) trajectories in the state space, resulting

in no overall change in the state path OABC from that just considered for a

constant load. This, of course, results in the same creep buckling response

observed earlier for the constant load.

The state path of Fig. 10 relates to the loading history of insert b in

Fig.. 7. Here, we begin to see evidence of state recovery. The load variations now produce stress reversals in bar 2 and the state point trajectories

are not just vertical (elastic) lines as before, but now follow curved paths

as the state point penetrates into the fourth quadrant (oo<o) corresponding

to a reversal of stress. The state recovers with the stress reversal, resulting in a relatively softer state (smaller 03) upon reloading. Corresppndingly, the creep rate is increased in response to each load cycle. Evidence of increased creep rate following reloading is apparent in curve b(Fig. 7)

Load histories involving larger stress reversals produce increased state

recovery. Figure 11 shows the state path in bar 2 for the fully reversed

load depicted in insert c (Fig. 7). Here, the effect of dynamic recovery at

each unloading/reloading cycle returns the state point well back into the

primary creep regime. As observed earlier, clear evidence of the rejuvenat.ion of primary creep after each loading cycle is seen in curve c, with the

:tot,al effect of diminishing the critical buckling time by about 30%.

As the period of application of the rapid loading cycles (TO = 0.15)

.was chosen quite arbitrarily, we now investigate the effect of the frequency

of load cycles. Figure 12 shows the buckling response curves H(T) for load

13
"" histories involving a full reversal, i.e., o0 = -35 MPa to o0 = +35 MPa (as

in insert c Fig. 7), with periods TO, T0/2, t0/4 and T0/8 (the constant load

response curve is also shown for reference). We see the pronounced effect of

more frequent (shorter period) stress reversals. Reversals with period t0/8

reduce the time to buckling by more than a factor of 3.

Next, we examine the influence of load reductions of extended duration

on the creep buckling time. The loading histories considered are shown in

the inserts of Fig. 13. Case a (i.e., insert a and response curve a)

\ ..' ' ', >

amounts to one of those considered previously (insert a of Fig. 7) where

the time duration at the reduced load o0 = 0 is effectively zero. This history, as noted earlier, produces no change in the buckling response over

the constant load case.

Insert b depicts the history where the load, having been applied for

period' TO, is abruptly removed (at a rate |o0| = 96 MPa/sec) and held at

zero for a comparable period TO. The sequence is then repeated. This history reduces the critical time as shown in curve b, by about 15%. Here, the

time T includes only that time in which the load o0 = -35 MPa is applied.

As before, this behavior is best visualized in the state space (02, <*2)»

Fig. 14. State recovery is observed as, in time, the state point moves at

zero stress toward smaller 02, e.g., on the first load reduction, from point

D to E. Reloading returns the state point to a softer state than before the

load reduction and correspondingly to a higher creep rate. Repetition of

the sequence thus causes acceleration of the creep buckling process.

Insert c of Fig. 13 shows a loading history in which the time at zero

stress is now increased a hundredfold to 100t0. The corresponding response

curve c shows a further decrease in the critical time, here about 30/t. The

related state point trajectory shown in Fig. 15 is qualitatively similar

14
to that of Fig. 14 but now shows significantly increased recovery with the

hundredfold increase in hold time at the reduced stress.

Comparison with Predictions by Norton Law

In this section we compare the results presented above with those for

identical loading histories based on a classical creep law of the NortonBailey type, i.e.,

e

P

 = B ON sgn (o) (25)

A roughly equivalent representation was obtained by calculating the steady

state creep rates for various stress levels using equations (l)-(4) and

"fitting" equation (25) to these "data" by choosing optimal values of B and

N i,n a least squares sense. This process yielded the following

B = 2.03 x 10-19

N = 9.375

•p ,

which are consistent with units of a in MPa and e in h"1. Using equation

(25), the critical time to buckling corresponding to the constant load case

of curve a Fig. 6, turned out to be t1 = 7.8 h, slightly greater than the

o

earlier reference t0 = 6 h. Thus, in the calculated results presented here

(i.e., Fig. 16), the nondimensional time T is obtained by normalization with

respect to t1. The time plotted is that for which the load o0 = -35 MPa is

o

applied. On this basis, calculations of creep buckling response n(t) for

all of the loading histories presented earlier have been included in Fig.

16. As expected, the time to buckling is completely unaffected by any of

the variable loading histories. This is because the classical Norton-Bailey

respresentation (equation (25)) fails to account for state recovery in any

form.

15
DISCUSSION

Dynamic recovery, as illustrated in Fig. 1 and by the state point

trajectories of Figs. 10 and 11, is characterized in the present constitutive model through the dual analytical forms of the evolutionary law, equations (2). This description is consistent with the viewpoint of Onat [17] in

representing inelastic behavior through the specification of analytically

different mathematical forms corresponding to various regions of the state

space/ Here, different analytical forms are specified depending on whether

the state point lies in the first (third) quadrant oo>o or the fourth

(second) quadrant oa<o of the state space. In effect, this permits the

state point to rapidly recover upon reversing the stress, even in the absence of significant reversed inelastic strain (Fig. 1). This formulation

is intended as an idealization of a physical process whereby the dislocation

structure (or the associated structure of internal stress) is abruptly altered with a reversal of the applied stress, because previously immobilized

dislocations are remobilized on their slip planes [18]. As the inelastic

state variable a is taken as an averaged, phenomenological measure of the

dislocation microstructure (or its associated internal stress state), it too

should reflect a rapid change as the stress is reversed.

The specific functional form of the first of equations (2) is made in

accordance with the experimental results of Mitra and McLean [19]. The second equation (2) can be considered an analytical continuation of the first,

evaluated at a0 (small a) in the first quadrant of the state space, into the

second quadrant; or with symmetrical response in tension and compression, at

-aQ in the third quadrant into the fourth quadrant. This amounts to a

highly idealized representation of the underlying physical process but,

16
.nevertheless, captures the essential feature that abrupt microstructural

'•' .'

rearrangements occur with stress reversals.

The path of the state point in the fourth quadrant oo<o in Figs. 11 and

12, and thus the extent of recovery, depends on the total strain rate and,

critically, on the parameter a0. Ideally, a0, or more comprehensively a function of o and o replacing it, should be determined from experiments in which

the recovery of creep rate or flow stress under stress reversals is measured

directly and not inferred indirectly from available monotonic tensile, creep

and/or relaxation data as was done here. Nevertheless, the present representation is believed to be adequate and consistent with the objective of demonstrating the strong influence of state recovery in creep buckling. A more

comprehensive description of dynamic recovery, in the same spirit as that described, and the relevant experimentation are topics of continuing research.

Thermal recovery, as depicted in Fig. 2 and in the state paths shown in

Figs. ,14 and 15, is manifest in the second (negative) term of the BaileyOrow.an evolutionary equations (2). In the applications considered here the

, state point recovers in time under constant (zero) reduced stress, giving rise

to an increased creep rate on reapplication of stress (Fig. 2). Physically,

, this macroscopic behavior is associated with thermally activated, diffusion

controlled microscopic processes such as climb of edge dislocations, allowing

dislocations, in time, to bypass immobilizing obstacles, thus producing a

softer state (smaller a). The important material parameters in equations (2)

are R, m and B. Ideally, these parameters are determined from both creep data

and information obtained from stress or strain transient dip tests [6] that

provide a direct measurement of thermal recovery. Again, as these data were

not readily available for the alloy Narloy-Z, the pertinent parameters were

inferred indirectly from available data. As discussed earlier, this approach

17
although not optimal, is considered consistent with the present objectives.

CONCLUSIONS

We have examined the creep buckling response of an idealized (Shanley)

column under some special variable loadings. The two types of loading considered amount to superpositions of the following load sequences on a constant applied load:

1) rapid cyclic unloading/reloading sequences involving stress reversals, and

2) cyclic loading that includes intermittent reductions of load for

extended periods of time (at temperature).

Although the sandwich column model used is highly idealized, the constitutive model is quite comprehensive, in as much as it incorporates a representation of dynamic and thermal (state) recovery. There is substantial

experimental evidence that many structural alloys embody internal mechanisms

at elevated temperature that allow inelastic strain rates to increase (recover) following periods of hardening. In particular, this is believed to be

true for the representative copper alloy Narloy-Z characterized here.

The loading sequences examined are not intended to represent prototypical loading histories for any particular structural component; instead, they

were chosen to best illustrate the generic influence of both dynamic and

thermal recovery on structural behavior in the presence of a creep induced

instability. It is expected that qualitatively similar behavior will accompany more realistic variable loading conditions, (and more realistic structures), and that neglect of state recovery effects in such cases will lead

similarly to nonconservative predictions of the critical time to creep

buckling.

18
We now list the conclusions drawn from this study:

• State recovery (dynamic and thermal) can have a significant effect

on creep buckling behavior, i.e., on the critical time.

• Failure to account for state recovery in the constitutive equations

can lead to nonconservative predictions of the critical buckling

time under variable loading.

• A classical Norton-Bailey type creep law as commonly used in creep

^buckling analyses, does not account for state recovery and therefore

may significantly over predict the time to creep buckling under

variable loading.

•••'. It is important that constitutive models which are to be used in

creep buckling analyses involving variable loading allow for recovery

• , ' i • . " •

effects, and furthermore, that the characterization tests used for

determining the pertinent material parameters include direct measure-

•""'•• ments of- state recovery.

19
REFERENCES

1 Pugh, C.E. and Robinson, D.N., "Modified Auxiliary Procedures for

Cyclic Creep and Relaxation," ORNL-5281, 1976, pp. 17-40.

2 Robinson, D.N., et.al, "Constitutive Equations for Describing HighTemperature Inelastic Behavior of Structural Alloys," Proceedings o1

Specialist Meeting on High-Temperature Structural Design Technology of

3 jSartory, W.K., "Analysis of the ORNL Two-Bar Thermal Ratchetting Test

SBRI," ORNL-5683. 1980, pp. 9-13.

4 Robinson, D.N., "Additions and Modifications to Constitutive Equations

in RDT Standard F9-5T," ORNL-5622, 1979, pp. 2-11.

5 Kubo, S., "Effects of Creep Recovery and Hardening on the Stress and

Strain-Rate Fields Near a Crack Tip in Creeping Materials," Elastic-

: Plastic Fracture: Second Symposium, Vol.* I - Inelastic Crack Analysis,

A5TM STP 803, ASTM, 1983, pp. 1-594, 1-614.

6 Robinson, D.N., and Swindeman, R.W., "Unified Creep/Plasticity Constitutive Equations for 2 1/4 Cr-1 Mo Steel at Elevated Temperature,"

ORNL TM-8444, 1982.

7 Hoff, N.J., "A Survey of the Theories of Creep Buckling," Proc. of the

;•- Third U.S. Nat. Congr. of Applied Mechanics. ASME, Vol. 3, 1958, pp.

29-49.

8 Hayman, B., "Some Observations on Creep Buckling Analysis," 2nd International Conference on Structural Mechanics in Reactor Technology.

Vol. 5, Part. L. Berlin, Germany, 1973, pp. 1-12.

9 Hoff, N.J., "Rules and Methods of Stress and Stability Calculations in

the Presence of Creep," Jour, of Applied Mechanics, Vol. 45, 1978,

pp. 669-675.

10 Libove, C., "Creep Buckling of Columns," Jour. Aeronautical Sciences,

Vol. 19, No. 7, 1952, pp. 459-467.

11 Jahsman, W.E., "Creep Stability of a Column with Coupled Geometric Imperfection and Material Behavior Effects," Creep In Structures, Ed. 0.

Hult, IUTAM Symposium, Gothenburg, Springer-Verlag, 1970, pp. 360-369.

12 Hoff, N.J., and Levi, I.M., "Short Cuts in Creep Buckling Analysis,"

Int. J. Solids Structures. Vol. 8, 1972, pp. 1103-1114.

13 Shanley, F.R., Weight-Strength Analysis of Aircraft Structures, Dover,

NY, 1960.

20
14 Esposito, J.J., and Zabora, R.F., "Thrust Chamber Life Prediction, Vol.

I - Mechanical and Physical Properties of High Performance Rocket Nozzle Materials," NASA CR-134806. 1975.

15 Conway, J.B., Stentz, R.H., and Berling, J.T., "High-Temperature, LowCycle Fatigue of Copper-Base Alloys for Rocket Nozzles: Part I - Data

Summary for Materials Tested, NASA CR-134908, 1975.

16 Kachanov, L.M., Creep Theory, Moscow, 1960.

17 Onat, E.T., "Representation of Inelastic Behavior," Creep and Fracture

of Engineering Materials and Structures, Ed. B. Wilshire and D.R.J.

Owen, Proc. of International Conf. held at Swansea, Pineridge Press,

Swansea, U.K., 1981.

18 Orowan, E., "Causes and Effects of Internal Stresses," Internal

Stresses and Fatigue In Metals, General Motors Symposium, Elsevier,

Amsterdam, 1959, pp. 59-80.

19 Mitra, S.K., and McLean, D., "Work Hardening and Recovery in Creep,"

Proc. Roy. Soc. (London) 295(a), 1961, pp. 288-299.

21
Fig. 1 Schematic representation of dynamic recovery. In the

absence of stress reversals, the stress relaxations

over a fixed time ab,cd and ef show successive

hardening. After the stress reversal fgh, stress

relaxation hj shows evidence of (creep) softening, i.e.

evidence of recovery of state (see [I]).

22
*»:•• "

AT

Pig. 2 -Schematic representation of thermal recovery. Response

in an interupted creep test typically exhibits

relatively sr.iall strain recovery £r but measurable

softening ( depending on the interval AT ) following a

stress reduction, i.e.recovery of state (see [2]).

23
250

200

150

•«

M

100

50

O O.2 O.4 O.6

strain (%)

250 -

200 -

i

150 -

100 -

0.2 0.4 0.6

strain (%)

0.8

Fig. 3 Comparison of a) tensile data [14] and, b) predictions

for Narloy Z at various temperatures. Strain rate is .002/s.

24
C TO

o o

•o

O) C

i. O •

O.t- TO

end.

TO <D z:

•o o

C -D <M

TO 0) t-t

JZ

i~i o vi

in 4-> t-

«-H re

C/)

«B i/» o»

*J V) t-

«» O «-»

•o «- i/>

C O)

o • c

TO O I.

X O TO

TO O 4->

E

i— i—• *

QJ •-_ -

— II/I ft IS>

at oo ai

«o

«»- INI

O

<D

C O

o — «*-

O U O

•»- TO

t- Z OJ

TO Ol

25
u(t)

S / / / '// ///

b)

Fig. 5 Geometry of the Shanley sandwich column,

26
1.00

0.80 -

;. BO -

0.40 -

0.20 -

0.00 -

0.00 0.20 O.U O 0.80 0.80 1.00

Fig. 6 Nondimensional displacement versus time for initial

imperfection s of H = 0.02 (a), 0.05 (b)., 0.1 (c) and

0.2 (d).

 v

27
1.00

O.B O 1.00

Fig. 7 Nondimensional displacement versus time Cor loading

histories depicted in inserts a),b) and c).

28
ooU1r*

CO±>oCoo

oCM

CU OJto ' 4J <O

«§!

& <S>

C -fJ

CSJ

to

oa

- a

^ ..

.?. « a> o ^ "* w o

oo

CO

CO

oo

aooo

oo

oo

oao

<\JI

oo

ooo«I

29
'OCo

oo

- ui

COr"jj

<fl oo oJCO CM u» 0 -H 1

0

2"

H

to «M

CM

CO

OO

- a0a00

oo

iaa

i0orw

CM

C

o

(\>

oo

oa

OJJ

a*n c -u o CO JJ

CTi

30
oo

Ioo

Ioa

IOoa*rxi

Oaui cvOCMu»Oa

- oooUJ

coa, • tn n o °- u •£ O 4J a, a,

4J-. a3 0 O 0

CM « £

O a •*

(TJ CJO C3

1

fO C w T3 (0

oo

oa*

•HDu

C\J o

31
ooOCM2Oo

- oooin

oo

oo

Iao

oCM

OO

IOo1ooo"^IIooa' toI

CO2LWO -

P

0

.

£=>

QJ CT

1

4J r

1

(0 ^

jjw -u &c •*-•

-H W 5 Q) o w

"w ••-*

a>0 CT> (0 C

a,-^

w *o(0

01 O

JJ ,-<

«3

4J O

32
>_i t

; *

•~

 ca

, o

C 0) W

•H c; "o

S o 0 —" 1 . *-

>

co n <u

i—I —U

o IB

<= W TJ

•H u C

1^2

s"

E

> o ^

Ul

(y *^» **•* *

C"0° O p* ^^ ^ ^^ ^ ° aJ L

?

W G

1 c

•^ Q US 'O u -u *

2

l^, CO C

^1 C fl

(0 O O C ,*^ ^ O J-> ^

"

W 14-1 O O

og•S o c; c

s 'a o c u-t O ^-J 2 0) OW * 01 •H

CL, 33
e

n3wUV0 >/) QJ -H

• H 0)

-O 4Ju

^H -H

ra a c o o 'o •HO W C 0 O -H £ 1-1 •H O tJ Jj c w O -H

34
- I/Io- oo°. - 10 O OQoa

IS)

oaoo

Iooo

Iooo

ao

w01uucCJ JJa(0wIO

! 4J

Cb

inc

(3

O4J

CM

D3

5
OaaCM

CM

Oaaovn

oo

Ioo

Toaioo

OO

COainoOu

It)

0) x)

tf 4JCO3 * O 4J CM ro o &« (0 »n <y c XI -H (3aj o

C/5 JJ

Oo

CM

D

36
o

 u

•2 g o o ao c U i-l o o > -po S (3 O o o -^ « 10 — i to Cu fl CO rH -I C«3 -H C 0 O 3

•-«

(0 CO

C 0)

•H O

t3 4J

C 01

o -^

z s:

ooo

37
1. Report No.

NASA CR-175094

2. Government Accession No.

4. Title and Subtitle

Effects of State Recovery on Creep Buckling Under

Variable Loading

3. Recipient's Catalog No.

5. Report Date

April 1986

6. Performing Organization Code

7. Author(s)

D.N. Robinson and S.H. Arnold

8. Performing Organization Report No.

None

10. Work Unit No.

9. Performing Organization Name and Address

University of Akron

Department of C1v1l Engineering

Akron, Ohio 44135

11. Contract or Grant No.

NAG 3-379

12. Sponsoring Agency Name and Address

National Aeronautics and Space Administration

Washington, D.C. 20546

13. Type of Report and Period Covered

Contractor Report

14. Sponsoring Agency Code

505-60-12

15. Supplementary Notes

Final report. Project Manager, Daniel J. Gauntner, Structures Division, NASA

Lewis Research Center, Cleveland, Ohio 44135. D.N. Robinson, University of

Akron and NASA Lewis Research Center; S.M. Arnold, University of Akron.

16. Abstract

 ::

Structural alloys embody Internal mechanisms that allow recovery of state with

varying stress and elevated temperature, I.e., they can return to a softer state

-following periods of hardening. Such material behavior 1s known to strongly

Influence structural response under some Important thermomechanlcal loadings, for

example that Involving thermal ratchettlng. Here, we Investigate the Influence

of dynamic .and thermal recovery on the creep buckling of a column under variable

loading. The column 1s taken as the Idealized (Shanley) sandwich column. The

constitutive model, unlike the commonly employed Norton creep model, Incorporates

-a representation of both dynamic and thermal (state) recovery. The material

parameters of; the constitutive model are chosen to characterize Narloy Z, a representative copper alloy used 1n thrust nozzle liners of reusable rocket engines.

Variable loading histories Include rapid cyclic unloading/reloading sequences and

Intermittent reductions of load for extended periods of time; these are superimposed on a constant load. The calculated results show that state recovery

significantly affects creep buckling under variable loading. Failure to account

for state recovery in the constitutive relations can lead to nonconservatlve

predictions.of the critical creep buckling time.

7. Key Words (Suggested by Author(s))

Creep buckling; Recovery; V1scoplastlcity; Variable loading;

High temperature

18. Distribution Statement

Unclassified - unlimited

STAR Category 39

9. Security Classif. (of this report)

Unclassified

20. Security Classif. (of this page)

Unclassified

21. No. of pages

38

22. Price"

A03

*For sale by the National Technical Information Service, Springfield, Virginia 22161
National Aeronautics and

Space Administration

Lewis Research Center

Cleveland, Ohio 44135

Official Business

Penalty for Private Use $300

SECOND CLASS MAIL

ADDRESS CORRECTION REQUESTED

Postage and Fees Paid

National Aeronautics and

Space Administration

NASA-451

NASA