- Email: [email protected]
Creep, relaxation and cyclic behavior of a beam using a state-variable constitutive model
Nuclear Engineering and Design 65 (1981) 411-421 North-Holland Publishing Company
411
CREEP, R E L A X A T I O N A N D CYCLIC B E H A V I O R OF A B E A M U S I N G A S T A T E - V A f i I A B L E CONSTITUTIVE MODEL
T.J. D E L P H Department of Mechanical Engineering and Mechanics, Lehigh University, Bethlehem, PA 18015, USA
Received 15 February 1981
Numerical solutions are presented for the time-dependent inelastic behavior of a Bernoulli-Euler beam under a variety of time-dependent moment loadings. The inelastic constitutive representation employed is the recently developed state-variable constitutive law of Robinson. Robinson's equations have several novel features, and these are examined within the context of the behavior of a beam. Despite the fact that the beam may contain relatively complex inhomogeneous stress fields, the overall behavior is shown to be analogous to that observed in simple uniaxial tests.
1. Introduction One of the most basic problems in describing the high temperature behavior of metals is the formulation of an appropriate set of constitutive equations. This problem becomes particularily acute when it becomes necessary to deal with loading histories involving periods where the loads are changing fairly rapidly interspaced with periods during which the loads are relatively constant. Unfortunately, such loading histories are commonly encountered in the operation of high temperature pressure vessel and piping systems.At present, the most widely used constitutive representation [l] makes use of the time-independent theory of classical plasticity to describe short-term deformation, while relying upon time-dependent classical theories of creep to describe long term deformation. Thus the inelastic strain is taken as the sum of a time-independent component and a time-dependent component. Although this approach works quite well for a number of important loading histories, there are others for which it is notably unsuccessful. Recently, however, a number of new constitutive theories have been introduced which reject this conventional representation and instead treat the inelastic strain as being totally time-dependent. Some representative examples are the work of Ponter and Leckie [2], Hart [3], and Miller [4], with a more comprehensive bibliography being given by Krieg [5]. Although these theories may differ considerably among themselves, they often make use of one or more variables of state and are thus 0029-5493/81/0000-0000/$02.50
sometimes known as 'state variable' theories, or else as 'unified' theories from their ability to model at least in a qualitative sense both short-term plastic and long-term creep behavior. Typically, the state-variable theories seem to offer significant advantages over the conventional representation, in that they are able to model at least qualitatively, such phenomena as thermal recovery, creepplasticity interactions, and cyclic hardening. These phenomena tend to be handled rather awkwardly, if considered at all, in the conventional constitutive representation through the use of various ad hoc rules [1]. The principal disadvantages of the new theories are that most are not yet fully tested or developed, and that they seem to be more difficult to incorporate into large finite-element structural analysis codes than the usual constitutive laws. In particular, for most of these theories, precise methods for determining required material constants are not available and recourse is often had to trial-and-error fitting of experimental data. Furthermore the governing equations may often predict extremely high rates or exhibit mathematically 'stiff' behavior and require considerable care in their time integration. Those problems are discussed in somewhat more detail by Krieg [5]. In the present work, we consider a state-variable constitutive theory which has recently been proposed by Robinson [6] in conjunction with developmental work on the Liquid Metal Fast Breeder Reactor (LMFBR) program. This work was prompted by the inability of the standard constitutive representation to describe
© 1981 N o r t h - H o l l a n d
412
T.J. Delph / Creep, relaxation and cyclic behavior of a beam
various aspects of the complex deformation behavior of the ferretic stainless steel 2J Cr-1Mo. Robinson's theory is based to some extent on the earlier work of Lagneborg [7] and Rice [8], but is particularly notable for the use of the state space concepts set forth by Onat [9]. These concepts have led to the development of a number of auxiliary rules which modify the form of the governing equations depending on the location in state space of any particular material point. These auxiliary rules appear to be unique to Robinson's model, and show promise of allowing in certain cases a more accurate representation of actual material behavior than would be the case if the unmodified governing equations applied equally to all regions of the state space. The intent of the present work is to investigate the inelastic behavior of a simple Bernoulli-Euler beam using Robinson's constitutive model. Aside from the practical importance of the beam as a structural member, this problem is of interest in that the beam, though essentially one-dimensional in the Bernoulli-Euler formulation, may contain quite complicated inhomogeneous states of stress. Hence its behavior under various time-dependent loadings may serve to give some indication of the behavior of more complex members. Additionally, the one-dimensional nature of the problem makes it amenable to relatively simple numerical techniques, without the need to resort to cumbersome and expensive finite-element methods. We note that a similar problem has been treated by Chang et al. [10] using Hart's constitutive model [3], though considering loading histories somewhat less complicated than those which will be analyzed herein.
2. Analysis In this section we give the equations governing the time-dependent deformation of a Bernoulli-Euler beam under the influence of a time-dependent bending moment. We then specialize these equations for three particular loading histories of interest. These are deformation under a constant (or piecewise-constant) applied moment, deformation at a constant rate of curvature, and relaxation of moment for a constant fixed value of curvature. If, as is often done in plate and shell theory, the applied moment M(t) is considered to represent a generalized stress and the curvature 3'(0 a generalized strain, then these three loading histories correspond respectively to generalizations of the creep, constant strain rate extension, and relaxation tests common in uniaxial materials testing. Fig. 1 shows the geometry of the beam which we
M(t)
h
h
M(t)
" --7-
Fig. 1. Beam geometry. assume to be of rectangular cross-section. According to Bernoulli-Euler beam theory, the total longitudinal strain in the beam is given by CT(t ) = y ' / ( t ) ,
(1)
where y is measured from the neutral axis (fig. 1) and )'(t) is the time-dependent curvature. We assume that the total longitudinal strain can be expressed as the sum of an elastic and an inelastic strain component, c T = c E + q,
(2)
where the elastic strain component is given by Hooke's Law ~E = o / e .
(3)
We further assume that the inelastic strain component is governed by a constitutive equation having the general form
~I = f ( o , a ) ,
(4)
where the superposed dot indicates differentiation (total or partial) with respect to time. Here a is a variable of state governed by a =g(o,a)
(5)
As pointed out in the Introduction, the inelastic strain e I is taken to be totally time-dependent and is assumed to contain, or model, both short-term 'plastic' strain and long-term 'creep' strain. Assuming symmetry about the neutral axis, we have from equilibrium considerations that the resultant moment is
M( t) = 2b fohyO(y,t ) d y.
(6)
Also from eqs. (1), (2), (3), and (4),
(~ = E[ y~ -- f( a,a ) ].
(7)
We now wish to consider some particular loading histoties.
2.1. Constant applied moment Assume that a constant bending moment of magnitude M 0 is applied to the beam at t--= 0. Differentiating
T.J. Delph / Creep, relaxation and cyclic behavior of a beam eq. (6) with respect to time and making use of eq. (7), we have h
2
fo' [ y "~-- yf( o,a)] d y = 0 .
(8)
Solving for the curvature rate,
~,( l) = (3 /h3) fohyf( o,a) d y.
(9)
413
2.2. Constant curvature rate We now assume that a time-varying moment M(t) is applied to the beam in such a manner that the beam curvature varies at a constant rate, %. Eq. (7) thus becomes d = E[y'~ o - - / ( o , a ) ] .
(15)
Differentiating eq. (6) with respect to time and using eq. (15), we obtain
Eqs. (9) and (7) then give
O(y,t)=E[(3y/h3)f;yf(o,a)dy--f(o,a)].
(10)
)(,I( l ) = EI~[o - 2eb fohyf( o,a ) d y.
(16)
Eqs. (5), (9) and (10) now constitute a set of three coupled non-linear equations whose solution yields the deformation history of the beam. The instantaneous response of the beam is presumed to be elastic, so that the appropriate initial conditions are given by the elastic solution
We nondimensionalize as follows. Let "/m be some representative value of the curvature (in the subsequent section it will be taken as the maximum curvature reached in a cyclic moment-curvature simulation.) Then define y ' = y/"}tm,
M ' = M/ElYm,
o(y,O) = M o y / I
a' = a/EhYm
e' I = e i / h ' g m .
y(O) = M o / e I ,
(11)
where I is the area moment of inertia of the rectangular cross-section, E is Young's modulus, and M 0 = M(0). Furthermore we take a ( y , 0 ) = 80, where 80 is the value of the state variable a existing in a 'virgin' material, that is, a material which has undergone no prior inelastic straining. Since the state variable a is in a sense a quantitative measure of the prior inelastic history of the beam, its initial value for a virgin material is typically zero or quite small. To place these three equations in nondimensional form, we define
)7=y/h,
~I = £1// ( noh// EI ),
7 = y / ( M o / E I ),
0=
o / ( M o h / I ),
S = a~ ( M o h / I ) . Thus we obtain ~=g(6,S),
03)
subject to the initial conditions
Here
we have
g/(Moh/I).
,7(0) = l,
Eqs. (5), (15), and (16) then become
a' = g ' ( o ' , ~ ' ) , 0' =37"/,6 - - f ' ( o ' , a ' ) ,
(18)
Yo' -- 3 f01fif' d )7, h,/' -- "" subject to the initial conditions a'()7,0) = 86,
0'()7,0) = 0,
M'(0) = 0.
(19)
loaded at a constant rate of curvature until some value of curvature y = Ym has been reached. The curvature is then held constant at this value and the applied bending moment allowed to relax. The governing equations for this condition may be derived from eqs. (18) by simply setting ?0 = 0. The appropriate initial conditions are the values a', o', and M' at the end of the constant curvature rate loading phase.
2.3. Constitutive equations
o = 3)7£ )Tf(o,a ) d ) 7 - f ( 6 , ~ ) ,
S ( y , 0 ) = go,
(17)
Here )7 is as defined by eq. (12) and f' =f/h'rm, g' = g/EhYm. Here we assume that the beam has been (12)
~ = 3fol)Tf(O,S ) d)7,
o' = o/EhYm,
~ ()7,0) =)7.
taken f = f / ( M o h / E I
(14)
) and g =
The equations developed in the preceding section are valid for any constitutive law having the general form of eqs. (4) and (5). We now wish to specialize to the particular set of constitutive relations proposed by Robinson [6]. These relations are formulated in terms of an arbitrary multiaxial stress state, but in the present
T.J. Delph / Creep, relaxation and cyclic behavior of a beam
414
case, when the only longitudinal stress component is non-zero, they reduce to the following form. F O'- ')/2(20/3 -- a ) ; F>0, 2/,~x---- 0,
F~0
o ( 2 o / 3 - a) > 0 , or
(F~O,o(2o/3--a)~O);
S ( o / o s ) F ( ~ - ' ) / 2 ( 2 0 / 3 -- a), F~> O,
oa ~ O,
o ~ os; (20)
21~Hq/( 3Vr3741a/KI)~ -R(
3v~Ia/KI)m-P-'a,
~=
Ot > OtO,
era ~>0;
(21)
2gH4,/( 3¢~7~la0/KI) a _ R ( 3¢-37~lao/K])m B a,
a ~OtO,
oa<0; where F= 3(20/3
- ~)2/(4K2)
- 1.
(22)
The quantities ~,H,R,n,fl,m,K, and s o in these equations are parameters, possibly functions of temperature, which characterize a given material. Here o is the longitudinal component of the stress tensor and a the corresponding component of the state variable tensor. The equation given in the third of eq. (20) is not present in Robinson's original formulation [6]. It is introduced here as a minor modification in order to avoid numerical difficulties associated with the (idealized) discontinuity in inelastic strain rate present in the original model when crossing the line o = 0 in the o-a plane. In the present work we have taken S(o/o~) = 3( o / o s)2 _ 2( o / o s)3
(23)
and %=K. We note that S ( 0 ) = S ( 1 ) = 1, and that d S / d o vanishes both at o - - 0 and o--%. Hence this modification simply allows the inelastic strain rate to increase in a smooth fashion from zero to some 'full-scale' value given by the first of (20). Typically this increase occurs over a small stress range ( K is on the order of 5-7 MPa), so that the predictions of the original equations are essentially unaltered, but the numerical difficulties occasioned by the sharp discontinuity in strain rate present in the original version are removed.
3. Numerical results
In this section we will present some numerical results for the response of a Bernoulli-Euler beam governed by Robinson's constitutive model to piecewise-constant bending moments, to a cyclic moment-curvature history, and to a relaxing moment with constant fixed curvature. The equations governing these various situations have been derived in the previous section. The numerical method used for their solution was the well-known Adams-Bashforth-Moulton algorithm, a fourth-order accurate predictor-corrector scheme for the solution of first-order differential equations. The spatial integrals involved in the governing equations were evaluated by means of ten-point Gaussian quadrature. This technique effectively converted the governing equations into a system of twenty-one coupled non-linear ordinary differential equations, two each for the stress o and the state variable a at each of the ten Gauss points plus an additional equation for either the time-varying moment or the curvature. Since 80' the initial value of a, is typically quite small, eq. (21) predicts quite high values for ti in the neighborhood of t = 0 and also in the second and fourth quadrants of the o-~ plane, where the second of eq. (21) holds. To maintain sufficient accuracy in the numerical calculations, it was found necessary to use a time step size on the order of 10 5_ 10 8 hours in these regions. These extremely small time steps in turn necessitated the use of an automatic time-step control scheme, wherein the time step size was halved when an appropriately defined relative error measure exceeded some upper limit and doubled whenever the error measure became sufficiently small. The error measure used was (relative error)= [(yc -Yp)/Yc[ where yp was the predicted value of the dependent variable at a particular time step and Yc the corrected value. The error measure was calculated for each of the twenty-one equations at each time step and the step size halved when the error in any equation exceeded 1 × 10-4. When the error in any six equations was less than 1 × 10 7, the step size was doubled. This scheme turned out to be extremely successful. In the worst case with an initial step size of 10 -8 hours it was found possible to integrate the governing equations for the beam for times out to several thousand hours in less than 20 seconds of CPU time on a CDC 6400 computer. The material constants used in the constitutive law (eqs. (20) and (21)) were chosen to typify the uniaxial behavior of a particular heat of the ferretic stainless steel 2¼Cr-lMo at a temperature of 566°C (1050°F). At the moment, no set procedure exists for the de-
415
T.J. Delph / Creep, relaxation and cyclic behavior of a beam
termination of these constants other than trial-and-error fits to experimental data. Such fits were carried out by Robinson [11] to data taken from constant strain rate extension and creep tests. The resulting values are/x = 1.06)< 108 MPa-hr, R = 2 . 2 5 ) < 10 -5 hr i, H = 3 . 0 ) < 10 - 4 hr -1, K = 6 . 9 0 MPa, N = 6 . 2 5 , M = 4 . 5 , /3= 1, and a 0 = 7.96 × 10 -3 MPa. In practice the solutions were found to be insensitive to the initial value of a, 80, so long as 80 was sufficiently small. Hence there is some allowable arbitrariness in the selection of 80, and in the present work we follow Robinson [11] in taking 80 = a 0. It is of course also possible to select 80 = 0 (in which case the second of eq. (21) applies initially), but as noted, the solutions are insensitive to 80 for sufficiently small values. The beam dimensions were taken to be b -- h = 2.54 cm, and the value of Young's modulus was E = 1.462 )< 105 MPa. 3.1. Piecewise-constant moment
The first loading history we consider is one in which a bending moment of constant magnitude is applied to the beam at t -- 0, reversed in sign at t = 1000 hrs., and then reversed in sign again at t = 2000 hrs. The value of the applied moment here, and in subsequent loading histories of this type, was taken to be M 0 = 1356 N -- M. Since the moment is piecewise-constant, the governing equations are given by (13). However it is necessary to restart the integration of eq. (13) upon each reversal of moment using as initial conditions the values of curvature, stress, and state variable just prior to reversal of moment, and taking into account the changes in curvature and stress under moment reversal due to elasticity '~ of the beam. Fig. 2 shows the nondimensional inelastic curvature as a function of time for the first 3000 hours. Here we define the inelastic curvature to be the total curvature less the value of curvature due to the elastic strains, i.e. 3'i = ~ ' T - M 0 / E I . Despite the fact that the stress at each individual location in the beam is far from constant with time, the inelastic curvature vs. time response shown in fig. 2 bears a close qualitative resemblance to a typical inelastic strain vs. time plot obtained from a piecewise-constant uniaxial stress creep test. In particular, we note the existence of well-defined primary and secondary phases analogous to the primary and secondary creep phases observed in uniaxial tests. It may also be noted that the primary phase reappears subsequent to each moment reversal, a feature also present in uniaxial simulations. Under a strictly constant moment loading it may be expected that the beam will tend to asymptotically
f
V-
!
~
30100*(hrs)
7'
> O
5
_¢2 ~ 3 w z
0 -I'
0
I000
2000
3000
TIME (hrs) Fig. 2. Nondimensional inelastic curvature vs. time under reversed moment loading.
approach a steady-state condition in which the distributions of stress and of the state variable over the beam cross-section will remain constant with time. This expectation has rigorously been verified by Ponter [12], who demonstrated that, for a class of constitutive relations of which Robinson's is a member, the existence of an asymptotic steady state is guaranteed under constant loading. Thus in the present case, Robinson's model would always predict both a primary and a steady-state secondary phase for any value of applied moment. It may also be noted that, on initial loading and immediately subsequent to each moment reversal, the inelastic curvature changes quite rapidly over a short period of time. This very rapid short-term straining arises from the ability of Robinson's constitutive theory to model, at least qualitatively, the effects arising from conventional, time-independent plasticity theory. The difference here is, of course, that plasticity theory would predict instantaneous straining, whereas in Robinson's theory, an equivalent plastic strain is accumulated over a very short time interval. Fig. 3 shows the calculated stress distribution in the beam in the initial elastic state ( t = 0 ) and at 1000, 2000, and 3000 hours just prior to moment reversal. It can be seen that the inelastic constitutive law acts to considerably modify the elastic stress distribution with the passage of time. The stress distributions shown here are qualitatively quite similar to results presented elsewhere [10], [13] for this problem using different constitutive relations. This qualitative similarity appears
416
T.J. Delph / Creep, relaxation and cyclic' behavior of a beam 1.5
ZOlOO
/
oI
0 -I 4- ~
~ / - t = 2 0 0 0 hrs 0.8 1
\
oot (hrs) / - t : 3000 hrs
ooo
t.O-
9=0.574
t=lOOehrsJ ~
ooI
/i
~=0.067
t=ON
0.4
-0.50
0.2
,
-I.0 -0.8 -0.6 -0.4 -0.2
0.2
0,4
0.6
0.8
0"125
[~
o.1:
0.50
[.0
Fig. 3. Stress distributions at the end of 1000, 2000, and 3000 hours prior to moment reversal.
to be due to the strong constraint placed upon the stress distribution by the equilibrium equation (6). It is of interest to note that the stress distributions at 1000, 2000, and 3000 hours differ little from each other (apart from a sign difference), despite the two reversals of moment. From this it may be inferred that the beam tends to fairly rapidly 'shakedown', or to approach a stable limit cycle in which further periodic reversals of moment would lead to a periodically-repeating stress distribution. Further evidence for the existence of a stable limit cycle may be found in fig. 4, which shows the trajectories followed in the 6 - ~ plane (state plane) at the points 37=0.067, 0.574, and 0.987. The arrows indicate the direction of increasing time. The points of moment reversals show clearly as vertical lines in the plane. The significant fact to be noted from fig. 4 is that the location of the state points at these three locations (and presumably at all other locations as well) is virtually identical at the end of 1000 and 3000 hours. In statevariable constitutive laws such as the one under consideration, the future behavior of a material at any time is uniquely determined by the current values of the state variables (in this case o and a) and the current loading. Since the beam appears to have almost the same state after 3000 hours as after 1000 hours, we may expect that future reversals of moment at 1000 hour intervals would cause each point on the beam to follow a closed trajectory in the state plane akin to the ones shown in fig. 4. We now consider a loading history in which the moment M 0 is applied for 1000 hours, removed for 1000 hours, and subsequently reapplied. The inelastic curvature vs. time response to this loading history is shown in
Fig. 4. State plane trajectories at )7= 0.067, 0.574 and 0.987.
fig. 5. Of particular interest here is the reappearance of a primary phase after reapplication of the moment at t -- 2000 hours. This behavior is analogous to the wellknown phenomenon of thermal recovery observed in uniaxial creep tests subjected to a similar loading history. It should again be pointed out, however, that the stress in each individual fiber of the beam is not constant with time, as would be the case with a typical creep test, but rather changing with time in a fairly complex fashion. Despite this fact, the curvature vs. time response shown in fig. 5 is qualitatively similar to a typical strain vs. time response obtained from a uniaxial test with a similar loading history. It should be noted, however, that no anelastic strain recovery is predicted during the period of time in which the beam is unloaded. Anelastic strain recovery is, in fact, specifically excluded from Robinson's model through the inequality in the second of (20), which requires the inelastic strain rate to vanish whenever the state point (o,a) falls below the line 2 o / 3 - a = 0 in the state plane. Observed anelastic strain recovery in 2) C r - 1M 0 steel typically does not exceed a quarter of the elastic strain. Hence it is not a particularly significant feature of the deformation behavior of this metal and may usually be neglected. Fig. 6 shows the stress distribution immediately after removal of the moment at t = 1000 hours and im-
T.J. Delph / Creep, relaxation and cyclic behavior of a beam 20
1.0
/
1000 2000 5000
>12o
_o <[ .J bJ _z 4-
o
,°o-/ "•0.987
I
uJ 16
y =0.574 ~.
t=O - / J
Y
0
t
y= 0 , 0 6 7 - ~ .
t=O J
I
i
1000
I
L
2000
A -J
I -0.25
-0.50
i
417
1 0.25
I 0.50
3000
TIME (hrs)
Fig. 5. Nondimensional inelastic curvature vs. time under moment loading which is applied, removed, and then reapplied. mediately prior to reapplication of the load at t = 2000 hours. It can be seen that some small redistribution of the stress occurs during this period in the outer fibers. Concurrently, a slight decrease in beam curvature occurs during this period, although its magnitude is too small to be noticeable in fig. 5. This decrease in curvature is brought about by relaxing residual stresses, and not, as noted, by any anelastic strain effects. Fig. 7 shows the trajectories described in the state plane by the variables 8 and ff at the points .~ = 0.067, and 0.574, and 0.987. During the thermal recovery period (t = 1000-2000 hours) it may be noted that the nondimensional state variable ~ in general decreases in magnitude while the stress ~ remains relatively constant. This effect accounts for the reappearance of the primary phase upon reapplication of the moment (fig. 6), since
~ ~
I.OT \x
t =IOO0+ hrs J
-0.4
-0.5
-0.2
:o I ~'~-~ 0 . 8 ~
-0.1
~ ' ~ I ) s rht ( IOO0 2000 3000
0
0.1
0.2
0.5
0.4
¢r
Fig. 6. Stress distributions immediately subsequent to removal of moment and immediately prior to reapplication of moment.
~°11~ IOOO
V--'-I t (hr$) 3OOO
--0.5
Fig. 7. State plane trajectories at )7= 0.067, 0.574 and 0.987.
the decrease in ~ will, through eqs. (13) and (20), result in a value of ? somewhat higher than that existing prior to removal of the moment. Conventional representations of creep behavior, such as those based on strain-hardening laws, lack the ability to model the thermal recovery phenomenon by being unable to predict the reappearance of the primary phase upon reloading. They thus tend to underpredict the strains which would be accumulaied in such a loading history. The inclusion of the thermal recovery phenomenon, at least in a qualitative sense, represents one of the advantages typically possessed by state variable constitutive theories over conventional representations. The last piecewise-constant moment history to be considered is one in which the moment M 0 is applied for 1000 hours and then reduced in magnitude by a factor of two for an additional 9000 hours. Fig. 8 shows the nondimensional inelastic curvature versus time response for this loading history. Here we note the existence of the so-called hesitation period between t = 1000-10000 hours which is observed in uniaxial tests under similar loading histories [6]. Here, after a decrease in the applied load, the inelastic strain remains constant, or nearly constant, for a period of time before slowly starting to increase again. This effect is modeled in Robinson's constitutive equations through the inequality in the second of eq. (20). This inequality sets
418
T.J. Delph / Creep, relaxation and cyclic behavior of a beam 0.4
I0-
/
u -I.0 I
I-
[
-0.5
-
//'
0.5 7
1.0
w z
-0.2-
o 0.5I0"~1 0
04,0
I t(hrs) I0000
I000
L,. 2000
4000
6000
8000
0.4 h
I0000
TIME (HRS)
Fig. 8. Nondimensional inelastic curvature vs. time under moment loading which is applied for 1000 hours, then reduced by half for 9000 hours. the inelastic strain rate to zero when the state point fails below the line 2 6 / 3 - 8 -- O, as occurs in a reduction of loading. This behavior is to be contrasted to that yielded by a conventional creep representation, which would predict merely a reduction of strain rate under these circumstances.
Fig. 9. Nondimensional moment versus curvature at constant curvature rate for two complete cycles.
y = __+3.94× 10 -3 cm ~ for two complete cycles. At the outer beam fiber, this corresponds to a strain rate magnitude of 0.30 hr ], with strain limits of c = ± 0 . 0 1 . It can be seen from fig. 9 that the beam appears to rapidly approach a steady-state limit cycle, and that the limit cycle seems to be essentially achieved after the first full cycle. Fig. 10 shows the stress distribution after two full cycles, which appears qualitatively rather similar to
3.2. Constant curvature rate [,0
We now consider the case in which a time-varying moment is applied to the beam in such a way as to cause the beam to deform at a constant rate of curvature. The governing equations for this case are given by eqs. (18). Here we may draw an analogy with the uniaxial constant strain rate test. The analogy in this case is in fact quite a close one, since from eq. (1) we note that a constant beam curvature rate implies a constant strain rate at any particular location in the. beam, with the strain rate varying linearly from location to location. Fig. 9 shows the nondimensional moment vs. curvature response for the case in which the beam is loaded at a constant curvature rate of % = 0.118 c m - n h r - 1 and cycled at this rate between the curvature limits
i
0.8
¸
0.6
¸
i
9 0.4
0.2
O,
0.1
0.2 cr
0.3
i
Fig. 10. Nondimensional stress distribution at the end of two full cycles of constant curvature rate cycling.
T.J. Delph / Creep, relaxation and cycfic behavior of a beam
that shown in fig. 3. As noted earlier, each beam fiber in a constant curvature rate simulation deforms at a constant strain rate. Hence the state-plane trajectory described by any given fiber is identical to one described in a uniaxial constant strain rate simulation. For a detailed description of this trajectory, we refer to [6]. 3.3. Relaxation of moment
Finally, we consider a loading history in which the beam is initially deformed at a constant curvature rate as described in the previous section. However now when the positive curvature limit is reached, the curvature is held constant. The bending moment required to maintain this value of curvature may now be calculated from eqs. (18) with "t0 = 0. Fig I 1 shows the nondimensional moment as a function of time, where t = 0 corresponds to the start of the constant curvature period. This loading history is, of course, analogous to the uniaxial relaxation test, in which a specimen is held at a fixed value of strain and the load required to maintain this value is measured. Again, the analogy here is quite a close one, since fixed value of curvature implies a fixed value of strain at each individual location in the beam, but one which varies linearly from location to
0.30
i
" ,oo 0.20
M'
o
0
L 0
I I000 T I M E [hrs)
2000
Fig. I 1. Nondimensional moment versus time with curvature held constant (relaxing moment).
419
location. We note that the moment versus time history shown in fig. 11 is qualitatively quite similar to typical stress versus time histories obtained from uniaxial relaxation tests. In the case of relaxation, Robinson's equations do not yield strictly steady-state solutions [6], although near-steady state behavior may be observed in fig. (l 1) for large times. Hence we should expect the nondimensional moment M ' to approach zero over a sufficiently great period of time.
4. Discussion In the foregoing we have investigated the timedependent inelastic behavior of a Bernoulli-Euler beam under various time-dependent moment loadings. The time-dependent inelastic behavior was assumed to be described by a recently proposed constitutive model due to Robinson, with a set of constants appropriate to the behavior of 21 C r - l M 0 steel at 565°C (1050°F). The governing equations were then derived in the form of a set of coupled, first-order differential equations by means of the standard Bemoulli-Euler beam theory. It should be noted that alternative formulations of the governing equations are possible. Here we have chosen to write the equations in what is termed the 'rate' formulation by Hayhurst and Krzeczkowski [14] as opposed to the alternate 'absolute' formulation. The difference between the two is principally that the rate formulation makes use of the equilibrium equation (eq. (16) in the present case) in rate form, while in the absolute formulation, the equilibrium equation is applied in standard form at every point in time. Although either formulation would serve for the present case, the rate formulation was chosen because it tends to unify to some extent the governing equations for the piecewiseconstant moment and constant curvature rate loading histories. However as noted by Hayhurst and Krzeczl~owski [14], this formulation suffers from the fact that only the rate of the equilibrium equation is satisfied, and not the equilibrium equation itself. Hence the calculated stresses might, for sufficiently large times, show a tendency to deviate from an equilibrium distribution. In the present study, the calculated stresses at each time step were substituted into the equilibrium equation (6), and the resulting integral evaluated numerically to insure that the calculated stresses did, in fact, represent an equilibrium distribution. No tendencies to deviate from equilibrium were noted, although this might have been the case had longer times or more complex loading histories been considered.
420
T.J. Delph / Creep, relaxation and cyclic behavior of a beam
The governing equations were solved numerically with the aid of the Adams-Bashforth-Moulton algorithm. The high rates predicted by Robinson's constitutive model in certain regions of the state plane made it necessary to use extremely small time steps in these regions, which in turn necessitated automatic time step control. The resulting numerical scheme was quite successful, obtaining solutions to the governing differential equations out to several thousand hours with only very modest amounts of computational effort. The AdamsBashforth-Moulton technique is well known and possesses excellent accuracy and stability properties. It does not, however, seem to have been previously applied to problems involving time-dependent material behavior. Based on the present results, it would appear to be a promising candidate for implementation into a finite-element code in order to handle more complex problems. Such an effort is now underway. However the technique has the drawback that it is expensive of computer storage, a fact which limits somewhat the size of the problems that can be handled. There are several conclusions which may be drawn from the results of the analyses. One is that if the moment is considered as a generalized stress and the beam curvature as a generalized strain, then the results are qualitatively very similar to those obtained from stress-strain or strain-time plots of uniaxial data. This is perhaps not too surprising in the cases where the beam was cycled at a constant curvature rate or subjected to a relaxing moment. Here, as noted previously, a rather close analogy exists between the corresponding uniaxial situations. However, no such close analogy exists in the case of piecewise-constant moment loading, which corresponds to a uniaxial piecewise-constant stress creep test. Despite the fact that the stress in each beam fiber varies with time in a complex manner, the curvature vs. time response still bears a strong qualitative resemblance to the strain-time response of a uniaxial specimen under an analogous stress history. Another interesting conclusion is that, at least for the loading histories considered, Robinson's constitutive equations appear to lead to stable, limit-cycle solutions under cyclic loadings. This is of course apparent in the cyclic curvature simulation, but we also note the results of the pieccwise-constant moment simulations (figs. 2 7). These all involve loading histories in which the moment is maintained at a constant value for lO00 hours, reversed or reduced for lO00 hours, and then reapplied with the original sign and magnitude for an additional lO00 hours. In all three cases considered the stress distribution within the beam at the end of 3000 hours, as well as the distribution of the state variable a,
was found to be virtually identical to that existing at 1000 hours, just prior to reduction or reversal of load. Thus we might expect that periodic reductions or reversals of load will lead to periodically-repeating solutions. In particular, we note definite signs of the existence of a stable limit cyclic in the case of reversed moment loading (fig. 4). Unfortunately there exist at present no theorems analogous to the shakedown theorems of classical plasticity which enable us to be certain of the existence of stable limit cycles for time-dependent constitutive relations of the sort exemplified by Robinson's model. However on the basis of the numerical simulations shown here and elsewhere [6], such stable, or at least nearly stable, cycles do appear to be a feature of Robinson's model. The existence of stable limit-cycle solutions to periodic loadings, though often taken for granted, is not a trivial property. In the absence of creep or fatigue damage, real materials often display such stable behavior. However this behavior may not always be present in the behavior of inelastic constitutive equations. In summary, Robinson's constitutive model seems to yield predictions for the inelastic behavior of a beam which are within at least reasonable qualitative accord with physically expected behavior under a variety of time-varying loadings. They thus represent a potentially attractive constitutive representation for the deformation of metals at elevated temperatures.
Acknowledgement The author is grateful to Dr. D.N. Robinson of Oak Ridge National Laboratory for many helpful discussions.
References [I] RDT Standard F9-5T, Guidelines and Procedures for Design of Nuclear System Components at Elevated Temperature (U.S. Dept. of Energy, May 1974) Ch. 4. [2] A.R.S. Ponter and F.A. Leckie, J. Engrg. Mater. Tech. 98 (1976) 47-51. [3] E.W. Hart, J. Engrg. Mater. Tech. 98 (1976) 193-202. [4] A. Miller, J. Engrg. Mater. Tech. 98 (1976) 97-113. [5] R.D. Krieg, SMiRT-4, San Francisco, CA, 1977, M6/4. [6] D.N. Robinson Oak Ridge National Laboratory, ORNL/TM- 5969 (Oct. 1978). [7] R. Lagneborg, Met. Sci, J. 6 (1972) 127-133. [8] J.R. Rice, J. Appl. Mech. 37, (1970) 728-737. [9] E.T. Onat, and F. Fardshisheh, Oak Ridge National Laboratory ORNL-4783 (August 1972).
T.J. Delph/ Creep, relaxation and cyclic behavior of a beam
[10] J.C. Chang, R.H. Lance and S. Mukerjee ASME J. Pressure Vessel Tech. 101 (1979) 305-310. [11] D.N. Robinson, Oak Ridge National Laboratory, ORNL5540 (November 1979). [12] A.R.S. Ponter, J. de M6chanique 15 (1976) 527-542.
421
[13] R.K. Penny and D.L. Marriott, Design for Creep (McGraw-Hill, New York, 1971). [14] D.R. Hayhurst and A.J. Krzeczkowski, Comput. Meths. Appl. Mech. Engrg. 20 (1979) 151-171.
411
CREEP, R E L A X A T I O N A N D CYCLIC B E H A V I O R OF A B E A M U S I N G A S T A T E - V A f i I A B L E CONSTITUTIVE MODEL
T.J. D E L P H Department of Mechanical Engineering and Mechanics, Lehigh University, Bethlehem, PA 18015, USA
Received 15 February 1981
Numerical solutions are presented for the time-dependent inelastic behavior of a Bernoulli-Euler beam under a variety of time-dependent moment loadings. The inelastic constitutive representation employed is the recently developed state-variable constitutive law of Robinson. Robinson's equations have several novel features, and these are examined within the context of the behavior of a beam. Despite the fact that the beam may contain relatively complex inhomogeneous stress fields, the overall behavior is shown to be analogous to that observed in simple uniaxial tests.
1. Introduction One of the most basic problems in describing the high temperature behavior of metals is the formulation of an appropriate set of constitutive equations. This problem becomes particularily acute when it becomes necessary to deal with loading histories involving periods where the loads are changing fairly rapidly interspaced with periods during which the loads are relatively constant. Unfortunately, such loading histories are commonly encountered in the operation of high temperature pressure vessel and piping systems.At present, the most widely used constitutive representation [l] makes use of the time-independent theory of classical plasticity to describe short-term deformation, while relying upon time-dependent classical theories of creep to describe long term deformation. Thus the inelastic strain is taken as the sum of a time-independent component and a time-dependent component. Although this approach works quite well for a number of important loading histories, there are others for which it is notably unsuccessful. Recently, however, a number of new constitutive theories have been introduced which reject this conventional representation and instead treat the inelastic strain as being totally time-dependent. Some representative examples are the work of Ponter and Leckie [2], Hart [3], and Miller [4], with a more comprehensive bibliography being given by Krieg [5]. Although these theories may differ considerably among themselves, they often make use of one or more variables of state and are thus 0029-5493/81/0000-0000/$02.50
sometimes known as 'state variable' theories, or else as 'unified' theories from their ability to model at least in a qualitative sense both short-term plastic and long-term creep behavior. Typically, the state-variable theories seem to offer significant advantages over the conventional representation, in that they are able to model at least qualitatively, such phenomena as thermal recovery, creepplasticity interactions, and cyclic hardening. These phenomena tend to be handled rather awkwardly, if considered at all, in the conventional constitutive representation through the use of various ad hoc rules [1]. The principal disadvantages of the new theories are that most are not yet fully tested or developed, and that they seem to be more difficult to incorporate into large finite-element structural analysis codes than the usual constitutive laws. In particular, for most of these theories, precise methods for determining required material constants are not available and recourse is often had to trial-and-error fitting of experimental data. Furthermore the governing equations may often predict extremely high rates or exhibit mathematically 'stiff' behavior and require considerable care in their time integration. Those problems are discussed in somewhat more detail by Krieg [5]. In the present work, we consider a state-variable constitutive theory which has recently been proposed by Robinson [6] in conjunction with developmental work on the Liquid Metal Fast Breeder Reactor (LMFBR) program. This work was prompted by the inability of the standard constitutive representation to describe
© 1981 N o r t h - H o l l a n d
412
T.J. Delph / Creep, relaxation and cyclic behavior of a beam
various aspects of the complex deformation behavior of the ferretic stainless steel 2J Cr-1Mo. Robinson's theory is based to some extent on the earlier work of Lagneborg [7] and Rice [8], but is particularly notable for the use of the state space concepts set forth by Onat [9]. These concepts have led to the development of a number of auxiliary rules which modify the form of the governing equations depending on the location in state space of any particular material point. These auxiliary rules appear to be unique to Robinson's model, and show promise of allowing in certain cases a more accurate representation of actual material behavior than would be the case if the unmodified governing equations applied equally to all regions of the state space. The intent of the present work is to investigate the inelastic behavior of a simple Bernoulli-Euler beam using Robinson's constitutive model. Aside from the practical importance of the beam as a structural member, this problem is of interest in that the beam, though essentially one-dimensional in the Bernoulli-Euler formulation, may contain quite complicated inhomogeneous states of stress. Hence its behavior under various time-dependent loadings may serve to give some indication of the behavior of more complex members. Additionally, the one-dimensional nature of the problem makes it amenable to relatively simple numerical techniques, without the need to resort to cumbersome and expensive finite-element methods. We note that a similar problem has been treated by Chang et al. [10] using Hart's constitutive model [3], though considering loading histories somewhat less complicated than those which will be analyzed herein.
2. Analysis In this section we give the equations governing the time-dependent deformation of a Bernoulli-Euler beam under the influence of a time-dependent bending moment. We then specialize these equations for three particular loading histories of interest. These are deformation under a constant (or piecewise-constant) applied moment, deformation at a constant rate of curvature, and relaxation of moment for a constant fixed value of curvature. If, as is often done in plate and shell theory, the applied moment M(t) is considered to represent a generalized stress and the curvature 3'(0 a generalized strain, then these three loading histories correspond respectively to generalizations of the creep, constant strain rate extension, and relaxation tests common in uniaxial materials testing. Fig. 1 shows the geometry of the beam which we
M(t)
h
h
M(t)
" --7-
Fig. 1. Beam geometry. assume to be of rectangular cross-section. According to Bernoulli-Euler beam theory, the total longitudinal strain in the beam is given by CT(t ) = y ' / ( t ) ,
(1)
where y is measured from the neutral axis (fig. 1) and )'(t) is the time-dependent curvature. We assume that the total longitudinal strain can be expressed as the sum of an elastic and an inelastic strain component, c T = c E + q,
(2)
where the elastic strain component is given by Hooke's Law ~E = o / e .
(3)
We further assume that the inelastic strain component is governed by a constitutive equation having the general form
~I = f ( o , a ) ,
(4)
where the superposed dot indicates differentiation (total or partial) with respect to time. Here a is a variable of state governed by a =g(o,a)
(5)
As pointed out in the Introduction, the inelastic strain e I is taken to be totally time-dependent and is assumed to contain, or model, both short-term 'plastic' strain and long-term 'creep' strain. Assuming symmetry about the neutral axis, we have from equilibrium considerations that the resultant moment is
M( t) = 2b fohyO(y,t ) d y.
(6)
Also from eqs. (1), (2), (3), and (4),
(~ = E[ y~ -- f( a,a ) ].
(7)
We now wish to consider some particular loading histoties.
2.1. Constant applied moment Assume that a constant bending moment of magnitude M 0 is applied to the beam at t--= 0. Differentiating
T.J. Delph / Creep, relaxation and cyclic behavior of a beam eq. (6) with respect to time and making use of eq. (7), we have h
2
fo' [ y "~-- yf( o,a)] d y = 0 .
(8)
Solving for the curvature rate,
~,( l) = (3 /h3) fohyf( o,a) d y.
(9)
413
2.2. Constant curvature rate We now assume that a time-varying moment M(t) is applied to the beam in such a manner that the beam curvature varies at a constant rate, %. Eq. (7) thus becomes d = E[y'~ o - - / ( o , a ) ] .
(15)
Differentiating eq. (6) with respect to time and using eq. (15), we obtain
Eqs. (9) and (7) then give
O(y,t)=E[(3y/h3)f;yf(o,a)dy--f(o,a)].
(10)
)(,I( l ) = EI~[o - 2eb fohyf( o,a ) d y.
(16)
Eqs. (5), (9) and (10) now constitute a set of three coupled non-linear equations whose solution yields the deformation history of the beam. The instantaneous response of the beam is presumed to be elastic, so that the appropriate initial conditions are given by the elastic solution
We nondimensionalize as follows. Let "/m be some representative value of the curvature (in the subsequent section it will be taken as the maximum curvature reached in a cyclic moment-curvature simulation.) Then define y ' = y/"}tm,
M ' = M/ElYm,
o(y,O) = M o y / I
a' = a/EhYm
e' I = e i / h ' g m .
y(O) = M o / e I ,
(11)
where I is the area moment of inertia of the rectangular cross-section, E is Young's modulus, and M 0 = M(0). Furthermore we take a ( y , 0 ) = 80, where 80 is the value of the state variable a existing in a 'virgin' material, that is, a material which has undergone no prior inelastic straining. Since the state variable a is in a sense a quantitative measure of the prior inelastic history of the beam, its initial value for a virgin material is typically zero or quite small. To place these three equations in nondimensional form, we define
)7=y/h,
~I = £1// ( noh// EI ),
7 = y / ( M o / E I ),
0=
o / ( M o h / I ),
S = a~ ( M o h / I ) . Thus we obtain ~=g(6,S),
03)
subject to the initial conditions
Here
we have
g/(Moh/I).
,7(0) = l,
Eqs. (5), (15), and (16) then become
a' = g ' ( o ' , ~ ' ) , 0' =37"/,6 - - f ' ( o ' , a ' ) ,
(18)
Yo' -- 3 f01fif' d )7, h,/' -- "" subject to the initial conditions a'()7,0) = 86,
0'()7,0) = 0,
M'(0) = 0.
(19)
loaded at a constant rate of curvature until some value of curvature y = Ym has been reached. The curvature is then held constant at this value and the applied bending moment allowed to relax. The governing equations for this condition may be derived from eqs. (18) by simply setting ?0 = 0. The appropriate initial conditions are the values a', o', and M' at the end of the constant curvature rate loading phase.
2.3. Constitutive equations
o = 3)7£ )Tf(o,a ) d ) 7 - f ( 6 , ~ ) ,
S ( y , 0 ) = go,
(17)
Here )7 is as defined by eq. (12) and f' =f/h'rm, g' = g/EhYm. Here we assume that the beam has been (12)
~ = 3fol)Tf(O,S ) d)7,
o' = o/EhYm,
~ ()7,0) =)7.
taken f = f / ( M o h / E I
(14)
) and g =
The equations developed in the preceding section are valid for any constitutive law having the general form of eqs. (4) and (5). We now wish to specialize to the particular set of constitutive relations proposed by Robinson [6]. These relations are formulated in terms of an arbitrary multiaxial stress state, but in the present
T.J. Delph / Creep, relaxation and cyclic behavior of a beam
414
case, when the only longitudinal stress component is non-zero, they reduce to the following form. F O'- ')/2(20/3 -- a ) ; F>0, 2/,~x---- 0,
F~0
o ( 2 o / 3 - a) > 0 , or
(F~O,o(2o/3--a)~O);
S ( o / o s ) F ( ~ - ' ) / 2 ( 2 0 / 3 -- a), F~> O,
oa ~ O,
o ~ os; (20)
21~Hq/( 3Vr3741a/KI)~ -R(
3v~Ia/KI)m-P-'a,
~=
Ot > OtO,
era ~>0;
(21)
2gH4,/( 3¢~7~la0/KI) a _ R ( 3¢-37~lao/K])m B a,
a ~OtO,
oa<0; where F= 3(20/3
- ~)2/(4K2)
- 1.
(22)
The quantities ~,H,R,n,fl,m,K, and s o in these equations are parameters, possibly functions of temperature, which characterize a given material. Here o is the longitudinal component of the stress tensor and a the corresponding component of the state variable tensor. The equation given in the third of eq. (20) is not present in Robinson's original formulation [6]. It is introduced here as a minor modification in order to avoid numerical difficulties associated with the (idealized) discontinuity in inelastic strain rate present in the original model when crossing the line o = 0 in the o-a plane. In the present work we have taken S(o/o~) = 3( o / o s)2 _ 2( o / o s)3
(23)
and %=K. We note that S ( 0 ) = S ( 1 ) = 1, and that d S / d o vanishes both at o - - 0 and o--%. Hence this modification simply allows the inelastic strain rate to increase in a smooth fashion from zero to some 'full-scale' value given by the first of (20). Typically this increase occurs over a small stress range ( K is on the order of 5-7 MPa), so that the predictions of the original equations are essentially unaltered, but the numerical difficulties occasioned by the sharp discontinuity in strain rate present in the original version are removed.
3. Numerical results
In this section we will present some numerical results for the response of a Bernoulli-Euler beam governed by Robinson's constitutive model to piecewise-constant bending moments, to a cyclic moment-curvature history, and to a relaxing moment with constant fixed curvature. The equations governing these various situations have been derived in the previous section. The numerical method used for their solution was the well-known Adams-Bashforth-Moulton algorithm, a fourth-order accurate predictor-corrector scheme for the solution of first-order differential equations. The spatial integrals involved in the governing equations were evaluated by means of ten-point Gaussian quadrature. This technique effectively converted the governing equations into a system of twenty-one coupled non-linear ordinary differential equations, two each for the stress o and the state variable a at each of the ten Gauss points plus an additional equation for either the time-varying moment or the curvature. Since 80' the initial value of a, is typically quite small, eq. (21) predicts quite high values for ti in the neighborhood of t = 0 and also in the second and fourth quadrants of the o-~ plane, where the second of eq. (21) holds. To maintain sufficient accuracy in the numerical calculations, it was found necessary to use a time step size on the order of 10 5_ 10 8 hours in these regions. These extremely small time steps in turn necessitated the use of an automatic time-step control scheme, wherein the time step size was halved when an appropriately defined relative error measure exceeded some upper limit and doubled whenever the error measure became sufficiently small. The error measure used was (relative error)= [(yc -Yp)/Yc[ where yp was the predicted value of the dependent variable at a particular time step and Yc the corrected value. The error measure was calculated for each of the twenty-one equations at each time step and the step size halved when the error in any equation exceeded 1 × 10-4. When the error in any six equations was less than 1 × 10 7, the step size was doubled. This scheme turned out to be extremely successful. In the worst case with an initial step size of 10 -8 hours it was found possible to integrate the governing equations for the beam for times out to several thousand hours in less than 20 seconds of CPU time on a CDC 6400 computer. The material constants used in the constitutive law (eqs. (20) and (21)) were chosen to typify the uniaxial behavior of a particular heat of the ferretic stainless steel 2¼Cr-lMo at a temperature of 566°C (1050°F). At the moment, no set procedure exists for the de-
415
T.J. Delph / Creep, relaxation and cyclic behavior of a beam
termination of these constants other than trial-and-error fits to experimental data. Such fits were carried out by Robinson [11] to data taken from constant strain rate extension and creep tests. The resulting values are/x = 1.06)< 108 MPa-hr, R = 2 . 2 5 ) < 10 -5 hr i, H = 3 . 0 ) < 10 - 4 hr -1, K = 6 . 9 0 MPa, N = 6 . 2 5 , M = 4 . 5 , /3= 1, and a 0 = 7.96 × 10 -3 MPa. In practice the solutions were found to be insensitive to the initial value of a, 80, so long as 80 was sufficiently small. Hence there is some allowable arbitrariness in the selection of 80, and in the present work we follow Robinson [11] in taking 80 = a 0. It is of course also possible to select 80 = 0 (in which case the second of eq. (21) applies initially), but as noted, the solutions are insensitive to 80 for sufficiently small values. The beam dimensions were taken to be b -- h = 2.54 cm, and the value of Young's modulus was E = 1.462 )< 105 MPa. 3.1. Piecewise-constant moment
The first loading history we consider is one in which a bending moment of constant magnitude is applied to the beam at t -- 0, reversed in sign at t = 1000 hrs., and then reversed in sign again at t = 2000 hrs. The value of the applied moment here, and in subsequent loading histories of this type, was taken to be M 0 = 1356 N -- M. Since the moment is piecewise-constant, the governing equations are given by (13). However it is necessary to restart the integration of eq. (13) upon each reversal of moment using as initial conditions the values of curvature, stress, and state variable just prior to reversal of moment, and taking into account the changes in curvature and stress under moment reversal due to elasticity '~ of the beam. Fig. 2 shows the nondimensional inelastic curvature as a function of time for the first 3000 hours. Here we define the inelastic curvature to be the total curvature less the value of curvature due to the elastic strains, i.e. 3'i = ~ ' T - M 0 / E I . Despite the fact that the stress at each individual location in the beam is far from constant with time, the inelastic curvature vs. time response shown in fig. 2 bears a close qualitative resemblance to a typical inelastic strain vs. time plot obtained from a piecewise-constant uniaxial stress creep test. In particular, we note the existence of well-defined primary and secondary phases analogous to the primary and secondary creep phases observed in uniaxial tests. It may also be noted that the primary phase reappears subsequent to each moment reversal, a feature also present in uniaxial simulations. Under a strictly constant moment loading it may be expected that the beam will tend to asymptotically
f
V-
!
~
30100*(hrs)
7'
> O
5
_¢2 ~ 3 w z
0 -I'
0
I000
2000
3000
TIME (hrs) Fig. 2. Nondimensional inelastic curvature vs. time under reversed moment loading.
approach a steady-state condition in which the distributions of stress and of the state variable over the beam cross-section will remain constant with time. This expectation has rigorously been verified by Ponter [12], who demonstrated that, for a class of constitutive relations of which Robinson's is a member, the existence of an asymptotic steady state is guaranteed under constant loading. Thus in the present case, Robinson's model would always predict both a primary and a steady-state secondary phase for any value of applied moment. It may also be noted that, on initial loading and immediately subsequent to each moment reversal, the inelastic curvature changes quite rapidly over a short period of time. This very rapid short-term straining arises from the ability of Robinson's constitutive theory to model, at least qualitatively, the effects arising from conventional, time-independent plasticity theory. The difference here is, of course, that plasticity theory would predict instantaneous straining, whereas in Robinson's theory, an equivalent plastic strain is accumulated over a very short time interval. Fig. 3 shows the calculated stress distribution in the beam in the initial elastic state ( t = 0 ) and at 1000, 2000, and 3000 hours just prior to moment reversal. It can be seen that the inelastic constitutive law acts to considerably modify the elastic stress distribution with the passage of time. The stress distributions shown here are qualitatively quite similar to results presented elsewhere [10], [13] for this problem using different constitutive relations. This qualitative similarity appears
416
T.J. Delph / Creep, relaxation and cyclic' behavior of a beam 1.5
ZOlOO
/
oI
0 -I 4- ~
~ / - t = 2 0 0 0 hrs 0.8 1
\
oot (hrs) / - t : 3000 hrs
ooo
t.O-
9=0.574
t=lOOehrsJ ~
ooI
/i
~=0.067
t=ON
0.4
-0.50
0.2
,
-I.0 -0.8 -0.6 -0.4 -0.2
0.2
0,4
0.6
0.8
0"125
[~
o.1:
0.50
[.0
Fig. 3. Stress distributions at the end of 1000, 2000, and 3000 hours prior to moment reversal.
to be due to the strong constraint placed upon the stress distribution by the equilibrium equation (6). It is of interest to note that the stress distributions at 1000, 2000, and 3000 hours differ little from each other (apart from a sign difference), despite the two reversals of moment. From this it may be inferred that the beam tends to fairly rapidly 'shakedown', or to approach a stable limit cycle in which further periodic reversals of moment would lead to a periodically-repeating stress distribution. Further evidence for the existence of a stable limit cycle may be found in fig. 4, which shows the trajectories followed in the 6 - ~ plane (state plane) at the points 37=0.067, 0.574, and 0.987. The arrows indicate the direction of increasing time. The points of moment reversals show clearly as vertical lines in the plane. The significant fact to be noted from fig. 4 is that the location of the state points at these three locations (and presumably at all other locations as well) is virtually identical at the end of 1000 and 3000 hours. In statevariable constitutive laws such as the one under consideration, the future behavior of a material at any time is uniquely determined by the current values of the state variables (in this case o and a) and the current loading. Since the beam appears to have almost the same state after 3000 hours as after 1000 hours, we may expect that future reversals of moment at 1000 hour intervals would cause each point on the beam to follow a closed trajectory in the state plane akin to the ones shown in fig. 4. We now consider a loading history in which the moment M 0 is applied for 1000 hours, removed for 1000 hours, and subsequently reapplied. The inelastic curvature vs. time response to this loading history is shown in
Fig. 4. State plane trajectories at )7= 0.067, 0.574 and 0.987.
fig. 5. Of particular interest here is the reappearance of a primary phase after reapplication of the moment at t -- 2000 hours. This behavior is analogous to the wellknown phenomenon of thermal recovery observed in uniaxial creep tests subjected to a similar loading history. It should again be pointed out, however, that the stress in each individual fiber of the beam is not constant with time, as would be the case with a typical creep test, but rather changing with time in a fairly complex fashion. Despite this fact, the curvature vs. time response shown in fig. 5 is qualitatively similar to a typical strain vs. time response obtained from a uniaxial test with a similar loading history. It should be noted, however, that no anelastic strain recovery is predicted during the period of time in which the beam is unloaded. Anelastic strain recovery is, in fact, specifically excluded from Robinson's model through the inequality in the second of (20), which requires the inelastic strain rate to vanish whenever the state point (o,a) falls below the line 2 o / 3 - a = 0 in the state plane. Observed anelastic strain recovery in 2) C r - 1M 0 steel typically does not exceed a quarter of the elastic strain. Hence it is not a particularly significant feature of the deformation behavior of this metal and may usually be neglected. Fig. 6 shows the stress distribution immediately after removal of the moment at t = 1000 hours and im-
T.J. Delph / Creep, relaxation and cyclic behavior of a beam 20
1.0
/
1000 2000 5000
>12o
_o <[ .J bJ _z 4-
o
,°o-/ "•0.987
I
uJ 16
y =0.574 ~.
t=O - / J
Y
0
t
y= 0 , 0 6 7 - ~ .
t=O J
I
i
1000
I
L
2000
A -J
I -0.25
-0.50
i
417
1 0.25
I 0.50
3000
TIME (hrs)
Fig. 5. Nondimensional inelastic curvature vs. time under moment loading which is applied, removed, and then reapplied. mediately prior to reapplication of the load at t = 2000 hours. It can be seen that some small redistribution of the stress occurs during this period in the outer fibers. Concurrently, a slight decrease in beam curvature occurs during this period, although its magnitude is too small to be noticeable in fig. 5. This decrease in curvature is brought about by relaxing residual stresses, and not, as noted, by any anelastic strain effects. Fig. 7 shows the trajectories described in the state plane by the variables 8 and ff at the points .~ = 0.067, and 0.574, and 0.987. During the thermal recovery period (t = 1000-2000 hours) it may be noted that the nondimensional state variable ~ in general decreases in magnitude while the stress ~ remains relatively constant. This effect accounts for the reappearance of the primary phase upon reapplication of the moment (fig. 6), since
~ ~
I.OT \x
t =IOO0+ hrs J
-0.4
-0.5
-0.2
:o I ~'~-~ 0 . 8 ~
-0.1
~ ' ~ I ) s rht ( IOO0 2000 3000
0
0.1
0.2
0.5
0.4
¢r
Fig. 6. Stress distributions immediately subsequent to removal of moment and immediately prior to reapplication of moment.
~°11~ IOOO
V--'-I t (hr$) 3OOO
--0.5
Fig. 7. State plane trajectories at )7= 0.067, 0.574 and 0.987.
the decrease in ~ will, through eqs. (13) and (20), result in a value of ? somewhat higher than that existing prior to removal of the moment. Conventional representations of creep behavior, such as those based on strain-hardening laws, lack the ability to model the thermal recovery phenomenon by being unable to predict the reappearance of the primary phase upon reloading. They thus tend to underpredict the strains which would be accumulaied in such a loading history. The inclusion of the thermal recovery phenomenon, at least in a qualitative sense, represents one of the advantages typically possessed by state variable constitutive theories over conventional representations. The last piecewise-constant moment history to be considered is one in which the moment M 0 is applied for 1000 hours and then reduced in magnitude by a factor of two for an additional 9000 hours. Fig. 8 shows the nondimensional inelastic curvature versus time response for this loading history. Here we note the existence of the so-called hesitation period between t = 1000-10000 hours which is observed in uniaxial tests under similar loading histories [6]. Here, after a decrease in the applied load, the inelastic strain remains constant, or nearly constant, for a period of time before slowly starting to increase again. This effect is modeled in Robinson's constitutive equations through the inequality in the second of eq. (20). This inequality sets
418
T.J. Delph / Creep, relaxation and cyclic behavior of a beam 0.4
I0-
/
u -I.0 I
I-
[
-0.5
-
//'
0.5 7
1.0
w z
-0.2-
o 0.5I0"~1 0
04,0
I t(hrs) I0000
I000
L,. 2000
4000
6000
8000
0.4 h
I0000
TIME (HRS)
Fig. 8. Nondimensional inelastic curvature vs. time under moment loading which is applied for 1000 hours, then reduced by half for 9000 hours. the inelastic strain rate to zero when the state point fails below the line 2 6 / 3 - 8 -- O, as occurs in a reduction of loading. This behavior is to be contrasted to that yielded by a conventional creep representation, which would predict merely a reduction of strain rate under these circumstances.
Fig. 9. Nondimensional moment versus curvature at constant curvature rate for two complete cycles.
y = __+3.94× 10 -3 cm ~ for two complete cycles. At the outer beam fiber, this corresponds to a strain rate magnitude of 0.30 hr ], with strain limits of c = ± 0 . 0 1 . It can be seen from fig. 9 that the beam appears to rapidly approach a steady-state limit cycle, and that the limit cycle seems to be essentially achieved after the first full cycle. Fig. 10 shows the stress distribution after two full cycles, which appears qualitatively rather similar to
3.2. Constant curvature rate [,0
We now consider the case in which a time-varying moment is applied to the beam in such a way as to cause the beam to deform at a constant rate of curvature. The governing equations for this case are given by eqs. (18). Here we may draw an analogy with the uniaxial constant strain rate test. The analogy in this case is in fact quite a close one, since from eq. (1) we note that a constant beam curvature rate implies a constant strain rate at any particular location in the. beam, with the strain rate varying linearly from location to location. Fig. 9 shows the nondimensional moment vs. curvature response for the case in which the beam is loaded at a constant curvature rate of % = 0.118 c m - n h r - 1 and cycled at this rate between the curvature limits
i
0.8
¸
0.6
¸
i
9 0.4
0.2
O,
0.1
0.2 cr
0.3
i
Fig. 10. Nondimensional stress distribution at the end of two full cycles of constant curvature rate cycling.
T.J. Delph / Creep, relaxation and cycfic behavior of a beam
that shown in fig. 3. As noted earlier, each beam fiber in a constant curvature rate simulation deforms at a constant strain rate. Hence the state-plane trajectory described by any given fiber is identical to one described in a uniaxial constant strain rate simulation. For a detailed description of this trajectory, we refer to [6]. 3.3. Relaxation of moment
Finally, we consider a loading history in which the beam is initially deformed at a constant curvature rate as described in the previous section. However now when the positive curvature limit is reached, the curvature is held constant. The bending moment required to maintain this value of curvature may now be calculated from eqs. (18) with "t0 = 0. Fig I 1 shows the nondimensional moment as a function of time, where t = 0 corresponds to the start of the constant curvature period. This loading history is, of course, analogous to the uniaxial relaxation test, in which a specimen is held at a fixed value of strain and the load required to maintain this value is measured. Again, the analogy here is quite a close one, since fixed value of curvature implies a fixed value of strain at each individual location in the beam, but one which varies linearly from location to
0.30
i
" ,oo 0.20
M'
o
0
L 0
I I000 T I M E [hrs)
2000
Fig. I 1. Nondimensional moment versus time with curvature held constant (relaxing moment).
419
location. We note that the moment versus time history shown in fig. 11 is qualitatively quite similar to typical stress versus time histories obtained from uniaxial relaxation tests. In the case of relaxation, Robinson's equations do not yield strictly steady-state solutions [6], although near-steady state behavior may be observed in fig. (l 1) for large times. Hence we should expect the nondimensional moment M ' to approach zero over a sufficiently great period of time.
4. Discussion In the foregoing we have investigated the timedependent inelastic behavior of a Bernoulli-Euler beam under various time-dependent moment loadings. The time-dependent inelastic behavior was assumed to be described by a recently proposed constitutive model due to Robinson, with a set of constants appropriate to the behavior of 21 C r - l M 0 steel at 565°C (1050°F). The governing equations were then derived in the form of a set of coupled, first-order differential equations by means of the standard Bemoulli-Euler beam theory. It should be noted that alternative formulations of the governing equations are possible. Here we have chosen to write the equations in what is termed the 'rate' formulation by Hayhurst and Krzeczkowski [14] as opposed to the alternate 'absolute' formulation. The difference between the two is principally that the rate formulation makes use of the equilibrium equation (eq. (16) in the present case) in rate form, while in the absolute formulation, the equilibrium equation is applied in standard form at every point in time. Although either formulation would serve for the present case, the rate formulation was chosen because it tends to unify to some extent the governing equations for the piecewiseconstant moment and constant curvature rate loading histories. However as noted by Hayhurst and Krzeczl~owski [14], this formulation suffers from the fact that only the rate of the equilibrium equation is satisfied, and not the equilibrium equation itself. Hence the calculated stresses might, for sufficiently large times, show a tendency to deviate from an equilibrium distribution. In the present study, the calculated stresses at each time step were substituted into the equilibrium equation (6), and the resulting integral evaluated numerically to insure that the calculated stresses did, in fact, represent an equilibrium distribution. No tendencies to deviate from equilibrium were noted, although this might have been the case had longer times or more complex loading histories been considered.
420
T.J. Delph / Creep, relaxation and cyclic behavior of a beam
The governing equations were solved numerically with the aid of the Adams-Bashforth-Moulton algorithm. The high rates predicted by Robinson's constitutive model in certain regions of the state plane made it necessary to use extremely small time steps in these regions, which in turn necessitated automatic time step control. The resulting numerical scheme was quite successful, obtaining solutions to the governing differential equations out to several thousand hours with only very modest amounts of computational effort. The AdamsBashforth-Moulton technique is well known and possesses excellent accuracy and stability properties. It does not, however, seem to have been previously applied to problems involving time-dependent material behavior. Based on the present results, it would appear to be a promising candidate for implementation into a finite-element code in order to handle more complex problems. Such an effort is now underway. However the technique has the drawback that it is expensive of computer storage, a fact which limits somewhat the size of the problems that can be handled. There are several conclusions which may be drawn from the results of the analyses. One is that if the moment is considered as a generalized stress and the beam curvature as a generalized strain, then the results are qualitatively very similar to those obtained from stress-strain or strain-time plots of uniaxial data. This is perhaps not too surprising in the cases where the beam was cycled at a constant curvature rate or subjected to a relaxing moment. Here, as noted previously, a rather close analogy exists between the corresponding uniaxial situations. However, no such close analogy exists in the case of piecewise-constant moment loading, which corresponds to a uniaxial piecewise-constant stress creep test. Despite the fact that the stress in each beam fiber varies with time in a complex manner, the curvature vs. time response still bears a strong qualitative resemblance to the strain-time response of a uniaxial specimen under an analogous stress history. Another interesting conclusion is that, at least for the loading histories considered, Robinson's constitutive equations appear to lead to stable, limit-cycle solutions under cyclic loadings. This is of course apparent in the cyclic curvature simulation, but we also note the results of the pieccwise-constant moment simulations (figs. 2 7). These all involve loading histories in which the moment is maintained at a constant value for lO00 hours, reversed or reduced for lO00 hours, and then reapplied with the original sign and magnitude for an additional lO00 hours. In all three cases considered the stress distribution within the beam at the end of 3000 hours, as well as the distribution of the state variable a,
was found to be virtually identical to that existing at 1000 hours, just prior to reduction or reversal of load. Thus we might expect that periodic reductions or reversals of load will lead to periodically-repeating solutions. In particular, we note definite signs of the existence of a stable limit cyclic in the case of reversed moment loading (fig. 4). Unfortunately there exist at present no theorems analogous to the shakedown theorems of classical plasticity which enable us to be certain of the existence of stable limit cycles for time-dependent constitutive relations of the sort exemplified by Robinson's model. However on the basis of the numerical simulations shown here and elsewhere [6], such stable, or at least nearly stable, cycles do appear to be a feature of Robinson's model. The existence of stable limit-cycle solutions to periodic loadings, though often taken for granted, is not a trivial property. In the absence of creep or fatigue damage, real materials often display such stable behavior. However this behavior may not always be present in the behavior of inelastic constitutive equations. In summary, Robinson's constitutive model seems to yield predictions for the inelastic behavior of a beam which are within at least reasonable qualitative accord with physically expected behavior under a variety of time-varying loadings. They thus represent a potentially attractive constitutive representation for the deformation of metals at elevated temperatures.
Acknowledgement The author is grateful to Dr. D.N. Robinson of Oak Ridge National Laboratory for many helpful discussions.
References [I] RDT Standard F9-5T, Guidelines and Procedures for Design of Nuclear System Components at Elevated Temperature (U.S. Dept. of Energy, May 1974) Ch. 4. [2] A.R.S. Ponter and F.A. Leckie, J. Engrg. Mater. Tech. 98 (1976) 47-51. [3] E.W. Hart, J. Engrg. Mater. Tech. 98 (1976) 193-202. [4] A. Miller, J. Engrg. Mater. Tech. 98 (1976) 97-113. [5] R.D. Krieg, SMiRT-4, San Francisco, CA, 1977, M6/4. [6] D.N. Robinson Oak Ridge National Laboratory, ORNL/TM- 5969 (Oct. 1978). [7] R. Lagneborg, Met. Sci, J. 6 (1972) 127-133. [8] J.R. Rice, J. Appl. Mech. 37, (1970) 728-737. [9] E.T. Onat, and F. Fardshisheh, Oak Ridge National Laboratory ORNL-4783 (August 1972).
T.J. Delph/ Creep, relaxation and cyclic behavior of a beam
[10] J.C. Chang, R.H. Lance and S. Mukerjee ASME J. Pressure Vessel Tech. 101 (1979) 305-310. [11] D.N. Robinson, Oak Ridge National Laboratory, ORNL5540 (November 1979). [12] A.R.S. Ponter, J. de M6chanique 15 (1976) 527-542.
421
[13] R.K. Penny and D.L. Marriott, Design for Creep (McGraw-Hill, New York, 1971). [14] D.R. Hayhurst and A.J. Krzeczkowski, Comput. Meths. Appl. Mech. Engrg. 20 (1979) 151-171.