micromechanical modelling of creep using distributed parameters

ArIo MProlturgir. Vol 29. pp 283 to 292 Pergamon Press Ltd 1981 Printed m Great Britain

MICROMECHANICAL MODELLING OF CREEP USING DISTRIBUTED PARAMETERS

RlSHI RAJt and A. K. GHOSH

Rockwell International Science Center. 1049 Camino DOS Rios. Thousand Oaks. CA 91360. U.S.A.

(Receic)ed I Seprerrrher 1980)

Abstrnct-Microstructural parameters such as diffusion coefficients and grain size usually have distributed vaiues in a polycrynal. However. micromechanical constitutive equations for power-law creep and diffusional creep assume that these parameters are uniform. In this paper it is shown that the use of distributed parameters can lead to a transition from power-law creep to diffusional creep which is spread over several orders of magnitude in strain rate. It also leads to the generation of internal stress and anelasticity. Two specific cases, a bimodal distribution of grain size and a fwo phase structure. are anaiysed in detail. A graphical analytical technique which is developed can be extended to other distribution functions; but even the simple bimodal structures are relevant to engineering materials such as austcnitic stainless steels and m-8 titanium alloys. This paper primarily concerns steady state creep and includes a brief discussion on the transient problem. Further developments will be given in a subsequent paper.

R&M&-Les paramttrcs microstructuraux teis que les coeficients de diffusion et la tailte des grains pr&entent une r&partition de valcurs dans un polycristai. Ccpendant, Ies Equations constitutives de la micromtcanique pour le fluage xion une loi en puissance et pour fluage de diffusion, supposent que ces paramttres sont uniformes. Dans cette article, nous montrons qu’une rtpartition de ces paramitres peut provoquer un ktaiement de la tcmpCrature de transition entre ie fluage seion une loi de puissance et ie fluage de diffusion, sur piusieurs ordres de grandeur de la vitesse de dtformation. Une teiie ripartition pcut igalement conduire B des contraintes inremes et B de I’antlasticitC. Nous analysons en d&ail deux cas particuliers, une r&partition bimodale de la taille des grains et une structure diphasbe. La technique d’analvse graphique que nous dkveioppons peut itre appfiqute B d’autres fonctions de r&partition; toutef& mime Ies structures bimodales simples intkressent ies mat~riaux de l’ingtnieur. tets que les aders inoxydables aust&nitiques et les aliiages de titane a-)?. Cet article concerne essentieiiement Ie fluage stationnaire et il contient une courte discussion des phtnomtnes transitoires. Nous prisenterons dcs dtveioppements ultCrieurs dans un autre article.

ZusammeafassuagMikrostrukturelie Parameter wie DiffusionskoefEzienten und KorngrSBe weisen in eintm Polykristail iiblicherweise eine Verteifung auf. In mikromechanischen Grundgieichungen ftir das Potenzgesetz- und Diffusionkriechen werden diese Parameter dagegen als einheitiich angenopmen. In dieser Arbcit wird gezeigt, da0 die An~hme ciner Vcrteiiung in den Parameterwerten tinen Ubergang vom Pota~g~t~~~hen zum ~ffusio~k~hen erzeugen kann, der sich iiber etliche Gr~~nordnun- gen in der Dehnungsrate erstreckt. Diem Ubergang Nhrt aulkrdcm zu inneren Spannungen und zu Anciastizit&t. Zwei besondere Fllle, eine bimodale Verteilung der KorngrijBen und eine zweiphasige Struktur, werden ausNhrlich untersucht. Ein grafisch-anaiytisches Verfahren wird entwickeit : es IiiDt sich oufandere Verteiiungsfunktionen erweitern. Die einfachen bimodalen Strukturen jedoch sind schon anwendbar bei WerkstolTen wie LB. austenitischen StPhlen und +Titaniegierungen.

~ODU~ON

The problem analysed in this paper can be most simply understood as follows. Let us consider two additive mechanisms of deformation in polycrystals: namely power law creep [l] and diffusional creep [Z-J. The accepted constitutive equation for steady store creep then is that

t On leave from the Department of Materials Science and Engineering. Cornell University. Ithaca. NY 14853, U.S.A.

where IE and Q are the strain and stress in a uniaxial test, or the equivalent strain and stress under multi- axial stress state, d is the (perfect) equiaxed grain size and D, and Db are the volume and boundary self- diffusivities; other parameters have their usual mean- ing&2].

The grain size distribution in real polycrystals is not monoiithic, as is assumed in the derivation of the second term in equation (I), but exhibits diversity depending upon the processing history of the material. According to equation (1X then, different parts of the polycrystal would tend to creep at different rates as dictated by the local grain size. Since compatabiiity would require uniform rate of strain

283

284 RAJ AND GHOSH: MICROMECHANICAL MODELLING OF CREEP

everywhere, internal stresses would develop and the stresses will re-distribute until a uniform rate of deformation is indeed achieved. The internal stress distribution would depend on the distribution of the grain size.

It is easily appreciated that a distribution in the effective diffusion coefficients in the polycrystal, as. for example, in a two phase microstructure, will lead, phenomenologically. to the same effects as described above.

The purpose of this paper is not to solve the com- pletely general problem of distributed parameters, but to present an approximate solution in order to emphasize the physical effects arising solely from using distributed parameters. These effects are illustrated by considering duplex microstructures, e.g., bimodal grain size, or two phase alloys, and by using a graphical technique for analysis. The technique, however, can be easily extended to more complex distributions, as illustrated in Appendix I. More exact, numerical results are also presented and compared with limiting stress-strain rate behavior which is obtained graphically.

We have considered here only a limited portion of the deformation map [3]: that which straddles across the power law creep and Cable creep mechanisms. In this region, the stresses are low and the temperature ranges from moderate, to high. It was chosen because of its engineering importance. Structural materials, e.g., austenitic stainless steels, are often used and tested under these conditions, and superplastic form- ing of fine grain alloys, e.g., 31-f!l titanium, is also con- tained in this regime. Considering only these two mechanisms. equation (1) simplifies to the following:

where k,, kb, and 3 are normalized parameters such that:

and

a_-d G'

(3)

D,Gb k,=A-

kT ’ (4)

and

14aG6Db kbskT. (5)

In equation (2), R refers to the atomic volume.

DUPLEX MKRmUCTURES

As a first step we shall consider the steady state stress-strain rate response of duplex microstructures. In one case, a polycrystal of single phase material will be assumed to have a bimodal grain size distribution.

This is often not far from reality in polycrystals that contain particles (or pores) in the grain boundaries. Pinning of grain boundaries can lead to exaggerated grain growth under certain conditions [4] which essentially results in a bimodal grain structure. Stain- less steels. which contain stabilized carbide precipi- tates, often develop such a microstructure as shown in the pictures in Fig. 1. Figure 1 (a) is representative of a regular mill annealed structure while Fig. I(b) shows an extreme example of duplex grain size which can result when a heavily worked stainless steel is solution annealed. A histogram of the chord lengths of ‘line intercepts’ for the regular structure is shown in Fig. 2. Although the histogram provides only a qualitative description of the grain size distribution [5]. it does serve to show that the grain sizes can be separated by quite a large factor (about 4 in this instance).

The other type of a bimodal structure is one in which the diffusion coefficients have a bimodal distribution because the structure is an alloy of two phases. A good example is the titanium alloys that can be superplastically deformed in the z-/l region. We shall show that the stress-strain rate behavior, in this instance, can be quite sensitive to the volume fraction of each phase. The lattice diffusivities of the phases are assumed to be different. The difference between the effective grain boundary diffusivities will depend upon the number of like and unlike interfaces formed at the grains of either phase. Although this can be calculated if the volume fractions are known, we shall assume that the more important effect is the difference between the lattice diffusivities. The boundary diffusivities will be assumed to be equal.

The material parameters in equations (2)-(5). e.g., k,, k, R, n, and d will be qualified by a subscript 1 or 2 according to the bimodal properties. For the single phase material all parameters except d, and d2 will be the same. In the two phase structure, d, # d2 and k,, # kp2, but all other parameters will be assumed to be equal. Actually, the equations which are developed in the next section allow greater generality, but these simplifications are made for the sake of the numerical analysis. The volume fractions of 1 and 2 will be rep resented by t’i and v2; of course t’i + v2 = 1.

STEADY STATE CREEP

Equations

In the steady state, compatibility would require that both regions 1 and 2 creep at the same rate, which leads us to the following equations:

and

ai - (‘5)

and that

n (7)

RAJ AND GHOSH: MICROMECHANJCAL MODELLtNG OF CREEP 285

Fig. 1, Examples of a regular mill-annealed structure of 316 stainless steel (a), and the structure after annealing heavily cold worked material (b). Both have duplex grain size.

ci = 6, - EC2 (8)

where d is the measured strain rate. Here Z, and 5, are the equival~t stresses experienced in the two regions of the bi-modal structure. They are unknowns in equations (6) and (7) but are related to the applied stress, Cs,, through the equation for equilibrium:

6, = Z,r, -+ a2v2

Equations (3) through (6) will yield solutions for P, Z,, and i?a once CTi, and the material parameters have been prescribed.

The above equations can easily be extended to higher order distributions by asserting that < = II = . . (I c2 = fj = c,$ = etc. and that ZW = Z~iviiv,.

It should be recognized that equations (6)-(9f are an approximate description of the problem at hand. For example, the real behavior should depend not just upon the volume fraction, but also on the mor- phologies of regions 1 and 2. It would be necessary to carry out finite element ~lcuiations[6~ to obtain a full solution to the problem, the simple analysis presented in this paper is meant to provide only an approximate solution.

Although the Bow behavior can be exactly ealcu- lated with the use of equations (Q-o-(), a graphical technique has been developed For a quick estimate of the stress-strain rate curve. The technique depends


::

i

$ 0.4

8

g

0.5 L

0.3

0.2

0.1

0 0 L-l I

10 20 30 40 50 60

AVERAGE INTERCEPT LENGTH (urn)

Fig. 2. A histogram of the chord lengths when measuring grain size by the intercept method for the structure in

Fig. l(a).

upon the simple rule that if the creep resistance of the two regions is different, then the approximate strain- rate can be obtained by asssuming that deformation is controlled by the species having the higher creep resistance; except that the creep rate will be enhanced by a stress-enhancement factor equal to the inverse of the volume fraction of the species supporting the applied stress. The use of the method is illustrated for the case of the bimodal grain size and the dual phase microstructure.

Bimodal grain size

Let the two grain sizes be d, and dz. where d, < dl, and their volume fractions t’, and ut respectively. The graphical construction is shown in Fig. 3(a). At high stress, both grains deform by power law creep; this is the region AB. At B, the smaller grains change their mechanism to diffusional creep, but the larger grains continue to deform by the (slower) power law creep. Therefore in region DB, the asymptotic behavior is given by FH, which is parallel to DB but is shifted by a stress enhancement factor l/u2 according to the rule mentioned above. At D, the larger grains also switch to diffusion creep, however, since the larger grains have the higher creep resistance the asymptote KF is shifted with respect to ED by a stress enhancement factor l/t+ The asymptotic behavior is therefore given by KF-FH-HB-BA.

All that is needed to find the asymtotes is to define the position of points B and D. These are detined by the condition that power law creep and diffusional creep, (i.e., the first and the second term in equation (2) make equal contribution to the strain rate. Therefore, the stress and strain rate at the transition points B and D, which we denote by a, and O,, will be given by:

k& to = k,a; = - a0 d3

(10)

a,, can be eliminated from equation (10) to express i0 simply in terms of the material parameters:

It follows that the distance between D and B depends upon d, and d, and will be given by (&,2)r,/(&,I)B = (d2/dl)“““- ‘), as shown in Fig. 3(a).

The asymptotic behavior was compared to the numerically computed results using equations (6). (7). (8) and (9). For the purposes of the calculation it was assumed that:

dJd, = 11.65 VI = 0.9 L‘2 = 0.1 n=5

Arbitrary units were used, and the first transition point B was represented by 5 = 1 and go = 10e3. The exact curve in thick solid line. It deviates from the asymptotic behavior only near the transition points by a factor of not more than 2.5 in strain-rate.

The strain rate sensitivity of the material, m, defined as m = Alog Z/Alog i, can be calculated from the computed curve. The results are shown at the bottom of Fig. 3. The thick line corresponds to the curve in Fig. 3a. The other curves represewing other ratios of dz/dl are included to show how the distribution of the grain size affects the m vs i behavior.

The notable feature of Fig. 3(b) is that the bimodal grain size spreads the transition region over several orders of magnitude in strain rate: about two orders of magnitude if d&i, = 2.5, but nearly five orders of magnitude when dz/d, = 11.65.

The graphical technique described here can be extended to higher order distributions. An example is given in Appendix I.

Two phase structure

Once again, the asymptotic stress-strain-rate behavior can be constructed in the following steps:

(a) Construct the power law creep lines for each phase.

(b) Mark the transition to diffusional creep for each phase.

(c) Enhance the creep rates of the slower moving species by adding a stress-enhancement factor equal to the inverse of the volume fraction of that species.

An example is shown in Fig. 4. For the dual structure we assume that kD2 > k,, and that d, > d, while the other material parameters remain the same. Lines ABC represent the stress strain rate for phase 1 and DEF for phase 2. The lateral distance between the transition points BE is given by io2/i0, where each of &, is defined by equation (11). The lateral shift

0.0’

/ E

I-

i

0-l 0.001

1

0.8

0.6

E

RAJ AND GHOSH: MICRUMECHA~ICAL MODELLING OF CREEP

3 10.7 10-6 10-5 10-4 10-a 10-Z 70”’ 1

1 (AR% UNITSI

10-8 10-7 10-6 10-6 10-4 to-3 10-2

c (ARE%. IJNtTSl

Fig. 3. Log o YS log 4 curve for a bimodal grain size in a single phase material. The slope of the curve, 1~1. is plotted as a function of log i in the bottom figure. The effect of changing the d,!d, ratio is also shown

in the bottom picture.

bctwcen lines AB and DE is simply (&,z/k,l). Having done this, the stress ~h~~rn~t factors are imple- mented and the asymptotic behavior HK-KL-LM- MN is obtained.

For the numerical calculation the foilowing valuts wm assumed:

k,dk,, = 10’ dl/dI = 3,4

C 1 = 0.3

v2 = 0.7

n, =??*=5

I 1 I, I 1 I i 1 i 10-e m-7 10-6 10-S 104 10-3 10-2 10-l 1 ‘10

c (AR%. UNITS)

1.0

0.8

0.6

E

0.4

0.2

0 1

i

“1 ‘ “2 = 5

Fig. 4. Log u YS log < curve for a twb phase strucmre in which the tatticc diffusivitirs differ by a laeror of IO*. The phase with the higher diflusivity (2) is also assumed to have the larger grain size, The volume fraction ratio r2/r1 is kept tixed at 0.7&.X3. The slope. m, Cs plotted at the bottom where the in!%xnce of

changing dl;d, is afso shown.

AM other parameters in equations (5) and f?) were cft/dI = 7 and about two orders d ma~itude when assumed to be equal. Again the asymptotes are a &dl = 1.5, good approximation to the exact calculation.

The m vs P curve corresponding to Fig. 4(a) is shown by the thick solid line in Fig. +b). For com-

TRANSIENT BEHAWOR

parison, the influence of different values of d,/d, is The use of distributed parameters would lead to a included. Again the transition region spreads out over type of transient which would not occur if the par- several orders of magnitude in strain rate: it spans ameters were single valued, The transient would arise about five orders of magnitude in strain rate when simply because of the variation in the creep resistance

RAJ AND GHOSH: MICROMECHANICAL MODELLING OF CREEP 289

TIME

Fig. 5. A sketch of the expected stress relaxation behavior when the displacement is held fixed after reaching the steady state. Nonlinear relaxation. as would occur in the region BA in Fig. 4. will lead to ‘a’ whereas linear viscoelastic behavior. e.g. in region HKL in Fig. 4. should lead to ‘b’. The stress relaxation behavior. therefore. will be

strain-rate dependent.

in different parts of the microstructure. This would require a redistribution of internal stress such that the creep, rates approach equality in all regions. An ap ::..oximate treatment of this problem for a bimodal ~::ucture is presented here. More detailed work is reported elsewhere [73.

The approximate magnitude of the anelastic strain can be calculated by assuming that, at steady state, the stress is supported entirely by one type of grains. If the other grains are assumed to be stress-free then they are equivalent to holes, which will serve to increase the complies of the material?. If I‘ is the volume fraction and E is Young’s modulus of the grains which have to support all the applied load then the anelastic strain. h. will be approximately given by:

t Since the grains will relieve their load in a time-dependent manner. the increase in compliance will be ‘anelastic.’

The second term in equation (12~ is the time indepen- dent. elastic strain. The notable feature of equation (12) is that if r is considerably smaller than I then h can be several times the elastic strain.

It should be remembered that equation (12) represents only that anelastic strain which arises from the use of distributed parameters. There are other mechanisms. such as grain boundary sliding. which will further decrease the effective modulus: the increase in Ae will be given by the geometric rather than the arithmetic sum of the anelastic moduli arising from the different mechanisms.

The relaxation time of the anelastic strain will be determined by the mechanism by which the ‘weaker’ grains will deform to relieve their load. In the region MN in Fig. 4 for example, grains of typ 2 will deform by power law creep to shed their load. while in region HKL the type 1 grains will deform by diffusional creep to give up their load. The stress-relaxation behavior would, therefore, be quite different in these two regions. A likely form of the stress-relaxation curves in these two regions is sketched in Fig. 5. They represent the relaxation in stress when the displacement is held fixed after reaching the steady state. Curve a is typical of stress relaxation for a non-linear creep mechanism while curve b represents linear viscoelastic behavior. A more detailed theoretical and experimental analysis of the transient problem is given in Ref. [i’].

DISCUSSION

The important feature of the results is that the transition from power law creep to diffusional creep is likely to depend on the distribution of the microstructural parameters such as the grain size and the diffusion coefficients. In all cases the transition will become broader (in strain-rate) than if only the average or idealized parameters were used. For the distributions we have used here, the transition region can

I I I I

ld - 1

304 STAINLESS STEEL - 666K

I I I I

104 10-z 10’2 10-l

I W h-l)

Fig. 6. Creep data for steady state creep in 316 stainkss steel [II]. The wide spread probably implies heat to heat variations in the microstructure. There is qualitative resemblance between the data and the shape

of the curve shown in Fig. 3.

290 RAJ AKD GHOSH: MICROMECHANICAL MODELLING OF CREEP

dA,,E, = -W - T = 1179K TeSAWV

1”‘ 19’ vJ*

STRAIN RATE IS’)

Fig 7. Experimental log u log a data for a superplastic TidAl-4V alloy [9]. with qualitative resemblance with the shape of the curve in Fig. 4.

span 3 or 4 orders of magnitude in i instead of less not be realistic. Nevertheless, in austenitic stainless than l/Z an order magnitude which is obtained if the steels the bimodal grain structure is often found. and average parameters are used. the superplastic z-/I titanium alloys consist of a fine

The bimodal microstructural distributions we have grained, two phase, microstructure. More rigorous considered here do have some realism, although the comparison with experiment must await further spread in the parameters we have used may or may experimentation since grain size distributions have

1

0

0

E

0

0

.o -

.8 -

.6 -

A -

I.2 -

o-

TRI-MODAL G.S.

9 Y

215 8:: 5 0.1

n-6

10” 10.’ 10= 10-y 10’ W’ 1o‘L lo” 1 i IARB. UNITSTS)

Fig. 8. The influence of a broader distribution is to smooth out the variation in m vs log i. as shown here b! comparing the curves for trimodal and bimodal grain size distribution.

RAJ AND GHOSH: MICROMECHANICAL MODELLING OF CREEP 291

not been traditionally reported along side creep CONCLLSIONS measurements in the literature. For the sake of comparison, however. a collection of creep data on 316

The use of distributed material parameters. e.g.. dif- f

stainless steel [8] is reproduced in Fig. 6. It is at least usion coefficients and grain size, can lead to a broad

qualitatively in agreement with the shape of the curve transition from power law creep to diffusional creep.

shown in Fig. 3 and Fig. Al in the Appendix. Simi- Near the transition it is possible that mixed deforma-

larly, the shape of the curve for a two phase material. tion may occur where the large grains deform by

given in Fig 4, resembles the actual stress-strain-rate power-law creep, but the smaller grains deform by d’ff 1

curve for a Ti-6A1-4V [9] alloy shown in Fig. 7. This usional creep. This may appear as a threshold-like

hh raises the possibility that the flow pattern of super-

avior in the 0 vs i curve. The distributed par-

plastic deformation depends upon the distribution of ameters will also lead to an internal stress which may

material parameters. rather than on any fundamen- cause significant anelastic strains. More work. both

tally new deformation mechanism. theoretical and experimental, is needed to study the

In real microstructures the distributions would be engineering importance of these concepts.

more complex than the simple bimodal structures Acknowledgemenr-This work was supported by the Inde- considered here. The graphical and the numerical pendent Research and Development Program at Rockwell

technique developed in this paper can be extended to International Science Center. We wish to thank Dr G.

higher order distributions. An example is given in the Subbaraman for providing the stainless steel specimens.

Appendix. The general effect of broadening the distri-

bution would be to reduce the amplitude of the undu- APPENDIX I

lation in ths m vs i curves such as those in Figs 3 and Graphical consrruction for a nlore general disrrihurton ./kc- ,ion

4. For example, in Fig. 8 it is shown that the influence of using the trimodal instead of the bimodal distribu- Here it is illustrated that the graphical technique de-

tion is to make m less sensitive to the strain rate. It scribed in the text can be extended to higher order distri-

follows that in most experimental situations the maxi- butions. For example. consider a single phase material with four types of grain sizes. d, < d2 c d3 < d,. and with

mum and minimum in the m vs i curves will not volume fractions c,. rl. r3 and L‘, with the condition that

appear as distinct as suggested by the curves for bi- ZL) = 1.

modal distributions; it will be more diffuse because First, the line for power-law creep is constructed which is

the actual distributions will be continuous rather than AB (and its extension shown as a dotted line). The tran- sitions to diffusional creeo then occur at B. C. D and E for

discrete. the grain size d,, d,. d3 and d,. respectively. Now in retion

n-4

/ / : X:f

6 0.2 2 0.1

I I I I I I I

10-8 10-7 104 10-a 10-4 4 (AM. UNITS)

10-3 10-z 10-l 1

Fig. AI. Graphical construction for a multi-modal grain size distribution for a single phase material.


BC. grains 1 will unload because they have lower creep resistance, hence there will be a stress enhancement factor I/& + rs + tr) which gives the line GF. Similarly in DC. grains 3 and 4 will support the load which will cause a stress enhancement of l/(rs I- c,). and so on. Thus the asymptotic behavior JI-IH-HG-GF-FB-BA is obtained. The exact numerically calculated curve is included. The values of the parameters were as follows: dt = 1. d2 = 3. d3 = 6. d, = 9, and rt = 0.4. rs = 0.3. rs = 0.2 and L:, = 0.1; n = 4 and k, and k, were chosen such that the first transition (point B) occurred at a = I and e = 10-s.

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R. Raj and M. F. Ashbv. Meralf. Trans. A. 2. t 113 (1971). M. F. Ashby. Acra metaft. 20. 887 11972). C. Zener. auoted bv C. S. Smith. Trans. metalf. Sot. A.I.M.E. Si5. p. I5 i1949). J. W. Cahn and R. 1. Fullman. J. Metals’Trons titerall. Sot. AIME, p. 610 (1956). J. Jinoch. S. Ankern and H. Margolin, Morer. Sri. Engng. 34,203 (1978). A. K. Ghosh and R. Raj. Acra metal/. in press. V. K. Sikka. Oakridge National Laboratory. Prelimi- nary Report: (1978). T. 1. Mackay. C. F. Yolton. R. F. Malone and S. M. L. Sastry. Interim Report. Contract F33615-C-5198. April 1979. Douglas Aircraft Company, Long Beach. CA 98046.

micromechanical modelling of creep using distributed parameters

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