The influence of anelasticity on the measurement of internal stress in creep

Scripta METALLURGiCA

Vol. 14, pp. 215-218, 1980 Printed in the U.S.A.

Pergamon Press Ltd. All rights reserved.

~"HE INFLUENCE OF ANELASTICITY ON THE MEASURemENT OF INTERNAL STRESS IN CREEP

Fo Dobe~ Institute of Physical ~etallurgy, Czechoslovak Academy of Sciences 616 62 Brno, Czechoslovakia (Received November 13, 1979) (Revised December 13, 1979) After a sudden reduction of applied stress during steady-state creep below a certain critical value the period of negative creep rate has been observed° According to the contemporary opinions this effect arises from two principal sources: (i) the backward motion of dislocations under the action of internal stress generated by neighbouring dislocations or other obstacles (I) (ii) straightening of bowed dislocations as a result of dislocation line tension (2)° We term the deformation which is due to the bowing of dislocation segments pinned at their terminal points as anelastic deformation. Anelastic and plastic deformations are therefore distinguished on the basis of the type of dislocation motion° It should be pointed out that from the phenomenological viewpoint, the mechanism sub (i) is also anelastic (3). The dip test technique for the measurement of internal stress is based on the assumption of activity of the former mechanism (4,5)° The reliability of the dip test technique is evidently sensitive to the degree of activity of the latter mechanism, ioeo of anelastic deformation during negative deformation° Experimental separation of the contributions of both mechanisms is not possible at present° A certain progress in the solution of this problem could perhaps be reached by those physical models which specify in detail both the distribution of dislocation segment lengths and the stress acting on them° One of these models is the stochastic model of creep (6,7), which will be used for the calculation of anelastic deformation in the present paper° The stochastic model of creep is based on the balance of thermal activations of strain tmit, eogo dislocation segments° If nd~ is the number of dislocation segments which require for their motion activation energy in an interval from to ~+~m, the time derivative of the frequency function n is (6)

dn

=

~

~ L nJz ~ v... ( _ ~r ,/ k T ) ,

,~

~ is the attempt frequency (~=I011 s -I (8)) I ~ is the magnitude where ~ ==~/~, # of the c~ange in activation ener~j during successfull actlvation and kT has its usual meaning. Confining our considerations to the steady-state creep where ~ n / ~ =0, we get (7)

The integration constants E I and K~ can be calculated by means of measured macroscopic quantities: the steady-state creep rate ~ and the dislocation density in steady-state ~ o These quantities are given by the relations

215 0036-9748/80/020218-04502.00/0 Copyright (c) 1980 Pergamon Press Ltd.

216

INTERNAL STRESS IN CREEP

Vol. 14, No. 2

f:f tz ,

(4)

where ~ is the length of the Burgers vector and ~ is the length of th~ dislocation segment° To determine the relation between the segment length ~ and the activation energy ~ we use the following approach: Activation energy is a function of the link length and the local effective stress

where ~ is the activation energy in the stress-free state. As in the previous work (7) we take ~ as the same for all obstacles and as equal to the activation enthalpy of volume self diffusion (6). This is in accordance with models for the motion of jogged screw dislocations° The local effective stress is given as the difference between the applied stress ~ and the local value of the internal stress. We assume that the dominant component of internal stress is that arising from neighbouring dislocations i.e. we deal with pure metals and solid solutions. The local effective stress is then (9,10) ~

: ~"

;

~-

(6)

where ~ is the shear modulus and ~ is a geometric constant (=0.2).From eqso (5) and (6) we can determine the relation between ~ and ~ which is necessary for calculations of the integrals in eqs. (3) and(4). Knowing the steady-state values of the creep rate and of the dislocation density, we can calculate by means of eqso (2) to (6) the corresponding constants ~ and K~. I I i Dislocation se~nents bow out under the action of local effective stress. If the bowings are small, the area swept during Fe - Ni -Er [123 ~ " bowing is (11) g73 K

//e//~/e

This equation is justifiable! if~ for angle between bowed dislocatxon and its .stress-free position, 8~$/n8. The angle Is glven by

Jy"

n-

~J

• STEADYSTATE

I'(/) ,<

AFTER STRESS REDUCTION (FROM (5 : lg6MPa)

-4~)

I

I

100

200

Using eqs. (5) and (6) we get for maximum bowing (i.e. for u =0)

0~,,

300

@o

=

~I,""

(9)

Inserting for ~,the values of activation enthalpy of volume self diffusion we get for~m4x uppermost values around 0.4 (sin Applied stress dependence of 0.4=0o389), which means that the used calculated anelastic deformation approximation does not deviate appreciably from the exact solution. The total anelastic deformation following from the bowing of dislocation segments is APPLIED STRESS ~ [MPo ] FIG. 1

-- I n @

~'~d~

•

(10)

An example of calculations of ~ for austenitic steel (12) is given in Fig. l. Calculations were realised on a digital computer. Integrations in eqso(3),(4) and (10) were performed in the limits 0 to ~ . For transitigns between normal and shear stresses and deformations the relations 6 =£~t & =F/~ were used.

Vol. 14, No. 2

INTERNAL STRESS IN CREEP

217

If the applied stress is reduced sufficiently quickly, the internal stress immediately after this stress reduction will be unchanged° This means that after the stress change only the first term on the right-hand side of eqo(7) is changing° The dependence of anelastic deformation after such stress change for one value of the original applied stress is given by dotted line in the Fig° Io (Individual points represent anelastic deformations in steady-state for given applied stresses@) From the figure as well as from eqo (7) it follows that the anelastic deformation after the stress change could reach even negative values, ioeo dislocation segments bow in the opposite senseoWith a further delay on reduced stress, internal stress will decrease - as a consequence of recovery - and anelastic deformation will, therefore, increase° This is in agreement with the experimentally observed behaviour (13): After the complete unloading at first there is observed a gradual decrease of deformation, which is after a certain time relieved by the small increase of deformation° The changes of anelastic deformation calculated by means of the stochastic model are approximately two orders of magnitude smaller than the changes of elastic deformation at the same stress changes° On the other hand, experimentally measured decrements of deformation after the stress change are nearly equal to or greater than corresponding decrements of elastic deformation (1315)o This fact points out that the negative deformation observed after the stress decrement is mostly due to the mechanism sub (i)j that is to the backward motion of dislocations under the effect of internal stress° The dip test technique consists of measurement of the creep rate immediately after the stress change° By means of the above introduced procedure we can calculate not the rate of anelastic deformation but the "equilibrium" value of anelastic deformation corresponding to the effective stress at the moment° In accordance with others (16) we assume that the rate of anelastic deformation is proportional to the difference between the instantaneous value of the anelastic deformation ~ and its "equilibrium" value ~

d~ Y,~

_

f',q6-,~ ~r

~

(11)

where tr is constant° (It is the relaxation time of anelastic deformation°) Knowing the relaxation time ~r ~ it is possible to calculate from eqo(11) by means of magnitudes of anelastlc deformation determined by the above introduced procedure the rate of anelastic deformation immediately after the stress change° The total creep rate after the stress change is calculated by addition of this contribution with the contribution following from dislocation motion, ioeo with the rate of plastic deformation@ ~~hl"s other contribution can also be determined from the stochastic model, namely by integration of eqo(3) with a new activation energy @' in the exponential term

u'~, u 4. ,,'x ~ { ~ ,

(12)

where ~ is the decrement of shear stress° This procedure is discussed elsewhere in greater detail (7)@ Analogously with the dip test technique, it is possible to find the particular stress decrement which is followed by a creep rate e ~ l to zero@ This stress decrease is designated as the mean effective s t r e s s T and the corresponding residual stress as the apparent internal stress ~ o The results of calculation of the apparent internal stress by means of the stochastic model are given in Figo2a as a function of relaxation time ~r o At short relaxation times the calculated internal stress approaches the applied stresso In practice, at very short relaxation times the anelastic deformation is realised so quickly that it cannot be measured° The exact values of relaxation time #r are not available at present@ (Relaxation time for the limiting case of dislocation bowing by climb is calculated by Reynolds et ale (17)o) Therefore, we attempt to assess the maximum influence of anelasticit~ on the measurement of internal stress° Regarding the construction of presen, experimental devices for measurement of creep rate, it is

218

INTERNAL STRESS IN CREEP

Vol. 14, No. 2

not possible to measure a creep rate after the stress change credibly before a certain time interval, which at the best is several seconds, say 5So (Approximately the same time was necessary for unloading in experiments of Gibbons et alo (141, while A hlquist et alo (i8) needed for internal stress measurements time intervals of SOs°) The greatest rate of anelastic deformation at this moment is for relaxation time also equal to 5So The examples of internal stress calculation without the influence of anelasticity (~r = oo) and with its maximum influence (~=~r=Ss) are given in Fig° 2b, again for an austenitic steel (12)o From the fi~zre it follows that the anelasticity can significantly influence the measured values of the apparent internal stress° Nevertheless, even at the maximum influence of anelasticity the measured internal stress is, for the greater part, due to the dislocation motion under the influence of local internal stresses@ References AoAoSolomon and WoDoNix, Acts Met° 18, 863 (1970)o G°JoLloyd and RoJoMcElroy, Acts Met@ 22, 339 (1974)o J@PoPoirier, Acts Met. 25, 913 (1977) GoB@Gibbs, Philo ~ago 13, 317 (1966)o CoNoAhlquist and WoDoNix, Scripts Met° 3, 679 (1969)o PoFeltham, Physo star@ Solo 30, 135 (1968)o FoDobe~, Acts Met° in press@ U@FoKOCkS, Ao~(;oArgon and MoFoAShby, Progro Mat@ SCio 19, 124 (1975)o GoSaada, Electron Microscopy and Strength of Crystals, po651o Interscience, New York (1963)o 10o RoLagneborg, B@HoForsen and J@Wiberg, Creep Strength in Steel and High Temperature Alloys° The Metals Society, London (1974)@ 11o FoRoNoNabarro, Theory of Crystal Dislocations, p@ 117o Clarendon Press, Oxford (1967)o 12o KoKuchaPov~, AoOrlovd and Jo~adek, Kovov~ Mater° 13, 598 (1975)o 13o MoPahutov~, Jo~adek and PoRy~, Scripts Met° 11, 1061 (1977) o 14o To B@ Gibbons , VoLupinc and Do~{cLean, Met° Scio 9, 437 (1975)o 15@ V@Lupinc and F@Gabrielli, Mat° Scio h~ugngo 37, 143 (1979)o 16@ JoHoGittus, Scripts Met@ 6, 247 (1972)o 17o GoLoReynolds, WoB@Beer~ and BoBurton, Met@ Sci@ 11, 213 (1977)o 18o CoN@Ahlquist and WoDoNix, Acts Met° 19, 373 (1971)@

1o 2@ 3@ 40 5o 60 7@ 8@ 9o

o

I

I

I

I

i

I

J

I

I

Fe-Ni Cr 9"73 WITHOUT ANELASTICITY

u~300 U"}

MAXIMUM INFLUENCE OF ANELASTICITY

---

I.U m,." l*M')

•

MEASURED[12]

//

_J

z~ 2o0 n.IJJ

(::,*

// //

•

~ lOO

0

_~6 ~-4 L-2 ~0 12 14 ~6 10 10 10 10 10 10 10 RELAXATION TIME tr [S]

I

I

I

100

200

300

APPLIED STRESS G [MPo]

FIG@ 2 Dependence of internal stress on a) relaxation time and b) applied stress@