Self-consistent stress relaxation model

Short Notes K149

phys. stat. sol. (a)% ~ 1 4 9 (1987) Subject classification: 62.20

Pakistan Steel Mills Corporation, Karachi (b), and University of Karachi (cf Self-consistent Stress Relaxation Model

BY S.M. RAZA (a), N. FAROOQUI (a), S.B.H. ABIDI (a), S.A. RAZA (b), and S. M. M. R. NAQVI (c)

Anomalous work hardening behaviour /1/ and its influence on the

Department of Physics, University of Baluchistan, Quetta 1) (a),

Introduction stress sensitivity of the relaxation rate in polycrystalline metals at low temperatures /2/ explained much of the unresolved problems /3 to 7/. In particular, our model on stress relaxation at low temperatures /8/ seems quite reasonable. This is verified from the experimental results on copper single crystals obtained by Dotsenko and Landau /9/. However, earlier studies on polycrystalline nickel /lo, 11/ have shown that the logarithmic stress relaxation prevails at low temperatures. Ray and Mallik /12/ have confirmed this observation and the validity of the expression used for determining the slope of the stress relaxation curves. Recently, Hamersky and Trojanova /13/ have analyzed the shape of the relaxation curves under similar assumptions to those in the case of the creep. Their pre- dictions a re based on qualitative interpretations of the results which are, however, not adequately defined. Our observations about upward curvature in the stress sensitivity of the relaxation rate in polycrystalline nickel at 295 K and the downward curvature in polyerystalline copper and nickel below 77 K /2/, con- firm the experimental results of Hamersky and Trojanova /13/.

In this short note, we will consider a simple stochastic model of lwarithtnic creep /14/, and suggest a more modified relationship for calculating the activation energy. The self-consistent stress relaxation testing will correspond to con- tinuous strain hardening of the specimen during stress relaxation. Hence, the strain rate sensitivity of the relaxation rate for each stress-relaxation curve during work hardening, i.e. s relar' can be calculated from

d d kT - - = - = S v* ' 'relax creep -

1 ) Quetta, Pakistan.

K150 physica status solidi (a) 100

Fig. 1. Made1 diagram of the Instron machine with load cell C and specimen S

A X

where V* is the activation volume. Model W e a re considering the specimen S in the

tensile machine, with C denoting the elastic cell. In Fig. 1, the machine is just assumed to have stopped, with the separation of the plates (shaded) fixed. Due

to relaxation of the specimen (which will creep somewhat), the force will drop in the course of time, so indicating the relaxation. Now, a s the separation of the plates is fixed, the strain of the specimen following stoppage of the machine (t = 0) will lead t o an elongationA1 = lA&$ where A f = E - E~ and 1 being the length of the specimen (we neglect changes of 1 with time) which we regard as constant:

A1 = ~ A E . (2 1 Now, i f M is the spring constant of the cell,- then

Af = MAX, (3)

whereAx is the change in length of the cell due toAf. W e have

A X = -A1 ( 4)

as the plates maintain a fixing spacing, so that

(A1 E ) 1AE =Af/M =AuA/M; A U = 0 - 0 , (5) 0

where A is the cross-sectional area of the specimen. Thus, from (5)

Equation (6) then suggests that the creep rate 6 is not instrument independent and a function of the relaxation rate; it wil l depend on the sprillg constant of the cell and the specimen dimensions (l/A). Thus, E ' is not the same as the elastic modulus of the specimen, as would be the case i f one considered that the over- all deformation of the specimens were zero. However, ipl + ie1 = 0 * ipl=

- 6 = - a /E. Hence, el

Short Note

t f = - E 6 .

K151

( 7)

In order to differentiate (7) from (6), we rewrite (6) as

&-WEE, (8 1

where I 1 M - M l E =arEand a = - EA - T x '

Thus to determinecc, we need to know the beam constant M, the elastic modulus E of the specimen, and its dimensions 1 and A. Now, considering the paper by Buckle and Feltham /14/, the transient creep is expressed (by their equation (12)) as

t

0 8 = s creep

(by their equation (1 7)) by and 'creep

S creep = 2.3 k T d /(uo - mkT)X1, (11)

where x is the adiabatic coefficient of work hardening and 1 corresponds to linear. Taking ipl = kcreep and using (7), in (1 0) and (1 1) we have

1 dU - kflf(0) - d b E dt (t + to)(uo - mkT)X1 - 2.3dln(t +to)

--- -

= -d(O) kTp = - srel&

where E

@ = {u -mkT)y+ * 0

Considering our case for stress relaxation, i.e. (S), w e can modify (13) by writing

For temperatures T, such that mkT <cud we have

2.3kTd(O)arE = s relax

uOX

2.3kTU(O)wE - 2.3kTor(O) - M 1 xa - - =3 uo -

relax S 'relax X

K152

(from (9)). Using (1) in (15), we have

physica status solidi (a) 100

Equation (15) is the same as usual except that it has an a in the numerator. If we assume that T is so low that m k T a ud then from (1 4) we have

Large x will lead to small p and hence (from (12) and (18)) t o low relaxation rate and for small x , to higher relaxation rate. For 'large' strains, the x - values for truly polycrystalline (compressive) specimens will be higher than (for similar t values) for coarse grained tensile wire specimens as the case may be in polycrystalline copper at 15 K (see. Fig. 3 and 4 in /l/); therefore ~ c o m p < pten (the relaxation rate for compressive specimens is relatively less than the relaxation rate for tensile specimens). Apparently, this looks reasonable from experimental results; but this is not true; in fact, otherwise the relaxation rate mechanism would become different for tensile and compressive specimens of the same material and at the same higher strain.

the spring constant (load cell) which must be taken into account for accurate calculations of the relaxation rate and hence the activation energy. This is why the activation volume for each of the stress relaxation curve is the important parameter for more accurate calculations of activation energies. Activation volume and its dependence on adiabatic work hardening, effective stress level, and temperatures can be reasonably estimated from the analysis of the stress relaxation curves in different stages of work-hardening . Thus, we have shown that the geometrical dimensions of the specimen and the spring constant itself have a great influence on the accurate measurement of the activation energy in relaxation rate processes.

What seems important is the geometrical dimensions of the specimens and

Special thanks are due to Mr. Abdul Rauf for typing this manuscript.

Fteferenc e s

/l/.S.M. RAZA and M.Z. BUTT, J. Mater. Sci. Letters 3 629 (1984). /2/ S.M. RAZA and M.Z. BUTT, J. Mater. Sci. Letters 3 955 (1984).

Short Notes K153

/3/ S.M. RAZA, J. appl. Phys. 5 296 (1984). /4/ S.M. RAZA, Acta metall . 1285 (1985). /5/ S.M. RAZA, N. FAROOQUI, S .B .H. ABIDI, and A.K. KHAN, Fizika

/6/ S.M. RAZA and S.A. RAZA, J. Mater. Sci. Letters 5 25 (1986). /7/ S.M. RAZA, Nucleus 5 15 (1985). /8/ S .M. RAZA, S.B .H. ABIDI, and N. FAROOQUI, Appl. Phys. Letters 3

/9/V.I. DOTSENKOand A.I. LANDAU, Mater. Sci. Engng. 3 101 (1976).

357 (1984).

34 (1985).

/lo/ P. FELTHAM, F r e i b e r g e r Forschungshefte I 3 3 29 (1965). /11/ P. FELTHAM, G. LEHMANN, and R. MOISEL, Acta metall . 5 1305

/12/ K.K. RAY and A.K. MALLIK, Mater . Sc i . Engng. /13/ M. HAMERSKY and 2. TROJANOVA, Czech. J. Phys. B35, 292 (1985). /14/ V.A. BUCKLE and P. FELTHAM, Metal Sc i . 5 541 (1975).

(1969). 59 (1983).

(Received September 4, 1986; i n rev ised fo rm January 21, 1987)

Documents

Self-consistent stress relaxation model