PRBVDECE RNI AND'V'D TH THRA ACIATO RAT EQUTIO BY R. W. ARMSTRONG. S 5 Oad" (if 4dc~tRap 4e4ta'ie l/ce4 u I

Transcription

1 PRBVDECE RNI AND'V'D TH THRA ACIATO RAT EQUTIO BY R. W. ARMSTRONG (if 4dc~tRap 4e4ta'ie l/ce4 u I S 5 Oad"4 1966

2 Relation Between the Petch "Friction,' Stress and the Thermal Activation Rate Equation* R. W Armstrong A e? Brown Universit The following discussion is addressed towards establishing a connection between two alternative constitutive equations that have been used in the past to describe the temperature (and strain rate) dependence of the ield stress of iron. 1 Heslop and Petch proposed that the temperature dependence of the ield stress of iron was primaril determined b the intrinsic lattice resistance to crstal dislocation movement, i.e. the Peierls-Nabarro stress. Their experimental measurements for this "friction" stress, a were expressed in one form as a B exp (-T) (1) where T is the absolute temperature and B and 8 are experimental constants. It was pointed out that the strain rate has a large effect on a and this influence enters equation (1) implicitl through the parameter 82,3. However, for a fixed strain rate, (1) is relativel eas to evaluate and it ma be used as well to describe measurements obtained for other materials *This stud was supported at Brown Universit b the Advanced Research Projects Agenc and at the Solid State Division of the Oak Ridge National Laborator through a Research Participant Appointment b the Oak Ridge Associated Universities. I *W D Sov1 813

3 -2- Conrad6 has shown that the results from a considerable number of studies of the plastic ielding of iron and steel ma be expressed in the relationship # 2kT CoT +- 1.n (2) where 9 is the tensile strain rate, k is Boltzmann's constant, and Uo, V, and 9 are-parameters emploed in the thermal activation rate analsis. The parameters emploed in equation (2) have some direct theoretical basis, e.g. U is an activation energ associated with the rate controlling process involved in dislocation movement, V is the activation volume through which work is done and e is a product of several factors: a geometric factor (relating tensile strain rate and shear strain rate), the dislocation densit, the area swept out in dislocation movement between obstacles, and the vibrational frequenc of the dislocation line. Experiments have shown that V is itself a function of, at least, o, and this seems reasonable on the basis of dislocation theor. To relate the parameters emploed in (1) and (2), it ma first be noted that at T =, the value of B is directl obtained as B = V V (3) where V is the value of V at T =. For iron, U = 8.8 x 1-13 ergs, as given b Conrad 6, and B 1.8 x 1 dnes/cm 2, as determined from the data of Heslop and Petch Substitution of these values in (3) gives v L 9.6 x 1-2cm, and this value compares favorabl with the lowest value of V = 1.2 x 1-22cm3 estimated b Conrad.

4 -3- To evaluate 8, the right hand sides of (1) and (2) are equated and the terms rearranged in the form 1 io 2kT - I n [ + -VB In ](4) Now, based on the work of Conrad 6, tpical values, at T = 19 K and = -4 sec-1 for the additional parameters in (4), are V = 3.4 x 1-22 cm3 * 8-1 and e 5 x 1 sec". This value of T is selected as the median tempera- ture for the range (8-3 K) over which most measurements have been made and the value of V is also the median value obtained, b Conrad for this temperature range. Using the preceding estimates, it ma be seen that < 2kT so that 8 ma be expanded in series, taking into account, also 4 U > 2kT I 6 VB~ VBn - )I(6 to give k_ In In (- (7) e For lower or higher temperatures, or different strain rates, it occurs that the change in a% and, hence V is such that (7) should still hold ver well. Also because V increases as the temperature increases (because of the variation of at ), it appears that these changes ma largel counteract one another in (7) to give a constant vajef,lb"ai ALT),a813

5 -4- as indicated b the experimental basis for (1). This point is further examined below b comparison between the experimental value of taken from (1) and that derived from (7). The value of obtained from the experimental data of Heslop and Petch, at 6 = 1-4 sec ", is 1.43 x 12 OK -. Emploing the preceding values given for U and o the value of the second term on the right hand side of (7) is obtained b difference as I f n (---) = 9.7 x 1- OK (8) At 19 K, the value of (8) obtained directl from U, V and B is 6.5 x 1-3 OK-1. Since (8) should not var with temperature, then at fixed e, An ft ] 1 {n--) ft (2 I AT + -- AV (9) T VB T 2 VB TV From (9), the variation in (8) for the temperature interval (19-8)OK and (19-3) K is x 1-3,K-1. respectivel, using for the limiting temperatures the values of V = 2.1 x 1-22 and 4.6 x 1-22 cm3 taken from Conrad. In light of the varied experimental data and the estimates involved in all the quantities emploed, the variation given b (9) is considered to indicate that the second term on the right hand side of (7) ma be approximated b~a constant value, o. Taking = 6.5 x 1-3 K, then (7) gives a value of = 1.1 x 1-2 K-1, which compares favorabl with the value of 1.43 x 1 OK determined b Heslop and Petch. Thus (1) ma be finall rewritten k Sexp + U n ( ) T] (1) O

6 -5- Note that in a numerical evaluation of a the smaller value of B deter- mined from the thermal activation rate parameters would largel compensate for the smaller a value given above. In (1), therefore, the Petch "friction" stress has been expressed fairl directl in terms of the thermal activation rate analsis parameters. A definite connection between the equations (1) and (2) is established. In conclusion, some brief comments should perhaps be made concerning the usefulness of the foregoing analsis. It shows an explicit influence of the strain rate on the parameter 8, in agreement with the previous suggestion b Heslop and Petch 2 and Petch 3 that this parameter is strain rate dependent. The analsis offers a further indication of the relative self-consistenc of the large amount of data collected until the present time on this aspect of the deformation of iron, a view promote4 initiall b the work of Conrad6 References 1. J. Heslop and N, J. Petch, Phil. Mag. 1, 866 (1956). 2, J. Heslop and N. J. Petch, Phil. Mag. 3., 1128 (1958). 3. N, J. Petch, Fracture, Ed., B. L. Averbach, D. K. Felbeck, G. T. Hahn and D. A. Thomas, Technolog Press, Boston, 1959, p P. Feltham and G. J. Cople, Acta Met, 8, 542 (196). 5. R. W. Armstrong, Phil. Mag. 9., 163, (1964). 6. H. Conrad, J. Iron St. Inst. 198, 364, (1961).