A model for the strain-rate dependence of yielding in Ni 3 Al alloys
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I.
INTRODUCTION
THE anomalous increase in yield stress with temperature of certain LIz alloys (such as Ni3AI) has been well described by a mechanism through which strengthening is derived by the cross slip of screw dislocation segments from octahedral to cube cross-slip planes.[I,Z] These crossslipped segments then act as pinning points for the octahedral portions of the dislocation line. Cross slip is assumed to be thermally activated, and as temperature increases, the density of pinning points rises which produces the observed increase in flow stress. In addition, the state of stress (as influenced by crystallographic orientation) has also been shown to influence the cross-slip process. [I ,Z,3] In general, trends in the effect of strain rate are expected to be opposite to those of temperature. Following this reasoning, a higher flow stress should be observed for lower strain rates in Ni 3AI-ordered alloys. Experimentally, it has been observed that the flow stress of single crystals is independent of strain rates for the strain rates at which testing has been done.[4,5,6] This observation is one of the more perplexing deformation characteristics of these materials. In this paper, a conceptual model based upon kinetic considerations is proposed to address the apparent strain-rate independence of Ni3AI materials.
II.
MODEL DEVELOPMENT
[1]
critical resolved shear stress on primary octahedral glide plane; b = magnitude of the Burger's vector;
1'111
=
ARNAUD de BUSSAC and GRAHAM WEBB, Graduate Students, and STEPHEN D. ANTOLOVICH, Professor and Director, are with the Mechanical Properties Research Laboratory, School of Materials Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0245. Manuscript submitted May 31, 1990. METALLURGICAL TRANSACTIONS A
Thus, the yield strength is linearly related to the density of cross-slipped segments. The "equilibrium" density of pinning points was defmed as the number of pinning points N rnax(T, 1') that would occur if an infinite amount of time were allowed for a given temperature and stress. The rate at which crossslip segments form to reach the equilibrium value can be expressed as N. = dN = -1/ dt Ie
[
(1 - x) exp
(
pin -H-) kT
- x exp ( -
HunPin)] kT
[2]
where Ie = critical length of cross-slipped segment; 1/ = frequency of attempts; x = fraction of dislocation line that IS cross slipped; H pin = activation enthalpy* for cross slip; Hunpin = activation enthalpy for reverse cross slip; k = Boltzman' s constant; and T = absolute temperature. *Enthalpy was used instead of Gibbs free energy, since entropy effects would obviously be small.
In Ni3AI materials, the creation of numerous pinning points on the screw dislocation line increases the drag stress and causes the dislocation to remain straight while in motion. The applied force per unit length (Tb) is balanced by the drag force due to these pinning points:
where
Co = drag force per pinning point; and N = number of pinning points per unit length.
The activation enthalpies defined in Eq. [2] describ
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