Modeling of Themally activated Dislocation Glide AND Plastic Flow
through Local obstacles
Masato Hiratani*, Hussein M. Zbib*, and Moe A. Khaleel**
*School of Mechanical and Materials Engineering, Washington State
P.O. Box 642920, Sloan 201, Spokane St., Pullman, WA 99164-2920, *[email protected],
**Pacific Northwest National Laboratory, P.O. BOX 999, 902 Battle
Blvd, Richland, WA 99352, [email protected]
ABSTRACT A unified phenomenological model is developed to study the
dislocation glide through weak obstacles during the first stage of
plastic deformation in metals. In this model, effects of thermally
activated breakaways of dislocations from obstacle arrays are
estimated analytically while dynamical properties of dislocations
during the flight process are obtained numerically by the discrete
dislocation dynamics (DD). Setting typical situations for copper
sample, the model reproduces transient from obstacle-controlled motion
at low stress regions to drag control motion at high stress regions.
INTRODUCTION: The interaction between dislocations and local obstacles
has been an essential research topic to determine the roles of
microstructures on macroscopic plastic the flow and hardening of
materials for long time. In single crystals at up to room temperatures
during the initial deformation stage, the system considered has been
rather simple: small number of dislocations on a few glide planes with
randomly distributed local obstacles such as stacking fault
tetrahedron (SFT), vacancy loops, or precipitates with short-range
stress fields. In such a system, dislocation motion is driven by
thermal activations at low stresses below the critical resolved shear
stress c, and hence, temperature elevation usually accompanies
softening of materials. It has been recognized, however, that
hardening could be found at very low temperatures in several dilute
alloy systems [Suzuki 1995], metals under strong magnetic fields,
etc. These anomalies are envisaged due to dynamical effects during
dislocation flight between consecutive metastable configurations,
which are usually overshadowed by thermal activation processes. The
previous phenomenological unified model by Hiratani et al. (2001) can
reproduce the aforementioned various features of the dislocation
motion successfully, but the model is based a number of approximation:
the dislocation line tension is negligibly small and it proceeds as a
straight line during the flight, dynamical effects are considered
solely at the collision with obstacles while real flights show complex
both forward and lateral motions, and unzipping dislocation-obstacle
bound configurations. The objective of this work is to improve the
model by evaluating the flight quantities using discrete dislocation
dynamics (DD) [Rhee et al. 1998], which should provide more
PROCEDURES, RESTULTS AND DISCUSSION: We estimate the average waiting
time tw spent in the metastable configurations based on generalized
Friedel relations analytically, whereas the average flight distance and
run-time tr are obtained from DD simulations, approximating the
average dislocation velocity in a form of
where indicate the averaged quantity. In addition, we select
an empirical form of tw as
where b is the magnitude of Burgers vector, D Debye frequency, lp the
average pinning distance, Go Helmholtz free energy, i are numerical
factors of order of unity, and p and q are fitting parameters. The
local quantities such as average pinning distance or obstacle strength
are related to the shear stress by the Friedel relations. We choose
one type of SFTs as local obstacles and the fitting parameters p and q
are estimated by calculating interaction energy between an infinite
straight dislocation and a SFT analytically. It turns out that Go is
relatively large, more than 3eV even in case of SFTs of size of just
10b placed off the glide plane, and dislocations become practically
immobile when the local dislocation force on the obstacle F is less
than 0.3Fc where Fc is the obstacle strength.
Flight quantities tr and are evaluated by DD simulations (Fig. 1).
Here is defined as a/Lx where a is the average swept area after an
activation event and Lx is the length of simulation cell side along
which a dislocation is oriented. The DD code is modified to include an
effective dislocation mass m so that the equation of motion is given
where the effective mass m is approximated as , is the mass
density, dislocation line direction, M is mobility,
is the total drag summed over all contributions [Al'shitz,
1992] from phonons and electrons, and and are the
stress fields formed by local obstacles and other dislocations. Copper
is selected as a model material and the Peierls stress is regarded as
a constant internal stress. According to the current main theories,
the relaxation time , which is a monotonically decreasing
function of temperature T, is estimated to be around 200ps at T10K
while around 9ps at T325K for an edge dislocation. Underdamped
conditions at the collisions are satisfied over a wide range of
temperature, i.e. T150K for low stress regions at 0.3c and T200K
at c in case that SFTs are scattered with average spacing 250b
and c1.86MPa. Some data, i.e. a, are obtained and the inertial
bypasses are evidenced during the simulations. About 70 data sets of
tr and are collected from multiple runs by changing the M (3.01047.0105/Pa.s)
and (04MPa). The corresponding temperature ranges from about 10K
to 325K, and data are extracted by non-linear fitting from simulation
results and by interpolations for those in the intermediate region.
The average dislocation velocity v(,T) obtained from Eqns. (1) and
(2) with the extracted data are shown in Fig.2. Non-linear and linear
stress dependencies are reproduced below and above c along with
reversal of the temperature dependence at transient regions. Fig.3
illustrates the competitions between thermal activation process and
dynamical flight process near but below c, which result in anomalous
negative temperature sensitivity for high velocities. The coupling of
two processes, therefore, is essential to describe the complex
dislocation behavior near the critical stress.
Fig.1 Dislocation percolating on
(111) plane with SFTs.
Fig.2 Stress dependence of the average velocity at various
Fig.3 Temperature dependence of the flow stress at various velocities.
Acknowledgement: The support from the Pacific Northwest National
Laboratory is greatly appreciated.
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