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

University

P.O. Box 642920, Sloan 201, Spokane St., Pullman, WA 99164-2920, *[email protected],

**[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

realistic information.

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

(1)

where indicate the averaged quantity. In addition, we select

an empirical form of tw as

(2)

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

as

(3)

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

temperatures.

Fig.3 Temperature dependence of the flow stress at various velocities.

Acknowledgement: The support from the Pacific Northwest National

Laboratory is greatly appreciated.

REFERENCES:

Al'shitz V. I., 1992, “The Phonon-Dislocation Interaction and its Role

in Dislocation Dragging and Thermal Resistivity” in Elastic Strain and

Dislocation Mobility. ed. V. L. Indenbom and J. Lothe, 31 chapt.11,

Elsevier Science Publishers, North-Holland.

Landau A. I., 1980, “The Effect of Dislocation Inertia on the

Thermally Activated Low-Temperature Plasticity of Materials. I.

Theory”, Phys. Stat. Sol. (a) 61, 555-563.

Hiratani M. and Nadgorny E. M., 2001, “Combined Model of Dislocation

Motion with Thermally Activated and Drag-Dependent Stages”, Acta

Mater. in press.

Rhee M., Zbib H. M. and Hirth J. P., 1998, “Models for Long/Short

Range Interactions in 3D Dislocation Simulation”, Modeling &

Simulations in Maters. Sci. Eng. 6, 467-492.

Suzuki T., 1995, Quantum Theory of Dislocation Motion in Metals. Proc.

Symp. Micromech. Adv. Mater. TMS, 9-15.