Plasticity in Polycrystalline Thin Films: a 2D Dislocation Dynamics Approach
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Plasticity in Polycrystalline Thin Films: a 2D Dislocation Dynamics Approach Lucia Nicola1 , Erik Van der Giessen1 and Alan Needleman2 Netherlands Institute for Metals Research/Dept. of Applied Physics, University of Groningen, Nyenborgh 4, 9747 AG Groningen, The Netherlands 2 Division of Engineering, Brown University, Providence, RI 02912, USA 1 The
ABSTRACT Thermal stress evolution in polycrystalline thin films is analyzed using discrete dislocation plasticity. Stress develops in the film during cooling from a stress-free configuration due to the difference in thermal expansion coefficient between the film and its substrate. A plane strain formulation with only edge dislocations is used and each grain of the polycrystal has a specified set of slip systems. The film–substrate interface and the grain boundaries are impenetrable for the dislocations. Results are presented for two film thicknesses, with higher hardening seen for the thinner films. INTRODUCTION The high strength which characterizes polycrystalline thin films on substrates has experimentally been found to depend both on the film thickness and on the grain size [1, 2]. In bulk metal the yield stress usually depends on the grain size d through the Hall-Petch relation d −1/2 : the grain size acts as a constraint on the length of dislocation pile-ups at grain boundaries. In thin films on substrates, the film thickness gives an additional constraint to dislocation activity. Mainly because of the difficulty in performing suitable mechanical tests on thin films with controlled grain size, the dependence on both film thickness and grain size in polycrystalline films are not yet fully understood. The discrete dislocation plasticity framework adopted here permits the two size effects to be studied independently; for space reasons, however, only the film thickness effect is addressed in this paper. MODEL The polycristalline film is modeled as a two-dimensional infinitely long array of rectangular grains of thickness h and width d (Fig. 1). Each grain (γ) is characterized by three sets of slip systems, with slip planes inclined with respect to the film-substrate interface by: φ γ1 = φγ , φγ2 = φγ + 60◦ and φγ3 = φγ + 120◦ . The film is taken to be perfectly bonded to a semi-infinite substrate with a different value of the coefficient of thermal expansion α. During thermal loading, the substrate constrains the film’s expansion in the x1 –direction (see Fig. 1). The substrate behaves elastically, but the stress in the film is partially relaxed by the motion of dislocations, which nucleate and glide on the slip planes in the grains. The grain boundaries as well as the interface between film and substrate are taken to be impenetrable barriers for the dislocations. The plane strain boundary value problem formulation and numerical method follow that in [4], where thermal stress evolution in single crystal films was analyzed. A more complete description of the methodology and additional references are in [4].
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