The effect of cracks on the superconducting transport current in thin films: The analogy with two-dimensional elasticity
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Simulations of arrays of resistively shunted Josephson junctions containing a crack of uncoupled junctions indicate that the crack can distort the supercurrent flow and provide a nucleation site at the crack tip for the formation of superconducting vortices at applied currents below the critical current of the homogeneous material. An analogy is established between the supercurrent distribution in two dimensions and the stress field distribution around the crack for antiplane mechanical loading. The analogy is used to show that the supercurrent distribution can be described analytically in terms of a Westergaard function used in elasticity theory. In addition, using a correspondence between the forces acting on a vortex and a crystal dislocation, models for screw dislocation emission from a crack tip are transposed to describe vortex emission from a crack tip. These lead to predictions for the pinning force required to prevent dissipation by vortex emission from the crack tip, as well as for the size of a vortex zone ahead of the crack for different values of the ratio of the applied current to the pinning force.
I. INTRODUCTION All the high-temperature superconductors, as well as a number of other materials having attractive superconducting transport properties, such as Nb 3Sn and the Chrevel phases, are brittle. As a result, cracking of these materials can occur relatively easily under a variety of different types of stresses. For instance, cracking can occur as a result of thermal expansion mismatch with a substrate, such as has been analyzed by Olsson etal. for [110] oriented YBa 2 Cu 3 0 7 on [110] SrTiO3 substrates.1 Cracking can also occur in response to stoichiometry-induced changes in lattice parameters constrained by a substrate, as has been noted for BKB films grown on MgO substrates by Hellman et al.2 Similarly, the changes in the molar volume of yttrium barium cuprate, as it is oxidized from its tetragonal to orthorhombic phase, and the associated anisotropic thermal expansion produce strains in polycrystalline material that can induce microcracking.3'4 Although cracking can, in principle, be avoided by appropriate microstructural design, it may be impractical. For instance, as with other examples of microcracking,5'6 there exists a critical size below which microcracking is not favored energetically, but so far it has been impractical to make polycrystalline materials with a grain size well below this value. It is thus appropriate to consider the effect of cracks on the attainable transport critical current and microwave losses in the superconducting state, in addition to their more traditional impact on mechanical properties and reliability. In this paper we calculate the effect of cracks on the superconducting transport current, extending two J. Mater. Res., Vol. 8, No. 7, Jul 1993
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previously published analyses for an isolated crack. 7'8 In doing so, we also consider interaction between cracks, an effect that may be especially important for m
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