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Near-tip field for diffraction on square lattice by rigid constraint

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Abstract

The displacement field, due to diffraction of a time harmonic lattice wave on square lattice, near the tip of a semi-infinite rigid constraint is investigated. A rigorous proof, supported by numerics, establishes that the exact solution of semi-infinite rigid constraint diffraction provides an approximation in ℓ 2 of the solution near a tip for finite rigid constraint diffraction. A closed-form expression for the displacement at lattice sites ahead of, as well as adjacent to, the rigid constraint tip is provided. A low-frequency approximation of the normalized shear force in ‘horizontal’ bonds, ahead of the semi-infinite rigid constraint, captures the classical edge diffraction singularity. It is demonstrated, with graphical illustrations, that at low frequency, the normalized shear force in ‘vertical’ bonds along the constraint approach the continuum approximation as expected.

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  1. Department of Mechanical Engineering, Indian Institute of Technology Kanpur, Kanpur, 208016, U.P., India

    Basant Lal Sharma

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  1. Basant Lal Sharma
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Sharma, B.L. Near-tip field for diffraction on square lattice by rigid constraint. Z. Angew. Math. Phys. 66, 2719–2740 (2015). https://doi.org/10.1007/s00033-015-0508-z

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  • DOI: https://doi.org/10.1007/s00033-015-0508-z

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