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Topography of Metallic Surfaces Subjected to Plastic Strain: Roughness Correlations and Eigenvalue Spectra



Joseph B. Hubbard, Mark R. Stoudt, Lyle E. Levine


We employ scanning confocal laser microscopy (SLCM) to obtain topographic images of the surface of an aluminum alloy subjected to various levels of uniaxial strain. These images are discretized into a square array of pixels, each of which is then assigned a numerical value corresponding to the deviation of surface height relative to an averaged background. The result is the generation of a set of real, non-Hermitian n x n matrices, which are diagonalized to produce a collection of spectra, each of which consists of n complex eigenvalues. These eigenvalue spectra are observed to change systematically as the degree of plastic strain is varied. We employ a block-randomization algorithm that destroys all correlation between pixels in patches of size 2m in order to assess the effects of spatial correlations in relative height on these spectra. As 2m approaches n, we observe a pronounced tendency for the normalized eigenvalues to migrate towards the interior of a disc centered at the origin in the complex plane, and the distribution within this disc tends to become uniform as n becomes large. Moreover, the modulus of the largest eigenvalue corresponds closely to the standard deviation of the height probability density function (Rq); this result is in close accord with a theorem, proved by Geman, on the spectral radius of large random matrices.
Journal of Applied Physics


Eigenvalue spectra, non-Hermitian matrices, random matrices, spatial correlation, surface roughness, topographic analysis


Hubbard, J. , Stoudt, M. and Levine, L. (2007), Topography of Metallic Surfaces Subjected to Plastic Strain: Roughness Correlations and Eigenvalue Spectra, Journal of Applied Physics, [online], (Accessed May 24, 2024)


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Created January 10, 2007, Updated February 17, 2017