Roughness Exponents, Microstructure, Correlation Length, and the Possible Origin of Selfaffine Fracture
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Roughness Exponents, Microstructure, Correlation Length, and the Possible Origin of Selfaffine Fracture. M. Hinojosa1, J. Aldaco1, R. Rodríguez2, U. Ortiz1. 1 FIME, Universidad Autónoma de Nuevo León, Mexico. 2 Inst. Tecnológico y de Estudios Superiores de Monterrey, Monterrey, Nuevo León, Mexico. Abstract The self-affine character of the fracture surfaces of metals, polymers and ceramics has been well documented over the past two decades. It has been established that these surfaces are selfaffine objects characterized by so called ‘universal’ roughness exponents independent of the microstructure and the loading conditions. Here we show that the self-affine correlation length is closely associated with the microstructure heterogeneities. We also explore the possibility of the existence of attractor values that govern the fracture process, as opposed to universal exponents. The possible origin of this behavior is also briefly discussed. Introduction The problem of linking length scales is particularly important in the modeling of crack propagation and fracture of heterogeneous materials, particularly when one takes into account the self-affine character of these phenomena, which manifest at all relevant scales, from the atomic to the macroscopic levels. In the past twenty years great advances in the understanding of the fracture process have been made by the concurrent development and application of Fractal Geometry, Atomic Force and Tunneling Microscopy, as well as other techniques. Mandelbrot [1] unveiled the fractal character of fracture surfaces and suggested that the fractal dimension is related to macroscopic mechanical properties. The current scenario [2] can be outlined as follows: fracture surfaces manifest statistical invariance through an affine transformation: (x, y, z)→(bx, by, bζz), ζ quantifies the roughness of the surface and can be evaluated by a variety of methods [3] knowing that for a self-affine surface the typical height h(r) scales as rζ. Fracture surfaces are self-affine up to a characteristic length called the correlation length, ξ. For rapid crack propagation it has been postulated [4] the existence of a somehow ‘universal’ value of the roughness exponent, ζ ≈ 0.78-0.8 regardless of the material, microstructure or load conditions. For slow crack propagation, such as that resulting from fatigue tests, and particularly when the analysis is performed at small enough length scales, ζ appears to have another ‘universal’ value of 0.5 [5]. This state of affairs implies that the self-affine character of fracture surfaces is independent of the mechanical properties. Within the theoretical framework of ‘line model’ [6, 7], a fracture surface can be modeled as the trace left by the line of the crack front as it advances. The morphology of the line thus determines the morphology of the fracture surface. The results obtained so far [6] include qualitative agreement with the existence of two characteristic roughness exponents but no clear relationship between the microstructure and the roughness
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