11.7 Material Characterization

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Material properties description is another application for wavelet analysis. Intuition suggests that

multiscale analysis is a natural way of describing microstructure or material heterogeneity. Various, in

fact, are the examples of multiscale microstructures, such as porosity distributions in ceramics, defects,

dislocations, grain boundaries, and pores. It is important, however, to understand how information at

different scales is related, and whether large or small scales affect macroscopic material properties such as

deformation, toughness, and electrical conductance. Further interest towards a multiscale description of

material properties is motivated by the need of alternatives to the standard finite element method (FEM).

11-18 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

The latter, although capable in principle, cannot simulate the actual behavior of materials such as

aluminum alloys, where pores may attain a size up to 500 mm and inclusions may attain a size of 3 to

6 mm in diameter. Further, in FEM-based methods, the constitutive response of the material at increasing

scales is not the result of microstructural analysis at smaller scales, but it is rather assumed on the basis of

macroscopic experiments.

Willam et al. [107] have performed multiresolution homogenization based on a recursive Schur

reduction method in conjunction with the Haar wavelet transform. The method allows coarse-grained

parameters, such as Young’s modulus of elasticity, to be extracted from fine-grained properties at the

meso- and microscales. Also, progressive elastic degradation can be modeled, which initiates at a quite

fine scale and evolves into a macroscopic zero stiffness at the continuum level.

Frantziskonis [108] has focused on stationary and isotropic porous media. The geometry of porous

media is generally described in terms of a fundamental function, defined as unity for spatial locations in

the matrix, and as zero for locations in the pores or flaws. At a solid-flaw interface, the porous medium is

represented mathematically through a local jump in the fundamental function. It has been found that

such a jump can be captured by a wavelet transform, as long as the finest scale is small enough relative to

the size of the pores. From this fact, a relationship between the energy of the wavelet transform of the

porous medium, and the variance and the correlation distance of the solid phase can be derived. In the

presence of heterogeneous materials, with multiscale porosity, the role of porosity at each scale has been

identified through the variation of the energy of the wavelet transform as a function of scale. Peaks of the

energy reveal the dominant scale in determining macroscopic properties of the materials, such as

mechanical failure. Specifically, a biorthogonal spline with four vanishing moments has been employed

as a wavelet basis. The results obtained have been subsequently extended in a second study, addressing the

crack formation in an aluminum alloy with distributed pores and inclusions [109]. The problem,

implemented for a one-dimensional solid, is tackled by wavelet transforming the flexibility function,

assumed to vary along the longitudinal axis of the one-dimensional solid. The relationship between the

energy of the wavelet transform and the variance of the flexibility is used to detect the dominant scale in

the crack-formation process.

Note that an application of a two-dimensional wavelet transform has been described in Ciliberto et al.

[110] for porosity classification on carbon fiber-reinforced plastics.

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