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Heterogeneity and Implicated Surface Effects: Statistical, Fractal Formulation and Relevant Analytical Solution
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The aim of this paper is to examine the implications of material heterogeneity on brittle material response and on relevant surface effects. Statistical and fractal concepts are used for this purpose. In the statistical formulation the displacement gradients of the micro-medium are considered to be random fields characterized by stationary exponential or Gaussian auto-covariance and by the relevant correlation length or scale of fluctuation. Through Taylor series expansion around the mean of the random field, an important analogy is found between the statistical formulation and the micro-structural theory, originally introduced by Mindlin, where higher order gradients of deformation appear in the constitutive equation. The analogy is valid only when fluctuations are small, so that some simplifications are allowed. It is found that the so-called internal length appearing in the micro-structural formulation is analogous to the correlation length in the statistical one. In the statistical approach there are no extra boundary conditions in the formulation as is the case when higher order gradients are introduced. However, what is known as "conditioning" of the field at the boundaries effects its behavior near/on them. The statistical approach can provide further information in the form of higher order moments not captured by the gradient theory. Material/structure response is strongly dependent on the aforementioned scale. Its effect is most pronounced near the boundaries of a structure where its role on surface related phenomena is paramount. In order to study heterogeneity at a hierarchy of scales, i.e. absence of characteristic length, complex disorderly system, fractal concepts and relevant power decay laws are considered. The formulation introduces the fractal dimension of the heterogeneous displacement gradient of the micro-medium, a length describing the overall size of the structure, and the lower cutoff of the scaling law. The physical interpretation of the lower cutoff is the lower limit of applicability of the power scaling law. Mathematically it is important since in this case the fractal can be "followed" in the spatial domain. Similarly to the statistical case, an analogy between the fractal formulation and gradient theories is identified. No extra boundary conditions appear in the fractal formulation. However, there are still open questions with respect to the behavior of a fractal after conditioning, as is the case on boundaries. The analytical solution of a relevant surface for the gradient, statistical and fractal formulation is presented. The solution was obtained through symbolic computations by computer because the analytical work is tedious and error prone. The analytical solution provides significant insight into the problem of heterogeneity and skin effects in brittle materials, internal length estimation, and the role of fractal scaling properties. Finally, the concepts introduced herein are discussed with respect to experimental information and numerical implementation.
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