We study theoretically and experimentally the effect of long-range correlations in the material microstructure on the stress concentration in the vicinity of the notch tip. We find that while in a fractal continuum the notch-tip displacements obey the classic asymptotic for a linear elastic continuum, the power-law decay of notch-tip stresses is controlled by the long-range density correlations. The corresponding notch-size effect on fracture strength is in good agreement with the experimental tests performed on notched sheets of different kinds of paper. In particular, we find that there is no stress concentration if the fractal dimension of the fiber network is D≤d-0.5, where d is the topological dimension of the paper sheet.

In this work the crack initiation stress of notched specimens of filter paper is studied. The paper in the microstructure has a random array in their fibers while macroscopically it behaves anisotropically. The self-affine crack mechanics is used to study the size effect in the tensile behavior of this kind of paper under the presence of several conditions of geometrical notches. While in the traditional fracture mechanics the crack initiation stress is a material parameter when is reached a critical level at the crack tip, in the self-affine crack mechanics, depends moreover of the resulting tortuosity of the crack. Four geometrical arrangements in two sizes we considered: centered circular notch, centered lineal notch, sided circular notches and without notch at 10 and 300 mm width with a relation 2a/w = 0.25 under the same loading conditions. In this, the without notch specimens present the higher stress, all other notched specimens presented a similar crack initiation stress about 1 % of difference among them, and the crack growth is not affected by the geometry of notch. In spite of this difference, no one of the specimens reach the theoretical stress concentration of 3 such as predicted the classical stress theory.

In this work the self-affine crack pattern in Filter Paper sheets is studied. This paper has a well-defined anisotropy of mechanical properties associated with visible preferable orientation of fibers in the machine direction. Fracture behavior is in essence brittle, the rupture lines have self-affine invariance, and the stresses ahead of the straight notch follow a power-law behavior. The roughness exponent value is of H = 0.50 0.01, different from the suggested universal value H = 0.8. The classical theory has demonstrated that, in materials such as metal, there is a relationship between the size and the starting crack stress, which does not happen in this material. The tests show that the starting crack stress from stress-strain behavior curves remains stable for the different specimen sizes w and crack length size. Moreover, different types of geometric groove, circular and linear, and without a crack, were tested and show almost the same behavior.

In the study of mechanical properties of materials the microstructure of a material is usually subjected to some kind of homogenization; however, there are materials in which the microstructural disorder must be considered. This disorder manifests itself in the fracture resistance of materials. Some empirical experimental studies and various types of models (based on variations in mass per unit area) have been made to relate the effect of the disorder during crack propagation with the macroscopic resistance of the material, but the absolute-density/mass projections have not been a good descriptor to extrapolate the behavior of the material between its microstructure and the macroscale since it is difficult to determine the porosity and the net trajectory of the fibers. The physical phenomenon of the instability of the crack propagation of interest in the present work occurs on a meso-scale, where the microstructure of the materials can be characterized only statistically and has been established as the range in which the bridge can exist between the micro and macro behavior of this kind of materials. By the Digital Image Correlation Technique the crack propagation is followed based on the displacements produced locally by the arrangement of the fibers in front of the crack tip of paper, as a material model. At the beginning of the load process is observed a smooth trace in the peak local deformation corresponding with the elastic part of the stress-curve; after, when the stress-curve starts to deflect, the peak local-deformation trace change in its slope and it becomes intermittent, this behavior is attributed to the local conditions of material. Finally, it observed that the local deformation is a good descriptor for the crack extension.