Size effects on nominal tensile strength of concrete structures: multifractality of material ligaments and dimensional transition from order to disorder

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The nominal tensile strength of concrete structures is constant for relatively large sizes, whereas it decreases with the size for relatively small sizes. When, as usually occurs, the experimental investigation does not exceed one order of magnitude in the scale range, a unique tangential slope in the bilogarithmic strength versus size diagram is found. On the other hand, when the scale range extends over more than one order of magnitude, a continuous transition from slope −1/2 to zero slope may appear. This means that for smaller scales a self-similar distribution of Griffith cracks is prevalent, whereas for larger scales the disorder is not visible, the size of the defects and heterogeneities being limited. In practice there may be a dimensional transition from disorder to order. The assumption of multifractality for the damaged material microstructure represents the basis for the so-called multifractal scaling law. This is a best-fit method that imposes the concavity of the bilogarithmic curve upwards, in contrast to the size effect law of Bažant. The relevant results in the literature for ranges in scale extending over more than one order of magnitude are analysed.