Effetti di scala sulla resistenza a trazione dei materiali

The dissertation analyses the scale effects on the tensile strength of materials. By the term scale effects it is meant the variation in a mechanical property as a function of structural size. In particular, it has been observed by numerous investigators that the nominal tensile strength of many materials decreases with increasing size of the specimen tested. This phenomenon is more evident in disordered materials, that is, materials that are macroscopically heterogeneous and damaged. On the basis of Weibull's statistical theory and the principles of Linear Elastic Fracture Mechanics, a self-similarity distribution for defect size is presented (Chapter 4). With this distribution the length of the most critical defect is taken to be proportional to the linear size of the specimen. It is shown that the assumption of self-similarity represents the instance of maximum disorder that can be encountered in real materials, and it supplies, in a strength-size bilogarithmic plane, a linear scaling law with an inclination of -1/2, corresponding to the power of the stress singularity envisaged by LEFM. This formulation contains the fractal concept of self-similarity, even though it is limited to maximum defect dimension. In order to consider the real nature of the micro-structure of the materials, a more complex fractal model is presented (Chapter 5) in which the property of self-similarity is extended to the entire population of defects. This topological law, based on fractal theory and on the so-called renormalisation procedure, states that in order to obtain a nominal constant strength for the material it is necessary to refer to surface areas with non integer physical dimensions. For disordered materials, such as for instance concrete and rocks, renormalised tensile strength is given by a force acting on a surface having a fractal dimension lower than 2. The dimensional decrease, always comprised in the [0, 1/2] range, represents self-similar vacancies in the undamaged section associated with the presence of pores, voids, defects, cracks, aggregate and inclusions, and it approaches the 1/2 limit only for extremely brittle and disordered materials, as is assumed, incidentally, in statistical approaches. As a rule, the scale variation taken into consideration in experimental investigations does not exceed one order of magnitude. In such circumstances, it is only possible to determine a single tangential inclination in the bilogarithmic diagram. Only by taking into account scale variations higher than one order of magnitude it proves possible to detect the transition from disordered to ordered conditions, and a continuous transition from -1/2 to zero inclination may be seen to appear. In physical reality, the peak load resistant section can be viewed as multifractal, of dimension 1.5 on small scales and dimension 2 at large scales. This clearly shows a transition from the extreme disorder that is associated with small scales, where a self-similar distribution of Griffith cracks predominates, to the extreme order of large scales, where the disorder of the microstructure is no longer visible, on account of the limited dimensions of the defects and heterogeneities. The assumption of multifractality for the microstructure of the damaged material (Chapter 7) is the basis of the so-called Multifractal Scaling Law (MFSL). Such law consists of an approximation method which imposes the concavity of the bilogarithmic curve facing upward, which contradicts Bazant's size effect law (SEL). To verify this scaling law and to determine experimentally the variation in nominal tensile strength and fracture energy, a totally innovative testing set-up has been created, involving the use of three servo-controlled jacks (Chapter 5). The interaction of the three jacks, arranged in L formation, makes it possible to centre instant by instant the resultant of the load with the respect to the undamaged section even in the presence of cracks which make the latter asymmetrical. The main goal of this instrumentation is to determine the parameters of the concrete subjected to uniform tension, eliminating any secondary bending effect which may affect the results and lead to erroneous explanations of the scale effect.