The statistical characterization of glass strength: from the micro- to the macro-mechanical response

Abstract

This work discusses an innovative micromechanically-motivated statistical characterization of the strength of annealed and heat-treated glass, with the aim of achieving a reliable definition of the mechanical properties of this material for structural applications also taking into account the effects of natural aging, thus avoiding either unsafe or redundant design.

The 2-parameter Weibull distribution, based upon the weakest-link-in-the-chain model, is by far the most common statistics to interpret the glass strength. However, its universal use has to be questioned, because comparison with the experimental evidence indicates that it cannot accurately interpret the left-hand-side tail of the population. This has very strong implications for structural design where the tails play the decisive role, because only very low probabilities of failure are tolerated. Starting from the observation that many experimental campaigns recorded in the technical literature seem to indicate that the measured strength of float glass cannot fall beyond a certain limit, even when the material is heavily damaged either naturally or artificially, we present arguments to support this conjecture. The lower bound can be attributed to factory production controls for marketed glass, which tends to discard the material that does not meet severe aesthetic and optical requirements. Since these are associated with the presence of surface cracks, the control should indirectly limit their depth and size, thus providing a threshold for the average strength according to well-established models in fracture mechanics. How the potential interaction between pre-existing cracks and cracks added by abrasion may affect the corresponding stress intensity factor is further discussed, demonstrating that the variation is in any case limited, and, consequently, the lower bound for the strength may be reduced, but not annihilated, by natural aging.

For a better interpretation of the aforementioned hypothesis, the connection between an assumed statistics for the crack-size population and the population of macroscopic strengths is further established. If one assumes a statistical distribution \textit{\`{a} la} Pareto to interpret the variability of crack depths, the 2-parameter Weibull distribution is obtained for the strengths. Remarkably, the effect of an upper-truncation of the population of crack lengths, possibly consequent to factory production controls, provides a left-truncated Weibull distribution that excellently fits most of the experimental results for annealed float glass, at least when it is the air-side under tensile stress in bending. Other generalized Weibull statistics, either bounded or unbounded, have been considered for the sake of comparison. For each of them, proper expressions for data re-scaling, to take into account the effects of size, type of applied stress and subcritical crack propagation, are provided.

Certainly, aging in the form of corrosion or abrasion can produce a variation of the defectiveness scenario originally present on the glass surface. The established correlation between the micro-defects and macroscopic strength allows to predict how the latter can be affected by variations in the defectiveness scenario. In particular, corrosion is assumed to be equivalent to the removal of a thin surface layer of glass, which reduces the depth of the micro-cracks, whereas abrasion consists in adding new defects to the pre-existing ones. This approach is used to interpret the difference in strength experimentally-measured at the tin-side with respect to the air-side, by assuming that the contact with the tin bath and the steel rollers during manufacturing is equivalent to a mild abrasion process.

This micro-macro approach is also employed to discuss the mechanical properties of heat-treated glass, where a compressive eigenstress is permanently induced by heating and successive cooling. Several experimental campaigns, recorded in the technical literature, have demonstrated that the characteristic strength of heat-treated glass can be much higher than the simple sum of the pristine material characteristic strength and the characteristic value of the prestress. A micromechanically-motivated cumulative probability function for the population of strengths of heat-treated glass is proposed, which accounts for the statistical interference between the mechanical properties of pristine glass and the eigenstress. This model justifies and confirms the experimental findings without having to refer to other phenomena like crack annealing, which certainly plays a significative role but is not essential. The statistical interference is deeply affected by the applied state of stress: the benefic effect of the heat-treatment is maximum when the applied stress is uniaxial, minimum when this is uniform equibiaxial.

All the previous findings are applied to the calibration of material partial factors to be used in the semi-probabilistic (level I) method of design, in order to guarantee the target probability of failure established by structural standards. The calibration is made on paradigmatic case-studies, comparing the results with those obtainable with full-probabilistic (level III) methods of design. For annealed glass, the partial material factors calculated from the proposed statistics are much lower than those obtainable with a 2-parameter Weibull model. Moreover, heat-treated glass reveals a quite surprising strength capacity, which seems to have been hardly appreciated before. An ad hoc experimental campaign has been conducted to corroborate the theoretical findings.

The refined statistical models presented here, motivated on the basis of micro-mechanical considerations, should increase the confidence in the structural applications of glass, achieving its optimal exploitation and, thus, improving its competitiveness on the market with respect to other construction materials.

This work discusses an innovative micromechanically-motivated statistical characterization of the strength of annealed and heat-treated glass, with the aim of achieving a reliable definition of the mechanical properties of this material for structural applications also taking into account the effects of natural aging, thus avoiding either unsafe or redundant design.

The 2-parameter Weibull distribution, based upon the weakest-link-in-the-chain model, is by far the most common statistics to interpret the glass strength. However, its universal use has to be questioned, because comparison with the experimental evidence indicates that it cannot accurately interpret the left-hand-side tail of the population. This has very strong implications for structural design where the tails play the decisive role, because only very low probabilities of failure are tolerated. Starting from the observation that many experimental campaigns recorded in the technical literature seem to indicate that the measured strength of float glass cannot fall beyond a certain limit, even when the material is heavily damaged either naturally or artificially, we present arguments to support this conjecture. The lower bound can be attributed to factory production controls for marketed glass, which tends to discard the material that does not meet severe aesthetic and optical requirements. Since these are associated with the presence of surface cracks, the control should indirectly limit their depth and size, thus providing a threshold for the average strength according to well-established models in fracture mechanics. How the potential interaction between pre-existing cracks and cracks added by abrasion may affect the corresponding stress intensity factor is further discussed, demonstrating that the variation is in any case limited, and, consequently, the lower bound for the strength may be reduced, but not annihilated, by natural aging.

For a better interpretation of the aforementioned hypothesis, the connection between an assumed statistics for the crack-size population and the population of macroscopic strengths is further established. If one assumes a statistical distribution \textit{\`{a} la} Pareto to interpret the variability of crack depths, the 2-parameter Weibull distribution is obtained for the strengths. Remarkably, the effect of an upper-truncation of the population of crack lengths, possibly consequent to factory production controls, provides a left-truncated Weibull distribution that excellently fits most of the experimental results for annealed float glass, at least when it is the air-side under tensile stress in bending. Other generalized Weibull statistics, either bounded or unbounded, have been considered for the sake of comparison. For each of them, proper expressions for data re-scaling, to take into account the effects of size, type of applied stress and subcritical crack propagation, are provided.

Certainly, aging in the form of corrosion or abrasion can produce a variation of the defectiveness scenario originally present on the glass surface. The established correlation between the micro-defects and macroscopic strength allows to predict how the latter can be affected by variations in the defectiveness scenario. In particular, corrosion is assumed to be equivalent to the removal of a thin surface layer of glass, which reduces the depth of the micro-cracks, whereas abrasion consists in adding new defects to the pre-existing ones. This approach is used to interpret the difference in strength experimentally-measured at the tin-side with respect to the air-side, by assuming that the contact with the tin bath and the steel rollers during manufacturing is equivalent to a mild abrasion process.

This micro-macro approach is also employed to discuss the mechanical properties of heat-treated glass, where a compressive eigenstress is permanently induced by heating and successive cooling. Several experimental campaigns, recorded in the technical literature, have demonstrated that the characteristic strength of heat-treated glass can be much higher than the simple sum of the pristine material characteristic strength and the characteristic value of the prestress. A micromechanically-motivated cumulative probability function for the population of strengths of heat-treated glass is proposed, which accounts for the statistical interference between the mechanical properties of pristine glass and the eigenstress. This model justifies and confirms the experimental findings without having to refer to other phenomena like crack annealing, which certainly plays a significative role but is not essential. The statistical interference is deeply affected by the applied state of stress: the benefic effect of the heat-treatment is maximum when the applied stress is uniaxial, minimum when this is uniform equibiaxial.

All the previous findings are applied to the calibration of material partial factors to be used in the semi-probabilistic (level I) method of design, in order to guarantee the target probability of failure established by structural standards. The calibration is made on paradigmatic case-studies, comparing the results with those obtainable with full-probabilistic (level III) methods of design. For annealed glass, the partial material factors calculated from the proposed statistics are much lower than those obtainable with a 2-parameter Weibull model. Moreover, heat-treated glass reveals a quite surprising strength capacity, which seems to have been hardly appreciated before. An ad hoc experimental campaign has been conducted to corroborate the theoretical findings.

The refined statistical models presented here, motivated on the basis of micro-mechanical considerations, should increase the confidence in the structural applications of glass, achieving its optimal exploitation and, thus, improving its competitiveness on the market with respect to other construction materials.