A generalized exponential function model of the integrity loss of stones subjected to laboratory weathering cycles

Ultrasonic measurements are widely used as NDT method in historical structures for evaluation of stone material strength properties. Actual stone degradation in situ is estimated based on various laboratory tests which allow researchers to correlate the number of artificial ageing cycles of stone specimens with ultrasonic wave velocity measured on these specimens. Paper presents the results obtained for granite, marble, limestone, travertine and sandstone which undergone various cyclic ageing tests including freezing and thawing, high temperature and salt crystallisation. Analysis of the obtained results shows that independent of the stone type tested and independent of the ageing test applied a rate of change of the stone elastic properties is described by an ordinary differential equation which solution is an exponential law analogues to the Newton’s law of cooling. The degradation function model can be used for further research on expected residual strength and dynamics of the heritage materials degradation processes.

Generalized Probability Functions

From the integration of nonsymmetrical hyperboles, a one-parameter generalization of the logarithmic function is obtained. Inverting this function, one obtains the generalized exponential function. Motivated by the mathematical curiosity, we show that these generalized functions are suitable to generalize some probability density functions (pdfs). A very reliable rank distribution can be conveniently described by the generalized exponential function. Finally, we turn the attention to the generalization of one- and two-tail stretched exponential functions. We obtain, as particular cases, the generalized error function, the Zipf-Mandelbrot pdf, the generalized Gaussian and Laplace pdf. Their cumulative functions and moments were also obtained analytically.