Use of Descartes Folium Equation for Deriving a Relation between Total Aperture of Fractures after Uniaxial Compression and Strain Parameters of Different Rocks Exhibiting Negative Total Volumetric Strains

The axial, crack and total volumetric strains, porosity, elastic constants, crack damage stresses, uniaxial compressive strengths, as well as fracture apertures and number of fracture traces in rock samples surface after compression were defined for different chalk, basalt, dolomite, granite, limestone and sandstone samples exhibiting negative total volumetric strain at failure. It is established that the total (summed) aperture of vertical fractures obtained on the lateral surface of rock sample is related to three characteristic strain parameters: axial strain at the onset of negative total volumetric strain, axial failure strain and negative total volumetric strain at failure. The relation is based on Descartes folium equation, where the length of the loop of folium is equal to axial strain coordinate at the onset of negative total volumetric strain. This relation shows that the total aperture increases according to power law with increasing difference between axial failure strain and axial strain at the onset of negative total volumetric strain. Simultaneously, an increase in this difference leads to an increase in the value of negative total volumetric strain at failure. It is found that a direct correlation between total aperture of fractures and negative total volumetric strain at failure is relatively weak. Nevertheless, total aperture of fractures tends to increase with increasing absolute value of negative total volumetric strain at failure. It is revealed that there is no connection between the number of fracture traces and negative total volumetric strain at failure.