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The thesis takes a closer look at the laboratory determination and study of the mechanical behavior of the discontinuity on rocks and specifically that of the Thassos’ Marble. The material was subjected to processing and shaping before being sheared and compressed for its mechanical characteristics to be found. From the results of both tests the angle of the internal friction φ and the cohesion c is determined. The mechanical characteristics are determined, in the case of solid rocks, by the intensive condition within the material, which is created by the triaxial compression, when the shear stress is high enough to cause the transition of the material from the brittle state to the ductile one. In the case of the direct shearing, when stress is applied, only certain parts of the discontinuity come into contact. Τhe developed normal stress on those parts of the material is so high that it can be considered to cause a case of deformation equivalent to that in the brittle-ductile transition zone of the triaxial compression. It is, therefore, understood that under these conditions similarities between the two tests occur. In both cases the deformation leads to a change in the behavior of the material from brittle to ductile. Initially, the direct shear stress tests were conducted on a range of normal stresses which were defined by the supervising professor. The tests done with the high normal stresses are expected to result in a stable shear stress in reference to the relative shear displacement. After the direct shear stress tests, a series of triaxial compression tests with variable shear stresses were performed. After finishing the tests, processing to the acquired measurements was conducted and the mechanical characteristics were calculated, according to the Mohr-Coulomb failure criterion and the new criterion of the shear strength (Papaliangas et.al, 1995a). These results and the initial measurements made during the testing are both presented in the designated chapter.