Issue 41

Ch. F. Markides et alii, Frattura ed Integrità Strutturale, 41 (2017) 396-411; DOI: 10.3221/IGF-ESIS.41.51 407 D ISCUSSION AND CONCLUDING REMARKS he displacement field developed in a transversely isotropic disc under diametral compression was determined analytically and in closed form. The main advantage of the present solution is the nature of the analytic expressions, which are relatively easily programmable, providing an overview of the deformed shape either of an element of the disc (see Figs.4(a, b)) or for the disc as a unit. To highlight the specific feature of the solution the deformed shape of a “fictitious” “transtropic” disc is plotted in Figs.6(a, b, c), in juxtaposition to the respective isotropic one, for three characteristic configurations, i.e., ϕ o =30 o (Fig.6a), ϕ o =45 o (Fig.6b), and ϕ o =60 o (Fig.6c). In all cases the transtropic disc is characterized by E =40 GPa, ν =0.30, E ΄=10 GPa, ν ΄= ν yx =0.12, ν xy =0.48 and the isotropic one by E = E ΄=40 GPa, ν = ν ΄=0.30. Again the length of the contact arcs is determined from the respective contact problem and the magnitude of the load imposed is “unrealistically” high for the deformed shape to be clearly distinguished from the undeformed one. It is again easily concluded that the deformation of the “transtropic” disc is asymmetric with respect to the loading line. This “degree of asymmetry” depends both on the anisotropy ratio, δ , and also on the value of angle ϕ o , although this depe- ndence is extremely complicated and by no means monotonous. Moreover, it is interesting to emphasize the fact that for the “transtropic” disc the diameters parallel and normally to the loading direction do not remain linear segments after deformation, but rather they become curved segments, contrary to what happens for the isotropic disc, for which the specific diameters (as expected) remain linear segments. The curvature of the deformed diameters depends (according to an extremely complicated manner) on quite a few parameters, the most important of which are: The orientation of the diameter with respect to the loading line, the distance from the disc center, and (obviously) on the orientation of the load with respect to the planes of isotropy (angle ϕ o ). Analogous conclusions are drawn for the diameters (symmetrically placed with respect to the loaded diameter) connecting the starting and ending points of the loaded arcs. Namely, while in the case of the isotropic disc these diameters exhibit completely symmetric deformation, for the “transtropic” disc the de- formation is asymmetric following the general distortion of the deformed configuration as a whole. Moreover, worth mentioning is the diversification in the values of ω o (corresponding to the half-contact length) between the isotropic and the “transtropic” disc, as well as, among the three deformed configurations of the “transtropic” disc itself; namely, while for the isotropic disc ω ο is constantly equal to 40.04 o for all three values of angle ϕ o (30 o , 45 o , 60 o ), this is not the case for the “transtropic” disc where the relevant values of ω ο read respectively as 58.46 o , 63.89 o and 72.22 o (due to the successive decrease of the “transtropic” disc stiffness for increasing ϕ o -values). (a) Figure 6 : (a) Deformed versus undeformed (black dashed line) configuration of a circular disc under parabolic diametral compression, for an isotropic- (green line) and a “transtropic”- (red line) material, for ϕ o =30 o . T