Stress singularities, annihilations and invisibilities induced by polygonal inclusions in linear elasticity

Notches, wedges, cracks, sti?eners, inclusions and defects in plane elastostatics are known to generate singular stresses and limit the overall strength of a composite material. In the present thesis, after showing experimentally that the singular stress ?eld predicted by the linear elastic solution for the rigid inclusion model can be generated in reality and with great accuracy within a material, attention is devoted then in achieving the out-of-plane response of an in?nite plane containing polygonal and hypocycloidal-shaped voids and rigid inclusions subject to generalized remote loading conditions. The analytical solution obtained for the case of polygonal inclusions shows some unexpected and interesting features such as an in?nite set of geometries and loading conditions exist for which not only the singularity is absent, but the stress vanishes (annihilates) at the corners. Thus the material, which even without the inclusion corners would have a ?nite stress, remains unstressed at these points in spite of the applied remote load. Moreover, similar conditions are determined in which a star-shaped crack or sti?ener leaves the ambient stress completely unperturbed, thus reaching a condition of ‘quasi-static invisibility’. The solution in closed-form is also obtained for the case of hypocycloidalshaped voids and rigid inclusions, showing that cusps may in certain conditions act as stress reducers, situations for which the stress at the cusp tip in the presence of the inclusion is smaller than in the case when the inclusion is absent. Ph.D. Thesis – Summer Shahzad vThe obtained solutions provide closed-form expressions for Stress Intensity Factors and Notch Stress Intensity Factors at varying the inclusion geometry and of loading conditions, fundamental quantities in de?ning criteria of fracture initiation/propagation or inclusion detachment. The ?ndings of stress annihilation, stress reduction and inclusion invisibility de?ne optimal loading modes for the overall strength of a composite and are useful in the design of ultra-resistant materials.