This study analyses the increase of crack propagations instabilities (pop-in) with irradiation during fracture toughness testing of an aluminium alloy 6061 alloy after neutron irradiation. The aim is to identify if the crack propagation instabilities are related to microstructural heterogeneities, i.e. weak zones, or if they are linked to a competition between testing machine stiffness and tearing modulus. A mechanical analysis is performed comparing the testing machine stiffness to the tearing moduli of the irradiated materials. For this purpose an experimental program is carried out in the hot cells facility at CEA Saclay to conduct mechanical and microstructural studies, after irradiation in a Material Testing Reactor (mean conventional thermal fluence: 6.8 x 10$^{21}$ $n_{th.conv}$.cm$^{-2}$. The material hardening is quantified in this study using tensile tests after the heat treatments. The effect of this hardening on crack propagation instabilities is measured by fracture toughness tests on irradiated and precracked CT12.5 samples. Damage mechanisms are studied by SEM fractography and classical ductile fracture micromechanisms are found for all conditions without notable differences between non irradiated and irradiated materials. The evolution of toughness with irradiation and subsequent heat treatments is therefore attributed to the matrix hardening/softening and its effect on nucleation of voids on brittle second phase particles as evidenced by the same authors for classical age hardening of the alloy. To investigate experimentally if the crack propagation instabilities are related to the tearing moduli, heat treatments of the irradiated material are applied that soften the material. It is shown that with softening, the number of crack propagation instabilities decreases and eventually the pop-ins disappear. The instability criteria using J-Δa R-curves of the toughness tests are applied and it is concluded that the increase in pop-ins with irradiation is resulting from the decrease of the material tearing modulus (and its interaction with the machine stiffness). The toughness test results of the irradiated material of the present study are compared with those of the literature and show very good agreement.