This paper focuses on examining the uniaxial behaviour of concrete materials incorporating rubber particles, obtained from recycled end-of-life tyres, as a replacement for mineral aggregates. A detailed account of a set of material tests on rubberised concrete cylindrical samples, in which fine and coarse mineral aggregates are replaced in equal volumes by rubber particles with various sizes, is presented. The experimental results carried out in this investigation, combined with detailed examination of data available from previous tests on rubberised concrete materials, show that the rubber particles influence the mechanical properties as a function of the quantity and type of the mineral aggregates replaced. Experimental evaluation of the complete stress-strain response depicts reductions in compressive strength, elastic modulus, and crushing strain, with the change in rubber content. Enhancement is also observed in the energy released during crushing as well as in the lateral strain at crushing, primarily due to the intrinsic deformability of the interfacial clamping of rubber particles which leads to higher lateral dilation of the material. The test results and observations enable the definition of a series of expressions to estimate the mechanical properties of rubberised concrete materials. An analytical model is also proposed for the detailed assessment of the complete stress-strain response as a function of the volumetric rubber ratio. Validations performed against the material tests carried out in this study, as well as those from previous investigations on rubberised concrete materials, show that the proposed models offer reliable predictions of the mechanical properties including the full axial and lateral stress-strain response of concrete materials incorporating rubber particles.