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This study presents a series of discrete element method (DEM) simulations consisting of mixtures of stiff (sand) and soft (rubber) particles, subjected to monotonic triaxial shearing under constant volume at very small strains where pure elasticity governs the behaviour of the samples. The elastic shear moduli of the simulated pure sand and pure rubber samples were first calibrated to values reported in previous experimental works. Sand-rubber mixtures were then simulated with a focus on small-strain stiffness to examine the role of rubber content on the prevailed micro-mechanisms of the samples. The macro-mechanical response of the numerical mixtures showed a decrease in the elastic shear modulus and the deviatoric stress as the soft particle content increased, in line with observations from laboratory tests. Micro-scale information including coordination number, fabric tensor and normal contact force anisotropy was obtained for all tests and the contribution of each type of contact, i.e. sand-sand, rubber-sand or rubber-rubber, in the overall response of the samples, was analysed. The contact force network in the mixtures changed from being sand-dominated to rubber-dominated, with the presence of an intermediate zone in between rubber and sand particles forming a stable contact force network mainly by sand-rubber contacts. Each type of contact was seen to contribute differently to the deviatoric stress in the system as the rubber content increased.