<p>In this paper, a three-dimensional non-uniform discretization peridynamics (NUDPD) with constant horizon is proposed for simulating cracking behaviours. First, the spatial geometric relationship between the discretization subdomain and the spherical horizon of source material points is defined in the proposed method. Then, the discretization subdomain volume of the corresponding family material point is obtained through the Monte-Carlo method, which eliminates the need for volume-correction. Next, conservation of energy, linear momentum and angular momentum of the proposed method are proved. Finally, the proposed method is applied to simulate several numerical cases under three-dimensional condition, and the numerical results obtained by the proposed method are in good agreement with those obtained by traditional peridynamics or other methods, verifying the ability of the proposed method. Moreover, the proposed non-uniform discretization peridynamics (NUDPD) significantly improves the low computational efficiency of traditional peridynamics under three-dimensional conditions.</p>