Critical-Block Envelopes for Rock Slopes with Periodic Joint Systems: Explicit Phase, Block and Mode Search

Conventional wedge analyses evaluate a block selected before calculation and do not directly resolve competition among finite blocks generated by repeated discontinuities. This study develops a deterministic phase-aware benchmark for an idealized class of rock slopes containing two planar, fully persistent, equally periodic joint systems and restricted to translational closed-contact sliding. The computational novelty is not the rigid-block equilibrium itself, which follows standard single-plane and wedge mechanics, but the explicit treatment of a relative joint-family phase, exhaustive finite-block enumeration at each searched phase, and nested minimization over phase, block and sliding mode. Computational geometry enumerates finite exposed blocks satisfying a prescribed release filter; standard rigid-block equilibrium then evaluates kinematically admissible J1, J2 and two-plane wedge sliding with hydrostatic loading and unilateral effective-contact admissibility. Controlled benchmarks recover the prescribed-wedge limit and demonstrate a non-monotonic spacing-dependent lower envelope; after strengthened global phase auditing, the controlled spacing sweep is J2-controlled at all five tested spacings. A 128-case deterministic database, a 32-case adaptive audit, independent prescribed-polyhedron checks, an end-to-end phase-space stress test, enlarged-domain checks, and external wedge benchmarks support implementation consistency within the adopted model class. States that reach the unilateral contact boundary are flagged separately as ot and require opening/separation analysis rather than assignment of a closed-contact factor of safety. The framework is therefore intended as a transparent deterministic benchmark for two fully persistent periodic joint sets, not as a general model of non-persistent, opening, rotational, toppling, or deformable rock-slope behavior.