IPS-A — Confined NRL Tube Mechanics: Pressure-Volume Response of Elastic Lay-Flat Tubes in Fluid-Saturated Cellular Matrices

The Internal Plumbing System (IPS) of the Adaptive Matrix Ecosystem consists of lay-flat NRL tubes embedded within a water-saturated elastic cellular matrix. No existing literature addresses the mechanics of elastic tube inflation in this specific configuration: a thin membrane tube, hydrostatically equilibrated with its surroundings, expanding asymmetrically (upward only) against hydrostatic overburden, bio-grip elastic restraint, and membrane strain energy. This paper derives the governing equations from first principles. The cracking pressure law, P_crack(D) = (3/2)·ρ·g·D = 14.715·D kPa, is monotonically increasing with depth; there is no crossover depth at which opening becomes easier. The pressure-volume characteristic follows a flat-bottom, circular-arc-top geometry with extreme early compliance (cubic dependence of membrane pressure on displacement) and a 656-fold pressure ratio between initial cracking and the NRL stretch limit. Bio-grip coupling is modelled as a Winkler elastic foundation; its contribution to cracking pressure is negligible but becomes significant during large-displacement expansion. The elastic bending resistance of the 0.10 mm NRL wall is six to ten orders of magnitude below the hydrostatic terms and is negligible. Total pump power for a 15-metre levee-scale IPS is 29.4 watts. This paper provides the mechanical foundation for five subsequent papers addressing cardiovascular architecture, layer separation, AMW transit, capillary growth, and predictive structural adaptation.