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, Pcrack (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.
James Danenberg (Thu,) studied this question.