XYY’ Calculus from Soil Consolidation to All Dynamics

Abstract

Terzaghi (1923) brought differential calculus in geotechnical engineering by using heat (diffusion) equation in consolidation. But the complexities of soil settlement phenomenon (combined hydrodynamic + creep) with unknown drainage and loading and its evaluation by usual time (X) centric consolidation methods exposed that classical Newton-Leibniz calculus is fundamentally incomplete. It forces to see 3D (XYY') dynamic reality on 2D (XY & XY') flat planes ignoring the most vital (YY') phase plane that is the X (usually Time) invariant geometric DNA of the dynamic system. For example: DNA of Terzaghi’s 1D Eq on semi-log U-ln(dU/dT) or U-lnU’ plot is Symmetrical S shaped curve, on U-U’ plot is straight line for U=60-100% and Straight line again on U-(1/U’) plot for U=0-40% [12-32]. Similarly, DNA of Sine wave is circle, Exponential function is straight line, Catenary is Hyperbola, Parabola is Parabola etc. As XY and XY' planes are hinged around X-axis, the classical calculus fails when X is unknown and that is usual case in the field, forensic geotechnics or sinking infrastructures on soft soils. The XYY' calculus with its YY' plane analysis was developed in mid 1990s and extensively used to overcome these limitations in soil consolidation. However, it is universally applicable in other sciences also like diffusion, creep, wave, thermodynamics, electro-dynamics, economics, epidemiology, mechanics, oscillations etc Instead of borrowing mathematical laws from physics, heat, hydraulics, mechanics etc Geotechnical Engineering is now in a position to give them back a very useful mathematical tool.