**Problem statement**

A soil under load does not assume an instantaneous deflection under that load, but settles gradually at a variable rate. The settlement is caused by a gradual adaptation of the soil to the load variation, known as soil consolidation

**Objectives**

Develop extension for Terzaghi model to the three-dimensional case, and the establishment of equations valid for any arbitrary load variable with time

**Previous/Related Research**

Terzaghi (1925)

Terzaghi applied soil consolidation concepts to the analysis of the settlement of a column of soil under a constant load and prevented from lateral expansion. The remarkable success of this theory in predicting the settlement for many types of soils has been one of the strongest incentives in the creation of a science of soil mechanics. Terzaghi's treatment, however, is restricted to the one-dimensional problem of a column under a constant load.

**Assumption**

The following basic properties of the soil are assumed: (1) isotropy of the material, (2) reversibility of stress-strain relations under final equilibrium conditions, (3) linearity of stress-strain relations, (4) small strains, (5) the water contained in the pores is incompressible, (6) the water may contain air bubbles, (7) the water flows through the porous skeleton according to Darcy's law.

**Methodology**

This paper consists of:

- the mathematical formulation of the physical properties of the soil
- the number of constants necessary to describe these properties
- physical interpretation of these various constants
- fundamental equations for the consolidation
- an application for one-dimensional problem corresponding to a standard soil test
- simplified theory for the case most important in practice of a soil completely saturated with water
- mathematical tool for the calculation of the settlement without having to calculate any stress or water pressure distribution inside the soil.

**Results**

Establish the differential equations for the transient phenomenon of consolidation, i.e., those equations governing the distribution of stress, water content, and settlement as a function of time in a soil under given loads

**Strong & Weakness**

(+) 3D case of soil consolidation (poroelasticity)

(-) isotropy material, incompressible fluid

**Notes**

**General Equation Governing Consolidation**

$$ G\nabla^2u+\frac{G}{1-2\nu}\frac{\partial \epsilon}{\partial x}-\alpha \frac{\partial \sigma}{\partial x}=0 $$

$$ G\nabla^2v+\frac{G}{1-2\nu}\frac{\partial \epsilon}{\partial y}-\alpha \frac{\partial \sigma}{\partial y}=0 $$

$$ G\nabla^2w+\frac{G}{1-2\nu}\frac{\partial \epsilon}{\partial z}-\alpha \frac{\partial \sigma}{\partial z}=0 $$

$$ \nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}$$

Introducing Darcy's law governing the flow of water in a porous medium

$$ V_x = -k \frac{\partial \sigma}{\partial x}; V_y = -k \frac{\partial \sigma}{\partial y}; V_z = -k \frac{\partial \sigma}{\partial z}; $$

$V_x, V_y, V_z$ volume of water flowing per second and unit area through the face of this cube perpendicular to the $x, y, z$ axis

If we assume the water to be incompressible the rate of water content of an element of soil must be equal to the volume of water entering per second through the surface of the element:

$$ \frac{\partial \theta}{\partial t} = -\frac{\partial V_x}{\partial x}-\frac{\partial V_y}{\partial y}-\frac{\partial V_z}{\partial z} $$

$$k\nabla^2 \sigma = \alpha \frac{\partial \epsilon}{\partial t} + \frac{1}{Q} \frac{\partial \sigma}{\partial t} $$

The four differential equations are the basic equations satisfied by the four unknowns *$u, v, w, \sigma$*

*$E, G, \nu$* = Young’s modulus, shear modulus and Poisson’s ratio