Mohr Circle for Hydrostatic Pressure

When a soil sample is subjected to hydrostatic pressure, it means that the pressure acting on the soil is uniform and equal in all directions. Imagine the soil being completely submerged in a fluid; the pressure at any point within the fluid at a certain depth is equal in all directions. In terms of stress, this implies a very specific stress state within the soil sample.

Stress State in Soil Sample

In a state of hydrostatic pressure \(\sigma\), the normal stresses acting on any plane passing through a point are equal in magnitude, and importantly, the shear stresses on these planes are zero. This is a crucial characteristic. Mathematically, we can express the normal stresses along the principal axes as:

\[ \sigma_x = \sigma_y = \sigma_z = \sigma \]

And the shear stresses in all directions are zero:

\[ \tau_{xy} = \tau_{yz} = \tau_{zx} = 0 \]

Since the shear stresses are zero on all planes, it means that every plane is a principal plane, and the normal stresses are the principal stresses.

Principal Stresses and Mohr Circle Properties

The Mohr circle is a graphical tool used in geotechnical engineering to represent the state of stress at a point within a material. It plots normal stress (\(\sigma\)) on the x-axis and shear stress (\(\tau\)) on the y-axis.

For a two-dimensional stress state, the center and radius of the Mohr circle are determined by the principal stresses. In a three-dimensional case, we typically consider the largest and smallest principal stresses to define the main Mohr circle for analysis, or we consider circles for different planes (e.g., \(\sigma_1\) and \(\sigma_3\)).

• The center of the Mohr circle (C) on the normal stress axis is given by:
• \[ C = \frac{\sigma_1 + \sigma_3}{2} \]
• The radius of the Mohr circle (R) is given by:
• \[ R = \frac{\sigma_1 - \sigma_3}{2} \]

Here, \(\sigma_1\) is the major principal stress and \(\sigma_3\) is the minor principal stress. In a hydrostatic condition, all principal stresses are equal.

Mohr Circle Calculation

Given that the soil sample is under hydrostatic pressure \(\sigma\), all principal stresses are equal to \(\sigma\). So, we have:

• \(\sigma_1 = \sigma\)
• \(\sigma_2 = \sigma\)
• \(\sigma_3 = \sigma\)

Now, let's calculate the center and radius of the Mohr circle:

• Center (C):
• \[ C = \frac{\sigma_1 + \sigma_3}{2} = \frac{\sigma + \sigma}{2} = \frac{2\sigma}{2} = \sigma \]
• Radius (R):
• \[ R = \frac{\sigma_1 - \sigma_3}{2} = \frac{\sigma - \sigma}{2} = \frac{0}{2} = 0 \]

Mohr Circle Representation

A Mohr circle with a radius of zero means that the circle degenerates into a single point. This point is located on the normal stress (\(\sigma\)) axis at a distance equal to the center of the circle from the origin.

The coordinates of this point would be \((\sigma, 0)\).

This single point graphically represents that for any orientation of a plane through the point in the soil sample, the normal stress will always be \(\sigma\), and the shear stress will always be zero.

Therefore, the Mohr circle for any point in the soil sample subjected to a hydrostatic pressure \(\sigma\) is a point at a distance \(\sigma\) from the origin on the normal stress axis.