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Liquid pressure inside the cell causes deformation of the surrounding medium. The amount of
deformation can be quantified by modification of formula found in Equation 7, where the
deformation (Y), produced by a uniform pressure (P), acting on a circular area, (R) radius, on the
surface of a material with modulus of elasticity (E) and Poisson’s ratio (
ν
), is given by:
At the center of the cell:
Y=
2 PR (1-
ν
2
)
E
Equation 10 - Deformation at the Center
At the edge of the cell:
Y=
4 PR (1-
ν
2
)
π
E
Equation 11 - Deformation at the Edge
The difference being:
PR (1-
ν
2
) (2 – 4/
π
)/E
Equation 12 - Difference in Deformation
The above formulas apply to pressures acting on a free surface. However, in the confined case,
Y, at the edge of the cell, can be assumed to be nearly zero. Therefore, Y, at the center, is
assumed to be the same as shown in Equation 12.
If the average Y across the cell is assumed to be half this value, and if the deformation of the
medium on either side of the cell is assumed to be the same, then the average total expansion of
the cell is given by:
Y = 0.73 PR (1-
ν
2
) x 0.5 x 2/E = 0.73 PR (1-
ν
2
)/E
Equation 13 - Average Total Expansion of the Cell
Equating Equation 9 and Equation 13 gives:
P (D/G + 0.73 R (1-
ν
2
)/E) = KD
Equation 14 - Combined Equations
