The relationship between the entropy change and the extension can be simplified for biaxial tension of a sphere in much the same way as has already been seen for uniaxial tension. Consider a square piece of membrane, with initial unstretched side L0.
For biaxial tension:
λ1 = λ2, and λ1λ2λ3 = 1
therefore:
putting this into the equation relating entropy change to extension ratios:
From this, the work done per unit volume on stretching is:
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(1) |
The work done on the square membrane is then the work done per unit volume multiplied by the area of the piece of membrane, L02, and the thickness, t0:
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(2) |
If we now make an incremental change to the extension ratio, δλ, the amount of work needed to make this incremental extension is:
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(3) |
where F is the force and L0δλ is the change in extension. By re-arrangement of (3) it follows that:
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(4) |
Therefore, from (4) and (2):
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(5) |
Rearranging (5), the force per unit length is given by:
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(6) |
The total force acting over a cross sectional area of a sphere is given by Pπ2, where P is the internal pressure: this must be equal to the force per unit length acting around the circumference
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(7) |
from which, using the relationship that 
, we
have:
which gives us the predicted variation of pressure with increasing λ:
Graph of pressure vs extension ratio
This predicts that there will be a maximum value of pressure at an extension ratio of about 1.38, i.e. this is when the stiffness is a maximum. At high extension ratios (>2.5) however, the finite extensibility effect becomes apparent and the pressure becomes larger than predicted.
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