POLYFLOW - How to evaluate the local stretch rate (third invariant of the rate-of-deformation tensor)?

Whereas the local stretch rate is available as a variable of a user defined function , this quantity can not be selected as a post-processor of POLYDATA, and so can no be visualized via the post-processor.

If the user asks for the calculation of the local shear rate (second invariant) and of the components of the rate-of-deformation tensor D, is is possible to evaluate the local stretch rate. If one defines the second and third invariants of D as:
II (secod invariant) = 2D:D = (g-dot)^2
III (third invariant) = det(D),
the local stretch rate e-dot can be defined as:
e-dot = 12 III / II
From this definition, one recovers the actual value of the stretch rate in a sinple alongationan uniaxial flow.
This can be defined through the mechanism of custom field functions in FLPOST or FIELDVIEW. However, care must be taken when calculating the determinant of D. Let us define the individual components of D as Dij (thus D11, D12, D13, etc.). The determinant is obtained as follows:
- 2D planar flow: III = 0
- 2D axisymmetric flow: III = D33.(D11.D22 - D12^2)
- 2.5D and 3D flows: III = D11.D22.D33 + 2.D12.D13.D23 - (D11.D23¨2 + D22.D13^2 + D33.D12^2)

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