Relation between Shear and Elongational Viscosities

For uniform uniaxial elongational deformation along axis 1, the tensor of the velocity gradients, taking into account the condition of incompressibility, can be written in the form 

If we exclude the pressure from the relation for the stresses (9.58) under the considered uniaxial deformation, we can obtain an expression for the extensional stress 

1 The earlier history of the investigation of elongational flow can be found in the monograph by Petrie (1979). 

The above formulae determine the elongational viscosity coefficient 

The viscosity coefficients 77 and A are functions of the first invariant of the tensor of additional stresses 

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