Non-Linear Effects in Oscillatory Shear Motion

From the methodical point of view, it is very interesting to consider the nonlinear terms of the stresses under oscillatory shear velocity gradients, which it is convenient to write in the complex form 

Akers and Williams (1969), calculating non-linear terms, noticed that the stresses are real quantities, which are determined through real quantities. That is why we ought to remember that formulae always contain the real parts of complex quantities, so that one has to bear in mind that Vik means vik + k ik), where the operation of complex conjugation is denoted by the bar above the symbol. 

Assuming that the flow is described by the set of equations (9.3) and (9.4), one can use equations (9.14) for arbitrary time dependence of velocity gradient, to obtain for oscillatory simple shear the solution in the form 

Since all non-oscillatory terms in the solution are now omitted, we shall determine the non-zero components of the stress tensor according to equation (9.3) 

Expression (9.16) determines the non-linear dynamic viscosity and dynamical modulus. The first difference of the normal stresses 711-022, defined by expressions (9.17), oscillate with a frequency twice that of velocity gradients (Akers and Williams 1969). 

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