Jaumann derivative

J. D. Goddard and C. Miller, An Inverse for the Jaumann Derivative and Some Applications to the Rheology of Viscoelastic Fluids, Rheol. Acta, 5, 177-184 (1966). 

There are plenty of covariant derivatives of the tensor among which the Jaumann derivative has the simplest form. Indeed, expressions (8.24) and (8.25) are followed by the relation... 

Here, and 2 the first and second stress difference coefficient functions, and the derivative of the strain rate is the Jaumann derivative, which is related to a frame of reference that translates and rotates with the local velocity of the fluid (this relationship can be numerically evaluated from the deformation and vorticity tensors). 

It is important to note that there are other types of time derivatives which also transform as a tensor from convected to fixed coordinates. One particular time derivative that has received particular attention by rheologists is the so-called Jaumann derivative, which was suggested first by Zaremba (1903) and later reformulated by other investigators (DeWitt 1955 Fromm 1947). The Jaumann derivative 2)/2)r of a second-order tensor a,-, is defined as... 

Comparison of Eq. (3.17) with (3.6) shows that the use of the Jaumann derivative of a in the classical Maxwell model gives rise to the material functions that are quite different in form as compared to when the contravariant components of the convected derivative of a are used in the classical Maxwell model. It is of great interest to... 

Note that when using the Jaumann derivative one obtains the identical expressions for the material functions, regardless of whether the covariant or contravariant components of tensors a and d are employed. 

For steady-state shear flow, the Jaumann derivative given by Eq. (2.111) reduces to... 

Equation (3. 141), first derived by Jaumann (1911), determines the local rate of entropy production by summing four distinctive contributions as a result of the products of flows and forces ... 

The mechanical elastic part is related to the Jaumann objective stress rate through Hooke s law. For the plastic parts, a general framework of non-associated plasticity is adopted in order to limit dilatancy. In that case, the plastic flow rate is derived from a plastic potential... 

Jaumann s derivative ordinary total derivative trace of the tensor, trx = Xu, vorticity tensor transpose of the tensor Vv... 

Here SqT/Dt represents an objective part of the time derivative of the second-order tensor T (the proof is similar to (2.170) as shown below). For example if the rotation tensor R and the spin tensor W are used instead of Q and Q, we can introduce the Zaremba-Jaumann rate as follows ... 

Note that the time derivative of Cauchy stress a is not objective, but the Zaremba-Jaumann raleSKrlDt is objective. 

Use of Eq. (3.33) into Eq. (3.25) yields the material functions that are exactly the same as Eq. (3.17) obtained from the ZFD model, Eq. (3.15). However, as will be shown, the Giesekus model with a = 1 is not the same as the ZFD model. Substituting the following relationship between the upper convected derivative and the Jaumann... 

---