Objectivity (frame Invariance) - Objectivity Rates

Objectivity Rates

It was shown above that even if a displacement field is objective, the velocity field is not. An objective vector and an objective tensor usually do not conserve their objectivity through time differentiation as demonstrated below:

Objectivity rates are modified material derivatives that allows to have an objective time differentiation. Before presenting some examples of objectivity rates, certain other quantities need to be introduced. First, the spatial velocity gradient is defined as:

where is a symmetric tensor and is a skew tensor called the spin tensor. For a given, and are uniquely defined. The Euclidean transformation for the spatial velocity gradient can be written as:

\ \begin{align} l^* & = \dot{F}^*(F^*)^{-1} \\ & = (\dot{Q}F+Q\dot{F})(QF)^{-1} \\ & = (\dot{Q}F+Q\dot{F})F^{-1}Q^T \\ & = \dot{Q}FF^{-1}Q^T + Q\dot{F}F^{-1}Q^{-1} \\ & = \dot{Q}Q^T + QlQ^{-1} \\ & = \Omega + QlQ^{-1}. \end{align}

Substituting in the above equation, one can obtain two following relations:

Substituting the above result in the previously obtained equation for the rate of an objective vector, one can write:

\ \begin{array}{rrcl} & \dot{u}^* &=& \dot{Q}u + Q\dot{u} \\ \Rightarrow & \dot{u}^* &=& (w^*Q - Qw)u + Q\dot{u} \\ \Rightarrow & \dot{u}^* &=& w^*u^* - Qwu + Q\dot{u} \\ \Rightarrow & (\dot{u}-wu)^* &=& Q(\dot{u}-wu) \\ \Rightarrow & \bar{u}^* &=& Q\bar{u}, \end{array}

where the co-rotational rate of the objective vector field is defined as:

and represents an objective quantity. Similarly, using the above equations, one can obtain the co-rotational rate of the objective second-order tensor field :

\ \begin{array}{rrcl} & (\dot{A}-wA+Aw)^* &=& Q(\dot{A}-wA+Aw)Q^T \\ \Rightarrow & \bar{A}^* &=& Q\bar{A}Q^T. \end{array}

This co-rotational rate second order tensor is defined as:

This objective rate is known as the Jaumann-Zaremba rate and it is often used in plasticity theory. Many different objective rates can be developed and the reader should refer to Holzapfel 2000 for more details.

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