## Displacement and Strain: Material Time Derivative

### Material Time Derivative

For time dependent deformations, the material time derivative, is the derivative of scalar, vector or tensor fields as a function of the material point with respect to time. Let and represent the reference and the deformed configurations of a body embedded in . Let represent the image of the vector at a certain point in time . Let and and be the velocity and the velocity gradient fields. Let , and be a scalar, a vector and a tensor field respectively. I.e., , and . Then, the material time derivative of these fields, denoted with a being , or is the total derivative of these quantities as follows:

Notice that is a scalar valued field, is a vector valued field, while is a tensor valued field. The components of these quantities are given by:

In case where the fields are functions of the reference configuration, i.e., , and are the respective scalar, vector and tensor fields, then, the material time derivative of these fields, denoted with a being , or is the total derivative of these quantities. However, since is fixed, the only variable that appears in the derivation is time:

And the components of these quantities are given by:

#### Example

In the following example, the temperature scalar field is used to illustrate the above concepts. Assuming a plate of width 4 units and height of 2 units is rotating around the origin with a speed of rad/second. The spatial distribution of the temperature in the medium is given by units of temperature. If we assume that the speed of the plate is slow enough such that the plate assumes the temperature of the medium, then the material time derivative of the temperature, which is the rate of change of the temperature of the material points on the plate, is non zero. Here are the calculations:

The position function at time is equal to:

where

The spatial velocity field is given by the following vector function:

The spatial gradient of temperature field is given by the following vector function:

The material time derivative of the temperature is given by the following scalar function:

The following tool draws the velocity vector field, the spatial gradient of temperature vector field and the contour plot of the material time derivative of the temperature. Change the value of between and units to see the effect on the position of the plate and the calculated fields.