The relationship between the physical coordinate and computational intrinsic coordinates
system for a brick element is given by the matrix equation:

(1)
$$\left[\begin{array}{c}\frac{\partial {\Phi}_{I}}{\partial \xi}\\ \frac{\partial {\Phi}_{I}}{\partial \eta}\\ \frac{\partial {\Phi}_{I}}{\partial \zeta}\end{array}\right]=\left[\begin{array}{ccc}\frac{\partial x}{\partial \xi}& \frac{\partial y}{\partial \xi}& \frac{\partial z}{\partial \xi}\\ \frac{\partial x}{\partial \eta}& \frac{\partial y}{\partial \eta}& \frac{\partial z}{\partial \eta}\\ \frac{\partial x}{\partial \zeta}& \frac{\partial y}{\partial \zeta}& \frac{\partial z}{\partial \zeta}\end{array}\right].\left[\begin{array}{c}\frac{\partial {\Phi}_{I}}{\partial x}\\ \frac{\partial {\Phi}_{I}}{\partial y}\\ \frac{\partial {\Phi}_{I}}{\partial z}\end{array}\right]={F}_{\xi}.\left[\begin{array}{c}\frac{\partial {\Phi}_{I}}{\partial x}\\ \frac{\partial {\Phi}_{I}}{\partial y}\\ \frac{\partial {\Phi}_{I}}{\partial z}\end{array}\right]$$

Hence:

(2)
$$\left[\frac{\partial {\Phi}_{I}}{\partial {x}_{i}}\right]={F}_{\xi}^{-1}.\left[\frac{\partial {\Phi}_{I}}{\partial \xi}\right]$$

Where,
${F}_{\xi}$
is Jacobian matrix.

The element strain rate is defined as:

(3)
$${\dot{\epsilon}}_{ij}=\frac{1}{2}\left(\frac{\partial {v}_{i}}{\partial {x}_{j}}+\frac{\partial {v}_{j}}{\partial {x}_{i}}\right)$$

Relating the element velocity field to its shape function gives:

(4)
$$\frac{\partial {v}_{i}}{\partial {x}_{j}}={\displaystyle \sum _{I=1}^{8}\frac{\partial {\Phi}_{I}}{\partial {x}_{j}}}\cdot {v}_{iI}$$

Hence, the strain rate can be described directly in terms of the shape
function:

(5)
$${\dot{\epsilon}}_{ij}=\frac{1}{2}\left(\frac{\partial {v}_{i}}{\partial {x}_{j}}+\frac{\partial {v}_{j}}{\partial {x}_{i}}\right)={\displaystyle \sum _{I=1}^{8}\frac{\partial {\Phi}_{I}}{\partial {x}_{j}}}\cdot {v}_{iI}$$

As was seen in Velocity Strain or Deformation Rate, volumetric strain rate is calculated separately by volume variation.

For one integration point:

(6)
$$\frac{\partial {\Phi}_{1}}{\partial {x}_{j}}=-\frac{\partial {\Phi}_{7}}{\partial {x}_{j}};\text{\hspace{1em}}\frac{\partial {\Phi}_{2}}{\partial {x}_{j}}=-\frac{\partial {\Phi}_{8}}{\partial {x}_{j}};\text{\hspace{1em}}\frac{\partial {\Phi}_{3}}{\partial {x}_{j}}=-\frac{\partial {\Phi}_{5}}{\partial {x}_{j}};\text{\hspace{1em}}\frac{\partial {\Phi}_{4}}{\partial {x}_{j}}=-\frac{\partial {\Phi}_{6}}{\partial {x}_{j}}$$

F.E Method is used only for deviatoric strain rate calculation in A.L.E and Euler
formulation.

Volumetric strain rate is computed separately by transport of density and volume
variation.