# Strain Rate

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)
${\stackrel{˙}{\epsilon }}_{ij}=\frac{1}{2}\left(\frac{\partial {v}_{i}}{\partial {x}_{j}}+\frac{\partial {v}_{j}}{\partial {x}_{i}}\right)$
Relating the element velocity field to its shape function gives:(4)
$\frac{\partial {v}_{i}}{\partial {x}_{j}}=\sum _{I=1}^{8}\frac{\partial {\Phi }_{I}}{\partial {x}_{j}}\cdot {v}_{iI}$
Hence, the strain rate can be described directly in terms of the shape function:(5)
${\stackrel{˙}{\epsilon }}_{ij}=\frac{1}{2}\left(\frac{\partial {v}_{i}}{\partial {x}_{j}}+\frac{\partial {v}_{j}}{\partial {x}_{i}}\right)=\sum _{I=1}^{8}\frac{\partial {\Phi }_{I}}{\partial {x}_{j}}\cdot {v}_{iI}$

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

For one integration point:(6)
$\frac{\partial {\Phi }_{1}}{\partial {x}_{j}}=-\frac{\partial {\Phi }_{7}}{\partial {x}_{j}};\text{ }\frac{\partial {\Phi }_{2}}{\partial {x}_{j}}=-\frac{\partial {\Phi }_{8}}{\partial {x}_{j}};\text{ }\frac{\partial {\Phi }_{3}}{\partial {x}_{j}}=-\frac{\partial {\Phi }_{5}}{\partial {x}_{j}};\text{ }\frac{\partial {\Phi }_{4}}{\partial {x}_{j}}=-\frac{\partial {\Phi }_{6}}{\partial {x}_{j}}$

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.