Strain Rate

The relationship between the physical coordinate and computational intrinsic coordinates system for a brick element is given by the matrix equation:(1)
[ ∂ Φ I ∂ ξ ∂ Φ I ∂ η ∂ Φ I ∂ ζ ] = [ ∂ x ∂ ξ ∂ y ∂ ξ ∂ z ∂ ξ ∂ x ∂ η ∂ y ∂ η ∂ z ∂ η ∂ x ∂ ζ ∂ y ∂ ζ ∂ z ∂ ζ ] . [ ∂ Φ I ∂ x ∂ Φ I ∂ y ∂ Φ I ∂ z ] = F ξ . [ ∂ Φ I ∂ x ∂ Φ I ∂ y ∂ Φ I ∂ z ]
Hence:(2)
[ ∂ Φ I ∂ x i ] = F ξ − 1 . [ ∂ Φ I ∂ ξ ]

Where, F ξ is Jacobian matrix.

The element strain rate is defined as:(3)
ε ˙ i j = 1 2 ( ∂ v i ∂ x j + ∂ v j ∂ x i )
Relating the element velocity field to its shape function gives:(4)
∂ v i ∂ x j = ∑ I = 1 8 ∂ Φ I ∂ x j ⋅ v i I
Hence, the strain rate can be described directly in terms of the shape function:(5)
ε ˙ i j = 1 2 ( ∂ v i ∂ x j + ∂ v j ∂ x i ) = ∑ I = 1 8 ∂ Φ I ∂ x j ⋅ v i I

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

For one integration point:(6)
∂ Φ 1 ∂ x j = − ∂ Φ 7 ∂ x j ;   ∂ Φ 2 ∂ x j = − ∂ Φ 8 ∂ x j ;   ∂ Φ 3 ∂ x j = − ∂ Φ 5 ∂ x j ;   ∂ Φ 4 ∂ x j = − ∂ Φ 6 ∂ 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.