dE ij
dt
¼ D ij
2.10.5 Relation Between the Spatial Gradient of Velocity
Tensor, L,and the Deformation Gradient Tensor, F
The definition of deformation gradient tensor F is given by
dx ¼ F Á dr
or
dx i ¼
∂x i
∂r j
∂r j
The rate of change of the deformation gradient is _
F . The spatial gradient of
velocity L is given by
dv ¼ L Á dx
Hence,
L ij ¼ v i,j ¼
∂v i
∂x j
If we write time derivative of F ¼
∂x i
∂r j
_
F ¼
d
dt
∂x i
∂r j
¼
∂_ x i
∂r j
¼
∂_ x i
∂x m
∂x m
∂r j
_
F ¼ L Á F and L ¼ _
F Á F
À1
d
dt
∂x i
∂r j
¼
∂v i
∂x m
∂x m
∂r j
∂v i
∂x m
¼
d
dt
∂x i
∂r j
∂r j
∂x m
Relations between Euler strain rate E
à , the rate of deformation tensor D, and the
spatial gradients of velocity L tensors can be given by
56
2 Stress and Strain in Continuum