# Kinematic Description

- The velocity strain (or rate of deformation)
(1) $${D}_{ij}=\frac{1}{2}\left(\frac{\partial {v}_{i}}{\partial {x}_{j}}+\frac{\partial {v}_{j}}{\partial {x}_{i}}\right)$$ - The Green strain tensor (Lagrangian strain tensor) measured with respect to initial
configuration
(2) $${E}_{ij}=\frac{1}{2}\left(\frac{\partial {u}_{i}}{\partial {X}_{j}}+\frac{\partial {u}_{j}}{\partial {X}_{i}}+\frac{\partial {u}_{k}}{\partial {X}_{i}}\frac{\partial {u}_{k}}{\partial {X}_{j}}\right)$$ - The Almansi strain tensor (Eulerian strain tensor) measured with respect to deformed
configuration
(3) $${E}^{A}{}_{ij}=\frac{1}{2}\left(\frac{\partial {u}_{i}}{\partial {x}_{j}}+\frac{\partial {u}_{j}}{\partial {x}_{i}}-\frac{\partial {u}_{k}}{\partial {x}_{i}}\frac{\partial {u}_{k}}{\partial {x}_{j}}\right)$$

All three measures of strains can be related to each other and can be used with any type of mesh.

## Velocity Strain or Deformation Rate

$l$ and $L$ are respectively the dimensions in the deformed and initial configurations. Furthermore, the integral in time for a material point does not yield a well-defined, path-independent tensor so that information about phenomena such as total stretching is not available in an algorithm that employs only the strain velocity. Therefore, to obtain a measure of total deformation, the strain velocity has to be transformed to some other strain rate.

## Green Strain Tensor

$C={F}^{T}F$ and $B=F{F}^{T}$ are called respectively right and left Cauchy-Green tensor.

Where, $l$ and $L$ are respectively the dimensions in the deformed and initial configurations.

and $H$ symmetric.

The polar decomposition theorem is important because it will enable to distinguish the straining part of the motion from the rigid body rotation.

式 26 allows the computation of $H$ , and 式 27 of $R$ .

As the decomposition of the Jacobian matrix $F$ exists and is unique, $H$ is a new measure of strain which is sometimes called the Jaumann strain. Jaumann strain requires the calculation of principal directions.

if second order terms are neglected.

So for 式 32 and 式 33, when rigid body rotations are large, the linear strain tensor becomes non-zero even in the absence of deformation.

The preceding developments show that the linear strain measure is appropriate if rotations can
be neglected; that means if they are of the same magnitude as the strains and if these are of
the order of 10^{-2} or less. It is also worth noticing that linear strains can be used
for moderately large strains of the order of 10^{-1} provided that the rotations are
small. On the other hand, for slender structures which are quite in extensible, nonlinear
kinematics must be used even when the rotations are order of 10^{-2} because, if you are
interested in strains of 10^{-3} - 10^{-4}, using linear strain the error due to
the rotations would be greater than the error due to the strains.

Large deformation problems in which nonlinear kinematics is necessary, are those in which rigid body rotation and deformation are large.