Nonlinear finite elements/Kinematics

Strain Measures in three dimensionsEdit

 
The motion of a body

Initial orthonormal basis:

 

Deformed orthonormal basis:

 

We assume that these coincide.

MotionEdit

 

Deformation GradientEdit

 

Effect of  :

 

Dyadic notation:

 

Index notation:

 

The determinant of the deformation gradient is usually denoted by   and is a measure of the change in volume, i.e.,

 

Push Forward and Pull BackEdit

Forward Map:

 

Forward deformation gradient:

 

Dyadic notation:

 

Effect of deformation gradient:

 

Push Forward operation:

 
  •   = material vector.
  •   = spatial vector.

Inverse map:

 

Inverse deformation gradient:

 

Dyadic notation:

 

Effect of inverse deformation gradient:

 

Pull Back operation:

 
  •   = material vector.
  •   = spatial vector.
ExampleEdit
 
Push forward and pull back

Motion:

 

Deformation Gradient:

 

Inverse Deformation Gradient:

 

Push Forward:

 

Pull Back:

 

Cauchy-Green Deformation TensorsEdit

Right Cauchy-Green Deformation TensorEdit

Recall:

 

Therefore,

 

Using index notation:

 

Right Cauchy-Green tensor:

 

Left Cauchy-Green Deformation TensorEdit

Recall:

 

Therefore,

 

Using index notation:

 

Left Cauchy-Green (Finger) tensor:

 

Strain MeasuresEdit

Green (Lagrangian) StrainEdit

 

Green strain tensor:

 

Index notation:

 

Almansi (Eulerian) StrainEdit

 

Almansi strain tensor:

 

Index notation:

 

Push Forward and Pull BackEdit

Recall:

 

Now,

 

Therefore,

 

Push Forward:

 

Pull Back:

 

Some useful resultsEdit

Derivative of J with respect to the deformation gradientEdit

We often need to compute the derivative of   with respect the the deformation gradient  . From tensor calculus we have, for any second order tensor  

 

Therefore,

 

Derivative of J with respect to the right Cauchy-Green deformation tensorEdit

The derivative of J with respect to the right Cauchy-Green deformation tensor ( ) is also often encountered in continuum mechanics.

To calculate the derivative of   with respect to  , we recall that (for any second order tensor  )

 

Also,

 

From the symmetry of   we have

 

Therefore, involving the arbitrariness of  , we have

 

Hence,

 

Also recall that

 

Therefore,

 

In index notation,

 

Derivative of the inverse of the right Cauchy-Green tensorEdit

Another result that is often useful is that for the derivative of the inverse of the right Cauchy-Green tensor ( ).

Recall that, for a second order tensor  ,

 

In index notation

 

or,

 

Using this formula and noting that since   is a symmetric second order tensor, the derivative of its inverse is a symmetric fourth order tensor we have