Nonlinear finite elements/Kinematics - volume change and area change

Volume changeEdit

Consider an infinitesimal volume element in the reference configuration. If a right-handed orthonormal basis in the reference configuration is   then the vectors representing the edges of the element are

 

The volume of the element is given by

 

Upon deformation, these edges go to   where

 

or,

 

Therefore, the deformed volume is given by

 

Now,

 

Hence,

 

Recall that from conservation of mass we have

 

Therefore, an alternative form of the conservation of mass is

 

Distortional component of the deformation gradientEdit

For many materials it is convenient to decompose the deformation gradient in a volumetric part and a distortional part. This is particularly useful when there is no volume change in the material when it deforms - for example in muscles, rubber tires, metal plasticity, etc.

Let us assume that the deformation gradient can be decomposed into a volumetric part and a distortional part, i.e,

 

Then

 

Since there is no volume change due to the pure distortion, we have

 

If   we have

 

Therefore the distortional component of the deformation gradient is given by

 

We can use this result to find the distortional components of various strain and deformation tensors. For example, the right Cauchy-Green deformation tensor is given by

 

If we define its distortional component as

 

we have

 

Area change - Nanson's formulaEdit

Nanson's formula is an important relation that can be used to go from areas in the current configuration to areas in the reference configuration and vice versa.

This formula states that

 

where   is an area of a region in the current configuration,   is the same area in the reference configuration, and   is the outward normal to the area element in the current configuration while   is the outward normal in the reference configuration.

Proof:

To see how this formula is derived, we start with the oriented area elements in the reference and current configurations:

 

The reference and current volumes of an element are

 

where  .

Therefore,

 

or,

 

or,

 

So we get

 

or,