Nonlinear finite elements/Stress and strain in one and two dimension

Stresses and strain in one dimensionEdit

1-D strain measuresEdit

  • Engineering strain :
 
  • Natural/Logarithmic/True strain:
 

Relation between engineering and true strain:

 
  • Green (Lagrangian) strain :
 
  • Almansi-Hamel (Eulerian) strain :
 

1-D stress measuresEdit

  • Engineering/Nominal stress:
 
  • Cauchy/True stress:
 

Relation between engineering and true stress (no volume change):

 

1-D stress-strain relationsEdit

  • True stress - Green strain:
 
  • True stress - True strain:
 

ExampleEdit

 
Stretching of a bar
 
 
  • Assume incompressible material.
 or : }
  • Equilibrium.
 : 

where

 

Stress-strain relation 1Edit

 

Then,

 

and

 

Highly nonlinear in  .

StiffnessEdit

Stiffness = change in equilibrium equation due to change in position.

 

Now,

 

Therefore,

 
 
 

Initial stress/Geometric stiffness

 

Stress-strain relation 2Edit

 

Then,

 

and

 

Highly nonlinear in  .

 

StiffnessEdit

 

Initial stress/Geometric stiffness:

 

Strain measures in two dimensionsEdit

 
Strains in two dimensions

Small strainsEdit

 

For 90  rotation,

 

Then strains are:

 

Rotation should not lead to non-zero strains!


Finite strainsEdit

 
Green strain in two dimensions

For 90  rotation,

 

Then,

 

Green strain (1-D)Edit

 

In 2-D:

 

Now,

 

Therefore,

 

Similar for   and  .