Nonlinear finite elements/Stress and strain in one and two dimension
Stresses and strain in one dimension
edit1-D strain measures
edit- Engineering strain :
- Natural/Logarithmic/True strain:
Relation between engineering and true strain:
- Green (Lagrangian) strain :
- Almansi-Hamel (Eulerian) strain :
1-D stress measures
edit- Engineering/Nominal stress:
- Cauchy/True stress:
Relation between engineering and true stress (no volume change):
1-D stress-strain relations
edit- True stress - Green strain:
- True stress - True strain:
Example
edit- Assume incompressible material.
- or : }
- Equilibrium.
- :
where
Stress-strain relation 1
editThen,
and
Highly nonlinear in .
Stiffness
editStiffness = change in equilibrium equation due to change in position.
Now,
Therefore,
Initial stress/Geometric stiffness
Stress-strain relation 2
editThen,
and
Highly nonlinear in .
Stiffness
editInitial stress/Geometric stiffness:
Strain measures in two dimensions
editSmall strains
editFor 90 rotation,
Then strains are:
Rotation should not lead to non-zero strains!
Finite strains
editFor 90 rotation,
Then,
Green strain (1-D)
editIn 2-D:
Now,
Therefore,
Similar for and .