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

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Stresses and strain in one dimension[edit]

1-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]

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

where

Stress-strain relation 1[edit]

Then,

and

Highly nonlinear in .

Stiffness[edit]

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

Now,

Therefore,

Initial stress/Geometric stiffness

Stress-strain relation 2[edit]

Then,

and

Highly nonlinear in .

Stiffness[edit]

Initial stress/Geometric stiffness:

Strain measures in two dimensions[edit]

Strains in two dimensions

Small strains[edit]

For 90 rotation,

Then strains are:

Rotation should not lead to non-zero strains!


Finite strains[edit]

Green strain in two dimensions

For 90 rotation,

Then,

Green strain (1-D)[edit]

In 2-D:

Now,

Therefore,

Similar for and .