# Numerical solving of elastomechanic problems

1. Jun 2, 2008

### Lojzek

I want to make a simple program for numerical solving of elastomechanic problems. The basic task it should perform is the calculation of deformation of a two dimensional homogeneous, isotropic body in the field of gravity and/or deformation under external force.
The basic idea for each time step is:

1. define the displacement in a finite number of points
2. calculate the strain tensor using finite diferences of displacement
3. calculate stress tensor (Hooke's law)
4. use stress tensor and second Newton's law to predict acceleration of each point
5. increase the speed and displacement at each point

The boundary condition should be either a fixed displacement or stress tensor.
Alternative method would be minimization of elastic energy (however this would only give the stationary state, not a movement towards it).

Questions:

Do I need to calculate derivatives of displacement with second order accuracy? (I seems so, since the acceleration depends on the changes of strain tenzor) If this is the case, how should I achieve second order accuracy near the boundary?
I planned to use central diferences (second order accuracy) and a square mesh, but it seems that on the boundaries one would have to know the displacement two steps away from the last calculated point: (but there is only one boundary condition).

Please let me know if you know a good book or internet site about numerical methods for solving this type of problems.