Eq. of motion of elastic 2D finite element

[hilbert2]

A simple question about elasticity theory/finite element method:

Suppose I have a tetragonal 2D piece of a linear isotropic elastic material, that has Young's modulus ##E## and Poisson's ratio ##\nu##. The vertices of the tetragon are at positions ##\textbf{x}_{1}##, ##\textbf{x}_{2}##, ##\textbf{x}_{3}##, ##\textbf{x}_{4}##. In its unstrained state, the element is a square with side length ##L##.

What are the equations of motion of the nodal points (vertices of the tetragon). What is the potential energy, ##V(\textbf{x}_{1},\textbf{x}_{2},\textbf{x}_{3},\textbf{x}_{4})## of the element?

Thanks for any help

 

[AlephZero]

This is unanswerable, unless you specify the nodal variables, element shape functions, integration rules, allowable stress states (e.g. plane stress or plane strain), what variational principle is used in the formulation, and probably a few more things I've forgotten.

I'm not sure what you mean be "equations of motion," unless you also specify how the mass properties are calculated (which might be different from the stiffness properties)

If you do specify all of the above, the answer is then a (probably long and tedious) plug-and-chug exercise.

Quadrilateral structural elements are where finite element mechanical analysis changes from a neat and tidy mathematical exercise in function approximation, to "welcome to the real world, and it's not necessarily a pretty sight"

(One thing I forgot and just remembered: just saying "the vertices are x1, x2, x3, x4" isn't enough to define the element geometry. Some people would take the diagonals of the element to be from x1 to x3 and x2 to x4, others from x1 to x4 and x2 to x3.)

 

[hilbert2]

Thanks for replying. I think I have to spend some more time on studying elasticity theory.

In the problem I was thinking about, all motion is constrained to happen in the same plane. I was thinking of coding a simulation like this with C++: http://wismuth.com/elas/elasticity.html . That was why I was asking.