# Eq. of motion of elastic 2D finite element

• hilbert2
In summary, the conversation discusses a question about elasticity theory and the finite element method. The question specifically asks about the equations of motion and potential energy of a tetragonal 2D piece of linear isotropic elastic material with specific parameters and dimensions. However, the answer cannot be provided without more specific information about the nodal variables, element shape functions, integration rules, allowable stress states, variational principle, and element geometry. The conversation also mentions the challenge of working with quadrilateral structural elements and the potential difficulties in coding a simulation using C++.
hilbert2
Gold Member
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

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.)

Last edited:
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.

## 1. What is the equation of motion for elastic 2D finite elements?

The equation of motion for elastic 2D finite elements is derived from Newton's second law of motion, which states that the sum of all external forces acting on an object is equal to the mass of the object multiplied by its acceleration. In the case of a 2D finite element, this equation is expressed as F = ma, where F is the vector of external forces, m is the mass matrix, and a is the vector of nodal accelerations.

## 2. How is the mass matrix calculated in the equation of motion for elastic 2D finite elements?

The mass matrix in the equation of motion for elastic 2D finite elements is calculated by integrating the product of the nodal shape functions and the material density over the element's domain. This results in a matrix that relates the nodal accelerations to the forces acting on the element.

## 3. What are the primary assumptions made in the equation of motion for elastic 2D finite elements?

The primary assumptions made in the equation of motion for elastic 2D finite elements are that the material is linearly elastic, the element is in a state of plane stress or plane strain, and the displacements within the element are small enough to be considered linear.

## 4. Can the equation of motion for elastic 2D finite elements be used for non-linear materials?

No, the equation of motion for elastic 2D finite elements is only applicable for linearly elastic materials. For non-linear materials, more complex equations of motion must be used, such as those based on the principle of virtual work.

## 5. How does the equation of motion for elastic 2D finite elements account for damping effects?

The equation of motion for elastic 2D finite elements can be modified to account for damping effects by adding a damping matrix to the mass and stiffness matrices. This damping matrix is typically proportional to the mass or stiffness matrix and the damping coefficient, which is a measure of the material's ability to dissipate energy.

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