FEM EquationElasticity

Description
This equation describes the mechanical properties of rigid bodies.

For info about the math of the equation, see the Elmer models manual, section Linear Elasticity.

Usage

 * 1) After adding an Elmer solver as described here, select it in the tree view.
 * 2) Now either use the toolbar button [[Image:FEM_EquationElasticity.svg|24px]] or the menu.
 * 3) Change the equation's solver settings or the general solver settings if necessary.

Solver Settings
For the general solver settings, see the Elmer solver settings.

The elasticity equation provides these special settings:
 * : If the principal angles should be calculated.
 * : If all stresses should be calculated.
 * : If strains will be calculated. This will also calculate the stresses, even if or  is false.
 * : If stresses should be calculated. Compared to the Tresca yield criterion and the principal stress will not be calculated.
 * : See the Elmer manual for more info.
 * : If mesh can be deformed. This is by default true and must be set to false for eigenfrequency analyses.
 * : If displacements or forces are set. thereby is automatically used.
 * : Considers the geometric stiffness of the body.
 * : Computation of incompressible material in connection with viscoelastic Maxwell material and a custom.
 * : Compute the viscoelastic material model.
 * : Uses model lumping.
 * : File to save the results from the model lumping.
 * : If true becomes a stability analysis (buckling analysis). Otherwise a modal analysis is performed.
 * : See the Elmer manual for more info.
 * : The variable for the elasticity equation. Only change this if is set to true in accordance to the Elmer manual.

Eigenvalues:
 * : If an eigen analysis should be performed (calculation of eigenmodes and eigenfrequencies).
 * : Should be true if the eigen system is complex. it must be false for a damped eigen value analyses.
 * : Computes residuals of the eigen value system.
 * : Set a damped eigen analysis. Can only be used if  is Iterative.
 * : Selection of which eigenvalues are computed. Note that the selection of Largest* cause an infinite run for recent Elmer solver (as of August 2022).
 * : Convergence tolerance for iterative eigensystem solve. The default is 100 times the.
 * : The number of the highest eigenmode that should be calculated.

Equation:
 * : Computes solution according to the plane stress situation. Applies only for 2D geometry.

Constraint Information
The elasticity equation takes the following constraints into account if they are set:


 * [[Image:FEM_ConstraintFixed.svg|32px]] Constraint fixed
 * [[Image:FEM_ConstraintDisplacement.svg|32px]] Constraint displacement
 * [[Image:FEM_ConstraintForce.svg|32px]] Constraint force
 * [[Image:FEM_ConstraintInitialTemperature.svg|32px]] Constraint initial temperature
 * [[Image:FEM_ConstraintPressure.svg|32px]] Constraint pressure
 * [[Image:FEM_ConstraintSelfWeight.svg|32px]] Constraint self weight
 * [[Image:FEM_ConstraintSpring.svg|32px]] Constraint spring

Eigenmode Analysis
To perform an eigenmode analysis (calculation if the eigenmodes and eigenfrequencies), you need to
 * 1) Set : to true
 * 2) Set : to false
 * 3) Set : to the highest number of eigenmodes you are interested in. The smaller this number the shorter the solver runtime since higher modes can be omitted from computation.
 * 4) Add a fixed constraint and set at least one face of the body as fixed.
 * 5) Run the solver.

Note: If you use more than one CPU core for the solver, you cannot use Umfpack, the only direct method for parallel solving is MUMPS. Also note that iterative solving is not recommended for eigenmode analysis. Therefore either only use one CPU core or install the MUMPS module to Elmer.

Buckling Analysis
To perform a buckling analysis, you need to do the same as for an Eigenmode Analysis, and additionally:
 * Set to true

Results
The available results depend on the solver settings. If none of the settings was set to true, only the displacement is calculated. Otherwise also the corresponding results will be available. If was set to true all results will be available for every calculated eigenmode.

If was set to true, the eigenfrequencies are output at the end of the solver log in the solver dialog and also in the document SolverElmerOutput that will be created in the tree view after the solver has finished.

Note: The eigenmode displacement $$\vec{d}$$ vector has an arbitrary value since the result is

$$\quad \vec{d} = c\cdot\vec{u} $$

whereas $$\vec{u}$$ is the eigenvector and $$c$$ is a complex number.