On the interpretation of strain energy and strain energy density contours in a finite element software

Introduction

Very often in finite element analysis software, one finds the plots available for strain energy and / or strain energy density following the completion of analysis.

Sometimes, the engineer wonders what do these plots physically signify and how to interpret these plots and for what reason?

The aim of this write up is to illustrate an interpretation of these plots in a static and dynamic analysis and the options a finite element analyst could consider in his design through the interpretation of the numerical values in the plots.

Strain energy plots following a static finite element analysis (stress critical design criteria)

Definition of strain energy:

When a body is deformed, work is done. The energy used up is stored in the body as strain energy [for an elastic body, the energy may be regained by allowing the body to relax].

That is: strain energy by definition is the energy that is stored in a body due to deformation.

 

How to interpret the strain energy plots?

This strain energy when plotted per element following a finite element analysis gives the analyst a design guideline i.e. a higher strain energy area requires a higher strength and hence one could add more material into the structure in regions of higher strain energy and less material into the structure into regions of less strain energy.

Thus, one can clearly note at this point that whilst doing a static stress analysis, use of strain energy would be directly related to stress.

Consequently, there would be no obvious advantage to using strain energy over stress in a static analysis.

 

Strain energy density plots in a modal analysis

When you are doing an eigen frequency analysis, the frequencies are what you are looking for, the displacements are arbitrary, these are normalized in different ways, and different FEM tools (software’s) have different norms. That MEANS that it is impossible to calculate actual displacements with such analysis. Because it only gives you the frequency at which the mode occurs, and the shape of the mode. The actual displacements depend on the input excitation.

Similarly, stress results from modal analysis are meaningless in terms of reported quantities; the

distribution of stresses does provide valuable insight into structural properties.

Therefore, in a modal analysis, analysts use strain energy density, as it will give them insight into what elements on the model have the greatest deformation normalized to the element volume.

Optimization of the model could therefore involve an evaluation of the strain energy density for several modes, and removing material from the structure with low strain energy density and adding material to the structure in regions of high strain energy density.

This is how a typical optimization analysis is performed on a mode

 

Other uses of strain energy plots for assessing a finite element solution

Other uses of strain energy plots in a finite element solution include checking the convergence of a problem.

 

The best way to visualize whether a solution is converged is to plot one solution variable over the iterations that have been run.  This plot should show the solution converging on a single value.

Convergence can be checked using many different solution variables.  These include:  strain energy, maximum displacements, maximum stresses, etc.  However, the best variable to use is strain energy.  This will provide the smoothest convergence plots.  Checking convergence based on maximum stress values is generally not recommended.  The values of maximum stress is a local measure; meaning that this value occurs at a local point and does not apply over a large region. 

Therefore, the value of maximum stress may not change in a smooth fashion as the accuracy of the solution is increased.

In the plot above, maximum stress in the x-direction is plotted with respect to iteration.  Notice that the curve is not smooth as it approaches the converged value.   If the solution had only been run over five iterations, then it would look like the solution was converged.  This could cause the solution to be incorrect by a large percentage.  In the plot below, strain energy is plotted with respect to iteration on the same model.  Notice that the curve is much smoother.

The reason is that the strain energy can be calculated over an entire region / whole structure where as the stress plotted above is at a single point.

Error estimators based on the strain energy approach have been available in different finite element software’s

Other uses of strain energy density contours

Further, the strain energy density contours are in some cases used for the prediction of the crack growth path which is a pre-requisite for the estimating the final shape of broken solids and structures. The predicted trajectory of the crack during unstable propagation is assumed to coincide with the minimum of the strain energy density function according to the strain energy density criterion. The degree of the crack path stability depends on the sharpness of the strain energy density oscillations.