What is Strain Formulation?

Strain formulation is sought-after in nonlinear analysis as the model/component of interest that undergoes plastic deformation. We have already studied that the explicit solver follows courant stability i.e critical time must be less than that of the critical length of the element. Under such situations, when plastic deformation happens, there are a lot of chances of large deformation happening in the model. When large deformation happens, the element's lengths tend to reduce, which in turn puts the stress on the solver to take care of such elements too.  If the solver considers those elements then the stability of solving further is going to be a big question.  

Now, basically, our solver is going to take the deformation in the previous case as a reference and it is going to calculate the same strain and deformation for the next step. As mentioned earlier, if this stability is reduced then solving for the next step is going to be a difficult task. This particular issue is taken care of by what is called strain formulation. 

Strain Formulation

Strain formulation can be divided into two: Large strain formulation and Small strain formulation. The explicit solver follows a large strain formulation. 

Large strain formulation is by default used in the RADIOSS calculation. Now let’s see what is a large deformation. Large deformation theory or large strain theory is defined as the deformation of a solid body in which the displacements created by the force are assumed to be larger and creates permanent deformation. Basically, the term ‘large’ is considered if the deformation is higher than the component’s thickness.

The phenomena like negative volume generation, large distortion, and excessive displacement come under large deformations. Large strain formulation is best suited for nonlinear, elastoplastic behavior and small strain formulation is suited for linear elastic behavior.

Small Strain Options

Small strain options are used in shell property definition in RADIOSS. In the same shell property, we select Ismstr=2 which is ‘full geometric nonlinearities with possible small strain formulation activation in Radioss Engine’, this is recommended property in a crash and also if not explicitly given, radioss takes this option only. As the term means, whenever there is a situation of very large deformation, a small strain flag can be activated. Now, to understand how this small strain works, let's get into a stress-strain relationship.

We know the difference between engineering stress-strain and true stress-strain. To give a quick answer, the engineering stress-strain curve depends on the original cross-section and the gauge length of the specimen and the true stress-strain curve depends on the instantaneous cross-section and gauge length of the specimen.

The large strain formulation results from incremental strain computation i.e stress and strains in large strain formulation are true stress and true strains. When a very large deformation happens in the case of negative volume, it leads to a very high decrease in the time step, and the stability of the solution reduces. At those times, the solver automatically takes that kind of deformation as a small strain formulation where it converts the true stress-strain curve to engineering stress-strain values which are going to be very less than the true stress-strain values. By doing this, the large deformations are reduced to a larger extent and the stability of the solution is confirmed. 

To make it very simple, explicit analysis follows large strain formulations. In large strain formulation, the material data are taken as true stress-strain values by the solver. Whenever, a very large deformation happens which results in negative volumes, it affects the solver’s stability condition resulting in a reduction in the time step. To avoid these problems, a small strain formulation is important. The small strain formulation, at large deformations, converts the material values as engineering stress-strain values, which results in better solving of the solver engine. Hence, we give freedom to RADIOSS to jump between large strain and small strain formulation at times of crisis. 

Things to Remember

It is important to note that a small strain formulation is not recommended for crash analysis tools for shell elements. The reason is very simple: shell elements during the crash undergo bending. Small strain formulations do not cover the bending of elements. If we look into the Ismstr option in the /PROP/SHELL card we get a better understanding: 

• If Ismstr=1 or 3, then the whole simulation will be carried out in small strain formulation but the credibility of the results is not ensured. 
• In the same way, if Ismstr=1 or 3 then the material data is taken as engineering stress-strain. 
• If Ismstr=4, RADIOSS will not switch to a small strain formulation whenever required. 
• If Ismstr=10, then it takes only a small strain formulation for hyperelastic materials

To conclude, the small strain formulation helps to keep the stability of the model under control during times of duress.