Figure. Linear elastic and Semi-elastic-plastic material
ProTOp contains special finite elements that make it possible to simulate elastic-plastic material behavior with very high efficiency. In fact, the special semi-plastic elements are practically as fast as elastic elements. It should be noted, however, that the semi-plastic elements are intended for monotonic loading only.
Special semi-plastic elements are engaged by ProTOp automatically, if their corresponding material is declared as semi-plastic and the associated yield stress value is set greater than zero.
There are two situations when elastic-plastic behavior engagement is useful:
The first reason is the usual one and needs no special explanation. What looks to be somewhat confusing, however, is the second reason. This second reason is related to the effects of strain energy minimization performed on linearly elastic model. Namely, as it turns out, in certain special situations useful stress reduction can be achieved by engaging the so called virtual plasticity.
Topology optimization can be extremely efficient in reducing/removing stress concentrations. However, in the regions with fixed geometry (e.g. in the proximity of a prescribed fixed circular hole), stress levels may remain high even after optimization.
The figure below illustrates a part of the optimized connecting rod. The design depicted on the left side was obtained by using a linearly elastic material model. One can see that the stress concentration is extremely high at the whole boundary. As a consequence the stress levels at the outer boundary drop so much that the material gets removed too aggressively. This is not a flaw of the optimizer - it is a flaw of the linear elastic material model.
Figure. Linear elastic and Semi-elastic-plastic material
To improve this situation, one of the most promising approaches seems to be to engage elastic-plastic material behavior. In this way stress concentrations (e.g., at the hole boundary) can be reduced significantly (see figure, right side), which leads to a beneficial material redistribution by the optimizer. Note that in such situations the new computed topologies typically exhibit lower stress concentrations even after switching back to a linear elastic material model.
In this procedure, elastic-plastic material behavior is engaged by making use of a virtual yield stress that has nothing in common with the actual yield stress level. The idea, namely, is just to let the material flow in order to obtain a better material redistribution. The virtual yield stress level has to be chosen in such a way, that the material will redistribute to the desired level. Typically, this can be done only by monitoring the optimization progress and setting the corresponding parameters accordingly.
Let us illustrate this with an example problem. Let the task be to optimize the topology of the structure shown on the following figure, where color levels indicate von Mises stresses.
Figure. The example problem: structure with supports and loads (left); the problematic high stress detail is marked with a dashed rectangle (right).
The structure exhibits excessive stress concentrations at the domain boundary, marked by a dashed rectangle. A closer look of the problematic detail is shown below.
Figure. The problematic detail with excessive stress concentrations.
Let us now run topology optimization by employing elastic material. The result is shown on the following figure.
Figure. Optimization result obtained by using elastic material.
Note that the optimizer has done its job flawlessly and the structural strain energy is at minimum. What is problematic here is that because of the domain boundary, the high stress concentrations of about AMStressMax = 380 could not be removed. What can help here is virtual plasticity. By engaging it, one can reduce such stress concentrations on the account of slightly worse strain energy of the structure.
Let us now engage the semi-plastic material behavior, and let us set the virtual yield stress to 160. By running further optimization cycles the material is just redistributed as shown in the figure below.
Figure. Optimization result obtained by using semi-plastic material; virtual yield stress was set to 160
One can see that the optimizer significantly redistributed the material. As expected, the stresses also fell to about AMStressMax = 160. Note however that this stress state would be the actual stress state only if the material would actually yield at 160.
Let us now suppose our actual material would not yield at all. This means that our 'semi-plastic optimal design' would actually exhibit stresses obtained with linear elastic analysis. By running linear analysis the last obtained result exhibits stress levels as shown below.
Figure. Stress state of the 'semi-plastic optimal design', computed by linear elastic analysis (no material yielding).
It turns out that the maximal stresses are around AMStressMax = 260. This is a substantial improvement compared to the purely elastic optimal design, which exhibited maximal stresses around 380. Thus, utilizing virtual plasticity might be useful even if the material will actually not yield at all.
NOTE. Engaging semi-plastic material behavior might be useful even if the material will actually not yield at all.