 # What’s the Differences Between Stress Concentration and Stress Singularity

Some engineers who are new to structural simulation often confuse the concepts of stress concentration and stress singularity. Both of them manifest as high stress in localized regions, but they have some fundamental differences. Stress concentration occurs when there is a large stress gradient due to poor transition of dimensions in the structure. However, it can be mitigated by refining the mesh or using submodeling techniques in software like ANSYS to achieve convergence of stress results and obtain accurate verification values. On the other hand, stress singularity is caused by more complex factors such as structural, material, and boundary condition discontinuities. Once stress singularity occurs, it cannot be eliminated by refining the mesh.

Let’s analyze the differences between stress concentration and stress singularity through specific examples.

### 1. Stress Concentration The above image shows a classic model with fixed constraints applied to face 1 and a vertically downward force applied to face 2. Based on experience, the highest stress is expected to occur at the corner angle between the two walls because maximum stress usually occurs at surfaces. Therefore, the maximum stress should be distributed on the rounded corner surface of the angle, and the calculated stress distribution confirms this. However, according to finite element theory, when creating a finite element model, the more refined the discretization of the model, the closer the calculated results will be to the true values. This means that mesh independence checks should be performed by refining the mesh sufficiently to obtain converged stress results. If the entire model is refined with a fine mesh, it can become computationally challenging. In this case, the submodeling technique, as described in the “Submodeling Analysis Technique,” can be used to reduce the computational burden when dealing with very fine meshes. The table below shows the results of the mesh independence check, and it can be observed that the stress results are almost converged when the mesh size is around 0.5 mm. Therefore, this mesh size is reasonable. Stress concentration issues encountered in practical engineering are similar to the example above. Stress concentration problems mainly involve two aspects: accuracy and structural optimization. Accuracy refers to ensuring the correctness of calculations by achieving result convergence through mesh independence checks. Structural optimization aims to minimize the degree of stress concentration by modifying structurally unreasonable areas. For example, in the structure of the L-shaped bracket mentioned earlier, under the same boundary conditions, if the corner radius of the two arms is increased, the maximum stress obtained after convergence will decrease.

### 2. Stress Singularity

Now let’s consider another model, where the size of the model is reduced for computational efficiency compared to the previous model. Additionally, the rounded corners at the angle are removed, and the same boundary conditions are applied for calculation. The results of the mesh independence check are shown below. It can be observed that as the mesh becomes finer, the stress results become higher and tend to diverge. In other words, as the mesh size decreases, the rate of increase in stress results becomes greater, and the slope of the mesh size-stress curve keeps increasing. We refer to this kind of result as stress singularity. The reason for stress singularity is primarily the complete absence of transition in this abrupt structural change. In the earlier explanation of stress concentration, although there were structural changes, they were continuous, resembling a mathematical curve where transitions occur at every point, and derivatives can exist at each point. However, this kind of abrupt structural change without any transition is like a mathematical step function that does not have derivatives at turning points. In finite element analysis, all such discontinuities cause stress singularity, and stress at the singularity cannot be accurately calculated because the stress results become more divergent as the mesh is refined.

Besides the structural discontinuities mentioned above, material discontinuities can also cause stress singularity. Take the following model as an example: a cantilever beam with one half made of a material with an elastic modulus of 200 GPa, and the other half made of a different material with an elastic modulus of 100 GPa. The two sections of the beam are connected at a common node. The left end face of the cantilever beam is fixed, and a vertical downward force is applied to the right end face. At the interface between the materials, there will be a sudden change in stress, resulting in different stress results on both sides of the interface at the same location. Therefore, special attention should be paid to stress results at different material interfaces in engineering applications. In cases where there are sharp contacts between structures, stress singularity phenomena can also occur. Imagine poking a steel plate with a needle. In reality, due to the high strength of the steel plate, applying manual force alone will not cause plastic deformation on the surface of the plate. However, in finite element analysis, the stress results obtained at the contact point between the needle tip and the steel plate are likely to exceed the yield strength of the steel plate. This is because it is a stress singularity, and the stress results obtained are not representative of the actual situation.

Stress singularity can also occur at constrained locations. For example, in simulating tensile tests, fractures occur in the parallel length section rather than in the clamped section. This is because the clamped section has a larger cross-sectional area and therefore experiences lower stress compared to the parallel length section. However, in finite element models, if a fixed constraint is applied at one end of the clamped section and tension is applied at the other end, when the mesh is dense enough, the maximum stress will appear on the face with the fixed constraint, as shown in the figure below. This is caused by stress singularity, so when modeling and simulating tensile tests, the application of boundaries requires careful consideration. Stress singularity in engineering simulations is commonly caused by constraints. For example, in structures with bolt connections, models are often assembled using node coupling to simplify the model. In addition, fixed constraints are applied using node coupling at installation holes. These locations are prone to stress singularity, and the stress results at these locations should be carefully evaluated. Overloading should not be easily determined based on stress results obtained from stress singularities.

When stress singularity occurs, two issues need to be considered. First is structural optimization, especially in structures subjected to vibration conditions. If stress concentration and stress singularity coexist, efforts should be made to avoid them as much as possible, as fatigue is highly sensitive to stress concentration. The second issue is accuracy. Generally, pure stress singularity does not require further efforts for accuracy, as it is almost impossible to obtain accurate results in such cases.

Good Luck!   