Today, plastic components are often assessed using deterministic FEA. This means fixed input values are defined for the material, geometry, and load, and the simulation provides a clear result based on those assumptions. In real production, however, component behavior is not that clear-cut. Material properties vary, loads can fluctuate, and in short-fiber-reinforced plastics, fiber orientation, fiber length, and local microstructure are strongly influenced by the manufacturing process.
In practice, this uncertainty is often handled through safety factors. That is robust, but it frequently leads to conservative designs and, as a result, unnecessary material use. Probabilistic simulations take a different approach: they include scatter directly and therefore provide a more realistic statement about the probability of failure. The drawback is the computational effort. For many industrially relevant plastic components, especially in series production, a comprehensive probabilistic calculation using conventional methods has so far been difficult to justify economically.
This is exactly where Pro4AI comes in. In the project “Pro4AI - Probabilistic Prognosis of Product Properties by Artificial Intelligence,” PART Engineering is working with the IPF - Leibniz Institute of Polymer Research to develop AI-supported methods that make probabilistic strength assessment and limit load analysis significantly faster and more practical. The aim is to accelerate time-consuming simulation steps by replacing them with trained and validated surrogate models.
A second important point is the distinction between local strength and global load-bearing capacity. Exceeding a local material limit does not automatically mean that the entire component fails. For component assessment, it is therefore also crucial to determine when a load-bearing cross section actually loses its capacity. Figure 1 shows this difference between a local strength limit and a global limit load.
Strength: When Does a Short-Fiber-Reinforced Plastic Fail?
Short-fiber-reinforced plastics are mechanically complex because their behavior depends not only on the matrix material, but also on fiber content and fiber orientation. For a realistic assessment, a representative volume element, or RVE, is used. This RVE represents a small but statistically representative section of the microstructure.
In principle, several failure mechanisms can occur in short-fiber-reinforced plastics: fiber failure, failure of the fiber-matrix bond, or failure of the matrix. For the materials considered in the project - PA6+GF15, PA6+GF30, PBT+GF15, and PBT+GF30 - the critical fiber length is clearly above the measured average fiber length. The critical lengths are approximately 680 µm for PBT, approximately 480 µm for dry PA6, and approximately 920 µm for conditioned PA6. Dominant fiber failure is therefore rather unlikely.
The fiber-matrix bond is difficult to characterize directly. With the typically good bond between fiber and matrix, however, its strength is in the same order of magnitude as the matrix strength. For the methodology pursued here, matrix failure is therefore considered the governing mechanism. To evaluate the local loading of the matrix, a strain-based utilization factor is used. At a value of 0, the matrix is unloaded; at a value of 1, matrix failure is likely.
However, one point is important: a single locally overloaded point does not yet cause component failure. By comparing experimentally determined failure stresses of the composite material with RVE simulations, it was concluded that approximately 20% of the matrix volume in the RVE must reach a utilization factor greater than 1 before the RVE - and therefore the composite material - is assessed as failed. This criterion is shown in Figure 2.
Initial validation was performed using PBT-GF15 as an example. The calculated critical stress levels show good agreement with the experimentally determined failure stresses. Figure 3 shows this comparison.
From the RVE to Fast Prediction: An ML Model for Local Failure
For industrial application, a direct FEM-based RVE simulation in every element of a component model is not practical. The computational effort would be far too high. For this reason, the project is developing a machine learning (ML) surrogate model that reproduces the behavior of the RVE simulation while calculating orders of magnitude faster.
The FFT-based solver Amitex was used to generate the training data. This reduced the computation time for an RVE with an edge length of 0.3 mm and a voxel resolution of 1 µm from approximately 3 hours in an FEM calculation to about 2 minutes. The FFT-based RVE simulations also show very good agreement with the experimental data, as shown in Figure 3.
The RVE geometries were created in the eigenspace of the orientation tensor. In total, 18 different orientation tensors were considered. Because the eigenspace of the orientation tensor does not necessarily match the eigenspace of the strain tensor, it is not sufficient to discretize the strain tensor only in this eigenspace. A farthest-point sampling algorithm was used to reduce the number of relevant strain tensors to 200. In addition, five RVE configurations were generated as ensembles for each orientation tensor in order to capture the statistical distribution of failure initiation. In total, this resulted in 18,000 RVE simulations, or 3,600 training data sets.
An XGBoostRegressor was used as the ML method. The input variables include the first two eigenvalues of the orientation tensor, the triaxiality of the stress tensor, and the angles between the first and second eigenvectors of the orientation and strain tensors. The target variables are the mean value and standard deviation of the averaged von Mises matrix stress at the time of failure. Figure 4 (left) shows that the mean value of the failure stress in particular is predicted very well; larger deviations still occur in the prediction of the standard deviation (right).
The model quality is very high for the mean value of the average failure stress: for both training and validation data, the coefficient of determination R² is above 97%, while the mean absolute percentage error (MAPE) remains below 4%. For the standard deviation, the MAPE values are significantly higher, while R² is still around 84 % to 87%.
Limit Load: Global Load-Bearing Capacity Instead of Only Local Strength
In addition to local strength assessment, the project is also developing an ML surrogate model for limit load prediction. The background is that limit load analyses usually require nonlinear simulations with elastic-plastic material models up to the range of plastic collapse. Such calculations are computationally intensive and are not always economical in everyday development work. At the same time, load-bearing capacity is decisive for the assessment of many components because it describes global failure.
For the model, a component database was created to cover the broadest possible range of geometries. For different material and load case combinations, an isotropic linear-elastic, ideally plastic FE analysis was performed. This produced around 100 data sets. The limit loads were determined using the component yield curve.
One challenge is that FEM results are available as spatial field quantities. To allow an ML model to process this information, the data space was reduced. Figure 5 shows the principle of this dimensionality reduction and the path from the FE result to the ML model.
For dimensionality reduction, numerous statistical characteristics of the stress distribution in the component were calculated, for example mean value, standard deviation, minimum, maximum, median, and the 25% and 75% percentiles. These were supplemented by dimensionless parameters describing component geometry and deformation state, as well as physically motivated limit load multipliers. All parameters are determined in the first linear-elastic calculation increment of the isotropic elastic-ideally plastic FE analysis. In total, around 190 parameters were defined.
These parameters are not used directly as input variables. First, the data sets are normalized, scaled, and clustered. A principal component analysis (PCA) is then applied. The resulting component scores serve as compact predictors. The first component often describes the basic trend of the target variable - in this case the limit load - while subsequent components capture additional secondary effects.
For this first model version, an ordinary least squares (OLS) regressor was used as the ML method. With the currently limited data volume of around 100 data sets, this is more appropriate than more data-hungry methods such as neural networks or XGBoost. In addition, under the assumption of linear relationships, an OLS regressor can extrapolate comparatively well. In a first training run, the significant PC scores were identified using ANOVA; in a second run, only these components were considered. The model also provides a prediction interval, here with a 95% error probability, allowing the model uncertainty to be assessed.
Figure 6 compares the limit loads predicted by the ML model with the values determined by FEA. The results of a leave-one-out cross-validation are shown. The agreement is very good: for the validation, R² is 93.98% and the MAPE is 3.26%. The exemplary comparison with experimentally determined limit loads also shows very good agreement between model prediction, FEA, and experiment.
Conclusion and Outlook
The results so far show that ML surrogate models offer a realistic opportunity to significantly accelerate computationally intensive simulations for strength and limit load assessment of plastic components. This creates an approach that connects two worlds: the physical significance of detailed simulations and the speed of data-driven models. This is particularly relevant for short-fiber-reinforced plastics, because scatter in the microstructure and anisotropic behavior can often be considered in conventional design only with substantial effort.
Pro4AI demonstrates how AI methods can make strength verification of plastic components faster and more practical. The current models describe both micromechanical failure mechanisms in the RVE and the global limit load of components with practical accuracy.
However, the work is not yet complete. The anisotropic limit load analysis, the improved description of scatter in the RVE prediction, and robust integration into industrial workflows will continue to be developed. After successful further development and validation, the project results are intended to be integrated into the PART software S-Life Plastics at a later stage. The goal is to provide users with fast, probabilistic, and practical assessment of plastic components in the future.
Project and Funding
The Pro4AI project is being carried out in cooperation with the IPF - Leibniz Institute of Polymer Research, Dresden. This project is funded by the Federal Ministry for Economic Affairs and Energy (BMWE) on the basis of a decision by the German Bundestag.
Author: Dr. Wolfgang Korte is Managing Director at PART Engineering GmbH, Bergisch Gladbach, Germany