1. Introduction

1. Objective of the Study

2. Materials and Methods

3. Results

3.1. Characterization of Compressive Behavior in Foam: Analysis and Curve Fitting

The Figure 3 presents the stress-strain curves obtained from uniaxial compression tests conducted on five different foam specimens, each tested 8 times under the same compression conditions. These tests were performed using the DIN 53577 Compression Load Deflection (CLD) methodology, a widely recognized standard in Europe for assessing the compressive behavior of foam materials. Each foam specimen, with dimensions of 200mm x 200mm x 200mm, was subjected to eight repeated compressive tests to evaluate the consistency and reliability of its mechanical properties. The larger specimen size used in these tests, compared to the standard dimensions (100mm x 100mm x 50mm), provides a more comprehensive understanding of the foam's behavior under significant loading conditions. The resulting stress-strain curves illustrate the characteristic deformation behavior of the foam, capturing the initial elastic response, the plateau associated with cell collapse, and the final densification stage.

The stress-strain curves of the foam demonstrate a consistent progression through distinct regions. Initially, all curves exhibit a linear increase in stress with strain, reflecting elastic behavior where the foam deforms proportionally to the applied stress. This initial section, known as the Linear Region, aligns with Region 1 where cell walls bend without collapsing. Following this, a noticeable plateau appears in the curves, marking the Plateau Region. In this middle section, corresponding to Region 2, the foam’s cells begin to collapse, resulting in significant deformation with minimal increase in stress. As strain continues to increase, the curves show a sharp rise in stress during the Densification Region, indicating that the foam is undergoing densification in Region 3. At this stage, the cell walls are fully compressed, leading to a considerable increase in material stiffness.

Across multiple tests, the foam demonstrates good reproducibility, with curves for each specimen closely overlapping. This consistency is crucial for ensuring reliable performance in applications. However, slight deviations are observed in some tests, such as Test 8, particularly within the densification region. These minor variations might be attributed to small differences in foam structure or test conditions. Overall, the foam’s compressive behavior remains consistent across different iterations, reinforcing its reliability.

When comparing specimens, the stress-strain behavior is notably similar across all five samples, indicating uniform material properties. This consistency is essential for practical applications, ensuring reliable performance. Although the point where densification begins varies slightly between specimens, this variation is likely due to differences in foam density or minor manufacturing inconsistencies. The tested specimens, being significantly larger than the standard size specified in DIN 53577 (200mm x 200mm x 200mm), might exhibit different stress distributions. Nonetheless, the overall similarity in behavior suggests that the increased size did not introduce significant variability in the results.

Figure 4. Stress-strain curves of foam specimens demonstrating the linear, plateau, and densification regions in: (a) specimen 1, (b) specimen 2, (c) specimen 3, (d) specimen 4 and (e) specimen 5.

Figure 4. Stress-strain curves of foam specimens demonstrating the linear, plateau, and densification regions in: (a) specimen 1, (b) specimen 2, (c) specimen 3, (d) specimen 4 and (e) specimen 5.

The graphs offer a thorough understanding of the compressive behavior of the foam specimens. The foam exhibits consistent performance across various specimens and test iterations, with clear linear, plateau, and densification regions evident in the stress-strain curves. This high level of reproducibility underscores the foam’s reliable and predictable mechanical properties under compression. Consequently, the foam appears well-suited for applications requiring uniform compressive strength and deformation characteristics. Further analysis could involve calculating precise values for the modulus of elasticity in the linear region, the plateau stress, and the strain at which densification commences, providing more quantitative insights into the material's properties. Ultimately, the goal is to develop an equation for the characteristic curve based on these extensive tests to better describe the foam's behavior.

The curves illustrate consistent performance across different specimens and test iterations, despite some observed hysteresis in the initial iteration of each specimen, where higher stress-strain values were recorded. This indicates that, while the tests are reproducible, the first iteration may slightly deviate, suggesting a need to account for hysteresis when developing the characteristic curve equation.

To cluster these signals and calculate the most representative signal of the largest group—assuming the initial signals are discarded—k-means and k-medoids are used, and the results compared without prior knowledge of the number of clusters. The steps in Python are as follows: i) Data Preparation: The signals are organized in a matrix, with each row representing one signal, ii) Calculation of Statistics: Statistical measures such as mean, variance, etc., can be used, or the entire signals can be directly employed, iii) Application of Clustering Algorithms: k-means and k-medoids are applied for clustering. To determine the optimal number of clusters, the elbow method or the silhouette criterion can be used, and iv) Comparison of Results: The outcomes of both methods are evaluated and compared, similar to what is done in [11,12].

With the focus on clustering the signals based on their compressive stress-strain characteristics next steps are followed. The first major step is determining the optimal number of clusters. To cluster signals based on their compressive stress-strain characteristics, the initial task is to determine the optimal number of clusters. This is achieved through the use of two key methods: the "elbow method" and silhouette scores. The elbow method involves evaluating the sum of distortions across a range of cluster numbers (k). Distortion is quantified as the average distance between each data point and the nearest cluster center. By plotting these distortions against the number of clusters, a distinct "elbow" shape often emerges. This elbow represents a point where increasing the number of clusters yields diminishing returns in reducing distortion, suggesting the optimal number of clusters. In parallel, silhouette scores are computed for each value of k. These scores measure the extent to which each data point is similar to points in its own cluster compared to points in other clusters. A higher silhouette score indicates better-defined clusters. By plotting silhouette scores, the peak of the curve helps to identify the value of k that results in the most coherent clustering structure. Together, the elbow method and silhouette scores provide a robust approach to selecting the number of clusters, ensuring that the clustering solution is both effective and interpretable. In the Figure 6 these results are shown to visually identify the optimal k, where the elbow in the distortion curve and the peak in the silhouette score indicate the most appropriate cluster count.

Figure 5. Optimal number of clusters determined using the elbow method and silhouette scores.

Figure 5. Optimal number of clusters determined using the elbow method and silhouette scores.

Once the optimal number of clusters is determined, both K-means and K-medoids clustering algorithms are applied. The clusters are visually represented in Figure 6 using a 2D projection of the signals obtained through Principal Component Analysis (PCA).

For each cluster, the Figure 8 plots all signals in grey, highlighting the mean signal and the medoid (central point) of the clusters in red and blue, respectively. The process helps identify groups of signals with similar behavior, as well as representative signals for each group, which is crucial for understanding the underlying patterns in the data.

In the final step a polynomial regression is performed to model the relationship between strain and stress for the mean signal of the second cluster as it can be seen in Figure 9.

4. Conclusions

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