Integrating deep learning, fractal analysis to construct the relationship between microstructure and tensile strength of polypropylene foams

Qi Chen; Tianyu Xie; Zhao Yang; Zhipeng Yang; Anping Wang; Wei Gong

doi:10.25259/AJC_93_2024

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Original Article

2025

:18;

932024

doi:

10.25259/AJC_93_2024

Integrating deep learning, fractal analysis to construct the relationship between microstructure and tensile strength of polypropylene foams

Qi Chen^a,b, Tianyu Xie^c, Zhao Yang^a, Zhipeng Yang^c, Anping Wang^d, Wei Gong^a,b,*

aSchool of Mathematical Science, Guizhou Normal University, Guiyang, Guizhou 550025, China

bSchool of Materials and Architectural Engineering, Guizhou Normal University, Guiyang, Guizhou 550025, China

cSchool of Mechanical and Electrical Engineering, Guizhou Normal University, Guiyang, Guizhou 550025, China

dKey Laboratory for Information System of Mountainous Area and Protection of Ecological Environment of Guizhou Province, Guizhou Normal University, Guiyang, Guizhou 550025, China

*Corresponding author: E-mail address: gongw@gznu.edu.cn (W. Gong)

Received: 2024-10-17, Accepted: 2025-07-03, Epub ahead of print: 2025-09-15, Published: 2025-10-15

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Abstract

In this study, with the help of deep learning, we use scanning electron microscope (SEM) images of polypropylene (PP) foams as the original dataset, and fuse the pre-trained VGG16 network into the UNet model, which realizes the feature migration and parameter sharing and effectively reduces the training cost. During the model training process, we also combined binary cross-entropy loss and focus loss to further optimize the training effect. Experimental results show that the improved UNet model performs well in automatically identifying and classifying the microstructure of PP foams. In terms of Precision (P), Mean Intersection over Union (MIoU), and F1-score of pore recognition, they reach 95.25%, 85.54%, and 95.63%, respectively. These values are significantly improved compared to the original UNet model as well as GRFB-UNet, UNext, ParaTransCNN, scSE-UNet, and Swin-UNet models. In addition, we determined the fractal dimension of the samples by the area-perimeter method and found that the samples had significant fractal characteristics. Gray correlation analysis showed that the complexity of pore structure was positively correlated with fractal dimension. The combined Gibson-Ashby strength modeling and fractal analysis demonstrate that both porosity and fractal dimension significantly affect tensile properties. Increased porosity reduces tensile strength, while higher fractal dimensions similarly lead to lower tensile strength.

Keywords

Deep learning

Fractal dimension

Microporous polypropylene

Tensile strength

UNet

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1. Introduction

Polypropylene (PP) is widely used in traditional applications such as thermal insulation, sound absorption, and cushion packaging due to its unique lightweight, porous structure, and tunable physicochemical properties [1]. In recent years, through functional modification, it has also demonstrated promising potential in emerging fields like electromagnetic interference (EMI) shielding, sensing devices, and smart responsive materials [2-4]. Furthermore, it plays an important role in specialized applications, including adsorption separation, medical encapsulation, and composite membrane materials [5-7]. As the demand for lightweight materials grows, low-density polymer foams with superior properties are gaining increasing attention [8-10]. The macroscopic properties of polypropylene foams are influenced by multiple factors, and these properties are not solely dependent on the polymer’s inherent characteristics (e.g., crystallinity), the material components (e.g., filler content), and the foam’s relative density, but also on parameters such as cell size, shape, distribution, cell density, and open-cell ratio [7,11-15]. Therefore, characterizing and extracting the structural parameters of PP foam cells is an important research task.

Currently, the majority of studies utilize SEM to acquire the microstructure of materials and employ software like Image J Pro to measure the size, count, and distribution of pores. This method is simple and low-cost, but the statistical volume is large, and the manual measurement of the data is subject to human factors, which makes the accuracy low and the process time-consuming [16,17]. In addition, various investigators have suggested employing Otsu’s thresholding alongside edge detection methods and local thresholding segmentation to delineate the pores and matrix within SEM/TEM imagery [18]. However, when dealing with SEM/TEM images that have severe noise and extremely uneven grayscale backgrounds, these methods do not perform optimally [19]. Consequently, precisely distinguishing between pores and matrices in highly intricate SEM/TEM images continues to be a formidable challenge. Alternative approaches, including X-ray tomography, enable 3D structural analysis of porous materials without causing damage, though they tend to be more costly and time-consuming [20,21]. Therefore, to analyze the microstructural parameters of foamed materials more efficiently and accurately, computational material science methods are beginning to receive attention, and automated SEM/TEM image analysis methods have great potential to help researchers get rid of heavy and repetitive work [22,23].

In the study of mechanical properties, the mechanical properties of polymer foams are mainly determined by physical experiments and numerical simulations [24]. Numerical methods have become an important tool for characterizing complex geometries as an important alternative to experimental measurements due to their advantages such as high accuracy, low cost, and efficiency [25]. For example, Fereidoon and Taheri analyzed the effect of microstructure on the energy absorption properties of open-cell polymer foams by the 3D finite element method using the Abaqus software and explored the effect of bubble size and density on the stress-strain behavior of open-cell PP foams [26]. On the other hand, Trauth et al. used a two-step Hashin-Shtrikman mean-field homogenization method to investigate the effective elastic properties of polymer composites, verifying the accuracy of their results experimentally [27]. However, in these methods, the particles/pores are usually reduced to inclusions with uniform or random size distribution and described in terms of volume fraction or total number, which differs from actual porous composites. In real structures, the size distribution of particles or pores is random, and this simplified approach becomes computationally expensive when the volume fraction Vp or total number N of particles or pores is high (Vp > 50% or N > 100). More challenging is the difficulty for existing numerical methods to effectively take into account the effect of microscale features on macroscopic properties when the geometries and distributions of pores in porous materials have significant differences.

In recent years, deep learning has demonstrated strong potential for applications in material characterization, performance prediction, and microstructure design [28-30]. Therefore, the combination of deep learning and fractal geometry opens up new paths for porous materials research. Since AlexNet’s breakthrough in 2012, deep learning has become a core technology for computer vision tasks such as image segmentation. It automatically learns features from large amounts of image data through a data-driven approach, optimizing algorithm performance beyond traditional SEM image segmentation methods. Models such as Full Convolutional Network (FCN) and UNet are prominent in medical image segmentation, and UNet is favored for its encoding-decoding structure and jump connections. Meanwhile, deep learning-based research has achieved remarkable results in microscopic image (SEM/transmission electron microscopy-TEM) segmentation [31], structure prediction and design [32], structure-property correlation analysis [24,33], and structural characterization [34] in material science, especially in the field of image segmentation. For example, Bishwas Pokharel et al. [35] have studied the segmentation of SEM images of activated charcoal using DeepLab V3 Plus, DeepLab V3, and Feature Pyramid Network models, and found that DeepLab V3 Plus has the best segmentation results. Peng Shi et al. [36] have introduced an upgraded version of the UNet model that is capable of autonomously segmenting intricate metallographic images, even with limited training data. The results indicate that this method has significant advantages in handling metallographic images with low contrast, blurred edges, and complex structures. It is also worth mentioning that Jorge Torre et al. [30] successfully achieved automatic segmentation and characterization of SEM images of polymer porous materials by a Mask R-CNN deep learning model, which proves the accuracy and efficiency of Mask R-CNN in the characterization of polymeric foam materials. In addition, experiments have confirmed that random targets usually follow fractal geometrical laws [37]. The fractal approach is particularly applicable in describing the microscopic features of materials that are irregular, inhomogeneous, and exhibit self-similar or self-affine structures at a certain scale. It goes beyond the traditional paradigm of describing 1D lines, 2D surfaces, 3D solids, and even 4D spacetime, and is closer to describing the real properties and states of complex systems, which is more in line with the diversity and complexity of objective things [37,38]. The fractal dimension is considered to be an important parameter to quantify the surface complexity. In addition, the complex structure of porous composites can be effectively decomposed by self-replication and concisely quantified by the fractal dimension. Despite the widespread application of deep learning in the segmentation of material microscopic images, there have been no reports on the automatic recognition and segmentation of pores in SEM images of PP foams, nor have there been any reports on the research of the mechanical properties of polyolefin foams based on fractal theory. This indicates that further research is needed in these areas.

In summary, to establish a workflow for PP foam materials that covers “Processing-Microstructure-Mechanical,” this paper aims to develop an efficient and accurate method for analyzing the cellular structure of PP foam using deep learning. First, SEM was used to characterize the microstructure of PP foams in detail. Then, different deep neural network models were used to process SEM images to achieve accurate recognition and segmentation of the pores and matrix, and the segmented PP foam material pore structure images were measured and analyzed to calculate key topological structural parameters such as pore density, average pore diameter, shape factor, pore distribution index, and fractal dimension, and compared with manual measurement results to verify the accuracy of the model. Finally, the effects of the pore structure parameters on the tensile properties of the materials were investigated in depth using fractal analysis and Gibson-Ashby strength model.

2. Materials and Methods

2.1. Image acquisition

General-purpose plastic PP was selected as the matrix material, and the lightweight material was prepared by injection molding and molded chemical foaming, and the detailed processing method of the PP foamed material is described in the reference [39]. Liquid nitrogen was applied to freeze section the material, and a scanning electron microscope (SEM) (TM-4000 Plus) was used to characterize the material; 300 SEM images were obtained at different magnifications with an image size of 1280×960 pixels.

2.2. Image annotation and dataset construction

In this study, the Labelme software was used to manually annotate the pores in SEM images. To improve the labeling accuracy as much as possible, the pores in 300 original section images (image size of 1280×960 pixels) were labeled. The images were first enlarged in Labelme software to fit the edges of the pores as closely as possible for polygonal point labeling. Finally, the images were reduced to the original size and saved as true value data, as shown in Figure 1(a-d).

(a, c) SEM micrograph; (b, d) Example of manual override annotation illustration with Labelme.

2.3. Data enhancement

To enhance the accuracy of the pore recognition model and to mitigate overfitting during the training process, a large dataset is typically necessary for support [40]. However, obtaining new data is often time-consuming and costly, so data augmentation becomes a necessity. In this paper, we employ the following data augmentation techniques: ±45 degree random rotation of the original image, horizontal flip, 20% pixel random cropping of the edge region, and 0.8-1.2 times luminance linear transformation. The 300 original images were used to expand the dataset to 1500 images by the above data enhancement techniques. Given that each image contains about 150 pore samples on average, after data enhancement, we obtained about 220,000 pore samples, which are not only sufficient in number but also diverse, and partitioned the standard dataset into training, validation, and testing sets according to the 8:1:1 ratio.

2.4. UNet network and improved algorithm

UNet is a U-shaped segmentation network proposed based on fully convolutional networks, which is the most commonly used image segmentation algorithm in recent years [41,42]. In this paper, we propose an improved version of UNet, which retains the core advantages of UNet while optimizing other key aspects. In this study, the VGG16 network is used as the backbone network to enhance the model’s feature extraction capability, as the target detection task faces the challenges of small target size, large number of targets, and high noise interference. In the training stage, the model combines a binary cross-entropy loss function and Focal Loss, and this combination of loss functions not only accurately measures the difference between the model predictions and the actual labels but also pays special attention to samples that are difficult to classify, thus significantly improving the accuracy of image segmentation. The structure of the improved UNet segmentation model has been detailed in Figure 2. The original UNet encoder was entirely replaced with the first 13 convolutional layers of VGG16 (with fully connected layers removed), forming a new encoder-decoder architecture.

2.4.1. Feature extraction module

The SEM images of PP foams pose a considerable challenge to the image segmentation task due to the large number of pores with non-uniform distribution, as well as significant differences in contrast and brightness. In addition, the convolutional operation of the UNet network in the encoding stage may cause redundancy in feature extraction, which not only affects the model’s accurate interpretation of the image content but also reduces the efficiency of training. To address these challenges, this study has designed an improved image segmentation algorithm that integrates the strengths of VGG16 and UNet. VGG16 is designed to decrease the parameter count by substituting larger convolutional filters with a series of 3×3 convolutional kernels while maintaining the network performance. The architecture of the VGG16 network has been depicted in Figure 3 [43]. This design simplifies the network structure and increases the training speed. Even though VGG16 is a shallower network, it still maintains efficient feature extraction with fewer parameters. To further enhance the model’s generalization ability, this paper employs the VGG16 model weights pre-trained on the ImageNet dataset, which significantly improves the training efficiency of the UNet and ensures the accuracy of the model through the migration learning approach.

2.4.2. Loss function optimization

Different from the traditional object semantic segmentation, the pores in polypropylene foams are complex, with a considerable number of tiny pores, uneven distribution of pore sizes, and small intervals between pores. Given the above characteristics, choosing an appropriate loss function is crucial for deep learning image segmentation tasks. The BCE loss function is commonly used in binary classification problems. It provides a concise, effective, and straightforward method for measuring the difference between the predicted results and the true values in binary classification tasks, thus improving the accuracy of the model and the stability of the training process. However, the BCE loss function alone is ineffective when there are category imbalances and hard-to-categorize samples [44]. Focal loss performs better when dealing with category imbalance and hard-to-categorize samples [45]. Therefore, this study combines BCE loss with focal loss to address the problem of category imbalance and challenging categorized instances. The introduction of focal loss reduces the impact of easy-to-identify samples on the loss and increases the weight of difficult-to-identify samples in the loss, which enhances the model’s ability to identify a small number of categories. Specifically, focal loss adds a conditioning factor ${(1 - p_{i})}^{γ}$ , γ is a moderating parameter that is used to reduce the impact of easily identifiable samples on the loss while enhancing the significance of the loss for samples that are hard to categorize. When p_i is close to 1, it indicates that the sample is easy to categorize, and the adjustment coefficient approaches 0, leading to a smaller loss; conversely, when p_i is small, indicating the sample is difficult to categorize, the adjustment coefficient approaches 1, which results in an increased loss.

BCE loss and Focal loss are calculated as Eqs. (1) and (2), respectively.

(1)

B C E = - \frac{1}{N} \sum_{\dot{i} = 1}^{N} (y_{i} \log (p_{i}) + (1 - y_{i}) log (1 - p_{i}))

(2)

F o c a l = - α {(1 - p_{i})}^{γ} l o g (p_{i})

Where: p_i signifies the likelihood of predicting a particular category; y_i signifies the genuine classification of the ith sample; the α parameter is utilized to equalize the significance of positive and negative samples, thereby addressing their imbalance issue. The γ parameter is employed to modify the influence of difficult and straightforward samples, aiming to resolve the challenge posed by varying sample complexity. We set γ = 2 and α = 0.5. The combined loss function is calculated using Eq. (3).

(3)

\begin{matrix} L o s s = B C E + F o c a l \\ = \sum_{i = 1}^{N} (\begin{matrix} - \frac{1}{N} \sum_{\dot{i} = 1}^{N} (y_{i} \log (p_{i}) + (1 - y_{i}) log (1 - p_{i})) \\ + (- α {(1 - p_{i})}^{γ} l o g (p_{i})) \end{matrix}) \end{matrix}

3. Results and Discussion

3.1. Parameterization

The hardware configuration for the experiments consists of an Intel Core i5-12600KF processor and an NVIDIA GeForce RTX 3060 graphics card. The software environment includes CUDA 11.7, and the development environment is Windows 11. The network model is implemented using PyCharm and PyTorch 2.0. The training is conducted with an Adam optimizer, a batch size of 2, an initial learning rate of 0.0001, and a total of 100 training epochs.

3.2. Evaluation metrics

To evaluate the accuracy of the modified UNet in segmenting pore structures in SEM images, we used a variety of evaluation metrics, including Precision (P), Accuracy (ACC), Recall (R), Intersection over Union (IoU), Mean Intersection over Union (MIoU), F1-score, and Dice coefficient [40]. Their use in image segmentation is extensive, with the definitions provided as follows.

ACC is the most basic evaluation metric, which is measured by calculating the percentage of correctly labeled pixels to the total number of pixels. P refers to the proportion of retrieved instances that are related to the anomalous data, which reflects the model’s ability to identify the anomalous data. R is the ratio of the number of correctly identified anomalous samples to the total number of true anomalous samples, which measures the model’s ability to identify all the anomalous data. The F1-score is the reconciled mean of precision and recall, which combines the two, and the robustness of the classification model is positively correlated with the F1-score. They are calculated by Eqs. (4-7), respectively.

(4)

A C C = \frac{T P + T N}{T P + T N + F P + F N}

(5)

P = \frac{T P}{T P + F P}

(6)

R = \frac{T P}{T P + F N}

(7)

F 1 = \frac{2 (P \times R)}{P + R}

In the classification evaluation metrics, True Positive (TP) signifies the count of pores accurately identified by the model; False Positive (FP) represents the instances where the model mistakenly classifies the background as pores; and False Negative (FN) denotes the cases of pores that exist but are overlooked by the model.

IoU serves as a criterion for assessing the predictive accuracy of image segmentation models. It measures the degree of overlap between the segmented regions predicted by the model and the actual segmented regions. Mean Intersection over Union (MIoU), on the other hand, is an arithmetic average of the IoU values of all categories to provide an overall performance evaluation metric. The calculation formulas are Eqs. (8) and (9).

(8)

I o U (A, B) = \frac{T P}{T P + F P + F N} = \frac{(A \cap B)}{(A \cup B)}

(9)

M I o U = \frac{1}{k + 1} \sum_{i = 0}^{k} \frac{T P}{F N + F P + T P}

Where A represents the segmentation predicted by the model, and B represents the actual segmentation. To quantitatively assess the model segmentation effect, the IoU is used as the model accuracy test index, and the higher its value indicates the better the segmentation effect, where IoU is considered as the most referential index for assessing the model segmentation effect [41].

The Dice coefficient is a measurement for assessing the similarity between two samples, widely used especially in the performance evaluation of image segmentation tasks. The higher the value of the Dice coefficient, the greater the similarity between the two samples. The calculation method for the Dice has been detailed in Eq. (10).

(10)

D i c e (A, B) = \frac{2 T P}{2 T P + F P + F N} = \frac{2 (A \cap B)}{(A) + (B)}

3.3. Comparing the UNet-VGG16 and UNet models

In this study, we introduce an improved calculation method, called UNet-VGG16, based on UNet by introducing the pre-trained VGG16 network as the backbone network for feature extraction. To investigate the effect of this improvement on model training and accuracy, we trained the original UNet and the improved UNet pore recognition models separately under the same conditions. Through comprehensive metrics evaluation and comparison, we find that the improved UNet improves 2.34% and 1.84% in the two key metrics, namely, F1-score and MIoU, compared with the original UNet, respectively. This not only verifies the effectiveness of the improved algorithm but also further confirms its positive effect in enhancing the pore identification and segmentation accuracy. In addition, Figure 4 clearly shows how the loss function changes during training for both the original UNet model and the UNet-VGG16 model. The improved UNet demonstrates a more rapid descent in its loss function and achieves a lower loss value within an equivalent number of iterations, indicating superior convergence performance of the UNet-VGG16 model.

The loss curve graph of the UNet-VGG16 and UNet on the training sets.

To further evaluate the performance of the two techniques for analyzing the pore structure of PP foams, four sets of images (A-1, A-2, A-3 and A-4 in Figure 5) were randomly selected for detailed analysis. The image segmentation and recognition are performed by these two models and compared with the manually labeled images. Figure 5 shows that the original UNet model has more errors in pore recognition, especially in the region marked by yellow circles. In contrast, the consistency between the improved model and the manual labeling is significantly improved, which indicates that the model obtained by the improved algorithm training can more accurately analyze the pore structure of PP foams, providing more reliable data support for research and application in related fields. In summary, the improved UNet algorithm not only performs better in terms of convergence, but also is more accurate in terms of segmentation and recognition accuracy of pores. Using this enhanced algorithm, the trained model can identify and evaluate the pore structure in polypropylene foams more efficiently, providing an effective analytical tool for the characterization of polyolefin foam materials.

Pore identification using UNet-VGG16 and UNet As visually documented by the yellow dashed circles, the original UNet model exhibits significantly higher error rates in pore identification (consistent with source annotations).

3.4. Performance comparison with other deep learning models

In this study, we perform a comprehensive comparison of UNet-VGG16 with several deep learning models specialized in micro-image segmentation, including UNet, GRFB-UNet (an improved model that embeds the group feeling field block (GRFB) structure into the underlying UNet network [46]), UNext, ParaTransCNN, scSE-UNet and Swin-UNet. To ensure the fairness and accuracy of the comparison, all models involved in the comparison were tested under the same experimental conditions, including the use of the same hardware equipment, image resolution, parameter configuration, and training process. The experimental outcomes are presented in Table 1, and UNet-VGG16 outperforms other methods in several key performance indicators. Specifically, the UNet-VGG16 model achieved excellent results in MIoU, P and, F1-score, reaching 0.8554, 0.9525 and, 0.9563, respectively. These data fully demonstrate the efficient performance of UNet-VGG16 in microscopic image segmentation tasks. Compared with existing SEM image segmentation methods based on the DeepLabv3+ model reported in the literature, the method proposed in this paper achieves a significant improvement of 8.0 percentage points in the key performance metric ACC [47]. The data in Table 1 further confirm the excellent performance of UNet-VGG16 in the task of segmenting the pores of polypropylene foams, with the best segmentation results among all the compared models. These results not only highlight the potential application of UNet-VGG16 in the field of microscopic image segmentation, but also show that the improved UNet model by combining the pre-trained VGG16 network can effectively enhance the segmentation accuracy and model performance.

Table 1. The metrics’ results of different deep learning models.

Methods	ACC	Dice	Recall	MIoU	Precision	F1
UNet	0.9144	0.9083	0.9484	0.832	0.9276	0.9379
GRFB-UNet	0.9225	0.9171	0.9644	0.845	0.9416	0.9529
UNext	0.9207	0.9057	0.9147	0.8284	0.9693	0.9412
ParaTransCNN	0.896	0.8932	0.9056	0.81	0.9191	0.9321
scSE-UNet	0.874	0.866	0.89	0.765	0.9208	0.9099
Swin-UNet	0.855	0.8453	0.8705	0.734	0.8409	0.8544
UNet-VGG16	0.9425	0.9217	0.96	0.8554	0.9525	0.9563

3.5. Robustness evaluation

Robustness is an important criterion for evaluating the ability of a model to maintain stable performance in the presence of abnormal data. In practice, 2D images of PP foams obtained by SEM often have large variability in brightness and contrast. To test the model’s ability to adapt to these variations in image properties, we conducted pore recognition experiments on images with different brightness and contrast conditions. The experimental data has been displayed in Figure 6, revealing that fluctuations in brightness and contrast have little effect on the model’s recognition accuracy, with pixel accuracies consistently above 0.93. This experimental result highlights the significant robustness of the model trained using UNet-VGG16. This robustness has the advantage of reducing the model’s dependence on the quality of the input image, which gives the model the potential for a wider range of applications when analyzing the pore structure of polyolefin foam materials.

Effect of brightness and contrast on segmentation effect.

3.6. Comparative analysis of deep learning and manual measurement

The pore structure of the PP foam obtained from the segmentation was measured and analyzed, and the structural parameters such as equivalent pore diameter, average equivalent pore diameter, and shape factor were calculated by Eqs. (11-13) [12,14,48].

(11)

D_{i} = \sqrt{\frac{4 S}{π}}

(12)

{\bar{D}}_{n} = \frac{1}{n} \sum_{1}^{n} D_{i}

(13)

I = \frac{P^{2}}{4 π S}

where: $D_{i}$ is the equivalent diameter of individual pores, µm; ${\bar{D}}_{n}$ is the number-averaged pore diameter, µm; I represents the shape factor of the pore. In general, a higher shape factor of a pore indicates a more irregular shape of the pore, with a shape factor of 1 for a sphere. S and P represent the area and boundary length of individual pores, µm² and µm, respectively.

To test the effectiveness of the UNet-VGG16 model in resolving the pore structure of PP foams, we randomly selected a slice image and studied it in detail. Through manual measurements and automatic measurements of the UNet-VGG16 model, the indicators of the pore structure of PP foams were calculated, including the equivalent diameter distribution, the shape factor distribution of the pores, and the cumulative percentage curves. As shown in Figure 7, by comparing the original image (a) with the image after segmentation processing (b), it is evident that deep learning technology excels in identifying the pores in PP foam material. Histograms (c) and (e) in Figure 7 clearly show the relative frequency distribution of the equivalent pore size and shape factors, while (d) and (f) visualize the trend of the cumulative distribution of the equivalent pore size and shape factors. Through these charts, it can be observed that although there may be minor differences between automatic and manual measurements for the values of pore diameter and shape factor, the improved UNet algorithm shows a high degree of consistency with manual measurement results. This not only confirms the model’s accuracy but also demonstrates its potential and practical value in automatically analyzing the pore structure parameters of materials.

(a) SEM micrograph; (b) Segmented result; (c, d) Comparison of pore diameter distribution and (e, f) shape factor distribution between manual measurement and deep learning segmentation method.

To further validate the stability and accuracy of the deep learning segmentation method, we randomly selected five images and measured them manually and by deep learning to compare the average aperture size and the number of pores identified. As shown in Figure 8, the comparison between the two methods revealed their similarity. The relative errors in the number of pores identified were all less than 10% and the relative errors in the average pore size were all less than 5% compared to manual measurements. Specifically, Figure 8(a) indicates that the number of pores identified by the UNet-VGG16 model is slightly more than that of the manual measurement, which may be due to the neglect of some minor pores during the manual measurement process, thus causing a slight deviation. Furthermore, Figure 8(b) shows that the pore diameter predicted by the UNet-VGG16 algorithm is slightly smaller than the actual measured value, but this deviation is not significant and has a limited impact on the overall analysis. These results further validate the effectiveness of the improved version of the algorithm in analyzing the pore structure of PP foam materials. In terms of efficiency, the automated image analysis technique using the UNet-VGG16 algorithm can accurately measure nearly 100 pores per image in less than a minute, compared to manual measurement methods that typically take about an hour. Thus, in terms of speed, the technique is approximately 60 times faster than traditional methods.

Comparison of (a) mean pore size and (b) number of pores between manual measurement and deep learning segmentation method.

3.7. Fractal analysis of the pore structure of PP foam

To accurately describe the shape of the pores and their distribution characteristics quantitatively, we used the fractal dimension to deeply analyze the shape of the pores and their distribution characteristics. The area and perimeter data of the sample’s pores were obtained from SEM images, and the slit island method was subsequently applied to calculate their fractal dimension. The slit island technique, also known as area-perimeter analysis, is a commonly used method for evaluating the fractal characteristics of particles and pores. Based on the results of the literature [38,48], the perimeter of a planar pore presents a clear functional link with its area, which can be mathematically expressed by Eqs. (14,15). According to Eq. (15), the specific analysis steps are as follows: firstly, the SEM image is segmented to extract the perimeter (P) and area (S) of the pore, and then the slope is obtained by linear fitting through the Log P-Log S relation, and finally twice of the slope is used as the fractal dimension. When correlation coefficients (R² > 0.9), the system is considered to conform to the fractal characterization.

(14)

P^{1 ∕ D_{f}} \propto S^{1 ∕ 2}

(15)

\log P = C + 0.5 D_{f} \log S

Here, S and P correspond to the area and perimeter of the pore, respectively, while C is a constant. D_fdenotes the fractal dimension, whose value range is usually between 0 and 2. In general, higher fractal dimensions imply a higher complexity of the pore structure. Figures 9(a) and (b) demonstrate the results of calculating the fractal dimension of pores for different samples. Log S and Log P show a strong correlation with all correlation coefficients greater than 0.9, indicating that all the samples are characterized by significant fractal features.

(a) and (b) present the fractal dimension of pores for different samples.

3.8. Relationship between pore structure parameters and fractal dimension of PP foams

In this paper, we carried out the computational analysis of the parameters of the pore structure for six samples obtained through the injection chemical foaming process. The calculated parameters mainly include the pore distribution index, average shape factor, pore density, and average pore diameter. The definitions of the pore distribution index and pore density and their calculation methods are described below.

The definition of the cell distribution index (CDI) is similar to the concept of dispersion. In this paper, using an analogous method to that of number-mean molecular weight and weight-mean molecular weight, we calculated the “number-mean pore diameter ( ${\bar{D}}_{n}$ )” and “weight-averaged pore diameter ( ${\bar{D}}_{d}$ )” [49]. CDI, as a key parameter to characterize the pore morphology, is given in Eqs. (16,17), which reveals the distribution of pore sizes from the minimum to the maximum as well as the intermediate values, thus reflecting the dispersion of the pore sizes. As the pore distribution expands, the CDI value will rise accordingly. If the CDI value reaches 1, it indicates uniformity in pore size, meaning that all pores are of the same size.

(16)

{\bar{D}}_{d} = \frac{\sum_{1}^{n} D_{i}^{2}}{\sum_{1}^{n} D_{i}}

(17)

C D I = \frac{{\bar{D}}_{d}}{{\bar{D}}_{n}}

The density of the specimen was determined using Archimedes’ principle, and the porosity and pore density were calculated by Eqs. (18,19) [50].

(18)

v_{f} = 1 - \frac{ρ_{f}}{ρ}

(19)

N_{c} = {(\frac{n M^{2}}{A})}^{\frac{3}{2}} \times (\frac{1}{1 - v_{f}})

Where v_f is the pore volume fraction, %; ρ_fand ρ are the densities of the foamed sample and polypropylene, respectively, g/cm³. N_c is the density of the pores, cells/cm³; n is the number of pores in the SEM image, pcs; M is the magnification; and A is the selected statistical area in the electron microscope image of the foamed sample, cm².

Finally, we explored the interrelationships between fractal dimensions and other structural parameters through gray correlation analysis, the results of which are presented in Table 2. The analysis results show that the fractal dimension has an extremely high correlation with the average shape coefficient of the pores and the pore distribution index, which are 0.928 and 0.966, respectively. This indicates that the higher the dispersion of the pore structure and the more irregular the shape, the more the structural complexity of the pore structure increases, and the corresponding fractal dimension increases as well. However, the pore density and average pore size have relatively small effects on the fractal dimension.

Table 2. Grey relation index between pore structure parameters and fractal dimension.

Pore structure parameters	Sample						Grey relation index
Pore structure parameters	1	2	3	4	5	6	Fractal dimension
N_c (cells/cm³)	9.42×10⁵	3.15×10⁶	1.83×10⁶	3.04×10⁵	1.5×10⁶	3.66×10⁶	0.552
Mean shape factor	1.253	1.317	1.517	1.57	1.279	1.277	0.928
D̅ (µm)	78.76	47.22	63.6	108.42	56.86	42.15	0.688
CDI	1.237	1.197	1.313	1.345	1.169	1.157	0.966
Fractal dimension	1.062	1.069	1.115	1.123	1.074	1.061

3.9. Relationship between pore structure parameters and tensile strength of PP foams

According to the ASTM D638-10 standard, we performed tensile tests at a rate of 50 mm/min, and the experimental results obtained were taken from the average of five independent measurements [51]. The data in Table 3 show a corresponding decreasing trend in tensile strength with increasing porosity. Nevertheless, the correlation between tensile strength and pore density and average pore size is not significant. For example, by comparing Sample 3 and Sample 4, we found that the tensile strength of Sample 3 did not show an increasing trend despite the fact that the average pore size of Sample 3 was much smaller than that of Sample 4 and the pore density was an order of magnitude higher. Thus, the tensile strength of PP foams is primarily affected by porosity and also secondarily affected by the distribution and shape of the pores.

Table 3. Pore structure parameters and mechanical properties of samples.

Pore structure parameters	Sample
Pore structure parameters	1	2	3	4	5	6	7
v_f(%)	36.9	36.3	29.6	24.6	20	17.8	40.8
N_c (cells/cm³ )	9.42×10⁵	3.15×10⁶	1.83×10⁶	3.04×10⁵	1.5×10⁶	3.66×10⁶	1.25×10⁵
Mean shape factor	1.253	1.317	1.517	1.57	1.279	1.277	1.524
$\bar{D}$ (µm)	78.76	47.22	63.6	108.42	56.86	42.15	112.38
Fractal dimension	1.062	1.069	1.115	1.123	1.074	1.061	1.117
CDI	1.237	1.197	1.313	1.345	1.169	1.157	1.389
Tensile strength (MPa)	16	16.5	17.34	19	22.55	23.17	15.78

The Gibson-Ashby model provides a mathematical equation for predicting the elastic modulus and strength of porous materials [52]. In this model, σ represents the strength of the porous material and ρ its density; E_s and ρ_s represent the modulus of elasticity and density of the matrix material, which are 37.5 MPa and 0.9 g/cm³, respectively, whereas C and n are computational parameters, which depend on the microstructure of the foamed material. The parameter n is more dependent on the geometry and distribution of pores, but determining this parameter is usually more difficult, as it needs to be obtained from complex microstructural images [53]. According to the literature, n values generally range from 1 to 4, while for materials with closed pore structures, n values are usually in the range of 1 to 2 [54]. Based on the above analysis of the correlation between the fractal dimension (D_f) and other pore structure parameters, it can be seen that D_freflects the distribution, shape, and topology of pores to some extent. On 2D SEM images, D_fvalues are usually between 1 and 2 [38]. Therefore, we tried to extend the Gibson-Ashby strength model (Eq. 20) to relate the parameter n to D_f. By fitting and analyzing the experimental data, as shown in Figure 10, we derived an extended empirical formula for the relationship between the tensile strength of PP foams and the pore fractal dimension (Eq. 21). In this equation, the parameter n can be directly replaced by the fractal dimension D_f, which enables a quantitative assessment of the heterogeneity of the pore structure. D_f indirectly reflects the effect of the pore geometry on the load transfer paths (i.e., the physical significance of the original parameter n), but it should be noted that this substitution is only applicable to the porous system with fractal characteristics, and the traditional Gibson-Ashby parameter n is still recommended for non-fractal structures.

(20)

\frac{σ}{σ_{s}} = C {(\frac{ρ}{ρ_{s}})}^{n} = C {(1 - ν_{f})}^{n}

(21)

\frac{σ}{σ_{s}} = C {(\frac{ρ}{ρ_{s}})}^{D_{f}} = 0.73 {(1 - v_{f})}^{D_{f}}

Fitted curves of tensile strength ratio versus porosity and fractal dimension.

Where: C was obtained by the least squares fitting method based on the porosity, fractal dimension, and tensile strength data of six samples. When the value of C is set to 0.73, the correlation coefficient R²=0.9912, the model has a good prediction of the tensile strength of PP foams. According to Eq. (21), the theoretical tensile strength of sample 7 was calculated to be 15.25 MPa, with an error of 3% compared to the experimental value of 15.78 MPa, which verifies the reliability of the method. It should be noted that the current model does not cover high-porosity systems (>0.4), and mechanical extrapolation for ultra-threshold pores may have limitations.

4. Conclusions

In this paper, SEM images of PP foam were segmented based on the improved UNet network model, and the pre-trained VGG16 network is migrated to the encoder part of the UNet model to realize feature migration and parameter sharing, and reduce the training cost. In the training stage, a composite loss function of BCE+Focal was introduced into the network to solve the difficult classification problem of image samples. Automatic extraction and computation of the features of the pore structure of polypropylene micro-foam is achieved. The improved UNet segmentation model is compared with UNet, GRFB-UNet, UNext, ParaTransCNN, scSE-UNet models and Swin-UNet. The results show that the improved UNet method exhibits significant advantages in polypropylene foam characterization. The method achieved excellent performance metrics: the MIoU reached 0.8554, the precision (P) was 0.9525, and the F1 score was as high as 0.9563. Despite the segmentation challenges posed by the pore cross-bracing, the method was able to achieve accurate segmentation and measurement of the pore structure, and the resulting pore characterization parameters were in high agreement with those of manual measurements. Notably, the method significantly reduces the analysis time cost and improves work efficiency. In addition, the fractal dimension of the samples was calculated using the area-perimeter method, and it was found that all the samples showed significant fractal characteristics. The relationship between the fractal dimension and the structural parameters of the pores was further analyzed by means of gray-scale correlation analysis, and it was found that the fractal dimension was positively correlated with the average shape factor and distribution index of the pores. The higher fractal dimension reflects the complexity, wide distribution, and irregular shape of the pore structure. Finally, the effects of the pore structure parameters on the mechanical properties of PP foams were deeply analyzed by combining the fractal dimension number and the Gibson-Ashby strength model. The modified model can accurately predict the tensile strength of PP foams. The results show that the tensile strength of PP foams is significantly dependent on porosity; the larger the porosity, the lower the tensile strength of the material. The fractal dimension also has a certain effect on the tensile strength; the larger the fractal dimension, the more complex the pore structure, and the tensile strength will be reduced. However, there is still some ambiguity in the segmentation results for highly overlapping pores or boundary regions with complex topologies, which may have a slight impact on the accuracy of porosity calculations. Although this study has achieved breakthrough results, there are still technical limitations: the current algorithm produces fuzzy regions when dealing with highly overlapping pore structures, which affects the characterization accuracy. To address this problem, the follow-up study will use multimodal data fusion technology to systematically construct a composition-structure-property cross-scale correlation model by integrating scanning electron microscope images, 3D CT data, compositional analysis results, and mechanical property test data. This research method will deepen the understanding of the structure-property relationship of materials and provide new theoretical guidance and technical support for material property optimization.

Acknowledgment

Guizhou Provincial Graduate Student Research Fund Project [2024YJSKYJJ187]; National Natural Science Foundation of China Post-Subsidy Program, Qianshi Xinmiao [2021]27; Centralized Guided Local Funds Project, Qiankehe Zhongcuidi [2024] No. 036; Scientist Workstation Project, Qiankehe Platform KXJZ[2024]028; National Natural Science Foundation of China, Grant/Award Number: 52063008; Guizhou Provincial Achievement Transformation Project, Grant/Award Number: Qiankehe Achievement [2023] General No. 052; the Natural Science Foundation Funthe Natural Science Foundation Funding Projects of Guizhou Province, Grant/Award Number: No. ZK[2021]050; Natural Science Research Project of Guizhou Provincial Department of Education, Grant/Award Number: No. QJJ[2023]011.

CRediT authorship contribution statement

Qi Chen: Conceptualization, Methodology, Formal analysis, Investigation, Writing - original draft. Tianyu Xie: Formal analysis. Zhao Yang: Methodology. Zhipeng Yang: Formal analysis, Supervision. Anping Wang: Supervision, Writing - review & editing. Wei Gong: Project administration, Supervision, Writing - review & editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Declaration of Generative AI and AI-assisted technologies in the writing process

The authors confirm that there was no use of artificial intelligence (AI)-assisted technology for assisting in the writing or editing of the manuscript and no images were manipulated using AI.

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1. Introduction

2. Materials and Methods

2.1. Image acquisition
2.2. Image annotation and dataset construction
2.3. Data enhancement
2.4. UNet network and improved algorithm

3. Results and Discussion

3.1. Parameterization
3.2. Evaluation metrics
3.3. Comparing the UNet-VGG16 and UNet models
3.4. Performance comparison with other deep learning models
3.5. Robustness evaluation
3.6. Comparative analysis of deep learning and manual measurement
3.7. Fractal analysis of the pore structure of PP foam
3.8. Relationship between pore structure parameters and fractal dimension of PP foams
3.9. Relationship between pore structure parameters and tensile strength of PP foams

4. Conclusions

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[1] Janjhi F.A., Chandio I., Janwery D., Vatanpour V., Castro-Muñoz R.. A review on hydrophobic electrospun nanofibers-based materials and membranes for water treatment: Challenges, outlook, and stability. Separation and Purification Technology. 2025;353:128370. https://doi.org/10.1016/j.seppur.2024.128370
[Google Scholar]

[2] Chen Y., Yu J., Ling Y., Yao S., Jiang X., Hu D., Wang H., Zhao L.. Comprehensive analysis of mechanical properties of microcellular polypropylene: Experiment and simulation. Polymer Testing. 2022;116:107812. https://doi.org/10.1016/j.polymertesting.2022.107812
[Google Scholar]

[3] Yang J., Liao X., Li J., He G., Zhang Y., Tang W., Wang G., Li G.. Light-weight and flexible silicone rubber/MWCNTs/Fe₃O₄ nanocomposite foams for efficient electromagnetic interference shielding and microwave absorption. Composites Science and Technology. 2019;181:107670. https://doi.org/10.1016/j.compscitech.2019.05.027
[Google Scholar]

[4] Shi M., Feng C.P., Tu Y.L., Shi G.S., He P.Y., Zhang Y., Zhang J., Li J., Guo S.. Visualization of deep convolutional neural networks to investigate porous nanocomposites for electromagnetic interference shielding. ACS Applied Materials & Interfaces. 2023;15:22602-22615. https://doi.org/10.1021/acsami.3c04557
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[5] Liu Y., Guan Y., Lin J., Li Y., Zhai J., Li X.. Mold-opening foam injection molded strong PP/CF foams with high EMI shielding performance. Journal of Materials Research and Technology. 2022;17:700-712. https://doi.org/10.1016/j.jmrt.2022.01.049
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[6] Castro-Muñoz R.. A critical review on electrospun membranes containing 2D materials for seawater desalination. Desalination. 2023;555:116528. https://doi.org/10.1016/j.desal.2023.116528
[Google Scholar]

[7] Siddiqui S.A., Yang X., Deshmukh R.K., Gaikwad K.K., Bahmid N.A., Castro-Muñoz R.. Recent advances in reinforced bioplastics for food packaging – A critical review. International Journal of Biological Macromolecules. 2024;263:130399. https://doi.org/10.1016/j.ijbiomac.2024.130399
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Integrating deep learning, fractal analysis to construct the relationship between microstructure and tensile strength of polypropylene foams

Abstract

Keywords

Deep learning

Fractal dimension

Microporous polypropylene

Tensile strength

UNet

1. Introduction

2. Materials and Methods

2.1. Image acquisition

2.2. Image annotation and dataset construction

2.3. Data enhancement

2.4. UNet network and improved algorithm

2.4.1. Feature extraction module

2.4.2. Loss function optimization

3. Results and Discussion

3.1. Parameterization

3.2. Evaluation metrics

3.3. Comparing the UNet-VGG16 and UNet models

3.4. Performance comparison with other deep learning models

3.5. Robustness evaluation

3.6. Comparative analysis of deep learning and manual measurement

3.7. Fractal analysis of the pore structure of PP foam

3.8. Relationship between pore structure parameters and fractal dimension of PP foams

3.9. Relationship between pore structure parameters and tensile strength of PP foams

4. Conclusions

Acknowledgment

CRediT authorship contribution statement

Declaration of competing interest

Declaration of Generative AI and AI-assisted technologies in the writing process

References

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