Automated detection of deformation mechanisms in re-entrant honeycomb auxetics using machine learning

Abstract

The intricate geometrical configuration of an auxetic structure enables high energy dissipation capacity at the expense of a highly nonlinear mechanical response. Under external stimuli, complicated deformation mechanisms emerge which dictate the extent of energy dissipation. Recently, the new ‘plastic hinge tracing’ method (Dhari et al., 2021) was introduced to detect such deformation mechanisms for elastoplastic porous materials. This approach is however subjective and cumbersome since it requires monitoring several plastic regions in consecutive deformed configurations. The present study innovatively extends this method by implementing machine learning (ML) techniques for objective detection of deformation modes in auxetics. To this end, a logistic regression ML model was developed to classify the deformation modes of a re-entrant honeycomb structure. The proposed procedure could successfully detect four out of the six deformation modes (‘X’, distorted ‘X’, ‘V’, and ‘V + Z + V’) using the training datasets generated by the finite element analysis and image labels created by K-means clustering algorithm. The success of the proposed automated approach lays the foundation for identifying the deformation mechanisms of other auxetics and porous materials with plastic deformations.

Keywords

machine learning (ML)logistic regression deformation modes auxetics plastic hinges

Introduction

Auxetics

The peculiar mechanical behaviour of auxetics is attributed to their negative Poisson’s ratio (Karimi et al., 2023), which is often induced by internal geometrical instabilities and sometimes material contrast, for example, in composites (Asthana et al., 2024; Hall and Javanbakht, 2021; Javanbakht et al., 2018, 2020b) and layered structures (Javanbakht et al., 2019). One consequence of this behaviour in auxetics is their high energy absorption (Hassani et al., 2024) which has found interesting applications in biomedical implants (Ebrahimzadeh Dehaghani et al., 2024) and aerospace components (Alderson and Alderson, 2007; Evans and Alderson, 2000) among others. Based on the internal movement of auxetics, three basic groups are recognised: re-entrant, chiral, and rotating rigid structures (Kolken and Zadpoor, 2017); among the first category is the re-entrant honeycomb (RH) structure which is created by inverting the side walls of an ordinary honeycomb geometry, see Figure 1.

Figure 1.

Various geometrical patterns for auxetics: (a) four main cell types: arrowhead, re-entrant, chiral and rotational; and (b) geometry of a symmetrical RH cell is characterised by cell width (r), arm length (l) wall thickness (t), and wall angle (θ). By adjusting these paremeters, the deformation mechanisms and consequential energy absorption characteristics of the RH structure are controlled.

Deformation mechanisms

The deformation mechanism of RHs, like many other structures, depends on geometrical variation (Hassani and Javanbakht, 2020, 2021; Hu et al., 2018; Qi et al., 2020), material variation (Javanbakht et al., 2020a), nature of the load (Wan et al., 2004), boundary conditions (Javanbakht et al., 2017), and the loading direction (Dhari et al., 2020) among others. In the literature, the deformation mechanisms are categorised based on their resemblance to the letters of the English alphabet (e.g., ‘X’, ‘y’, ‘Z’, ‘V’, Bi-‘v’ shapes) or simple geometries (e.g., diamond ⋄) (Dong et al., 2019; Zhang et al., 2015). The approach is rather equivocal and subjective to interpretation; for instance, the ‘X’ and Bi-‘v’ modes are nearly identical except for the middle separating space in the latter, see Figure 2. In addition, there exist deformation modes that do not necessarily feature a certain letter or shape, rendering their systematic detection cumbersome.

Figure 2.

Examples of different deformation modes observed in a RH structure under compressive loading. The deformation modes depend on the loading angle. The V, V + Z + V and X modes are observed during perpendicular loading whereas y, distorted X (X′), and horizontal V $(>)$ modes are observed under inclined loads. Adapted from (Dhari et al., 2020).

Plastic hinge tracing method

Recently, new deformation modes were reported for RHs under inclined loads (Dhari et al., 2020), which further complicates the visual detection of deformation mechanisms. Towards a systematic approach, the plastic hinge tracing (PHT) method (Dhari et al., 2021) proposes to monitor the evolution of plastic hinges for detecting any emerging deformation patterns—especially in structured materials. In this approach, high-strain regions of the structure are marked as plastic hinges while ignoring the elastic parts. In the subsequent loading steps, the evolution of these points form plastic paths, which distinguish the highly deformed regions from others. For instance in Figure 2, red points mark regions with an equivalent plastic strain greater than 10% and dashed lines show the plastic paths, see (Dhari et al., 2021) for more details. Although successful in identifying new modes, this method lacks objectivity and requires several iterations in reviewing the movement of plastic regions. Thus, automating the detection procedure seems to be an attractive extension of the method.

Machine learning for structures

ML offers a collection of techniques for classifying data and making predictions by modelling their relations in a variety of fields including computer vision, speech recognition, and natural language processing among others (Hall et al., 2023; LeCun et al., 2015). ML algorithms rely on data instead of explicit relations, and thus they become instrumental where conventional algorithms are challenging to develop. They are categorised into two main groups: supervised learning employs the ground truth knowledge as labels whereas unsupervised learning relies on clustering algorithms to identify patterns in data (Russell et al., 2010). In terms of application, ML has also proven its versatile role in structural analysis in predicting deformations (Wang et al., 2021), peak loads (Dennis and Rigby, 2023; Zahedi and Golchin, 2022), damage and failure (Ibrahim et al., 2011; Ngo et al., 2018), optimisation (Chang et al., 2022), risk mitigation (Pannell et al., 2022), and material characterisation (Liu and Aldrich, 2022). More specifically in the context of texture analysis, image processing has been used to classify fracture modes (Endo et al., 2022) and to distinguish between ductile and brittle fractures in fractography (Naik and Kiran, 2019). The present work uses image processing and ML to identify the deformation modes of auxetic materials.

Aim

As a novel application, it would be interesting to challenge ML in detecting the deformation mechanisms of structures at the mesoscopic level. This study aims to enhance the objectivity of the PHT method by introducing a novel ML-based methodology which can objectively analyse the deformation modes of RHs under various loadcases. The study is organised as follows: first, a detailed methodology is provided which combines the finite element (FE) method (Javanbakht and Öchsner, 2017, 2018), PHT, image processing techniques, and k-means clustering to train a the model. Secondly, the performance of the enhanced method is discussed in detecting the complex deformation modes of RHs under various loading angles. Finally, the study concludes the findings and provide some directions for future studies. In addition, a summary of the notation and algorithms are provided in the appendix.

Methodology

An unsupervised learning model was developed using Python to categorise the deformation mechanisms of the RH structure. To this end, nine steps should be taken, under three main stages: I. finite element analysis and image pre-processing; II. clustering; and III. training and evaluation, see Figure 3.

Figure 3.

Main steps of the proposed methodology and their sub-steps used in the study. Initially, FE analysis is performed and images at every 0.15% strain intervals are extracted and enhanced. The images are then clustered and used in the model for training and identification of deformation modes.

FE analyses

FE prototype

A RH structure (an array of 7 × 8 unit cells) was chosen as the prototype to perform quasi-static FE analyses, see Figure 4(a). The geometry of the structure was discretised into quadrilateral plane stress elements (CPS4) with an out-of-plane thickness of 50 mm. The elastoplastic behaviour was modelled as a rate-independent material with isotropic hardening (stainless steel: Young’s modulus of E = 187.5 GPa, Poisson’s ratio of ν = 0.3, yield strength of σ_y = 1085.5 MPa, and ultimate strength of σ_u = 1593 MPa), see (Dhari et al., 2021) for further details.

Figure 4.

FE model details and validation: (a) FE mesh and cell geometry of the RH structure (all dimensions are in millimetres) and (b) force-displacement validation graph. The RH structure is a 7 × 8 array of the unit cell which was discretised into four-node plane stress (CPS4) elements. The resulting FE mesh is loaded from the top and fixed at the bottom. Adapted from (Dhari et al., 2021).

FE simulations

A pushover static analysis was conducted on the structure to capture the mechanical response under a range of loading angles, θ ∈ {0°, 5°, …, 20°}. A rigid plate was used to apply displacement-controlled loading (1 × 10⁻³ mm vertically) to the top edge of the structure with a fixed boundary condition at the bottom edge. The deformed configurations of the RHs were exported in 0.15% strain intervals as a series of images, each corresponding to a load increment. In total, 293, 366, 347, 386 and 455 frames were obtained for 0°, 5°, 10°, 15° and 20°, respectively. The extracted images were pre-processed, augmented, and enhanced to create the datasets for the ML algorithm.

FE validation

FE prototype was validated against the experimental results provided in the literature (Remennikov et al., 2019), see Figure 4(b). Relative percentage errors below 4% were obtained for various quantities:

• simulated peak force of 1.35 kN versus the 1.3 kN experimental value (3.9% relative error),

• simulated energy absorption of 0.064 mJ versus 0.065 mJ (relative error of 3.17%), and

• densification strain of 45.9% was obtained in the simulations versus 47.4% in experiments (relative error 3.27%).

Overall, simulated values were in good agreement with the benchmarks and the force-deformation graph agreed reasonably with the experimental one.

Preparing the region of interest

Greyscaling and thresholding the frames

The goal of image pre-processing is to extract the region of interest (RoI) from a sequence of images $D_{0}$ and reduce their size to (128,128) in order to reduce the computational cost of the algorithm, see Algorithm 1. For a given square image of isolated plastic hinges, I (x, y) is defined as the greyscale intensity of the image I at point (x, y). Greyscale intensity is computed as the average intensity for each red, green and blue channels. By applying an inverse binary threshold to the greyscaled images, the background was removed and only the plastic hinges remained:

I (x, y) = \{\begin{cases} 0 & if I (x, y) < τ \\ 255 & otherwise \end{cases},

(1)

where τ is the threshold value which can be safely chosen towards the higher end of the greyscale intensity range around (e.g., τ = 200).

RoI detection

The RoI was the centred and scaled bounding box of plastic hinges (i.e., the smallest rectangle containing all the plastic hinges), see Figure 5. By summing up the intensities of the rows (U) and columns (V), the bounding box was detected:

\begin{align} U & = {[\begin{matrix} \sum_{c = 0}^{w - 1} I (0, c) & \sum_{c = 0}^{w - 1} I (1, c) & \dots & \sum_{c = 0}^{w - 1} I (h - 1, c) \end{matrix}]}^{T}, \end{align}

(2a)

\begin{align} V & = [\begin{matrix} \sum_{r = 0}^{h - 1} I (r, 0) & \sum_{r = 0}^{h - 1} I (r, 1) & \dots & \sum_{r = 0}^{h - 1} I (r, w - 1) \end{matrix}], \end{align}

(2b)

Figure 5.

Illustration of the detection process for the RoI (enclosed in the blue box). The RoI refers to the smallest rectangle that contains all the plastic hinges present in the image. The deformed configuration is scanned for RoI, centred, and scaled.

Where □^T is the transpose operator. The bounding box (R) was represented by a set of ordered pairs containing the top-left (x₁, y₁) and bottom-right (x₂, y₂) coordinates:

R = \{\begin{cases} \{(x 1, y 1), (x 2, y 2)\} & if \exists i | U (i) \neq 0 \land V (i) \neq 0 \\ \{(0,0), (w - 1, h - 1)\} & otherwise \end{cases},

(3)

where empty rows and columns were cropped:

\begin{align} (x_{1}, y_{1}) & = (\underset{i}{\arg \min} {U (i) \neq 0}, \underset{i}{\arg \min} {V (i) \neq 0}), \end{align}

(4a)

\begin{align} (x_{2}, y_{2}) & = (\underset{i}{a r g m a x} {U (i) \neq 0}, \underset{i}{a r g m a x} {V (i) \neq 0}) . \end{align}

(4b)

When no plastic hinges were detected (i.e., a blank image), the RoI was set to be the whole image without any cropping.

RoI scaling and centring

To ensure that our deformation detection algorithm remains invariant to the size of plastic hinges, the RoI was scaled to a fixed size (128 × 128 pixels) and centred. The scaling matrix A_s is defined as

A_{s} = [\begin{matrix} s & 0 & 0 \\ 0 & s & 0 \end{matrix}],

(5)

where s = 128/max ({x₂ − x₁, y₂ − y₁}) is the scale factor. For blank images, s = 128/h where h is the number of rows in the image. Next, an affine transformation was used to centre the image:

A_{t} = [\begin{matrix} 1 & 0 & Δ x \\ 0 & 1 & Δ y \end{matrix}],

(6)

where the translation matrix (A_t) contains the number of pixels for in-plane translation, that is, Δx = −x₁ + ⌊x₁ + x₂/2⌋, Δy = −y₁ + ⌊y₁ + y₂/2⌋. The resulting standardised image sequences were stored in a dictionary

D_{2}

Augmentation

Image transformation

To improve the clustering efficiency, the dataset was augmented using a series of random transformations (e.g., rotation, scaling, and mirroring) which were applied to randomly-selected images, see Algorithm 2. The following linear transformations were used:

1. A random rotation was applied using the rotation matrix (A_r):

A_{r} = [\begin{matrix} - \sin θ & \cos θ & \sin θ \cdot c_{x} + (1 - \cos θ) \cdot c_{y} \\ \cos θ & \sin θ & (1 - \cos θ) \cdot c_{x} - \sin θ \cdot c_{y} \end{matrix}],

(7)

where θ ∈ [−π/18, π/18] is the rotation angle and (c_x, c_y) = (⌊w/2⌋, ⌊h/2⌋) denote the rotation axis.

2. A random scaling was achieved by two operations: firstly, scaling was applied using the scaling matrix (A_s) with a random scaling factor (s ∈ [0.85, 1.15]); then, translation was applied using the translation matrix (A_t) to recentre. The first operation scales the image from the top-left corner, and the second operation re-centres the RoI to the image centre:

A_{s} = [\begin{matrix} s & 0 & 0 \\ 0 & s & 0 \end{matrix}], A_{t} = [\begin{matrix} 1 & 0 & Δ x \\ 0 & 1 & Δ y \end{matrix}],

(8)

where Δx = c_x (1 − s)/2 and Δy = c_y (1 − s)/2.

3. Horizontal (A_⇄) or vertical (A_↑↓) mirroring was applied using their respective transformation matrices:

A_{⇄} = [\begin{matrix} 1 & 0 & 0 \\ 0 & - 1 & 2 c_{y} \end{matrix}], A_{↑ ↓} = [\begin{matrix} - 1 & 0 & 2 c_{x} \\ 0 & 1 & 0 \end{matrix}]

(9)

These linear transformations were applied to the RoIs in a way that while limiting the distortion of the plastic hinges, an adequate number of variations were created. This was carried out by limiting the rotation and scaling to ±5% and ±15% of the values in the raw dataset, respectively. In addition, to enhance the detection of any symmetric deformation modes, mirroring was applied. As a result, the size of the dataset was increased from 1, 847 to 10, 000 frames which was stored in dictionary

D_{2}

Enhancement

Several morphological operations may be applied at this stage to obtain clear images and reduce noise. The choice of enhancement operations depends on the geometry of the structure, density of its cells, and the density of plastic hinges to be detected. Herein, sufficiently clear images were obtained by applying image dilation and blurring, see Algorithm 3.

Image dilation

Morphological dilation was applied to increase the fullness of plastic hinge regions for better detection during clustering. A square (5 × 5 pixels) structuring element (set dilation, K_d) was applied to each frame (I) using the Minkowski addition:

I \leftarrow I \oplus K_{d} \equiv ⋃_{m \in K_{d}} I_{m},

(10)

where I_m is the translation of I by m, that is, the dilation of I by m is the loci of the points whose centre sweeps inside the image I.

Gaussian blurring

Image blurring was achieved by convolving the image with the Gaussian function:

K_{G} (i) = α \exp (- \frac{{(i - \frac{t - 1}{2})}^{2}}{2 σ^{2}}),

(11)

where t is the kernel size, σ is the standard deviation, and α is the scaling factor such that

Σ_{i = 0}^{t - 1} K_{G} (i) = 1

, i ∈ {0, …, t − 1}. Convolving every image with K_G resulted in a blurred image which was finally resized to 64 × 64.

Clustering and balanced augmentation

Since the detection of deformation mechanism is a subjective process, obtaining ground truth labels is limited; thus, before employing a detection algorithm, clustering should be carried out. To this end, each 64 × 64 image (I) from the set of images $(D_{3})$ was reshaped into a row of greyscale data with 4096 elements. After detecting the optimum number of clusters, Lloyd’s technique (naïve k-means clustering) (Lloyd, 1982) was applied to assign the closest-possible label to each image. Principal component analysis (PCA) was applied to the resulting set of images $(D_{4})$ , reducing the dimensionality of data. The clustering step was finalised by applying a balanced augmentation which resulted in the final set of images $(D_{5})$ .

Optimum number of clusters

The Calinski-Harabasz (Caliński and Harabasz, 1974) and elbow (Satopaa et al., 2011) methods were selected to calculate the optimum number of clusters among the set (k ∈ {2, …, 20}), see Figure 6(a). The Calinski-Harabasz score is the ratio of the inter- to intra-cluster variance; since the variance of the dataset is limited, the score decreases only by increasing the number of clusters. Thus, it failed in identifying the optimal number of clusters. Alternatively, distortion score was plotted versus the number of clusters to determine the optimal number of clusters using the elbow method. In this method, an ‘elbow’ or a ‘cut-off point’ on the curve is determined where increasing number of clusters are no longer worth the additional cost of analysis, that is, the change in the gradient of distortion becomes insignificant to the increase of the number of clusters. Using the elbow method, k = 8 was selected as the optimal number of clusters, see Figure 6(b).

Figure 6.

Estimating the optimal number of clusters: (a) Calinski-Harabasz metric, and (b) distortion metric (the elbow method). The Calinski-Harabasz method failed to identify the optimal number of clusters due to limited variance of the dataset. However, the elbow method returned 8 as the optimal number of clusters to perform cluster analysis.

k-means clustering

Using the optimum number of clusters, the k-means algorithm is initialised at the first iteration (t = 1) with an equal number of random mean values $(μ_{0}^{(1)}, μ_{1}^{(1)}, \dots, μ_{k - 1}^{(1)})$ , each representing the centroid of a cluster. At every iteration t, each observation x_i is first assigned to the cluster $y_{i}^{(t)}$ :

[\begin{matrix} x_{0} & x_{1} & \dots & x_{n - 1} \end{matrix}] \to [\begin{matrix} y_{0}^{(1)} & y_{1}^{(1)} & \dots & y_{n - 1}^{(1)} \\ y_{0}^{(2)} & y_{1}^{(2)} & \dots & y_{n - 1}^{(2)} \\ ⋮ & ⋮ & \dots & ⋮ \\ y_{0}^{(t)} & y_{1}^{(t)} & \dots & y_{n - 1}^{(t)} \end{matrix}],

(12)

where

y_{i}^{(t)}

is the cluster assignment of x_i at iteration t; it renders the L²-norm minimum:

y_{i}^{(t)} = \underset{j}{\arg \min} {‖x_{i} - μ_{j}^{(t)}‖}_{L^{2}}^{2},

(13)

where

μ_{j}^{(t)}

is the mean of cluster j at iteration t. Assignment of an observation updates the mean value of the cluster. The clustering algorithm was repeated until the cluster assignments did not change or the maximum number of iterations was reached. The final cluster assignments were used to assign pseudo-labels (C₀, C₁, …, C₇) to each image. The labels were then used to train the classifier. Algorithm 4 summarises these steps.

Data visualisation using PCA

PCA is used to transform high-dimensional data into lower dimensions such that the variance of the data is maximised (Tipping and Bishop, 1999). The output of PCA is a set of principal components: the first principal component has the largest possible variance whereas the second principal component is orthogonal to the first principal component and has the second largest variance. In addition, the dimensionality reduction was used to illustrate the detected clusters of deformation modes in a 2D plot, see Figure 7.

Figure 7.

Results of the PCA and k-means clustering for the given set of images from FE simulations. The PCA step reduced the dimensionality of the data from 4096 to 2, which allowed us to visualise the data in a 2D scatter plot. In the main figure, data points are clustered into eight groups using the k-means method where colour codes denote deformations modes C₀ to C₇. In the subfigure, the projected data points onto a 2D space by the PCA model and colour-coded according to the loading angle in the simulations (E_0° to E_20°).

Balanced augmentation

Each cluster was balanced to increase efficiency, see Algorithm 5. Augmentation of the datasets was carried out by adding minor variations, that is, some transformations (rotation, scaling, and translation) along with adding noise (step 6). Consequently, each cluster was expanded to 3, 000 images, that is,. $D_{5}$ with a total of 24, 000 training frames.

Pre-training, training, and testing

At the final stage of the methodology, the ML-based model was trained, validated, and tested, see Algorithm 6.

Pre-training and training

In pre-training, the dataset $(D_{5})$ was shuffled and partitioned into training (60%), validation (20%), and test (20%) datasets. A logistic regression model (Cox, 1958) was trained using the training and validation datasets. The ML-based model used a linear function to calculate the log odds of the outcome and was transformed using the logistic function (Figure 8):

1. Given is a set of observations X, each with N = 4096 features, and a set of labels Y as assigned by k-means clustering:

\begin{align} X & = {[\begin{matrix} x_{0}, x_{1}, \dots, x_{n - 1} \end{matrix}]}^{T}, \end{align}

(14a)

\begin{align} Y & = {[\begin{matrix} y_{0}, y_{1}, \dots, y_{k - 1} \end{matrix}]}^{T}, \end{align}

(14b)

Figure 8.

Architecture of the logistic regression model, showing the flow of data from input to output through various stages. The logistic regression model was trained using the training and validation datasets.

Where n is the number of observations and k is the number of classes in the labelled dataset.

2. The model was initialised for training with a set of weights (w) and biases (b):

\begin{align} W & = [\begin{matrix} w_{0}^{(0)} & w_{0}^{(1)} & \dots & w_{0}^{(N - 1)} \\ w_{1}^{(0)} & w_{1}^{(1)} & \dots & w_{1}^{(N - 1)} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ w_{k - 1}^{(0)} & w_{k - 1}^{(1)} & \dots & w_{k - 1}^{(N - 1)} \end{matrix}], \end{align}

(15a)

\begin{align} B & = {[\begin{matrix} b_{0}, b_{1}, \dots, b_{k - 1} \end{matrix}]}^{T} . \end{align}

(15b)

3. The input vector is transformed into log odds (Z) using the following affine map:

z_{k} = \sum_{i = 0}^{N - 1} w_{k}^{(i)} \cdot x^{(i)} + b_{k} \Rightarrow Z = {[\begin{matrix} z_{0}, z_{1}, \dots, z_{k - 1} \end{matrix}]}^{T},

(16)

where z_i is the log-odd of y_i.

4. The logistic function was used to compute the probabilities of the cluster labels:

{\hat{y}}_{k} = \frac{e^{z_{k}}}{\sum_{i = 0}^{k - 1} e^{z_{i}}} \Rightarrow \hat{Y} = {[\begin{matrix} {\hat{y}}_{0}, {\hat{y}}_{1}, \dots, {\hat{y}}_{k - 1} \end{matrix}]}^{T},

(17)

where

{\hat{y}}_{i}

is the probability of y_i. The logistic function returns a value between 0 and one and is used to predict the probability of the cluster labels in the context of a binary classification problem.

5. The loss (L) was calculated as the cross-entropy function between the predicted and actual labels using the sum of the product of the actual label to the log of its predicted label, that is, $- \sum_{i = 0}^{k - 1} y_{i} \cdot \log ({\hat{y}}_{i})$ . It was used to update the weights and biases using stochastic gradient descent.

6. To build a single integrated model, the predicted probabilities were passed through the softmax function, ensuring that the sum of the outputs across all deformation classes was one. The softmax function extends the concept of logistic regression to handle multiple classes:

softmax (\hat{y_{i}}) = \frac{\exp (\hat{y_{i}})}{\sum_{j} \exp (\hat{y_{j}})} .

(18)

The final prediction is given by

\hat{y}

as the class with maximum probability

{\hat{y}}_{k}

such that

{\hat{y}}_{k}

corresponds to C_k.

Results and discussion

Detected deformation mechanisms

Six deformation modes were reported for inclined loading of RH structures in the literature for loading angles of 0°, 5°, 10°, 15° and 20°. Apart from the four previously-detected modes during uniform compression, that is, horizontal ‘V’ $(>)$ , vertical ‘V’, ‘X’, and ‘y’; two new localised modes (distorted X, and V + Z + V) were recently reported for inclined compressive loading (Dhari et al., 2021). The k-means clustering could distinguish and label eight distinct deformation mechanisms (C₀ − C₇) in an automated process, see Figure 7. Namely, instead of assigning a certain deformation mechanism to its correct cluster, the detection process was done without supervision.

Clustering quality

In Table 1, automated clustering results are visually matched to the reported deformation modes from FE analyses. For most loading angles, deformation mode ‘

>

’ is activated followed by ‘X’ and ‘X′’ (distorted X) modes in sequence. FE results indicate that the ‘X’ and ‘X′’ modes are dominant in the RH structures under inclined loads. This is supported by the clustering data showing that the clusters (C₁, C₃, and C₄) collectively amounting to more than 28.85% of the total.The C₅, C₆, and C₇ clusters denote the early stages of plastic hinge formation and do not characterise any deformation modes (Mode 0, M0), see Table 1. This can be attributed to the limited performance of k-means on the data with a large variety of size and density in clusters and clustering outliers (Celebi et al., 2013). However, the model could successfully cluster ‘V’ (C₀), ‘X’ (C₁), and ‘V + Z + V’ (C₂) modes despite their visual similarities (Mode 1, M1). On the other hand, the distorted ‘X’ mode was included in two clusters (C₃ and C₄) simultaneously (Mode 2, M2). The clustering algorithm was unable to distinguish between these two clusters because of visual similarity. Namely, the data points in these clusters appear similar enough to be part of both clusters. Finally, it is worth mentioning that the horizontal V mode could not be detected using the current model.

Table 1.

Observed deformation types and respective detected clusters.

Addressing the unbalanced dataset

The clusters comprised of 3.31%, 3.88%, 5.51%, 10.23%, 14.74%, 19.71%, 20.52%, and 22.1% of the 10,000 samples from the first augmentation and thus, the dataset was unbalanced. This links back to the induced deformations modes under various loads as there was no intention to create the same number of images for each deformation mechanism. Training data using unbalanced data disadvantages the performance of the model; therefore, a second dataset of images was created using balanced augmentation to train the model.

Training and testing

Multiple models were trained with three different learning rates using a fixed batch size of 64 for 100 epochs, see Table 2. The models showed an accuracy of 47.97%, 80.91% and 88.77% on the validation set for the learning rates of 0.00001, 0.0001, and 0.001, respectively. The model with the highest accuracy (the learning rate of 0.001) was selected for further analysis. The selected model jumped to 80% accuracy in the first 10 epochs and then slowly improved to reach 85% accuracy by the 30^th epoch; by 100^th epoch, an accuracy of 88% was slowly reached. The performance can be enhanced by using a larger dataset containing more deformation mechanisms. For the test dataset, the chosen model had an accuracy of 88.08%, demonstrating good efficacy. Obviously, it cannot distinguish between the modes that were not discovered by k-means clustering. Using a higher number of epochs might improve the trained model; nevertheless, there will be a chance of overfitting.

Table 2.

Learning rates and accuracy of predictions using the validation and test datasets; the highlighted model was selected for label prediction.

η	Accuracy on validation dataset (at n^th epoch).										Test (%)
η	10	20	30	40	50	60	70	80	90	100	Test (%)
0.00001	29.66%	32.50%	34.41%	36.56%	38.54%	40.02%	41.85%	44.20%	46.27%	47.97%	46.29
0.00010	55.95%	64.91%	73.33%	76.62%	78.39%	79.22%	79.68%	80.08%	80.43%	80.91%	79.83
0.00100	80.81%	83.35%	85.31%	86.10%	86.62%	87.43%	87.85%	88.16%	88.45%	88.77%	88.08

Label prediction

The predictions of a semi-trained model for sample images are listed in Table 3. In each column, the probability of belonging a single input image to the detected labels C₀–C₇ is calculated where a probability equal to one provides 100% assurance in the prediction. Therefore, non-zero possibility values indicate the diverse weighting of labels and some degree of confidence in the detection procedure. The label with the highest probability is highlighted in grey within each column, which indicates the best match based on model’s predictions. This predicted label is then compared with the actual label determined by the k-means algorithm. The correct predictions (highlighted in green) occur when the actual and predicated labels match. Seven out of 10 samples were predicted correctly by the algorithm, the three failed cases are highlighted in red.

Table 3.

Sample predictions of the semi-trained model for ten cases; highest probabilities are highlighted in grey, correct detections in green, and incorrect matches in red. The model with the learning rate of 0.001 was found to be the most accurate one.

Confusion matrix

The learning progress of the ML model can be traced using the confusion/error matrix in Table 4. The first column shows a 97% efficiency in predicting the C₀ label, which can be occasionally misidentified as C₁. Except that, the model does not confuse C₀ with any other labels. The lowest efficiency (78%) occurs in predicting the C₅ label because of the resemblance of the ‘X’ and the distorted ‘X’ modes. The algorithm can predict most of the labels with an accuracy higher than 90% indicating the effectiveness of the ML model.

Table 4.

Confusion matrix at the end of training. The algorithm can predict most of the labels with high accuracy $(> 90 %)$ , indicating the effectiveness of the ML model.

		Predicted labels
		C ₀	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆	C ₇
Actual labels	C ₀	0.97133407	0.02866593	0.00000000	0.00000000	0.00000000	0.00000000	0.00000000	0.00000000
	C ₁	0.02360044	0.95334797	0.02085620	0.00109769	0.00000000	0.00109769	0.00000000	0.00000000
	C ₂	0.00000000	0.02220957	0.92255125	0.03929385	0.01025057	0.00569476	0.00000000	0.00000000
	C ₃	0.00457404	0.00457404	0.00743282	0.82618639	0.07089766	0.08519154	0.00114351	0.00000000
	C ₄	0.00109890	0.00000000	0.00164835	0.01697793	0.89505495	0.03406593	0.00000000	0.00000000
	C ₅	0.00000000	0.00278396	0.02115813	0.00596252	0.07293987	0.78396437	0.01948775	0.00000000
	C ₆	0.00164384	0.00054795	0.00328767	0.00000000	0.00712329	0.09260274	0.84876712	0.03835616
	C ₇	0.00550055	0.00000000	0.00055006	0.00000000	0.00000000	0.00110011	0.04565457	0.94664466

Concluding remarks

Significance and contribution

The present study develops a novel methodology to address an important deficiency of the PHT method, that is, the lack of objectivity in detecting deformation modes. The k-means clustering and logistic regression ML algorithms were used to classify the deformation mechanisms and develop the KNet1 model in Python. This algorithm extends the application of PHT method towards autonomous deformation detection in structured materials; characterising deformation modes has important applications in structural health monitoring (e.g., real-time monitoring (Miller et al., 2023)) and inverse design methods (e.g., customising failure mode or tailoring energy absorption).

Conclusion

Using the developed algorithm, the deformation images were classified into eight clusters with an accuracy of 88.08% on the test set. The imbalance in cluster populations can be observed due to the dominance of the recurring deformation modes during inclined compressive tests. The algorithm successfully identified four out of six deformation mechanisms (‘X’, distorted ‘X’, ‘V’, and ‘V + Z + V’ modes). The two remaining modes ‘y’ and ‘ $>$ ’ cannot be detected because their samples in the original dataset lacked diversity. This is attributed to the imbalance of dataset and the shortcoming of k-means in clustering them. Nevertheless, in the absence of ground truth and despite the limitations of the dataset, the ML model provides an automated unsupervised labelling for a more objective detection of the deformation mechanisms in RH structures.

Future work

The limitations from k-means can be overcome by refining image processing techniques, optimising k-means parameters, and exploring alternative clustering methods such as DBSCAN or hierarchical clustering. In addition, the sizing and density of clusters can be improved by a more robust algorithm. More importantly, the introduced methodology can be used to detect the deformation modes of any elastoplastic structured material without any size restrictions. A challenge for this algorithm (and likewise the PHT method) would be detecting the deformation modes of non-porous complicated geometries like scutoids (Dhari and Patel, 2022).

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Gaurav Singh

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