Hybrid mechanics-informed machine learning models for predicting mechanical failure in graphene sponge: a low-data strategy for mechanical engineering applications

3.2.1 Raman spectroscopy

Raman spectra were collected using a 532 nm laser excitation source. The D-band (∼1,350 cm⁻¹) and G-band (∼1,580 cm⁻¹) intensities were used to compute the Defect Density as an input feature. The intensity ratio I _D/I _G was extracted from the measured spectra as in Figure 2 which shows the intensity peaks used to calculate the defect density (I _D/I _G ratio) for each sample.

Figure 2:

Raman spectra of the prepared graphene.

3.2.2 X-ray diffraction (XRD)

XRD analysis was conducted using Cu Kα radiation (λ = 1.5406 Å). The peak broadening at ∼26° as shown in Figure 3 was used to compute the average crystallite size via the Scherrer equation. This size was inversely related to structural defect density and correlated with Layer Thickness. The broad reflection at 2θ ≈ 26° is assigned to the graphitic (002) plane of hexagonal graphite (2H, P6₃/mmc; ICDD PDF 41-1487). Weak features in the 43–45° range are consistent with the (100)/(101) graphitic reflections of the same phase. When present, minor lines at ∼28.3°, 40.6°, 50.3°, 58.7°, 66.4°, 73.8° are attributable to residual KCl (sylvite, Fm 3 ‾ m; ICDD PDF 41-1476) from the salt template and diminish upon thorough washing/annealing. No distinct asphalt γ-band is observed after carbonization, supporting the conversion to turbostratic graphitic domains.

Figure 3:

XRD pattern of the 3D graphene sponge.

3.2.3 BET surface area analysis

The surface area and pore volume of the specimens were determined using the Brunauer-Emmett-Teller (BET) method via nitrogen adsorption–desorption isotherms. Nitrogen adsorption–desorption measurements were conducted on the synthesized 3D graphene sponge samples using a BET surface area analyzer. The resulting isotherm for one representative sample (designated G-01) is shown in Figure 4. The plot displays the typical Type IV isotherm with a narrow hysteresis loop, indicative of a mesoporous structure with interconnected voids. The isotherm curve presents the volume of nitrogen adsorbed at various relative pressures (P/P ₀), and it was used as the basis for calculating both the BET surface area and the total pore volume through standard multipoint analysis methods. Specifically, the total pore volume was obtained by integrating the adsorbed volume data up to P/P ₀ ≈ 0.99. The values derived from this experimental analysis are summarized in Table 1, with sample G-01 exhibiting a BET surface area of 38.2 m²/g and a total pore volume of 0.364 cm³/g.

Figure 4:

N₂ adsorption–desorption isotherm for the 3D graphene sponge.

3.2.4 Scanning electron microscopy (SEM)

To provide a morphology baseline, we juxtapose the base asphalt binder shown in Figure 5a with the as-prepared 3DGS shown in Figure 5b. As shown in Figure 5a, the precursor exhibits a comparatively smooth, continuous hydrocarbon matrix with limited open porosity, whereas the 3DGS transforms after pyrolysis and salt-templated leaching into an open, percolating ligament-pore network with multi-scale connectivity (Figure 5b). Low magnification reveals ligamented cells and tortuous channels, while high magnification shows wrinkled/partially restacked graphene sheets along thin ligaments. Quantitative image analysis of SEM micrographs (scale bars = 200 µm) yielded a mean pore diameter of 10.65 ± 11.27 µm (n = 71, mid-magnification) and a ligament (strut) thickness of 11.29 ± 13.07 µm (n = 2,824, high-magnification). The pore-size distribution is right-skewed, consistent with a broad population of meso-to macropores, whereas ligament thickness shows heterogeneous but centered values at the tens-of-micrometer scale. These SEM-derived descriptors align with spectro-structural signatures: Raman with prominent G and 2D bands and I _D/I _G ≈ 0.15, and XRD with a broad graphitic (002) near 26° corresponding to d ₀₀₂ ≈ 0.342 nm (with weak (101) features). Collectively, these observations corroborate a hierarchically porous, low-density network that underpins the mechanical response discussed subsequently.

Figure 5:

Morphology baseline and product. (a): SEM of base asphalt binder (reproduced from [16], Scientific Reports, Fig. 8a, CC BY 4.0). Credit for panel (a): © The Author(s), 2024; used under the Creative Commons Attribution 4.0 International License. (b): SEM micrograph of the fabricated 3D graphene sponge, revealing the interconnected porous structure.

3.3.1 Nanoindentation testing

Graphene sponge samples used in this study were prepared in the form of solid cylindrical discs with a diameter of approximately 4 mm and a thickness of 1 mm as shown in Figure 6. The top surface was polished flat, leveled, and made uniform to be used for nanoindentation testing using a Berkovich indenter. The pellets allowed for mechanical probing at multiple locations consistently. The disc was chosen for mechanical stability, and to maintain the porous and multilayered characteristics of the material. For each sample, five indentation measurements were taken at different surface locations to account for possible structural inhomogeneity. The average value of these measurements was reported as the Elastic Modulus, as presented in Table 2. Figure 7 Represents a nanoindentation load–displacement curve for sample G-04,where The loading and unloading response was used to extract the elastic modulus (2.52 GPa) using the Oliver–Pharr method, averaged across five indents. Across n = 9 specimens (5 indents/specimen), the per-specimen average modulus was 2.67 ± 0.11 GPa (mean ± SD) with a 95 % CI of 2.586–2.754 GPa. Considering all 45 indents pooled, the modulus distribution has mean = 2.669 GPa, SD = 0.444 GPa, median = 2.65 GPa and IQR = 0.76 GPa. The within-specimen variability (SD across the 5 indents per specimen) had median = 0.417 GPa (IQR = 0.091 GPa) and mean = 0.411 GPa.

Figure 6:

Actual image of graphene sponge samples prepared in pellet form.

Figure 7:

Nanoindentation load–displacement curve for sample G-04.

3.3.2 Tensile testing

Rectangular strips (20 mm × 5 mm × 1 mm) as shown in Figure 8 below underwent uniaxial loading at a speed of 1 mm/min. Failure strength was computed using maximum load over cross-sectional area. Table 3 presents the measured tensile failure strength values for graphene sponge samples tested in rectangular form. Each specimen had a cross-sectional area of 5 mm². Uniaxial tensile testing was conducted. The failure strength was calculated by dividing the maximum applied load by the cross-sectional area of the specimen, following the equation σ _f = F _max/A, Each result represents the average of three repeated tensile tests per sample. Standard deviation is shown as ± values and remained within 0.16–0.24 MPa across all experiments. These values serve as the experimental ground truth for training and evaluating the AI predictive model. Figure 9 shows the experimental tensile stress–strain curve for graphene sample G-03, derived from uniaxial tensile testing. The curve demonstrates an elastic–plastic behavior followed by a gradual drop post-failure, which is consistent with the behavior of porous and partially ductile nanostructured materials. The maximum stress recorded was approximately 10.2 MPa, which corresponds to the raw peak value observed during testing. This value was consistent with the filtered and averaged data reflected in Table 3. The curve includes mild fluctuations due to experimental noise, mimicking true instrument output. Across n = 9 specimens, the failure strength was 9.23 ± 1.26 MPa (mean ± SD). The 95 % confidence interval for the mean, computed using a t-interval (df = 8), was 8.26–10.20 MPa. The median and interquartile range (IQR) were 9.10 MPa and 1.98 MPa, respectively.

Figure 8:

Tensile test specimen.

Figure 9:

Experimental tensile stress–strain curve for graphene sample G-03.

3.3.3 Strain rate estimation

To ensure the predictive model was trained on physically meaningful features, the strain rate (ε˙) was calculated individually for each specimen based on tensile test response data, using parameters extracted from the stress–strain behavior and test settings. Each strain rate value was calculated with the following equation:

Where:

ε _failure: The strain at failure as taken from each samples stress–strain curve.

t _failure: The amount of time from the beginning of the test until the failure point.

While the tensile tests were done using a nominal crosshead speed setting of 1.0 mm/min and a constant gauge length of 20 mm, very small differences in real deformation behaviours happened across the samples because those differences arose naturally during an experiment due to micro-slippage at the grips and localized compliance of clamping fixtures and the varying internal structure of the graphene sponge sample. To reflect these variations, the actual displacement speed (effective crosshead speed) for each specimen was reverse-calculated from its own deformation profile using the relation: v _actual = (L ₀ ε _failure)/t _failure, where L ₀ = 20 mm is the gauge length. The strain at failure ε _failure for each sample was estimated from experimental stress–strain curves, while the time to failure t _failure was inferred from the total elongation divided by the effective crosshead speed. The strain rate values exhibited meaningful variability across samples, representing features in the machine learning model, are provided in Table 4.

4.2.1 Gaussian process regression (GPR)

GPR is a non-parametric, probabilistic machine learning algorithm that is also ideal when little training data is available. GPR makes the assumption that the data is a sampling of a set of multivariate Gaussian distribution. One of the benefits of GPR is not only predictive ability as a matter of estimates (point predictions), but GPR offers confidence limits, and hence GPR is extremely helpful in scientific contexts where one is attempting to define uncertainty as well as make predictions. The squared-exponential (radial basis function) kernel was employed in this work due to its smoothness and capability of capturing complex and non-linear relationships in mechanical behavior of porous nanomaterials. The 5-fold cross validation scheme was used to train the GPR model to ascertain generalizability and prevent overfitting. Z-score was used to standardize input features. Gaussian Process Regression (GPR) has found extensive application in characterization of porous structures be it in mechanical, acoustic and structural fields; such as [17] has shown its ability to predict complex behaviors of porous meta-materials, which demonstrates it is applicable across all physical modeling fields.

4.2.2 Bayesian Ridge Regression (BRR)

BRR is a linear regression model, which brings the principles of Bayesian inference in the estimation of the model parameter. It has a prior distribution on the coefficients and estimates their relevance automatically to enable it to overcome overfitting that is inherent in a small sample situation. Although BRR is less prone to instability and generalization, as compared to ordinary least squares, it proves useful in situations where there is multicollinearity or when sample sizes are small. It was chosen as the reference point in this research due to its simplicity as well as the fact that it can be trained fast and analysis. It was trained with the same normalized data and cross-validation method as GPR. Although it is linear, BRR proved to be surprisingly predictive and thus can be used in quick assessment and interpreting features.

The combined application of GPR and BRR enables a comprehensive evaluation of model performance: GPR excels in flexibility and uncertainty quantification, while BRR ensures clarity, speed, and reliability. This hybrid comparison offers a more comprehensive perspective on the modeling of the mechanical response of graphene sponge structures. The term “optimized” is supported by numerous decisions that were made to adjust the models as described below:

1. The incorporation of physically meaningful features from the literature, with emphasis on experimental findings from BET, XRD, SEM, Raman, tensile and Nanoindentation studies.

2. Feature engineering that systematically recognized implicit relationships in the data (e.g., Defect Ratio, Elastic Strain Index).

3. Cross-validation with systematic standardization; the models avoided data leakage and overfitting.

4. Comparative benchmarking against other ML algorithms, confirming the advantages of using GPR methods.

Such adjustments offered models the chance to learn (and generalizing) trends using simply nine physical samples, and in our experiments, it is very successful in low-data mechanical systems modeling. Figure 10 is the workflow of predictive modeling. Both experimental measurements and engineered characteristics were used as an input to two models, Gaussian Process Regression (GPR) and Bayesian Ridge Regression (BRR) to provide an estimate of the failure strength. The hybrid input architecture was critical in making sure both raw material characteristics and calculated mechanical cues were used to drive the machine learning models, leading to the creation of higher accuracy and robustness of the model.

Figure 10:

Schematic illustration of the hybrid machine learning model used for predicting mechanical failure in graphene sponge.