Introduction

Fracture mechanics focuses on studying the initiation, propagation, and instability failure patterns of materials, and it has been extensively applied across various engineering domains1,2. In rock engineering, the application of fracture mechanics primarily serves two distinct and contrasting objectives: 1) to prevent crack growth and maintain the structural integrity and safety of excavated rock structures, including tunnels and subway stations; 2) to explore methods that promote efficient crack propagation for the efficient extraction of resources like geothermal energy and shale gas. Regardless of the purpose, gaining insights into the fracture characteristics of rocks is crucial.

Dolomite is widespread in southwestern China, and the probability of encountering dolomite strata is very high during underground engineering projects such as tunnel excavation and subway construction. For example, the Jixin tunnel on the Chengdu-Kunming railway3,4 and the Fangjiawan tunnel in Shanxi province5 are both undergoing construction through dolomite strata. Understanding the fracture characteristics of dolomite is vital to guarantee the safety and stability of tunnels, subway stations, and other underground projects during construction and operation.

To determine the fracture properties accurately, many auxiliary techniques have emerged6,7,8,9,10. Among them, the digital image correlation (DIC) technique is an advanced application within the field of particle tracking, where the tracking targets are sub-images with unique patterns11. By comparing speckle patterns captured at different times by a high-speed camera, the specimen displacement and strain can be obtained12,13. Due to this technique has the advantages of not requiring auxiliary optical devices, easy installation and operation of experimental equipment and the ability to accurately calculate the specimen deformation, it enjoys widespread adoption and is very popular in the study of material fracture mechanics14,15,16,17,18,19.

Fracture toughness, reflecting the ability to resist crack propagation, is the widely used fracture parameter in rock engineering20,21. Due to the tensile strength (ft) being significantly lower than the compression strength for rock materials, the mode I fracture toughness is most common, and its use is most frequent. At present, various specimen configurations are employed to measure this fracture toughness, such as chevron bend22, short rod22, cracked chevron notched Brazilian disc23,24,25,26 and semi-circular bend (SCB)27. Among the numerous specimen configurations available, the SCB specimen has the advantages of simple sample preparation and convenient testing procedures, making it widely used in rock28,29,30,31, asphalt concrete32,33 and fiber-reinforced concrete34,35,36,37. The mode I fracture toughness (KI) of the SCB specimen can be calculated as follows27,31,32,33,34,35,36,37:

$$\left\{ \begin{gathered} K_{I} = Y\frac{{P\sqrt {\pi a} }}{2RB} \hfill \\ Y = - 1.297 + 9.516\frac{S}{2R} - \left( {0.47 + 16.457\frac{S}{2R}} \right)\frac{a}{R} + \left( {1.071 + 34.401\frac{S}{2R}} \right)\left( \frac{a}{R} \right)^{2} \hfill \\ \end{gathered} \right.$$
(1)

where B, S and R represent the thickness, span and radius of the SCB specimens, respectively. For materials where fracture process zone (FPZ) can neglected (such as silicate glasses), by substituting peak load (Pmax) and initial crack length (a0) into Eq. (1), the true fracture toughness (KIc) can be obtained. However, for materials where the FPZ cannot be neglected, substituting Pmax and a0 into Eq. (1) will only obtain the apparent fracture toughness (KIf), because when the load reaches Pmax, the crack has already propagated, and the crack length is no longer equal to a0, but rather to ac.

Currently, to determine KIc, there are mainly two methods for calculating ac: 1) Using techniques such as the DIC and numerical simulations to determine ac by observing the specimen deformation28,38; 2) Obtaining ac by solving the relationship between theoretical dimensionless compliance and experimental compliance29,30,39,40. Although the methods for calculating KIc based on the SCB specimens are already available, there are only a few scholars to calculate KIc when they studied the fracture properties of rock materials28,38. Scholars tend to calculate KIf of materials such as rock31, asphalt concrete32,33 and fiber-reinforced concrete34,35,36,37, and have generally reached a consensus: the measured KIf41,42,43,44 is size-dependent and has s strong relationship with ft45.

It should be emphasized that scholars have conducted independent studies on the size effect of KIf and the relationship between KIf and ft. Currently, there are no scholars to study the relationship between KIf and ft considering the influence of specimen size. The possible reasons of which maybe are: 1) it is difficult and costly to obtain the samples in site. For example, it is nearly impossible to directly measure fracture toughness by obtaining a large number of rock samples through continuous sampling that meets experimental requirements46. 2) the experimental process is complex. Not only are the tests for obtaining ft required, but also those for calculating the fracture toughness of various specimen sizes. 3) it is difficult to select suitable data analysis techniques. This study will become simple if a data analysis technique can be selected that allows the use of previous data to construct a dataset including specimen size, ft, and fracture toughness, and to predict the fracture toughness.

Artificial neural network (ANN), as a data processing technique that rapidly developed in the 1980s, can directly study from examples, respond correctly or nearly correctly to incomplete tasks, generate generalizable results from new problems, and extract information from poor-quality or noisy data. Consequently, ANN has emerged as a highly powerful tool for solving numerous engineering problems, especially in cases where data is complex or insufficient. Over the past decade, ANN has been successfully applied in fields such as fracture mechanics of cement-based materials47,48,49, prediction of material fracture toughness50,51,52, material fatigue53, and concrete technology54,55.

In this study, based on the DIC technique, three-point bending (3-pb) tests were conducted on dolomite SCB specimens. Based on the data of the full-field deformation measured by the DIC, the calculation methods for mid-span deflection (δs) and crack mouth opening displacement (CMOD) were given, and the load–displacement curve, the evolution of crack opening displacement and strain concentration were investigated. Meanwhile, a comparison was made between two fracture toughness (KIf and KIc). In addition, by constructing two datasets using the diameter (D), span (S), initial crack length (a0), tensile strength (ft) and fracture toughness (KIf or KIc), three backpropagation artificial neural network (BP-ANN) models were established: two for predicting KIf and one for predicting KIc, all of which were utilized to analyze size-related variations in fracture properties.

Experimental program

Experimental materials

The rock samples were collected in Zhaotong City, Yunnan Province. To determine the sample type, X-ray diffraction (XRD) tests were employed on the samples. The analysis revealed that the mineral composition of the samples consisted of 85.2% dolomite, 13% calcite, and 1.8% quartz. Which confirmed that the collected rock samples are dolomite. The elastic modulus (E) of those samples is about 67.71 GPa.

The ISRM-suggested geometries for SCB specimens specify several key dimensions. Firstly, the specimen thickness (B) must be larger than 0.8R, where R represents the specimen radius. Secondly, the initial crack length (a0) should range from 0.4R to 0.6R. Lastly, the distance between support points (S) needs to be between R and 1.6R. Therefore, the collected samples were designed as semi-circular disks. The specimens including the initial cracks were made using waterjet cutting, and the specimen size is shown in Table. 1.

Table. 1 Specimen size.

Experimental process

Before conducting the tests, the surface of each specimen was treated by spraying speckles. After the speckle production was completed, the specimen was placed on the loading machine (RMT-301). The supports chosen were of the roller-in-groove type, with a roller diameter of 2 mm. Two DIC cameras (the camera resolution and frame rate are 4872 × 3248 pixels and 3 fps respectively) were positioned on the front side of the specimen to record the specimen deformation during the test. 3-pb tests were performed on the specimens at a loading rate of 0.0001 mm/s (see Fig. 1).

Fig. 1
figure 1

Test setup.

Experimental results

Deformation and crack propagation of specimen

The full-field horizontal displacement contour of the D-1 specimen at pre-peak 90% Pmax is shown in Fig. 2. Obviously, the left part of the specimen moves in the negative x-axis, while the right part moves in the positive x-axis. This indicates that as the test process, the crack in the specimen are gradually widening.

Fig. 2
figure 2

Horizontal displacement contours.

Taking the D-1 specimen as an example, the trajectory of the crack propagation is shown in Fig. 3. Obviously, the crack emerges from the initial crack tip and extends toward the loading point. However, the crack did not propagate straight up along the plane of the initial crack. This is mainly because the rock material is heterogeneous, and the strengths of different mineral particles are different. Therefore, the crack tends to bypass the mineral particles with higher strength and propagate in a wavy pattern. After the peak, the crack rapidly develops, and the specimen suddenly fractures, indicating that the dolomite specimens are brittle, and the 3-pb tests of the SCB specimens do not provide good control over crack propagation.

Fig. 3
figure 3

Trajectory of the crack propagation.

Load–displacement curves

Measuring the CMOD and δs is a challenge by arranging sensors on the specimen surface, given their small size. Therefore, scholars tend to obtain the relationship between the actuator displacement (δa) and applied load (P). Figure 4a displays the P-δa curve of the D-1 specimen. At the pre-peak stage, the P-δa curve is not linear. The slope appears to increase slightly as the test progresses. This may be due to insufficient contact between the specimen and the loading equipment, as well as between the specimen and the supports during the initial stages of loading. When the load reaches the peak point, it suddenly drops vertically. This phenomenon occurs in both sandstone56 and granite57.

Fig. 4
figure 4

Load–displacement curves, (a) P-δa curve, (b) P-CMOD and P-δs curves.

Utilizing the DIC allows for the determination of CMOD by computing the difference in horizontal displacement between two specified points, which are located at each side of the initial crack tip (see points C and E on the SCB specimen in Fig. 4a):

$$CMOD = u_{C} - u_{E \, }$$
(2)

The P-CMOD curve of the D-1 specimen, displayed in Fig. 4b, demonstrates the correlation between the actual specimen deformation and load. Unlike the P-δa curve, a nonlinear segment (d-e segment) was observed in the pre-peak P-CMOD curve. It appears that the specimen starts to crack at point d, and then the crack stable propagates until it reaches point e (i.e. the peak point), whereupon the specimen undergoes rapid failure.

Based on the DIC technique, δs can be indirectly calculated by selecting four points on the specimen surface (such as the points B, C, E and F in Fig. 4a):

$$\delta_{s} = \frac{{v_{C} - v_{E} }}{2} + \, \frac{{v_{C^{\prime}} - v_{E^{\prime}} }}{2}$$
(3)

Figure 4b displays the P-δs curve of the D-1 specimen. The variation pattern of the P-δs curve is consistent with that of the P-CMOD curve, but it differs greatly from the variation pattern of the P-δa curve. This discrepancy arises because δa may be influenced by factors such as the gap between the support and the specimen, which prevents it from accurately reflecting the true specimen deformation.

Previous scholars generally believed that rock materials are in an elastic (linear) state before reaching the peak by analyzing the P-δa curve. However, this conclusion may have been misled by δa, which fails to accurately reflect the specimen deformation. In contrast, the variation patterns of the P-CMOD and P-δs curves indicate that there is a nonlinear segment in rock materials before reaching the peak. In the subsequent analysis, we attempt to explain the reasons for this nonlinearity through an analysis of the strain and displacement fields.

Crack opening displacement

The horizontal displacement at the initial crack tip along the line MN (see Fig. 4a) was extracted using the DIC, as displayed in Fig. 5. When a relatively small load (such as pre-peak 55% Pmax) was applied to the specimen, a linear change in horizontal displacement was observed along the line MN, suggesting that the specimen is undergoing elastic deformation. At pre-peak 95% Pmax, a noticeable displacement jump occurs, suggesting that a crack has formed. At Pmax, the horizontal displacement jump becomes more significant, indicating that the crack has propagated upwards.

Fig. 5
figure 5

Horizontal displacement along the line MN.

Measuring crack tip opening displacement at the peak (CTODc) is crucial as it provides insights into the local failure characteristics58. As shown in Fig. 5, CTODc of the SCB specimen can be determined by measuring the displacement difference between the points located at approximately ± 2.27 mm from the crack tip (from x = -11.14 mm to x = -6.61 mm), and this value is about 0.009 mm. Lin et al.59 also determined CTODc within a range of approximately ± 2 mm from the crack tip, whereas Ji et al.58 conducted their measurements within a narrower range of approximately ± 1 mm. The above analysis shows that when calculating crack tip opening displacement, the selection of positions on both sides of the crack needs to be determined based on the starting and ending points of the horizontal displacement jump.

By extracting the horizontal displacements of points on the vertical lines EG and CH (see Fig. 4a) located on both sides of the initial crack, the crack propagation can be assessed. As shown in Fig. 6, along the positive y-axis, the opening displacement gradually decreases. Compared to the opening displacement at pre-peak 55% Pmax, the opening displacement at Pmax is significantly larger, and the slope change point of the y-u curve shifts forward, indicating that the specimen has already cracked and the crack has propagated upwards.

Fig. 6
figure 6

Crack opening displacement along the y-axis.

Strain concentration zone

By observing the evolution of the full-field strain, the crack propagation also can be judged. According to the bilinear softening law60,61, The strain threshold (εc) can be determined as follows:

$$\varepsilon_{c} = f_{t} /E$$
(4)

where ft can be calculated as follows62:

$$K_{If} = 0.125f_{t}$$
(5)

where KIf is the apparent fracture toughness, and it is about 1.10 MPa. By substituting KIf into Eq. (5), ft can be obtained and is about 8.78 MPa. By substituting E and ft into Eq. (4), εc can be obtained and is about 0.0001. Figure 7 displays the full-field strain of the D-1 specimen and the horizontal strains of points on the line MN. Obviously, as the test progresses, a strain concentration area starts to emerge and gradually extends upward, indicating that the specimen has cracked before reaching Pmax. The strain concentration observed in the D-1 specimen at pre-peak 95% Pmax is significantly stronger than that at pre-peak 75% Pmax, indicating that the specimen has already cracked when the load is between 75% Pmax and 95% Pmax.

Fig. 7
figure 7

Horizontal strain of D-1 specimen.

The aforementioned analysis shows that the crack initiated and progressed before reaching Pmax, which is the likely main reason for the nonlinearity observed in the rock during this period.

Fracture toughness

(1) Based on theoretical derivation.

Figure 8 displays the KIf and KIc values for the D-1 specimen. Obviously, the KIf (1.08 MPa·m1/2) is lower than the KIc (1.86 MPa·m1/2), indicating that the fracture toughness estimated using the LEFM underestimates the inherent toughness of the rock. This underestimation arises because the crack initiated and propagated before reaching Pmax. In addition, the KIf and KIc values of sandstone SCB specimens28 were also plotted in Fig. 8, and consistent laws were found. The above findings indicate that using KIc is more accurate in practical engineering.

Fig. 8
figure 8

Fracture toughness.

(2) Based on DIC and simulation techniques

Based on the full-field displacement measured by the DIC and simulation techniques, the stress intensity factor (SIF) of the dolomite specimens can also be obtained. The calculation process is shown in Fig. 9. Firstly, the DIC technique is employed to acquire the displacement field data of the target loading level (such as Pmax). Subsequently, a numerical model is established based on the measured crack trajectory, and the displacement field data obtained by DIC is interpolated to obtain the displacement of each node in the numerical model. Finally, the SIF can be obtained using the contour integral method after assigning material parameters to the model and running it.

Fig. 9
figure 9

Calculation process of the fracture process based on DIC and simulation techniques.

The numerical model has a length of 30 mm and a height of 24 mm, and it was considered to be an isotropic and homogeneous elastic material, with E = 67.71 GPa. Since the influence of Poisson’s ratio on the calculation results of the model can be ignored63, it is selected as 0.2 here. The model comprises 5176 nodes and 5078 elements. Specifically, within the contour integration area, the element type is CPS3 with a maximum element edge length of 0.3 mm; in other areas, the element type is CPS4R with a maximum element edge length of 1 mm.

The SIF of the D-1 specimen at the Pmax was calculated, which can be regarded as the fracture toughness. Due to the crack trajectory of the D-1 specimen not being a straight line but instead developing in a zigzag manner, there exists within the specimen not only Mode I fracture toughness (KI) but also Mode II fracture toughness (KII). The value of the KI is about 2.20 MPa·m1/2, which has a 15.45% error compared to the KIc (1.86 MPa·m1/2) calculated using Eq. (1). This indicates that the KIc calculated using Eq. (1) can represent the inherent toughness of rock materials. The reasons for this relatively large error between the KI and KIc may be multifaceted, and two primary reasons are concluded as follows:

(1) Caused by the setting of the E in the model. Although the E used in the model is obtained through uniaxial compression tests (UCT), due to the significant discreteness of rock materials, there may be deviations between the E of specimens used for SCB tests and those used for UCT. When E = 60.94 GPa, KI = 1.98 MPa·m1/2, resulting in only a 6.06% error compared to the KIc.

(2) Caused by the tortuous propagation of the crack in the specimen. Due to the tortuous propagation of the crack, both the KI and KII exist in the specimen, and the fracture mode of the specimen can be regarded as a composite fracture. However, the use of Eq. (1) is based on the assumption that the crack is straight. The value of the KII is about 1.38 MPa·m1/2, indicating that the crack would continue to propagate, but its propagation direction would deviate from the initial crack direction.

Prediction of fracture toughness

Principles of the BP-ANN

The BP-ANN is a multi-layer neural network that employs the BP algorithm as its learning rule, exhibiting strong nonlinear mapping capabilities. It typically features an input layer, followed by one or several hidden layers and concludes with an output layer64, as shown in Fig. 10. Each layer of the BP-ANN contains multiple neurons, and there are no connections among neurons within the same layer.

Fig. 10
figure 10

Three-layer BP-ANN.

The training process of the BP-ANN consists of forward propagation (FP) stage and BP stage. During the FP stage, firstly, the neurons in the input layer transmit the input parameters to the neurons in the hidden layer. Subsequently, the neurons in the hidden layer compute the weighted sum of the received parameters, apply an activation function to this sum, and then pass the activated output to the neurons in the output layer. Ultimately, the neurons in the output layer compute the weighted sum of the received parameters, apply an activation function to this sum, and produce the final output of the network. In civil engineering, the widely used activation function, which can reflect the nonlinear characteristics of a problem, is as follows:

$$f_{j} = \frac{1}{{1 + \exp \left( { - \sum {w_{nj} } x_{n} + b_{j} } \right)}}$$
(6)

where wnj represents a connection weight; bj denotes a bias parameter. In the output layer, the actual output of the kth neuron (gk) can be obtained as follows:

$$g_{k} = \sum {v_{jk} } f_{j}$$
(7)

where vjk represents a connection weight. After one forward propagation is completed, the mean squared error (MSE) is used to determine whether to terminate the training process. If the MSE falls below the target error, it signifies the completion of the training process, and the trained BP-ANN possesses predictive capabilities. Otherwise, the BP-ANN enters the BP stage. During this stage, the MSE is propagated backward along the original path, and the weights across all layers are updated to reduce the MSE. At present, the primary methods used for updating weights include the Levenberg–Marquardt (LM) and Bayesian Regularization (BR) algorithms. Since the BR algorithm demonstrates robust generalization capabilities when dealing with challenging, small, or noisy datasets, and the LM algorithm offers advantages such as reduced number of iterations, fast convergence speed, high accuracy, and strong robustness, these two algorithms were employed for training the established BP-ANN model.

Data handling process

1) Establish of dataset

Previous studies have shown that the diameter (D)41,42,65, initial crack length (a0)66 and span (S)67 of the SCB specimen can affect KIf, while the thickness (B) has little influence on KIf66. To establish the BP-ANN model for predicting KIf that considers the influence of specimen size, a dataset (including D, a0, S, ft and KIf) was established68,69,70,71,72,73,74,75 (see Table A.1 in Appendix). In addition, to establish the BP-ANN model for predicting KIc that considers the influence of specimen size, a dataset (including D, a0, S, ft and KIc) was established28,68,76,77,78,79 (see Table A.2 in Appendix).

2) Establish of BP-ANN models.

Using the datasets presented earlier, two BP-ANN models (Models I and II) were established to predict KIf and KIc, respectively. It should be noted that although 174 and 59 sets of data are contained in Table A.1 and Table A.2 respectively, some data were randomly discarded when constructing Model I and Model II to prevent overfitting. Before training the established models, it is necessary to normalize the data input into the model. In this study, the following formula was used for data normalization:

$$x_{n}^{a} = 0.1 + \frac{{x_{n}^{b} - x^{\min } }}{{x^{\max } - x^{\min } }} \times 0.8$$
(8)

where xmax and xmin represent the maximum and minimum values of the selected vector before normalization, respectively; xa n and xb n represent the values before and after normalization, respectively.

The numbers of neurons in the input and output layers correspond to those of the input and output parameters. Therefore, when establishing the models, the numbers of neurons are set as 4 and 1, respectively. In addition, the hidden layers are set to a total of 1, since any engineering problem can be simulated effectively with a neural network that has one or two hidden layers48,49,50. In the hidden layer, the calculation for the neurons goes as follows:

$$s = \sqrt {q + w} + \gamma_{r}$$
(9)

where γr is a random parameter, and γr = 1 ~ 10; w and q denote the counts of the neurons in the output and input layers, respectively; s denotes the counts of the neurons in the hidden layer, and is set as 4 in this study.

Training effect

When training the model I, the LM and BR algorithms were used. When using the BR algorithm, the training and test sets contained 85 and 36 sets of data, respectively. When using the LM algorithm, the training, validation, and test sets contained 85, 18, and 18 sets of data, respectively.

Figure 11 displays the training effect of model I based on the BR algorithm. The slope of the fitted curve in the training set is 0.98, and the error bar chart exhibits a characteristic of being higher in the middle and lower on both sides, indicating that the predicted and the target outputs are highly correlated. The slope of the fitted curve in the test set is 1.00, suggesting that the trained model has strong generalization performance. Since the slope of the fitted curve based on all data is also 1.00, it can be concluded that the model I trained using the BR algorithm possesses predictive capability.

Fig. 11
figure 11

Fitting effect and error based on the BR algorithm, (a) Fitted result of training set, (b) Fitted result of test set, (c) Total fitted result, (d) Error bar chart.

Figure 12 displays the training effect of model I based on the LM algorithm. The slope of the fitted curve in the training set is 0.83, and the error bar chart exhibits a characteristic of being higher in the middle and lower on both sides, indicating that the predicted output aligns closely with the target output. The slope of the fitted curve in the test set is 0.78, indicating that the trained model has good generalization performance. Since the slope of the fitted curve based on all data is also 0.82, it can be concluded that the model I trained using the LM algorithm possesses predictive capability.

Fig. 12
figure 12

Fitting effect and error based on the LM algorithm, (a) Fitted result of training set, (b) Fitted result of test set, (c) Total fitted result, (d) Error bar chart.

When training the model II using the LM algorithm, the training, validation and test sets have 29, 15 and 15 sets of data, respectively, and the training effect is shown in Fig. 13. The slope of the fitted curve in the training set is 0.98, and the error bar chart exhibits a characteristic of being higher in the middle and lower on both sides, indicating a strong correlation between the predicted and target outputs. The slope of the fitted curve in the test set is 0.71, suggesting that the generalization ability of the trained model is moderate. This may be due to incomplete data or large data dispersion. However, since the slope of the fitted curve based on all data is 0.91, it can be considered that the trained model II has predictive capability.

Fig. 13
figure 13

Fitting effect and error based on the LM algorithm, (a) Fitted result of training set, (b) Fitted result of test set, (c) Total fitted result, (d) Error bar chart.

Parameters analysis

Since KIf is size-dependent, the size effect of KIf was investigated using the trained model I, as shown in Fig. 14a. When S, a0 and ft remain constant (S = 84 mm, a0 = 18.10 mm, ft = 7.00 MPa), as D increases, KIf gradually increases. Similar phenomena have been observed by Zhang75 and Ayatollahi80 et al. when they study the fracture properties of rocks. This indicates that in small specimens, the FPZ has not fully developed, and it varies with changes in specimen size. When D, a0 and ft remain constant (D = 100 mm, a0 = 18.10 mm, ft = 7.00 MPa), as S increases, KIf gradually decreases. Similar phenomena have also been observed by Tutluoglu et al.72 when they study the fracture properties of rocks. When D, S and ft remain constant (D = 100 mm, S = 84 mm, ft = 7.00 MPa), as a0 increases, KIf gradually decreases. Zhang75 and Zhao81 et al. have also observed similar phenomena in their studies on the fracture properties of rocks.

Fig. 14
figure 14

Effect of geometric dimensions, (a) Effect of specimen size on KIf, (b) Effect of D on the KIf-ft curve, (c) Effect of S on the KIf-ft curve, (d) Effect of a0 on the KIf-ft curve.

The impact of specimen size (including D, a0 and S) on the KIf-ft curve was investigated here. When D remains constant, KIf gradually increases as ft increases, which is consistent with the conclusions drawn by Whittaker82, Bachers83 and Zhang84; as D increases, the KIf-ft curve shifts upwards (see Fig. 14b). As S increases, the KIf-ft curve shifts downwards (see Fig. 14c). As a0 increases, the KIf-ft curve shifts downwards (see Fig. 14d).

Figure 15a displays the size effect of KIc. When S = 50 mm, a0 = 16 mm and ft = 7 MPa, KIc increases with the increases of D. When D = 70 mm, a0 = 16 mm and ft = 7 MPa, KIc decreases with the increases of S. When D = 70 mm, S = 50 mm and ft = 7 MPa, KIc decreases with the increases of a0.

Fig. 15
figure 15

Effect of geometric dimensions, (a) Effect of specimen size on KIc, (b) Effect of D on the KIc-ft curve, (c) Effect of S on the KIc-ft curve, (d) Effect of a0 on the KIc-ft curve.

How the specimen size (including D, a0 and S) influenced the KIc-ft curve was investigated here. When S = 85.48 mm and a0 = 18.90 mm, as D increases, the KIc-ft curve shifts upwards (see Fig. 15b). When D = 127.25 mm and a0 = 18.90 mm, as S increases, the KIc-ft curve shifts downwards (see Fig. 15c). When S = 85.48 mm and D = 127.25 mm, as a0 increases, the KIc-ft curve curve shifts downwards (see Fig. 15d).

Using the BP-ANN to predict fracture toughness holds great potential. However, it should be noted that the tensile strength and fracture toughness within the same set of data in the dataset are mostly not obtained from the same specimen. Due to the significant randomness in the internal structure and defects of rock specimens, this may lead to variations in tensile strength and fracture toughness among different specimens. These variations will directly affect the quality of the dataset, and subsequently affect the prediction performance of the model. Additionally, the size of the dataset is also a crucial factor. If the dataset is too small, the prediction model may fail to converge effectively85. Therefore, we must acknowledge that the accuracy of ANN model predictions heavily relies on the quality and size of the dataset.

Conclusions

In this study, based on the digital image correlation technique, dolomite specimens were tested, and the fracture properties of these specimens were investigated. Subsequently, apparent fracture toughness (KIf) and true fracture toughness (KIc) were predicted. The following are the main conclusions:

  • By analyzing the load-actuator displacement (P-δa) curve, previous studies believe that the rock remains in an elastic state before reaching the peak load (Pmax). However, this conclusion may have been misled by δa, as it includes the gap between the specimen and its supports, and therefore fails to accurately reflect the specimen’s deformation.

  • The pre-peak load-crack opening displacement curve exhibits nonlinearity. By analyzing both the crack opening displacement and strain concentration zone, it can be inferred that the nonlinearity observed before reaching the Pmax may be attributed to the emergence and propagation of cracks.

  • KIf (calculated based on the initial crack length) is significantly lower than KIc (determined by considering the effect of crack propagation), indicating that the fracture toughness estimated using the LEFM underestimates the inherent toughness of the rock.

  • After considering the size effect in establishing the BP-ANN models, the trained models demonstrated good predictive performance for both KIf and KIc, as well as strong generalization capabilities.

  • The trained models were used for parametric analysis, and the results show that as the span and initial crack length of the specimen increase, both KIc and KIf gradually decrease; while the KIf-ft and KIc-ft curves both shift downwards progressively.

  • Using the BP-ANN to predict the fracture toughness holds great potential. However, data dispersion arises because the tensile strength and fracture toughness values in the same set of data are not derived from identical specimens, due to potential variations in mechanical properties (such as these two parameters) across different specimens. This dispersion poses certain challenges for conducting this research using the BP-ANN.