Effects of Support Friction on Mixed-Mode I/II Fracture Behavior of Compacted Clay Using Notched Deep Beam Specimens under Symmetric Fixed Support

Abstract

This paper investigates the effects of support friction on mixed-mode I/II fracture behavior of compacted clay using notched deep beam (NDB) specimens under symmetric fixed support. Numerical models of 330 NDB specimens were established considering the crack inclination angle, crack length, support span, and support friction coefficient, and the normalized fracture parameters (Y_I, Y_II, and T*) of NDB specimens were calibrated. The numerical results showed that the values of Y_I, Y_II, and T* decreased at different degrees after considering the support friction. Notably, the support friction coefficient could significantly change the loading pattern at the crack tip. To verify this phenomenon, 12 compacted clay NDB specimens were prepared, and a mixed-mode I/II fracture test was performed under fixed support conditions; the phenomenon of asymmetric crack propagation was studied. The test data were processed using the numerical calibration results of Y_I, Y_II, and T* with and without consideration of friction. Afterward, the test data were compared and analyzed by combining the generalized maximum tangential stress (GMTS) and the maximum tangential stress (MTS) criteria. The analysis indicated that the real fracture characteristics of compacted clay NDB specimens could not be reflected when conducting mixed-mode I/II fracture tests under symmetric fixed support conditions if the test results were analyzed by Y_I, Y_II, and T* without considering support friction, as in previous studies.

1. Introduction

Cracks are a common hazard and one of the main causes of damage to all types of geotechnical structures [1,2]. For rockfill dams with clay core walls in water conservancy projects, cracks are a significant threat to the safety of the dam [3]. Due to the difference in the deformation modulus between the clay of the core wall and the stone material of the dam shell, the clay core wall can easily produce transverse or longitudinal cracks because of uneven deformation and dry shrinkage. Under complex stress conditions, the core wall cracks are at risk of further cracking, resulting in the hydraulic splitting of the core wall and threatening the safety of the dam [4].

Since Skempton et al. [5] suggested explaining the destabilization process of fracture-containing hard clay slopes using the fracture mechanics theory, more studies have been conducted on the fracture properties of clay materials. In terms of mode I fracture properties, Lakshmikantha et al. [6] and Lee et al. [7] investigated the mode I fracture toughness of compacted clay using compact tension (CT) specimens. CT specimens were inspired by a method recommended by the American Society for Testing and Materials (ASTM) [8] for metallic materials, which required holes to be drilled in the specimen for fixture attachment. The specimen preparation and test preparation were complicated, and the fracture was likely to occur at the fixture due to the low tensile strength of the clay material. Zhang et al. [9] followed the idea of direct tensile loading, and uniaxial tensile tests were conducted using unilateral open-joint rectangular clay specimens. The test operation was relatively simple, but it was easy to fracture at the end of the specimen when using epoxy vinyl to bond the end of the specimen. For compacted clay, the bending loading method is more convenient and widely used. Most scholars used single-edge notched beam (SENB) specimens [10,11,12,13,14] recommended by the ASTM [8] to carry out mode I fracture tests of compacted clay. Some scholars used semi-circular bend (SCB) specimens [15,16] recommended by the ISRM [17] to test the mode I fracture toughness of compacted clay. However, the mode I/II fracture properties of the clay were barely studied. Only Wang et al. [18] used a four-point bending method to perform a mixed-mode I/II fracture test on the SENB specimens of compacted clay. Aliha et al. [19] investigated the mixed-mode I/II fracture properties of air-dried compacted clay using asymmetric semi-circular bend (ASCB) specimens [20]. Since most cracks in the core walls of rockfill dams are under mixed stress, studying the mixed-mode I/II fracture in compacted clay is of great engineering importance [21,22].

Considering the convenience of preparing fracture specimens of compacted clay, scholars usually use non-circular SENB specimens for clay fracture tests. However, there is a large length-to-width ratio (greater than 4.5) for the SENB specimens recommended by the ASTM [8]. For compacted clay with low tensile strength, slight disturbance during the demolding and transfer of the specimen may damage the SENB specimen of the compacted clay [12,16]. The single-sided notched deep beam (NDB) specimen (i.e., SENB specimen with a length-to-width ratio of 2:1) proposed by Luo et al. [23] can effectively reduce the clay specimen disturbance before the test and save the test material [24].

In the past decades, researchers employed the three mentioned support types of I, II, and III for conducting three-point bending fracture tests on different materials and test specimens. According to Bahrami et al. [25], the types of support for loading in bending tests can be classified into three categories (type I, II, and III): roller support, roller in groove, and fixed support, respectively. The impact of supports on the fracture results has often not been considered, and there is very little research in the literature studying the effect of support friction. In recent years, some scholars have found that the supports used in three-point bending fracture tests can affect the cracking behavior of materials significantly. Bahrami et al. [25,26] investigated the effect of support type on the fracture toughness of polymethyl methacrylate and rock materials in SCB tests. Qin et al. [27] investigated the influence of friction on the determination of rock mode I fracture toughness using an SCB specimen. Pirmohammad et al. [28,29] discussed the effect of support type on the fracture toughness of asphalt concrete using SCB specimen at various temperatures. Bakhshizadeh et al. [30] provided an investigation of the impact of support type on the fracture mode of loading using an SCB specimen. Ayatollahi et al. [31] evaluated the effects of support friction on the mode I stress intensity factor and the fracture toughness values determined from SENB testing. It should be noted that the mentioned investigations have been considered under pure mode I and pure mode II, and no studies have been carried out so far concerning the effect of supports on mixed mode I/II fracture behavior in NDB tests.

Among the three mentioned support types, only type I has no friction between the support and the specimen [31]. However, this type of support is disadvantaged by the inconvenient fixation for clay fracture specimens with small peak loads [16], thus the fixed support was mostly used in previous clay fracture tests. The fracture parameters derived using the normalized stress intensity factor (SIF) without considering the support friction cannot reflect the real fracture properties of compacted clay [32].

Therefore, the current study considers the impact of support friction on the mixed mode I/II fracture behavior of compacted clay using the NDB specimen under symmetric fixed support. First, the frictional effect between the support and the specimen was considered, and the normalized K_I, K_II, and T-stress at the crack tip of the NDB specimen were numerically calibrated using the finite element method. Their variation in the crack inclination angle under different support friction coefficients was analyzed, demonstrating the effect of the support friction coefficient on the normalized K_I, K_II, and T-stress. Then, the mixed-mode I/II fracture test was performed on compacted clay using NDB specimens. The differences in the fracture properties of the compacted clay, with and without the consideration of support friction, were analyzed. With the generalized maximum tangential stress (GMTS) criterion [33], the differences between the mixed-mode I/II fracture properties of compacted clay under the two conditions and the theoretical values were discussed.

2. Numerical Model for NDB Specimens

2.1. Loading Principle of NDB Specimens

The elastic stress field at the crack tip can be expressed as a set of expansions of an infinite series, as shown in Equation (1) [34]:

$$\left\{\begin{array}{l}{\sigma }_{\theta }=\frac{1}{\sqrt{2\pi r}}\cos \frac{\theta }{2}\left(K_{\mathrm{I}}{\cos }^2\frac{\theta }{2}-\frac{3}{2}{K}_{\mathrm{II}}\sin \theta \right)+T{\sin }^2\theta +O(r^{1/2})\\ {\sigma }_r=\frac{1}{\sqrt{2\pi r}}\cos \frac{\theta }{2}\left[K_{\mathrm{I}}\left(1+{\sin }^2\frac{\theta }{2}\right)+K_{\mathrm{II}}\left(\frac{3}{2}\sin \theta -2\tan \frac{\theta }{2}\right)\right]+T{\cos }^2\theta +O(r^{1/2})\\ {\tau }_{r\theta }=\frac{1}{2\sqrt{2\pi r}}\cos \frac{\theta }{2}\left[K_{\mathrm{I}}\sin \theta +K_{\mathrm{II}}\left(3\cos \theta -1\right)\right]-T\sin \theta \cos \theta +O(r^{1/2})\end{array}\right.$$

(1)

where r and θ are the crack tip coordinates; the first term is the singular term (the r^−0.5 term); the second term is the non-singular term (the r⁰ term); and O(r^1/2) is the higher-order term, which is usually not considered because O(r^1/2) is much smaller than the first two terms.

The SIF can reflect the strength of the stress field around the crack tip. Although the non-singular stress term (T-stress) is relatively small, it has a certain influence on the initiation and development of the crack [33,35]. Therefore, the SIFs, K_I, K_II, and T-stress, are important parameters for studying mixed-mode I/II fracture problems.

As shown in Figure 1, for the NDB specimen proposed by Luo et al. [23], the specimen width is W, the specimen length is 2W, the crack length is a, the angle between the pre-fabricated crack and the loading direction is β, the support span is 2S, and the specimen thickness is B.

The K_I, K_II, and T-stress at the crack tip of the NDB specimen can be expressed as:

$$K_{\mathrm{I}}=\frac{P\sqrt{\pi a}}{2WB}{Y}_{\mathrm{I}}$$

(2)

$$K_{\mathrm{II}}=\frac{P\sqrt{\pi a}}{2WB}{Y}_{\mathrm{II}}$$

(3)

$$T=\frac{P}{2WB}{T}^{*}$$

(4)

where Y_I, Y_II, and T* are normalized factors of the mode I SIF, mode II SIF, and T-stress, respectively, and P is the applied load.

For the loading modes of the crack in the NDB specimen, when the crack inclination angle β of the NDB specimen is 0, K_I ≠ 0, K_II = 0, the crack is subjected to pure mode I loading. As the crack dip angle β gradually increases, the crack loading condition can be changed from the pure mode I to mixed mode I/II (K_I ≠ 0, K_II ≠ 0), and pure mode II loading (K_I = 0, K_II ≠ 0) is obtained when the crack dip angle β reaches a specific degree.

2.2. Finite Element Model of NDB Specimens

In previous studies, the fracture stress field calibration of fracture specimens usually excluded the support friction effect, and the normalized factors Y_I, Y_II, and T* were only functions influenced by β, a/W, and S/W. After considering the support friction, the normalized factors Y_I, Y_II, and T* became functions influenced by β, a/W, S/W, and f. Therefore, quantifying the effect of support friction on the stress field at the crack tip of NDB specimens is essential for accurately determining the fracture properties of the test material.

In this study, the finite element software ABAQUS was used to establish the numerical model for NDB specimens. As shown in Figure 2, the NDB specimen has a length of 2W = 75 mm, a width of W = 37.5 mm, a thickness of B = 30 mm thick, and an applied reference load of P = 1000 N. According to previous studies, the modulus of elasticity and Poisson’s ratio have no effect on the calibration of the stress field [36]. Therefore, the modulus of elasticity and Poisson’s ratio were taken as 1.65 GPa and 0.35, respectively.

To simulate the boundary conditions of the NDB specimen, the support at the bottom was completely fixed, and the loading point at the top was limited to moving in the vertical direction only. The entire specimen was loaded by applying a vertical downward concentrated force to the top loading point. To reflect the friction coefficient between the actual support point and the specimen, a discrete rigid body was used to model the top loading point and the bottom support [25,31], each of the three supports was modeled as a rigid part, and the frictional contact was regarded between the supports and the specimen. The mesh division model of the NDB specimen at β = 20°, a/W = 0.6, and S/W = 0.6 is illustrated in Figure 2.

The J-integral approach was adopted to calculate K_I and K_II, and the integral interaction method was used to calculate T-stress [23,24,30]. Due to the singularity of stress distribution near the crack tip, singular cells were arranged in the first ring of the crack tip, and the size of the singular cells was less than 1/100 of the crack length a. In addition, a 20-turn four-node quadratic integral quadrilateral cell was divided around the singular cell using a sweeping technique. The distance between the edge of the outermost ring cell and the crack tip was about 1/9 of the crack length, and the rest of the region was divided freely to produce quadrilateral cells. The K_I, K_II, and T-stress at the crack tip of the specimen can be obtained by numerical computation. Then, Y_I, Y_II, and T* can be back-calculated according to Equations (2)–(4).

In order to verify the reasonableness of the model, the crack tip stress field of the NDB specimen at f = 0 and a/W = 0.4 was calculated and compared with the results of Luo et al. [23], as shown in Figure 3. With a different S/W and different β, the Y_I obtained through numerical simulation is basically consistent with the Y_I patterns and magnitudes obtained from the literature [23], demonstrating the rationality of the proposed numerical model.

2.3. Calculation Scheme

To investigate the effect of support friction on the crack tip stress field of NDB specimens, the effect of the support friction coefficient f was added based on the conventional calibration work considering β, a/W, and S/W. The calculation scheme is shown in

Table 1. Calculation scheme.

S/W	a/W	f	β/°
0.6	0.4, 0.5, 0.6	0, 0.2, 0.4, 0.6, 0.8	0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50
0.8	0.4, 0.5, 0.6	0, 0.2, 0.4, 0.6, 0.8	0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50

, with 2 S/W, 3 a/W, 5 f, 11 β, and 330 models selected.

3. Effect of Support Friction on the Stress Field at the Crack Tip of NDB Specimens

3.1. Variation Curve of Y_I with β under Different Support Friction Conditions

Through the calculation of 330 numerical models, the K_I, K_II, and T-stress at the crack tip in each model were obtained. Then, the Y_I, Y_II, and T* values were obtained by back-calculating according to Equations (2)–(4).

As shown in Figure 4, Y_I gradually decreases with increasing β for any a/W, S/W, and f condition. When a/W and S/W are fixed, the slope of the Y_I–β curve gradually decreases with the increase in the friction coefficient f, and the influence of support friction on Y_I is more significant when β is small.

The comparison of the Y_I–β curves under different S/W conditions suggests that when a/W is fixed, a larger S/W leads to a larger Y_I, which is more sensitive to the influence of support friction. Comparing the Y_I–β curves under different a/W conditions, it can be concluded that when S/W is fixed, a larger a/W leads to a larger Y_I, which is more significantly affected by the support friction.

In addition, there is a critical dip angle β_II of the crack in a certain range between a/W and S/W. It makes Y_I = 0, indicating that the crack achieves pure mode II loading. However, at a large S/W and small a/W, pure mode II loading of cracks cannot be achieved, no matter how much β increases.

Interestingly, the critical crack inclination angle β_II corresponding to the pure mode II loading condition (Y_I = 0) of the NDB specimen changed significantly after considering the support friction, indicating that the support friction significantly impacts the loading state of the crack, which directly affects the scheme design of the mixed-mode I/II fracture test.

3.2. Variation Curve of Y_II with β under Different Support Friction Conditions

As shown in Figure 5, under any a/W, S/W, and f condition, as the crack inclination angle β increases, mode II load components start to appear, Y_II increases first and then decreases, and the Y_II–β relation curve is distributed in an asymmetric parabolic pattern.

Overall, when a/W and S/W are certain, increasing f only decreases Y_II, but does not affect the trend of Y_II–β curve, and the β corresponding to the maximum Y_II is also basically unchanged. In addition, as S/W or a/W increases, Y_II increases gradually, and it is more sensitive to the influence of support friction.

As seen in Figure 4, Y_I corresponding to the β that maximizes Y_II is not 0, which indicates that the Y_II of the crack is not the maximum value during pure mode II loading.

3.3. Variation Curve of T* with β under Different Support Friction Conditions

Extensive literature has shown that T-stress has a non-negligible role in crack initiation and expansion [20,37,38,39,40]. Therefore, the normalized T-stress (T^*) of NDB specimens was calibrated in this paper.

As seen in Figure 6, T^* gradually increases with increasing β, and then stabilizes or slightly decreases. Its value reveals that the T-stress gradually changes from compressive to tensile stress. As a/W increases, the Y_II~T* curve changes gradually from an approximately linear to a convex S-shaped curve, and the β corresponding to the maximum T* decreases gradually.

Similar to the rule in Figure 4 and Figure 5, T* is more significantly influenced by the support friction at a larger S/W or larger a/W.

3.4. Effect of Support Friction on Y_I

To further observe the influence law of the support friction coefficient on Y_I, the Y_I–f relation curve was drawn, as shown in Figure 7. Under different a/W, S/W, and β conditions, Y_I decreases gradually with the increase in f; when f increases to a certain extent, it basically has no effect on Y_I. This result is because the load P acting on the specimen is fixed in numerical calibration. As f increases, the friction between the support and the specimen increases, and the load that causes the specimen to fracture gradually decreases, thus decreasing K_I and Y_I.

3.5. Effect of Support Friction on Y_II

To further observe the influence law of the support friction coefficient on Y_I, the relation curve of Y_II–f under different cases was drawn, as shown in Figure 8. Overall, when other factors are certain, Y_II gradually decreases with the increase in f (except for β = 0° when Y_II is 0, since the crack is the pure mode I loading). Additionally, Y_II is more sensitive to support friction when S/W or a/W is larger. By comparing with Figure 7, it can be found that the influence of support friction on Y_I is significantly greater than that on Y_II.

3.6. Effect of Support Friction on T*

The T*–f relation curves for different a/W, S/W, and β conditions are shown in Figure 9. Overall, T* is not sensitive to f when a/W is small, regardless of the variation of β and S/W, and shows a horizontal trend. As a/W increases, the T*–f relation curve gradually shows a decreasing trend. For example, when a/W = 0.6, T* gradually decreases as f increases and tends to be stable after f reaches a certain value.

In addition, as a/W increases, the T*–f curve corresponding to β = 30° gradually approaches the T*–f curve corresponding to β = 40° and β = 50°, which is because increasing a/W leads to a smaller β corresponding to the maximum T*.

3.7. Effect of Support Friction on the Pure Mode II Crack Inclination Angle

As seen in Figure 4, the support friction has a significant effect on the normalized mode I SIF at the crack tip of the NDB specimen and directly affects the critical crack inclination angle β_II under pure mode II loading. Therefore, the critical crack inclination angle β_II of NDB specimens under different a/W, S/W, and f conditions is given according to the calculation scheme in this paper, as shown in

Table 2. Crack inclination β_II of the NDB specimen under pure mode II loading at S/W = 0.6.

a/W	βII/(°)
	f = 0	f = 0.2	f = 0.4	f = 0.6	f = 0.8
0.4	-	-	-	-
0.5	49	44	42	40	38
0.6	42	39	37	34	32

and

Table 3. Crack inclination β_II of the NDB specimen under pure mode II loading at S/W = 0.8.

a/W	βII/(°)
	f = 0	f = 0.2	f = 0.4	f = 0.6	f = 0.8
0.4	-	-	-	-
0.5	-	-	-	-
0.6	-	57	51	45	42

.

From Table 2, it can be concluded that the crack cannot achieve pure mode II loading at S/W = 0.6 and a/W = 0.4. When S/W = 0.6, and a/W = 0.5 and 0.6, the crack can achieve pure mode II loading in the range of f = 0–0.8, and the critical crack inclination angle β_II gradually decreases as f increases.

From Table 3, it can be deduced that the crack cannot achieve pure mode II loading at S/W = 0.8, and a/W = 0.4 and 0.5. When S/W = 0.8 and a/W = 0.6, the critical crack inclination angle β_II gradually decreases as f increases. In addition, with the support friction considered, pure mode II loading can be achieved in cases where it cannot be achieved originally, such as the case of a/W = 0.6, S/W = 0.8, and f = 0.

First, it is necessary to reduce the S/W or increase a/W to achieve the full range of the mixed-mode I/II fracture test for cracks from pure mode I to pure mode II based on the three-point bending test of NDB specimens, and reduce the critical inclination angle β_II for the convenience of specimen processing. Additionally, it can also be achieved by fixing the support and increasing the roughness of the support.

3.8. Effect of Support Friction on Mises Stress at the Crack Tip

Mises stress nephogram of a typical NDB specimen (a/W = 0.5, S/W = 0.6, β = 45°) with different friction coefficients is shown in Figure 10. With the increase in the friction coefficient f, the range of Mises stress profile at the crack tip gradually decreases and basically stabilizes after f reaches 0.4, which confirms the trend of the normalized fracture parameter mentioned above that first decreases and then stabilizes with the increasing friction coefficient (Figure 7, Figure 8 and Figure 9). In addition, the elliptical zone (i.e., red- and orange-colored zones) near the bottom support tends to rotate inside the specimen due to friction force at the contact points.

4. Mixed-Mode I/II Fracture Test of Compacted Clay Using NDB Specimens

4.1. Test Purpose

To investigate the influence of support friction on the mixed-mode I/II fracture behavior of compacted clay using NDB specimens, a mixed-mode I/II fracture test was conducted on compacted clay under symmetric fixed support conditions. The primary objectives of the test are as follows: (1) To evaluate the mixed-mode I/II fracture behavior of compacted clay utilized in this study; (2) To analyze the variation in mixed-mode I/II fracture parameters of compacted clay, with and without support friction; (3) To furnish test data for discussion on fracture criterion analysis.

4.2. Soil Parameters

The test clay was taken from a construction site in Chongqing, China. Its particle size distribution is as follows: 0.5–0.25 mm (0.5%), 0.25–0.1 mm (17.9%), 0.1–0.075 mm (41.6%), and <0.075 mm (40.0%). Its basic physical and mechanical properties are as follows: specific gravity G_S = 2.72, plasticity index I_P = 20.1, liquid limit W_L = 50.1%, plastic limit W_P = 30.0%, maximum dry density ρ_max = 1.68 g/cm³, optimal moisture content w_op = 16.6%, tensile strength σ_t = 44.3 kPa, cohesion c = 48.5 kPa, and internal friction angle φ = 16.3°.

4.3. Specimen Preparation Method and Loading Instrument

The specimen was prepared using a self-made compaction vessel with the dimensions of 100 × 50 × 80 mm. First, the dry density and moisture content of the compacted clay specimen were controlled to calculate the required mass of soil and water. In this test, a 95% degree of compaction was selected, the dry density of the compacted clay specimen was 1.6 g/cm³, and the optimum moisture content of 16.6% was taken. Then, the soil materials were mixed thoroughly, and the well-mixed soil materials were poured into a fresh-keeping bag in an airtight container to rest for 24 h. Second, the soil materials were poured into the compaction vessel, and rectangular specimens of 100 mm in length, 50 mm in width and 30 mm in thickness were obtained through layered compaction. Finally, the specimens were demolded, and then cutting equipment was used for crack preparation. The specimen preparation process is shown in Figure 11.

During the compaction, the configured soil materials were compacted in three layers. The quality of each layer was controlled as equally as possible, and each layer was fully scraped after compaction [16]. To facilitate the demolding of the specimens, a certain amount of lubricant was applied to the side walls and bottom of the compaction vessel before pouring in the soil material [16].

All tests for this study were conducted at the Key Laboratory for Hydraulic and Waterway Engineering of the Ministry of Education of Chongqing Jiaotong University. As shown in Figure 12, the self-developed DTV-3 soil fracture instrument was used, including a servo motor, load sensor, displacement sensor, telescopic rod, fixture, and data acquisition software. The maximum axial load and axial displacement were 3 kN and 40 mm, respectively, the precision was 0.1 N, and the loading rate ranged from 0.12 to 30 mm/min. In this NDB tests, displacement-controlled loading was adopted at a rate of 0.8 mm/min [16,41], and the bottom was a fixed steel support.

4.4. Friction Coefficient of the Compacted Clay–Steel Interface

The friction coefficient of the compacted clay–steel interface was determined using the method [41,42] shown in Figure 13. First, the prepared NDB specimen was placed in the middle of the steel plate. Then, the steel plate was slowly raised until the specimen began to slide. The lifting height, h, of the steel plate was tested. The friction coefficient f can be calculated according to the relationship between f and α (f = tanα), where α is the angle between the steel plate and the horizontal direction. According to the above method, the average f was 0.4 for the compacted clay NDB specimen and the steel plate under the steel support (the steel plate is of the same material as the steel support), i.e., the friction coefficient between the support and the specimen was regarded as 0.4 in this paper.

4.5. Test Scheme

According to the numerical calibration results in Section 3, NDB specimens with a/W = 0.6 and S/W = 0.6 (specimen size of 100 mm × 50 mm × 30 mm) were selected to perform the mixed-mode I/II fracture test.

To verify the effect of support friction on the test results, consistent with previous studies, support friction was not considered in designing the crack inclination angle of the specimens. According to the Y_I–β curve corresponding to f = 0 in Figure 4, six crack inclination angles were selected, namely β = 0° (pure mode I), 10°, 20°, 30°, 37°, and 42° (pure mode II). Two sets of parallel specimens, totaling 12 specimens, were considered.

To facilitate the processing and analysis of the test results, the normalized factors Y_I, Y_II, and T* of NDB specimens at a/W = 0.6 and S/W = 0.6 were compiled according to Figure 4, Figure 5 and Figure 6, as shown in Figure 14.

5. Test Results and Analysis

Figure 4 and Figure 5 show the results of the mixed-mode I/II fracture test of compacted clay without and with consideration of support friction, respectively. As illustrated in the

Table 4. Mixed-mode I/II fracture test results of compacted clay without considering support friction.

β (°)	Me	KI (kPa·m0.5)	KII (kPa·m0.5)	KI/KIC	KII/KIC	T (kPa)
0	1.00	19.01	0.00	1.00	0.00	−13.08
10	0.86	15.57	3.57	0.82	0.19	2.96
20	0.70	11.82	5.98	0.62	0.31	31.12
30	0.49	6.46	6.62	0.34	0.35	51.21
37	0.26	3.30	7.50	0.17	0.39	69.20
42	0.00	0.00	7.79	0.00	0.41	79.62

and

Table 5. Mixed-mode I/II fracture test results of compacted clay considering support friction.

β (°)	Me	KI (kPa·m0.5)	KII (kPa·m0.5)	KI/KIC	KII/KIC	T (kPa)
0	1.00	9.41	0.00	1.00	0.00	−24.45
10	0.76	7.52	3.03	0.80	0.32	−7.32
20	0.51	5.29	5.05	0.56	0.54	20.71
30	0.23	2.13	5.65	0.23	0.60	41.57
37	≈0.00	≈0.00	6.63	≈0.00	0.71	60.52
42	−0.22	−2.31	6.51	−0.25	0.69	64.78

, the combined relationship between K_I and K_II is represented by M^e. As shown in Equation (5), M^e = 0 corresponds to pure mode II, M^e = 1 corresponds to pure mode I, and M^e between 0 and 1 is compound loading.

$$M^{\mathrm{e}}=\frac{2}{\pi }{\tan }^{-1}\left(\frac{K_{{}_{\mathrm{I}}}^{}}{K_{\mathrm{I}\mathrm{I}}}\right)$$

(5)

As shown in Figure 4, when β = 0°, K_II is 0, the crack is the pure mode I loading and K_IC is 19.01 kPa·m^0.5. When β = 42°, K_I is 0, the crack is pure mode II loading, K_IIC is 7.79 kPa·m^0.5, and the fracture toughness ratio K_IIC/K_IC is 0.41.

As shown in Table 5, the mixed-mode I/II fracture characteristics of compacted clay change significantly after data processing using Y_I, Y_II, and T*, considering support friction.

When β = 0°, the crack is still under pure mode I loading and K_IC is 9.41 kPa·m^0.5; however, the value is 51% lower than the K_IC without considering support friction. When β = 42°, K_I is −2.31 kPa·m^0.5 and the crack is in the compression-shear-stress state, which does not belong to mixed-mode I/II fracture. Due to the support friction, the critical inclination angle β_II of the crack corresponding to pure mode II becomes smaller. When β = 37°, K_I is very small, approximately equal to 0, so it is considered to be pure mode II loading at this time. Correspondingly, K_IIC is 6.63 kPa·m^0.5, which is 15% lower than K_IIC without considering the support friction, and the fracture toughness ratio K_IIC/K_IC is 0.71, which is 72% higher than K_IIC/K_IC without considering the support friction. In addition, the M^e corresponding to different β values decreases after considering the support friction, indicating that the support friction increases the proportion of mode II loading at the crack tip of the NDB specimen.

The peak load measured in the three-point bending fracture test using fixed support exists in part to resist the friction between the support and the specimen. Therefore, the obtained K_I, K_II, and T must show different degrees of the decrease law when the test results are processed by using the previous numerical calibration results (Y_I, Y_II, T*) without considering the support friction.

The fracture phenomena of specimens at different prefabricated crack inclination angles are shown in Figure 15. Under pure mode I loading (β = 0°), the crack undergoes a self-similar expansion along the direction of the prefabricated crack. As β increases, the crack is deflected along the prefabricated crack direction, and the crack initiation angle θ₀ gradually increases. When β exceeds 30°, the crack initiation angle θ₀ varies little.

6. Discussion

The mixed-mode I/II fracture criterion is an important theory for judging whether the crack reaches the propagation condition and the direction of the crack propagation. The most commonly used criteria are the maximum tangential stress (MTS) criterion [43], the minimum strain energy density (MSED) criterion [44] and the maximum energy release rate (MERR) theoretical criterion [45,46]. Studies have shown that these classical criteria have large deviations in predicting the fracture behavior of materials under laboratory conditions [35,47]. Many scholars believe that the non-singular constant term T-stress of the Williams expansion term [19,38,42,47,48] accounts for the failure of conventional fracture mechanics theories to predict experimental results well. The widely adopted GMTS criterion, proposed by Smith et al. [33], considers that the stress field at the crack tip is jointly determined by K_I, K_II, and T-stress, the calculation principle of the GMTS criterion is given in Appendix A.

In order to compare the degree of agreement between the theoretical prediction values and the test values considering and not considering the support friction, the crack initiation angle of NDB specimens was theoretically predicted by the GMTS and MTS criteria for both cases considering and not considering the support friction, using the Y_I, Y_II, and T* of NDB specimens and the r_c of compacted clay. According to the tensile strength of compacted clay and the results of K_IC testing considering support friction, the r_c of the compacted clay used was calculated to be 7.2 mm using Equation (A8). The prediction result was compared with the experimental results, as shown in Figure 16.

The direction of the crack initiation angle was measured from fractured NDB specimens. First, to measure the angle θ₀ for each specimen, the crack front area of the fractured specimen was photographed. A tangent line was drawn from the tip of the crack along the direction of the crack initiation angle. Then, θ₀ was determined by measuring the angle between the original crack line and the tangent line.

From Figure 16, it can be concluded that there is a large deviation between the theoretical prediction curve of the crack initiation angle based on the classical MTS criterion and the experimental results, regardless of whether friction is considered. After the GMTS criterion is adopted, the theoretical prediction curve of the crack initiation angle better agrees with the experimental results.

A comparison of Figure 16a,b shows that the theoretical curve of the GMTS criterion (r_c = 7.2 mm) is closer to the experimental data point after considering the support friction. It is worth noting that the critical crack dip angle β_II corresponding to pure mode II loading becomes smaller after considering the support friction. Therefore, there is one less set of data points in Figure 16b than in Figure 16a.

Furthermore, the mixed-mode fracture envelope of NDB specimens was plotted based on the GMTS and MTS criteria for both cases considering and not considering the support friction, and was compared with the experimental results, as shown in Figure 17.

Similarly to the pattern in Figure 16, the theoretical curve based on the GMTS criterion is closer to the experimental results than that based on the MTS criterion, regardless of whether friction is considered. In addition, the degree of agreement between the theoretical curve based on the GMTS criterion and the experimental results is significantly improved after considering the support friction.

From Figure 16 and Figure 17, the following conclusions can be drawn: (1) The GMTS criterion can better reveal the mixed-mode fracture failure behavior of compacted clay under indoor laboratory conditions than the MTS criterion, due to the consideration of T-stress. (2) For materials with large r_c values, the support friction has a greater effect on the test results. (3) After considering the support friction, the degree of agreement between the theoretical curves based on the GMTS criterion and the experimental results is significantly improved, which indicates that the effect of support friction on mixed-mode I/II fracture of compacted clay cannot be ignored.

Above all, when the NDB specimen is adopted to conduct the mixed-mode fracture test of compacted clay, the normalized parameters (Y_I, Y_II, T*) considering the support friction should be used to deduce the fracture parameter of the compacted clay NDB specimen in the case of fixed steel support; otherwise, the calculated fracture parameters cannot reflect the real fracture properties of compacted clay.

7. Conclusions

In this paper, the normalized fracture factors (Y_I, Y_II and T*) of the NDB specimen crack were calibrated numerically by considering the crack inclination angle, crack length, support span, and support friction coefficient. Then, the mixed-mode I/II fracture test of compacted clay was performed using NDB specimens. The test results were analyzed by combining the GMTS criteria. The main conclusions are as follows:

• When the support friction is not considered, as β increases, Y_I decreases gradually, Y_II increases first and then decreases, and T* increases gradually. After considering the support friction, Y_I, Y_II, and T* all tend to decrease, but the trends of Y_I–β, Y_II–β, and T*–β curves are not affected. When the a/W or S/W is large, the stress concentration at the crack tip of the specimen is greater, resulting in a more significant effect of the support friction coefficient f on Y_I, Y_II, and T*.
• For the case where pure mode II loading can be achieved originally, increasing the friction coefficient f can effectively reduce the critical inclination angle β_II for pure mode II loading of crack initiation, which is beneficial for the crack prefabrication in the test. For some cases where pure mode II loading cannot occur originally, pure mode II loading can be achieved by increasing the friction coefficient f.
• The test data were processed and analyzed using Y_I, Y_II, and T* considering and not considering the support friction. After considering the support friction, the measured fracture parameters all decreased at different degrees, such as K_IC decreased by 51%, K_IIC decreased by 15%, and the fracture toughness ratio K_IIC/K_IC increased by 72%, indicating that the support friction increases the proportion of mode II loading at the crack tip.
• After considering the support friction, the degree of agreement between the theoretical curves based on the GMTS criterion and the experimental results is significantly improved, which indicates that the effect of support friction on the mixed-mode I/II fracture of compacted clay cannot be ignored.
• When the NDB specimen is used to conduct the mixed-mode fracture test of compacted clay, the normalized parameters (Y_I, Y_II, and T*) considering the support friction should be used to deduce the fracture parameter of the compacted clay NDB specimen in the case of fixed support.