The test soil was collected from a construction site in Chongqing. The maximum soil particle diameter was 0.075 mm, and Figure 1 showed the grading curve of soil. The soil particle specific gravity Gs was 2.72, the plasticity index was 20, the liquid limit was 50%, and the plasticity limit was 30%. Through compaction tests, the optimum moisture content of the clay was determined to be 17.58%, and the maximum dry density was 1.72 g cm^-3. The cement used was the P.O 42.5 ordinary Portland cement. Its insoluble content is 1.30%, the firing loss is 4.2%, the magnesium oxide content is 3.1%, the sulfur trioxide content is 1.8%, the specific surface area is greater than 300 square meters/kg, and the fineness of 80 μm square hole sieve is 8.7%.

The “dry specimen preparation' method was adopted, as shown in Figure 2 with the following specific preparation steps: 1) The qualities of soil, cement and water were calculated based on the sample size, moisture content and dry density. 2) After adding water to the clay powder and stirring, the mixed clay was sealed in a bag and allowed 24 h for full moisture diffusion before mixing with the cement powder to obtain cement soil. 3) The steel mould was installed, with a layer of petrolatum and a layer of cling film applied on its inner wall, and the cement soil was compacted layer by layer (Figure 2A). 4) The specimen was slowly pushed out of the mould using an ingot. 5) The demoulded specimen was wrapped in cling film and placed in a shade for curing (Figure 2B). 6) The inclination angle and length of the precast crack were marked on the cured specimen, and a crack with a width of 1 mm was formed with a cutting machine (Figure 2C).

To study the fracture failure mechanism of cement soil, different cement proportions and curing ages were considered in this experimental research. Specifically, the cement proportions were 5%, 10%, 15%, 20%, and 25%, and the curing ages were 1, 3, 5, and 7 days.

Table 1 shows the specific test schemes. Using the control variable method, Scheme 1 was designed to consider the effect of curing age, and Scheme 2 was designed to consider the effect of cement proportion. The crack angles of 0°, 10°, 20°, 30°, 40°, and 50° were selected, and the calculations were conducted based on the average peak load of 3 specimens. A total of 132 specimens were used.

The tests were performed using chevron-notched SCB (CNSCB) specimens, and the loads were applied through three-point bending (Figure 3). According to the ISRM recommendation (Kuruppu et al., 2014), the length-span ratio S/2R was 0.51, and the relative length of the crack a/R was 0.4. The specimens in this study had a radius R of 75 mm and a thickness B of 50 mm. The loading rate was 0.6 mm/min.

Using specimens with a cement proportion of 15% and a curing age of 3 days as an example, the typical cement soil failure modes in the mixed mode I-II fracture tests are shown in Figure 5. As can be observed, the propagations begin at the tips of the initial cracks. The specimen with the initial crack inclination angle α of 0° shows a mode I fracture, and the propagation is along the initial crack direction. Specimens with α = 10°–40° show mixed mode I-II fractures. The specimen with α = 50° shows a mode II fracture. The crack propagation deviates from the direction of the initial crack. The greater the inclination angle of the initial crack, the more significant the deviation of the propagation direction. According to the sketch, the crack growth is not along a uniform, straight line but a jagged-like line. This is due to the inhomogeneity of the manually prepared specimens, manifested as many particle granules of varying strength that are bypassed by the propagating cracks. Meanwhile, the original propagation path is restored under the action of stress, and this back-and-forth process leads to jagged cracks.

Figure 6A shows the variation patterns of the crack initiation angle, i.e., the angle between the initial crack and the extended crack, under different cement proportions. With the cement proportion being 5%–25%, the value of θ ₀ varies between 23.5° and 20.5° at the α of 10°. The value of θ ₀ varies between 42° and 39° at the α of 20°. The value of θ ₀ varies between 56.75° and 55° at the α of 30°. The value of θ ₀ varies between 71° and 67° at the α of 40°; the value of θ ₀ varies between 82.75° and 80.25° at the α of 50°. Overall, the variations of θ ₀ are within 4°, not exceeding the margin of error. Thus, it can be considered that the cement proportion has basically no effect on θ ₀. Therefore, the average values of θ ₀ were selected for the subsequent analysis, namely, 0°, 22.15°, 40.55°, 55.90°, 69.55°, and 81.05°, respectively.

Figure 6B shows the variation patterns of crack initiation angle under different curing ages. With the curing age being 1 day–7 days, the value of θ ₀ varies between 22.5° and 21.83° at the α of 10°; the value of θ ₀ varies between 41.25° and 35.75° at the α of 20°; the value of θ ₀ varies between 57.5° and 54.5° at the α of 30°; the value of θ ₀ varies between 69.83° and 65.75° at the α of 40°; the value of θ ₀ varies between 80.5° and 79.5° at the α of 50°. Overall, the variations of θ ₀ are within 5.5°, not exceeding the margin of error. Therefore, the curing age has basically no effect on θ ₀. Under different curing ages, the average values of θ ₀ are 0°, 22.19°, 38.88°, 55.96°, 69.83°, and 80.33°.

In summary, the effects of cement proportion and curing age. The mathematical relationship between its average values and M _e is modeled as Eq. 4, and the curve is plotted as shown in Figure 7. θ 0 = − 82.31 M e 2 + 4.9383 M e + 79.445 (4)

Figure 8 shows the relationship between the critical stress intensity factors K _If and K _IIf with the initial crack inclination angle α. With the gradual increase of the initial crack inclination angle, the critical stress intensity factor K _If gradually decreases while K _IIf gradually increases, and K _If and K _IIf also increase with the increase of cement proportion (Figure 8A) and curing age (Figure 8B). It can be observed that K _IIf = 0 corresponds to mode I fracture, at which time K _If is the fracture toughness K _IC of mode I fracture; K _If = 0 corresponds to pure mode II fracture, at which time K _IIf is the fracture toughness K _IIC of pure mode II fracture.

Figure 9 shows the K _IIf and K _If test value envelopes. Under different cement proportions, K _If and K _IIf gradually increase with the increase of cement proportion, and the envelopes are more inward at lower cement proportions (Figure 9A). Under different curing ages, K _If and K _IIf exhibit the same variation patterns as described above (Figure 9B).

Figure 10 shows the K _II/K _IC and K _I/K _IC test value envelopes. It can be observed that the envelopes under different cement proportions intersect, and the K _I/K _IC and K _II/K _IC envelopes are close to basically stable within certain intervals. With K _I/K _IC = 0, the K _II/K _IC value varies between 0.61 and 0.39. Other than the obvious deviated points in the figure, the rest are between 0.39 and 0.45 (Figure 10A). The envelopes under different curing ages also intersect, and the K _I/K _IC and K _II/K _IC envelopes are close to basically stable within certain intervals. With K _I/K _IC = 0, the K _II/K _IC value varies between 0.44 and 0.40, which is within the 0.40 to 0.44 range (Figure 10B). The reason for the above phenomenon is that the soil-cement heterogeneity is more significant under the influence of factors such as material mixing degree, curing temperature and test conditions.

Figure 11 shows the variation ranges of K _II/K _IC, K _I/K _IC, and crack initiation angle upon cement soil fracture failure obtained based on the test data. Based on the envelope variation intervals in Figure 11A, the upper and lower boundary functions of the K _II/K _IC value are fitted, i.e., the K _II/K _IC variation range of the cement soil with the cement proportion of 5%–25% and the curing age of 1 day–7 days. The variation range of θ ₀ under the K _II/K _IC and K _I/K _IC mixed stress states can be obtained based on Eq. 4, as shown in Figure 11B.

Soil fracture analyses are often based on rock fracture criteria, such as the MTS criterion, G criterion, and S criterion mentioned above. Additionally, Wang et al. (Suits et al., 2006) adopted a circular fracture criterion in their analysis of Nuozhadu clay. Their equation is as follows: K I K IC 2 + K II K IC 2 = 1 (5)

As shown in Figure 12, the crack initiation angle upon pure mode II fracture is 70.53° according to the MTS criterion, which is significantly different from the θ ₀ in this study. In contrast, the K _II/K _IC obtained in this study is between 0.39 and 0.45, and those according to the MTS criterion are between 0 and 0.87, which is significantly not consistent. With the S criterion, both the crack initiation angle and the envelope are related to μ, which is set to 0.3 in this study. At this time, the crack initiation angle is 82.34°, and the K _II/K _IC value is between 0 and 0.96. According to Figure 12, the envelope is still above that of the MTS criterion. Compared with the two criteria above, the K _II/K _IC value based on the circular fracture criterion is above that of the S criterion, and the theoretical and test results are significantly different. Therefore, describing the cement soil fracture failure mechanism with classical fracture criteria has certain limitations.

Considering the unsatisfactory results of the above classical fracture criteria, further analysis is conducted with the GMTS criterion. The stress field at the crack tip is as follows: σ r = 1 2 2 π r K I ⁡ cos θ 2 3 − ⁡ cos ⁡ θ + K II ⁡ sin θ 2 3 ⁡ cos ⁡ θ − 1 + T ⁡ cos 2 ⁡ θ + O r 0.5 σ θ = 1 2 2 π r cos θ 2 K I 1 + ⁡ cos ⁡ θ − 3 K II ⁡ sin ⁡ θ + T ⁡ sin 2 ⁡ θ + O r 0.5 τ r θ = 1 2 2 π r cos θ 2 K I ⁡ sin ⁡ θ + K II 3 ⁡ cos ⁡ θ − 1 − T ⁡ sin ⁡ θ ⁡ cos ⁡ θ + O r 0.5 (6)

Compared to the MTS criterion, the T-stress is included, and the critical size r _c of the crack tip micro-fracture zone is also taken into account. According to the GMTS criterion, the crack is initiated when the maximum tangential stress is reached. Then, we have: ∂ σ θ ∂ θ = 0 → K I ⁡ sin ⁡ θ 0 + K II 3 ⁡ cos ⁡ θ 0 − 1 − 16 3 T 2 π r c sin θ 0 2 cos ⁡ θ 0 = 0 (7) K IC = 1 2 cos θ 0 2 K I 1 + ⁡ cos ⁡ θ 0 − 3 K II ⁡ sin ⁡ θ 0 + T 2 π r c sin 2 ⁡ θ 0 (8)

Equation 8 can be normalized as: K IC K I = 1 2 cos θ 0 2 1 + ⁡ cos ⁡ θ 0 − 3 Y II Y I sin ⁡ θ 0 + T * Y I 2 r c a sin 2 ⁡ θ 0 (9) K IC K II = 1 2 cos θ 0 2 Y I Y II 1 + ⁡ cos ⁡ θ 0 − 3 ⁡ sin ⁡ θ 0 + T * Y II 2 r c a sin 2 ⁡ θ 0 (10)

As shown in Figure 13, the envelope of the MTS criterion is the outermost, while the envelope with a larger r _c is more inward under the GMTS criterion. Under different cement proportions and curing ages, the test data envelopes are all far smaller than those under the MTS criterion, indicating the insufficiency of the MTS criterion in explaining the cement soil fracture mechanism. In contrast, the GMTS criterion is basically consistent with the test results. Other than the discrete points with large deviations, the test points are basically near the r _c = 1 mm envelope.

As shown in Figure 14, the test values of θ ₀ under different cement proportions and curing ages are in the ranges of theoretical curves r _c = 1 mm to r _c = 0.1 mm, indicating that the r _c of the cement soil at this time is 0.1–1 mm. Considering the theoretical curves of K _II/K _IC and K _I/K _IC in Figure 13, the test values also fall near the r _c = 1 mm curve. In summary, the K _II/K _IC and K _I/K _IC values and θ ₀ values of cement soil under the GMTS criterion are near the r _c = 1 mm theoretical curve, indicating that the GMTS criterion can better describe the cement soil fracture failure mechanism.

Through inversion based on the θ ₀ test values, the theoretical value of r _c is 0.3 mm–1.9 mm. In essence, adding cement and changing the curing age alter the brittleness of the material, and the r _c corresponding to different cement proportions and curing ages should be different. Therefore, the r _c value of cement soil should not be a fixed value but within a recommended range.