Introduction

Carbonaceous mudstone is widely distributed in southwestern China and is often discarded as a poor roadbed fill due to its disintegration in water, causing great economic losses. Therefore, there is an urgent need to investigate the compressive strength characteristics (i.e., single-particle crushing strength and uniaxial compressive strength) of carbonaceous mudstone in water environments to provide reference information for the engineering application of carbonaceous mudstone.

During rainfall infiltration, the coarse particles in carbonaceous mudstone filler are broken into small particles, and the strength is significantly reduced to form a settlement in an embankment (Zhao et al. 2022). The above phenomenon describes the deterioration of single-particle crushing strength under water conditions. The crushing strength is an important mechanical property of rock materials. Ma et al. (2019) carried out single-particle breaking tests on individual coral particles to investigate the effect of particle size on crushing strength. Then, Kuang et al. (2021) further investigated the effect of the specimen size on crushing strength by investigating the mechanical properties of individual calcareous sands. The effect of size on the crushing strength decreases with increasing particle size. The factors influencing crushing strength include not only particle size but also particle morphology. Therefore, Qing et al. (2022) demonstrated the effect of grain shape on the crushing strength of different forms of limestone and reported that the Weibull modulus decreased with more irregular particle shape, as was also found by Meng et al. (2022). For the calculation of the particle crushing strength, Jaeger (1967) proposed the strength calculation model based on a sphere subjected to a pair of radial forces; subsequently, McDowell and Amon (2000) and Nakata et al. (1999) simplified the model proposed by Jaeger (1967) and applied the simplified model to investigate the single-particle crushing of sand.

The previous models require the highest possible sphericity of the test material, which is well-preserved by highly spherical sand particles (Wu et al. 2021). It was found that particle shape has a significant effect on crushing strength (Zhu and Zhao 2021; Wu et al. 2022a; Nie et al. 2020) with crushing strength decreasing with decreasing particle sphericity (Wang et al. 2021; Mellmann et al. 2013), as is particularly evident for particles with a much smaller short-axis size than the long-axis size and mid-axis size (Wang and Coop 2016; Zhao et al. 2015). Therefore, several studies have introduced shape parameters (i.e., Feret length, Feret width aspect ratio, and flatness) into the model to consider the effect of shape on crushing strength (Wang et al. 2022); unfortunately, obtaining these shape parameters is time-consuming, and the scientific validity of the introduction of shape parameters was not confirmed in previous work.

As a common material in engineering applications, soil–rock aggregates (SRAs) exhibit highly complex mechanical behavior (Wang et al. 2015, 2017; Wang and Li 2015; Zhu et al. 2023; Wu et al. 2022b). The mechanical behavior of SRAs is mainly controlled by the volume block proportion (VBP), the matrix strength, and the friction strength (Hu et al. 2015; Xu and Zhang 2021; Wang et al. 2020). Therefore, the strength of SRA is usually predicted based on these properties (i.e., VBP, matrix strength). Sonmez et al. (2006) derived a formula for predicting compressive strength based on matrix strength and VBP by fitting previously obtained compressive strength data. Based on this, Kalender et al. (2014) introduced parameter A to improve the model prediction accuracy and defined this parameter as the connection strength at the soil–rock interface (Zhang et al. 2020). Although these equations enable accurate predictions, they do not consider the strength of interparticle friction because the friction strength increases with increasing VBP (Hu et al. 2015).

In summary, previous studies usually tend to qualitatively analyze the effect of the particle shape on crushing strength, and few shape parameters are introduced into the crushing strength model; this approach fails to quantify the effect of the particle shape on crushing strength. Therefore, based on the existing model, this study introduces the shape factor to quantify the effect of the particle shape on crushing strength and provides a new method for calculating the crushing strength of irregular particles. The prediction of the uniaxial compressive strength (UCS) of SRA currently is based mainly on the matrix strength and fails to consider the contribution of the friction strength. Therefore, this study introduces the friction strength into the model and achieves improved model accuracy. Moreover, based on the relationship between the friction strength and crushing strength, a model of compressive strength based on the particle crushing strength was established to provide a new method for calculating the compressive strength of SRA.

Single-particle breakage tests under different water contents

Water content test on the grain fraction

Water content tests were carried out on the carbonaceous mudstones to determine a reasonable water content gradient. Based on previous research, improved test results can be obtained with grain sizes in the range of 2–20 mm (Oskooei et al. 2021; Xiao et al. 2020; Afifipour and Moarefvand 2014), and 5 mm is often used as the limit of soil and rock grain size in engineering (Medley and Lindquist 1995; Xu et al. 2007). Therefore, the disintegrated carbonaceous mudstone was divided into four different size ranges using a vibrating screen: 0–5 mm, 5–10 mm, 10–15 mm, and 15–20 mm. Grains greater than 5 mm were selected as the cutting ring sample, and grains less than 2 mm in size were incorporated as the cementing material. Figure 1 shows the process and conditions for the water content test of the grain groups, and the results of the test are plotted in Fig. 2.

Fig. 1
figure 1

Test procedure and conditions for determining the water content of the grain groups

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Fig. 2
figure 2

Saturated water content curves for different coarse grain contents

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Figure 2 shows that the saturated water content ωsat decreases with increasing grain dimension at the same level of coarse grain content mc, and the nonlinear relationship between the coarse grain content and saturated water content is obtained by fitting the test results. The parameter ϖ is the water content of the grain group, and t is the water absorption coefficient of the rock. Considering the ϖ different groups of grains, the water content of the single-particle breaking test was set to 6%, 7%, 8%, 9%, and 10% because the initial water content of 6% was convenient. Moreover, the dry state was also adopted to reveal the degradation effect of water on the crushing strength.

Single-particle breakage test

To provide accurate statistics on the particle crushing strength (Lim et al. 2004; Lu et al. 2002; Manso et al. 2018), 30 specimens were tested for each test condition in the laboratory. After drying in the oven, dry carbonaceous mudstone was obtained, and specimens were prepared by sealing the mixture of dry carbonaceous mudstone and water for 24 h.

Figure 3 shows four typical force–displacement curves of particle breaking. Here, Ff is defined as the peak corresponding to the displacement (N) of 0.4–0.6 mm, and Ft is defined as the maximum value of the peak (N). Figure 3 shows that the particles undergo two stages during loading: edge fracture and volumetric fracture. The edge fracture corresponds to the first peak point Ff, which is attributed to the concentration of stress at the edges of the particle, leading to the preferential occurrence of edge fracture. As the loading process continued, volume fracture was observed after exceeding the strength threshold (the second peak point Ft); this process was also observed in previous studies (Ishikawa and Miura 2011; Wang and Coop 2016; Zhao et al. 2015; Meng et al. 2019).

Fig. 3
figure 3

Typical force–displacement curves of the carbonaceous mudstone with 8% water content (Note: a, b, c, and d represent the force–displacement curves for particle sizes of 5–10, 10–15, and 15–20 mm, respectively)

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Regardless of the dimension, the force did not disappear after block fracture, and the particle remains compressible. As the test progresses, the curve passes through the lowest point and shows an upward trend, which is attributed to the internal defects of the fractured particle, and the strength increases with increased loading.

Crushing strength model considering shape effects

To obtain an analytical solution for the crushing strength of carbonaceous mudstone, a simplification of the force state of the carbonaceous mudstone is shown in Fig. 4. The simplification of the mechanical model includes the simplification of the force state and the change in the particle shape, and the mechanical model is ultimately simplified to an equal-thickness disc under a pair of concentrated loads.

Fig. 4
figure 4

Schematic diagram of the forces on the particles between the plates where q and F are the uniform load and force in the particle, respectively; a, b, and c are the lengths of the particles along the long, medium, and short axes, respectively; and Rd is the radius of a disc of equal height to a sphere with diameters R

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The force state is expressed by Eq. (1), i.e., the uniform load q is converted into force F at point O; therefore, the moment M is zero.

$$\text{F} = \sum_{{}_{i}}^{n}q({A}_{i})d{A}_{i},\text{\hspace{0.17em}}M=\sum_{i}^{n}\underset{{A}_{i}}{\int }q\left({A}_{i}\right)d{A}_{i}\cdot {d}_{i}\approx 0$$
(1)

where q(Ai) is the distributed load applied to area Ai (N/mm2), Ai is the unit area of the particle (mm2), and di represents the vertical distance of the load from the origin of the coordinates (mm).

The volume equivalence method is used for transforming the particle shape, which is reduced to a sphere of equivalent radius R, i.e., the particle volume VP (mm3) is equivalent to the sphere volume Vs (mm3).

$$R=\sqrt[3]{\frac{3{V}_{p}}{4\pi }}$$
(2)

Upon further transforming the shape to a disc of height 2R, the product of the scale factor n and the height of the disc 2R are the radius Rd of the disc.

$${R}_{d}=n\times 2R$$
(3)

Assuming that the volume of the particle and the disc of the same height are equal, the scale factor n in Eq. (3) can be calculated using the ratio Red/c.

$${V}_{P}={V}_{ed}=\pi {R}_{ed}^{2}\times c\Rightarrow n=\frac{{R}_{ed}}{c}=\sqrt{\frac{{V}_{p}}{\pi {c}^{3}}}$$
(4)

where Ved is the disc volume at the same height as a particle (mm3) and Red represents the disc radius at the same height as the particle (mm).

Based on the theoretical solution of an infinitely elastic plate (Wijk 1978), the model stress equation was obtained by using the stress boundary conditions and the principle of superposition as follows:

$${\sigma }_{f}={\sigma }_{{\text{inf}}}+{\sigma }_{ps}$$
(5)

where σf is the model stress (MPa), σinf is the stress solution for an infinite elastic plate (MPa), and σps is the stress special solution considering the stress boundary condition (MPa).

According to the Boussinesq solution, the maximum stress occurs under force. Therefore, the stress state under flat compression can be calculated according to Dougall (1906).

$$\begin{array}{l}{\sigma }_{{\text{inf}},x}\left(x=0,y=0,z\right)=\frac{F}{2\pi {R}^{2}}{\int }_{0}^{\infty }\frac{\varsigma d\varsigma }{2\varsigma +{\text{sinh}}2\varsigma }\left[\left(1+\frac{2}{\chi }\right){\text{cosh}}\left(\varsigma \frac{z}{R}\right){\text{sinh}}\left(\varsigma \right)\right.\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left.+\varsigma \left(\frac{z}{R}{\text{sinh}}\left(\varsigma \right){\text{sinh}}\left(\varsigma \frac{z}{R}\right)-{\text{cosh}}\left(\varsigma \right){\text{cosh}}\left(\varsigma \frac{z}{R}\right)\right)\right]\end{array}$$
(6)

where χ is Poisson’s number and ϛ is the circular area of the force.

For z = 0 and 3 ≤ χ ≤ 7, Eq. (6) can be simplified to Eq. (7), where the parameter of the infinite elastic plate λinf is 0.61–0.96 (Dougall 1906).

$${\sigma }_{{\text{inf}}}={\lambda }_{{\text{inf}}}\frac{F}{2\pi {R}^{2}}$$
(7)

For the stress solution of a finite elastic plate, the solution must be found from the stress generalization in terms of the boundary conditions. The generalized solution for the stresses in an elastic plate subjected to a pair of radial forces is given by (Dougall 1906):

$${\sigma }_{x}\left(x,y,z\right)=2\mu \left[\frac{{\partial }^{2}\theta }{\partial {x}^{2}}+\frac{\left(\lambda +3\mu \right)}{\lambda +\mu }\frac{{\partial }^{2}\varphi }{\partial {x}^{2}}+2z\frac{{\partial }^{3}\varphi }{\partial z\partial {x}^{2}}+\frac{\lambda }{\mu }\left(1-\frac{\left(\lambda +3\mu \right)}{\lambda +\mu }\right)\frac{{\partial }^{2}\varphi }{\partial {z}^{2}}\right]$$
(8)

where σx is the stress in the x-direction of the elastic plate (MPa), λ and µ are Lame’s constants, and θ and φ are the orientation parameters.

The potential functions of the parameters θ and φ are presented in Eqs. (9) and (10), respectively (Dougall 1906).

$$\theta =\frac{F\left(\chi +1\right)}{2\pi \chi E}\left[\sum_{\varsigma }\frac{{\text{sinh}}\left(\varsigma \right)+2\varsigma {\text{cosh}}\left(\varsigma \right)}{\varsigma \left({\text{cosh}}\left(2\varsigma \right)+1\right)}\times {G}_{0}\left(\varsigma \frac{\rho }{R}\right){\text{cosh}}\left(\varsigma \frac{\rho }{R}\right)-\frac{3}{4}{\text{ln}}\left(\frac{\rho }{2R}\right)\right]$$
(9)
$$\phi =\frac{F\left(\chi +1\right)}{2\pi \chi E}\left[\sum_{\varsigma }\frac{2\varsigma {\text{cosh}}\left(\varsigma \right)}{\varsigma \left({\text{cosh}}\left(2\varsigma \right)+1\right)}\times {G}_{0}\left(\varsigma \frac{\rho }{R}\right){\text{cosh}}\left(\varsigma \frac{\rho }{R}\right)-\frac{1}{2}{\text{ln}}\left(\frac{\rho }{2R}\right)\right]-\theta$$
(10)

where E is Young’s modulus, ρ is a coordinate variable under cylindrical coordinates, and G0 is the type III specific form of the Bessel function.

By combining Eqs. (8), (9), and (10) and considering the stress boundary conditions, the finite elastic plate stress solution can be obtained; moreover, the derivation of the process equation can be found in the study of Dougall (1906):

$${\sigma }_{ps}=\frac{F}{2\pi \chi {R}_{d}^{2}}$$
(11)

Substituting Eqs. (7) and (11) into Eq. (5), the solution σf to the finite disc stress subjected to a pair of concentrated loads can be derived as follows:

$${\sigma }_{f}=\frac{F}{2\pi }\left(\frac{{\lambda }_{{\text{inf}}}}{{R}^{2}}+\frac{1}{{R}_{d}^{2}\chi }\right)$$
(12)

For a sphere of radius R, Hiramatsu and Oka (1966) obtained the sphere crushing strength σs using Eq. (13). For 3 ≤ χ ≤ 7, the sphere parameter λs is 1.21–1.37.

$${\sigma }_{s}={\lambda }_{s}\frac{F}{2\pi {R}^{2}}$$
(13)

where σs is the crushing strength of the spherical particle (MPa).

By comparing Eq. (7) with Eq. (13), we can determine that the σinf/σs ratio is equal to the λinf/λs ratio, where the stress multiple λinf/λs is defined as the shape factor α

$$\alpha =\frac{{\lambda }_{{\text{inf}}}}{{\lambda }_{s}},\text{\hspace{0.33em}}0.44={\alpha }_{{\text{min}}}\le \alpha \le {\alpha }_{{\text{max}}}=0.79$$
(14)

where αmin is the lower limit of the shape factor and αmax is the upper limit of the shape factor.

Substituting Eqs. (2), (3), (4), and (14) into Eq. (12) gives the following equation. According to the literature (Hiramatsu and Oka 1966), the values of λs and χ in Eq. (15) are 1.37 and 7.0, respectively.

$$\sigma_f=\frac F{\left(2R\right)^2}\left(\alpha\lambda_s\frac2\pi+\frac1{2\pi\chi n^2}\right)=\frac F{\left(\frac{6V_P}\pi\right)^\frac23}\left(\alpha\lambda_s\frac2\pi+\frac{\pi c^3}{2\chi V_p}\right)$$
(15)

Moreover, Eq. (15) can be further simplified for αmax < α ≤ 1.0 as follows:

$$\sigma_f=\frac F{\left(2R\right)^2}\alpha\lambda_s\frac2\pi=\frac F{\left(\frac{6V_P}\pi\right)^\frac23}\alpha\lambda_s\frac2\pi$$
(16)

To determine the grain shape factor α, the particle sphericity classification method proposed by Zingg (1935) was adopted.

$$\alpha =\frac{EI+FI}{2}\text{\hspace{0.33em}\hspace{0.33em}}or\text{\hspace{0.33em}\hspace{0.33em}}\alpha =\sqrt{EI\times FI}$$
(17)

where EI is the elongation index and FI is the flatness index. FI and EI are calculated according to Eq. (18) (Zingg 1935).

$${\text{FI}}=\frac{\text{c}}{b},\text{\hspace{0.33em}}EI=\frac{b}{a}$$
(18)

Ten sets of characteristic size data are selected to classify the shape properties of carbonaceous mudstone under different test conditions, as shown in Fig. 5. It is found that flat and spherical are the main shapes of the carbonaceous mudstone particles.

Fig. 5
figure 5

Classification of carbonaceous mudstone shape properties

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Based on Eq. (18), the data in Fig. 6 indicate that Eqs. (15) and (16) can be used to calculate the crushing strength of carbonaceous mudstone particles with sizes ranging from 5 to 10 mm. For carbonaceous mudstones with lengths of 10–15 mm and 15–20 mm, Eq. (15) can be used to calculate the crushing strength.

Fig. 6
figure 6

Percentage of shape factors α at different intervals

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To indirectly demonstrate that Eqs. (15) and (16) are reasonable, Eq. (19) was used as the calibration criterion for the crushing strength (ASTM 2001), and the results obtained using Eq. (19) were compared to the results obtained using Eqs. (15) and (16) and to the results obtained using the equation reported by McDowell and Amon (2000).

$$\sigma =\frac{F}{{D}_{e}^{2}}$$
(19)

where De is the equivalent diameter of the particle (mm) (De2 = 4bd/π).

Figure 7 shows the crushing strengths obtained from the different models. The particle crushing strengths calculated by Eqs. (15) and (16) are close to the values calculated by Eq. (19), and the values calculated by McDowell and Amon (2000) and by Eq. (19) are significantly different. Therefore, Eqs. (15) and (16) can be used to estimate the single-particle crushing strength.

Fig. 7
figure 7

Comparison of the particle crushing strengths of the carbonaceous mudstone specimens

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Weibull statistical analysis

Since the Weibull statistical distribution is an effective tool for studying particle breaking, an analysis of the survivability of carbonaceous mudstone under different test conditions based on the crushing strength σ was conducted to obtain the Weibull modulus m, and the probability of survival Ps at the characteristic crushing strength σc (MPa) was calculated by Eq. (20) (McDowell and Amon 2000).

$${P}_{s}\left(\sigma \ge {\sigma }_{c}\right)=\frac{{n}_{0}}{{N}_{0}}$$
(20)

where n0 is the number of broken particles at σ ≥ σc and N0 is the total number of particles under the same test conditions.

Figure 8 shows the survival probability curves of the particles with the sizes of 5–10, 10–15, and 15–20 mm under different water contents.

Fig. 8
figure 8

Probability of survival curves for different sizes and water contents (Note: a, b, c, d, e, and f show the survival probability curves for different grain sizes at 0%, 6%, 7%, 8%, 9%, and 10% water content, respectively)

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The curves in Fig. 8 show a negative correlation between the probability of survival and grain size because large carbonaceous mudstone has a wider distribution of internal defects and breaks at the same crushing strength. Additionally, the distance between curves increased continuously for the water content greater than 8%, which was most likely related to the high water content.

According to the Weibull distribution theory, the Weibull modulus m is calculated by (Weibull 1951; McDowell and Amon 2000):

$$P\left(V\right)=1-F\left(V\right)={e}^{-{\left(\frac{\sigma }{{\sigma }_{37\%}}\right)}^{m}}\Rightarrow {\text{ln}}\left({\text{ln}}\left(\frac{1}{{{P}}\left(V\right)}\right)\right)=m{\text{ln}}\left(\frac{\sigma }{{\sigma }_{37\%}}\right)$$
(21)

where P(V) is the survival probability function, F(V) is the failure probability function, and σ37% is the characteristic crushing strength for the survival probability of 37% (MPa).

Figure 9 shows the experimental data points and fitted lines for different test conditions. The experimental data generally satisfied the Weibull statistics, and the Weibull modulus m was obtained by fitting the experimental values with Eq. (21). The Weibull modulus m of carbonaceous mudstone ranges from 1.31 to 3.03, with a higher value indicating that the crushing strength can converge to a constant value. For a given grain size range, the Weibull modulus m exhibited a negative correlation with the water content and changed significantly from 8 to 9% water content. This suggests that the increase in the water content does not lead to a monotonic decrease in the Weibull modulus m so that the Weibull modulus m may increase when the water content reaches a characteristic value, which is the optimal water content of the carbonaceous mudstone. Moreover, there is no uniform relationship between the Weibull modulus m and the grain scale. In this study, two relationships between the Weibull modulus m and the grain scale are found: in the first case, the Weibull modulus m first increases and then decreases with increasing grain scale, and in the second case, the Weibull modulus m decreases with increasing grain scale, occurring mainly at water contents ranging from 8 to 9%. Furthermore, the linear fit of the Weibull distribution for the different test conditions appeared to deviate from the Weibull distribution at the first and last ends (McDowell and Amon 2000), which was most significant at 8% and 10% water contents, indicating the effect of increasing water content on the linear fit of the Weibull distribution.

Fig. 9
figure 9

Characteristic crushing strength and Weibull modulus of particles with different particle sizes and different water contents (Note: a, b, c, d, e, and f represent the Weibull linear fitting for water contents of 0%, 6%, 7%, 8%, 9%, and 10%, respectively)

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To compare the differences between the Weibull modulus m and the characteristic crushing strength σ37% under different test conditions, the numerical calculation results presented in Fig. 9 are summarized in Table 1. Table 1 shows that the characteristic crushing strength of the particles is negatively correlated with the water content for the same size, and the crushing strength linearly decreases with increasing water content (Fig. 10).

Table 1 Characteristic crushing strength and Weibull modulus of carbonaceous mudstone samples with different grain sizes
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Fig. 10
figure 10

Relationships between characteristic crushing strength and water content for different grain sizes

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The characteristic crushing strength decreases by 7.40% in the dry state and decreases by 41.02% when the water content increases to 10%, indicating that the size effect becomes increasingly significant with increasing water content. For the Weibull modulus, there is no evident relationship between the particle size and water content. Moreover, the ratio of strength η was used to characterize the water-weakening effect to study the influence of water on crushing strength (Manso et al. 2021). According to the data presented in Table 1, the ratio η decreases with increasing water content from 6 to 10% and is negatively correlated with the particle size at the same water content.

Using the calculation of the average crushing strength σavc (MPa) by McDowell and Amon (2000) and combining with the fitting function in Fig. 10, a relationship between the water content and the average crushing strength is obtained.

$${\sigma }_{\text{avc}}=(\kappa \omega +{b}_{0})\Gamma \left(1+\frac{1}{m}\right)$$
(22)

where κ and b0 are the fitting parameters.

Taking the value of Г(1 + 1/m) as 0.9, Eq. (22) is further simplified as follows:

$${\sigma }_{\text{avc}}=0.9\left(\kappa \omega +{b}_{0}\right)$$
(23)

To verify the validity of Eq. (23), the average crushing strength σavc calculated by Eq. (23) was compared with the measured average crushing strength σav (MPa), with the results presented in Fig. 11. The measured average crushing strength presented a nonlinear decrease with increasing water content, and the decrease in the measured average crushing strength reached the maximum value at 6–8% water content and then slowed, indicating that the effect of water content on the average crushing strength has characteristic value and that its influence will be weakened when this characteristic value is reached. Moreover, the predicted average crushing strength has a linear relationship with the water content, which is attributed to the linear character of Eq. (23). The measured and predicted values show strong linear correlation, with a goodness of fit R2 greater than 0.95. The maximum difference between the predicted and measured values is 13.04%, which is lower than that in the previous research ranging from 14 to 20% (McDowell and Amon 2000; Meng et al. 2019); thus, Eq. (23) is reasonable for calculating the average crushing strength of particles under different water contents.

Fig. 11
figure 11

Comparison of the predicted and measured average crushing strengths of carbonaceous mudstone (Note: a, b, and c represent the average crushing strength under different water contents for grain sizes of 5–10, 10–15, and 15–20 mm, respectively. The error values are marked in light yellow and light red with a 10% error threshold, and the same applies for the following figures)

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Uniaxial compression strength prediction model considering the particle crushing strength

The existing model for predicting the UCS mainly depends on the matrix cementation strength of the UCS (Kalender et al. 2014), as shown in Eq. (24).

$${\sigma }_{SRA}=\frac{A-{A}^\frac{VBP}{100}}{A-1}{\sigma }_{matrix}$$
(24)

where σSRA is the UCS of the SRA (MPa), σmatrix indicates the bonding strength of the matrix (MPa), and A is a model parameter and ranges from 0.1 to 500. Indeed, A can be any positive number not equal to 1.0, and the range of A is extended to be greater than 0 in this paper.

Equation (24) shows the nonlinear relationship between the UCS and the VBP; however, the exponential function does not fit the data well. Based on the observed sinusoidal variation characteristics of the data, a sine function is introduced:

$${\sigma }_{SRA}={\text{sin}}\left(\frac{\pi }{2}\times \frac{A-{A}^\frac{VBP}{100}}{A-1}\right){\sigma }_{matrix}$$
(25)

Figure 12 shows the nonlinear fitting of the measured data and the values calculated using the models. Both models satisfy the boundary conditions well. For Eq. (24), the difference between the calculated and measured values is large, several fitting curves do not pass through many data and approximately overlap, and the nonlinear fitting is poor. For Eq. (25), the nonlinear fit passes through the data points, and the fitting curves reflect the trend of the data very well with no overlap, which is a satisfactory result. To quantify the level of fit, the R2 goodness-of-fit statistic was used as the standard of evaluation. The R2 of Eq. (25) is much closer to 1.0, indicating that the predicted UCS of Eq. (25) is more consistent with the measured values. The parameter A in Eq. (25) is reduced by a factor of 4–9 and ranges from 0.1 to 1.21, which is attributed to the sinusoidal function. However, Eqs. (24) and (25) should be modified to reflect the effect of frictional strength on the UCS of the SRA during compression.

Fig. 12
figure 12

Comparison of calculated and measured values of UCS for different prediction models (Note: a and b represent the models proposed by Kalender et al. (2014) and the model presented in this study, respectively)

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As discussed above, Eq. (25) can be improved by considering the compressive characteristics of the SRA and the boundary conditions of the prediction model of the UCS.

$${\sigma }_{SRA}={\text{sin}}\left(\frac{\pi }{2}\times \frac{A-{A}^\frac{VBP}{100}}{A-1}\right){\sigma }_{matrix}+{\text{sin}}\left(\frac{\pi }{2}\times \frac{A-{A}^{1-\frac{VBP}{100}}}{A-1}\right){\sigma }_{fric}$$
(26)

where σfric is the frictional strength of the SRA (MPa).

The frictional strength σfric, between particles is related to the normal stress σn (MPa). Since the normal stress must be less than or equal to the crushing strength of the particle, the relationship between the normal stress σn and the average crushing strength of the particle σavc is given by

$${\sigma }_{fric}\propto {\sigma }_{n}\propto {\sigma }_{avc}^{{k}_{0}},0<{k}_{0}<1.0$$
(27)

where k0 is a parameter less than 1.0.

By combining Eqs. (26) and (27), the relationship between the average crushing strength σavc and the UCS of the SRA σSRA can be established as follows:

$$\sigma {}_{SRA}={\text{sin}}\left(\frac{\pi }{2}\times \frac{A-{A}^\frac{VBP}{100}}{A-1}\right){\sigma }_{matrix}+{\text{sin}}\left(\frac{\pi }{2}\times \frac{A-{A}^{1-\frac{VBP}{100}}}{A-1}\right){\left(\sum_{i=1}^{N}{\sigma }_{avc,i}{\left(\frac{VBP}{100}\right)}_{i}\right)}^\frac{1}{N}$$
(28)

where N is the number of grain groups and is equal to 3.0 and σavc,i is the average crushing strength corresponding to different grain groups (MPa).

Unconfined compression tests (UCTs) were performed on graded carbonaceous mudstones with different water contents, and a mass block proportion (MBP) was calculated for coarse particles (particle size greater than 5 mm) to determine the effectiveness of Eq. (28). Figure 13a shows the particle size distribution curves for different mass block proportions (MBPs). The interrupted area in the figure is due to the failure to consider the content of 2–5 mm grain groups, and this content was assigned proportionally to grain groups larger than 5 mm (such as 5–10 mm, 10–15 mm, and 15–20 mm). The dimensions of the remolded cylinder sample, which consisted of carbonaceous mudstone with a size less than 2 mm, were 100 mm × 200 mm. To avoid the effect of specimen inhomogeneity on the test results, two reference specimens were tested under each condition.

Fig. 13
figure 13

Particle size distribution curves and stress–strain curves of the SRA samples (Note: a is the grain size grading curve with different MBPs. b, c, d, e, and f are the stress–strain curves for different MBPs)

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Figure 13b–f shows the stress–strain curves of graded carbonaceous mudstones with different MBPs at 8% water content, and the variation trends of the curves under other conditions are approximately similar. According these figures, a higher MBP leads to a lower fracture strength of the specimen at the same level of breaking strain, and the peak strengths are reduced by 2.19%, 28.42%, 63.91%, and 69.41%. Figure 13a shows that with increasing MBPs, the particle size distribution curve shifts to lower size values, i.e., the fine grain content decreases, and the strength of the “soil–rock” interface decreases. Analysis of the shape of the stress–strain curve shows that it is divided into three phases, with each corresponding to a specific mechanical state. For an MBP of 0, the range of curve phase I is significantly larger than that of phase II, indicating that the specimen has a greater compressive capability and a faster process that corresponds to brittle damage. For MBP of 20–80%, phase II range is larger, showing that the damage mode has changed from brittle damage to plastic expansion, and this change is attributed to the increase of coarse particles that intensifies the interparticle dislocation and to the weakening of the overall structural properties of the specimen.

Figure 14a–e shows the mean and relative standard deviation (RSD) of the UCS under test conditions. The number next to the line reflects the increase in the compressive strength, and the VBP in the figure can be calculated from the relationship between the MBP and the density of the sample. At a given same water content, the peak strength decreases with increasing VBP, corresponding to a negative growth rate. For the same VBP conditions, the peak strength is negatively correlated with the water content. For a VBP ranging from 30 to 50%, the compressive strength increases the least for the water content ranging from 6 to 9%, indicating that the mechanical properties of the materials change strongly within a certain range and that the effect of the coarse grain content on the mechanical properties begins to increase. The relative standard deviation (RSD) also shows that the compressive strength error is small at low VBP. For VBP greater than 40%, the compressive strength error increases, and the maximum RSD corresponds to the maximum VBP. This difference may be due to the increase in coarse particles leading to more diverse contact patterns between particles, which gives rise to a significant difference in the UCS values of the specimens.

Fig. 14
figure 14

Peak strengths and RSDs under different test conditions (Note: a, b, c, d, and e show the peak strengths and RSDs at different VBPs for water contents of 6%, 7%, 8%, 9%, and 10%, respectively)

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In Fig. 15a–e, the errors of the predictions obtained using Eq. (28) relative to the measured values are small, the predicted values and measured values exhibit good linear correlation, and the goodness of R2 is close to 1.0. The error bars are likely to be correlated with the VBP, because a larger VBP generally corresponds to a larger error. This is attributed to the increase in the factor influencing the compressive strength, so that the compressive strength determined solely by the VBP calculation is not adequate, leading to an increase in the difference between the calculated and test results. Although the maximum error is 56.92% in Fig. 15b, for most stages, the relative error of the predicted values falls within 0 to 30%, indicating that the overall accuracy in predicting the UCS of SRA by using the single-particle crushing strength is high. Moreover, parameter A in Eq. (28) is less than 0.1 and is negatively correlated with the water content, which is related to the decrease of the UCS by water.

Fig. 15
figure 15

Measured and predicted uniaxial compressive strengths under different test conditions (Note: a, b, c, d, and e represent uniaxial compressive strengths at different VBPs for 6%, 7%, 8%, 9%, and 10% water content, respectively)

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To reveal the contribution of the matrix strength σmatrix and single-particle average crushing strength σavc (or friction strength σfric) to the UCS, the relationship between the sine function of Eq. (26) (or Eq. (28)) and VBP was analyzed and is shown in Fig. 16. The different strength factors show the opposite trend: whereas one rises, the other one decreases. On the left of the demarcation line, the strength factor for the matrix strength is greater than that for the single-particle crushing strength, and the opposite is true to the right of the demarcation line. The VBP on the demarcation line is 50%, and the two strength factor curves intersect on the demarcation line, indicating that the different types of strength (as matrix strength and friction strength) theoretically contribute equally to the UCS of SRA when the VBP is 50%. However, the true assignment of strength is very complex and is related to the dimension and arrangement of the large particles within the specimen.

Fig. 16
figure 16

Strength factor curves under different test conditions

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Conclusion

This study aimed to investigate the effect of water content on the strength of soft rock. Single-particle crushing tests and uniaxial compression tests were carried out on carbonaceous mudstone under different water content conditions (i.e., 6%, 7%, 8%, 9%, and 10%). By introducing the shape factor, a model of single-particle crushing strength was developed considering the effect of particle shape, which provides a new method for estimating the crushing strength of nonspherical particles. Furthermore, a model was established to predict the compressive strength of the mixture based on the single-particle crushing strength, and the model was demonstrated to achieve better performance in predicting the UCS of SRA.

  • 1.The calculated crushing strengths considering the particle shape are consistent with the reference values, indicating that the model reflects the effect of the particle shape on the crushing strength. Notably, the shape factor in the model can be defined in more complicated manner (including sphericity and concavity) to improve the model accuracy.

  • 2.Increasing the water content of a material decreases its Weibull modulus, increasing the discrete strength. Moreover, the average crushing strength model based on the Weibull distribution fits the measured values well, reflecting the effect of water content on crushing strength.

  • 3.The model of UCS based on the single-particle crushing strength fits the measured values well. In addition, this model can further derive the contribution of the particle crushing strength to the UCS of the aggregate, and the variation in this contribution can be expressed by a sinusoidal function.