Introduction

Slate is very typical of laminated rocks in nature, and it contains different angles of joints and weak surfaces, resulting in differences in its mechanical properties. It is inevitable to encounter such unfavorable geological environments for construction safety in various projects1,2. The modulus of elasticity, as one of the essential mechanical parameters, also varies with different physical structures of rocks. Therefore, a dynamic damage constitutive model of the effect of joint inclination on elastic modulus under impact loading for laminated slate can more accurately predict the dynamic mechanical response of slate, which is crucial for assessing the stability and safety of slate and guiding construction.

Many researchers studied on the elastic modulus of rocks3,4,5,6. The modulus of elasticity of rocks is an index reflecting the difficulty of their elastic deformation, which is usually affected by different rock compositions7, pore spaces8, temperature conditions9,10,11, water content12,13, and stress conditions14,15,16. It has also been found that the characteristic jointing surfaces in layered rocks affect the elastic modulus of the rocks. Using numerical simulation, Hu et al.17,18 found a power function relationship between the elastic modulus and the number of parallel joints and a negative exponential relationship with the rock dimensions. They gave their unique mathematical models. Shu et al.19 explored the effects of inclination, intersection angle, spacing, and strength of rock joints on the rock tensile Young’s modulus. Gaur et al.20 established a progressive damage model based on damage initiation and propagation. Then, they calculated the composite’s Young’s modulus for impact on the hybrid laminate at different strain rates. It has also been shown that the effect of the inclination angle of the joints of rocks on the compressive strength and elastic modulus of rocks is the most significant21,22, and the difference in the inclination angle of the joints leads to the different elastic modulus of rocks. Therefore, it is necessary to investigate the quantitative relationship between joint inclination and elastic modulus of laminated slate under impact loading.

In addition, there are many research results on the dynamic damage constitutive model that can reflect the dynamic mechanical behavior of rocks. Zhao et al.23 considered the effect of damage on the dynamic load strength of rocks. They proposed a simplified modified overstress model constitutive relationship by applying continuous damage and statistical strength theories. Ye et al.24 and Du et al.25 considered the strain rate effect on rocks to establish the damage ontology model of rocks from different perspectives. Yu et al.26 conducted dynamic impact experiments on shale specimens experiencing different high-temperature thermal damage, obtained the evolution of shale dynamic characteristics with temperature, and constructed a dynamic damage constitutive model of shale considering thermal damage and compaction stage. Li et al.27 developed a dynamic damage constitutive model for brittle materials such as rocks, which can better reflect the dynamic mechanical behavior of rocks under high perimeter pressure. The above dynamic damage constitutive models were developed for non-laminated rocks. Fan et al.28 developed a dynamic damage constitutive model for laminated sandstones by conducting triaxial dynamic compression tests with different impact dynamic loads and the same triaxial static loads, taking into account the inclination angle of the layers, dynamic and static loads. Based on the ZWT model and damage fracture mechanics, Yang et al.29 established the damage variables of the jointed rock body considering the macroscopic and microscopic coupling. Then, they established the dynamic principal model of the jointed rock body. The above scholars established a dynamic constitutive model for laminated rocks, which can reflect the dynamic response of the rock under impact loading at the joint surface. However, they did not consider the change of the physical properties of the rock itself, i.e., the change of the elastic parameters, which is brought about by the angle of the joints.

Therefore, this paper takes slate as the research object, explores the quantitative relationship between joint inclination and elastic modulus through the results of Hopkinson compression rod impact tests on slate with different joint inclinations, uses statistical damage theory to simulate the damage evolution law of slate under impact loading, and establishes a dynamic damage constitutive model that can reflect the change of elastic modulus with the change of joint angle. Finally, the reasonableness of the model in this paper is proved by the comparative analysis of test and theoretical curves.

Dynamic mechanical properties test of slate

Test equipment and method

The tests were conducted using the Hopkinson Punch Rod ALT100 test system. It consists of a launching chamber, a punch, an incident bar, a transmitting bar, an energy absorption device, and a data acquisition system, as shown in Fig. 1. As shown in Table 1, the diameter of the bar and bullet is 50 mm, the length of the incident bar and transmissive bar is 2000 mm, and the length of the bullet is 600 mm. During the test, the impact air pressure is provided by a pressurized air pump to drive the bullet. This test was all carried out at an impact air pressure of 0.5 MPa (impact velocity of about 22.88 m/s) to avoid the effect of the magnitude of the air pressure on the slate’s elastic modulus. The bullet impacts the incident bar at a certain speed and generates an incident pulse. The stress wave reaches the rock specimen through the projectile rod and causes high-speed deformation under the action of the stress pulse, which realizes the simulation of rock impact. Here, because the length of the stress pulse is much larger than the thickness of the slate specimen, the stress wave inside the specimen can be considered equal everywhere after multiple reflections. Therefore, it is reasonable to assume that the stress and strain inside the specimen are uniformly distributed. In the whole process, the incident, transmitted, and reflected waves are amplified by the strain amplifier and then analyzed and calculated systematically by the relevant supporting software. Finally, the slate specimen’s stress, strain, and average strain rate can be obtained.

Fig. 1
Fig. 1
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Simplified schematic of Hopkinson’s lever system.

Table 1 Test parameters of the SHPB test system.
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The test rock samples were taken from a natural slate tunnel, and the test rock samples with joint inclinations θ of 0°, 30°, 45°, 60°, and 90° were obtained by sampling the rock body in situ from different angles, as shown in Fig. 2. The joint angle of inclination θ here refers to the angle between the direction of the joint and the radial direction. Cylindrical standard specimens of slate with a diameter of 50 mm and a height of 25 mm were made according to the International Society of Rock Mechanics standards. The rock samples for the same set of tests were taken from the same rock so that the experimental results would not be too discrete due to differences in the properties of the rock samples. The tested samples were divided into 5 groups according to the different angles of the joints for rock impact tests respectively, with 10 specimens in each group.

Fig. 2
Fig. 2
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Slate specimen.

Test results and analysis

The stress–strain curves of laminated slate under different joint inclinations are the macroscopic manifestation of its destructive process, which can reflect the changing trend of laminated slate in different stages. Here, a set of specimen impact loading test data with typical characteristics is selected for analysis and calculation. The stress–strain curve can be obtained as shown in Fig. 3, and the corresponding peak strength \(\sigma_{{{\text{z}}\max }}\) and maximum strain rate are shown in Table 2.

Fig. 3
Fig. 3
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Dynamic stress–strain curves at different joint inclinations.

Table 2 Impact compression properties of slate specimens at different bedding dip angles.
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As can be seen in the Fig. 3, the stress–strain curve of the slate will go through the following stages under impact loading:

(1) Initial nonlinear stage: The stress–strain curve presents an upward concave shape in the early loading stage. Due to the many pores and microcracks inside the slate, in this stage, the external force is mainly used to overcome the compression of the pores inside the rock and the closure resistance of the microcracks. At this time, the micropores, microcracks, and other defects in the slate are gradually closed under pressure. The strain increases faster while stress grows relatively slowly.

(2) Elastic stage: At the early loading stage, the stress–strain curve rises nearly straight. At this time, the slate under different joint inclinations shows elastic properties, and the defects, such as microcracks and pores inside the slate, have not yet begun to expand significantly. The deformation of rocks is mainly elastic deformation caused by the stretching and compression of interatomic bonds. With the increase of stress, the strain increases linearly accordingly, and the elastic modulus remains relatively stable.

(3) Plastic stage: When the stress reaches a certain degree, the stress–strain curve deviates from the straight line and enters the yield stage. In this stage, the microcracks inside the slate begin to expand and connect gradually, and the deformation of the slate is no longer completely elastic. The rate of increase of stress gradually slows down while the strain continues to increase. At this time, irreversible plastic deformation begins in the slate, and its ability to resist deformation gradually decreases. When the joint inclination angle is 0°, the direction of the laminated structure of the slate is perpendicular to the direction of impact, which makes the rock better able to withstand the stress and less likely to undergo sudden damage when impacted, and thus has enough time to form a plastic platform. In addition, a secondary increase in stress occurs at a joint inclination angle of 0°. This is because slate, as a natural metamorphic rock, has differences in mechanical properties between its internal layers. At this time, when the joint surface is perpendicular to the loading direction, with the gradual increase of stress, the softer slate layer will be the first to reach its bearing limit, and then damage occurs, which is shown as the first peak in the stress–strain curve. When the softer slate layer is damaged, the stress continues to be transferred and encounters the next harder layer. Since the harder layer has a greater load-bearing capacity and can withstand greater stresses, the stress–strain curve goes through a downward phase after the first peak and then rises again as the harder layer begins to carry the load. As loading continues, the stresses increase until the harder layer also reaches its damage limit, at which point the stress–strain curve reaches a second, higher peak. When the joint inclination angle is not 0°, the structural inhomogeneity of the slate increases and different parts of the rock respond differently to the stress. Under the impact, the transmission path of stress in the rock becomes complicated, and it is easy to produce stress concentration at the joints or structurally weak parts, which leads to faster destruction of the rock, so there is not enough time to form an apparent plastic platform.

(4) Destruction stage: when the stress reaches the ultimate strength of the slate, the slate enters the destruction stage. At this time, the stress–strain curve decreases rapidly, the rock is damaged, and the internal cracks expand and penetrate rapidly, resulting in the loss of bearing capacity of the slate. In this stage, the strain of the slate may increase sharply while the stress decreases sharply. As can be seen from the figure, the curve will also have apparent hysteresis when the joint inclination is 0°. This is because slate at other joint inclinations tends to break down along the joint surface or in a specific weak direction. However, at 0°, the slate jointing direction is perpendicular to the stress direction, and the damage of the slate at this time is mainly due to the development and closure of microcracks. When the stress increases, the microcracks gradually expand. When the stress decreases, some microcracks may close, thus forming a hysteresis curve. For the stress–strain curves at other joint angles, there is volatility at the later stages, which is caused by the multiple reflections of energy waves between the laminae during the impact load traversing the layers.

Table 2 lists the impact compressive strength and maximum strain rate of slate specimens at different nodal inclinations. By analyzing the peak stress data of the slate under different joint inclination angles, we found that the joint inclination angle has a significant effect on the peak stress of the slate. When the joint inclination angle is 0°, the joint surface is perpendicular to the loading direction, and at this time, the joint surface is mainly subjected to positive stresses, which enables the slate to withstand higher stresses. As the angle of joint inclination increases to 30°, 45°, and 60°, the angle between the joint surface and the loading direction changes, and the shear stress on the joint surface gradually increases during the loading process. The joint surface is the weak surface inside the slate, which is prone to relative sliding and misalignment, making the stress concentration phenomenon inside the slate intensify, and microcracks are more likely to sprout and expand near the joint surface, thus reducing the peak stress of the slate. When the angle of the joints is 90°, the joint surface is parallel to the loading direction, the weakening effect of the joints on the slate to resist the pressure is relatively small, and the slate mainly relies on the interaction between the internal mineral particles to withstand the load, so that the peak stress of the slate is higher.

In slate impact testing, the strain rate of the specimen is of great significance. The strain rate reflects the degree of deformation of the material per unit of time, and in the impact test, it reflects the degree of speed of loading of the impact load. There are two main reasons for the different strain rates under the same air pressure. As can be seen from Table 2, there are different strain rates under the same air pressure, which is mainly due to the following two reasons. On the one hand, different slate specimens have different internal mineral compositions, microstructures, and natural defects, which leads to differences in the transfer and response of the material to impact energy even under the same air pressure loading conditions. On the other hand, although the same air pressure is used, the sealing of the impact device, the stability of the gas flow, and other factors may have subtle differences, which will lead to fluctuations in the actual impact loads applied to the specimens.

Generally speaking, the higher the strain rate, the more likely the specimen will be damaged at the same gas pressure. A high strain rate means that the impact load is applied to the specimen in a very short period, and the stresses within the slate do not have time to be evenly distributed, resulting in a stress concentration. This stress concentration tends to trigger the rapid initiation and expansion of microcracks in the weak areas of the material.

We recorded the impact damage process of the slate with a high-speed camera. Figure 4 shows the impact damage process of slate specimens at different joint angles. At a joint inclination angle of 0°, the joint surface is perpendicular to the loading direction, and the damage begins with the formation of through cracks with the loading direction. Due to the existence of the joint surface, the cracks are hindered to a certain extent in the process of expansion, which makes the crack expansion relatively stable, but will eventually form through cracks leading to the destruction of the slate. At a joint inclination angle of 30°, the stress in the loading direction causes tensile and shear deformation of the slate, leading to crack sprouting along the loading direction. At the same time, there is an angle between the joint surface and the loading direction, and the joint surface is subjected to shear stresses during the loading process, and cracks are also generated in the direction of the joint surface. At a nodal angle of 45°, cracks are mainly generated along the nodal surface and expand rapidly. At this joint angle, the shear stress on the joint face reaches a large value, which causes the slate to be damaged in a short time due to the expansion of cracks on the joint face. At a joint inclination angle of 60°, cracks are generated mainly along the joint faces. Similar to the 45° angle, the cracks continue to expand on the joint faces, gradually leading to the loss of the bearing capacity of the slate. When the angle of nodal inclination is 90°, the nodal surface is parallel to the loading direction, at this time, the nodal surfaces can not slide with each other, the stress is concentrated near the nodal surface, and the cracks continue to develop along the nodal surface, which ultimately leads to the destruction of the slate.

Fig. 4
Fig. 4
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Impact damage of slate specimens at different bedding dip angles.

Quantitative relationship between joint inclination and slate elastic parameters

The relationship between joint inclination and elastic modulus of laminated rocks under impact loading needs quantitative analysis in the existing slate damage constitutive model. It is not accurate enough to calculate the modulus of elasticity under different joint inclinations, which often use the same value. Therefore, obtaining the relationship between elastic modulus and joint inclination from the stress–strain curve is particularly important. To study the effect of joint inclination on the modulus of elasticity, it is first necessary to determine the modulus of elasticity of slate at different joint inclinations. Usually, the actual modulus of elasticity of slate at different joint inclinations can be determined from the slope of the linear part of the stress–strain curve, i.e., the elastic region obtained from the impact test of slate. Since the tests in this experiment are all-natural slate, there are differences in its internal mineral composition, microstructure, and the existence of natural defects, resulting in the mechanical properties of different specimens not being the same, which makes the test data inevitably discrete. Therefore, we made a preliminary screening of the data and selected five groups of data with less discrete degrees and more concentrated distributions under each joint angle. Then, we linearly fit the elastic modulus of slate under different joint inclinations, and the fitting results are shown in Fig. 5.

Fig. 5
Fig. 5
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Fitted curves of slate elastic parameters at different joint angles.

The regression function between joint inclination angle \(\theta\) and elastic modulus E can be obtained by fitting as follows:

$$E\left( \theta \right) = 11.38\sin \left( {0.0675\theta - 5.1} \right) + 44.85$$
(1)

where E is the modulus of elasticity of the slate in GPa; \(\theta\) is the joint inclination angle, with a range of \(0^{ \circ } \le \theta \le 90^{ \circ }\).

The joint inclination affects the modulus of elasticity and shows significant anisotropy characteristics under different joint inclinations. The modulus of elasticity decreases and then increases with the increase of joint inclination, which belongs to the typical “U”-shaped anisotropic curve. Under the impact load, the elastic modulus of the slate shows a trend of first decreasing and then increasing with the joint inclination angle. When the joint inclination angle is 0°, the joint surface is perpendicular to the loading direction, at this time, the stress of the slate in the loading direction is mainly borne by the rock body, and the overall strain is relatively small because the joints will not produce large deformation and damage. According to the definition of elastic modulus equal to stress divided by strain, at this time the stress is larger and the strain is smaller, so the elastic modulus is larger. As the angle of nodule inclination increases from 0° to 60°, the angle between the nodule surface and the loading direction changes, the shear stress on the nodule surface gradually increases, and the strength of the rock sample decreases. Sliding deformation is more likely to occur at the joints, and the strain of the slate increases under the same impact load. Since the modulus of elasticity is inversely proportional to the strain, the increase of strain leads to the gradual decrease of the modulus of elasticity, and it reaches the minimum at 60°. When the angle of joints gradually increases from 60° to 90°, the loading direction is gradually parallel to the joint surface, and the damage of the rock samples is gradually controlled by the mechanical properties of the rock itself, which is no longer determined by the joints alone. The strength of the rock itself is greater than that of the joint surface, so the overall strength of the rock sample increases. At the same time, in this angular change range, the growth of strain is smaller than that from 0° to 60°, which makes the ratio of stress to strain increase, i.e., the modulus of elasticity gradually increases again.

For rocks, the arrangement and density of internal particles directly affect the size of the elastic modulus. When subjected to external forces, the interaction between the particles makes the rock more resistant to deformation. However, for laminated slate, the arrangement of particles at the joint surface is relatively loose, the pore space between the particles is ample, and the degree of compactness is low, which may lead to uneven distribution of stress when subjected to impact loading, thus affecting the magnitude of the modulus of elasticity. Here, the angle between the nodal direction and the force direction is \(\alpha\), as shown in Fig. 6.

Fig. 6
Fig. 6
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Schematic diagram of the angle between the joint orientation and the direction of the force.

(1) At \(\alpha = 0^{^\circ }\), the deformation of slate is mainly axial, and the weak surface of joints does not undergo closed deformation and shear deformation; the joint surface is parallel to the direction of force, and the deformation of the joint surface on the rock as a whole is less hindered. At this point, the rock relies mainly on the elastic deformation of the intact rock portion to resist the external force when the force is applied, so the value of the modulus of elasticity is larger;

(2)At 0° < \(\alpha\) < 90°, the deformation of the slate includes axial deformation, closing deformation, and shear slip deformation of the weak face of the joints. With the gradual increase of joint inclination, the elastic modulus will first decrease and then increase, and the closer \(\alpha\) is to 52.5°, the smaller the elastic modulus is. The interaction between the joint surface and the force direction makes the rock more prone to deformation, leading to decreased elastic modulus. When the joint inclination is close to a certain angle, the joint inclination may make the bearing capacity of the rock in a specific direction increase, and the modulus of elasticity will recover again;

(3) At \(\alpha = 90^{^\circ }\), the deformation of the slate has axial deformation and closed deformation of the weak surface of the joints. The pores between the slate joints will be compacted under external force with relatively good integrity and high elastic modulus. At this time, the joint surface and the force direction of the angle perpendicular to the joint surface of the overall mechanical properties of the rock weakening effect are relatively weak, and its modulus of elasticity is relatively large.

The theoretical value of the modulus of elasticity at each joint inclination \(\overline{E}_{{\text{t}}}\) can be calculated by Eq. (1), which is compared with the experimental value \(\overline{E}_{{\text{e}}}\). The results with mean relative error \(\delta\) are shown in Table 3.

Table 3 Comparison of elastic modulus results under different joint inclination angles.
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The fit of the regression model to the dataset is usually assessed using the indicator R2, which is the proportion of variance in the response variable that can be explained by the predictor variables in the model, taking values in the range of [0,1] The closer R2 is to 1, the better the fit is, which can be calculated by Eq. (2) to obtain R2 = 0.95508.

$$R^{2} = 1 - \frac{{\sum {\left( {y_{i} - \hat{y}_{i} } \right)^{2} } }}{{\sum {\left( {y_{i} - \overline{y}_{i} } \right)^{2} } }}$$
(2)

where \({\text{y}}_{{\text{i}}}\) is the actual value of the dependent variable, \({\hat{\text{y}}}_{{\text{i}}}\) is the predicted value of the dependent variable, and \({\overline{\text{y}}}_{{\text{i}}}\) is the mean value of the dependent variable.

In multivariate linear regression, since the increase in the number of independent variables leads to an increase in R2, it is also necessary to adjust the R2 using Eq. (3), and the calculation gives Adjusted R2 = 0.94866.

$$A{\text{djusted}}R^{2} = 1 - \frac{{{{\left( {1 - R^{2} } \right)} \mathord{\left/ {\vphantom {{\left( {1 - R^{2} } \right)} {\left( {n - 1} \right)}}} \right. \kern-0pt} {\left( {n - 1} \right)}}}}{{\left( {n - k - 1} \right)}}$$
(3)

where n is the number of observations, k is the number of independent variables.

After analyzing the fitting formula, the elastic modulus and joint inclination angle are well fitted, and the whole conforms to the change rule of sinusoidal function. According to the fitting formula, the mathematical model of slate elastic modulus and joint inclination can be proposed:

$$E\left( \theta \right) = A{\text{sin}}\left( {\omega \theta + \phi } \right) + {\text{b}}_{0}$$
(4)

where b0, A, \(\omega\), \(\phi\) are the fitting parameters, taking the values of A = 11.38 GPa; b0 = 44.85 GPa; \(\omega\) = 0.069 rad/s; \(\phi\) = -5.1 rad, respectively.

Where b0 is the central tendency of the elastic modulus to vary with joint inclination, it represents the average or baseline value of the elastic modulus when the joint inclination fluctuates over the entire range of variation. The central value shows the approximate range of elastic modulus of slate in general. It can also be used as a reference to evaluate the relative magnitude of elastic modulus at different joint inclinations. A represents the maximum fluctuation of elastic modulus with joint inclination, which reflects the maximum degree of deviation from the central value of elastic modulus of slate during the variation of joint inclination. A is larger, which implies a larger difference in the elastic modulus at different joint inclinations. \(\omega\) and \(\phi\) represent the angular frequency and initial phase in the sinusoidal function, respectively. \(\omega\) determines the rate of change of elastic modulus with joint inclination in this model. The larger the \(\omega\) is, the more easily the elastic modulus responds to the slight change in joint inclination angle. It can reflect the sensitivity of the slate elastic modulus to changes in joint inclination. While \(\phi\) reflects the relative position offset of the elastic modulus and the standard sine function in the initial joint inclination state, the value \(\phi\) needs to be derived through specific experiments, and \(\phi\) may be different under different loading conditions. Hence, \(\phi\) in this paper only represents the slate under a single impact load and does not consider the case of axial and peripheral pressures. Readers need to pay attention to the conditions of use.

Laminated slate damage constitutive modeling

Dynamic damage constitutive model based on a component combination method

The study of the mechanical properties of slate under external force is of great theoretical significance in guiding engineering practice. Layered jointed rocks through geological evolution and multiple periods of complex tectonic movement, containing different orders of randomly distributed microporous and macro-structural surfaces, anisotropic characteristics are significant. Therefore, slate’s mechanical properties differ from those of other non-laminated rocks. The test results show that the mechanical properties of slate have nonlinear characteristics, and selecting a reasonable damage ontology model to describe the stress–strain relationship of slate has always been a research difficulty. In the following, the corresponding statistical damage constitutive model of slate will be established to describe the mechanical behavior of slate more accurately.

Slate has an apparent laminated structure. When the laminated rock is subjected to external forces, sliding between layers may occur, leading to interlayer shear damage of the rock, and the pressure on the rock will also lead to the closure of the pores between the rock bodies, thus generating the phenomenon of volume contraction, as shown in Fig. 730.

Fig. 7
Fig. 7
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Microstructure of slate.

Many scholars believe the slate damage constitutive model can be simplified into different parts. The damage deformation component of the rock mass, the component considering the closure deformation of the nodal structural surface, and the component considering the shear-slip deformation can be connected in series to simulate the total deformation of the slate31. Then, a viscous body can be connected in parallel to simulate the dynamic strain-rate effect of the rock body subjected to the impact action. Finally, the dynamic damage constitutive slate model can be obtained by combining the components, as shown in Fig. 8.

Fig. 8
Fig. 8
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Dynamic compression damage constitutive model of slate.

In this case, the stresses on the three series elements under impact loading are \(\sigma_{{1}}\), \(\sigma_{{2}}\), and \(\sigma_{{3}}\), and the strains are \(\varepsilon_{1}\), \(\varepsilon_{2}\), and \(\varepsilon_{3}\), respectively, and the following relationship exists:

$$\sigma_{z} = \sigma_{{1}} = \sigma_{2} = \sigma_{3}$$
(5)
$$\varepsilon_{{\text{z}}} = \varepsilon_{1} + \varepsilon_{2} + \varepsilon_{3}$$
(6)

(1) Damage and deformation components of rock mass.

Slate belongs to brittle materials, and its stress damage belongs to centralized damage; statistics for centralized damage of such a nature are often portrayed using the lognormal distribution, the structural resistance probability distribution of the internal structure of the rock can better reflect the cumulative damage and fracture toughness law, the lognormal distribution as a classic distribution commonly used in engineering reliability analysis has vast applicability32. Therefore, this paper uses the lognormal distribution to represent the statistical distribution of the rock strain limit, and the joint lognormal distributions have two kinds of base in natural logarithm e and base in scientific count 10. In this paper, a is used as a general derivation, \(l{\text{og}}_{{\text{a}}} \varepsilon \sim N\left( {\mu_{0} ,S_{0} } \right)\) is set, and its probability density function is:

$$P(\varepsilon ) = \frac{1}{{\varepsilon \ln {\text{a}}S_{0} \sqrt {2\pi } }}\exp \left[ { - \frac{1}{2}\left( {\frac{{l{\text{og}}_{{\text{a}}} \varepsilon - \mu_{0} }}{{S_{0} }}} \right)^{2} } \right]_{,} \cdot \varepsilon > 0$$
(7)

Where \(\mu_{0}\) and \(S_{0}\) are lognormal distribution parameters, a takes the natural logarithm e.

For a brittle material like rock, we usually use the strain induced by loading to analyze its damage. The rock has discontinuities and carries a random distribution of particles. To calculate the damage variables of the rock under loading at a particular strain level D. The initial number of rock micropores is recorded as \(N_{0}\). The number of rock micropores destroyed at a particular strain level is \(N_{{\text{d}}}\). Then, the damage variable D can be expressed as:

$$D = \frac{{N_{d} }}{{N_{0} }}_{,} \cdot \left( {0 \le D \le 1} \right)$$
(8)

The magnitude of the damage variable D can reflect the percentage of damaged rock micropores inside the rock material; when D = 0, the inside can be regarded as an undamaged state, and when D = 1, the rock is completely destroyed. When the strain of the rock under loading goes from 0 to \(\varepsilon_{{\text{r}}}\), the number of damaged rock micropores is:

$$\begin{gathered} N_{d} = N_{0} \int_{0}^{{\varepsilon_{1} }} {P\left( \varepsilon \right)} d\varepsilon \hfill \\ = N_{0} \int_{0}^{{\varepsilon_{1} }} {\frac{1}{{\varepsilon S_{0} \sqrt {2\pi } }}{\text{exp}}\left[ { - \frac{1}{2}\left( {\frac{{\ln \varepsilon - \mu_{0} }}{{S_{0} }}} \right)^{2} } \right]} d\varepsilon \hfill \\ = N_{0} \Phi \left( {\frac{{\ln \varepsilon_{{1}} - \mu_{0} }}{{S_{0} }}} \right) \hfill \\ \end{gathered}$$
(9)

where \(\Phi \left( {\frac{{\ln \varepsilon_{{1}} - \mu_{0} }}{{S_{0} }}} \right)\) is the distribution function of the standard normal distribution.

From Eq. (8) and Eq. (9) there is:

$$D = \Phi \left( {\frac{{\ln \varepsilon_{1} - \mu_{0} }}{{S_{0} }}} \right) = 1 - \Phi \left( {\frac{{\mu_{0} - \ln \varepsilon_{1} }}{{S_{0} }}} \right)$$
(10)

Based on the Lemaitre strain equivalence hypothesis and generalized Hooke’s law, the rock damage eigen structure equation can be obtained:

$$\varepsilon_{1} = \frac{\sigma }{{\left( {1 - D} \right) \cdot E}} = \frac{\sigma }{{E \cdot \Phi \left( {\frac{{\mu_{0} - \ln \varepsilon_{1} }}{{S_{0} }}} \right)}}$$
(11)

(2) Elements with closed deformation of nodal structure surfaces and elements with shear-slip deformation.

The forces on the slate under impact loading are shown in Fig. 9.

Fig. 9
Fig. 9
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Force diagram of slate under impact load.

Slate with a joint inclination of \(\theta\) is subjected to a uniaxial impact load \(\sigma_{{\text{z}}}\), which generates stresses \(\sigma_{\theta }\) and shear stresses \(\tau_{\theta }\) at the joint faces, which can be expressed as:

$$\left\{ \begin{gathered} \sigma_{\theta } = \sigma_{2} \cos^{2} \theta + \sigma_{y} \cos \theta \sin \theta \hfill \\ \tau_{\theta } = \sigma_{3} \cos \theta \sin \theta - \sigma_{y} \cos^{2} \theta \hfill \\ \end{gathered} \right.$$
(12)

The equations for the nodal surface closure deformation and nodal surface shear-slip deformation are shown in Eq. (13) 33:

$$\left\{ \begin{gathered} \varepsilon_{2} = \varepsilon_{j0} \left[ {1 - \exp \left( { - \frac{{\sigma_{2} \cos^{2} \theta + \sigma_{y} \cos \theta \sin \theta }}{{E_{j0} \varepsilon_{j0} }}} \right)} \right]\cos \theta \hfill \\ \varepsilon_{3} = \frac{{\left( {\sigma_{3} \cos \theta \sin \theta - \sigma_{y} \cos^{2} \theta } \right)\sin \theta }}{{K_{s0} L}} \hfill \\ \end{gathered} \right.$$
(13)

where \(\varepsilon_{j0}\) is the maximum closed strain of the structural surface, taken as the strain value of the compression-dense section under the compressive loading of the slate with an inclination angle of 0°; \(E_{j0} = 1978MPa\) represents the closed modulus of elasticity of the structural surface of the joints, and \(E_{{{\text{s}}0}} = {{2GPa} \mathord{\left/ {\vphantom {{2GPa} m}} \right. \kern-0pt} m}\) represents the tangential stiffness of the structural surface of the joints32.

(3) Dynamic strain rate effects in slate.

The viscous body expresses the dynamic strain rate effect of slate under impact loading, and the stress expression of the viscous body is as follows:

$$\sigma^{\prime} = \eta \frac{{d\varepsilon^{\prime}}}{dt}$$
(14)

where \(\eta\) represents the viscous coefficient of the sticky pot is 0.1 MPa·s, \(\varepsilon_{{\text{z}}}\) represents the axial strain of the specimen, where the dynamic load strain rate \(\varepsilon^{\prime}\) is a constant corresponding to the impact load at different velocities, which is taken as 10/s here.

From the relation of parallel connection of elements, it can be obtained that the total stress of the rock is shared by the two parts connected in parallel, i.e.:

$$\sigma = \sigma_{{\text{z}}} + \sigma^{\prime}$$
(15)

Based on the laminated rock constitutive assemblage relationship, the stress–strain expression for the dynamic damage constitutive model of nodular slate containing strain rate can be obtained as follows:

$$\varepsilon_{{\text{z}}} = \frac{{\sigma - \eta \varepsilon^{\prime}}}{{{\text{E}} \cdot \Phi \left( {\frac{{\mu_{0} - \ln \varepsilon_{1} }}{{S_{0} }}} \right)}} + \varepsilon_{j0} \left[ {1 - \exp \left( { - \frac{{\left( {\sigma - \eta \varepsilon^{\prime}} \right)\cos^{2} \theta }}{{\Phi \left( {\frac{{\mu_{0} - \ln \varepsilon_{1} }}{{S_{0} }}} \right)K_{j0} \varepsilon_{j0} }}} \right)} \right]\cos \theta + \frac{{\left( {\sigma - \eta \varepsilon^{\prime}} \right)\sin^{2} \theta \cos \theta }}{{\Phi \left( {\frac{{\mu_{0} - \ln \varepsilon_{1} }}{{S_{0} }}} \right)K_{s0} L}}$$
(16)

Finally, Eq. (1) can be brought into Eq. (16) to obtain a dynamic damage constitutive model for slate considering the variation of elastic modulus with joint inclination as follows:

$$\varepsilon_{{\text{z}}} = \frac{{\sigma - \eta \varepsilon^{\prime}}}{{{\text{E}}\left( \theta \right) \cdot \Phi \left( {\frac{{\mu_{0} - \ln \varepsilon_{1} }}{{S_{0} }}} \right)}} + \varepsilon_{j0} \left[ {1 - \exp \left( { - \frac{{\left( {\sigma - \eta \varepsilon^{\prime}} \right)\cos^{2} \theta }}{{\Phi \left( {\frac{{\mu_{0} - \ln \varepsilon_{1} }}{{S_{0} }}} \right)K_{j0} \varepsilon_{j0} }}} \right)} \right]\cos \theta + \frac{{\left( {\sigma - \eta \varepsilon^{\prime}} \right)\sin^{2} \theta \cos \theta }}{{\Phi \left( {\frac{{\mu_{0} - \ln \varepsilon_{1} }}{{S_{0} }}} \right)K_{s0} L}}$$
(17)

Solution of distribution parameters

Using the peak point method to determine the statistical parameters can reduce the calculation error, which has been used in our previous study32, and such a method has a clear physical meaning and universality. Therefore, this paper introduces the peak point method to determine the distribution parameters \(\mu_{0}\) and \(S_{0}\). It is assumed that the peak point of the stress–strain curve is point M, as shown in Fig. 10.

Fig. 10
Fig. 10
Full size image

Stress–strain curve of rock under compression.

According to the theory of extremum of multivariate functions, if the derivative at the peak point M of the stress–strain curve is 0, then we have:

$$\,\left. {\sigma_{1} = \,\sigma_{M} ,\varepsilon_{1} = \varepsilon_{M} ,\,\frac{\partial \sigma }{{\partial \varepsilon }}} \right|_{{\varepsilon_{1} = \varepsilon_{M} }} = \,0$$
(18)
$$\Phi \left( {\frac{{\mu_{0} - \ln \varepsilon_{{\text{M}}} }}{{S_{0} }}} \right) = \frac{{\sigma_{M} }}{{E\varepsilon_{M} }}$$
(19)

Setting \(Y = \frac{{\mu_{0} - {\text{ln}}\varepsilon_{M} }}{{S_{0} }}\), there is:

$$\Phi \left( Y \right) = \frac{{\sigma_{M} }}{{E\varepsilon_{M} }}$$
(20)

Substituting the stress–strain at the peak point M of the stress–strain curve of the rock under compression in Eq. (20) yields \(\Phi \left( Y \right)\), and then the value of Y can be found through the standard normal distribution function table.

$$\frac{{\Phi \left( {\frac{{\mu_{0} - \ln \varepsilon_{{\text{M}}} }}{{S_{0} }}} \right) - \varphi \left( {\frac{{\mu_{0} - \ln \varepsilon_{{\text{M}}} }}{{S_{0} }}} \right)}}{{S_{0} = 0}}$$
(21)

where \(\varphi\) is the probability density function of the standard normal distribution:

$$\varphi \left( t \right) = \frac{1}{{\sqrt {2\pi } }}\exp \left( { - \frac{{t^{2} }}{2}} \right)_{,} \cdot {\text{t}} \in R$$
(22)

From the association of Eqs. (20),(21), and (22)it can be calculated that:

$${\text{exp}}\left( { - \frac{{Y^{2} }}{2}} \right) = S_{0} \sqrt {2\pi } \frac{{\sigma_{M} }}{{E\varepsilon_{M} }}$$
(23)

Taking logarithms on both sides of Eq. (23) yields:

$$- \frac{{Y^{2} }}{2} = \ln S_{0} + \ln \left[ {\sqrt {2\pi } \frac{{\sigma_{M} }}{{E\varepsilon_{M} }}} \right]$$
(24)

From the probability density function of the standard normal distribution, the expressions of the parameters in the intrinsic equation of the damage of the rock mass under impact loading can be obtained:

$$\left\{ \begin{gathered} S_{0} = {\text{exp}}\left\{ { - \frac{{Y^{2} }}{2} - {\text{ln}}\left[ {\sqrt {2\pi } \frac{{\sigma_{M} }}{{E\varepsilon_{M} }}} \right]} \right\} \hfill \\ \mu {}_{0} = YS_{0} + \ln \varepsilon_{M} \hfill \\ \end{gathered} \right.$$
(25)

However, the modulus of elasticity under different joint inclinations will be different. To make the derived distribution parameter more accurate, the modulus of elasticity is corrected here, and the statistical parameters of slate considering the change of modulus of elasticity with joint inclination can be obtained by bringing Eq. (1) into Eq. (25) as follows:

$$\left\{ \begin{gathered} S_{0} = {\text{exp}}\left\{ { - \frac{{Y^{2} }}{2} - {\text{ln}}\left[ {\sqrt {2\pi } \frac{{\sigma_{M} }}{{E\left( \theta \right) \cdot \varepsilon_{M} }}} \right]} \right\} \hfill \\ \mu {}_{0} = YS_{0} + \ln \varepsilon_{M} \hfill \\ \end{gathered} \right.$$
(26)

Model validation

In order to verify the reasonableness of the above model, the test curves are compared and analyzed here with the theoretical curves in this paper. Among them, the corresponding statistical distribution parameters of the slate can be derived from the results of Eq. (26), as shown in Table 4.

Table 4 Statistical distribution parameter table of damage constitutive model.
Full size table

Finally, the stress–strain relationship curves of slate with joint inclination angles of 0°, 30°, 45°, 60°, and 90° can be calculated from Eq. (17) and compared with the results of the dynamic impact test of laminated slate, as shown in Fig. 11. The computational analysis shows that the average percentage error for a joint inclination angle of 0° is 15.19%; for a joint inclination angle of 30° the average percentage error is 19.63%; for a joint inclination angle of 45° the average percentage error is 18.59%; for a joint inclination angle of 60° the average percentage error is 8.65%; and for a joint inclination angle of 90° the average percentage error is 12.67%. The overall error of this model is in the acceptable range for all kinds of projects. Therefore, the damage principal model we established can provide a reference value for reflecting the mechanical behavior of slate in the compression-dense and elastic phases under impact loading. At the same time, the establishment method and validation process of the model can also be used to enrich the research results of the related rock damage principal relationship.

Fig. 11
Fig. 11
Full size image

Comparison of stress–strain test and theoretical curve under different joint inclination angles.

Conclusion

  • The elastic modulus of slate is analyzed under different angles of joints through the results of Hopkinson’s compression rod impact experiment. With the increase of joint angle, the modulus of elasticity decreases and then increases, which belongs to the typical “U” type change.

  • When \(\alpha = 0^{^\circ }\), the slate mainly relies on the elastic deformation of the intact rock part to resist the external force, and the modulus of elasticity is larger; when 0° < \(\alpha\) < 90°, the modulus of elasticity decreases and then increases with the gradual increase of \(\alpha\) and the modulus of elasticity is the smallest when the angle of nodal inclination is 52.5°; when \(\alpha = 90^{^\circ }\), the weakening effect of the nodal surface on the overall mechanical properties of the rock is relatively weak, and the modulus of elasticity is also relatively larger. The modulus of elasticity is also relatively large.

  • The mathematical model of modulus of elasticity and joint angle was established by fitting the impact experimental data, and the fitted curve conforms to the rule of sinusoidal function. The accuracy of the fitting formula is 94.86% after error calculation.

  • The dynamic damage constitutive model of slate was established by introducing the lognormal distribution and considering the quantitative relationship of elastic modulus under different angles of joints, and the accuracy of the model was more than 80% by comparing the experimental and theoretical calculations.