1 Introduction

With the continuous socio-economic development and rapid industrialisation, the problem of air pollution has become more and more prominent, and the effect of acid rain on rock damage caused by environmental pollution has attracted the attention of many scholars. As a non-homogeneous material, the mechanical properties of rocks are strongly influenced by the environment in which they are found, and the damage mechanisms are complex [1, 2]. On the basis of the energy principle and the fracture theory, a damage ontology model is derived for rock masses that do not penetrate the nodal point [3]. A damage-intrinsic model for the whole rock failure process has been developed using the triaxial test method, based on the Griffith strength criterion and assuming that the rock micro-element strength follows a Weibull distribution [4]. A triaxial compression test was carried out on quartz sand as the test specimen and a statistical damage-intrinsic model for the rock was proposed based on a modified Mohr–Coulomb strength criterion [5].

Natural rock contains a large number of joints and fractures, the presence of which affects the mechanical properties of the rock. Model tests were carried out with rock-like materials to derive the shape of the damage and the strength of the rock mass under different jointing conditions [6]. Derivation of an damage constitutive model for regularly fractured rock masses based on the yield theory of rocks [7]. In order to investigate the damage evolution of fractured rock masses under water pressure, the crack evolution pattern of rock masses was analysed by numerical tests and indoor tests [8]. A combination of numerical simulations and experiments was used to study the damage characteristics and crack propagation pattern of the jointed rock mass [9]. The discrete unit method was used to model the type of fracture penetration in fractured rock masses under compression-shear conditions and to describe the law between fracture dip and fracture extension [10]. The strength and damage properties of fractured rock masses under uniaxial compression have been investigated through extensive experimental studies, and the variation of rock mass strength with fracture inclination has been obtained [11]. Mechanical tests were carried out on six different types of prefabricated fissure-like rock specimens at different circumferential pressures to analyse the variation in fissure inclination and peak rock strength under triaxial compression conditions [12]. Park et al. performed uniaxial compression tests on gypsum specimens and classified the crack development into winged cracks, coplanar secondary cracks and oblique secondary cracks [13]. Shao et al. developed an constitutive model of water–rock interaction in saturated and unsaturated rock based on elastic–plastic damage theory [14]. An constitutive model for rocks under coupled mechano-chemical effects was proposed by Min et al. [15]. Li introduced a combined chemical and stress damage variable and developed a statistical damage model taking into account compression-density effects [16]. Damage to fractured rock is a relatively complex damage evolution process influenced by a combination of microscopic cracking and macroscopic fracturing. The Weibull statistical damage model for microscopic defects has been widely used to study the mechanical properties of rock masses [17,18,19,20,21].

In view of this, this paper takes red sandstone as the research object, and makes the fissure inclination angle 0°, 30°, 45° and 60° specimens, and carries out acid corrosion maintenance on some of the specimens, after the maintenance is completed, carries out uniaxial compression test to study the mechanical deterioration properties and damage mode of red sandstone under the effect of acid corrosion. Based on the damage model and fracture mechanics model to establish the constitutive equations of sandstone under combined chemical–mechanical loading, the test curves are compared with the theoretical curves to verify the applicability of the parameters.

2 Fractured rock uniaxial compression test

2.1 Experimental programme and results

The red sandstone sample with an internal friction angle of 28.5°and a Poisson's ratio of 0.25 used in this paper is from Yunnan Province, China. Red sandstone with a uniform texture and no macroscopic cracks was selected as the bedrock and coring was carried out using a core drill. Once the sandstone was cored, the cored sample was processed using a rock cutter to a standard cylindrical sample of 50 mm diameter and 100 mm height in accordance with the International Society for Rock Mechanics (ISRM) test protocol. The two end faces of the sample were levelled using a rock grinder and finally, using the centre of the cylinder as a reference point, cracks 15 mm long and 2 mm wide were cut with a rock cutter at 0°, 30°, 45° and 60° inclination to the horizontal. To ensure the reliability of the test results, six complete specimens and six specimens with different crack inclinations were made, three of which were placed in a pH = 2 hydrochloric acid solution for maintenance. Calibrates the pH measurement of the solution every 24 h, ensuring that the acidity reaches a pre-set level, eliminating the effect of solution volatility on the pH. The test apparatus is a ZTRE-210 microcomputer controlled triaxial rock testing system, as shown in Fig. 1. The test data were collected and collated, and the mechanical parameters of the specimens tested by uniaxial compression are given in Table 1.

Fig. 1
figure 1

Microcomputer controlled triaxial rock testing system

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Table1 Mechanical parameters of test piece
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2.2 Effect on peak compressive strength of fracture inclination

The average peak compressive strength of the fractured sandstone under normal and acid corrosion conditions is shown in Fig. 2. The average peak compressive strength of the complete sandstone under normal conditions is 70.51 MPa. The average values of peak compressive strength for fracture inclination angles α of 0°, 30°, 45°and 60°are 68.58 MPa, 63.31 MPa, 60.95 MPa and 69.21 MPa respectively. Compared to the complete specimen, the peak compressive strength decreases by approximately 2.74% at a crack inclination of 0°; at a crack inclination of 30°, the peak compressive strength continues to decrease by approximately 10.21%; at a crack inclination of 45°, the peak compressive strength is the lowest and decreases by approximately 13.56%; at a crack inclination of 60°, the peak compressive strength of the specimen decreases by approximately 1.84% compared to the complete specimen.

Fig. 2
figure 2

Relationship between fracture inclination and peak stress

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The average peak compressive strength of the complete sandstone after corrosion with pH = 2 acid solution was 66.47 MPa. The average peak compressive strengths were 61.86 MPa, 58.68 MPa, 55.55 MPa and 64.29 MPa for fracture inclination angles α of 0°, 30°, 45°and 60°respectively.Compared to the complete specimen, the peak compressive strength decreases by approximately 6.94% at a crack inclination of 0°; at a crack inclination of 30°, the peak compressive strength continues to decrease by approximately 11.72%; at a crack inclination of 45°, the peak compressive strength is the lowest and decreases by approximately 16.43%; at a crack inclination of 60°, the peak compressive strength of the specimen decreases by approximately 16.43% compared to the complete specimen.

The peak strength of fractured rock specimens with different fracture inclination angles decreases after corrosion by acid solutions with pH = 2. The peak strength of fractured rock specimens under normal and acid corrosion conditions decreases and then increases with increasing fissure inclination angle and is lowest at 45°fracture inclination angle and does not differ much from the peak strength of complete specimens at 60°fracture inclination angle.

2.3 Effect on peak strain of fracture inclination

Peak strain averages for fractured sandstone under normal and acid corrosion conditions are shown in Fig. 3. The average peak strain for complete sandstone under normal conditions is 3.29 × 10–3. The average values of peak strains were 3.53 × 10–3, 2.98 × 10–3, 2.85 × 10–3 and 3.26 × 10–3 for fracture inclination angle α of 0°, 30°, 45° and 60°, respectively. Compared to the complete specimen, the peak strain decreases by approximately 4.96% at a crack inclination of 0°; at a crack inclination of 30°, the peak strain continues to decrease by approximately 15.50%; at a crack inclination of 45°, the peak strain is the lowest and decreases by approximately 20.16%; at a crack inclination of 60°, the peak strain of the specimen decreases by approximately 4.56% compared to the complete specimen.

Fig. 3
figure 3

Relationship between fracture inclination and peak strain

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The average peak strain of the complete sandstone after corrosion with pH = 2 acid solution was 3.72 × 10–3. The average values of peak strain for fracture inclination angles α of 0°, 30°, 45°and 60°were 3.53 × 10–3, 2.98 × 10–3, 2.85 × 10–3, 3.26 × 10–3 respectively. Compared to the complete specimen, the peak strain decreases by approximately 5.02% at a crack inclination of 0°; at a crack inclination of 30°, the peak strain continues to decrease by approximately 19.98%; at a crack inclination of 45°, the peak strain is the lowest and decreases by approximately 23.39%; at a crack inclination of 60°, the peak strain of the specimen decreases by approximately 12.37% compared to the complete specimen.

The peak strains of rock specimens with different fracture inclination angles increase after corrosion by acid solutions with pH = 2. The peak strains of fractured rock specimens under normal and acid corrosion conditions decrease and then increase with increasing fracture inclination and are lowest at a fracture inclination of 45°.

2.4 Effect on elastic modulus of fracture inclination

The average elastic modulus of the fractured sandstone under normal and acid corrosion conditions is shown in Fig. 4. The average elastic modulus of the complete sandstone under normal conditions is 28.10 GPa. The average values of elastic modulus for fracture inclination angles α of 0°, 30°, 45°and 60°are 27.51 GPa, 27.01 GPa, 25.49 GPa and 27.98 GPa respectively. Compared to the complete specimen, the elastic modulus decreases by approximately 2.11% at a crack inclination of 0°; at a crack inclination of 30°, the elastic modulus continues to decrease by approximately 3.87%; at a crack inclination of 45°, the elastic modulus is the lowest and decreases by approximately 9.30%; at a crack inclination of 60°, the elastic modulus of the specimen decreases by approximately 0.43% compared to the complete specimen.

Fig. 4
figure 4

Relationship between fracture inclination and elastic modulus

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The average elastic modulus of the complete sandstone after corrosion with pH = 2 acid solution was 24.58 GPa. The average elastic modulus were 23.17 GPa, 21.44 GPa, 20.12 GPa and 24.52 GPa for fracture inclination angles α of 0°, 30°, 45°and 60°respectively. Compared to the complete specimen, the elastic modulus decreases by approximately 6.77% at a crack inclination of 0°; at a crack inclination of 30°, the elastic modulus continues to decrease by approximately 13.17%; at a crack inclination of 45°, the elastic modulus is the lowest and decreases by approximately 19.05%; at a crack inclination of 60°, the elastic modulus of the specimen decreases by approximately 1.31% compared to the complete specimen.

The elastic modulus of fractured rock specimens with different fracture inclination angles decreases after corrosion by acid solutions with pH = 2. The elastic modulus of fractured rock specimens under normal and acid corrosion conditions decreases and then increases with increasing fissure inclination angle and is lowest at 45°fracture inclination angle and does not differ much from the elastic modulus of complete specimens at 60°fracture inclination angle.

3 Fracture characteristics under uniaxial compression

From the above test results, it can be seen that there are differences in the mechanical properties of fractured rocks under different fracture inclinations, and the prefabricated fractures and fracture expansion of the specimens under uniaxial compression make the anisotropy of the specimens more significant. The different loading methods will directly affect the way the specimens are stressed and their fracture patterns will be changed. In uniaxial compression, there are three main types of cracks in rock specimens: wing cracks, coplanar secondary cracks and oblique secondary cracks, as shown in Fig. 5.

Fig. 5
figure 5

Crack propagation mode under compression

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When the crack inclination is 0°, the prefabricated crack is compressed during the compression process, and the specimen cracks develop symmetrically, with wing cracks and oblique secondary cracks developing at both ends of the crack, extending up and down the specimen. The damage to the specimen is mainly due to the shear damage caused by the joint formation and development of the wing cracks and oblique secondary cracks, and the damage characteristics are shown in Fig. 6.

Fig. 6
figure 6

Failure characteristics of 0° specimen

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The specimens with 30° crack inclination under normal maintenance show different development patterns of wing cracks, coplanar secondary cracks and oblique secondary cracks during compression. Secondary cracks are more developed than winged cracks, with coplanar secondary cracks passing through oblique secondary cracks. The damage to the specimen is mainly due to shear damage, which causes secondary crack development, and vertical cleavage damage after the crack has penetrated.

After corrosion by the acid solution, the wing cracks and secondary cracks are fully developed, and the wing cracks of the specimen expand and penetrate upward and downward during the compression process, and the coplanar secondary cracks penetrate with the oblique secondary cracks. The damage to the specimen is mainly caused by the progressive development of wing cracks due to tensile damage, the progressive development of secondary cracks due to shear damage and the vertical cleavage damage after crack penetration, the damage characteristics are shown in Fig. 7.

Fig. 7
figure 7

Failure characteristics of 30° specimen

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The fracture inclination is 45° specimen, during the compression process the overall damage to the specimen is more severe, the rock fragments and rock chips are detached from the parent rock over a large area. During compression, wing cracks are more developed and extend upwards and downwards, and coplanar secondary cracks and oblique secondary cracks are strongly developed to form penetrations. The main damage to the specimen is the continuous development of secondary cracks caused by shear damage and vertical cleavage damage after crack penetration.

After corrosion by the acid solution, the secondary cracks in the specimen are more fully developed than the wing cracks, and the wing cracks in the specimen expand upward and downward during the compression process, and the coplanar secondary cracks penetrate with the oblique secondary cracks. The main damage to the specimen is shear damage and the macroscopic manifestation is the continuous development and expansion of secondary cracks, the damage characteristics of which are shown in Fig. 8.

Fig. 8
figure 8

Failure characteristics of 45° specimen

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Specimens with a fracture inclination of 60° show the strongest development of coplanar secondary cracks during compression, with more developed oblique secondary cracks and almost no development of wing cracks. The vertical shear cracks of the specimens are more frequent, and the coplanar secondary cracks are associated with the prefabricated fractures, the main damage mode being shear damage.

The crack formation and damage pattern of the specimens corroded by the acid solution with a fracture inclination of 60° is generally consistent with that of the specimens not corroded by the acid solution, and the damage characteristics are shown in Fig. 9.

Fig. 9
figure 9

Failure characteristics of 60° specimen

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4 Damage constitutive model of fractured rock mass

Uniaxial compression tests were carried out on fractured rock after corrosion with acid solutions. The damage consisted of three main components: chemical damage Dc caused by corrosion with acid solutions, damage Dm caused by randomly distributed microscopic defects in the fractured specimens during loading, and damage Df caused by macroscopic cracks during loading.

The deterioration of the mechanical properties of the sample can be characterised by parameters. The elastic modulus, an important property of mechanical properties, is easily obtained before and after the sample has been corroded by the acid solution. Define the chemical damage Dc of the rock as follows:

$${\text{D}}_{{\text{c}}} = {1} - \frac{{{\text{E}}_{{\text{c}}}}}{{{\text{E}}_{{0}} }},$$
(1)

where E0 is the pre-corrosion elastic modulus and Ec is the post-corrosion elastic modulus.

There are a large number of microscopic defects inside the rock, based on the statistical damage mechanics approach to the study, assuming that the defects inside the rock specimen obey some random distribution, the degree of damage is quantified by the micro-element strength satisfying the Weibull distribution function, and the corresponding statistical damage model of the rock is established. In this paper, the damage model is based on the D-P strength criterion degree and Weibull distribution [22]. Assuming that the micro-element strength of the specimen obeys the Weibull distribution, its probability density function is:

$${\text{P(F)}} = \frac{{\text{m}}}{{\text{F}}_{{0} }}\left( {\frac{{\text{F}}}{{\text{F}}_{0}}} \right)^{{\text{m}}-1} {\text{exp}}\left[ { - \left( {\frac{{\text{F}}}{{\text{F}}_{{0}}}} \right)^{\text{m}}} \right],$$
(2)

where P(F) is the distribution function of rock micro-element strength; F is the random distribution variable of micro-element strength; m and F0 are distribution parameters.

Assuming that the number of micro-elements that have broken the ring under load is n and the total number of micro elements is N, the statistical damage variable Dm is expressed as:

$${\text{D}}_{{\text{m}}} = \frac{{\text{n}}}{{\text{N}}}$$
(3)

When loaded to a certain strain F level, the number of micro-elements with broken rings is:

$${\text{n}} = \int_{{0}}^{{\text{F}}} {\text{NP(F)dF}} = {\text{N}}\left\{ {{1} - {\text{exp}}\left[ { - \left( {\frac{{\text{F}}}{{{\text{F}}_{{0}} }}} \right)^{{\text{m}}} } \right]} \right\}$$
(4)

It can be obtained that the statistical damage evolution equation of rock with micro-element strength as the damage evolution control variable under load is:

$${\text{D}}_{{\text{m}}} = {1} - {\text{exp}}\left[ { - \left( {\frac{{\text{F}}}{{{\text{F}}_{{0}} }}} \right)^{{\text{m}}} } \right]$$
(5)

Therefore, under uniaxial compression, the constitutive equation with microscopic damage is:

$$\sigma_{1} = {\text{E}}_{{\text{c}}} \varepsilon_{1} \left( {{1} - {\text{D}}_{{\text{m}}} } \right) = {\text{E}}_{{\text{c}}} \varepsilon_{1} {\text{exp}}\left[ { - \left( {\frac{{\text{F}}}{{{\text{F}}_{{0}} }}} \right)^{{\text{m}}} } \right]$$
(6)

According to Drucker–Prager (D–P) strength criterion [23], the distribution variable F is expressed as:

$${\text{F}} = \left( {{\upalpha }_{{0}} {\mathbf{I}}_{{\mathbf{1}}} + \sqrt {{\mathbf{J}}_{{\mathbf{2}}} } } \right){\text{ E}}_{{\text{c}}} \frac{{\varepsilon_{1} }}{{\sigma_{1} }}$$
(7)

where I1 is the first invariant of the stress tensor and J2 is the second invariant of the stress deviation; α0 is the strength coefficient related to the internal friction angle of rock. Wherein I1, J2 α0 expression is:

$$\begin{aligned} {\mathbf{I1}} &= \sigma 1\\ {\mathbf{J2}}& = {\sigma_{1}^{2}}/3 \\ {\upalpha }_{0} &= {\text{sin}}\varphi/\sqrt{9 + \text{3sin}^{2}\varphi }\end{aligned}$$
(8)

Combining formula (7) and formula (8) gives:

$${\text{F}} = \frac{{1}}{{\sqrt {3} }}{\text{E}}_{{\text{c}}} \left( {{1} + \frac{{{\text{sin}}\varphi }}{{\sqrt {{3} + {\text{sin}}^{{2}} \varphi } }}} \right)\varepsilon_{1}$$
(9)

At the highest point of the peak stress–strain curve, according to the boundary conditions:

$$\frac{{{\text{d}}\sigma_{{\text{p}}} }}{{{\text{d}}\varepsilon_{{\text{p}}} }} = 0$$
(10)

Combining the above equations yields:

$${\text{m}} = 1/{\text{ln(E}}_{{\text{c}}} \varepsilon_{{\text{p}}} {/}\sigma_{{\text{p}}} {)}$$
(11)
$${\text{F}}_{{0}} = {\text{F}}_{{\text{p}}} {\text{m}}^{{\text{1/m}}} = {\text{F}}_{{\text{p}}} \left[ {1/{\text{ln(E}}_{{\text{c}}} \varepsilon_{{\text{p}}} /\sigma_{{\text{p}}} {)}} \right]^{{{\text{ln(E}}_{{\text{c}}} \varepsilon_{p} {/}\sigma_{{\text{p}}} {)}}}$$
(12)

When the rock mass is studied, the original fractures in its distribution are considered as macroscopic defects, and the original damage state of the intact rock is used as the reference damage state, and the macroscopic defects are considered as macroscopic damage Df. Based on the theory of strain energy under uniaxial stress, the incremental strain energy Ui due to macroscopic cracks can be expressed as [24]:

$${\text{U}}_{{\text{i}}} = {\text{V}}\left[ {\frac{{\sigma_{1}^{2} }}{{{\text{2E}}_{{\text{c}}} \left( {{1} - {\text{D}}_{{\text{f}}} } \right)}} - \frac{{\sigma_{1}^{2} }}{{{\text{2E}}_{{\text{c}}} }}} \right],$$
(13)

where V represents the specimen volume.

Based on the stress intensity factor calculation method in fracture mechanics, for plane strain problems, the expression for the incremental strain energy of a rock specimen caused by a crack is:

$${\text{U}} = \frac{{{1} - {\upnu }^{{2}} }}{{{\text{E}}_{{\text{c}}} }}\int_{{0}}^{{\text{A}}} {\left( {{\text{K}}_{{\text{I}}}^{{2}} + {\text{K}}_{{{\text{II}}}}^{{2}} } \right)} {\text{dA}},$$
(14)

where KI and KII are the stress intensity factors of the wing crack tip type I and II respectively and A is the single crack zone.

Both equations represent the incremental elastic strain energy due to macroscopic fracture. According to the principle of energy equivalence, this gives:

$$\frac{{{1} - {\upnu }^{{2}} }}{{{\text{E}}_{{\text{c}}} }}\int_{{0}}^{{\text{A}}} {\left( {{\text{K}}_{{\text{I}}}^{{2}} + {\text{K}}_{{{\text{II}}}}^{{2}} } \right)} {\text{ dA}} = {\text{V}}\left[ {\frac{{\sigma_{1}^{2} }}{{{\text{2E}}_{{\text{c}}} \left( {{1} - {\text{D}}_{{\text{f}}} } \right)}} - \frac{{\sigma_{1}^{2} }}{{{\text{2E}}_{{\text{c}}} }}} \right]$$
(15)

From the above equation, the macroscopic damage variable Df for fractured rock is given by:

$${\text{D}}_{{\text{f}}} = {1} - \left[ {{1} + \frac{{{2}\left( {{1} - {\upnu }^{{2}} } \right)}}{{{\text{V}}\sigma_{1}^{2} }}\int_{{0}}^{{\text{A}}} {\left( {{\text{K}}_{{\text{I}}}^{{2}} + {\text{K}}_{{{\text{II}}}}^{{2}} } \right)} {\text{ dA}}} \right]^{{ - {1}}}$$
(16)

For uniaxial compression, the positive and shear stresses at the fracture surface are:

$$\begin{gathered} \sigma_{n} \left( {\sigma {,}\alpha } \right) = \sigma_{{1}} {\text{cos}}^{{2}} \alpha \hfill \\ \tau_{n} \left( {\sigma {,}\alpha } \right) = \frac{{\sigma_{{1}} }}{{2}}{\text{sin2}}\alpha \hfill \\ \end{gathered}$$
(17)

where α is the fracture inclination angle.

In the case of closed cracks, the slip drive in uniaxial compression is not influenced by cohesion and the slip drive on the face of the crack is as follows:

$$\tau_{eff} = \left\{ {\begin{array}{*{20}c} {0,} & {{\text{tan}}\alpha < {\text{tan}}\varphi } \\ {\tau_{n} - \mu \sigma_{n} {,}} & {{\text{tan}}\alpha \ge {\text{tan}}\varphi } \\ \end{array} } \right.$$
(18)

where φ is the angle of internal friction and μ = tanφ.

Wing cracks are caused by local tensile stresses at the tip of the joint due to frictional sliding of the joint surface. According to Lee et al. [25], as the load increases, the wing crack expands along the direction of maximum tensile stress, approximately 70.5°, as shown in Fig. 10. Considering the direction of wing crack expansion, the stress intensity factors KI and KII at the nodal tip are modified as follows:

$$\left\{ {\begin{array}{*{20}c} {{\text{K}}_{{\text{I}}} = - \frac{{{\text{2a}}\tau_{eff} {\text{sin}}\theta }}{{\sqrt {{\uppi }\left( {{\text{l}} + {\text{l}}^{*} } \right)} }} + \sigma_{n} \left( {\sigma {,}\alpha + \theta } \right)\sqrt {\pi {\text{l}}} } \\ {{\text{K}}_{{{\text{II}}}} = - \frac{{{\text{2a}}\tau_{eff} {\text{cos}}\theta }}{{\sqrt {\pi ({\text{l}} + {\text{l}}^{*} )} }} - \tau_{n} \left( {\sigma {,}\alpha + \theta } \right)\sqrt {\pi {\text{l}}} } \\ \end{array} } \right.$$
(19)

where a is half the crack length; l is the extension length of the crack; θ is the extension angle; and l* = 0.27a is introduced so that KI and KII are non-singular for l = 0.

Fig. 10
figure 10

Schematic diagram of wing crack growth model

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Consider the critical state of wing crack extension, i.e. at l = 0, the strength factors KI and KII are given by:

$$\left\{ {\begin{array}{*{20}c} {{\text{K}}_{{\text{I}}} = - \frac{{{\text{2a}}\tau_{eff} {\text{sin}}\theta }}{{\sqrt {\pi ({\text{l}} + {\text{l}}^{*} )} }}} \\ {{\text{K}}_{{{\text{II}}}} = - \frac{{{\text{2a}}\tau_{eff} {\text{cos}}\theta }}{{\sqrt {\pi ({\text{l}} + {\text{l}}^{*} )} }}} \\ \end{array} } \right.$$
(20)

Combining Eqs. (16) and (20), an expression for the macroscopic damage to fractured rock specimens can be obtained as:

$${\text{D}}_{{\text{f}}} = {1} - \left[ {{1} + \frac{{{29}{\text{.63cos}}^{{2}} \alpha \left( {{\text{sin}}\alpha - {\text{cos}}\alpha {\text{tan}}\varphi } \right)^{{2}} {\text{B}}^{*} {\text{a}}^{{2}} \left( {{1} - \nu^{{2}} } \right)}}{{\pi {\text{V}}}}} \right]^{{ - {1}}}$$
(21)

where B* is the equivalent fracture depth.

Based on the theory of damage mechanics, the definition of damage variables is the basis of the damage model. In the study of rock specimens, macroscopic damage is defined as the initial damage to the prefabricated fractured rock node fractures, and the initial damage does not change during the loading process. Fracture propagation is caused by the continuous development and expansion of microscopic cracks during loading.

According to Lemaitre's assumptions, macro–micro damage is coupled based on the strain equivalence principle, assuming that the elastic modulus of a rock with macroscopic damage, a rock with microscopic damage only, a rock with macro–micro damage only and a rock with no damage at all in uniaxial compression are Em, Ef, Ep and E0 respectively. The strain expressions are as follows:

$$\left\{ {\begin{array}{*{20}c} {\varepsilon_{p} = \varepsilon_{m} + \varepsilon_{f} - \varepsilon_{0} } \\ {\frac{\sigma }{{{\text{E}}_{{\text{p}}} }} = \frac{\sigma }{{{\text{E}}_{{\text{m}}} }} + \frac{\sigma }{{{\text{E}}_{{\text{f}}} }} - \frac{\sigma }{{{\text{E}}_{{0}} }}} \\ \end{array} } \right.$$
(22)

Based on the Lemaitre equivalence variation principle, the following expression can be obtained:

$$\left\{ {\begin{array}{*{20}c} {{\text{E}}_{{\text{p}}} = {\text{E}}_{{0}} \left( {{1} - {\text{D}}_{{\text{p}}} } \right)} \\ {{\text{E}}_{{\text{m}}} = {\text{E}}_{{0}} \left( {{1} - {\text{D}}_{{\text{m}}} } \right)} \\ {{\text{E}}_{{\text{f}}} = {\text{E}}_{{0}} \left( {{1} - {\text{D}}_{{\text{f}}} } \right)} \\ \end{array} } \right.$$
(23)

The combination of expression (22) and expression (23) is:

$${\text{D}}_{{\text{p}}} = {1} - \frac{{\left( {{1} - {\text{D}}_{{\text{m}}} } \right)\left( {{1} - {\text{D}}_{{\text{f}}} } \right)}}{{{1} - {\text{D}}_{{\text{m}}} {\text{D}}_{{\text{f}}} }}$$
(24)

If we define the state of the fractured rock specimen after acid corrosion as the first damage state and the state under uniaxial compressive loading as the second damage state, the intrinsic relationship of the fractured rock body under acid corrosion is as follows:

$$\sigma_{1} = {\text{E}}_{{0}} \left( {{1} - {\text{D}}} \right){\upvarepsilon }_{{1}} = {\text{E}}_{{0}} \left( {{1} - {\text{D}}_{{\text{c}}} } \right)\left( {{1} - {\text{D}}_{{\text{p}}} } \right)\varepsilon_{1} ,$$
(25)

where D is the total damage variable for the uniaxial compression test after corrosion of the fractured rock by the acid solution. Combining Eq. (24) and Eq. (25) we obtain:

$${\text{D}} = {1} - \left( {{1} - {\text{D}}_{{\text{c}}} } \right)\frac{{\left( {{1} - {\text{D}}_{{\text{m}}} } \right)\left( {{1} - {\text{D}}_{{\text{f}}} } \right)}}{{{1} - {\text{D}}_{{\text{m}}} {\text{D}}_{{\text{f}}} }}$$
(26)

5 Verification of constitutive model

In order to verify the adequacy of the present constitutive model, the specimen with the peak stress closest to the average value was selected for analysis based on the results of the compression test of the specimen. The parameters m and F0 were solved by the peak stress method proposed in Sect. 4. The corresponding mechanical parameters are shown in Table 2.

Table 2 Mechanical parameters of rock specimen under different conditions
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From the values of the solved distribution parameter m it can be seen that the same type of rock specimen decreases in m value and becomes less brittle after corrosion by acid solution. The value of the distribution parameter F0 varies regularly with the strength of the rock sample; the greater the strength of the rock, the greater the value of F0. The theoretical curves were obtained by substituting the solved distribution parameters into the damage constitutive equation and comparing them with the uniaxial compression test curves. Figure 11 shows a comparison of stress–strain curves for fractured rock under normal maintenance conditions and Fig. 12 shows a comparison of stress–strain curves for fractured rock under acid solution maintenance conditions.

Fig. 11
figure 11

Stress–strain experimental and theoretical curves for specimens with different fracture inclinations under normal conditions

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Fig. 12
figure 12

Stress–strain experimental and theoretical curves for specimens with different fracture inclinations under acid corrosion

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As can be seen from Figs. 11 and 12, the damage constitutive model established in this paper can better reflect the stress–strain relationship between the stress-elastic deformation stage and the plastic yielding stage of rock specimens containing fractures. The experimental curves for the early compressive and post-peak stress stages in fractured rock specimens differ slightly from the theoretical curves because the damage values in the theoretical curves are obtained on the basis of Lemaitre equivalent strains, where the strain due to the stress acting on the damaged rock specimen is equal to the strain due to the actual stress acting on the undamaged rock specimen. The rock sample is closed by microcracks during the initial compaction stage, a stage in which the rock material becomes more resistant to deformation and can be considered as a reverse damage process. However, the present constitutive model considers the damage caused by the expansion of new cracks during loading and does not take into account the suppression effect of the initial micro-crack closure. The fractured rock specimens have more pronounced brittle damage in the post-peak phase, with no stable peak residual strength.

6 conclusion

The peak stress, peak strain and elastic modulus of specimens at 0°, 30°, 45° and 60° crack inclination under the same maintenance conditions, with the change of crack inclination, the peak stress, peak strain and elastic modulus all show regular changes, with the increase of crack inclination first decreasing and then increasing, with the most significant change at 45° inclination.

Peak stress, peak strain and elastic modulus of the unfractured specimens were higher than those of the fractured specimens under normal and acidic solution conditions. The peak stress and elastic modulus tended to decrease and the peak strain tended to increase after acid solution maintenance, indicating that the strength and stiffness of the specimens were weakened to different degrees and the ductility of the specimens increased after acid solution maintenance under the same conditions.

Based on the three main types of cracks in uniaxial compression, by observing and analysing the fracture characteristics of specimens with different crack inclination angles under normal and acidic conditions, with the increase of the crack inclination angle, the change of damage shape tends to gradually change from shear damage dominated to tensile-shear composite damage and finally back to shear damage dominated. After acid corrosion, the fracture characteristics of specimens with the same crack inclination are basically the same, the damage pattern of the fractured rock specimens is more influenced by the inclination angle and less influenced by acid corrosion.

Based on the Lemaitre strain equivalence principle, the macroscopic fracture model of single-fracture sandstone and the Weibull distribution statistical damage model of micro-fracture were combined to derive the macroscopic coupled damage variables of the specimens, while the acid corrosion effect was considered to couple the above damage variables and derive the damage constitutive model under uniaxial compression.

The damage constitutive model curves and test curves of the fractured rock mass under normal and acid corrosion conditions have a high degree of agreement, which can effectively reflect the coupling effect of acid corrosion damage, microscopic damage and macroscopic damage. The mechanical characteristics of the specimens at different stages of uniaxial compression are analysed and the theoretical-test stress–strain curves are combined to show that the established damage constitutive model has good reliability in the elastic deformation and plastic yielding stages.