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BY 4.0 license Open Access Published by De Gruyter Open Access October 18, 2022

Fission characteristics of heavy metal intrusion into rocks based on hydrolysis

  • Feng Cheng , Zhi-hua Huang , Chun-hui Su EMAIL logo , Ai-jun Chen , Jun-hua Chen and Di Wu
From the journal Open Geosciences


The mechanism of hydrolysis and extension diffusion of heavy metal pollution elements infiltrated into rock is analyzed by the theory of ion hydrolysis displacement. The hydrolysis properties of typical elements such as cadmium, zinc, lead, and copper are verified by convective dispersion model, and the diffusion law and fission characteristics of heavy metal with different hydrolysis constant are discussed. A three-dimensional constitutive relation model of rock extension diffusion surface is established by combining viscoelastic monomer model with a damage monomer model. Considering the influence of diffusion coefficient, hydrolysis constant, deformation factor, and other parameters, the rationality of the test results and model fitting results of heavy metal invading rock are verified. The results show that the replacement rate of colloidal mineral elements in rock varies with different hydrolysis constant, when the hydrolysis constant is large, the extension diffusion rate in rock is large; otherwise, the extension diffusion rate is small. Constitutive relation curves of polluted rock with different lithologies are in good agreement with the fitting results of the combination model under the influence of the same test conditions and the same parameters.

1 Introduction

The safety protection problem of heavy metal intrusion into rock slope caused by metal mining has been puzzling the mining development. Especially, the problem of mine slope instability caused by the infiltration of heavy metal contaminated water has been unable to be effectively solved for a long time. The reason is the fission of the colloidal mineral structure caused by the invasion of heavy metal elements. The effect cannot be effectively identified, which leads to the destruction of rock mass structure and the reduction of mechanical properties, which affects the stability and safety of the slope. On the basis of this, this article intends to carry out the research on the fission response law, which can effectively identify the microstructure damage of rock mass under heavy metal intrusion and solve the problem of unreasonable calculation in the process of slope protection design by determining reasonable calculation parameters.

In the field of micro damage mechanics of rock mass, there are many micro influencing factors that destroy mineral colloid, among which the influence of heavy metal intrusion is the most direct and the most destructive. The hydrolysis displacement of the rock microstructure is the main cause of fission failure. However, due to the lack of research in this field by scholars at home and abroad, and the relatively long and complex hydrolysis process, the research on the hydrolysis failure process and mechanism after heavy metal ions intrusion has not made great progress.

Hydrolysis and hydration of heavy metals intruded into rocks can lead to the damage of rock mass. In the research of rock mass damage mechanics, some results have been obtained on the invasion and destruction mechanism under the heavy metal ions hydrolysis. Guo et al. [1] carried out the rock permeability test by pushing rock mass to study the primary pore hydration, which gained the relationship between time and thrust. Yang et al. [2] published the ion exchange law of rock mass that was verified by the simulation test method of diagenetic fracture, structural fracture, and weathering fracture. Du et al. [3] studied the exchange process of coordination molecules, and solvent molecules in rock mass by ion leaching test of rock cemented minerals and expounded the failure mechanism of cemented minerals with fracture degree. Gong et al. [4] conducted the mineral colloid hydration test of mudstones and sandstones. The test results showed that the greater the ionic hydration constants, the smaller the ion radius and the larger the charge number.

The previous research results show that the destroyed rock mineral structure is the fundamental cause of rock mass damage. However, the mechanism of mineral structure destruction has not been clearly explained [3,5]. With respect to the effect of the heavy metal ions on rock mass mineral hydrolysis, a number of conclusions of the surface body structure damage have been come up with by some scholars [4,6] by using the permeation test considering exfoliation of intrusion surface. However, the research findings of the hydrolysis effect of contaminated rocks on heavy metal invasion have not been reported [2,7]. A series of experiments and numerical studies on hydrolysis equilibrium and dynamic micro invasion characteristics have been carried out by foreign scholars [3,8], and the dynamic change process of colloidal structure initiation, diffusion, and destruction of mineral materials in rock and soil after invasion has been obtained. However, most of the studies focus on the action category caused by the single hydrolysis reaction under macro conditions, especially the study on the relationship between micro structure and the volume change of rock and soil under the invasion of heavy metals. The internal mechanism of invasion failure and strength reduction of rock and soil has not been explained from the micro perspective of hydrolysis replacement. The main reason is that the hydrolysis of rock and soil has timeliness and is limited by the testing equipment. It is difficult to observe and study the microstructure damage.

In this article, to further reveal the influence law of heavy metal intrusive elements on rock fission characteristics under hydrolysis, the three-dimensional combined damage model of intrusion control equation and extended diffusion surface is established according to acid-base proton theory, convection–dispersion characteristics of microporous media and monomer model, and studies on fission and strength properties of heavy metal invaded rocks under hydrolysis.

Based on the hydrolysis mechanism and extension diffusion mechanism of heavy metal pollution elements penetrating into rocks, a three-dimensional constitutive relationship model is proposed by combining a viscoelastic monomer model with a damage monomer model. The influence of diffusion coefficient, hydrolysis constant, and deformation factor on the displacement rate and extension diffusion rate of colloidal mineral elements in rocks is determined. The problem of stability calculation of rock slope with heavy metal intrusion is solved, which provides important calculation parameters and theoretical basis for the safety design of rock slope.

2 Theoretical method

The hydrolysis of heavy metals into rocks is the key factor for the loss of mineral ions. This study shows that the exchange process meets the chemical principle of ion replacement. According to the hydrolysis displacement of mineral microstructure, the control equation of hydrolysis suitable for rock microstructure is explored, and then the hydrolysis displacement migration model is established to obtain the key parameters, which provides basic data and theoretical support for the establishment and calculation of the intrusion fission constitutive model.

2.1 Establish the intrusion control equation

2.1.1 Analysis of hydrolysis of rock microstructure

According to the principle of hydrolysis and hydration, the hydration membrane formed by colloidal mineral ions dissolving in water expands the space for water molecules, the aggregation effect between ions and water molecules forms the energy of interaction, which makes water molecules move along with ions. The original molecular structure of water will be destroyed when the generated energy is greater than the hydrogen bonding energy between molecules. The energy layer of colloidal particle is divided into two layers according to their bond energy. The energy of the inner layer is larger and is not affected by external factors such as temperature, and hence, the ions and water molecules are firmly combined and do not have the ability to move independently [9]. For the outer layer, the attraction of ions and water molecules is weaker because of the greater intermolecular distance. Because of the influence of external factors such as temperature, ions and surrounding water molecules can separate from the energy matrix and constantly exchange with external molecules. When the hydrolysis constant is greater than that of its own element, hydration will occur [2,10]. The results show that the ion’s hydration reduces the free water molecule numbers and increases the volume of ions. The conductivity and activity coefficient of electrolytes between adjacent water molecules and ions are changed. That is, the dielectric constant is reduced, and the stable crystal structure between mineral particles is destroyed, as shown in Figure 1.

Figure 1 
                     Arrangement of mineral wafer structure in rock and soil.
Figure 1

Arrangement of mineral wafer structure in rock and soil.

2.1.2 Hydrolysis displacement transport model

The heavy metal elements intrude into the rock mass and displace with mineral ions in the rock mass under the action of hydration. Because the invasive elements are highly reactive and toxic, it is easy to react with ions with low coordination potential energy [3,11]. The adsorption potential produced by heavy metals in contact with mineral ions in rock mass enhances the hydrolysis ability, increases the thickness of the double diffusion layer and the molar concentration in the hydration membrane, and reduces the ion potential energy of mineral colloids, resulting in hydrolysis displacement reaction. In this way, the hydrolytic displacement reaction occurs. The law of migration and diffusion is shown in Figure 2.

Figure 2 
                     Diffusion and transport model of heavy metal ions in hydration membrane.
Figure 2

Diffusion and transport model of heavy metal ions in hydration membrane.

According to the regularity of hydration and hydrolysis displacement of crystal structure in the rock mass, the invasion of heavy metals into rock mass increases the diffusion repulsion force of mineral element ions, which breaks the mechanical equilibrium of the colloidal structure. The hydration expansion between adjacent wafers causes the mechanical equilibrium state to be readjusted, which leads to that expansion, and extension strain occurs on the rock expansion fracture surface. The hydrolysis process satisfies the acceptance mechanism of acid–base proton theory [4], as shown in equation (1).

(1) M ( H 2 O ) m n + H 2 O = M ( OH ) ( H 2 O ) m 1 ( n 1 ) + + H 3 + O .

According to equation (1), it can be seen that the hydrolysis ability of metal ions is determined by the bond energy. The radius, charge, and potential of metal ions are the most important factors affecting the induction and acceptance of coordination water molecules H 3 + O , which is usually expressed in terms of hydrolysis constants PKm. The empirical formula is as follows [11,12]:

(2) PKm = 19.04 r 3.65 Z + 3.56 M 0.74 ( Z 2 / r ) + 1.16 .

In this equation, r is the ion radius, M is the electron shell number, Z is the charge num, and Z 2/r is the on-potential.

From equation (2), the stability of mineral elements in rock-soil mass is determined by mineral ions. In general, the hydrolysis constants of mineral elements are small and stable, and the hydrolysis itself will not cause structural changes. Table 1 presents the hydrolysis constants of typical metal ions obtained from experiments [4,13] and calculations.

Table 1

Test and calculation of hydrolysis constant of metal ions

Ion name Radius (r) Cycle (M) Electric charges (Z) Ionic potential (Z 2/r) PKm
Calculation value Test value
K+ 1.071 3 1 0.934 27.89 5.07
Na + 1.164 3 1 0.859 29.72 4.23
Mg2+ 0.862 3 2 4.640 17.52 6.18
Ca2+ 1.144 4 2 3.497 27.29 5.41
Al3+ 0.675 3 3 13.333 3.88 3.93
Zn2+ 0.89 4 2 4.494 21.72 7.32
Hg2+ 1.167 6 2 3.428 34.90 5.94
Pb2+ 1.331 6 2 3.005 38.34 9.51
Fe2+ 0.925 4 2 4.324 22.51 8.06
Cd2+ 1.643 6 1 0.859 49.52 13.61
Cu2+ 0.876 4 2 4.566 21.40 6.86
Cr3+ 0.755 4 3 11.921 10.00 2.82

From Table 1, elements with larger hydrolysis constants are Pb, Cd, and Zn, which are typical elements of contaminated rock-soil mass. The colloidal mineral elements in the rock mass are mostly composed of Ca, K, Mg, and Na, whose respective hydrolysis constants are small compared with Pb, Cd, and Zn. When heavy metals such as Pb and Cd intrude into rock mineral colloid, hydrolysis will occur, and the original ions such as Ca and Mg are replaced to form new bonds – OH and Zn. Cu invasion hydrolyzed and replaced Mg ions. As the equilibrium of cemented structure is broken, the loss of metal elements in rock minerals accelerates, which gives rise to the shrinkage of intrusive structural planes and deepening of the degree of cracking. This leads to integrity damage, and the mechanical properties decrease.

2.1.3 Extended diffusion control equation

The bond energy between mineral elements is in high coordination bond energy because of the large cementation force of rock microstructures. The process of hydrolysis and replacement is relatively slow. Because of the invasion of heavy metal elements, the rock mass medium has porous characteristics, which can be represented by a convection–dispersion model [3,11], as shown in equation (3):

(3) R d C t = κ 2 C x 2 ν C x .

In this equation, R d is the retardation coefficient, which can be determined by the empirical value according to the type of rock, and the value range of R d = 1 ∼ 2.5; C is the concentration of invasive heavy metal pollutants; κ = PKm is the diffusion coefficient, which is determined by permeation test; and ν is the extended diffusion rate.

R d and κ in equation (3) can be taken as constants. Equation (3) can be simplified into the following expression:

(4) C t = κ 2 C x 2 ν C x .

In this equation, κ* = κ/R d , ν* = ν/R d .

Because of the selectivity of ion hydrolysis reaction, the invasion rate is basically the same in the micro invasion surface structure. In equation (4), it is necessary to determine the initial and boundary conditions of the destruction of cemented mineral elements by heavy metalpollution elements. It is assumed that the initial concentration of intrusive elements in heavy metal contaminated rocks is 0. In penetration tests, the concentration of intrusive ions is usually measured by a spectroscopic detector, which is used as a known value in calculation. Assuming that the initial concentration of intrusive elements in heavy metal contaminated rocks is C 0, the calculated boundary conditions can be expressed as follows:

(5) C ( 0 , t ) = C 0 ( t > 0 ) ,

(6) C ( x , 0 ) = 0 ( x 0 ) .

In fact, the intrusion of heavy metal contaminants into rock mass is affected by the retardation of mineral cementation force, and the intrusion distance and path are limited. Especially under the control of the adsorption of layer fissures, the time of extension and diffusion is limited, and the intrusion time limit satisfies the following expression:

(7) C ( , t ) = 0 ( t > 0 ) .

To obtain the relationship between the elongation diffusion rate and time, according to the initial conditions, boundary conditions, and intrusion time limits of the flow-dispersion model, equations (4)–(7) are derived and sorted, and the expression of the relationship between the concentration of heavy metal pollutants and time in intrusive rocks is obtained. The expression is as follows:

(8) C ( x , t ) = C 0 2 erfc x ν t 2 κ t + exp ν x κ erfc x + ν t 2 κ t .

where C 0 x, κ * , and ν * have the same meaning as in the previous text; t is time; and erfc is the error function commonly used in engineering.

To verify the relationship between elongation diffusion characteristics and elongation rate and time, limestone, limestone shale, and purple sandstone contaminated by heavy metals were selected as test objects, strength of which are medium. The common heavy metal pollution elements are lead, cadmium, copper, and zinc. Samples were assured of good integrity and uniformity. For the convenience of calculation, the average values of rock test results of three different lithologies were taken as the calculation parameters, as shown in Table 2.

Table 2

Calculating parameters

Element C 0 (mg L−1) R d (–) κ (cm2/s) ν (mm/s) t (/s)
Pb 1.62 × 10−3 1.28 1.91 × 10−5 1.63 × 10−4 6.35
Cd 1.41 × 10−3 1.21 1.72 × 10−5 2.51 × 10−4 18.76
Cu 1.39 × 10−3 1.25 1.76 × 10−5 2.56 × 10−4 38.92
Zn 1.43 × 10−3 1.21 1.75 × 10−5 2.52 × 10−4 158.21

Since the intrusive and permeable properties of rocks with different lithologies are different, there are some differences in the test results. The hydrolysis ability parameters of the intrusive elements presented in Table 1 are substituted into equation (8) to calculate the concentration of the intrusive heavy metals. The hydrolysis ability curves of the four elements are plotted according to the calculation results as shown in Figure 3.

Figure 3 
                     Relationship curve of hydrolysis performance of intrusive elements.
Figure 3

Relationship curve of hydrolysis performance of intrusive elements.

Figure 3 shows that the hydrolysis abilities of four typical metal elements are in the descending order of Cd2+, Pb2+, Cu2+, and Zn2+. The elongation rate is also proportional to the hydrolysis capacity, which increases with time, increasing the ion concentration and mass ratio. The elongation rate is the main factor affecting the elongation capacity. Therefore, the hydrolysis ability of pollutant ions determines the intrusion ability in rock fissures.

2.2 Constitutive model of intrusive fission

2.2.1 Basic assumptions

  1. It is assumed that the unit body is a viscoelastic body before hydrolysis occurs [11], and the relationship of σ a ε a conforms to the linear effective superposition principle [14].

  2. The damaged body of rock element is D a and the viscous cylinder block is η b , and there are no damage characteristics. The computational unit is a combination of damage and viscous characteristics. The combination mode of monomer model is in series, which was established by Du et al. [3] and Gong et al. [4].

  3. The elongation rate of hydrolysis is relatively slow [6,12], and the inertia effect can be neglected in constructing the constitutive relationship.

  4. Rock D a is linear elastomers before hydrolysis occurs. The anisotropic damage fission of D a obeys the law (m, a) of probability distribution. Fission parameter D can be written as follows [10]:

    (9) D = 1 exp [ ( ε a / α ) m ] ( ε a 0 ) .

    σ a ε a represents the constitutive relationship [7]:

    (10) σ a = E ε a ( 1 D ) ( ε a 0 ) .

    where E is the elastic modulus of rock.

  5. Constitutive relation of the sticky cylinder block η b is shown as follows [7,15]:

    (11) σ b = η d ε b / d t .

  6. Hydrolysis extends gradually along the fracture of the intrusion surface after heavy metals intrude into rocks. The rocks on both sides of the extension surface are regarded as nondestructive materials because of the slow extension speed. Cauchy stress is equivalent to effective stress, which conforms to Lemaitre strain effective principle [16,17]. That is, the stress–strain relationship of the extension of fracture surface in a rock mass can be obtained from the constitutive model of the viscoelastic body.

2.2.2 Constitutive model of hydrolysis damage Constitutive relation before extended diffuse-on damage

According to the basic assumptions of the monomer model in Section 2.2.1, the three-dimensional rock mass is basically intact before the intrusion and diffusion failure, and the constitutive monomer model before the failure is established. It can be expressed in the following equation:

(12) S i j = 2 G e i j ,

(13) σ m = 3 K ε m .

where S ij is stress; G is shear modulus; e ij is strain deviation tensor; and K is bulk modulus, which can be expressed by the following equation: K = E 2/3(1 − 2ν).

The relationship among e ij , S ij , and σ m is given as follows:

(14) σ i j = S i j + δ i j σ m .

The relations among e ij , ε m , and ε ij is given as follows:

(15) ε i j = e i j + δ i j ε m .

where δ ij is a Dirac symbol.

According to the basic assumption (5) given in Section 2.2.1, the rock is a viscoelastic body before it is destroyed, which conforms to the Lemaitre strain effective principle, and its constitutive relationship is given as follows:

(16) f ( d ) σ = g ( d ) ε .

By substituting equations (12) and (13) into (16), the three-dimensional extension equations before rock diffusion failure are obtained:

(17) f ( d ) S i j = 2 g ( d ) e i j ,

(18) f 1 ( d ) σ m = 3 g 1 ( d ) ε m .

In the equation, f(d) and g(d) are related to mineral fission, and f 1(d) and g 1(d) are related to mineral extension and compression, respectively.

Equations (17) and (18) are higher-order linear differential equations, satisfying the principle of effective superposition. They are analytical solutions of the higher-order linear differential equations. The stress boundary conditions of the element need to be determined according to the principle of stress superposition. The derivation process is as follows.

The contaminated rock has not been in the state of extended diffusion before the hydrolytic replacement of the contaminated rock. The initial stresses along the x-axis, y-axis, and z-axis at the initial time are S x0, S y0, and S z0, respectively. When t is zero, σ and ε are given by the following formula:

σ x ( t 0 ) = σ y ( t 0 ) = σ z ( t 0 ) = 0 , ε x ( t 0 ) = ε y ( t 0 ) = ε z ( t 0 ) = 0 .

When t is zero, the initial force in the x-axis direction is assumed to be σ r (t), and equations σ z = S z0 + σ r (t) and ε z = ε z0 + ε r (t). ε z0 can be obtained from the following equation:

(19) ε z 0 = 1 E 1 [ S z 0 ν ( S x 0 + S y 0 ) ] .

where E 1 is the elastic modulus of rock before hydrolysis.

Due to the relatively slow spreading process on the intrusion surface, the influence of inertia effect is negligible according to the basic assumption (3) given in Section 2.2.1. The initial stress of rock is nonzero, and the constitutive relationship in a three-dimensional state accords with Laplace’s theorem, as shown in equation (20).

(20) L [ h ( t + t 0 ) ] = h ¯ = 0 + e p t h ( t + t 0 ) d t L [ d h ( t + t 0 ) ] = p L [ h ( t + t 0 ) ] h ( t 0 ) = p h ¯ .

According to basic assumption (4) given in Section 2.2.1 and Laplacian transformation equations (20), (17), and (18) are transformed as follows:

(21) f ( p ) S ¯ i j = 2 g ( p ) e ¯ i j ,

(22) f 1 ( p ) σ ¯ m = 3 g 1 ( p ) ε ¯ m .

Comparing equation (12) with equations (21), (13), and (22), it is found that if V in equation (21) is replaced by s in equation (12) and B in equation (22) is replaced by A in equation (13), the meaning expressed by equations (12) and (21) is the same, and the meaning expressed by equations (13) and (22) is the same.

According to the basic assumption (1) in Section 2.1, the rock mass contaminated by heavy metals is viscous in the initial state and has damage characteristics due to hydrolysis replacement after intrusion. The stress–strain relationship accords with the equivalent characteristics of the viscous cylinder block and the damaged body. The concrete expressions are as follows:

(23) σ = σ a1 + σ b = σ a2 ε = ε 1 + ε 2 ε 2 = ε b .

where σ a1 is the stress of damaged body, σ b is the stress of viscose cylinder before heavy metal ion invades, σ a2 is the stress of extended injury assembly, ε 1 is strain of damaged body, ε 2 is the strain of assembly, and ε b is equivalent strain of assembly.

According to equation (10) in the basic assumption (4) in Section 2.2.1, the constitutive equation of linear elastomer, it can be seen that σ a1 and σ a2 can be expressed, respectively, as follows: σ a1 = E 1 ε 1(1 − D) and σ a2 = E 2 ε 2(1 − D).

Similarly, according to the constitutive relation of the viscous cylinder block equation (11), the expression σ b is as follows: σ b = ηdε 2/dt. Because of the continuity of hydrolysis after rock intrusion, the constitutive relation of viscoelastic rigid body and intrusion can be linked by stress–strain relationship. That is, the expression of constitutive relation model of combination can be obtained. Functional equations σ a1 = E 1 ε 1(1 − D)σ a2 = E 2 ε 2(1 − D) and σ b = ηdε 2/dt are substituted into equation (23), and the integral calculation is carried out. The constitutive relationship before hydrolysis is obtained as follows:

(24) η σ + [ E 1 ( 1 D ) + E 2 ( 1 D ) ] σ = E 2 ( 1 D ) η ε + [ E 1 ( 1 D ) ε ] .

where E 1 is the same as in the previous text and E 2 is the elastic modulus of rock mass after being extended and diffused.

According to the basic assumption (2) Section 2.2.1, the rock mass structure is still in the viscoelastic state in the early stage intruded by heavy metal pollutants, and the damage characteristics of rock mass are not considered at this time. When solving equation (24), elastic modulus E can be used instead of E(1 − D). The constitutive relation of viscoelastic body can be simplified as follows:

(25) η σ + ( E 1 + E 2 ) σ = E 2 ( η ε + E 1 ε ) .

Equation (25) is transformed into the Laplacian equation, and the result is expressed as follows:

(26) ( η p + E 1 + E 2 ) σ ¯ i = E 2 ( η p + E 1 ) ε ¯ i .

Comparing equations (26) and (21) by equivalent transformation, f(p) and g(p) can be obtained as follows:

(27) f ( p ) = η p + E 1 + E 2 ,

(28) g ( p ) = E 2 ( η p + E 1 ) .

By equivalent transformation of equations (21) and (22), S i j ¯ and can be obtained as follows:

(29) S ¯ i j = 2 E 2 ( η p + E 1 ) η p + E 1 + E 2 e ¯ i j , S ¯ = K e ¯ .

where K is the bulk modulus, which can be calculated by equation K = E 2/3(1 − 2ν) instead of equation g 1(p)/f 1(p).

The combination model has been established, and the specific problems are determined according to the initial conditions, which satisfy the superposition principle and strain-effective principle. The initial state satisfies the following boundary conditions: σ x = S x0, σ y = S y0, σ z = S z0 + σ r .

Hence, we obtain the following equations:

(30) S ¯ = S x 0 + S y 0 3 p + σ ¯ z 3 S ¯ z z = 2 σ ¯ z 3 S x 0 + S y 0 3 p .

By equivalent transformation of equations (29) and (30), equations (31) and (32) can be obtained.

(31) e ¯ = S x 0 + S y 0 9 K p + σ ¯ z 9 K ,

(32) S ¯ z z = 2 σ ¯ z 3 S x 0 + S y 0 3 p = 2 E 2 ( η p + E 1 ) η p + E 1 + E 2 ( ε ¯ z e ¯ ) = 2 E 2 ( η p + E 1 ) η p + E 1 + E 2 ε ¯ z S x 0 + S y 0 9 K p σ ¯ z 9 K .

Equation (32) is transformed by Laplace transform, and equation (33) is obtained.

(33) 6 ( η p + E 1 + E 2 ) + 2 E 2 ( η p + E 1 ) K σ ¯ z = 18 E 2 ( η p + E 1 ) ε ¯ z + 3 ( S x 0 + S y 0 ) η + E 1 + E 2 p 2 E 2 ( S x 0 + S y 0 ) K η + E 1 p .

Equation (33) is simplified to facilitate the practical application of the equation. Equation (34) is simplified as follows:

(34) σ ¯ z = 9 E 2 ( 3 K + E 2 ) η η + E 1 β η p + β ε ¯ z + S x 0 + S y 0 2 ( 3 K + E 2 ) η γ p + δ P + β .

The rock is intruded, of which the fission characteristics are continuous and satisfy the reverse transformation process of Laplace’s theorem. According to the time effect, the constitutive equation of rock surface intrusion can be obtained by inverse derivation of equation (34). The constitutive relation is given as follows:

(35) σ z ( t + t 0 ) = 9 K E 2 ( 3 K + E 2 ) η η ε z ( t + t 0 ) + E 1 β η 0 t ε z ( τ + t 0 ) e β ( t + t 0 τ ) d τ S x 0 + S y 0 2 ( 3 K + E 2 ) η [ γ + δ e β ( t + t 0 ) ] .

When the equation ε z (τ + t 0) = ε z0 + ε r (t) = ε z0 + Ct is satisfied, the following equation can be obtained from equation (35):

(36) σ z ( t + t 0 ) = 9 K E 2 ( 3 K + E 2 ) η η ( ε z 0 + ε r ( t ) ) + E 1 β η β ( ε z 0 + ε r ( t ) C ) e β t 0 E 1 β η β ( ε z 0 C ) e β ( ε r ( t ) C + t 0 ) + S x 0 + S y 0 2 ( 3 K + E 2 ) η γ + δ e β ( ε r ( t ) C + t 0 ) .

where σ z (t + t 0) S is the time–effect stress of the extended diffusion surface (t ≠ 0); ε z0 is the initial strain; S x0, S y0, and S z0 composite stress in three directions; β, γ, and δ can be expressed by the following equations:

(37) β = 3 K ( E 1 + E 2 ) + E 1 E 2 ( 3 K + E 2 ) η ,

(38) γ = 3 K ( E 1 + E 2 ) 2 E 1 E 2 β ,

(39) δ = 3 ( K 2 E 2 ) η γ .

Condition S x0 = S y0 = S z0 is satisfied, which indicates that in the initial stage of rock invasion by heavy metal pollutants, the hydrolysis has not yet occurred. The model only represents the constitutive relation before the extension and diffusion of the intrusion surface. Constitutive relation after extended diffuse-on damage

A large number of test results show that the rock surface under the static load generally shows shear yield failure [8,18], which conforms to the Coulomb criterion. According to the failure combination relationship of rock elements established by Bian et al. [19], the variable of extension diffusion surface failure is introduced as follows:

(40) D = 1 exp ε z α 1 + sin ϕ 1 sin ϕ 2 ν σ x 0 + σ y 0 2 ( E 2 α ) m .

where ν is Poisson ratio, ϕ is the internal friction angle, and m is the deformation factor, the value of which is related to the lithology of heavy metal contaminated rock and the stress state before failure.

In the initial stage of hydrolysis of heavy metal contaminants in the intrusive rock mass, the intruded area is still in the stress–strain elastic range. Liu et al. [5] and Parisio and Laloui [20] considered that there is no obvious stress damage on the intrusive rock surface at this time. When the hydrolysis of the intruded surface diffuses, obviously, the fission characteristics begin to appear gradually, and the intruded surface shows stress relaxation and damage. On the rock surface, when the hydrolysis diffuses, obviously, the fission characteristics begin to appear, and the intrusion surface is damaged due to stress relaxation. Therefore, the damage of the extended diffusion surface caused by the heavy metals hydrolysis after intrusion into rocks conforms to the stress–strain initiation criterion [17,20], which can be expressed by the evolution equation (41).

(41) D = 1 exp ε z α 1 + sin ϕ 1 sin ϕ 2 ν σ x 0 + σ y 0 2 ( E 2 α m when ε z > 1 + sin ϕ 1 sin ϕ 2 ν ( σ x 0 + σ y 0 ) 2 E 2 0 when ε z 1 + sin ϕ 1 sin ϕ 2 ν ( σ x 0 + σ y 0 ) 2 E 2 .

From the basic assumption (6) in Section 2.1, equation (41) conforms to Lemaitre’s strain equivalence principle. Equation (41) is transformed according to the time effect parameter theory. The traditional parameters E 1, E 2, and K in equation (36) are replaced by time-effect parameters E 1 [ 1 D ( t + t 0 ) ] E 2 [ 1 D ( t + t 0 ) ] and K [ 1 D ( t + t 0 ) ] of elastic modulus and bulk modulus, respectively. Then substituting the expressions of β, γ, and δ into equation (36), the constitutive relation model of the failure of extended diffusion surface under hydrolysis can be obtained. Equation (41) can be changed into:

(42) D ( t + t 0 ) = D = 1 exp ε z 0 + ε r ( t ) / α 1 + sin ϕ 1 sin ϕ 2 ν σ x 0 + σ y 0 2 ( E 2 α m C .

where mC are the same as earlier. In this article, the external factor C is mainly the influence of heavy metal intrusion, extension, and diffusion. Because the direction of the extension and diffusion path is uncontrollable, m is the average value, which is 0.3–0.8.

In the combined constitutive model of diffusion failure of extension surface established earlier, in addition to determining parameters such as E 1 , E 2 , m, α and η, parameters such as PKm, K, φ, ν, and R d should be determined according to rock lithology and hydrolysis capacity. The other parameters can be determined by referring to the traditional model parameters determination method.

3 Test method

3.1 Comparison of three-dimensional static load theory and experiments on diffusion damage of contaminated rock extension surface

According to the damage types of rock slope caused by common heavy metal pollution, the rock contaminated by typical heavy metal elements is selected for routine mechanical tests. According to the requirement of rock lithology, limestone shale, purple sandstone, and marl are selected as test objects, its density is 1.98 g/cm3, and the compressive strength of rock is 11.98 MPa. The appearance and dimension of the sample are made according to the international standard. To ensure that specimens have no obvious defects and meet the requirements of completeness, uniformity, and surface smoothness, it is necessary to inspect and grind the cracks of the specimens. The specimen size is 50 mm × 50 mm × 100 mm so as to meet the requirements of international standard specimens.

A modified electro-hydraulic servo material machine with visual function, G-Instron 1342, was adopted in the experiment. The pressure sensor of the electro-hydraulic servo material machine is controlled by computer in the whole process of the experiment, of which maximum output load is ±300 kN. DH-5932 data acquisition instrument is used to complete the data of extended diffusion surface. DH-3840 recorder is used for dynamic data recording. The whole test process is controlled in real time by computer, including data recording and information acquisition. The intrusive pollutants selected from the contaminated rock samples are typical heavy metal contaminated elements such as lead, zinc, cadmium, and copper. The hydrolysis performance parameters can be obtained from the curves shown in Figure 3. The three-dimensional combined stress of the test is set as S x0 = S yo = S z0. The results of the test data are presented in Table 3 when the values of S x0 , S yo , and S z0 is 25 MPa.

Table 3

The test results and the fitting curve parameters of combined constitutive relation model under confining pressures of S x0 = S y0 = S z0 = 25 MPa

Rock tratum PKm C (/10−6 s−1) E 1 (/GPa) E 2 (/GPa) ϕ (/(°)) ν (–) K (/GPa) t 0 (/s) η (/GPa s)) R d (–) κ (cm2/s) α (/10−3) m (–)
Marl① 9.51 1.15 × 103 23.82 24.7 39 0.17 12.49 35.00 0–500 1.24 1.86 × 10−3 3.72 0.5
Purplish red sandstone② 8.99 1.15 × 105 23.82 26.9 39 0.25 17.81 35.00 0–500 1.26 1.79 × 10−3 3.91 0.5
Grey shale③ 9.26 1.15 × 106 23.82 31.5 39 0.21 21.36 35.00 0–500 1.19 1.81 × 10−3 5.23 0.5
Marl① 8.89 1.24 × 103 24.12 25.7 41 0.17 12.49 40.00 500 1.35 1.91 × 10−3 3.74 0.5
Purplish red sandstone② 9.56 1.24 × 105 24.12 27.9 41 0.25 17.81 40.00 500 1.28 1.92 × 10−3 3.95 0.5
Grey shale③ 9.42 1.24 × 106 24.122 34.5 41 0.21 21.36 40.00 500 1.24 1.86 × 10−3 5.26 0.5
Marl① 8.97 1.33 × 103 24.74 23.7 43 0.17 12.49 38.00 500 1.23 1.75 × 10−3 3.74 0.7
Purplish red sandstone② 9.26 1.33 × 105 24.74 27.6 43 0.25 17.81 38.00 500 1.18 1.83 × 10−3 3.93 0.7
Grey shale③ 9.35 1.33 × 106 24.74 31.5 43 0.21 21.36 38.00 500 1.31 1.72 × 10−3 5.24 0.7
Marl① 9.58 1.48 × 103 25.23 24.8 38 0.17 12.49 45.00 900 1.32 1.84 × 10−3 3.73 0.7
Purplish red sandstone② 9.47 1.48 × 105 25.23 26.4 38 0.25 17.81 45.00 900 1.25 1.79 × 10−3 3.92 0.7
Grey shale③ 9.35 1.48 × 106 25.23 32.6 38 0.21 21.36 45.00 900 1.27 1.83 × 10−3 5.25 0.7
Marl① 9.63 1.52 × 103 26.18 24.3 39 0.17 12.39 37.00 900 1.16 1.91 × 10−3 3.77 0.9
Purplish red sandstone② 9.12 1.52 × 105 25.24 26.9 39 0.25 17.61 37.00 900 1.25 1.84 × 10−3 3.96 0.9
Grey shale③ 9.85 1.52 × 106 25.24 33.2 39 0.21 21.46 37.00 900 1.32 1.93 × 10−3 5.28 0.9

4 Discussion of results

In the traditional composite constitutive model, the parameters determine the reliability of the fitting results. In this article, the hydrolysis constant, diffusion coefficient, and rate parameters are used in the experiment and fitting process besides using the parameters of the traditional model. The traditional combination model has verified the consistency of parameters such as m, K, and η and obtained better fitting results based on the traditional model. Compared with traditional models, the parameters of hydrolysis performance were added to the three-dimensional model. To obtain the effect of parameters of hydrolysis performance, the fitting results, experimental results, theoretical results of different parameters were compared respectively.

4.1 The influence of diffusion coefficient κ

According to the servo test results of limestone shale, purple sandstone, and marl presented in Table 3 and the hydrolysis performance parameters of the three-dimensional constitutive model, the stress–strain relationship curves are drawn, as shown in Figures 46.

Figure 4 
                  The comparative curve of the experimental and theoretical fitting of the marauder coefficient of diffusion κ = 1.75 × 10−3 cm2/s.
Figure 4

The comparative curve of the experimental and theoretical fitting of the marauder coefficient of diffusion κ = 1.75 × 10−3 cm2/s.

Figure 5 
                  The comparative curve of the experimental and theoretical fitting of purple sandstone under the coefficient of diffusion κ = 1.85 × 10−3 cm2/s.
Figure 5

The comparative curve of the experimental and theoretical fitting of purple sandstone under the coefficient of diffusion κ = 1.85 × 10−3 cm2/s.

Figure 6 
                  The comparative curve of the experimental and theoretical fitting of ash shales under coefficient of diffusion κ = 1.95 × 10−3 cm2/s.
Figure 6

The comparative curve of the experimental and theoretical fitting of ash shales under coefficient of diffusion κ = 1.95 × 10−3 cm2/s.

From Figures 46, the experimental curves and model fitting curves show consistency in the overall trend under the influence of diffusion coefficients of contaminated rocks. At the low strain stage, because of the slow diffusion process, the rock mass still maintains its original characteristics, the diffusion coefficient has little effect, and the fitting curve is in good agreement with the theoretical curve. With the increase of diffusion, the peak value of high strain increases gradually, and the fitting curve deviates to a certain extent. The diffusivity of lead element has the most obvious effect. The fitting curves of the other three elements are in good agreement with theory curves.

4.2 The effect of hydrolysis performance parameters

Figures 79 show the results of the experiment and model fitting under the influence of hydrolysis, in which these key parameters such as hydrolysis constant PKm, elongation rate C, and deformation factor m, are taken into account. Figures 57 show that the experimental results are in good agreement with the theoretical model fitting curves under different values of hydrolysis performance parameters. The hydrolysis constant is determined by the average value, which has a very little impact on the fitting results. Deformation fact or m is affected by the path of elongation rate C, and the influence process is uncontrollable, which leads to the deviation of fitting results and deviation on stress-strain path curves. In addition, the fission path is not fully controlled by the existing test equipment yet, especially the variation of elongation rate and deformation parameters. Therefore, the existence of errors with the actual situation is inevitable, which affects the fitting results to some extent.

Figure 7 
                  The contrast curves between experimental and theoretical stress–strain curves of marlstone.
Figure 7

The contrast curves between experimental and theoretical stress–strain curves of marlstone.

Figure 8 
                  The contrast curves between experimental and theoretical stress–strain curves of purple sandstone.
Figure 8

The contrast curves between experimental and theoretical stress–strain curves of purple sandstone.

Figure 9 
                  The contrast curves between experimental and theoretical stress–strain curves of ash shale.
Figure 9

The contrast curves between experimental and theoretical stress–strain curves of ash shale.

5 Conclusion

  1. Hydrolysis replacement reaction occurs because metal ions intrude into rocks, which causes the loss of colloidal mineral ions and accelerates the extension and diffusion speed of rock intrusion surface. As a result, rock mass fractures develop, and integrity is destroyed, which leads to the reduction of mechanical strength.

  2. Hydrolysis properties, deformation parameters, and diffusion coefficient of metal elements determine the extension and diffusion properties of heavy metal contaminants in rocks. The elongation of different metal elements follows reaction time.

  3. A three-dimensional constitutive model of the extended diffusion surface is established by combining viscoelastic monomer model with a damage monomer model, on which the key parameters such as hydrolysis constant, elongation rate, deformation factor, and diffusion coefficient have influence. The model simulation results are in good agreement with the experimental results, which shows that the model is reasonable in describing the fission law of rock mass under intrusion action.

  1. Funding information: This work was supported by Natural Science Foundation of Guangxi autonomous region of China (No. 2019GXNSFAA245011). This work was supported by Special funds for talents of Guangxi autonomous region of China (No. 2016047).

  2. Conflict of interest: Authors state no conflict of interest.

  3. Data availability statement: All data generated or analysed during this study are included in this published article [and its supplementary information files].


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Received: 2019-11-20
Revised: 2022-07-20
Accepted: 2022-09-23
Published Online: 2022-10-18

© 2022 Feng Cheng et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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