The contact between two rough surfaces can be approximately equivalent to the contact between a rough surface and a rigid smooth flat plane (Majumder and Tien, 1990). From the W-M fractal function, it follows that the profile curve of a single asperity on the backfill surface before deformation can be expressed as (Berry and Lewis, 1980; Mandelbrot, 1985) z ( x ) = G D − 1 l 2 − D ⁡ cos π x l (1) where D is the fractal dimension of the rough surface, and 1 < D < 2; G is the characteristic length scale of the rough surface; l is the base length of an asperity; x is the horizontal distance from any point on the base to the tip of an asperity; and z(x) is the profile curve of a single asperity before deformation.

Figure 2 presents the deformation schematic diagram of a single asperity. The parameters δ, r, and a are the deformation at the tip of an asperity, the microcontact radius, and the microcontact area in Figure 2, respectively. Meanwhile, according to the M-B fractal model, the relationship between a and l can be simplified as l = a ^1/2. Then, the deformation and the curvature radius of an asperity are given as follows: δ = G D − 1 a ( 2 − D ) / 2 (2) R = 1 G D − 1 l − D π 2 (3) where R is the radius of curvature at the tip of an asperity.

The microcontact area and micro- contact load of an asperity depend on its deformation regime: Elastic, elastoplastic, or plastic.

1) Elastic Deformation

Considering Hertz contact theory (Johnson, 1985), the elastic microcontact area, the elastic microcontact area-load relation, and the maximum microcontact pressure of an asperity are a = π R δ (4) F e = 4 E ∗ 3 π G D − 1 a 3 − D 2 (5) P 0 = 2 E ∗ π ( δ R ) 1 2 (6) where E ^* is the composite elastic modulus of the interface, and E ∗ = ( 1 − μ 1 2 E 1 + 1 − μ 2 2 E 2 ) − 1 ; μ ₁, μ ₂, E ₁, and E ₂ are the Poisson’s ratios and the elastic moduli of the two microcontact materials, respectively; F _e is the elastic load of an asperity; and P ₀ is the maximum microcontact pressure during deformation.

The maximum microcontact pressure is 3/2 times the average microcontact pressure that arises during elastic deformation (Johnson, 1985), namely, P e = 2 3 P 0 = 4 E ∗ 3 π ( δ R ) 1 2 (7) where P _e is the average elastic microcontact pressure on an asperity.

Without taking friction into account, the critical average pressure of an asperity at the first yield is P e c = 1.1 σ y (8) where P _ec is the critical average pressure of an asperity, demarcating the elastic and elastoplastic microcontacts, and σ _y is the yield strength of the softer material.

Therefore, the critical microcontact deformation is δ e c = ( 3.3 σ y π 4 E * ) 2 R = ( 3.3 σ y 4 E * ) 2 π a D / 2 G D − 1 (9) where δ _ec is the critical microcontact deformation of an asperity, demarcating the elastic and elastoplastic deformations.

Then, δ e c δ = π ( 3.3 σ y 4 E * ) 2 ( a G 2 ) D − 1 (10)

When δ = δ _ec , the critical microcontact area of an asperity at the first yield is obtained as follows: a e c = G 2 [ 1 π ( 4 E * 3.3 σ y ) 2 ] 1 D − 1 (11) where a _ec is the critical microcontact area of an asperity, demarcating the elastic and elastoplastic microcontact areas.

2) Completely Plastic Deformation.

When an asperity undergoes completely plastic deformation, the microcontact area is equal to the truncated microcontact area (Johnson, 1985), that is, a = 2 π R δ (12)

At this time, β = E * r σ y R ≈ 30 (13)

Thereby, a p c = G 2 [ π ( E * ) 2 225 σ y 2 ] 1 D − 1 (14) δ p c = G [ π ( E * ) 2 225 σ y 2 ] 2 − D 2 D − 2 (15) where a _pc is the critical microcontact area of an asperity, which delimits the elastoplastic and plastic microcontact area, and δ _pc is the critical microcontact deformation of an asperity, which delimits the elastoplastic and plastic microcontact deformation.

When the average microcontact pressure is equal to 3σ _y, the asperity is in the completely plastic deformation (Johnson, 1985), namely, P p c = P p = 3 σ y (16) where P _pc is the critical microcontact pressure of an asperity, demarcating the elastoplastic and plastic microcontacts, and P _p is the average microcontact pressure of an asperity.

In summary, the critical microcontact areas (a _ec and a _pc ) and the critical microcontact deformations (δ _ec and δ _pc ) are independent of the radius of curvature at the tip of an asperity. These parameters relate to only the physical parameters of the materials and the fractal parameters of the surface. In other words, the critical microcontact area and critical microcontact deformation are unique for a rough surface. Then, the critical microcontact deformations are used to determine the microcontact area and microcontact load of an asperity during elastoplastic deformation.

3) Elastoplastic Deformation

When δ e c ≤ δ ≤ δ p c , the asperity is in elastoplastic deformation. Considering the proportion of elastic deformation and plastic deformation, the average microcontact pressure is expressed as (Zhao et al., 2000) P e p = 3 σ y − 1.9 σ y ( ln δ p c − ln ⁡ δ ln ⁡ δ p c − ln δ e c ) (17) where P _ep is the average elastoplastic microcontact pressure on an asperity.

The microcontact area and microcontact load of an asperity are a e p = π R δ + ( 2 π R δ − π R δ ) × [ − 2 ( δ − δ e c δ p c − δ e c ) 3 + 3 ( δ − δ e c δ p c − δ e c ) 2 ] = π R δ [ 1 − 2 ( δ − δ e c δ p c − δ e c ) 3 + 3 ( δ − δ e c δ p c − δ e c ) 2 ] (18) F e p = P e p ⋅ a e p = [ 3 σ y − 1.9 σ y ( ln ⁡ δ p c − ln ⁡ δ ln ⁡ δ p c − ln ⁡ δ e c ) ] × [ 1 − 2 ( δ − δ e c δ p c − δ e c ) 3 + 3 ( δ − δ e c δ p c − δ e c ) 2 ] ⋅ π R δ (19) where a _ep is the elastoplastic microcontact area of an asperity, and F _ep is the elastoplastic microcontact load of an asperity.

In summary, the different deformation regimes of an asperity can be determined by the critical deformation. When δ>δ _pc , an asperity is undergoing completely plastic deformation. When δ e c ≤ δ ≤ δ p c , an asperity is undergoing elastoplastic deformation. When δ<δ _ec , an asperity is undergoing elastic deformation.

The M-B fractal model clearly shows that the vertical deformation of an asperity gradually evolves from plastic to elastic deformation. The asperity volume is assumed to remain constant during loading (Hill, 1950), considering the energy conservation theorem. The asperity shape evolves from “tall and thin” to “short and fat”, as shown in Figure 3. The force p is the vertical external load, and the force T is the horizontal internal force. Thus, the horizontal deformation of an asperity gradually evolves from elastic to plastic.

The dashed box in Figure 3 is enlarged to analyze the interaction between two asperities during microcontact. The microcontact stress schematic diagram of the two asperities is drawn in Figure 4.

Combined with the Mohr-Coulomb criterion, the shear strengths of the microcontact interface, backfill and rock can be expressed as follows: τ w = c w + σ w ⁡ tan φ w (20a) τ 1 = c 1 + σ 1 ⁡ tan ( φ 1 + i ) (20b) τ 2 = c 2 + σ 2 ⁡ tan ⁡ φ 2 (20c) where τ _w and σ _w are the shear stress and the normal stress of the microcontact interface, respectively; τ ₁ and σ ₁ are the shear stress and the normal stress on the shear failure surface of the backfill, respectively; and τ ₂ and σ ₂ are the shear stress and the normal stress on the shear failure surface of the rock, respectively. Adhesion is defined as the ability of one material to adhere to the surface of another material (Luo et al., 2017). Cohesion is defined as the ability of adjacent parts of the same material to attract each other. Thus, c _w is defined as the adhesion of the microcontact interface; c ₁ and c ₂ are the cohesion of the backfill and rock, respectively; φ _w is the friction angle of the microcontact interface; φ ₁ and φ ₂ are the friction angles of backfill and rock, respectively; and i is the inclined angle of the asperities on the rock surface to impede the movement of the asperities on the backfill surface.

A unit on the microcontact interface in Figure 4 is taken for force analysis, as shown in Figure 5. According to Figure 5, the static equilibrium equations of the unit are established for different deformation regimes. The relationships of the strength parameters between the microcontact interface and backfill in different deformation regimes are solved.

1) Elastic Deformation

When the asperity on the backfill surface is undergoing elastic deformation, the normal stress on the microcontact interface is σ _w = σ ₁ = σ ₂ = P _e. The shear strength of the interface is mainly determined by the backfill, namely, τ w = τ 1 . Then, the cohesion and friction angle of the microcontact interface are { c w = c 1 φ w = φ 1 + i (21)

2) Elastoplastic Deformation

When the asperity on the backfill surface is undergoing elastoplastic deformation, the normal stress on the microcontact interface is σ _w = σ ₁ = σ ₂ = P _ep. The shear strength is mainly determined by the backfill and rock, namely, τ w = τ 2 − τ 1 . Then, the cohesion and friction angle of the microcontact interface are { c w = c tan ⁡ φ w = tan ⁡ φ 2 − tan ( φ 1 + i ) (22) where 0 < c < c 1 .

3) Completely Plastic Deformation

When the asperity on the backfill surface is undergoing completely plastic deformation, the normal stress on the microcontact interface is σ _w = σ ₁ = σ ₂ = P _p. The shear strength is mainly determined by the backfill and rock, namely, τ w = τ 2 − τ 1 . Then, the cohesion and friction angle of the microcontact interface are { c w = 0 tan ⁡ φ w = tan ⁡ φ 2 − tan ( φ 1 + i ) (23)

In different deformation regimes, the contact area and contact load are related to the deformation of the largest microcontact when the two rough surfaces are in contact.

1) Completely Plastic Deformation

If a L < a p c , all the asperities on the backfill surface are undergoing completely plastic deformation. At this time, the size distribution of the microcontact areas is as follows: n ( a ) = D 2 ψ ( 2 − D ) / 2 ( a L ) D / 2 ( a ) − ( D + 2 ) / 2 (24) where a _L is the largest microcontact area of the interface and n(a) is the size distribution of the microcontact areas under completely plastic and elastoplastic deformation.

Then, the plastic contact area and contact load can be given by A r p = ∫ 0 a p c n ( a ) a d a = D 2 − D ψ 2 − D D ( a L ) D 2 ( a p c ) 2 − D 2 (25) F c p = ∫ 0 a p c P p a n ( a ) d a = 3 σ y D 2 − D ψ 2 − D D ( a L ) D 2 ( a p c ) 2 − D 2 (26) where A _rp and F _cp are the contact area and contact load of the interface, respectively, during completely plastic deformation only.

2) Elastoplastic Deformation

The interface contact area and contact load during elastoplastic deformation are solved by the critical microcontact area and critical microcontact deformation of the different deformation regimes. If a p c ≤ a L ≤ a e c and δ p c ≤ δ ≤ δ e c , all the asperities on the backfill surface are undergoing elastoplastic deformation. The elastoplastic contact area and contact load are expressed as follows: A r e p = ∫ a p c a e c n ( a ) a d a = D 2 − D ψ 2 − D D ( a L ) D 2 [ ( a e c ) 2 − D 2 − ( a p c ) 2 − D 2 ] (27) F c e p = ∫ a p c a e c ∫ δ p c δ e c F e p n ( a ) d δ d a = ∫ a p c a e c ∫ δ p c δ e c [ 3 σ y − 1.9 σ y ( ln ⁡ δ p c − ln ⁡ δ ln ⁡ δ p c − ln ⁡ δ e c ) ] × [ 1 − 2 ( δ − δ e c δ p c − δ e c ) 3 + 3 ( δ − δ e c δ p c − δ e c ) 2 ] ⋅ π R δ ⋅ n ( a ) d δ d a = 2 ∫ a p c a e c ∫ δ p c δ e c [ 3 σ y − 1.9 σ y ( ln ⁡ δ p c − ln ⁡ δ ln ⁡ δ p c − ln ⁡ δ e c ) ] × [ 1 − 2 ( δ − δ e c δ p c − δ e c ) 3 + 3 ( δ − δ e c δ p c − δ e c ) 2 ] ⋅ a ⋅ D 2 ψ ( 2 − D ) / 2 ( a L ) D / 2 ( a ) − ( D + 2 ) / 2 d δ d a (28) where A _rep and F _cep are the contact area and contact load of the interface, respectively, during elastoplastic deformation only.

3) Elastic Deformation

If a L > a e c , all the asperities on the backfill surface are undergoing elastic deformation. The size distribution of the microcontact areas is as follows: n ( a ) = D 2 ( a L ) D / 2 ( a ) − ( D + 2 ) / 2 (29) where n(a) is the size distribution of the microcontact areas during elastic deformation.

Then, the plastic contact area and contact load are expressed as follows: A r e = ∫ a e c a L n ( a ) a d a = D 2 − D ( a L ) D 2 [ ( a L ) 2 − D 2 − ( a e c ) 2 − D 2 ] (30)

If 1 < D < 2 and D ≠ 1.5 , F c e = ∫ a e c a L F e n ( a ) d a = 4 E * D G D − 1 a L D 2 3 π ( 3 − 2 D ) [ ( a L ) 3 − 2 D 2 − ( a e c ) 3 − 2 D 2 ] (31a)

Similarly, if D = 1.5, F c e = ∫ a e c a L F e n ( a ) d a = 1 π E ∗ G 1 2 a L 3 4 ( ln ⁡ a L − ln ⁡ a e c ) (31b) where A _re and F _ce are the contact area and contact load of the interface, respectively, during elastic deformation only.

A contact strength model of the interface is established by analyzing the strength model parameters of a single asperity.

1) Elastic Deformation.

The shear strength of the interface can be expressed as follows: τ w e = c 1 + F c e A r e tan ( φ 1 + i ¯ ) (32) where i ¯ is the average inclined angle of the asperities on the rock surface and τ _we is the shear strength of the interface during elastic deformation only.

2) Elastoplastic Deformation

The shear strength of the interface can be given by τ w e p = c + F c e p A r e p [ tan ⁡ φ 2 − tan ( φ 1 + i ¯ ) ] (33)

where τ _we is the shear strength of the interface during elastoplastic deformation only.

3) Completely Plastic Deformation

The shear strength of the interface is τ w p = F c p A r p [ tan ⁡ φ 2 − tan ( φ 1 + i ¯ ) ] (34) where τ _wp is the shear strength of the interface during completely plastic deformation only.

Therefore, the total shear strength of the interface is τ w = c 1 + { F c e A r e tan ( φ 1 + i ¯ ) + F c e p A r e p [ tan ⁡ φ 2 − tan ( φ 1 + i ¯ ) ] + F c p A r p [ tan ⁡ φ 2 − tan ( φ 1 + i ¯ ) ] } (35) where τ _w is the total shear strength of the interface.

Eq. 35 indicates that the shear strength of the interface is determined by the adhesion and friction. To ensure the stability of the backfill-rock interface, the maximum adhesion (namely, c ₁) during deformation is taken as the total adhesion of the interface in Eq. 35. The total friction of the interface is the sum of all the frictions corresponding to different deformation regimes.

Meanwhile, the relationship among the parameters can be obtained as: ( P e + P e p + P p ) tan ⁡ φ w = P e ⁡ tan ( φ 1 + i ¯ ) + ( P e p + P p ) [ tan ⁡ φ 2 − tan ( φ 1 + i ¯ ) ] (36)