Numerical Analysis for the Progressive Failure of Binary-Medium Interface under Shearing

Binary-medium specimens were fabricated using the particle flow code, and the shear strength, dilatancy, and failure behavior of the binary-medium specimens with different bond strength ratios (0.25, 0.5, 0.75, and 1.0) under different normal stresses were studied. Numerical results show that the bond strength ratio and normal stresses considerably influence the shear strengths of binary-medium interface. Shear strength increases as the bond strength ratio and normal stress increase.)e dilation of interfaces with high bond strength ratios is more evident than those of interfaces with lower bond strength ratios, and the curves for the high bond strength ratio exhibit remarkable fluctuations during the residual stage. At increased normal stress and bond strength ratio, the peak dilation angle shows decreasing and increasing trends successively. In this study, the specimens exhibited three kinds of failure modes. In mode II, the sawtooth experienced shear failure, but some tensile cracks appeared on the interface of the binarymedium. In mode III, no sawtooth was cut off, indicating tensile failure on the interface. At a low bond strength ratio, damage or failure is mostly concentrated in the upper part of the model. Failure parts gradually transfer to the lower part of the model when the bond strength ratio and normal stress increase. Furthermore, evident tensile cracks occur on the interface. When the bond strength ratio reaches 1.0, the failure mode of the specimen gradually transforms from sheared-off failure to chip-off failure. )e number of microcracks in the specimens indicates that the lower the bond strength ratio, the more severe the damage on the specimens.


Introduction
Contact problem between different materials is often encountered in geotechnical engineering [1][2][3][4].Such problem is mostly encountered between concrete dam and rock foundation, anchor mortar and rock surface, and concrete pile and rock or soil [5][6][7].e contact interface between two different materials is usually called a binary-medium interface [8][9][10].e mechanical properties of the interface of a binary-medium are complicated because of the different materials involved.Furthermore, the binary-medium interface is a weak surface that has a significant impact on the stability of rock mass under complex stress environments.For example, parts on both sides of the interface rock or concrete medium produce relative sliding in a compressshearing stress state and damage the continuity and integrity of the material, thereby causing engineering damage under a long-term load action.Many studies on the interface in rock mass engineering start on the discontinuous surface of rock mass and most of them are based on the direct shear test under constant normal stress.
e results of the study showed that the mechanical properties of rock joints are affected by the normal stress, joint surface roughness, and inherent mechanical properties of rocks [11][12][13][14].Several current models for the estimation of joint shear strength exist, including the Coulomb model, Patton model, Barton model, and Byerlee model.Using these models, many researchers have made some amendments to obtain better results [15][16][17][18][19].
eoretical derivation and experimental research are the most common research methods of rock mechanics and engineering.In light of the continuous development and progress of computer technology in recent years, researchers have developed a variety of numerical calculation methods on the basis of existing theoretical results and successfully used them to analyze the stability of rock mass engineering.Among these methods, the nite element, boundary element, and discrete element have been generally recognized.In the past decade, the discrete element method has been the most common numerical calculation method used to analyze rock failure and widely applied in research on cracking and failure in brittle rock-like materials.Furthermore, results of numerical simulation and laboratory and eld experiments have shown good correlation [20][21][22][23].In this paper, the numerical analysis was performed on direct shear tests, in which di erent bond strength ratios under di erent normal stress were used.In addition, shear strength parameters and interface failure process under di erent normal stresses were analyzed from the microscopic view through discrete element numerical analysis method PFC2D.e results can be used as reference for the prediction of failure face and strength of rock-like mass engineering.

Numerical Model
e binary-medium interface is a regular sawtooth, and the uctuating angle is 25 °. Figure 1 shows its shear numerical model, the lower part of which is high-strength concrete.e upper part represents low-strength concrete.
e bond strength ratio (R b ) between the upper concrete (concrete A) and lower concrete (concrete B) is set to be 0.25, 0.5, 0.75, and 1.0.When the ratio is 1, the upper concrete and lower concrete have the same strength.In concrete with di erent proportions, the friction coe cient of the aggregate is basically consistent.In this paper, the particle ow code method was performed, and parallel-bond model was adopted.In this model, the particle is similar to ne sand, and the parallel bonds among particles are similar to cement.us, the particles are cemented into a solid mass.e direct shear test is used in the numerical simulations, and normal stress is applied to the upper part of the model at 1.0, 2.0, 3.0, and 4.0 MPa. e side wall of the upper part of the shear is supported by the rolling support boundary with the velocity of the wall, and the particles are maintained at 0.03 mm/(10 6 step.Meanwhile, the velocity from left to right is applied to the side wall of the lower part for transverse shear tests.In this study, the development of microcracks in the specimen was monitored by the sh program during the loading process. In the lower part of the concrete (Table 1), the normal bond strength between the particles is consistent with the tangent bond strength.According to the microparameters of the lower part of the concrete (Table 1), normal bond strength is consistent with tangent bond strengths among the particles.e stress-strain curves and failure modes of the concrete specimens under uniaxial compression with di erent strength ratios are shown in Figure 2. e curve 4 shown in Figure 2(a) is a stress-strain curve with a bond strength ratio of 1 and peak strength of 60 MPa, which is also the strength of the lower concrete with high strength.e peak strength of the upper concrete (concrete A) varies with the change in the bond strength ratio.At increased bond strength ratios, the peak compressive strength of the sample increases, but the elastic modulus change is not evident at increased bond strength.In failure modes under di erent bond strengths, shear failure modes are observed in the specimens under di erent bond strengths, and obvious shear bands are detected in the specimens (Figure 2(b)).

Shear Failure Characteristics of the Binary-Medium
Interface.Figure 3 shows the direct shear failures of binary-medium specimens with di erent bond strengths.e shear stress-displacement curve of the specimen with a bond strength ratio of 0.25 under a normal stress of 3.0 MPa and damages at di erent stages are shown in Figure 3(a).e shear curve is apparently at after the peak value, indicating that the sample is destroyed gradually after the peak value and residual strength are reached rapidly.e point a on the shear curve represents the internal damage of the specimen prior to the peak value.A certain tensile failure around the binary-medium interface is present, but the specimen has no structural failure, and the asperities have not been cut o .When the shear curve reaches point b during the declining stage, and the shear stress of the specimen decreases rapidly, the asperities at the upper part of the model have been cut o .e rapid decrease in shear stress indicates that the specimen has structural failure.e  shear stress-displacement curve has no apparent change in the residual stage as the shear simulation experiments are continued.Furthermore, the asperities on the upper part of the model are further sheared, and the straight shear surface is formed in the residual stage.Figure 3(b) shows the shear stress-displacement curve of the specimen with a bond strength ratio of 1.0 under a normal stress of 3.0 MPa and internal damage.Shear stress declines sharply after its peak, similar to the shear stressdisplacement curve shown in Figure 3(a).However, unlike the shear stress that tends to atten after decreasing (Figure 3(a)), the shear stress shown in Figure 3(b) is similar to a stepwise decline.In short, the uctuation of the curve in Figure 3(a) is more evident than that in Figure 3(b), mainly because the asperities are sheared after the peak, while the initiation and propagation of the tensile crack in the residual stage leads to apparent uctuations in shear stress.e higher the concrete strength, the more evident the uctuation of the shear curve with invariant normal stress, and the di erent types of failure modes considerably a ect strength parameters and shear curves.

Shear Stress-Displacement Curves and Strength
Parameters. Figure 4 shows the shear stress-displacement curves with di erent bond strengths and normal stresses.Figures 4(a)-4(d) are the shear stress-displacement curves for the bond strength ratio of 0.25, 0.5, 0.75, and 1.0, respectively.
e shear stress-displacement curves of the specimen nearly drop after they increase to the peak shear strength when shear displacement increases.e curves then atten.Moreover, the peak shear stress of the specimen 4 Advances in Civil Engineering increases when the normal stress for the model with the same bond strength ratio increases, and the curves are consistent before the peak shear stress.At increased bond strength ratio, the ductility of the curve becomes more evident after the peak shear.Furthermore, neither the bond strength of concrete A nor the normal stress has a considerable in uence on the residual stage of the curve.Peak shear displacement increases when the bond strength increases, while the increase in normal stress has no e ect on peak shear displacement.
Figure 5 shows the relationship of the variations in peak shear strength and bond strength with bond strength and normal stress.As shown in Figure 5(a), the peak shear strength shows a signi cant linear increase at increased normal stress and bond strength.Four straight lines can be obtained by tting the peak strength of the specimen with di erent bond strengths.Similar to rock mass, the shear strength of the binary-medium interface can also be expressed as follows: where τ is the shear strength, C is the cohesion of binarymedium interface, σ n is the normal stress, and φ is the friction angle of the interface.Figure 5(b) shows the relationship between interface shear strength and bond strength ratio (R b ).Similar to the peak shear strength depicted in Figure 5(a), shear strength in Figure 5(b) has a linear growth at increased bond strength ratio (R b ). Figure 5(c) shows the variations in cohesion C and slope k tanφ with bond strength ratio.At increased concrete bond strength ratio (R b ), both parameters considerably increase.Furthermore, the antishear capacity of the binarymedium interface increases when the bond strength ratio increases.e relationship between the shear strength parameters and bond strength ratios of the binary-medium interface, shown by ( 2) and (3), respectively, is obtained by tting the shear strength parameters: where C and φ are the cohesion and friction angles of interface, respectively, and R b is the bond strength ratio.e values of a 1 , a 2 , b 1 , b 2 , b 3 , and b 4 are 7.417, 0.29, 10.17, −3.9, 2.52, and −1.112, respectively, for this study.

Analysis of Dilatancy.
When the strength of concrete A increases, the shear strengths and shear-displacement curves of the specimen show evident changes.During direct shear, concrete A has a shear dilation trend, and the growth and trend of shear dilation are a ected by the strength of the material.For the analysis of the e ect of the upper concrete bond strength on shear dilation, the normal displacement of the upper wall is monitored and calculated during the simulation, and the shear dilation displacement-shear displacement curve is shown in Figure 6. e four curves in

Advances in Civil Engineering
Figures 6(a) and 6(d) show a signi cant growth trend as the shear displacement increases.e curve 4, representing the bond strength ratio of 1.0, is located above the other curves, whereas the curve representing the bond strength ratio of 0.25 is at the bottom.is result indicates that the strength of concrete A is low, and the asperities on concrete A are easily destroyed by normal and shear loads.us, large degrees of sliding along the serrated surface cannot occur and form shear dilation after the asperities are partially cut.When the bond strength ratio is 0.25, the shear expansion curves increase smoothly as shear displacement increases, which corresponds to the shear stress-displacement curve in Figure 4.
In the residual stress stage, shear stress tends to be gentle, indicating that shear dilation is caused by the broken pieces and resulting friction among them.ese pieces are produced by the overall destruction of the specimen.As the bond strength ratio increases, the failure module of the binary-medium interface is not only the shear or friction failure in concrete A but also some friction failures and tensile crack propagations in concrete B. us, the dilation curve uctuates at increased shear displacement, which is not as smooth as the bond strength ratio of 0.25.Notably, similar to the direct shear test, shear dilation decreases at increased normal stress and shear dilation reaches a minimum value at a normal stress of 4.0 MPa.
Similar to that of rock joint surface shear, di erent degrees of dilation similar to that of rock joint interface are observed during the shear test of binary-medium interface.
e degrees of dilation and dilation angle vary with bond strength ratio.e dilation angle of joint surface is usually calculated by where ψ is the dilation angle and Δu v and Δu h are the normal and shear displacement increment, respectively.e relationship between shear displacement and the tangent value of the dilation angle can be obtained after the normal displacement of each period is divided by shear displacement.us, we can extract the peak tangent value of the dilation angle of the di erent models with di erent normal stresses, and then we can deduce the change rule of the dilation angle.Figure 7 shows the change in the peak dilation angle of the binary-medium interface with the normal stress and the bond strength ratio.Figure 7(a) shows that the peak shear dilation angle of the binary-medium interface has a decreasing trend at increased normal stress as the strength of the normal constraints increases.us, the normal displacement is limited, and the peak shear dilation angle decreases gradually.e following formula is obtained by tting the peak dilation curve: where a and b are the tting coe cients and σ n is the normal stress.e tting parameters for the peak dilation angle with di erent normal stress values are shown in Table 2. Figure 7(b) shows the relationships among the peak dilation angle, bond strength ratio, and tting curve.Apparently, the peak dilation angle has a signi cant growth trend when the bond strength ratio increases.Increasing the bond strength ratio increases the strength of concrete A, thereby changing the failure mode.When the bond strength ratio is relatively small, the asperity can be cut o easily, and the shear dilation phenomenon is not evident.At increased bond strength ratio, the failure near the contact surface is changed from simple shear to wear failure, increasing the peak dilation angle gradually.e tting relation between peak dilation angle and bond strength ratio is shown as follows: where a 1 , b 1 , and b 2 are the tting coe cients and σ n is the normal stress.e tting parameters for the peak dilation angle with di erent bond ratios are shown in Table 3.

Shear Morphology Damage in Two Medium
Materials.e three typical failure modes of the two medium materials under di erent bond strength and normal stress values are shown in Figure 8. Figure 8(a) shows the distribution of displacement eld and microcrack of the sample with a bond strength ratio of 0.25 and normal stress of 1.0 MPa.Parts A, B, and C are selected for feature analysis, and their displacement elds are shown in the right part of Figure 8(a).
e gure shows that the failure is mainly concentrated in  Advances in Civil Engineering the upper part of the sample.e displacement eld of the sheared asperities is observed toward the right direction, while that of other part of the sample is upward through the displacement eld shown in the right part of Figure 8(a).e distribution of the displacement vectors with upward As mentioned above, the bond strength ratio and normal stress of the upper concrete in uence the failure of the model.Figure 9 shows the failure modes of the sample with di erent bond strength and normal stress values.As shown in Figure 9(a), the failure of samples shows the rst failure mode when the upper bond strength is 0.25.Regardless of the normal stress, the failure of the model is concentrated in the upper part of the model, but nearly no damage occurs in the lower part of the concrete.e damage near the contact surface becomes increasingly apparent at increased normal stress.e model in Figure 9(c) mainly belongs to the second failure mode, and some damages in the upper and lower parts of the contact surface of the model are observed.In addition, the damage of the upper part is more severe than that of the lower part, and some tension cracks are observed in the lower part of the model.e failure mode in Figure 9(b) is between the rst and second modes.e damages are mainly concentrated in the upper model, and some tensile cracks appear at the upper part of the model when the normal stress increases.However, some wear failures are found in the lower part of the model.
When the bond strength ratio is 1.0, the failure of the model mainly presents a chip-o failure along the interface of the binary-medium model.is failure is the third failure mode, which is a common failure mode of rock joint shear test.In this failure mode, the bond strength of the upper and lower parts of the model and the distribution of natural microcracks are consistent, and the tensile microcracks near the contact surface are also common.Figure 10 shows the distribution and quantity of microcracks in each model with di erent bond strength ratio and normal stress values.
e number of microcracks is the quantitative analysis parameter of the wear and failure of the interface and can represent the size and extent of the damage area in the model after shearing.Figure 10 shows that the number of microcracks shows a signi cant decreasing trend when the bond strength ratio of the model increases.e decreasing trend indicates that the damage area near the contact surface decreases gradually at increased bond strength ratio, as shown in the failure mode in Figure 9.Meanwhile, owing to the increased normal stress limits, the shear dilation of the model and internal microcrack specimens have obvious growth changes at the same bond strength ratio and increased normal stress.In addition, the damage near the binary-medium interface is more evident.is feature is similar to the failure characteristics of the rock shear experiment.numerical analysis method, we establish four coupling models of binary-medium materials with four different bond strength ratios and four different normal stresses.Moreover, the mechanical properties and failure modes are analyzed from the microscopic perspective and on the basis of the results of numerical simulation:

Conclusions
(1) e bond strength of the upper part of the model considerably influences the shear strength parameters of the binary-medium interface.e shear strength parameters apparently increase at increased bond strength ratio.Furthermore, the peak dilation angle shows a decreasing trend at increased normal stress.Conversely, the peak dilation angle shows an increasing trend at increased bond strength.
(2) ree kinds of failure modes are found for the binary-medium model during the direct shear test.e first type of failure mode is shear failure, in which the asperity of the upper concrete has a shearing-off failure.In mode II, some sawtooth experiences shear failure, but some tensile cracks appear on the interface of the binary-medium materials.In mode III, no sawtooth was cut off, mainly exhibiting tensile failure on the interface.When the bond strength ratio is low, the damage or failure is more severe and mainly appears as shear failure.Furthermore, at increased bond strength ratio, the degree of wear and failure near the contact surface is gradually reduced, and the failure mode is mainly manifested as chip-off and tension failure.

Figure 3 :
Figure 3: Shear stress and failure process for specimens with di erent bond strength ratios and normal stresses: (a) bond strength ratio of 0.25 and normal stress of 3.0 MPa; (b) bond strength ratio of 1.0 MPa and normal stress of 3.0 MPa.

Figure 5 :
Figure 5: Relationship between the shear strength parameters and bond strength ratio: (a) relationship of peak shear strength and normal stress; (b) relationship of peak shear strength and bond strength ratio; (c) relationship of C and k with bond strength ratio.

Figure 7 :
Figure 7: Trends of the peak dilation angle: (a) relationship of dilation angle and normal stress; (b) relationship of dilation angle and bond strength ratio.

Figure 8 :
Figure 8: Di erent kinds of failure modes and distribution of displacement eld: (a) bond strength ratio of 0.25 and normal stress of 1.0 MPa, (b) bond strength ratio of 0.75 and normal stress of 1.0 MPa, and (c) bond strength ratio of 1.0 and normal stress of 2.0 MPa.

e
purpose of this paper is to study the shear strength and failure characteristics of a binary-medium interface through the simulation of direct shear.Basing on the discrete element Normal stress of 1.0 MPa Normal stress of 2.0 MPa Normal stress of 3.0 MPa Normal stress of 4.0 MPa(d)

Figure 10 :
Figure 10: Number of microcracks for specimens with di erent bond strength ratios under di erent normal stresses.

Table 1 :
Microscopic parameters for rock mass.

Table 2 :
Fitting parameters for peak dilation angle with di erent bond strength ratios.

Table 3 :
Fitting parameters for peak dilation angle with di erent normal stress.