Research Article | Open Access
Analytical Solution of Interface Effect on the Strength of Combined Model Composed of Different Geologic Bodies
According to the special combined structure of surrounding rock in western mining area of China, a micromechanical model with variable parameters containing contact interface was proposed firstly. Then, the derived stresses in coal and rock near the interface were analyzed on the basis of the harmonized strain relation, and the analytical solutions with respect to stress states near the interface were drawn up. The triaxial compressive strength of coal and rock was further determined in case the contact interface was in the horizontal position. Moreover, effects of stiffness ratio, interface angle, and stress level on the strength of two bodies near the contact area were expounded in detail. Results indicate that additional stresses which have significant effect on the strength of combined model are derived due to the adhesive effect of contact interface and lithological differences between geologic bodies located on both sides. The interface effect on the strength of combined body is most associated with the stiffness, interface angle, and the stress level. These conclusions are also basically valid for three-body model and even for the multibody model and lay important theory foundation to guide the stability study of soft strata composed of different geologic bodies.
Roadway in western mining area of China is generally arranged in the relatively hard coal seam due to the unstable weakly cemented soft rock strata, so a unique combined structure of surrounding rock composed of coal and rock is formed. However, some mining disasters such as fall and weak shock in roof and terrible floor heave are frequently occurring. Actually, these disasters are closely related to the overall mechanical behavior of special structure of surrounding structure whose stability is determined by the comprehensive interaction between different geologic bodies. Mechanical behavior between different geologic bodies exhibits discontinuity and inhomogeneity due to their lithological differences and the contact effect which makes the combined body show abnormal mechanical behavior. Therefore, it has important engineering significance for predicting the stability of roadway to make clearly the interface effect on rock strength.
Currently, studies at home and abroad mainly focus on the joint surface which plays a key role in mechanical behavior and stability of rock mass. Among them, special attention was paid to the shear strength of the joint surface since the sliding failure theory of two-dimensional weak plane presented by  in 1960. At present, dozens of well-known shear constitutive models to predict the criteria of shear strength for interfaces have been proposed [2–4]. However, these models are two-dimensional and cannot reflect the real contact state of interface. Given this, in [5, 6], a statistical mathematical model describing the relation between the effective shear angle of interface element and the corresponding contact area through a large number of direct shear tests for joint surface was studied, and furthermore a three-dimensional shear model was established. On this basis, a modified peak strength model was put forward in  which can reflect the dilatancy effect considering the relationship between the shear dilatancy angle at peak and the characteristic of three-dimensional morphology. In addition, some constitutive models for the mechanical behavior of interface in normal and tangential directions were developed in [8, 9], and the degradation of interface asperities was discussed in . The above conclusions have achieved good results for the study of mechanical properties of joint in rock, but they are only suitable for a single rock medium. Actually, the lithological difference between geologic bodies on both sides of the contact surface has an important effect on the failure mode of combined body which has been proved by indoor test . Many scholars have studied the contact behavior between engineering body and geologic body. In [12, 13], the contact behavior between rock and concrete was systematically analyzed from experimental and theoretical aspects. Bimaterials and bibodies models for combined body composed of engineering body and geologic body were established in [14–16] considering the different contact characteristics. In [17, 18], the effect of contact behavior on the cracking of dam was analyzed. Up to now, some scholars have carried out studies on the mechanical behavior of combined model in underground engineering. For example, the stability of two body model composed of roof and coal was studied in . In [20, 21], the effect of contact angle on the integral strength of coal-rock combination model was analyzed. In , a double shear friction test for sandstone-coal composite sample was carried out by biaxial loading system, and the frictional sliding behavior of coal during impact buckling was analyzed. Based on a series of compressive tests, it is found that the integral strength of coal-rock combination model is between the strength of strong rock and weak coal, and the mechanical behavior of combined model is closely related to the monomer property as well as the combination style in [23, 24].
The above findings have studied in detail the integral mechanical behavior of combined model and the contact characteristics which show important significance for understanding the influence of interface effect on combined model. However, the results of interface effect on different mediums are only based on the macroscopic failure of combined model from test aspect, and quantitative theory is insufficient to measure the influence degree. Therefore, based on coal-rock combination model as main research object, we proposed a micromechanical model with variable parameters containing contact interface in this study. Then, the analytical analysis of interface effect on stress states of different geologic bodies on both sides was carried out in order to quantify the influence degree of contact interface on different rock mediums near contact area. All the drawn conclusions laid the theoretical foundation for further studying the stability of surrounding rock from the angle of combined model.
2. Mechanical Model of Interface Effect
From geological exploration data, mining area of western China is mainly located in soft rock strata as shown in Figure 1. Roof and floor with respect to the occurrence condition of coal seam are mainly weakly cemented soft rock as carbonaceous mudstone, fine sandstone, mudstone, silty mudstone, and so on . If roadway is excavated in coal seam, the stability of surrounding rock is explicitly determined by the combined structure of soft rock and coal. For convenience, the thickness of contact interface is neglected here. A microelement marked as A which contained soft rock, coal, and interface was selected in order to analyze the interface effect. Apparently, the model has the special characteristic of varying material parameters. Without loss of generality, the contact angle is set an arbitrary direction. The enlarged model is shown in Figure 2(a).
(b) Coordinate transformation
(c) Micromodel after transformation
Lithological difference between soft rock and coal will surely result in mutual restraint stress at the contact interface in order to keep the strain compatibility under loading. For inclined interface, both restraint normal stress and shear stress will be derived. For convenient analysis, the contact interface was firstly transformed to a horizontal direction through three-dimensional coordinate transformation and stresses also performed the corresponding transformation as shown in Figures 2(b) and 2(c).
With the help of the formula of rotated axes, namely, , the calculation results of stresses after transformation can be expressed as where , , and represent the direction cosine between the new coordinate axis and the old, respectively. The corresponding relationship is
3. Stress State Analysis of Coal and Rock Near Interface Area
3.1. Decomposition of Interfacial Mechanical Model
The contact interface between coal and rock is considered to have bonding strength. Let elastic modulus of soft rock and coal body be , , and let Poisson’s ratio be , , respectively. The difference of elastic constants will lead to different deformation in the two geological bodies under the same loading condition. Nevertheless, the same deformation must be produced near the interface due to the bonding condition, so it is inevitable for tensile and compressive stress as well as shear stress to be derived in the two different geological bodies near the interface area. This will result in a sharp discontinuity of stress and continuity of displacement in coal and rock body around the interface.
For the analysis of the derived stress, the mechanical model of microelement after coordinate transformation was divided into the following two parts by employing the superposition principal: one is only under the action of normal stress as shown in Figure 3(a) and the other is under shear stress presented in Figure 3(b).
(a) Action of normal stress
(b) Action of shear stress
3.2. Derived Stresses in Different Geologic Bodies under Normal Stress
Let be , , , and . Compressive stress is specified as positive later on. Only normal stresses will be derived in the plane of , , and , respectively, under the action of the original normal stresses. Figure 4(a) presented the distribution of total normal stress acting on the micromodel which is composed of the origin stress and derived stress as shown in Figures 4(b) and 4(c), respectively. According to the superposition principal, the relationships between them are where superscript and , respectively, stand for soft rock and coal body and subscript denotes the derived stress.
The harmonious relationship of longitudinal strain in the transverse plane () near the interface area was demonstrated in Figure 5 when the normal stress in three directions was applied, respectively.
(a) Action of
(b) Action of
(c) Action of
As shown in Figure 5(a), the soft rock and coal will transversely expand in and directions, respectively, under the action of . Thus, tensile strain will be produced in the two geologic bodies in each direction. Among them, the strain in the coal body is, respectively, written as and , and the strain in soft rock is denoted by and , respectively, where stands for the action direction of normal stress. Because of the assumed relations of and , comparison results of the transverse strain in the two geologic bodies should be and , respectively. However, these transverse stains must satisfy the harmonious relation according to the bonding interface. Thus, the final results of transverse strains in the two geologic bodies are denoted by and , respectively. For this reason, compressive stresses will be derived in coal body which are demonstrated by and in each direction. On the contrary, tensile stresses will be induced in soft rock which are accordingly written as and , respectively. Their relations should be and based on the equilibrium relationship. For convenience, the derived stresses in and direction are unified written as and , respectively.
Similarly, when is applied to the model individually as shown in Figure 5(b), derived compressive stress and tensile stress in and directions will be produced in coal body which are denoted as and , respectively. In contrast, derived compressive stress and tensile stress will be applied to soft rock in and directions. In addition, they should meet the following relation: and . As before, the derived stresses in each direction are written as and , respectively. Besides, the derived stresses in the two geologic bodies in and directions are denoted by and , respectively, under the action of as demonstrated in Figure 5(c).
From the above analysis, when one of the normal stresses is applied to the model, derived normal stresses will be induced in other two directions in the two geologic bodies near contact area. These derived stresses will distinctly affect the integral strength and failure mode of coal-rock combination model. As shown in Figure 5, the longitudinal strains should satisfy
By substituting the generalized Hooke’s law into (4), the derived normal stresses can be calculated as where
3.3. Derived Stresses in Different Geologic Bodies under Shear Stress
When the shear stresses are applied to the model individually as shown in Figure 3(b), no normal stress will be derived by neglecting the coupling effect between longitudinal strain and shear strain. On this condition, only shear stress which is related to the transverse deformation will induce derived shear stress. Therefore, only and in Figure 3(b) affect the interface effect. Figure 6 demonstrates the harmonious relationship of shear strain in transverse plane under the action of shear stress.
Let and , finding where denotes the shear modulus. Thus, the transverse shear strain in coal body and soft rock should satisfy . However, the two should keep harmonious relationship due to the transverse constraint, so unified final transverse shear strain is written as . And this means that derived shear stresses which are denoted by and will be induced in the two geologic bodies under the action of original shear stress and as shown in Figure 7.
Thus, the shear stresses on the model near contact area should be determined as
Substituting (8) into the relation and , then
3.4. Analytical Model of Stress States in the Two Geologic Bodies Near Contact Area
Similarly, we can obtain the stress state of coal body near contact area as where
From (11) and (13), the stress states in two geologic bodies are changed due to the bonding interface, and some of the stress components are not continuous because of the differences of deformation parameter. Let ; , , and are equal to 0, respectively, and no derived stress is induced in the model.
4. Influence of Interface Effect on the Model Strength
4.1. Strength of the Two Geologic Bodies Near Contact Area under Triaxial Compression
For convenient discussion, let inclined angle of contact interface be 0; find . Substituting the relation into (11), the stress state in soft rock is determined as And the stress state in coal body can be expressed as
It is evident the micromodel is changed as the principal element for each geologic body under this condition. Let the yielding conditions of coal and rock meet Mohr-Coulomb yielding criterion, respectively; namely, where and (, ). and , respectively, represent the friction angle and cohesion.
Geologic bodies at the far interface will be less affected by interface effect, and their strength under triaxial compression can be established according to (17) because there is no variation in their stress states. However, the stresses of geologic bodies near the contact area will be evidently changed due to the interface effect, so strength will be dramatically affected. If , then is the third principal stress , and the compression strength of soft rock near contact area is determined as
On the contrary, if , the third principal stress is related to , and the compression strength of soft rock near contact area is established as
Similarly, if , the compression strength of coal body near contact area is written as
And if , the compression strength is changed as
In fact, if the order of principal stress in combined model satisfies , the stresses in the two geologic bodies will constantly meet and due to the coefficient relationship of and . Thus, the compressive strengths of soft rock and coal can be directly calculated using (18) and (20), respectively.
4.2. Effect of Rock Stiffness on the Strengths of Two Geologic Bodies
The following basic parameters are set: MPa, , GPa, MPa, . Let the stiffness ratio between soft rock and coal be . Effect of rock stiffness on the strength of micro model is analyzed below by gradually increasing the rock modulus and keeping the coal modulus constantly.
Figure 8 demonstrates the strength variation of two geologic bodies near contact area with rock stiffness when and are set to 2 MPa and 3 MPa, respectively. In the results, and , respectively, stand for the strength of coal and soft rock which are not affected by contact interface. The corresponding affected strengths are distinguished by superscript and which are denoted by and . Thus, the variation of strengths is calculated as and . From Figure 8(a), the strength of coal body (weak body) under triaxial compression is greatly improved due to the interface effect, and the variation presents increasing trend with the rising of stiffness ratio . When , the strength of coal body will exceed that of soft rock near the contact area due to the action of derived stresses. On the contrary, strength of soft rock is decreased with the increasing of . When , tends to be steady and is approximately equal to the strength of coal body .
(a) Triaxial compression
(b) Uniaxial compression
The uniaxial compression strengths shown in Figure 8(b) exhibit the same changing trend with the results in Figure 8(a). When , the strength of soft rock near contact area is higher than that of coal body, but the results are just opposite when . Actually, if and , the transverse tensile strain of soft rock near contact area will be less than that of coal body, so tensile and compressive stress will be, respectively, derived in soft rock and coal body. The derived stresses change the stress of the two geologic bodies near contact area from uniaxial states to three-dimensional states which result in strength decreasing and enhancing soft rock and coal body, respectively.
4.3. Effect of Confining Pressure on the Strengths of Two Geologic Bodies
Figure 9 shows the changes of strength with confining pressure when and . The strengths of the two geologic bodies are all enhanced with the increasing of confining pressure, and it is seen to be even greater in coal body. The strength of soft rock in this area is significantly lower than the original strength . Strength under equal confining pressure is higher than that under unequal confining pressure from the results comparison of Figures 8(a) and 9.
4.4. Effect of Inclined Angle of Interface on the Strengths of Two Geologic Bodies
Changes of principal stresses of the two geologic bodies near the contact area under the action of original stress , , with inclined angle can be determined by employing (11) and (13). The following parameters are set: MPa, , GPa, MPa, , MPa, and MPa.
Changes of principal stresses in the two geologic bodies near contact area with inclined angle are illustrated in Figure 10 (), where dotted lines represent the original stresses. As shown in Figure 10(a), in soft rock remains unchanged, while and are slightly increased but do not change significantly with the inclined angle when . However, is rapidly increased, while increases first and then decreases with inclined angle and finally approaches the original stress when . According to M-C yielding criterion, rock strength is weakened in this region which is especially obvious when . However, changes of principal stresses in coal body show quite different trend as shown in Figure 10(b). When , remains unchanged, while decreases with inclined angle. In this angle scope, coal strength is enhanced when , while it is decreased when . When , and , respectively, show obvious increasing and decreasing trend, so coal strength is enhanced.
(a) Principal stresses of soft rock
(b) Principal stresses of coal body
According to the structure effect on the stability of surrounding rock in soft rock strata, we proposed a microelement model with varying parameters and contact interface. Then, the stress states and strength behavior of two different geologic bodies were analyzed. Some conclusions are drawn as follows.(1)The lithological differences between different geologic bodies and the interface effect can change the stress state of the combined model near the contact area which will explicitly affect the strength. Effect of stiffness ratio on the strength is evidently distinct under different inclined angles, and this means the stiffness effect shows obvious anisotropic characteristic. Stress states of combined model near the contact area are closely related to the stiffness ratio and inclined angle.(2)Results obtained in this study are not only applied to coal-rock combination model, but are also suitable for the combined model composed of different geologic bodies. For example, the typical combined structure of hard rock-soft rock-hard rock is very common in underground engineering. If the soft rock lies between hard rock, its strength will be enhanced due to the transverse constraint of above and below hard rock which is conducive to the stability of composite structure. However, it is unstable if soft rock is located on one side.(3)The conclusions of the study are applicable on fully bonded condition for the interface. Yet general bonded condition still exists in the case of engineering structure. For instance, the weak filling medium lies between different rock masses, or different geologic bodies overlap together directly. On these conditions, the interface show both interface effect and friction sliding effect which should be further studied.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This project is supported by the National Natural Science Foundation of China (Grant no. 51174128) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant no. 20123718110007).
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