Analytical Solution of Interface Effect on the Strength of Combined Model Composed of Different Geologic Bodies

1 State Key Laboratory of Mining Disaster Prevention and Control Cofounded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China 2 College of Mining and Safety Engineering, Shandong University of Science and Technology, Qingdao 266590, China 3 College of Civil Engineering and Architecture, Shandong University of Science and Technology, Qingdao 266590, China


Introduction
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 [1] in 1960.At present, dozens of wellknown shear constitutive models to predict the criteria of shear strength for interfaces have been proposed [2][3][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 [7] 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 [10].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 [11].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][15][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 [19].In [20,21], the effect of contact angle on the integral strength of coal-rock combination model was analyzed.In [22], 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.

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 [25].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).
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

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).

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 Contact interface demonstrated in Figure 5 when the normal stress in three directions was applied, respectively.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 Obviously, the coefficients satisfy   =   ,   =   , and   =   .The analytical expression of normal stresses in the two geologic bodies is deduced as follows by substituting (5) into (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 The shear stresses near contact area can be further determined by substituting ( 9) into (8); that is

Analytical Model of Stress States in the Two Geologic Bodies
Near Contact Area.Stress state of soft rock near contact area is confirmed as follows by substituting ( 7) and ( 10) into (1): consider where 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  =  = 1;   ,   , and   are equal to 0, respectively, and no derived stress is induced in the model.

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   = (1 + sin   ) / (1 − sin   ) and   = 2  / (1 − sin   ) ( = , ).  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  3 , and the compression strength of soft rock near contact area is determined as On the contrary, if    <    , the third principal stress  3 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  1 >  2 ≥  3 , 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.

Effect of Rock Stiffness on the Strengths of Two Geologic
Bodies.The following basic parameters are set:   = 1 MPa,   = 30 ∘ ,   = 1 GPa,   = 2 MPa,   = 40 ∘ .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  2 and  3 are set to 2 MPa and 3 MPa, respectively.In the results,   1 and   1 , 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   1 and   1 .Thus, the variation of strengths is calculated as Δ  1 =   1 −  1 and Δ  1 =   1 −   1 .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 Δ  1 presents increasing trend with the rising of stiffness ratio .When  > 2, the strength of coal body   1 will exceed that of soft rock   1 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  > 3,   1 tends to be steady and is approximately equal to the strength of coal body   1 .The uniaxial compression strengths shown in Figure 8(b) exhibit the same changing trend with the results in Figure 8(a).When  < 1.5, the strength of soft rock near contact area is higher than that of coal body, but the results are just opposite when  > 2.5.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.

Effect of Confining Pressure on the Strengths of Two
Geologic Bodies.Figure 9 shows the changes of strength with confining pressure when  = 2.5 and  2 =  3 .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   1 .Strength under equal confining pressure is higher than that under unequal confining pressure from the results comparison of Figures 8(a) and 9.

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  1 ,  2 ,  3 with inclined angle can be determined by employing (11) and (13).The following parameters are set:   = 1 MPa,   = 30 ∘ ,   = 1 GPa,   = 2 MPa,   = 40 ∘ ,  1 = 5 MPa, and  2 =  3 = 2 MPa.
Changes of principal stresses in the two geologic bodies near contact area with inclined angle are illustrated in Figure 10 ( = 2.5), where dotted lines represent the original stresses.As shown in Figure 10(a),  1 in soft rock remains unchanged, while  2 and  3 are slightly increased but do not change significantly with the inclined angle when  ≤ 30 ∘ .However,  1 is rapidly increased, while  2 increases first and then decreases with inclined angle and finally approaches the original stress when  > 30 ∘ .According to M-C yielding criterion, rock strength is weakened in this region which is especially obvious when  > 30 ∘ .However, changes of principal stresses in coal body show quite different trend as shown in Figure 10(b).When  ≤ 30 ∘ ,  1 remains unchanged, while  3 decreases with inclined angle.In this angle scope, coal strength is enhanced when 0 ≤  ≤ 20 ∘ , while it is decreased when 20 ∘ ≤  ≤ 30 ∘ .When  ≥ 50 ∘ ,  3 and  1 , respectively, show obvious increasing and decreasing trend, so coal strength is enhanced.

Conclusions
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.

Figure 1 :
Figure 1: Occurrence condition of coal seam in weakly cemented soft rock strata.

Figure 5 :
Figure 5: The transverse strain relationship near contact interface under normal stress.

Figure 6 :
Figure 6: The relationship of transverse shear strain near contact interface.

Figure 7 :
Figure 7: Distribution of shear stress near contact interface.

Figure 8 :Figure 9 :
Figure 8: Effect of rock stiffness on the strength of coal and rock, respectively, near contact interface.

Figure 10 :
Figure 10: Effect of inclined angle on principal stresses of rock and coal near contact interface.