Study on the Constitutive Model for Jointed Rock Mass

A new elasto-plastic constitutive model for jointed rock mass, which can consider the persistence ratio in different visual angle and anisotropic increase of plastic strain, is proposed. The proposed the yield strength criterion, which is anisotropic, is not only related to friction angle and cohesion of jointed rock masses at the visual angle but also related to the intersection angle between the visual angle and the directions of the principal stresses. Some numerical examples are given to analyze and verify the proposed constitutive model. The results show the proposed constitutive model has high precision to calculate displacement, stress and plastic strain and can be applied in engineering analysis.


Introduction
In the rock engineering, joints have significant effect on the stress-strain relationship of jointed rock mass. Generally speaking, there are two categories of approaches: the first method is that joint element is utilized to simulate jointed rock mass. The other method is that special constitutive model is utilized to simulate jointed rock mass.
Constitutive models for jointed rock masses are important for numerical modeling of the behavior of jointed rocks. Many constitutive models for rock joints, based on both empirical and theoretical approach, such as are summarized in [1]. The behavior of the joints is dependent on their sizes, because the scale dependence of surface roughness of the joints whose thresholds are a scaling parameter [2][3][4]. Some researchers took study on landslide problems and the dynamic frictional processes of the joints using theories of dynamic chaos and catastrophe for an analysis of the interactions between the fracture surfaces regarding friction, fracture stiffness and elastic materials for the jointed rocks [5]. Some researchers utilized joint factor to simulate jointed rock mass based on the finite element method [6]. Some researchers proposed the model for the equivalent elastic parameters of jointed rock mass [7,8]. Some researchers performed the modeling of dynamic rock fracture sliding using the state variable friction models. In the model, the shear stresses are the functions of both the sliding history and velocity. And the model represented the evolution of rate effects and the path-dependence of the frictional properties [9]. Some researchers utilized representative volume method to analyze nonlinear characteristics of one-way joint and the interaction of two-way orthogonal joints [10]. Some researchers reported new 3D constitutive models for rough rock fractures based on experimentally determined relations between the contact areas under normal loads and asperity inclination angles [11,12]. Some researchers established the model to calculate the physical parameters of jointed rock mass [13]. Some researchers established softening model for multijoints [14].
In this paper, the studies on elasto-plastic constitutive model for jointed rock mass are made. The influences of joints on the jointed rock mass are analyzed. Based on these studies, a constitutive model for jointed rock mass, which can consider anisotropic strength of jointed rock mass and anisotropic increase of plastic strain, is constructed. And then the numerical examples are performed to analyze and verify the proposed constitutive model.

The construction of constitutive model
Morh-Coulomb model is well-known model in geotechnical engineering application, including in rock engineering modelling and design. The basic concepts of the Mohr-Coulomb model suggest that the behaviors of a rock material are made up of two parts: a constant cohesion and a friction coefficient. And it can be described as where τ s is the shear strength, σ n is the normal stress, c is the cohesion, φ is friction angle. The parameters of this model are only two, and it is widely used due to the simple expression. But this model is based on the isotropy theory. It can only describe the isotropic material. And jointed rock mass is anisotropic material. The classical Morh-Coulomb model cannot describe the behaviors of jointed rock. So it need to be improved due to its limitations. Fig 1 shows rock bridges exist in jointed rock masses because of the non-persistent nature of joints. In order to calculate the decrease of strength of jointed rock masses in different directions, It defines the mechanical persistence ratio of rock mass as that the ratio of joint network on the shear failure path when jointed rock mass is sheared to damaged state along a certain direction [15]. Fig 2 shows that the mechanical persistence ratio k is calculated by where JL and RBR are the projection length of joints and rock bridges in the shear failure path respectively. β 0 is the visual angle, which can be used to express the direction of joints. It defines cohesion c β0 and friction angle φ β0 of jointed rock masses in direction of β 0 [16,17]as where c r and φ r are the cohesion and friction angle of rock bridges, respectively. c i and φ i are the cohesion and friction angle of joints. Thus, based on Mohr-Coulomb model, the yield strength criterion f can be given by where τ and σ are the shear stress and normal stress in direction of β 0 , respectively. The Mohr-Coulomb model is based on plotting Mohr's circle for states of stress at failure in the plane of the maximum and minimum principal stresses. According to Fig 3, we have where σ 1 and σ 3 are the maximum and minimum principal stresses, respectively. β is the intersection angle between β 0 and the directions of the maximum principal stresses σ 1 .  Thus, the yield strength criterion f in plane can be rewritten as in which From (8), through calculating df/dβ = 0, we can obtain the least angle β L in β when β min β β max and we have And the minimum value of σ 1 and σ 3 obey the function f min when β = β L , and we have According to Fig 3, we also have where σ m and τ m are the mean normal stress and the maximum shear stress, respectively. Thus, the yield strength criterion f in plane can be rewritten as ( The yield strength criterion f in plane is extended to three-dimensional yield strength criterion and we have ( Study on the Constitutive Model for Jointed Rock Mass in which where I 1 , J 2 and J 3 are the first invariant of stress tensor, the second and third invariant of deviatoric stress tensor, respectively. In plasticity theory, the strain increment can be decomposed into two parts where dε is the incremental strain tensor; dε e and dε p are the incremental elastic and plastic strain tensor, respectively. The stress-strain relationship is expressed as where dσ' is the incremental stress tensor; D ep is the elasto-plastic stiffness tensor. The elasto-plastic stiffness tensor is expressed as: in which where σ is the stress tensor; D e is the elasto stiffness tensor; n and n g are the loading and flow direction vectors, respectively; f and g are the yield and plastic potential functions, respectively. And in the model, the plastic potential function g is adopted as the same as the yield function f. The fluidity variable Λ can be expressed as The distinction between loading and unloading directions is described through the following criteria: Because the plastic strain will also increase in the process of reloading, the incremental plastic strain is where the symbol〈〉is defined as〈Λ〉 = Λ for Λ>0 and〈Λ〉 = 0 for Λ 0. It shows that the plastic strain will increase if jointed rock mass is in the state of loading. In other word, we have

Numerical implementation
The integral algorithm based on fully implicit backward Euler return mapping algorithm is adopted to calculate the updated stresses. The convergence rule is adopted according to the difference of updated stresses less than tolerance. Fig 4 shows  shows the strength parameters of the joint surface and the rock bridge. Fig 5 shows the persistence ratio, friction coefficient and cohesion of jointed rock mass at the visual angle β 0 . Through observing the results of Figs 5-8, they show that the yield strength criterion f in plane of jointed rock mass is not only related to the friction angle φ β0 and cohesion c β0 of jointed rock masses in direction of β 0 (the visual angle). The yield strength criterion f is also related to β (the intersection angle between the visual angle and the directions of the maximum principal stresses). The yield strength criterion f has the relation of φ β0 and c β0 only when β min β β max . The relation of β and β 0 is also important to the yield strength criterion f. The different relation of β and β 0 leads to different yield strength criterion f. In some special relation of β and β 0 , such as Fig 8 (c), the friction angle φ β0 and cohesion c β0 has no use for the yield strength criterion f. In other word, the persistence ratio k has no use for the yield strength criterion f in some special condition.

Jointed rock direct shear experiment and numerical simulation by proposed model
To verify proposed constitutive model, the compared results of proposed model and experiment are given. The rock mass samples containing joints are 0.3m×0.3m. The visual angles β 0 of joints  Table 2.
Through observing the results of Fig 11 and [19] in commercial software Abaqus, respectively. Table 3 shows the physical parameters of the rectangle foundation.

The numerical examples for a rectangle foundation of jointed rock mass
Through observing the results of Fig 14 and Fig 15, they show that the results of displacement and stress calculated by proposed constitutive model are close to that calculated by ubiquitous-joint constitutive model in commercial software Abaqus, which has been verified. The maximum relative errors of results of displacement and Mises stress calculated by proposed model are 1.77% and 15.25%, respectively.     Fig 21 shows two kinds of jointed rock mass, whose persistence ratios are different, are used to calculate. Table 5 shows the parameters of slope.

The numerical example for slope
Through observing the results of Fig 22 and Fig 23, they show that equivalent plastic strain is easy to develop along the direction, which has higher persistence ratio k. It is unfavorable to anti-slide stability if the visual angle β 0 , which has higher persistence ratio k, is similar to the angle of rock slope. And it shows the proposed constitutive model can consider the persistence ratio k in different visual angle β 0 .               Supporting Information