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Displacement discontinuities method (DDM) is convenient for and efficient at dealing with discontinuity problems such as fracture and fault. However, the known method of ease iteration is not available for nonlinear joint surface problems. This paper introduces Barton-Bandis nonlinear joint deformation failure criterion, figures out the propagation model of joint through the maximum energy release rate theory of rock fracture mechanics, and establishes three-dimensional nonlinear displacement discontinuity model for rock joint. This paper gives results of the joint propagation pattern and its distribution law under tension and compression with different parameters and side pressure coefficients via compiled program.

Displacement discontinuities method (DDM) is an indirect boundary elements method. By inputting relative displacement on fracture surface as unknown variables in seeking internal stress and strain of the model, DDM is convenient for and efficient at dealing with discontinuity problems. Compared to finite element method, DDM requires less variables so as to improve the calculating efficiency and accuracy and has its own advantages on joint propagation simulation.

In 1957, Rongved put forward his

In terms of joint propagation, Li applied two-dimensional direct and indirect boundary element method to simulate the nonlinear joint propagation [

In the geotechnical engineering, the initial stress state of the rock mass is a pressure state. The deformation characteristics of the discontinuities such as faults show a fairly complex nonlinearity. The simplification of the structural plane as an elastic fracture would ignore its vital character of contact. For rock discontinuity structural plane in compression shear condition, the closed condition of the structural plane should be considered (open, closed without slip, and closed slip). The known method of solving the iterative solution is only available to the linear deformation structural plane. As for the nonlinear structural plane model which conforms to the actual deformation characteristic of the fracture, it is necessary to consider not only the three closed conditions mentioned above, but also the iterative solution of the element nonlinear deformation.

This paper combines the Barton-Bandis normal, tangential deformation model and the Barton shear failure criteria to establish a nonlinear three-dimensional DDM. It invokes the maximum energy release rate theory of rock fracture mechanics to figure out the propagation pattern of a joint [

Bandis established a hyperbolic function of the natural joint normal fracture closure

where

Formulas of the initial stiffness

where

The normal stiffness

The relationship between shear stiffness

where

Peak shear stiffness

where

The Barton-Bandis rule was established on numerous shearing tests on rock joint [

where

When the normal stress is small and the weathering thickness of joint plane is less than 1 mm, the peak shear strength is controlled by the residual friction angle. The basic friction angle

The three-dimensional triangular displacement discontinuity element is shown in Figure

3D displacement discontinuity element.

For the triangular displacement discontinuity element, the distributions of the displacement are obtained as follows.

The distributions of the stress are obtained as follows:

where G is the shear modulus, v is Poisson’s ratio, and the kernel function I is as follows.

When the collocating point, being the center of triangular element, is not on the integration element, the integrals in (

The arbitrary discontinuities are separated into N elements. Applying the basic solutions mentioned above, the local coordinates of the joint are replaced by x, y, and z in the above formula:

where

Respectively, with the additional shear force

where

For stress analysis of cracks, the most important part is near the crack front. The research of fracture mechanics is limited to the displacement field and stress field in the vicinity of the crack front. The local coordinate established at the crack front is shown in Figure

Local coordinate at the crack front.

where

The definition formulas of stress intensity factor are as follows:

where

For the displacement discontinuity in the discrete crack front element in the three-dimensional DDM, it is expressed with

For practical purpose, the limits in (

We adopt the maximum energy release rate (G criterion) as the criteria for 3D compound joint propagation on plane. There are two hypotheses in the criteria: (1) the joint extends along the direction in which the maximum energy release rate is generated; (2) when the energy release rate reaches the critical value

For elastic media, the strain energy release rate is [

According to the energy release rate criterion, the direction of the joint propagation is the direction of the maximum value of the circumferential stress. In the normal plane of the joint frontier, the angle between the joint propagation direction and the normal n-axis, that is, the propagation angle

When

Considering type II crack and its effect, we substitute

where B is g empirical coefficient and it usually takes B = 1.0.

According to selected joint propagation criteria, the extended step length of any crack front should be proportional to its equivalent stress intensity factor Ke. Therefore, the extended step length of crack front is

where

The maximum step length of the initial step

where E stands for the coefficient; this paper takes E = 0.01.

The simulation of the three-dimensional joint propagation might be distorted. While assuming that the propagation occurs in the joint frontier plane, the crack front could be treated as a set of line segments. After a propagation occurs outside this line, a new set of leading edge segments would be generated [

The new and old joint front edge surface mesh.

Joint propagation requires stepwise load. In each step, it needs to determine whether the joint reaches the propagation conditions and calculate the propagated angle and length to complete propagation of the step. If the initial load is too small, the joint would not propagate. The smaller the load step is, the more accurate the simulation would be, while the more calculation would be required. For the hyperbolic nonlinear rock joint model, iterative solution is needed for its nonlinearity. The entire calculation process requires nested iteration to complete, as shown in Figure

Flowchart of joint propagation.

At any step of external loading or propagation within a step of the external loading, do a DDM analysis to obtain the displacement discontinuity components

If all elements are open, then calculate the strain energy release rate for all front elements and check for extension of the whole joint.

If there is any front element that extends, then add new elements to its front; go back to (a) and do a new round DDM analysis since geometry has changed; check all the elements again for their state of open or closed and repeat the process.

If there is no front element that extends, then if the external loading reaches the specified value, stop the simulation; otherwise, increase the external loading; go back to (a) and do a new round DDM analysis since the external loading has changed; check all the elements again for their state of open or closed and repeat the process.

If there is any element that is closed, then joint stiffness takes effect.

Calculate the joint force

If

If

Add the joint force components as outlined above to the traction on all the closed elements, leave the traction on the open elements unchanged, and do a new round DDM analysis to obtain the displacement discontinuity components

Check the tolerance of

If

If

If

When new crack front element appears, we need to determine whether the calculation of joint propagation should continue. If it is still required, update statistics of crack front element, and go back to (a). If not, terminate the calculation.

We use compiled software to calculate and analyze conditions in reference and have got the following comparison between result of joint propagation calculation and result in the reference, as shown in Figure

Trajectory of joint propagation under compression.

Simulation result

Experiment result

Although the actual joint in the rock is often a plane, its geometric boundaries would be normally irregular shapes. In order to facilitate the analysis of the universality, this paper takes the discoid joint as an example to carry out the simulation. The disc model joint angle is 45°, radius 100m, and shape as in Figure

Parameters of the joint propagation model.

Rock | Joint | ||||||

| |||||||

Elastic modulus ( | Poisson’s ratio | Shear coefficient ( | Shear coefficient ( | c ( | Maximum closure ( | Initial normal stiffness ( | |

| |||||||

4 | 0.2 | 0.01 | 0.2 | 0.05 | 38.6 | 1 | 15 |

3D nonlinear joint propagation process under compression.

The three-dimensional propagation under vertical compression 100MPa is shown in Figure

The joint propagation is carried out in the direction of the vertical pressure, and the propagation begins at the upper and lower ends of the joints. They, respectively, propagate upward and downward. The propagations at the upper and lower sides are the largest, and they decline to the middle, forming a wrap-like joint propagation. At the end of the propagation, variation of stress intensity factor of the joint front slows down and the joint propagation tends to be stable due to the three-dimensional wrap-like propagation shape.

The three-dimensional propagation process under the vertical of 100 MPa could be seen in Figure

3D nonlinear joint propagation process under tension.

The joint propagation process under tension differs from that under compression. Although both of them begin at the upper and lower ends, the upper end propagates downward and the lower end propagates upward. After three steps of propagation, the direction of joint propagation begins to incline to the horizontal direction, that is, perpendicular to the direction of tension. Like the compressing propagation, the final type presents warped shape and is eventually stabilized.

Joint cohesion c is usually small. When keeping other parameters’ values the same as those in Table

Comparison of joint propagation with different cohesion under compression.

The joint propagation peaks when c value is 0.05MPa, while it changes slightly when c value is 1MPa and 2MPa. It indicates that when c value increases to some extent, joint propagation would be limited, and the main differences of joint propagation would be reflected in propagation form. The outer edge shape of the upper and lower ends of the joint are basically the same, yet the propagation of the middle part is different. With the increase of c value, the propagation in the middle part in the vertical pressure direction is restricted, and the overall propagation of the joint tends to be stable.

Keeping other parameters’ values the same as those in Table

Comparison of joint propagation with different friction under compression.

The propagation distances change a little when

Using the parameters in Table

Comparison of joint propagation under compression with different lateral pressure coefficient in axial-Y.

When the lateral pressure coefficient

Comparison of joint propagation with different lateral pressure coefficient in axial-X under compression.

This paper introduces the Barton-Bandis nonlinear joint deformation failure criterion, finds out joint propagation pattern through stress intensity factor parameter in rock fracture mechanics, and establishes the three-dimensional nonlinear joint displacement discontinuity method model. We compare different propagations under tension and compression with different parameters and lateral pressure coefficients through program and get the following results.

The propagation results with vertical compression and tension shows that the direction of joint propagates is along the direction of the maximum pressure which is perpendicular to the direction of tension:

The propagation of the three-dimensional joint is the greatest at both ends of the pressure direction. The middle part gradually reduced into warped shape, and propagation tends to be stable.

The cohesion c has a great influence on the range of joint propagation. The larger the c value is, the smaller the range of joint propagation would be.

The effect of friction angle

With the increase of the lateral pressure coefficient, the joint propagation range in the corresponding direction gets obviously smaller. The magnitude of the lateral pressure, especially that of joint normal pressure, plays a vital role in joint propagation.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This work is financially supported by National Key R&D Program of China (2017YFC0806000), National Natural Science Foundation of China (41601574), and Chongqing Basic and Frontier Research Project (cstc2015jcyjBX0118).