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New constitutive model based on disturbed state concept for shear deformation of rock joints

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Abstract

The mechanical behavior and constitutive relation of rock joints have caught more and more attention in the field of geotechnical engineering. The disturbed state concept (DSC) theory offers a powerful tool for building a constitutive model to interpret the mechanical response of geomaterials. In this paper, a new constitutive model for joint shear deformation was developed based on the DSC theory. The characteristics of quasi-elastic phase, pre-peak hardening phase, peak shear strength, post-peak softening phase and residual strength during the whole process of joint shear deformation are considered in this model. In the framework of this shear constitutive model, the rock material was assumed to consist of two kinds of micro-units with different mechanical responses, namely the relatively intact unit and the fully adjusted unit. Subsequently, the DSC theory was used to connect the mechanical behavior of micro-units with the macroscopic joint shear deformation characteristics, and a disturbance factor was introduced to reveal the disturbed state evolution process inside the rock. In addition, the proposed DSC model was simple in form, less in parameters and reasonable in physical meaning. The model was cross-validated by experimental data of different kinds of natural joints and artificial joint replicas. Finally, the model is compared with existing models, and the model effectiveness is quantitatively evaluated through statistical indicators. The values of R2 are greater than 0.9, and the AAREP and RMSE of the proposed DSC model are closer to 0 than those of other models. The research results can provide a valuable reference for further understanding of shear deformation mechanism.

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Data availability

Some or all data, models or codes that support the findings of this study are available from the corresponding author upon reasonable request.

Abbreviations

A :

Nominal shear area

A 0 :

Area of a single unit

A RI :

Total area of rock units in RI state

A FA :

Total area of rock units in FA state

B :

Model parameter

C 1 :

Integration constants

C 2 :

Integration constants

D :

Disturbance factor

D ′ :

Disturbance rate

FA :

Fully adjusted state

k s :

Joint shear stiffness

M :

Represents the total number of data points

m :

Sum of the number of rock units in RI state

n :

Sum of the number of rock units in FA state

N :

Total number of rock units

RI :

Relatively intact state

α :

Fitting coefficients

β :

Fitting coefficients

λ :

Fitting coefficients

σ n :

Normal stress

τ :

Nominal shear stress

τ mea :

Measured shear stress value

τ pre :

Predicted shear stress value

τ r :

Residual shear stress

τ p :

Peak shear stress

τ RI :

Shear stress borne by rock units in RI state

τ FA :

Shear stress sustained by rock units in FA state

δ :

Shear displacement

δ p :

Peak shear displacement

δ m :

Shear displacement necessary to mate the joints

r :

DSC model parameter

ŋ :

DSC model parameter

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Acknowledgements

This paper gets its funding from Projects (42277175, 52104110) supported by National Natural Science Foundation of China; Project (2020JJ5715) supported by Hunan Provincial Natural Science Foundation of China; Hunan provincial key research and development Program (2022SK2082); and Hunan Civil Air Defense Research Project (HNRFKJ-2021-07). The authors wish to acknowledge these supports.

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  1. School of Resources and Safety Engineering, Central South University, Changsha, 410083, Hunan, China

    Shijie Xie, Hang Lin & Yifan Chen

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Correspondence to Hang Lin.

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Appendix

Appendix

In this appendix, we provide a detailed procedure for solving the basic equation of DSC theory for stress.

Before modeling the joint plane with the DSC theory, the following basic assumptions were made:

  1. (1)

    The joint plane is made up of innumerable rock units. Assuming that the total number of units is N (N →  + ∞) and the area of a single unit is A0, the nominal shear area A of the joint can be expressed as A = NA0.

  2. (2)

    The sum of the number of rock units m in RI state and n in FA state at any time is equal to the total number of rock units N, that is, N = m + n.

  3. (3)

    Before shearing, the rock units are all in RI state. Throughout the shearing of rock joints, the increment of rock units changing from RI state to FA state is gradually augmenting under shearing action. This process is rapid and irreversible. When the shearing process enters the residual phase, the rock units are all in FA state.

The application of external forces, including normal and shear direction, is a quasi-static process in each phase of the joint shear failure process. In view of the mechanical equilibrium along with the loading directions of shear stress, the mechanical equilibrium equation can be obtained as follows:

$$\tau A = \tau_{{\text{RI}}} A_{{\text{RI}}} { + }\tau_{{\text{FA}}} A_{{\text{FA}}}$$
(A1)

where τ and A represents nominal shear stress and corresponding nominal shear area, respectively, τRI refers to the shear stress borne by the rock units in RI state, ARI is the total area of these units, τFA denotes the shear stress sustained by the rock units in FA state and AFA is the total area of these units.

Underpinned by the previous assumptions, the three shear areas A, ARI and AFA can all be expressed as the product of the number of rock units and the area of micro-elements A0 in the corresponding state, i.e.,

$$\left\{ \begin{gathered} A = NA_{0} \hfill \\ A_{{\text{RI}}} = mA_{0} \hfill \\ A_{{\text{FA}}} = nA_{0} \hfill \\ \end{gathered} \right.$$
(A2)

Then substituting Eq. (A2) into Eq. (A1) yields:

$$\tau NA_{0} = \tau_{{\text{RI}}} mA_{0} { + }\tau_{{\text{FA}}} nA_{0}$$
(A3)

Divide both sides of Eq. (A3) by the NA0 term:

$$\tau = \tau_{{\text{RI}}} \frac{m}{N}{ + }\tau_{{\text{FA}}} \frac{n}{N}$$
(A4)

The relative contribution of these rock units in different states (RI or FA) to the joint macroscopic shear behavior could be defined as the disturbance factor D. The number of rock units in the FA state and their growth rate indicate the failure degree of rock materials [52]. As a result, the disturbance factor D can be defined as follows in terms of the proportion of rock units in the FA state to all rock units:

$$D = \frac{n}{N}$$
(A5)

Considering N = m + n, substitute Eq. (A5) into Eq. (A4), and Eq. (A4) can be converted into:

$$\tau = \tau_{{\text{RI}}} \left( {1 - D} \right){ + }\tau_{{\text{FA}}} D$$
(A6)

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Xie, S., Lin, H. & Chen, Y. New constitutive model based on disturbed state concept for shear deformation of rock joints. Archiv.Civ.Mech.Eng 23, 26 (2023). https://doi.org/10.1007/s43452-022-00560-z

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