Analysis of S Wave Propagation Through a Nonlinear Joint with the Continuously Yielding Model


Seismic wave propagation through joints that are embedded in a rock mass is a critical issue for aseismic issues of underground rock engineering. Few studies have investigated nonlinear joints with a continuously yielding model. In this paper, a time-domain recursive method (TDRM) for an S wave across a nonlinear Mohr–Coulomb (MC) slip model is extended to a continuously yielding (CY) model. Verification of the TDRM-based results is conducted by comparison with the simulated results via a built-in model of 3DEC code. Using parametric studies, the effect of normal stress level, amplitude of incident wave, initial joint shear stiffness, and joint spacing is discussed and interpreted for engineering applications because a proper in situ stress level (overburden depth) and acceptable quality of surrounding rock mass are beneficial for seismic stability issues of underground rock excavation. Comparison between the results from the MC model and the CY model is presented both for an idealized impulse excitation and a real ground motion record. Compared with the MC model, complex joint behaviors, such as tangential stiffness degradation, normal stress dependence, and the hysteresis effect, that occurred in the wave propagation can be described with the CY model. The MC model seems to underestimate the joint shear displacement in a high normal stress state and in a real ground motion excitation case.

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C s :

S wave velocity

CY model:

Continuously yielding model

E :

Young’s modulus


Joint normal stiffness exponent of the CY model


Joint shear stiffness exponent of the CY model

F :

Tangent modulus governing parameter of the CY model

K n :

Joint normal stiffness

K s :

Joint shear stiffness

R :

Joint roughness parameter of the CY model

r :

Loading direction index of the CY model

R s :

Reflection coefficient magnitude


Time-domain recursive method

T s :

Transmission coefficient magnitude

u s :

Shear displacement

u ps :

Plastic shear displacement

V :

Velocity of the joint incident side

V + :

Velocity of the joint transmit side

V Is(i) :

Incident wave velocity

V Rs(i) :

Reflection wave velocity

V Ts(i) :

Transmission wave velocity

Z s :

Impedance of the rock mass medium for S wave

a n and K ni :

Joint initial normal stiffness of the CY model

a s and K si :

Joint initial shear stiffness of the CY model

μ :

Poisson’s ratio

ρ :

Density of the rock mass

\(\emptyset\) :

Joint basic friction angle of the CY model

\({\emptyset_{m}^{i}}\) :

Joint initial friction angle of the CY model

\({\emptyset_{m}}\) m :

Joint friction angle

σ n :

Joint normal stress

τ :

Joint shear stress

τ m :

Joint shear strength


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We would like to acknowledge the reviewers and the editors for their comments and suggestions. The study was financially supported by the National Basic Research Program of China (No. 2015CB057905), National Key R&D Program of China (No. 2016YFC0401803), and the National Natural Science Foundation of China (Nos. 51409263, 41672319, 11472292).

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Correspondence to Zhen Cui.

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Cui, Z., Sheng, Q. & Leng, X. Analysis of S Wave Propagation Through a Nonlinear Joint with the Continuously Yielding Model. Rock Mech Rock Eng 50, 113–123 (2017).

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  • Wave propagation
  • Nonlinear joint
  • Time-domain recursive method
  • Continuously yielding model