Modelling the stress–strain curves of rocks using an improved spring model

Understanding the stress–strain behaviour of rocks is essential for ensuring the safe and stable operation of rock structures in engineering. The spring model is conceptually simple and can capture complete stress–strain curves, but the traditional form neglects the influence of confining pressure. In this study, an improved spring model is proposed that incorporates confining pressure effects through empirical relationships between model parameters and confining pressure. Implemented in MATLAB, the model successfully reproduces the stress–strain curves of red layer rocks under confining pressures of 1, 3 and 4 MPa, showing close agreement with experimental results in both the peak strength and post-peak behaviour. Parametric analysis reveals that the spring retention force factor primarily controls the post-peak stress drop, the diameter-to-height ratio influences the slope of the curve, and the mean maximum deformation determines the peak strength. Additionally, the model can simulate both Class I and Class II curves by adjusting the relative deformation capacities of the spring blocks, and it captures the transition from uniform to localized failure modes. These findings demonstrate that the improved spring model offers a simple yet effective tool for simulating rock mechanical behaviour under varying confining pressures.