Dynamic system modeling of rock damage evolution: A new type of nonlinear rock constitutive relation

Based on continuum damage mechanics and dynamical systems theory, this study proposes an improved damage theoretical model and describes the damage evolution behavior of rock under uniaxial compression from the perspective of nonlinear dynamical systems. Existing damage models often have difficulty representing the full deformation process in a physically consistent manner, particularly in the initial compaction stage. Building upon the logistic damage equation derived from the elastic modulus method and incorporating the model of insect population dynamics, this study establishes a unified dynamical equation coupling damage accumulation and damage suppression. Two coordinate translation formulas are introduced to calculate D according to the physical mechanisms governing compaction and postcompaction damage growth, from which a piecewise full range stress–strain constitutive relation is constructed. Utilizing experimental data from eight types of rocks, parameter fitting and model validation are performed using MATLAB. Results demonstrate that the model accurately captures the complete stress–strain response. Furthermore, through nonlinear dynamical systems analysis, the stable states, bifurcation behavior, and critical catastrophe conditions in damage evolution are intuitively revealed. This overcomes existing difficulties, such as the challenge of solving complex differential equations and the lack of intuitiveness in understanding parameter coupling effects, achieving a unification of the rock damage process from the macro to meso and back to macro scales. It reflects the profound mathematical dynamical nature underlying the mechanical behavior and establishes a bidirectional confirmation from mathematical logic to physical essence. This study provides a new theoretical framework and tool for rock damage mechanics, possessing significant theoretical importance and engineering applicability.