A Nonlinear Breakage Mechanics Model: From Extreme Entire Life Model to Breakage Evolution of Limestone Based on Separation of Helmholtz Free Energy under Cyclic Loading

Abstract

Rock structures and load conditions play a considerable role in the deformation and failure process of rocks. To explore the nonlinear mechanical mechanism of brittle rocks, this study presents the extreme entire life constitutive equation, breakage equation, and rock resistance breakage energy (RBE) equation of limestone specimens under unequal amplitude cyclic loads. The theoretical model is based on the calculus method and thermodynamic principle, and the Helmholtz free energy of a rock system after the action of a composite system is nonlinearly separated. The extreme entire life constitutive equation of limestone is deduced on the basis of analytical solutions of nonlinear separation parameters and breakage evolution, whereas the breakage equation is deduced on the basis of the principles of residual breakage energy and thermodynamics. We present the analytical solution of the material RBE, which reflects the ability of rocks to resist breakage under load. The model performance is verified by limestone samples with different prefabricated cracks under unequal amplitude cyclic loading. This research may lay the foundation for studies of nonlinear breakage models under different conditions. The study investigates a theoretical model derived from the coupling of nonlinear separation of Helmholtz free energy and breakage evolution. It is discovered that the entire lifespan of complex natural objects, caused by the timely evolution of breakage, can be represented by the nonlinear separation of Helmholtz free energy. An analytical solution for Helmholtz free energy considering nonlinearity and breakage is proposed. Additionally, an equation in the theoretical model is presented to reflect the law of constitutive mechanics governing the entire lifespan of brittle rock materials. Based on the principles of residual breakage energy and thermodynamics, we provide an analytical solution for material resistance breakage energy, which represents the ability of brittle rocks to resist breakage under load. The model is validated using limestone samples with different prefabricated cracks under unequal amplitude cyclic loading. This study establishes a foundation for exploring nonlinear breakage models across various conditions and offers theoretical guidance for enhancing the safety and stability of engineering structures.