Entire Life Theoretical Model of Limestone under Unequal Cyclic Loading Based on the Expanding Theory of Thermodynamic System Analysis

Abstract

Breakage, Helmholtz free energy, and nonlinearity are involved in many fundamental phenomena of complex systems across natural sciences. However, a mathematical equation that can express the entire life cycle of complex natural objects (such as rock-like quasi-brittle materials) is lacking. We expand the material body system to an isolated compound thermodynamic system to establish an innovative theoretical model induced by coupling nonlinear separation of Helmholtz free energy and breakage evolution that can be used to express the extreme entire life model of rock under unequal amplitude loading and unloading cycle. We gain crucial insights into the life essence of limestone, which is termed “Negative Dissipation.” This study first shows that the change in the mechanical properties of quasi-brittle materials caused by the timely evolution of breakage can be represented by the nonlinear separation of Helmholtz free energy and negative dissipation. An analytical solution to the nonlinear separation variables of Helmholtz free energy is provided by combining the method of solving nonlinear partial differential equations in mathematics and thermodynamic law. An analytical solution of Helmholtz free energy considering nonlinearity and breakage is proposed, and an equation that can reflect the constitutive mechanics law of the entire life cycle of rock in the theoretical model is presented. Theoretical results are consistent with the experimental data obtained from limestone samples with different prefabricated cracks. This original study provides a theoretical foundation for the life model of complex natural objects for nonlinear breakage and an early warning investigation of rocks under various unprecedented conditions.Practical ApplicationsThis study first found that the change in the mechanical properties of complex natural objects caused by the timely evolution of breakage can be represented by the nonlinear separation of Helmholtz free energy. We establish an innovative theoretical model induced by coupling nonlinear separation of Helmholtz free energy and breakage evolution. The innovative theory includes three parts: (1) an analytical solution to the nonlinear separation variables of Helmholtz free energy, (2) an analytical solution of Helmholtz free energy considering nonlinearity and breakage, and (3) an equation of the theoretical model that can reflect the constitutive mechanics’ law of the entire life of quasi-brittle rocks is presented for the first time. This original research result provides the foundation for a more in-depth life cycle of quasi-brittle rocks to develop the theoretical basis for the nonlinear breakage and early warning research of materials under various unprecedented conditions. We report a route to material microstructure composition, which may open an alternative pathway to quasi-brittle materials, which may, in turn, open a door into the mysterious world of science.