Generalized effective medium theory for elastic systems with correlated disorder - Escobar Agudelo, Jorge Mario [UNESP] (2025)
This work develops a generalized coherent potential approximation (CPA) framework to study rigidity transitions in elastic systems with correlated disorder, inspired by the mechanical properties of soft colloidal gels. By extending traditional effective medium theories, we incorporate spatial correlations into the probability distribution of disorder, enabling the analysis of more complex systems. The framework is implemented numerically using a triangular lattice as a test case, leveraging symmetries and approximations to simplify the self-consistency constraint. Our primary contribution is the formulation of a versatile theoretical tool capable of describing rigidity transitions in a wide range of disordered systems, including those with correlated disorder. As a demonstration, we apply the framework to a model inspired by colloidal gels, revealing that spatial correlations shift the critical parameter p_c for rigidity transitions but do not alter the coordination number z at which rigidity occurs, consistent with Maxwell's criterion (z_c=2d). Additionally, we observe scale invariance in the system, indicating universal behavior near the critical transition. These results validate the robustness of the generalized CPA and highlight its potential for studying complex materials such as granular networks, biological tissues, and mechanical metamaterials.