Fractal Operators and Convergence Analysis in Fractional Viscoelastic Theory

This study delves into the convergence of operators and the viscoelastic properties of fractal ladder and tree structures. It proves the convergence of fractal stiffness operators through operator algebra, revealing a fundamental connection between operator sequence limits and fractal operator algebraic equations. Our findings demonstrate that, as the hierarchical levels of these structures increase, their viscoelastic responses increasingly align with the fractional viscoelastic behavior observed in infinite-level fractal structures. We explore the similarity in creep and relaxation behaviors between fractal ladders and trees, emphasizing the emergence of ultra-long characteristic times in steady-state creep and pronounced tailing effects in relaxation curves. This research provides novel insights into the design of fractional-order viscoelastic structures, presenting significant implications for materials science and mechanical engineering.