The advanced rheological models of Andrade (1910) and Sundberg & Cooper (2010) are compared to the traditional Maxwell model to understand how each affects the tidal dissipation of heat within rocky bodies. We find both the Andrade and Sundberg-Cooper rheologies can produce at least 10$times$ the tidal heating compared to a traditional Maxwell model for a warm (1400-1600 K) Io-like satellite. Sundberg-Cooper can cause even larger dissipation around a critical temperature and frequency. These models allow cooler planets to stay tidally active in the face of orbital perturbations-a condition we term tidal resilience. This has implications for the time evolution of tidally active worlds, and the long-term equilibria they fall into. For instance, if Ios interior is better modeled by the Andrade or Sundberg-Cooper rheologies, the number of possible resonance-forming scenarios that still produce a hot, modern Io is expanded, and these scenarios do not require an early formation of the Laplace resonance. The two primary empirical parameters that define the Andrade anelasticity are examined in several phase spaces to provide guidance on how their uncertainties impact tidal outcomes, as laboratory studies continue to constrain their real values. We provide detailed reference tables on the fully general equations required for others to insert the Andrade and Sundberg-Cooper models into standard tidal formulae. Lastly, we show that advanced rheologies greatly impact the heating of short-period exoplanets and exomoons, while the properties of tidal resilience can mean a greater number of tidally active worlds among all extrasolar systems.
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