Dissipated energy function, hysteresis and precondition of a viscoelastic solid model

In this paper, energy dissipation for any (n+1)(n+1)th cycle of the standard viscoelastic solid model under the saw-tooth loading is discussed based on the Boltzmann superposition principle. The dissipated energy function Wn+1Wn+1 for the complete (n+1)(n+1)th cycle is obtained. Some results previously obtained are included in it. We also obtain the energy dissipation difference and the stress difference between two adjacent cycles, which imply that the energy dissipation for a full cycle decreases with an increase of the period and converges to a positive constant, and provide a beautiful interpretation for hysteresis and precondition from mathematical perspective.