Abstract
Phenomenological equation relating a non-linear response function to a time-dependent excitation function is presented, starting from the assumptions of causality, convergence and stationariness. When the excitation and the response are denoted by σ(t) and ε(t), respectively, the equation is
ε(t)=∫∞0σ(t-τ1)J1(τ1)dτ1+∫∞0∫∞0σ(t-τ1)σ(t-τ2)J2(τ1, τ2)dτ1dτ2+∫∞0∫∞0∫∞0σ(t-τ1)σ(t-τ2)σ(t-τ3)J3(τ1, τ2, τ3)dτ1dτ2dτ3+………,
where J1, J2, J3, …… are a series of the decay functions which characterize the excitation-response system. Fourier representation of the above equation (Eqs. (11) and (12) in the text) as well as several transformation relations (Eq. (13)) are derived. Applications of the theory to the case of a sinusoidal excitation and to the case of a step function-like excitation, especially with exponential decay functions, are developed. Finally, a critical examination of the adopted assumptions and a remark on the difference between a creep function and a recovery function are made.
