Macroscopic Models with Complex Coefficients and Causality

In this paper the causality of linear viscoelastic models with complex coefficients is examined. Such constitutive models have been found effective in describing the response of practical dampers and other dissipation devices used for seismic protection of structures. Complex-parameter viscoelastic models must be subjected only to complex-valued excitations that are analytic functions, i.e., their imaginary and real parts are related with the Hilbert transform. First, it is shown that the resulting force from complex parameter constitutive models is also an analytic signal. Subsequently, the analyticity of the impedances of constitutive models with complex-coefficients is investigated and it is found that under certain conditions they satisfy the Kramers-Kronig relations. These relations ensure that the differential operator used in the model is causal; however, the entire model (differential operator and analytic input) is noncausal, since the Hilbert transform needed to construct the analytic input requires information from the future. Finally, a general real-valued representation of these models is developed. Real-valued representations are needed when the analysis of the response is performed in the time domain using step-by-step integration techniques. Time-domain techniques are necessary when the proposed constitutive models describe devices which are incorporated in structures that exhibit nonlinear response. The equivalence between complex-valued and real-valued representations is shown through a practical example, and the noncausality of these models is analyzed.