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 ef fective in describing the response of practical dampers and other diss ipation devices used for seismic protection of structures. Complex-par ameter viscoelastic models must be subjected only to complex-valued ex citations 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 o f constitutive models with complex-coefficients is investigated acid i t is found that under certain conditions they satisfy the Kramers-Kron ig relations. These relations ensure that the differential operator us ed in the model is causal; however, the entire model (differential ope rator and analytic input) is noncausal, since the Hilbert transform ne eded to construct the analytic input requires information from the fut ure. 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 integ ration techniques. Time-domain techniques are necessary when the propo sed constitutive models describe devices which are incorporated in str uctures that exhibit nonlinear response. The equivalence between compl ex-valued and real-valued representations is shown through a practical example, and the noncausality of these models is analyzed.