2019/10/16 by Nicos Makris
Engineering · Physics and Astronomy · #Causality (physics) #Constitutive equation #Elasticity and Material Modeling #Function (biology) #Hilbert space #Meaning (existential) #Nonlinear system #Relation (database) #Thermoelastic and Magnetoelastic Phenomena #Vibration Control and Rheological Fluids #Viscoelasticity #physics.class-ph
paper · pdf · doi:10.1007/s11012-020-01137-x
published as Meccanica 2020
arxiv created 2019/10/16 · openalex created_date 2019/10/25 · openalex publication_date 2020/02/14 · arxiv updated 2020/11/17 · openalex updated_date 2026/08/05
This paper revisits the physical meaning of linear, time-domain constitutive models with complex parameters that have been presented in the literature and concludes that such models are not physically realizable. While complex-parameter phenomenological models (including those with complex-order time derivatives) may be efficient in capturing in the frequency domain the frequency-dependent behavior of viscoelastic materials over a finite frequency band, they do not possess physically acceptable time-response functions. The paper first reviews the intimate relation between the causality of a physically realizable constitutive model and the analyticity of its frequency-response function and explains that in theory it is sufficient to conduct a nonlinear regression analysis for estimating the model parameters either on only the real part or on only the imaginary part of its frequency-response function, given that they are related with the Hilbert transform. Consequently, the resulting model-parameters are real-valued; therefore, there is no theoretical justification to conduct the nonlinear regression analysis for estimating the model parameters in the complex space. The paper concludes with an example by showing that the relaxation modulus of the complex-coefficient Maxwell model is a divergent function at all positive times; therefore it is not a physically realizable constitutive model.