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Time-domain modeling of damping using anelastic displacement fields and fractional calculus. (English) Zbl 0943.74008
Summary: We develop a fractional derivative model of linear viscoelasticity based on the decomposition of the displacement field into an anelastic part and an elastic part. The evolution equation for the anelastic part is a differential equation of fractional order in time. By using a fractional-order evolution equation for the anelastic strain, the present model becomes very flexible for describing the weak frequency dependence of damping characteristics. To illustrate the modeling capability, we fit the model parameters to available frequency domain data for a high-damping polymer. By studying the relaxation modulus and the relaxation spectrum, we examine the physical meaning of material parameters of the present viscoelastic model. The use of this viscoelastic model in structural modeling is discussed, and the corresponding finite element equations are outlined, including the treatment of boundary conditions. The anelastic displacement field is mathematically coupled to the total displacement field through a convolution integral with a kernel of Mittag-Leffler function type. Finally, we give a time-step algorithm for solving the finite element equations, and discuss some numerical examples.
MSC:
74D05Linear constitutive equations (materials with memory)
74S05Finite element methods in solid mechanics