Fractional topology optimization of periodic multi-material viscoelastic microstructures with tailored energy dissipation

The microstructural configuration of a material affects its macroscopic viscoelastic behavior, which suggests that materials can be engineered to achieve a desired viscoelastic behavior over a range of frequencies. To this end, we leverage topology optimization to find the optimized topology of a multi-phase viscoelastic composite to tailor its energy dissipation behavior as a function of frequency. To characterize the behavior of each material phase, we use a fractional viscoelastic constitutive model. This type of material model uses differential operators of non-integer order, which are appropriate to represent hereditary phenomena with long- and short-term memory. The topology optimization formulation aims to find the lightest microstructure that minimizes the sum of squared loss modulus residuals for a given set of target frequencies. This leads to the design of materials with either maximized loss modulus for a given target frequency or tailored loss modulus for a predefined set of frequencies. We present several numerical examples, both in 2D and 3D, which demonstrate that the microstructural configuration of multi-phase materials affects its macroscopic viscoelastic behavior. Thus, if properly designed, the material behavior can be tailored to dissipate energy for a desired frequency (maximized loss modulus) or for a range of frequencies (tailored energy dissipation behavior).