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Over the past few decades, the prospect of energy generation from an oscillating piezoelectric patch has gained attention. A typical setup of this kind would be a piezoelectric patch mounted on a flexible plate that is exhibiting self-sustained flapping motion. Piezoelectric patches are generally multilayered consisting of piezoelectric, substrate and electrode layers placed on top of each other. In this paper, we first propose a quasi-monolithic formulation with exact interface tracking to simulate the fluid-multilayered structure interactions. We validate the proposed formulation, using an in-house solver, by considering a simple two-layered plate-like structure with identical material properties against a single-layered plate. We then use this formulation to study a two-layered flexible plate, wherein each layer of the plate has independent material properties. Systematic parametric simulations are conducted to understand the effect of difference in material properties between the two layers of a plate on the self-sustained flapping dynamics. The parametric simulations are performed at a Reynolds number Re=1000 by selecting different values of Young's modulus and density for each of the layers such that the average structure to fluid mass ratio (m)avg=0.1 and average non-dimensional bending stiffness (KB)avg=0.0005. Firstly, the effects of difference in elasticity between the two layers on the flapping amplitude, frequency, forces and vortex shedding patterns are investigated. Following this, the effect of such difference in elastic properties on the onset of flapping is investigated for a case with Re=1000, (m)avg=0.1 and (KB)avg=0.0008 and two distinct response regimes are observed: (a) fixed-point stable and (b) periodic LCO. Finally, we look into the effects due to the difference in the structural density between the two layers on the flapping dynamics of the plate.