  The practical applications of gas-filled encapsulated microbubbles involve
inherent nonspherical oscillations under acoustic fields. The gas-encapsulation
and encapsulation-liquid interfaces significantly affect the mechanics of the
bubbles, especially of smaller radii, and their consideration is vital for
mimicking the experimental setting. In this paper, we apply the interface
energy model [N. Dash and G. Tamadapu, J. Fluid Mech. 932, A26 (2022)] to
examine the nonspherical oscillations of an encapsulated microbubble with a
radius of $2$$\mu$m and $5$$\mu$m under an acoustic field. Using the Lagrangian
energy formulation, the coupled dynamical governing equations for spherical and
nonspherical modes are derived, incorporating the effects of interface energy
at the interfaces, shell elasticity, and viscosity. Through a perturbation
analysis based on the Krylov-Bogoliubov method of averaging, a set of
first-order differential (slow-time) equations is obtained to conduct
steady-state and conditional-stability analysis. The stability analysis helped
in determining the excitation pressure and frequency of the acoustic field
required for smaller radii bubbles to exhibit finite amplitude shape
oscillations. Direct numerical simulations of the governing equations revealed
that the parametrically forced even mode ($n=2$) excites even modes, while the
odd modes ($n=3$) excite both even and odd modes. For smaller radii bubbles, we
observe shape mode oscillations of finite non-zero amplitudes only in the
presence of interface parameters. The initial size-dependent interface
parameter and shell viscoelastic parameters are identified as the key
parameters that play a critical role in exhibiting finite shape mode
oscillations of the bubble.
