Summary of my research:

1. I have proposed a general higher-order theory for arbitrarily curved rod/pipe-like structures to analyze general wall deformation of the structure made of hyperelastic material along with its finite element model. First, a way to frame a general closed space curve via hybrid frame is put forth and then, a novel way of hybrid frame-based curvilinear cylindrical coordinate (CCC) system is introduced to map the complex geometries of any curved pipe. Similar Bishop- and Frenet-CCC systems can be obtained as a special case of the hybrid-CCC system. This theory has a potential application in the analysis of biological structures like arteries in the human body.

2. In the shell theory, I have proposed a general higher-order shell theory using Cartan's moving frame to analyze the large deformation of thin or thick shell structures made of isotropic compressible or incompressible hyperelastic materials. Weak-form finite element models of the proposed shell theories are also developed. The use of Cartan's moving frame makes the shell theory computationally very efficient than the customarily used natural covariant coordinate bases, especially in the case of the nonlinear material

3. In my research, for both curved rods/pipes and shell geometries, I have used the method of the exterior calculus in conjugation with the orthonormal (Cartan's) moving frame to obtain the kinematic relations which is proved to be very elegant and computationally efficient as compared to the classically used tensor method. This approach has not been reported in structural theories such as rods and shells before. The use of the orthonormal frame makes the mathematical formulation of complex constitutive relation relatively easier.

4. I would also like to highlight the methodology developed for the formulation of the theories as well as their finite element models for the general basis functions approximation of displacement field via matrix approach, which is also scalable to various kinds of problems in solid and fluid mechanics. For example, the Fourier basis and Taylor's or Legendre series for the rod theory and Taylor's series, Legendre, or Lagrange polynomial (for layered structure) for shell theory, can be chosen as suited in the problem at hand.  

5. In the context of the Cosserat continuum, higher-order theories for beams and plates for rotation gradient dependent and couple stress theories are developed, which can be used to analyze beams and plates with rigid embeddings in a relatively compliant matrix material.