Micropolar non-classical continuum theories for solids and fluids with rotational inertial physics

Abstract

In this thesis, we derive the conservation and balance laws (CBL) and the constitutive theories for micropolar non-classical continuum theories (NCCT) for thermoviscoelastic solids (TVES) without memory in Lagrangian description and thermoviscous (TV) fluids in Eulerian description based on internal or classical rotations ($_i\nunezbold{\Theta})$ and internal or classical rotation rates $(_i^r{\bar{\nunezbold{\Theta}}})$ due to skew symmetric part of deformation gradient tensor $(\nunezbold{J})$ and due to skew symmetric part of the velocity gradient tensor $(\bar{\nunezbold{L}})$. Thus, these micropolar NCCT for solids and fluids incorporate $\nunezbold{J}$ and $\bar{\nunezbold{L}}$ in their entirety in the derivation of the conservation and balance laws. Both NCCT derived here incorporate rotational inertial physics due to the presence of microconstituents in the derivation of CBL. For micropolar TVES, we consider small deformation, small strain physics. To our knowledge, the inclusion of rotational inertial physics in micropolar NCCT presented here is the first presentation of such a micropolar NCCT.The micropolar NCCT presented in this work for TVES considers two dissipation mechanisms. The first is due to strain rate appearing in the constitutive theory for the deviatoric Cauchy stress tensor. This mechanism is purely due to classical continuum mechanics (CCM). The second dissipation mechanisms is due to the rate of symmetric part of the internal rotation gradient tensor, appearing in the constitutive theory for the symmetric part of the Cauchy moment tensor. The first dissipation mechanisms is viscous. The second dissipation mechanisms is due to the micropolar non-classical physics and it accounts for the drag forces experienced by the microconstituents during deformation of the volume of matter. In the case of micropolar fluids, the dissipation mechanism is due to symmetric part of the velocity gradient tensor appearing in the constitutive theory for the deviatoric Cauchy stress tensor (CCM) and also due to the symmetric part of the rotation rate gradient tensor appearing in the constitutive theory for the symmetric part of the Cauchy moment tensor (NCCM). It is shown that the mathematical models for micropolar TVES and micropolar TV fluids consisting of the conservation and balance laws and constitutive theories have closure.It is established and demonstrated that micropolar NCCT with rotational inertial physics derived here permits coexistence of translational waves (deviatoric Cauchy stress waves) and rotational waves (Cauchy moment waves). In the case of micropolar TV fluids with rotational inertial physics, neither translational nor rotational waves can exist. This is due to the lack of elasticity in classical as well as non-classical physics, as a consequence the balance of linear momenta and balance of angular momenta are time dependent diffusion equations in translational and rotational velocities and are not wave equations.Simple model problems describing evolutions (IVPs) in micropolar TVES and micropolar TV fluids are considered and their numerical solutions are presented (computed using space-time coupled finite element method) to specifically illustrate the influence of rotational inertial physics on the resulting evolutions.