Non-Classical Continuum Theories for Solid and Fluent Continua

Abstract

This dissertation presents non-classical continuum theories for solid and fluent con- tinua. In these theories additional physics due to internal rotations and rotation rates arising from the Jacobian of deformation and the velocity gradient tensor as well as Cosserat rotations and rotation rates are considered. While the internal rotations and rotation rates are completely defined by the deformation physics, the Cosserat rota- tions and Cosserat rotation rates are additional degrees of freedom at a material point. The non-classical theories that only consider the internal rotations and the internal ro- tation rates are referred to as internal polar theories, while those that consider both are called polar or non-classical theories. The conservation and balance laws and the constitutive theories are derived for non- classical continuum theories. It is shown that these non-classical theories require mod- ifications of the balance laws used in classical continuum theories. In the presence of additional rotation and rotation rate physics in non-classical theories, the modifica- tions of the balance laws used in classical continuum theories are not sufficient to ensure equilibrium of the deforming matter. It is shown that these theories require the balance of moments of moments as an additional balance law. The constitutive theories for solid and fluent continua are derived using the conditions resulting from the entropy inequality and the representation theorem. Use of integrity in their deriva- tions ensures completeness of the resulting constitutive theories. Specific derivations and details of the constitutive theories for thermoelastic and thermoviscoelastic solids with and without memory are presented for small deformation, small strain physics. Detailed derivations of the constitutive theories for compressible as well as incom- iiipressible thermoviscous and thermoviscoelastic fluent continua are also presented. Retardation and/or memory moduli are derived for polymeric solids and fluids. The present theories are compared with published works, particularly with the microp- olar theories of Eringen, to highlight the significance and the thermodynamic consis- tency of the present work, as well as to contrast the differences.