Internal Polar Continuum Theories for Solid and Fluent Continua
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Classical continuum theories are useful in the study of a variety of problems of engineering and applied sciences. However, the emergence of new materials has provided the need for refined theories that account for certain features that are not accounted for in the classical continuum theories. Polar decomposition of the deformation gradient tensor into pure stretch and pure rotation tensors shows that the rotation tensor will in general vary from point to point. Similarly, polar decomposition of the velocity gradient tensor shows that the rate of rotation tensor will vary from point to point. It can also be shown that the strain and strain rate tensors used in classical theories of continuum mechanics do not depend on the rotation tensor or its gradients and therefore neglect the effect of changing rotations and rates of rotations between neighboring material points in Lagrangian description, and between neighboring locations in Eulerian description. Varying rotations and rates of rotations between neighboring material points will, if resisted by the continua, result in internal moments which are conjugate to these rotations and rates of rotations. These internal moments along with the conjugate rotations and rates of rotations will result in energy storage and dissipation, in addition to the energy storage and dissipation resulting from stress and its conjugate strain and strain rate. Based on this observation, it is necessary to modify the existing conservation and balance laws to include internal moments, which results in a more complete thermodynamic framework for solid and fluent continua. In this work, new conservation and balance laws are derived for solid and fluent continua that include internal moments which result from varying rotations and rotation rates. Also, constitutive theories are derived for the stress tensor, moment tensor, and heat vector, resulting in a complete mathematical model internal polar thermoelastic solids and internal polar thermoviscous fluids. This derivation does not rely on the introduction of external micro-rotations or stress couples as is done in the so called micro-polar or couple-stress theories. The theories presented here are therefore referred to as “internal polar continuum theories”, as they are derived using only internal measures of deformation and do not require introduction of external degrees of freedom. We also present a framework for obtaining approximate solutions to the mathematical models resulting from the new continuum theories. Numeric results are presented to show the affect of the internal polar theories presented here.
Non-classical continuum mechanics
rates of rotations
Powell, Michael Joseph (2016). Internal Polar Continuum Theories for Solid and Fluent Continua. Doctoral dissertation, Texas A & M University. Available electronically from