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Non-Classical Continuum Theories for Fluent Continua with Internal Rotation Rates
Long, Stephen William
Long, Stephen William
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Abstract
This thesis develops a non-classical continuum theory of fluent continua with internal rotation rates. In classical continuum mechanics (CCM) of fluent media only translational degrees of freedom exist at a material point. The development of this classical theory utilizes only the symmetric part of the velocity gradient tensor neglecting the skew-symmetric part. Omission of this skew-symmetric part in classical theory indicates that the internal rotations, i.e., those coming from the skew-symmetric part of the velocity gradient tensor, are not considered in the deformation physics. Should a fluent medium consist of micropolar microconstituents dispersed homogeneously throughout the parent material then inclusion of the internal rotations in the deformation physics is warranted.A non-classical continuum mechanics (NCCM) incorporating resistance to the gradients of the internal rotation rates leads to a conjugate Cauchy moment tensor similar to how the resistance of velocity gradients yields the conjugate Cauchy stress tensor. The development of all tensorial quantities is done in a basis independent manner throughout the text and one can recover the basis dependent forms by simple substitution of appropriate covariant and contravariant measures. The conservation and balance laws are derived for micropolar fluent media.The inclusion of another kinematic variable (the internal rotations) suggests the need for another balance law governing micropolar media. This law is called the Balance of Moment of Moments. Inductive reasoning for the inclusion of this balance law as well the details of its derivation is provided in Chapter 3. Utilization of this balance law indicates that the Cauchy moment tensor should be a symmetric tensor. In Chapter 4 a model problem study illustrates what having this balance law present or not being present has on the deformation of a micropolar medium.Chapter 5 and Chapter 6 are devoted to developing ordered rate constitutive theories for non-classical fluent media with internal rotation rates. Simplified linear constitutive theories are presented. This work provides a consistent framework for developing such theories and it can be shown that setting the material coefficients of all non-classical quantities to zero the governing equations in these NCCM theories identically recover the physics present in CCM theories.
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2024-01-01
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University of Kansas
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Mechanical engineering, Mechanics, Thermodynamics, Conservation and Balance Laws, Constitutive Theory, Continuum Mechanics, Fluid Mechanics