Abstract
ABSTRACTThis paper presents two specific thermodynamically consistent non-classical continuum theories for solid and fluent continua. The first non-classical continuum theory for solid continua incorporates Jacobian of deformation in its entirety in the conservation and the balance laws and the derivation of the constitutive theories. The second non-classical continuum theory for solid continua considers Jacobian of deformation in its entirety as well as the Cosserat rotations in the conservation and balance laws as well as the constitutive theories. The first non-classical continuum theory for fluent continua presented here considers velocity gradient tensor in its entirety. The second non-classical continuum theory for fluent continua considers velocity gradient tensor in its entirety as well as Cosserat rotation rates in the derivation of the conservation and balance laws and the constitutive theories. Since the non-classical continuum theories for solid and fluent continua considered here incorporate additional physics of deformation due to rotations and rotation rates compared to classical continuum mechanics, the conservation and balance laws of classical continuum mechanics are shown to require modification as well as a new balance law balance of moment of moments is required to accommodate the new physics due to rotations and rotation rates. Eringen’s micropolar, micromorphic and microstretch theories, couple stress theories and nonlocal theories are also discussed within the context of the non-classical theories presented here for solid and fluent continua. Some applications of these theories are also discussed.
Highlights
Solid continuaFollowing reference [67] quantities with an over-bar are quantities in the current (deformed) configuration (i.e., all quantities with over-bar are functions of coordinates xi and time t, the Eulerian description)
In classical continuum theories for a solid continua a material point has only three translational degrees of freedom
We have shown that internal rotation or rotation rate in the non-classical that exists in all isotropic, continuum theories incorporate physics due to ad J or W
Summary
When the gradients of displacements vary between neighboring material points, so do the internal rotations da J and likewise the Cosserat rotations ea γ may vary between the neighboring material points. When rotations ta r are resisted by the deforming matter conjugate moments are created. In the deforming matter total rotations ta r are conjugate to moment tensor which necessitates that on the boundary of the deformed volume there must exist a resultant moment. Based on the small deformation assumption, the deformed coordinates xi are approximately same as undeformed coordinates xi , the deformed tetrahedron T1 in the current configuration is close to its map T1 in the reference configuration. With this assumption all stress measures (first and second Piola-Kirchhoff stress tensors, Cauchy stress tensor) are approximately the same. In which σ is Cauchy stress tensor and m is Cauchy moment tensor (per unit area)
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