This paper presents a nonlinear micropolar nonclassical continuum theory (MPNCCT) for finite deformation, finite strain deformation physics of thermosviscoelastic solid medium with memory (polymeric micropolar solids)...This paper presents a nonlinear micropolar nonclassical continuum theory (MPNCCT) for finite deformation, finite strain deformation physics of thermosviscoelastic solid medium with memory (polymeric micropolar solids) based on classical rotations cΘand their rates. Contravariant second Piola-Kirchhoff stress and moment tensors, in conjunction with finite deformation measures derived by the authors in recent paper, are utilized in deriving the conservation and balance laws and the constitutive theories based on conjugate pairs in entropy inequality and the representation theorem. This nonlinear MPNCCT for TVES with rheology: 1) incorporates nonlinear ordered rate dissipation mechanism based on Green’s strain rates up to order n;2) also incorporates an additional ordered rate dissipation mechanism due to microconstituents, the viscosity of the medium and the rates of the symmetric part of the rotation gradient (of cΘ) tensor up to order n, referred to as micropolar dissipation or micropolar viscous dissipation mechanism;3) incorporates the primary mechanism of memory or rheology due to long chain molecules of the polymer and the viscosity of the medium by using the contravaraint second Piola-Kirchhoff stress tensor and its rates up to order m, resulting in a relaxation spectrum;4) incorporates second mechanism of memory or rheology due to nonclassical physics, interaction of microconstituents with the viscous medium and long chain molecules by considering rates of the contravariant second Piola-Kirchhoff moment tensor up to order m, resulting in relaxation of second Piola-Kirchhoff moment tensor. This results in another relaxation spectrum for the second Piola-Kirchhoff moment tensor due to microconstituents, referred to as micropolar relaxation spectrum consisting of micropolar relaxation time constants of the material. This nonlinear MPNCCT for TVES with memory is thermodynamically and mathematically consistent, and the mathematical model consisting of conservation and balance laws and the constitutive theories has closure and naturally reduces to linear MPNCCT based on infinitesimal deformation assumption. BMM is the essential balance law for all MPNCCT and is used in the present work as well. In the absence of this balance law, a valid thermodynamically and mathematically consistent nonlinear MPNCCT is not possible. The nonlinear MPNCCT based on rotations (cΘ+αΘ) and αΘ(ignoring cΘ) is not considered due to the fact that even the linear MPNCCT based on these rotations is invalid and is thermodynamically and mathematically inconsistent MPNCCT.展开更多
In 3M continuum theories,micro deformations greatly influence macro deformation physics.Thus,in 3M theories we need a mechanism of micro deformation from which macro deformation can be derived.The one way to accomplis...In 3M continuum theories,micro deformations greatly influence macro deformation physics.Thus,in 3M theories we need a mechanism of micro deformation from which macro deformation can be derived.The one way to accomplish this is to assume that a material point contains a deformable director,the deformation of the director is representation of the microconstituent deformation in the material point,thus in essence a material point is deformable in 3M theories as opposed to classical continuum mechanics,in which material points are rigid.The thermodynamic principles of classical continuum mechanics are assumed to hold for micro deformation of the microconstituents leading to‘integral-average’definitions for macro deformation that can be used in the thermodynamics of the matter at the macro level.The principal element of this theory is to derive deformation/strain measures for a material point based in a single deformable director representing the macro deformation physics in the material point.This derivation is the first paper in which nonlinear deformation/strain measures are established for the micro as well as macro deformation physics.It is shown in this paper that,in currently published works,only deformation measures are possible and not strain measures,which is also true in our derivation presented here.Reasons for this are explained in the paper.Only the rate of work conjugate pairs in entropy inequality establish whether any of these measures are strain measures or can be made strain measures by simple modifications.Second part of the paper is devoted to the evaluation of various linear micropolar theories in the published literature based on the following considerations:1)Are the linear form of the deformation measures derived in this paper utilized appropriately in the derivation of the theories?2)Is the adequacy of conservation and balance laws of classical continuum mechanics and need for their modifications and perhaps the need for a new balance law,addressed satisfactorily?This is necessary due to presence of new micropolar physics over and beyond classical continuum mechanics 3)Are the derivations of constitutive theories supported by the representation theorem?4)Are the conservation and balance laws and constitutive theories thermodynamically and mathematically consistent?5)Lastly,do the complete mathematical models have closure?展开更多
文摘This paper presents a nonlinear micropolar nonclassical continuum theory (MPNCCT) for finite deformation, finite strain deformation physics of thermosviscoelastic solid medium with memory (polymeric micropolar solids) based on classical rotations cΘand their rates. Contravariant second Piola-Kirchhoff stress and moment tensors, in conjunction with finite deformation measures derived by the authors in recent paper, are utilized in deriving the conservation and balance laws and the constitutive theories based on conjugate pairs in entropy inequality and the representation theorem. This nonlinear MPNCCT for TVES with rheology: 1) incorporates nonlinear ordered rate dissipation mechanism based on Green’s strain rates up to order n;2) also incorporates an additional ordered rate dissipation mechanism due to microconstituents, the viscosity of the medium and the rates of the symmetric part of the rotation gradient (of cΘ) tensor up to order n, referred to as micropolar dissipation or micropolar viscous dissipation mechanism;3) incorporates the primary mechanism of memory or rheology due to long chain molecules of the polymer and the viscosity of the medium by using the contravaraint second Piola-Kirchhoff stress tensor and its rates up to order m, resulting in a relaxation spectrum;4) incorporates second mechanism of memory or rheology due to nonclassical physics, interaction of microconstituents with the viscous medium and long chain molecules by considering rates of the contravariant second Piola-Kirchhoff moment tensor up to order m, resulting in relaxation of second Piola-Kirchhoff moment tensor. This results in another relaxation spectrum for the second Piola-Kirchhoff moment tensor due to microconstituents, referred to as micropolar relaxation spectrum consisting of micropolar relaxation time constants of the material. This nonlinear MPNCCT for TVES with memory is thermodynamically and mathematically consistent, and the mathematical model consisting of conservation and balance laws and the constitutive theories has closure and naturally reduces to linear MPNCCT based on infinitesimal deformation assumption. BMM is the essential balance law for all MPNCCT and is used in the present work as well. In the absence of this balance law, a valid thermodynamically and mathematically consistent nonlinear MPNCCT is not possible. The nonlinear MPNCCT based on rotations (cΘ+αΘ) and αΘ(ignoring cΘ) is not considered due to the fact that even the linear MPNCCT based on these rotations is invalid and is thermodynamically and mathematically inconsistent MPNCCT.
文摘In 3M continuum theories,micro deformations greatly influence macro deformation physics.Thus,in 3M theories we need a mechanism of micro deformation from which macro deformation can be derived.The one way to accomplish this is to assume that a material point contains a deformable director,the deformation of the director is representation of the microconstituent deformation in the material point,thus in essence a material point is deformable in 3M theories as opposed to classical continuum mechanics,in which material points are rigid.The thermodynamic principles of classical continuum mechanics are assumed to hold for micro deformation of the microconstituents leading to‘integral-average’definitions for macro deformation that can be used in the thermodynamics of the matter at the macro level.The principal element of this theory is to derive deformation/strain measures for a material point based in a single deformable director representing the macro deformation physics in the material point.This derivation is the first paper in which nonlinear deformation/strain measures are established for the micro as well as macro deformation physics.It is shown in this paper that,in currently published works,only deformation measures are possible and not strain measures,which is also true in our derivation presented here.Reasons for this are explained in the paper.Only the rate of work conjugate pairs in entropy inequality establish whether any of these measures are strain measures or can be made strain measures by simple modifications.Second part of the paper is devoted to the evaluation of various linear micropolar theories in the published literature based on the following considerations:1)Are the linear form of the deformation measures derived in this paper utilized appropriately in the derivation of the theories?2)Is the adequacy of conservation and balance laws of classical continuum mechanics and need for their modifications and perhaps the need for a new balance law,addressed satisfactorily?This is necessary due to presence of new micropolar physics over and beyond classical continuum mechanics 3)Are the derivations of constitutive theories supported by the representation theorem?4)Are the conservation and balance laws and constitutive theories thermodynamically and mathematically consistent?5)Lastly,do the complete mathematical models have closure?
