NCCT for Micropolar Solid and Fluid Media Based on Internal Rotations and Rotation Rates with Rotational Inertial Physics: Model Problem Studies

This paper presents model problem studies for micropolar thermoviscoelastic solids without memory and micropolar thermoviscous fluid using micropolar non-classical continuum theories (NCCT) based on internal rotations and rotation rates in which rotational inertial physics is considered in the derivation of the conservation and balance laws (CBL). The dissipation mechanism is due to strain rates as well as rotation rates. Model problems are designed to demonstrate and illustrate various significant aspects of the micropolar NCCT with rotational inertial physics considered in this paper. In case of micropolar solids, the translational and rotational waves are shown to coexist. In the absence of microconstituents (classical continuum theory, CCT) the internal rotations are a free field, hence have no influence on CCT. Absence of gradients of displacements and strains in micropolar thermoviscous fluid medium prohibits existence of translational waves as well as rotational waves even though the appearance of the mathematical model is analogous to the solids, but in terms of strain rates. It is shown that in case of micropolar thermoviscous fluids the BAM behaves more like time dependent diffusion equation i.e., like heat conduction equation in Lagrangian description. The influence of rotational inertial physics is demonstrated using BLM as well as BAM in the model problem studies.

Rotational Inertial Physics in Non-Classical Thermoviscous Fluent Continua Incorporating Internal Rotation Rates

In this paper, we derive non-classical continuum theory for physics of compressible and incompressible thermoviscous non-classical fluent continua using the conservation and balance laws (CBL) by incorporating additional physics of internal rotation rates arising from the velocity gradient tensor as well as their time varying rates and the rotational inertial effects. In this non-classical continuum theory time dependent deformation of fluent continua results in time varying rotation rates i.e., angular velocities and angular accelerations at material points. Resistance offered to these by deforming fluent continua results in additional moments, angular momenta and inertial effects due to rotation rates i.e., angular velocities and angular accelerations at the material points. Currently, this physics due to internal rotation rates and inertial effects is neither considered in classical continuum mechanics (CCM) nor in non-classical continuum mechanics (NCCM). In this paper, we present a derivation of conservation and balance laws in Eulerian description: conservation of mass (CM), balance of linear momenta (BLM), balance of angular momenta (BAM), balance of moment of moments (BMM), first and second laws of thermodynamics (FLT, SLT) that include: (i) Physics of internal rotation rates resulting from the velocity gradient tensor;(ii) New physics resulting due to angular velocities and angular accelerations due to spatially varying and time dependent rotation rates. The balance laws derived here are compared with those that only consider the rotational rates but neglect rotational inertial effects and angular accelerations to demonstrate the influence of the new physics. Constitutive variables and their argument tensors are established using conjugate pairs in the entropy inequality, additional desired physics and principle of equipresence when appropriate. Constitutive theories are derived using Helmholtz free energy density as well as representation theorem and integrity (complete basis). It is shown that the mathematical model consisting of the conservation and balance laws and constitutive theories presented in this paper has closure. Influence of new physics in the conservation and balance laws on compressible and incompressible thermoviscous fluent continua is demonstrated due to presence of angular velocities and angular accelerations arising from time varying rotation rates when the deforming fluent continua offer rotational inertial resistance. The fluent continua are considered homogeneous and isotropic. Model problem studies are considered in a follow-up paper.

Rate Constitutive Theories of Orders n and 1n for Internal Polar Non-Classical Thermofluids without Memory

In recent papers, Surana et al. presented internal polar non-classical Continuum theory in which velocity gradient tensor in its entirety was incorporated in the conservation and balance laws. Thus, this theory incorporated symmetric part of the velocity gradient tensor (as done in classical theories) as well as skew symmetric part representing varying internal rotation rates between material points which when resisted by deforming continua result in dissipation (and/or storage) of mechanical work. This physics referred as internal polar physics is neglected in classical continuum theories but can be quite significant for some materials. In another recent paper Surana et al. presented ordered rate constitutive theories for internal polar non-classical fluent continua without memory derived using deviatoric Cauchy stress tensor and conjugate strain rate tensors of up to orders n and Cauchy moment tensor and its conjugate symmetric part of the first convected derivative of the rotation gradient tensor. In this constitutive theory higher order convected derivatives of the symmetric part of the rotation gradient tensor are assumed not to contribute to dissipation. Secondly, the skew symmetric part of the velocity gradient tensor is used as rotation rates to determine rate of rotation gradient tensor. This is an approximation to true convected time derivatives of the rotation gradient tensor. The resulting constitutive theory: (1) is incomplete as it neglects the second and higher order convected time derivatives of the symmetric part of the rotation gradient tensor;(2) first convected derivative of the symmetric part of the rotation gradient tensor as used by Surana et al. is only approximate;(3) has inconsistent treatment of dissipation due to Cauchy moment tensor when compared with the dissipation mechanism due to deviatoric part of symmetric Cauchy stress tensor in which convected time derivatives of up to order n are considered in the theory. The purpose of this paper is to present ordered rate constitutive theories for deviatoric Cauchy strain tensor, moment tensor and heat vector for thermofluids without memory in which convected time derivatives of strain tensors up to order n are conjugate with the Cauchy stress tensor and the convected time derivatives of the symmetric part of the rotation gradient tensor up to orders 1n are conjugate with the moment tensor. Conservation and balance laws are used to determine the choice of dependent variables in the constitutive theories: Helmholtz free energy density Φ, entropy density η, Cauchy stress tensor, moment tensor and heat vector. Stress tensor is decomposed into symmetric and skew symmetric parts and the symmetric part of the stress tensor and the moment tensor are further decomposed into equilibrium and deviatoric tensors. It is established through conjugate pairs in entropy inequality that the constitutive theories only need to be derived for symmetric stress tensor, moment tensor and heat vector. Density in the current configuration, convected time derivatives of the strain tensor up to order n, convected time derivatives of the symmetric part of the rotation gradient tensor up to orders 1n, temperature gradient tensor and temperature are considered as argument tensors of all dependent variables in the constitutive theories based on entropy inequality and principle of equipresence. The constitutive theories are derived in contravariant and covariant bases as well as using Jaumann rates. The nth and 1nth order rate constitutive theories for internal polar non-classical thermofluids without memory are specialized for n = 1 and 1n = 1 to demonstrate fundamental differences in the constitutive theories presented here and those used presently for classical thermofluids without memory and those published by Surana et al. for internal polar non-classical incompressible thermofluids.

Internal Rotation of Cyclopentadienyl Rings in Ferrocene Derivatives

The relaxation time of several ferrocene derivatives was measured, and the internal rotation was discussed. For almost all the derivatives, the degree of the internal rotation was constant in spite of the different molecular weights. However, for (triphenylmethyl)ferrocene, the rotation of the unsubstituted ring would be slower due to the bulkiness of the substituent. Furthermore, the derivatives that have a hydroxyl- or acetyl group on the substituent were also discussed. Their rotation would be influenced by the location of these substituents.

Thermodynamic Consistency of Plate and Shell Mathematical Models in the Context of Classical and Non-Classical Continuum Mechanics and a Thermodynamically Consistent New Thermoelastic Formulation

Inclusion of dissipation and memory mechanisms, non-classical elasticity and thermal effects in the currently used plate/shell mathematical models require that we establish if these mathematical models can be derived using the conservation and balance laws of continuum mechanics in conjunction with the corresponding kinematic assumptions. This is referred to as thermodynamic consistency of the mathematical models. Thermodynamic consistency ensures thermodynamic equilibrium during the evolution of the deformation. When the mathematical models are thermodynamically consistent, the second law of thermodynamics facilitates consistent derivations of constitutive theories in the presence of dissipation and memory mechanisms. This is the main motivation for the work presented in this paper. In the currently used mathematical models for plates/shells based on the assumed kinematic relations, energy functional is constructed over the volume consisting of kinetic energy, strain energy and the potential energy of the loads. The Euler’s equations derived from the first variation of the energy functional for arbitrary length when set to zero yield the mathematical model(s) for the deforming plates/shells. Alternatively, principle of virtual work can also be used to derive the same mathematical model(s). For linear elastic reversible deformation physics with small deformation and small strain, these two approaches, based on energy functional and the principle of virtual work, yield the same mathematical models. These mathematical models hold for reversible mechanical deformation. In this paper, we examine whether the currently used plate/shell mathematical models with the corresponding kinematic assumptions can be derived using the conservation and balance laws of classical or non-classical continuum mechanics. The mathematical models based on Kirchhoff hypothesis (classical plate theory, CPT) and first order shear deformation theory (FSDT) that are representative of most mathematical models for plates/shells are investigated in this paper for their thermodynamic consistency. This is followed by the details of a general and higher order thermodynamically consistent plate/shell thermoelastic mathematical model that is free of a priori consideration of kinematic assumptions and remains valid for very thin as well as thick plates/shells with comprehensive nonlinear constitutive theories based on integrity. Model problem studies are presented for small deformation behavior of linear elastic plates in the absence of thermal effects and the results are compared with CPT and FSDT mathematical models.

Large Amplitude Motion in 9-Methylanthracene:High-Resolution Spectroscopy and Ab Initio Theoretical Calculation

CH3 internal rotation is one of the typical large amplitude motions in polyatomic molecules,the spectral analysis and theoretical calculations of which,were developed by Li-Hong Xu and Jon Hougen.We observed a Doppler-free high-resolution and high-precision spectrum of 9-methylanthracene(9MA)by using the collimated supersonic jet and optical frequency comb techniques.The potential energy curve of CH3 internal rotation is expressed by a six-fold symmetric sinusoidal function.It was previously shown that the barrier height(V6)of 9MA-d12 was considerably smaller than that of 9MA-h12[M.Baba,et al.,J.Phys.Chem.A 113,2366(2009)].We performed ab initio theoretical calculations of the multicomponent molecular orbital method.The barrier reduction by deuterium substitution was partly attributed to the difference between the wave functions of H and D atomic nuclei.

分子阻障内旋转配分函数的二点pade’逼近

一、分子阻障内旋转配分函数的计算在理论化学中一直为人们所研究,早在五十年代Pitzer就用数值方法进行了计算,但求得这一配分函数的近似解析表示式仍有重要意义,在文献中作者曾用Wigner-Kirkwood微扰展开法求得阻障内旋转配分函数的近似解析表示式,根据此式所作的数值计算结果表明:在温度较高,内旋转动惯量较大时和Pitzer的数值计算结果符合得甚好,但随着温度的降低偏离也就增大,计算Wigner-Kirkwood展开式的的高次项固然可以改善所得之配分函数,但此时计算烦杂,困难较大,本文则采用二点Pade’逼近法将Wigner-Kirkwood微扰展开式（包括一级量子改正项）作为高温展开式,并选用配分函数Q=Σe-tEi的前三项作为其低温展开式,然后用二点Pade’逼近进行内插,求得阻障内旋转配分函数的[5/5]Pade’逼近式,它在整个温度范围内均能给出良好的数值结果。

Observation of tiny polarization rotation rate in total internal reflection via weak measurements

In this paper,we examine the tiny polarization rotation effect in total internal reflection due to the spin–orbit interaction of light.We find that the tiny polarization rotation rate will induce a geometric phase gradient,which can be regarded as the physical origin of photonic spin Hall effect.We demonstrate that the spin-dependent splitting in position space is related to the polarization rotation in momentum space,while the spin-dependent splitting in momentum space is attributed to the polarization rotation in position space.Furthermore,we introduce a quantum weak measurement to determine the tiny polarization rotation rate.The rotation rate in momentum space is obtained with 118 nm,which manifests itself as a spatial shift,and the rotation rate in position space is achieved with 38 μrad∕λ,which manifests itself as an angular shift.The investigation of the polarization rotation characteristics will provide insights into the photonic spin Hall effect and will enable us to better understand the spin–orbit interaction of light.

Investigation on the potential barrier to internal rotation in molecules——Ab initio calculation and energy partition of the internal rotation barrier in complex molecule H_3N-BH_3

Based on the ab initio/6-31Gcalculation, the potential barrier to internal rotation in mol-ecule HN--BHhas been studed by means of PD/LSF atomic charge model and Buckingham（exp--6-1）energy partition method. The results indicate that the order of the contributions of the componentsto the total energy barrier △E is |△V|（electrostatic） >|△V| （charge transfer）>|△V| （exchangerepulsion）>|△V| （dispersion）. For △Vthere are maxima at θ= 30°and 90°, and a saddle atθ= 60°. There are good linear relationships for the total barrier △E, △Vand △Vwith cos3θrespectively, and the same for the dipole moment from PD/LSF model (μ) and that from abinitio calculation (μ) vs. cos3θ respectively.

Understanding the Conformational Interconversions of a Polymer Chain in a Liquid Environment at the Single-molecule Level

The conformational interconversions of polymer chains have been of great interest as a basic scientific issue.Single-molecule force spectroscopy(SMFS)is a powerful tool for molecular manipulation,which enables experimental studies on the single-chain behaviors of polymers.The SMFS results show that an individual polymer chain in a liquid environment may have similar properties to an ideal chain,which contradicts the traditional theoretical view.Herein,by taking into account the collisions of solvent molecules,the conformational interconversions of a single polymer chain in a liquid environment have been analyzed.The conformational interconversion frequency of a carbon-carbon bond of an alkane chain can be estimated by establishing the relationship between the internal rotation barriers of small molecules(monomers)and the corresponding macromolecules.Since the time scale of conformational interconversions of the polymer backbone is much shorter than that of SMFS experiments,most polymers with C-C backbones behave as ideal chains in liquid environments.

Non-classical continuum theories for solid and fluent continua and some applications

This paper presents two specific thermodynamically consistent nonclassical 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.