In this paper, we present the non-local nonlinear finite element formulations for the case of nonuniform rotating laminated nano cantilever beams using the Timoshenko beam theory. The surface stress effects are also taken into consideration. Non-local stress resultants are obtained by employing Eringen's nonlocal differential model. Geometric nonlinearity is taken into account by using the Green Lagrange strain tensor. Numerical solutions of nonlinear bending and free vibration are presented. Parametric studies have been carried out to understand the effect of non-local parameter and surface stresses on bending and vibration behavior of cantilever beams. Also, the effects of angular velocity and hub radius on the vibration behavior of the cantilever beam are studied.

Nonlocal nonlinear bending and free vibration analysis of a rotating laminated nano cantilever beam

The constitutive analog of the linear Maxwell viscoelastic fluid model with the explicit addition of couple stresses is derived from thermodynamic first principles. Furthermore, the concepts of the recently proposed “Consistent Couple Stress theory” are considered to fully determine the rheological description of the fluid model. Finally, the resulting linear momentum equations are used to investigate the couple stress modulated flow dynamics of a few simple flow configurations. The plots of the variations in the elastic behavior of the fluid due to the presence of couple stresses show that the couple stress acts in addition to the conventional force-stress to increase the diffusion of momentum and thereby enhance the dampening of the elastic deformations in the equivalent spring-dashpot interpretation of the linear Maxwell viscoelastic fluid.

Effect of couple stresses on the rheology and dynamics of linear Maxwell viscoelastic fluids

This paper presents a non-classical continuum theory for fluent continua in which the conservation and balance laws are derived by incorporating both internal rotation rates arising from the velocity gradient tensor and the rotation rates of the Cosserats. Specifically, in this non-classical continuum theory we have (1) the usual velocities (
 $$\bar{ \pmb {\varvec{v }}}$$
 ), (2) the three internal rotation rates (
 $${}_i^t\bar{ \pmb {\varvec{\Theta }}}$$
 ) about the axes of a fixed triad whose axes are parallel to the x-frame arising from the velocity gradient tensor $$(\bar{ \pmb {\varvec{L }}})$$
 that are completely defined by the antisymmetric part of the velocity gradient tensor, and (3) three additional rotation rates (
 $${}_e^t\bar{ \pmb {\varvec{\Theta }}}$$
 ) about the axes of the same triad located at each material point as additional three unknown degrees of freedom, referred to as Cosserat rotation rates. This gives rise to $$\bar{ \pmb {\varvec{v }}}$$
 and $${}_e^t\bar{ \pmb {\varvec{\Theta }}}$$
 as six degrees of freedom at a material point. The internal rotation rates $${}_i^t\bar{ \pmb {\varvec{\Theta }}}$$
 , often neglected in classical fluid mechanics, exist in all deforming fluent continua as these are due to velocity gradient tensor. When the internal rotation rates $${}_i^t\bar{ \pmb {\varvec{\Theta }}}$$
 are resisted by deforming fluent continua, conjugate moment tensor arises that together with $${}_i^t\bar{ \pmb {\varvec{\Theta }}}$$
 may result in energy storage and/or dissipation, which must be considered in the conservation and balance laws. The Cosserat rotation rations $${}_e^t\bar{ \pmb {\varvec{\Theta }}}$$
 also result in conjugate moment tensor that together with $${}_e^t\bar{ \pmb {\varvec{\Theta }}}$$
 may also result in energy storage and/or dissipation. The main focus of this paper is a consistent derivation of conservation and balance laws for fluent continua that incorporate the aforementioned physics and associated constitutive theories for thermofluids using the conditions resulting from the entropy inequality. The material coefficients derived in the constitutive theories are clearly defined and discussed.

Non-classical continuum theory for fluids incorporating internal and Cosserat rotation rates

This paper presents two specific thermodynamically consistent non-classical continuum theories for solid and fluent continua. The first non-classical continuum theory for solid continua incorporate...

https://tandfonline.com/doi/pdf/10.1080/19475411.2018.1530700

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

In the non-classical theories for fluent continua, the presence of internal rotation rates and their gradients arising due to the velocity gradient tensor necessitate existence of moment tensor. The Cauchy moment tensor acting on the faces of the deformed tetrahedron (derived using Cauchy principle) and the gradients of the rates of total rotations are a rate of work conjugate pair in addition to the rate of work conjugate Cauchy stress tensor and rate of strain tensor. It is well established that in such non-classical continuum theories the Cauchy stress tensor is non-symmetric and the antisymmetric components of the Cauchy stress tensor are balanced by the gradients of the Cauchy moment tensor, balance of angular momenta balance law. In the non-classical continuum theories incorporating internal rotation rates and conjugate Cauchy moment tensor that are absent in classical continuum theories, the fundamental question is “are the conservation and balance laws used in classical continuum mechanics sufficient to ensure dynamic equilibrium of the deforming volume of matter?" If one only considers conservation and balance laws used in classical continuum theories, then the Cauchy moment tensor is non-symmetric. Thus, requiring constitutive theories for the symmetric as well as non-symmetric Cauchy moment tensors. The work presented in this paper shows that when the thermodynamically consistent constitutive theories are used for the symmetric as well as antisymmetric Cauchy moment tensors, non-physical and spurious solutions result even in simple flows. This suggests that perhaps the additional conjugate tensors resulting due to the presence of internal rotation rates, namely the Cauchy moment tensor and the antisymmetric part of Cauchy stress tensor, must obey some additional law or restriction so that the spurious behavior is precluded. The paper demonstrates that in the non-classical theory with internal rotation rates considered here the balance of moment of moments balance law and the equilibrium of moment of moments are in fact identical. When this balance law is considered, the Cauchy moment tensor becomes symmetric, hence eliminating the constitutive theory for the antisymmetric Cauchy moment tensor and thereby eliminating spurious and non-physical solutions. The necessity of this balance/equilibrium law is established theoretically, its derivation is presented using rate considerations, and its necessity is also demonstrated by a model problem using thermoviscous incompressible fluid as an example. The findings reported in this paper hold for all fluent continua, compressible as well as incompressible.

Necessity of law of balance/equilibrium of moment of moments in non-classical continuum theories for fluent continua

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

/pdf/rate-constitutive-theories-of-orders-n-and-1n-for-internal-p3i3x68xxq.pdf

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

The paper presents constitutive theories for non-classical thermoviscoelastic fluids with dissipation and memory using a thermodynamic framework based on entirety of velocity gradient tensor. Thus, the conservation and the balance laws used in this work incorporate symmetric as well as antisymmetric part of the velocity gradient tensor. The constitutive theories derived here hold in coand contra-variant bases as well as in Jaumann rates and are derived using convected time derivatives of Green’s and Almansi strain tensors as well as the Cauchy stress tensor and its convected time derivatives in appropriate bases. The constitutive theories are presented in the absence as well as in the presence of the balance of moment of moments as balance law. It is shown that the dissipation mechanism and the fading memory in such fluids are due to stress rates as well as moment rates and their conjugates. The material coefficients are derived for the general forms of the constitutive theories based on integrity. Simplified linear (or quasi-linear) forms of the constitutive theories are also presented. Maxwell, Oldroyd-B and Giesekus constitutive models for non-classical thermoviscoelastic fluids are derived and are compared with those derived based on classical continuum mechanics. Both, compressible and incompressible thermoviscoelastic fluids are considered.

/pdf/ordered-rate-constitutive-theories-for-non-classical-54ecjaaq6v.pdf

Ordered Rate Constitutive Theories for Non-Classical Thermoviscoelastic Fluids with Internal Rotation Rates

This paper considers non-classical continuum theory for thermoviscous fluids without memory incorporating internal rotation rates resulting from the antisymmetric part of the velocity gradient tensor to derive ordered rate constitutive theories for the Cauchy stress and the Cauchy moment tensor based on entropy inequality and representation theorem. Using the generalization of the conjugate pairs in the entropy inequality, the ordered rate constitutive theory for Cauchy stress tensor considers convected time derivatives of the Green’s strain tensor (or Almansi strain tensor) of up to orders n e as its argument tensors and the ordered rate constitutive theory for the Cauchy moment tensor considers convected time derivatives of the symmetric part of the rotation gradient tensor up to orders n Θ . While the convected time derivatives of the strain tensors are well known the convected time derivatives of higher orders of the symmetric part of the rotation gradient tensor need to be derived and are presented in this paper. Complete and general constitutive theories based on integrity using conjugate pairs in the entropy inequality and the generalization of the argument tensors of the constitutive variables and the representation theorem are derived and the material coefficients are established. It is shown that for the type of non-classical thermofluids considered in this paper the dissipation mechanism is an ordered rate mechanism due to convected time derivatives of the strain tensor as well as the convected time derivatives of the symmetric part of the rotation gradient tensor. The derivations of the constitutive theories presented in the paper is basis independent but can be made basis specific depending upon the choice of the specific basis for the constitutive variables and the argument tensors. Simplified linear theories are also presented as subset of the general constitutive theories and are compared with published works.

https://www.mdpi.com/1099-4300/22/4/443/pdf

Ordered Rate Constitutive Theories for Non-Classical Thermofluids Based on Convected Time Derivatives of the Strain and Higher Order Rotation Rate Tensors Using Entropy Inequality.

Stephen W. Long

Papers

Necessity of law of balance/equilibrium of moment of moments in non-classical continuum theories for fluent continua

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

Ordered Rate Constitutive Theories for Non-Classical Thermoviscoelastic Fluids with Internal Rotation Rates

Ordered Rate Constitutive Theories for Non-Classical Thermofluids Based on Convected Time Derivatives of the Strain and Higher Order Rotation Rate Tensors Using Entropy Inequality.