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Thermodynamic consistency of nonclassical continuum theories for solid continua incorporating rotations

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Abstract

In this paper, we present derivations of three micropolar nonclassical continuum theories in which a material point always has displacements \( \pmb {\varvec{u }} \) as degrees of freedom. Additionally, (i) in the first nonclassical continuum theory (NCCT) we consider internal or classical rotations \({}_i \pmb {\varvec{\varTheta }}\) (known) due to skew symmetric part of the deformation gradient tensor; (ii) in the second NCCT, we consider both internal rotations \({}_i \pmb {\varvec{\varTheta }}\) and external or Cosserat or microrotations \({}_e \pmb {\varvec{\varTheta }}\) (unknown degrees of freedom); (iii) in the third NCCT, we consider Cosserat rotations \({}_e \pmb {\varvec{\varTheta }}\) only; hence, \({}_i \pmb {\varvec{\varTheta }}\) are neglected in this NCCT. We examine consistent choice of kinematic variables, modifications of conservation and balance laws of classical continuum theories (CCTs) due to the presence of new physics of rotations and determine whether the consideration of rotations requires additional balance laws. NCC theories derived here are examined for thermodynamic and mathematical consistency and are compared with published works. Model problem studies are also presented.

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Abbreviations

\(\bar{ \pmb {\varvec{x }}}\), \(\bar{x}_i\), \(\{\bar{x}\}\) :

Deformed coordinates

\({ \pmb {\varvec{x }}}\), \(x_i\), \(\{x\}\) :

Undeformed coordinates

\(\rho _{_{_{\!\!0}}}\) :

Reference density

\(\rho \) :

Density in Lagrangian description

\(\bar{\rho }\) :

Density in Eulerian description

\({\eta }\) :

Specific entropy in Lagrangian description

\(\bar{\eta }\) :

Specific entropy in Eulerian description

e :

Specific internal energy in Lagrangian description

\(\bar{e}\) :

Specific internal energy in Eulerian description

\({}_i \pmb {\varvec{\varTheta }}\), \({}_i\varTheta _i\), \(\{{}_i\varTheta \}\) :

Internal or classical rotations in Lagrangian description

\({}_e \pmb {\varvec{\varTheta }}\), \({}_e\varTheta _i\), \(\{{}_e\varTheta \}\) :

External or Cosserat or microrotations in Lagrangian description

\({}_t \pmb {\varvec{\varTheta }}\), \({}_t\varTheta _i\), \(\{{}_t\varTheta \}\) :

Total rotations in Lagrangian description

\( \pmb {\varvec{J }}\) :

Deformation gradient tensor in Lagrangian description

\({}_s \pmb {\varvec{J }}\) :

Symmetric part of deformation gradient tensor in Lagrangian description

\({}_a \pmb {\varvec{J }}\) :

Skew symmetric part of deformation gradient tensor in Lagrangian description

\({}^{\;d}\!\! \pmb {\varvec{J }}\) :

Displacement gradient tensor in Lagrangian description

\({}_{s}^{\;d}\!\! \pmb {\varvec{J }}\) :

Symmetric part of displacement gradient tensor in Lagrangian description

\({}_{a}^{\;d}\!\! \pmb {\varvec{J }}\) :

Skew symmetric part of displacement gradient tensor in Lagrangian description

\( {}^{\varTheta }\! \pmb {\varvec{J }} \) :

Rotation gradient tensor in Lagrangian description

\( {}^{{}_i\varTheta }\! \pmb {\varvec{J }} \) :

Internal rotation gradient tensor in Lagrangian description

\( {}_{\;s}^{{}_i \varTheta }\!\! \pmb {\varvec{J }} \) :

Symmetric part of internal rotation gradient tensor in Lagrangian description

\( {}_{\;a}^{{}_i \varTheta }\!\! \pmb {\varvec{J }} \) :

Skew symmetric part of internal rotation gradient tensor in Lagrangian description

\({}^{{}^{^r}\!\!\!{}_i\bar{\varTheta }}\!\!\!\!{}_s\bar{ \pmb {\varvec{J }}}\) :

Symmetric part of gradient of internal rotation rate tensor in Eulerian description

\( {}_{\;\,s}^{{}_i\varTheta }\!\overset{\,{{\textbf {.}}}}{ \pmb {\varvec{J }}} \) :

Rate of symmetric part of gradient of internal rotation tensor in Lagrangian description

\({{}^{^r}\!\!\!{}_i\bar{ \pmb {\varvec{\varTheta }}}}\) :

Internal rotation rate tensor in Eulerian description

\({}_i\overset{\,{{\textbf {.}}}}{ \pmb {\varvec{\varTheta }}}\) :

Internal rotation rate tensor in Lagrangian description

\( \pmb {\varvec{q }} \), \({q}_i\), \(\{q\}\) :

Heat vector in Lagrangian description

\(\bar{ \pmb {\varvec{q }} }\), \({\bar{q}}_i\), \(\{\bar{q}\}\) :

Heat vector in Eulerian description

\({ \pmb {\varvec{v }}}\), \({v}_i\), \(\{v\}\) :

Velocities in Lagrangian description

\(\bar{ \pmb {\varvec{v }}}\), \({\bar{v}}_i\), \(\{\bar{v}\}\) :

Velocities in Eulerian description

\( \pmb {\varvec{u }} \), \({u}_i\), \(\{u\}\) :

Displacements in Lagrangian description

\(\bar{ \pmb {\varvec{u }} }\), \({\bar{u}}_i\), \(\{\bar{u}\}\) :

Displacements in Eulerian description

\({ \pmb {\varvec{P }}}\) :

Average stress in Lagrangian description

\(\bar{ \pmb {\varvec{P }}}\) :

Average stress in Eulerian description

\( \pmb {\varvec{M }}\) :

Average moment in Lagrangian description

\(\bar{ \pmb {\varvec{M }}}\) :

Average moment in Eulerian description

\( \pmb {\varvec{\sigma }} \) :

, \(\sigma _{ij}\), \([\sigma ]\) Cauchy stress tensor in Lagrangian description

\(\bar{ \pmb {\varvec{\sigma }} }\) :

, \(\bar{\sigma }_{ij}\), \([\bar{\sigma }]\) Cauchy stress tensor in Eulerian description

\( {}_s \pmb {\varvec{\sigma }} \) :

Symmetric part of Cauchy stress tensor

\( {}_a \pmb {\varvec{\sigma }} \) :

Antisymmetric part of Cauchy stress tensor

\( {}^d_s \pmb {\varvec{\sigma }} \) :

Deviatoric part of the symmetric Cauchy stress tensor

\( {}^e_s \pmb {\varvec{\sigma }} \) :

Equilibrium part of the symmetric Cauchy stress tensor

\({\theta }\) :

Temperature in Lagrangian description

\({\bar{\theta }}\) :

Temperature in Eulerian description

k :

Thermal conductivity in Lagrangian description

p :

Thermodynamic or mechanical pressure in Lagrangian description

\(\bar{p}\) :

Thermodynamic or mechanical pressure in Eulerian description

\( \pmb {\varvec{g }} \), \({g}_i\), \(\{g\}\) :

Temperature gradient tensor in Lagrangian description

\(\bar{ \pmb {\varvec{g }} }\) :

, \({\bar{g}}_i\), \(\{\bar{g}\}\) Temperature gradient tensor in Eulerian description

\(\bar{ \pmb {\varvec{L }}}\) :

Velocity gradient tensor in Eulerian description

\(\bar{ \pmb {\varvec{D }}}\) :

Symmetric part of velocity gradient tensor in Eulerian description

CBL:

Conservation and balance laws

CCM:

Classical continuum mechanics

CCT:

Classical continuum theory

NCCT:

Nonclassical continuum theory

NCCM:

Nonclassical continuum mechanics

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Acknowledgements

First author is grateful for his endowed professorships and the department of mechanical engineering of the University of Kansas for providing financial support to the second author. The computational facilities provided by the Computational Mechanics Laboratory of the mechanical engineering department are also acknowledged.

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    K. S. Surana & S. S. C. Mathi

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Communicated by Andreas Öchsner.

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Surana, K.S., Mathi, S.S.C. Thermodynamic consistency of nonclassical continuum theories for solid continua incorporating rotations. Continuum Mech. Thermodyn. 35, 17–59 (2023). https://doi.org/10.1007/s00161-022-01163-y

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