Continuum mechanics with torsion

This paper is an attempt to introduce methods and concepts of the Riemann–Cartan geometry largely used in such physical theories as general relativity, gauge theories, solid dynamics to fluid dynamics in general and to studying and modeling turbulence in particular. Thus, in order to account for the rotational degrees of freedom of the irregular dynamics of small-scale vortexes, we further generalize our unified first-order hyperbolic formulation of continuum fluid and solid mechanics which treats the flowing medium as a Riemann–Cartan manifold with zero curvature but non-vanishing torsion. We associate the rotational degrees of freedom of the main field of our theory, the distortion field, to the dynamics of microscopic (unresolved) vortexes. The distortion field characterizes the deformation and rotation of the material elements and can be viewed as anholonomic basis triad with non-vanishing torsion. The torsion tensor is then used to characterize distortion’s spin and is treated as an independent field with its own time evolution equation. This new governing equation has essentially the structure of the nonlinear electrodynamics in a moving medium and can be viewed as a Yang–Mills-type gauge theory. The system is closed by providing an example of the total energy potential. The extended system describes not only irreversible dynamics (which raises the entropy) due to the viscosity or plasticity effect, but it also has dispersive features which are due to the reversible energy exchange (which conserves the entropy) between micro- and macroscales. Both the irreversible and dispersive processes are represented by relaxation-type algebraic source terms so that the overall system remains first-order hyperbolic. The turbulent state is then treated as an excitation of the equilibrium (laminar) state due to the nonlinear interplay between dissipation and dispersion.

[1]  S. Capozziello,et al.  f(T) teleparallel gravity and cosmology , 2015, Reports on progress in physics. Physical Society.

[2]  J. Stewart,et al.  On transient relativistic thermodynamics and kinetic theory. Il , 1979, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[3]  Haran Jackson,et al.  A numerical scheme for non-Newtonian fluids and plastic solids under the GPR model , 2018, J. Comput. Phys..

[4]  H. Ch. Öttinger,et al.  Beyond Equilibrium Thermodynamics , 2005 .

[5]  E. Toro.  Godunov Methods: Theory and Applications , 2001 .

[6]  Dima Bolmatov,et al.  Revealing the Mechanism of the Viscous-to-Elastic Crossover in Liquids. , 2015, The journal of physical chemistry letters.

[7]  S. Pope.  Turbulent Flows: FUNDAMENTALS , 2000 .

[8]  Olivier Pouliquen,et al.  A constitutive law for dense granular flows , 2006, Nature.

[9]  W. Israel.  Nonstationary irreversible thermodynamics: A Causal relativistic theory , 1976 .

[10]  Michael Dumbser,et al.  Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws , 2008, J. Comput. Phys..

[11]  G. E. Romero,et al.  Is Teleparallel Gravity Really Equivalent to General Relativity? , 2017, 1708.04569.

[12]  L. Rosati.  Derivatives and Rates of the Stretch and Rotation Tensors , 1999 .

[13]  An elastoplastic theory of dislocations as a physical field theory with torsion , 2001, cond-mat/0105270.

[14]  Miroslav Grmela,et al.  Time reversal in nonequilibrium thermodynamics. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  L. Rakotomanana-Ravelonarivo,et al.  Covariant gradient continua applied to wave propagation within defective material , 2014 .

[16]  J. G. Pereira,et al.  Teleparallel Gravity: An Introduction , 2012 .

[17]  Ilya Peshkov,et al.  On a pure hyperbolic alternative to the Navier-Stokes equations , 2014 .

[18]  Theodoros G. Soldatos,et al.  Correction: Corrigendum: Src activation by β-adrenoreceptors is a key switch for tumour metastasis , 2013, Nature Communications.

[19]  S. Stoupin,et al.  Thermally triggered phononic gaps in liquids at THz scale , 2016, Scientific Reports.

[20]  J. Ollitrault.  Relativistic hydrodynamics , 2007 .

[21]  R. Geroch,et al.  Dissipative relativistic fluid theories of divergence type. , 1990, Physical review. D, Particles and fields.

[22]  É. Cartan,et al.  On manifolds with an affine connection and the theory of general relativity , 1986 .

[23]  F. Hehl,et al.  General Relativity with Spin and Torsion: Foundations and Prospects , 1976 .

[24]  Yu. D. Fomin,et al.  Two liquid states of matter: a dynamic line on a phase diagram. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  G. Russo,et al.  Implicit-explicit Runge-Kutta schemes for stiff systems of differential equations , 2000 .

[26]  R. Bullough,et al.  Continuous distributions of dislocations: a new application of the methods of non-Riemannian geometry , 1955, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[27]  Dislocation theory as a 3-dimensional translation gauge theory , 2000, cond-mat/0006280.

[28]  C. Pfeifer,et al.  Teleparallel theories of gravity as analogue of nonlinear electrodynamics , 2017, 1711.09930.

[29]  J. W. Maluf,et al.  The teleparallel equivalent of general relativity , 2013, 1303.3897.

[30]  E. Scholz,et al.  E. Cartan’s attempt at bridge-building between Einstein and the Cosserats – or how translational curvature became to be known as torsion , 2018, The European Physical Journal H.

[31]  H. C. Öttinger,et al.  Stability analysis for a thermodynamically consistent model of relativistic fluid dynamics. , 2018, Physical review. E.

[32]  G. Maugin,et al.  Thermoelasticity with Dual Internal Variables , 2011 .

[33]  R. D. Mindlin.  Micro-structure in linear elasticity , 1964 .

[34]  Friedrich W. Hehl,et al.  Elie Cartan's torsion in geometry and in field theory, an essay , 2007, 0711.1535.

[35]  Ilya Peshkov,et al.  Thermodynamically consistent nonlinear model of elastoplastic Maxwell medium , 2010 .

[36]  E. I. Romensky,et al.  Thermodynamics and Hyperbolic Systems of Balance Laws in Continuum Mechanics , 2001 .

[37]  Y. Cai,et al.  Unified phonon-based approach to the thermodynamics of solid, liquid and gas states , 2015, 1512.07191.

[38]  The use of the teleparallelism connection in continuum mechanics , 2013, 1305.3477.

[39]  M. Grmela,et al.  Multiscale Thermo-Dynamics , 2018 .

[40]  I. Muller,et al.  Zum Paradoxon der Warmeleitungstheorie , 1967 .

[41]  M. Guzev,et al.  Thermomechanical model of elastic–plastic materials with defect structures , 2000 .

[42]  R. Toupin.  Elastic materials with couple-stresses , 1962 .

[43]  Michael Dumbser,et al.  Cell centered direct Arbitrary-Lagrangian-Eulerian ADER-WENO finite volume schemes for nonlinear hyperelasticity , 2016 .

[44]  A. Alú,et al.  Controlling sound with acoustic metamaterials , 2016 .

[45]  Tarasankar DebRoy,et al.  An improved prediction of residual stresses and distortion in additive manufacturing , 2017 .

[46]  K. Hayashi.  The Gauge Theory of the Translation Group and Underlying Geometry , 1977 .

[47]  S. Gavrilyuk,et al.  Extended Lagrangian approach for the defocusing nonlinear Schrödinger equation , 2018, Studies in Applied Mathematics.

[48]  M. Shliomis.  Hydrodynamics of a Liquid with Intrinsic Rotation , 1967 .

[49]  C. Boehmer,et al.  Soliton-like solutions based on geometrically nonlinear Cosserat micropolar elasticity , 2015, 1503.08860.

[50]  Derivatives of the stretch and rotation tensors , 1993 .

[51]  Michael Dumbser,et al.  A unified hyperbolic formulation for viscous fluids and elastoplastic solids , 2016, 1705.02151.

[52]  R. Toupin.  ELASTIC MATERIALS WITH COUPLE STRESSES, ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS , 1962 .

[53]  Mustafa Megahed,et al.  Metal additive-manufacturing process and residual stress modeling , 2016, Integrating Materials and Manufacturing Innovation.

[54]  I. Volovich,et al.  Theory of defects in solids and three-dimensional gravity , 1992 .

[55]  J. Stewart.  On transient relativistic thermodynamics and kinetic theory , 1977, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[56]  Michael Dumbser,et al.  High order ADER schemes for a unified first order hyperbolic formulation of continuum mechanics: Viscous heat-conducting fluids and elastic solids , 2015, J. Comput. Phys..

[57]  E. I. Romenski,et al.  On modeling the frequency transformation effect in elastic waves , 2011 .

[58]  A. Kosevich.  DYNAMICAL THEORY OF DISLOCATIONS , 1965 .

[59]  M. Berezovski,et al.  Waves in microstructured solids: a unified viewpoint of modeling , 2011 .

[60]  P. Morrison,et al.  Structure and structure-preserving algorithms for plasma physics , 2016, 1612.06734.

[61]  D. A. Dunnett.  Classical Electrodynamics , 2020, Nature.

[62]  Dominic G.B. Edelen,et al.  A Gauge Theory of Dislocations and Disclinations , 1983 .

[63]  Miroslav Grmela,et al.  Continuum mechanics and thermodynamics in the Hamilton and the Godunov-type formulations , 2017, Continuum Mechanics and Thermodynamics.

[64]  L. G. Margolin,et al.  Finite scale theory: The role of the observer in classical fluid flow , 2014 .

[65]  Olivier Pouliquen,et al.  Granular Media: Between Fluid and Solid , 2013 .

[66]  E. Cosserat,et al.  Théorie des Corps déformables , 1909, Nature.

[67]  I-Shih Liu,et al.  Relativistic thermodynamics of gases , 1986 .

[68]  C. Boehmer,et al.  Teleparallel theories of gravity: illuminating a fully invariant approach , 2018, Classical and Quantum Gravity.

[69]  J. Nye.  Some geometrical relations in dislocated crystals , 1953 .

[70]  Alireza Mazaheri,et al.  A first-order hyperbolic system approach for dispersion , 2016, J. Comput. Phys..

[71]  P. Steinmann.  A micropolar theory of finite deformation and finite rotation multiplicative elastoplasticity , 1994 .

[72]  Michael Dumbser,et al.  Theoretical and numerical comparison of hyperelastic and hypoelastic formulations for Eulerian non-linear elastoplasticity , 2018, J. Comput. Phys..

[73]  John C. Baez,et al.  Gauge Fields, Knots and Gravity , 1994 .

[74]  E. Kröner.  The Dislocation as a Fundamental New Concept in Continuum Mechanics , 1963 .

[75]  E. I. Romensky,et al.  Hyperbolic systems of thermodynamically compatible conservation laws in continuum mechanics , 1998 .

[76]  T. Nakano,et al.  Extended Translation Invariance and Associated Gauge Fields , 1967 .

[77]  C. Boehmer,et al.  Soliton solutions in geometrically nonlinear Cosserat micropolar elasticity with large deformations , 2018, Wave Motion.

[78]  Dimitris Drikakis,et al.  An Eulerian finite‐volume scheme for large elastoplastic deformations in solids , 2010 .

[79]  S. Godunov,et al.  Elements of Continuum Mechanics and Conservation Laws , 2003, Springer US.

[80]  T. Teichmann,et al.  Three‐Dimensional Problems of the Theory of Elasticity and Introduction to the Mechanics of a Continuous Medium , 1965 .

[81]  Manuel Jesús Castro Díaz,et al.  Well-Balanced High Order Extensions of Godunov's Method for Semilinear Balance Laws , 2008, SIAM J. Numer. Anal..

[82]  Werner Israel,et al.  Transient relativistic thermodynamics and kinetic theory , 1979 .

[83]  S. K. Godunov,et al.  Nonstationary equations of nonlinear elasticity theory in eulerian coordinates , 1972 .

[84]  S. Godunov,et al.  Systems of thermodynamically coordinated laws of conservation invariant under rotations , 1996 .

[85]  Michael Dumbser,et al.  High order ADER schemes for a unified first order hyperbolic formulation of Newtonian continuum mechanics coupled with electro-dynamics , 2016, J. Comput. Phys..

[86]  Stephen Lin,et al.  A fully coupled finite element formulation for liquid–solid–gas thermo-fluid flow with melting and solidification , 2018, Computer Methods in Applied Mechanics and Engineering.

[87]  L. Scriven,et al.  Theory of structured continua I. General consideration of angular momentum and polarization , 1963, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[88]  Y. Berezin,et al.  A model of non-equilibrium turbulence with an asymmetric stress. Application to the problems of thermal convection , 1995 .

[89]  A. Eringen.  Micromorphic description of turbulent channel flow , 1972 .

[90]  Arkady Tsinober,et al.  An informal conceptual introduction to turbulence , 2009 .

[91]  Neil J. Balmforth,et al.  Yielding to Stress: Recent Developments in Viscoplastic Fluid Mechanics , 2014 .

[92]  H. C. Öttinger.  Beyond Equilibrium Thermodynamics: Öttinger/Beyond , 2005 .

[93]  B. Temple,et al.  Causal dissipation for the relativistic dynamics of ideal gases , 2017, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[94]  S. Forest.  Micromorphic Media , 2020, Encyclopedia of Continuum Mechanics.

[95]  F. dell’Isola,et al.  Some Introductory and Historical Remarks on Mechanics of Microstructured Materials , 2018 .

[96]  S. Gavrilyuk.  Multiphase Flow Modeling via Hamilton’s Principle , 2011 .

[97]  Yi Liu,et al.  Hyperbolic Navier-Stokes Method for High-Reynolds-Number Boundary Layer Flows , 2017 .

[98]  T. Koivisto,et al.  On the covariance of teleparallel gravity theories , 2017, 1701.06271.

[99]  A. C. Eringen,et al.  Mechanics of Micromorphic Continua , 1968 .

[100]  Alain Goriely,et al.  Riemann–Cartan Geometry of Nonlinear Dislocation Mechanics , 2012 .

[101]  Hiroaki Nishikawa.  New-Generation Hyperbolic Navier-Stokes Schemes: O(1=h) Speed-Up and Accurate Viscous/Heat Fluxes , 2011 .

[102]  Y. M. Cho,et al.  Einstein Lagrangian as the translational Yang-Mills Lagrangian , 1976 .

[103]  P. Neff,et al.  A Review on Wave Propagation Modeling in Band-Gap Metamaterials via Enriched Continuum Models , 2016, 1609.01073.

[104]  M. Fecko.  Differential Geometry and Lie Groups for Physicists , 2006 .

[105]  Dima Bolmatov,et al.  Thermodynamic behaviour of supercritical matter , 2013, Nature Communications.