Geometrically exact Cosserat rods with Kelvin–Voigt type viscous damping

Abstract. We present the derivation of a simple viscous damping model of Kelvin–Voigt type for geometrically exact Cosserat rods from three-dimensional continuum theory. Assuming moderate curvature of the rod in its reference configuration, strains remaining small in its deformed configurations, strain rates that vary slowly compared to internal relaxation processes, and a homogeneous and isotropic material, we obtain explicit formulas for the damping parameters of the model in terms of the well known stiffness parameters of the rod and the retardation time constants defined as the ratios of bulk and shear viscosities to the respective elastic moduli. We briefly discuss the range of validity of the Kelvin–Voigt model and illustrate its behaviour for large bending deformations with a numerical example.

[1]  Zdeněk P. Bažant,et al.  Mechanics of solid materials , 1992 .

[2]  A. Becker.  A Review on Image Distortion Measures , 2000 .

[3]  D. Zupan,et al.  Dynamics of spatial beams in quaternion description based on the Newmark integration scheme , 2013 .

[4]  Alex H. Barbat,et al.  Dynamic analysis of beam structures considering geometric and constitutive nonlinearity , 2008 .

[5]  J. C. Simo,et al.  A finite strain beam formulation. The three-dimensional dynamic problem. Part I , 1985 .

[6]  Eitan Grinspun,et al.  Discrete elastic rods , 2008, ACM Trans. Graph..

[7]  Victor L. Berdichevsky,et al.  Variational Principles of Continuum Mechanics , 2009 .

[8]  S. Antman.  Nonlinear problems of elasticity , 1994 .

[9]  A. Zemitis.  On interaction of a liquid film with an obstacle , 2002 .

[10]  Nicole Marheineke,et al.  Systematic derivation of an asymptotic model for the dynamics of curved viscous fibers , 2008 .

[11]  J. Orlik.  Homogenization for viscoelasticity of the integral type with aging and shrinkage , 1998 .

[12]  L. Landau,et al.  statistical-physics-part-1 , 1958 .

[13]  F. T. Trouton,et al.  On the coefficient of viscous traction and its relation to that of viscosity , 1906 .

[14]  Horst W. Hamacher,et al.  Inverse Radiation Therapy Planning: A Multiple Objective Optimisation Approach , 1999 .

[15]  J. Linn.  On the Frame — Invariant Description of the Phase Space of the Folgar–Tucker Equation , 2004 .

[16]  Nicole Marheineke,et al.  Fluid-fiber-interactions in rotational spinning process of glass wool production , 2011 .

[17]  E. Vouga,et al.  Discrete viscous threads , 2010, ACM Trans. Graph..

[18]  Rakesh K. Kapania,et al.  On a geometrically exact curved/twisted beam theory under rigid cross-section assumption , 2003 .

[19]  V. Starikovicius,et al.  The multiphase flow and heat transfer in porous media , 2003 .

[20]  Miran Saje,et al.  The quaternion-based three-dimensional beam theory , 2009 .

[21]  Piero Villaggio,et al.  Mathematical Models for Elastic Structures , 1997 .

[22]  Walter Noll,et al.  Foundations of Linear Viscoelasticity , 1961 .

[23]  Johannes Gerstmayr,et al.  A continuum mechanics based derivation of Reissner’s large-displacement finite-strain beam theory: the case of plane deformations of originally straight Bernoulli–Euler beams , 2009 .

[24]  S. Kruse,et al.  On the Pricing of Forward Starting Options under Stochastic Volatility , 2003 .

[25]  A. Love.  A treatise on the mathematical theory of elasticity , 1892 .

[26]  Martin Arnold,et al.  Numerical aspects in the dynamic simulation of geometrically exact rods , 2012 .

[27]  Laurent Grisoni,et al.  Geometrically exact dynamic splines , 2008, Comput. Aided Des..

[28]  Axel Klar,et al.  Hierarchy of mathematical models for production processes of technical textiles , 2009 .

[29]  Alex H. Barbat,et al.  Static analysis of beam structures under nonlinear geometric and constitutive behavior , 2007 .

[30]  Martin Arnold,et al.  Integration of Nonlinear Models of Flexible Body Deformation in Multibody System Dynamics , 2014 .

[31]  S. Timoshenko,et al.  Theory of elasticity , 1975 .

[32]  H. Neunzert,et al.  Mathematics as a Technology: Challenges for the next 10 Years , 2004 .

[33]  Andrew J. Kurdila,et al.  『Fundamentals of Structural Dynamics』(私の一冊) , 2019, Journal of the Society of Mechanical Engineers.

[34]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[35]  Liping Liu.  THEORY OF ELASTICITY , 2012 .

[36]  G. Cowper.  The Shear Coefficient in Timoshenko’s Beam Theory , 1966 .

[37]  N. Ettrich.  Generation of surface elevation models for urban drainage simulation , 2005 .

[38]  V. Berdichevskiĭ.  On the energy of an elastic rod , 1981 .

[39]  V. Berdichevsky.  Structure of equations of macrophysics. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[40]  Lev Davidovich Landau,et al.  THE FUNDAMENTAL PRINCIPLES OF STATISTICAL PHYSICS , 1980 .

[41]  Robert Bohlin,et al.  Fast simulation of quasistatic rod deformations for vr applications , 2008 .

[42]  Anton Winterfeld,et al.  Application of general semi-infinite programming to lapidary cutting problems , 2008, Eur. J. Oper. Res..

[43]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[44]  Miran Saje,et al.  Quaternion-based dynamics of geometrically nonlinear spatial beams using the Runge-Kutta method , 2012 .

[45]  Ignacio Romero,et al.  A comparison of finite elements for nonlinear beams: the absolute nodal coordinate and geometrically exact formulations , 2008 .

[46]  Werner Wagner,et al.  Shear correction factors in Timoshenko's beam theory for arbitrary shaped cross-sections , 2001 .

[47]  J. Orlik.  Homogenization for contact problems with periodically rough surfaces , 2004 .

[48]  G. M.,et al.  A Treatise on the Mathematical Theory of Elasticity , 1906, Nature.

[49]  L. A. Starosel'skii,et al.  On the theory of curvilinear timoshenko-type rods , 1983 .

[50]  Peter Schröder,et al.  A simple geometric model for elastic deformations , 2010, ACM Trans. Graph..

[51]  Nicole Marheineke,et al.  Asymptotic model for the dynamics of curved viscous fibres with surface tension , 2009, Journal of Fluid Mechanics.

[52]  L. Mahadevan,et al.  Coiling Viscous Jets , 1996 .

[53]  Th. Hanne,et al.  Applying multiobjective evolutionary algorithms in industrial projects , 2006 .

[54]  P. Schröder,et al.  A simple geometric model for elastic deformations , 2010, SIGGRAPH 2010.

[55]  M. Krekel.  Optimal Portfolios With A Loan Dependent Credit Spread , 2002 .

[56]  H. Weiss.  Dynamics of Geometrically Nonlinear Rods: I. Mechanical Models and Equations of Motion , 2002 .

[57]  Olivier A. Bauchau,et al.  Flexible multibody dynamics , 2010 .

[58]  T. Hanne.  Eine Übersicht zum Scheduling von Baustellen , 2005 .

[59]  D. Kehrwald.  Parallel lattice Boltzmann simulation of complex flows , 2004 .

[60]  R. Wegener,et al.  On an asymptotic upper‐convected Maxwell model for a viscoelastic jet , 2012 .

[61]  Hans Irschik,et al.  Large deformation and stability of an extensible elastica with an unknown length , 2011 .

[62]  Sigrid Leyendecker,et al.  A discrete mechanics approach to the Cosserat rod theory—Part 1: static equilibria , 2011 .

[63]  Horst W. Hamacher,et al.  Inverse Radiation Therapy Planning: A Multiple Objective Optimisation Approach , 1999 .

[64]  Frank-Thomas Lentes,et al.  Three-dimensional radiative heat transfer in glass cooling processes , 1998 .

[65]  S. Antman.  Invariant Dissipative Mechanisms for the Spatial Motion of Rods Suggested by Artificial Viscosity , 2003 .

[66]  Ettore Pennestrì,et al.  Modeling elastic beams using dynamic splines , 2011 .

[67]  Ahmed A. Shabana,et al.  A nonlinear visco-elastic constitutive model for large rotation finite element formulations , 2011 .

[68]  Ertugrul Taciroglu,et al.  Much ado about shear correction factors in Timoshenko beam theory , 2010 .

[69]  H. Neunzert.  »Denn nichts ist für den Menschen als Menschen etwas wert, was er nicht mit Leidenschaft tun kann« , 2001 .

[70]  Dewey H. Hodges,et al.  Nonlinear Composite Beam Theory , 2006 .

[71]  Oliver Wirjadi,et al.  Survey of 3d image segmentation methods , 2007 .

[72]  Robert L. Taylor,et al.  Numerical formulations of elasto-viscoplastic response of beams accounting for the effect of shear , 1984 .

[73]  A. Naumovich.  On a Finite Volume Discretization of the Three-dimensional Biot Poroelasticity , 2006 .

[74]  H. Knaf.  Kernel Fisher discriminant functions – a concise and rigorous introduction , 2007 .

[75]  David Rubin,et al.  Introduction to Continuum Mechanics , 2009 .

[76]  M. Junk.  On the Construction of Discrete Equilibrium Distributions for Kinetic Schemes , 1999 .

[77]  O. Bauchau,et al.  Interpolation of finite rotations in flexible multi-body dynamics simulations , 2008 .

[78]  C. Petrie,et al.  Extensional viscosity: A critical discussion , 2006 .

[79]  H. Weiss,et al.  Dynamics of Geometrically Nonlinear Rods: II. Numerical Methods and Computational Examples , 2002 .

[80]  H. Lang,et al.  Multi-body dynamics simulation of geometrically exact Cosserat rods , 2011 .

[81]  M. Géradin,et al.  Flexible Multibody Dynamics: A Finite Element Approach , 2001 .

[82]  A. Unterreiter,et al.  Numerical evidance for the non-existing of solutions of the equations desribing rotational fiber spinning , 2007 .