Torsional vibrations of a column of fine-grained material: A gradient elastic approach

The gradient theory of elasticity with damping is successfully employed to explain the experimentally observed shift in resonance frequencies during forced harmonic torsional vibration tests of columns made of fine-grained material from their theoretically computed values on the basis of the classical theory of elasticity with damping. To this end, the governing equation of torsional vibrations of a column with circular cross-section is derived both by the lattice theory and the continuum gradient elasticity theory with damping, with consideration of micro-stiffness and micro-inertia effects. Both cases of a column with two rotating masses attached at its top and bottom, and of a column fixed at its base carrying a rotating mass at its free top, are considered. The presence of both micro-stiffness and micro-inertia effects helps to explain the observed natural frequency shift to the left or to the right of the classical values depending on the nature of interparticle forces (repulsive or attractive) due to particle charge. A method for using resonance column tests to determine not only the shear modulus but also the micro-stiffness and micro-inertia coefficients of gradient elasticity for fine-grained materials is proposed.

[1]  A. Giannakopoulos,et al.  Structural analysis of gradient elastic components , 2007 .

[2]  Mohammad Taghi Ahmadian,et al.  Torsion of strain gradient bars , 2011 .

[3]  S. Kramer.  Geotechnical Earthquake Engineering , 1996 .

[4]  B. Guzina,et al.  A physical perspective of the length scales in gradient elasticity through the prism of wave dispersion , 2013 .

[5]  R. D. Mindlin,et al.  Response of Damped Elastic Systems to Transient Disturbances , 1989 .

[6]  E. Aifantis,et al.  Gradient elasticity in statics and dynamics: An overview of formulations, length scale identification procedures, finite element implementations and new results , 2011 .

[7]  M. Lazar,et al.  Nonsingular stress and strain fields of dislocations and disclinations in first strain gradient elasticity , 2005, cond-mat/0502023.

[8]  D. Beskos,et al.  Static and Dynamic BEM Analysis of Strain Gradient Elastic Solids and Structures , 2012 .

[9]  Ching S. Chang,et al.  Wave Propagation in Granular Rod Using High-Gradient Theory , 1997 .

[10]  D. Beskos,et al.  Static Analysis of Gradient Elastic Bars, Beams, Plates and Shells , 2010 .

[11]  S. M. Hasheminejad,et al.  Size dependent torsional vibration of nanotubes , 2010 .

[12]  A. Giannakopoulos,et al.  Structural analysis using a dipolar elastic Timoshenko beam , 2013 .

[13]  B. Hardin,et al.  The Nature of Damping in Sands , 1965 .

[14]  Demosthenes Polyzos,et al.  Velocity dispersion of guided waves propagating in a free gradient elastic plate: application to cortical bone. , 2009, The Journal of the Acoustical Society of America.

[15]  Singiresu S. Rao.  Vibration of Continuous Systems , 2019 .

[16]  R. D. Mindlin.  MICROSTRUCTURE IN LINEAR ELASTICITY , 1999 .

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

[18]  G. Huber,et al.  Time-dependent behavior of fine-grained model material in resonant column experiments , 2004 .

[19]  Jinkyu Kim,et al.  Extended framework of Hamilton's principle for continuum dynamics , 2013 .

[20]  D. Beskos,et al.  Wave dispersion in gradient elastic solids and structures: A unified treatment , 2009 .

[21]  Ioannis Vardoulakis,et al.  Microstructure in linear elasticity and scale effects: a reconsideration of basic rock mechanics and rock fracture mechanics , 2001 .

[22]  S. P. Filopoulos,et al.  On the capability of generalized continuum theories to capture dispersion characteristics at the atomic scale , 2012 .

[23]  D. Beskos,et al.  Response of gradient-viscoelastic bar to static and dynamic axial load , 2004 .

[24]  Ioannis Vardoulakis,et al.  Dispersive Rayleigh-Wave Propagation in Microstructured Solids Characterized by Dipolar Gradient Elasticity , 2004 .

[25]  H. Georgiadis,et al.  Problems of the Flamant–Boussinesq and Kelvin Type in Dipolar Gradient Elasticity , 2007 .

[26]  D. E. Beskos,et al.  Static, stability and dynamic analysis of gradient elastic flexural Kirchhoff plates , 2008 .

[27]  J. Bleustein.  A note on the boundary conditions of toupin's strain-gradient theory , 1967 .

[28]  C. Lim,et al.  Free torsional vibration of nanotubes based on nonlocal stress theory , 2012 .

[29]  G. Huber,et al.  Resonant column experiments with fine-grained model material – evidence of particle surface forces , 2003 .

[30]  H. Askes,et al.  Four simplified gradient elasticity models for the simulation of dispersive wave propagation , 2008 .

[31]  J. Troutman.  Variational Principles in Mechanics , 1983 .

[32]  Demosthenes Polyzos,et al.  Bending and stability analysis of gradient elastic beams , 2003 .

[33]  D. Beskos,et al.  Dynamic analysis of gradient elastic flexural beams , 2003 .

[34]  Dimitrios I. Fotiadis,et al.  Derivation of Mindlin’s first and second strain gradient elastic theory via simple lattice and continuum models , 2012 .