Using symbolic computation in the characterization of frictional instabilities involving orthotropic materials

Mohamed A. Agwa; António Pinto da Costa

International Journal of Applied Mathematics and Computer Science (2015)

  • Volume: 25, Issue: 2, page 259-267
  • ISSN: 1641-876X

Abstract

top
The present work addresses the problem of determining under what conditions the impending slip state or the steady sliding of a linear elastic orthotropic layer or half space with respect to a rigid flat obstacle is dynamically unstable. In other words, we search the conditions for the occurrence of smooth exponentially growing dynamic solutions with perturbed initial conditions arbitrarily close to the steady sliding state, taking the system away from the equilibrium state or the steady sliding state. Previously authors have shown that a linear elastic isotropic half space compressed against and sliding with respect to a rigid flat surface may get unstable by flutter when the coefficient of friction μ and Poisson’s ratio ν are sufficiently large. In the isotropic case they have been able to derive closed form analytic expressions for the exponentially growing unstable solutions as well as for the borders of the stability regions in the space of parameters, because in the isotropic case there are only two dimensionless parameters (μ and ν). Already for the simplest version of orthotropy (an orthotropic transversally isotropic material) there are seven governing parameters (μ, five independent material constants and the orientation of the principal directions of orthotropy) and the expressions become very lengthy and literally impossible to manipulate manually. The orthotropic case addressed here is impossible to solve with simple closed form expressions, and therefore the use of computer algebra software is required, the main commands being indicated in the text.

How to cite

top

Mohamed A. Agwa, and António Pinto da Costa. "Using symbolic computation in the characterization of frictional instabilities involving orthotropic materials." International Journal of Applied Mathematics and Computer Science 25.2 (2015): 259-267. <http://eudml.org/doc/270145>.

@article{MohamedA2015,
abstract = {The present work addresses the problem of determining under what conditions the impending slip state or the steady sliding of a linear elastic orthotropic layer or half space with respect to a rigid flat obstacle is dynamically unstable. In other words, we search the conditions for the occurrence of smooth exponentially growing dynamic solutions with perturbed initial conditions arbitrarily close to the steady sliding state, taking the system away from the equilibrium state or the steady sliding state. Previously authors have shown that a linear elastic isotropic half space compressed against and sliding with respect to a rigid flat surface may get unstable by flutter when the coefficient of friction μ and Poisson’s ratio ν are sufficiently large. In the isotropic case they have been able to derive closed form analytic expressions for the exponentially growing unstable solutions as well as for the borders of the stability regions in the space of parameters, because in the isotropic case there are only two dimensionless parameters (μ and ν). Already for the simplest version of orthotropy (an orthotropic transversally isotropic material) there are seven governing parameters (μ, five independent material constants and the orientation of the principal directions of orthotropy) and the expressions become very lengthy and literally impossible to manipulate manually. The orthotropic case addressed here is impossible to solve with simple closed form expressions, and therefore the use of computer algebra software is required, the main commands being indicated in the text.},
author = {Mohamed A. Agwa, António Pinto da Costa},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {Coulomb friction; dynamic instabilities; surface solutions; orthotropic material},
language = {eng},
number = {2},
pages = {259-267},
title = {Using symbolic computation in the characterization of frictional instabilities involving orthotropic materials},
url = {http://eudml.org/doc/270145},
volume = {25},
year = {2015},
}

TY - JOUR
AU - Mohamed A. Agwa
AU - António Pinto da Costa
TI - Using symbolic computation in the characterization of frictional instabilities involving orthotropic materials
JO - International Journal of Applied Mathematics and Computer Science
PY - 2015
VL - 25
IS - 2
SP - 259
EP - 267
AB - The present work addresses the problem of determining under what conditions the impending slip state or the steady sliding of a linear elastic orthotropic layer or half space with respect to a rigid flat obstacle is dynamically unstable. In other words, we search the conditions for the occurrence of smooth exponentially growing dynamic solutions with perturbed initial conditions arbitrarily close to the steady sliding state, taking the system away from the equilibrium state or the steady sliding state. Previously authors have shown that a linear elastic isotropic half space compressed against and sliding with respect to a rigid flat surface may get unstable by flutter when the coefficient of friction μ and Poisson’s ratio ν are sufficiently large. In the isotropic case they have been able to derive closed form analytic expressions for the exponentially growing unstable solutions as well as for the borders of the stability regions in the space of parameters, because in the isotropic case there are only two dimensionless parameters (μ and ν). Already for the simplest version of orthotropy (an orthotropic transversally isotropic material) there are seven governing parameters (μ, five independent material constants and the orientation of the principal directions of orthotropy) and the expressions become very lengthy and literally impossible to manipulate manually. The orthotropic case addressed here is impossible to solve with simple closed form expressions, and therefore the use of computer algebra software is required, the main commands being indicated in the text.
LA - eng
KW - Coulomb friction; dynamic instabilities; surface solutions; orthotropic material
UR - http://eudml.org/doc/270145
ER -

References

top
  1. Adams, G. (1995). Self-excited oscillations of two elastic half-spaces sliding with a constant coefficient of friction, ASME, Journal of Applied Mechanics 62(4): 867-872. Zbl0913.73036
  2. Agwa, M. and Pinto da Costa, A. (2008). Instability of frictional contact states in infinite layers, European Journal of Mechanics, A: Solids 27(3): 487-503. Zbl1154.74347
  3. Agwa, M. and Pinto da Costa, A. (2011). Surface instabilities in linear orthotropic half-spaces with a frictional interface, ASME, Journal of Applied Mechanics 78(4), Paper 041002. 
  4. Batoz, J.-L. and Dhatt, G. (1990). Modélization des Structures par Éléments Finis. Vol. 1: solides élastiques, Hermès, Paris. 
  5. Ibrahim, R. (1994). Friction-induced vibration, chatter, squeal, and chaos, Part I: Mechanics of contact and friction, Part II: Dynamics and modeling, ASME, Applied Mechanics Reviews 47(7): 209-253. 
  6. Maple (2013). Maplesoft software company, http://www.maplesoft.com/. 
  7. Martins, J., Faria, L. and Guimarães, J. (1992). Dynamic surface solutions in linear elasticity with frictional boundary conditions, in R. Ibrahim and A.A. Soom (Eds.), FrictionInduced Vibration, Chatter, Squeal and Chaos, ASME, New York, NY, pp. 33-39. 
  8. Martins, J., Guimarães, J. and Faria, L. (1995). Dynamic surface solutions in linear elasticity and viscoelasticity with frictional boundary conditions, ASME, Journal of Vibration and Acoustics 117(4): 445-451. 
  9. Martins, J. and Raous, M. (Eds.) (2002). Friction and Instabilities, Springer, Vienna. 
  10. Pinto da Costa, A. and Agwa, M. (2009). Frictional instabilities in orthotropic hollow cylinders, Computers and Structures 87(21-22): 1275-1286. 
  11. Rand, O. and Rovenski, V. (2005). Analytical Methods in Anisotropic Elasticity with Symbolic Computational Tools, Birkhauser, Basel. Re-revised: 25 October 2014 Zbl1092.74002

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.