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.
@article{bwmeta1.element.bwnjournal-article-amcv25i2p259bwm,
author = {Mohamed A. Agwa and Ant\'onio Pinto da Costa},
title = {Using symbolic computation in the characterization of frictional instabilities involving orthotropic materials},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {25},
year = {2015},
pages = {259-267},
zbl = {1322.70013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv25i2p259bwm}
}
Mohamed A. Agwa; António Pinto da Costa. Using symbolic computation in the characterization of frictional instabilities involving orthotropic materials. International Journal of Applied Mathematics and Computer Science, Tome 25 (2015) pp. 259-267. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv25i2p259bwm/
[000] 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. | Zbl 0913.73036
[001] 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. | Zbl 1154.74347
[002] 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.
[003] Batoz, J.-L. and Dhatt, G. (1990). Modélization des Structures par Éléments Finis. Vol. 1: solides élastiques, Hermès, Paris.
[004] 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.
[005] Maple (2013). Maplesoft software company, http://www.maplesoft.com/.
[006] 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.
[007] 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.
[008] Martins, J. and Raous, M. (Eds.) (2002). Friction and Instabilities, Springer, Vienna.
[009] Pinto da Costa, A. and Agwa, M. (2009). Frictional instabilities in orthotropic hollow cylinders, Computers and Structures 87(21-22): 1275-1286.
[010] Rand, O. and Rovenski, V. (2005). Analytical Methods in Anisotropic Elasticity with Symbolic Computational Tools, Birkhauser, Basel. Re-revised: 25 October 2014 | Zbl 1092.74002