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April  2016, 9(2): 529-536. doi: 10.3934/dcdss.2016010

## Kinematical structural stability

 1 IBISC, UFRST-UEVE, 40, rue du Pelvoux CE 1455 Courcouronnes, 91020 Evry Cedex, France 2 South Britain University, LIMATB -UBS -Lorient Research Center, Rue de Saint Maudé - BP 92116, 56321 Lorient cedex, France, France, France

Received  November 2014 Revised  November 2015 Published  March 2016

This paper gives an overview of our results obtained from 2009 until 2014 about paradoxical stability properties of non conservative systems which lead to the concept of Kinematical Structural Stability (Ki.s.s.). Due to Fischer-Courant results, this ki.s.s. is universal for conservative systems whereas new interesting situations may arise for non conservative ones. A remarkable algebraic property of the symmetric part of linear operators may generalize this result for divergence stability but leading only to a conditional ki.s.s. By duality, the concept of geometric degree of nonconservativity is highlighting. Paradigmatic examples of Ziegler systems illustrate the general results and their effectiveness.
Citation: Jean Lerbet, Noël Challamel, François Nicot, Félix Darve. Kinematical structural stability. Discrete and Continuous Dynamical Systems - S, 2016, 9 (2) : 529-536. doi: 10.3934/dcdss.2016010
##### References:
 [1] D. Bigoni and G. Noselli, Experimental evidence of flutter and divergence instabilities induced by dry friction, Journal of the Mechanics and Physics of Solids, 59 (2011), 2208-2226. doi: 10.1016/j.jmps.2011.05.007. [2] V. V. Bolotin, Non-conservative Problems of the Theory of Elastic Stability, Pergamon Press, 1963. [3] N. Challamel, F. Nicot, J. Lerbet and F. Darve, Stability of non-conservative elastic structures under additional kinematics constraints, Engineering Structures, 32 (2010), 3086-3092. doi: 10.1016/j.engstruct.2010.05.027. [4] K. E. Gustafson and D. K. M. Rao, Numerical Range. The field of Values of Linear Operators and Matrices, Universitext, Springer, 1997. doi: 10.1007/978-1-4613-8498-4. [5] R. Hill, A general theory of uniqueness and stability in elastic-plastic solids, Journal of the Mechanics and Physics of Solids, 6 (1958), 236-249. doi: 10.1016/0022-5096(58)90029-2. [6] R. Hill, Some basic principles in the mechanics of solids without a natural time, J. Mech. Phys. Solids, 7 (1959), 209-225. doi: 10.1016/0022-5096(59)90007-9. [7] O. N. Kirillov and F. Verhulst, Paradoxes of dissipation-induced destabilization or who opened Withney's umbrella?, Z. Angew.Math. Mech., 90 (2010), 462-488. doi: 10.1002/zamm.200900315. [8] J. Lerbet, M. Aldowaji, N. Challamel, F. Nicot, F. Prunier and F. Darve, P-positive definite matrices and stability of nonconservative systems, Z. Angew. Math. Mech., 92 (2012), 409-422. doi: 10.1002/zamm.201100055. [9] J. Lerbet, M. Aldowaji, N. Challamel, F. Nicot, O. Kirillov and F. Darve, Geometric degree of nonconservativity, Math. and Mech. of Complex Systems, 2 (2014), 123-139. doi: 10.2140/memocs.2014.2.123. [10] T. Tarnai, Paradoxical behaviour of vibrating systems challenging Rayleigh's theorem, 21st International Congress of Theoretical and Applied Mechanics, Warsaw, 2004. [11] J. M. T. Thompson, 'Paradoxical' mechanics under fluid flow, Nature, 296 (1982), 135-137. doi: 10.1038/296135a0.

show all references

##### References:
 [1] D. Bigoni and G. Noselli, Experimental evidence of flutter and divergence instabilities induced by dry friction, Journal of the Mechanics and Physics of Solids, 59 (2011), 2208-2226. doi: 10.1016/j.jmps.2011.05.007. [2] V. V. Bolotin, Non-conservative Problems of the Theory of Elastic Stability, Pergamon Press, 1963. [3] N. Challamel, F. Nicot, J. Lerbet and F. Darve, Stability of non-conservative elastic structures under additional kinematics constraints, Engineering Structures, 32 (2010), 3086-3092. doi: 10.1016/j.engstruct.2010.05.027. [4] K. E. Gustafson and D. K. M. Rao, Numerical Range. The field of Values of Linear Operators and Matrices, Universitext, Springer, 1997. doi: 10.1007/978-1-4613-8498-4. [5] R. Hill, A general theory of uniqueness and stability in elastic-plastic solids, Journal of the Mechanics and Physics of Solids, 6 (1958), 236-249. doi: 10.1016/0022-5096(58)90029-2. [6] R. Hill, Some basic principles in the mechanics of solids without a natural time, J. Mech. Phys. Solids, 7 (1959), 209-225. doi: 10.1016/0022-5096(59)90007-9. [7] O. N. Kirillov and F. Verhulst, Paradoxes of dissipation-induced destabilization or who opened Withney's umbrella?, Z. Angew.Math. Mech., 90 (2010), 462-488. doi: 10.1002/zamm.200900315. [8] J. Lerbet, M. Aldowaji, N. Challamel, F. Nicot, F. Prunier and F. Darve, P-positive definite matrices and stability of nonconservative systems, Z. Angew. Math. Mech., 92 (2012), 409-422. doi: 10.1002/zamm.201100055. [9] J. Lerbet, M. Aldowaji, N. Challamel, F. Nicot, O. Kirillov and F. Darve, Geometric degree of nonconservativity, Math. and Mech. of Complex Systems, 2 (2014), 123-139. doi: 10.2140/memocs.2014.2.123. [10] T. Tarnai, Paradoxical behaviour of vibrating systems challenging Rayleigh's theorem, 21st International Congress of Theoretical and Applied Mechanics, Warsaw, 2004. [11] J. M. T. Thompson, 'Paradoxical' mechanics under fluid flow, Nature, 296 (1982), 135-137. doi: 10.1038/296135a0.
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