Modes of Stability Loss of Materials


Article Preview

Three problems of stability loss are investigated and corresponded buckling modes are discussed. The first one is the stability loss of a compressed transversely isotropic linearly elastic medium. The standard analysis based on the Hadamard condition is conducted to solve this problem. The critical compression could be uniquely defined from the bifurcation equations but not a wave length. So, the buckling mode remains generally indefinite. The second considered problem is the stability loss of a compressed half-space with a free surface. It could be shown that the waviness is localized near the free plane surface but as for an entire space the wave length and the buckling mode are indefinite. These problems are treated in linear and nonlinear statement. In linear approach the pre-buckling deformations are ignored. It is shown that for some values of parameters the linear approach leads not only to the numerical error but also to qualitatively incorrect results. The thisrd problem under investifation is the stability loss of an uniformly compressed plate lying on a soft elastic half-space. In this problem the wave length is uniquely defined. Using the nonlinear post-critical analysis it is shown that the buckling mode could be fully defined and is has a chessboard-like character.



Edited by:

Robert V. Goldstein, Dr. Yeong-Maw Hwang, Yeau Ren Jeng and Cho-Pei Jiang




S. Kashtanova et al., "Modes of Stability Loss of Materials", Key Engineering Materials, Vol. 528, pp. 89-99, 2013

Online since:

November 2012




[1] P.S. Ciarlet, Mathematical Elasticity. Amsterdam etc., North-Holland, (1988).

[2] P. Ye. Tovstik, The vibrations and stability of a pre-stressed plate on an elastic foundation, J. Appl. Math. Mech. 73 1 (2009) 77-87.

[3] N.F. Morozov, M.V. Paukshto, P.E. Tovstik: Stability of the surface layer under thermal, Mech. Solids. 1 (1998) 130-139.

[4] P.E. Tovstik, Local buckling of plates and shallow shells on an elastic foundation, Mech. Solids. 40 1 (2005) 120-131.

[5] N.F. Morozov, P.E. Tovstik: Stability of a homogeneous transversely isotropic linearly elastic, Doklady. Ran. 438 3 (2011) 1-5.

[6] P.E. Tovstik, The volume and the surface stability of a transversely isotropic material under compression. Vestnik St. Petersburg Univ. Ser. 1, V1(2010).

[7] N.F. Morozov, P.E. Tovstik: Bulk and surface stability loss of materials, multi-scaling of synthetic and natural systems with self-adaptive capacity, Taiwan, 2010, pp.27-30.

[8] N.F. Morozov, P.E. Tovstik: Volume and surface stability of transversely isotropic material. Advanced Problems in Mechanics, 38 summer school, St. Petersburg, 2010, pp.376-390.

[9] N.F. Morozov, P.E. Tovstik: Localized buckling modes of a compressed elastic media. 14 Int. Conf. Modern problems of compressed media, Rostov-na-Donu, 1 , 2010, pp.240-245.

[10] N.F. Morozov, P.E. Tovstik: Control of surface waviness. Advanced Dynamics and Model Based Control of Structures and Machines, Springer, 2011, pp.57-64.


[11] N.F. Morozov, P.E. Tovstik: Stability of a surface layer under force and termal loading. Mech. Solids. 45 6 (2010) 5-15.

[12] L.E. Panin, V.E. Panin: Effect of the chessboard, and mass transfer in interfacial media of organic and inorganic nature, Physical. Nanomechanics. 10 6 (2007) 5-20.


[13] N.F. Morozov, P.E. Tovstik: On Modes of Buckling for a Plate on an Elastic Foundation. Mech. Solids. 45 4 (2010)519-528.


[14] N. Bowden, S. Brittaln, A.G. Evans, J.W. Hutchinson, G.M. Whitesides: Spontaneous formation of ordered structures in thin films of mttals supported on elastomeric polymer. Letters to nature, 393, 1998, pp.146-149.

[15] P.E. Tovstik, A.L. Smirnov: Asymptotic Methods in the Buckling Theory of Elastic Shells. World Scientific. Singpore at al., p.347.

[16] M.A. Il'gamov, V.A. Ivanov, B.V. Gulin: Strength, Stability and Dynamics of Shells with an Elastic Filler, Moscow, Nauka, (1977).

[17] P.E. Tovstik: The reaction of a pre-stressed orthotropic foundation. Vestnik St. Petersburg Univ. Ser. 1. 4 (2006) 98-108.

[18] N.F. Morozov, P.E. Tovstik: Initial supercritical behavior of buckled transversely isotropic elastic medium. Vestnik St. Petersburg Univ. Mathematics, V1, 44-50, (2010).


[19] G.P. Cherepanov: The theory of thermal stresses in a thin bounding layer, J. Appl. Phys. 78 11 (1995).

[20] R.V. Goldstein, V.E. Panin, N.M. Osipenko and L.S. Derevyagina: Model of the formation of the fracture structure in a layer with hardened near-surface zones. Physical Nanomechanics. No. 6 (2005) 23-32.

[21] J.F. Dorris, S. Nemat-Nasser: Instability of a layer on a half space. Trans. ASME.  47 (1980) 304-312.