Bifurcation and Stability of Dissipative Systems by Q.S. Nguyen

By Q.S. Nguyen

The first subject matter matters the plastic buckling of constructions within the spirit of Hill’s classical strategy. Non-bifurcation and balance standards are brought and post-bifurcation research played via asymptotic improvement process in relation with Hutchinson’s paintings. a few contemporary effects at the generalized normal version are given and their connection to Hill’s normal formula is gifted. Instability phenomena of inelastic movement tactics reminiscent of pressure localization and necking are mentioned. the second one subject matter issues balance and bifurcation difficulties in internally broken or cracked colids. In brittle fracture or brittle harm, the evolution legislation of crack lengths or harm parameters is time-independent like in plasticity and ends up in an analogous mathematical description of the quasi-static evolution. balance and non-bifurcation standards within the feel of Hill may be back received from the dialogue of the speed response.

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By Q.S. Nguyen

The first subject matter matters the plastic buckling of constructions within the spirit of Hill’s classical strategy. Non-bifurcation and balance standards are brought and post-bifurcation research played via asymptotic improvement process in relation with Hutchinson’s paintings. a few contemporary effects at the generalized normal version are given and their connection to Hill’s normal formula is gifted. Instability phenomena of inelastic movement tactics reminiscent of pressure localization and necking are mentioned. the second one subject matter issues balance and bifurcation difficulties in internally broken or cracked colids. In brittle fracture or brittle harm, the evolution legislation of crack lengths or harm parameters is time-independent like in plasticity and ends up in an analogous mathematical description of the quasi-static evolution. balance and non-bifurcation standards within the feel of Hill may be back received from the dialogue of the speed response.

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20) look like what RA YLEIGH has studied in elastodynamics and called surface waves (see LOVE [44]): in that sense, one may say that the failure of the complementing condition allows for the existence of stationary surface waves. This has already been pointed out by THOMPSON [68] for traction boundary conditions ; these waves are however also possible under some circumstances for boundary conditions of place. 22), one gets a system of three algebraic equations and the boundary complementing conditions fails when non trivial solutions are possible for this system, that is if and only if its determinant is zero.

32, pp. 369-399, 1969. 69. TRIANTAFYLLIDIS, N. : Bifurcation phenomena in pure bending, J. Mech. Solids, 28, pp. 221-245, 1980. 70. , GUENOT, A. : Borehole instabilities as bifurcation phenomena, lnt. J. Rock. Mech. Min. Sei. & Geomech. Abstract, 25, pp. 159-170, 1988. 71. YOUNG, N. J. : Bifurcation phenomena in the plane compression test, J. Mech. 77-, 1976. APPENDIX : RIESZ-FREDHOLM THEORY FOR LINEAR OPERATORS In this appendix, classical results on Fredholm theory are gathered. The starring point is the situation which prevails in finite dimensional vector spaces.

32 A. Benallal, R. Billardon and G. 12 to Ql is denoted by V. This condition requires that for every vector k tangent to the interface at xo, the only solution of the linear boundary value problern div[ C! 1 (xo): Vv 1 ] = 0 everywhere in flJ 1 div[ C! 25) with bounded w 1 and w2, is the null solution v1 = v2 =0. The mechanical interpretation of this condition can be drawn in the same lines asthat for the complementing condition given above. lndeed, Iet xo be a point where this condition has failed.

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