Branching of Nonlinear Boundary Problem Solutions
Branching, i.e., the splitting of solutions into two or more branches, is considered to be the essential feature of nonlinear boundary problems solutions behaviour for deformation of thin shells. The types of branching and branching (singular) points are
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Branching of Nonlinear Boundary Problem Solutions
Abstract Branching, i.e., the splitting of solutions into two or more branches, is considered to be the essential feature of nonlinear boundary problems solutions behaviour for deformation of thin shells. The types of branching and branching (singular) points are distinguished as bifurcation points and limit points. Such special cases as multiple branching, symmetric bifurcation points, and isolated branches are considered as well. The static and energy criteria of stability and expected behaviour of shell structure are presented. The vector–matrix formulation of a bifurcation problem as the instrument to reveal the non-uniqueness of the solution is presented. The connection between the bifurcation problem for boundary and Cauchy problems is demonstrated. The properties of the correspondent Frechet matrix as the key parameters of branching are considered. The specifics of eigenvalues and eigenforms (obtained in the frameworks of linear theory) for most typical shell loading cases which demonstrate the buckling modes of behaviour—external pressure and axial compression—are presented.
3.1 Branching Patterns and Types of Singular Points The concept of elastic stability in the mechanics of deformable bodies is associated with the bifurcation of equilibrium forms or with buckling phenomenon. An axially compressed rod represents the first type of behaviour, the bending of a shallow arc, the second one. The theory of branching of nonlinear equations solutions offers a unified approach to model instability phenomena. Any solution (form of equilibrium) a(k) is considered to be a function of load parameter k; at a certain k = kcr (singular point) the solution splits into two or more branches a1(k), a2(k), …. a is a certain generalized coordinate of the system (say, a = kUk). Two types of singular points are distinguished:
N. I. Obodan et al., Nonlinear Behaviour and Stability of Thin-Walled Shells, Solid Mechanics and Its Applications 199, DOI: 10.1007/978-94-007-6365-4_3, Springer Science+Business Media Dordrecht 2013
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3 Branching of Nonlinear Boundary Problem Solutions
Fig. 3.1 Bifurcation point
cr
a2( )
a1( )
a
acr
Fig. 3.2 Limit point cr
a2( ) a1( )
acr
a
Fig. 3.3 Isolated branch
a3( ) a2( ) a1( ) a
– the bifurcation (branching) point (Fig. 3.1) where the initial solution (a1(k)) intersects another one (a2(k)); – the limit point (Fig. 3.2) where at the vicinity of k = kcr either no one solution exists (for k [ kcr), or two solutions exist (for k \ kcr, two solutions a1(k) and a2(k) have a common horizontal tangent at k = kcr). The special case of a postcritical pattern is presented in Fig. 3.3. It is possible that some solution does not intersect (bifurcate from) any other one but represents the isolated branch.
3.1 Branching Patterns and Types of Singular Points
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Fig. 3.4 Symmetric bifurcation point
a1( ) cr
a2( )
a cr
a
Fig. 3.5 Multiple branching
cr
a2 ( ) a 3( ) a m+1( ) a1( ) a cr
a
Sometimes the combined case is distinguishe
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