Boundary Problem of Thin Shells Theory
A boundary problem for thin elastic shells is formulated. The generally acceptable geometrical (straight normal) and physical (linear elasticity) hypotheses which underlie the relations are considered. Geometrical nonlinearity of shell deformation is take
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Boundary Problem of Thin Shells Theory
Abstract A boundary problem for thin elastic shells is formulated. The generally acceptable geometrical (straight normal) and physical (linear elasticity) hypotheses which underlie the relations are considered. Geometrical nonlinearity of shell deformation is taken into account. Equilibrium equations expressed via the displacements of shell middle surface are presented as governing relations. Tangent and bending boundary conditions along the arbitrary shell contour (free and clamped edge, free-hinge and fixed-hinge, elastic support) are formulated. A set of previously satisfied conditions (regularity of shell surface and material, piecewise continuity of shell contour and of boundary support parameters) is implemented and the concept of a generalized solution is introduced. The small perturbation of a vector-function of a generalized solution is considered, becoming the basis of investigation of non-uniqueness of the generalised solution, and of branching of the solutions.
2.1 General Concepts and Hypotheses Thin shells are the bodies bounded by two curvilinear surfaces placed in such a manner that the distance h between such surfaces is sufficiently less than any other specific overall dimensions Li ; i ¼ 1; 2. It is convenient to assign the spatial position of a shell point to its x ¼ ðx1 ; x2 ; x3 Þ coordinates referred to shell middle surface X (see Fig. 2.1). Here coordinate lines x1 ; x2 coincide with the middle surface curvature lines, and x3 is normal to these lines. A11 ; A22 ; A12 are the surface’s second fundamental form coefficients. D2 ¼ A11 A22 A212 . Contravariant components of the respective tensor are given by A11 ¼ A22 D2 ; 12 A ¼ A21 ¼ A12 D2 ; A22 ¼ A11 D2 ;
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_2, Springer Science+Business Media Dordrecht 2013
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2 Boundary Problem of Thin Shells Theory
Fig. 2.1 Shell element
x3 w
x1 u1 u2
h
x2
Li m
Ri
1 ¼ A11; x1 A22 þ A11; x2 A12 2A12 A12; x1 ; G 11 2D2 1 ¼ A22 2A12; x2 A22; x1 A12 A22; x2 ; G 22 2D2 A A 1 ¼ 22 11; x2 A12 A22; x1 ; 1 $ 2: 1 ¼ G G 12 21 2D2 Contour C bounds shell middle surface X. Its segments are denoted as Ci . The segments Ci are considered to be finite and possibly disconnected sets. Ci [ 0 means that contour segment Ci contains the connected piece of positive length. mk and mk are covariant and contravariant components of unit normal vector m to C (belonging to X). sk and sk are covariant and contravariant components of unit tangent vector to C. i denote middle surface curvature radii in xi directions i ¼ 1; 2. R ij are principal surface curvatures, i; j ¼ 1; 2. B Shell material is characterized by Young’s modulus EðxÞand Poisson’s ratio m. The existence of shell thickness parameter h=Ri 1 makes possible the transition from a three-dimensional model for the shell body to a two-dimensional model, presuming predefined strain-stre
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