Abstract
The paper presents a general methodology of introducing the shell-type variables which is based on the rotation constraint-equation (RC-equation). The RC-equation is proven to be equivalent to the polar decomposition of the deformation gradient formula, and the rotations which it yields are interpreted in terms of rotations of vectors of an ortho-normal basis. The deformation function and rotations are assumed as polynomials of the thickness coordinate ζ, and in this form used in the RC-equation. Solving this equation, we can express the coefficients of the quadratic deformation function in terms of the following shell-type variables: (a) the mid-surface position x 0, (b) the constant rotation Q 0, (c) the rotation vector ψ * for the ζ-dependent rotations, and (d) the normal components U 33 0 and U 33 1 of the right stretching tensor. This new methodology (i) ensures that all shell kinematical variables are consistent with the RC-equation, which is justified on 3D grounds, (ii) provides a general framework from which various Reissner-type hypotheses can be obtained by suitable assumptions. As an example, two generalized Reissner hypotheses are derived: one with two normal stretches, and the other with the in-plane twist and the bubble-like warping parameters.
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References
K.F. Chernykh, Nonlinear theory of isotropically elastic thin shells. Mekh. Tverdogo Tela 15 (1980) 148–159 (in Russian).
K.F. Chernykh, The theory of thin shells of elastomers. Adv. in Mech. 6 (1983) 111–147 (in Russian).
R. deBoer, Vector-und Tensorrechnung für Ingenieure. Springer, Berlin (1982).
W. Jaunzemis, Continuum Mechanics. MacMillan, New York (1967).
J. Makowski and H. Stumpf, Finite strains and rotations in shells. In: W. Pietraszkiewicz (ed.) Finite Rotations in Structural Mechanics. Springer, Berlin (1986) pp. 175–194.
R. Ogden, Non-Linear Elastic Deformations. Ellis Horwood, Chichester, UK (1984).
H. Parish, A continuum-based shell theory for non-linear applications. Internat. J. Numer. Methods Engrg. 38 (1995) 1855–1883.
R. Piltner, A derivation of thick and thin plate formulation without ad hoc assumptions. J. Elasticity 29 (1992) 133–173.
R. Piltner and D.S. Joseph, An accurate low order plate bending element with thickness change and enhanced strains. Comput. Mech. 27 (2001) 353–359.
J.C. Simo and D.D. Fox, On a stress resultant geometrically exact shell model. Part IV: Variable thickness shells with through the thickness stretching. Comput. Methods Appl. Mech. Engrg. 81 (1990) 53–91.
J.C. Simo, D.D. Fox and T.J.R. Hughes, Formulations of finite elasticity with independent rotations. Comput. Methods Appl. Mech. Engrg. 95 (1992) 227–288.
J. Stuelpnagel, On the parametrization of three-dimensional rotational group. SIAM Rev. 6(4) (1964) 422–430.
T.C.T. Ting, Determination of C 1/2, C-1/2 and more general isotropic tensor functions of C. J. Elasticity 15 (1985) 319–323.
K. Wisniewski, Finite rotation quadrilateral element for multi-layer beams. Internat. J. Numer. Methods Engrg. 44 (1999) 405–431.
K.Wisniewski and E. Turska, Kinematics of finite rotation shells with in-plane twist parameter. Comput. Methods Appl. Mech. Engrg. 190(8-10) (2000) 1117–1135.
K. Wisniewski and E. Turska, Warping and in-plane twist parameter in kinematics of finite rotation shells. Comput. Methods Appl. Mech. Engrg. 190(43-44) (2001) 5739–5758.
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Wisniewski, K., Turska, E. Second-Order Shell Kinematics Implied by Rotation Constraint-Equation. Journal of Elasticity 67, 229–246 (2002). https://doi.org/10.1023/A:1024974422809
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DOI: https://doi.org/10.1023/A:1024974422809