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On a stress resultant geometrically exact shell model. IV: Variable thickness shells with through-the-thickness stretching. (English) Zbl 0746.73016

In previous parts [see the foregoing entry ( Zbl 0746.73015)] of this work a nonlinear geometrically exact stress resultant shell model for the analysis of large deformations is presented. The finite element method is applied on the basis of classical kinematic hypothesis.
It is shown that ill-conditioning of the formulation can be circumvented by introducing a multiplicative decomposition of the director field into an inextensible vector part and scalar stretching part, which involves only a trivial modification of the weak form of the equilibrium equations.
The exact update procedure automatically ensuring the positive thickness stretch is presented. Numerical examples are presented that illustrate the effects of the thickness stretch. The performance of the proposed mixed interpolation, and the well-conditioned response exhibited by the present approach in thin-shell limit.

MSC:

74K15 Membranes
74S05 Finite element methods applied to problems in solid mechanics
74B20 Nonlinear elasticity

Citations:

Zbl 0746.73015
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References:

[1] Simo, J. C.; Fox, D. D., On a stress resultant geometrically exact shell model, Part I: Formulation and optimal parametrization, Comput. Methods Appl. Mech. Engrg., 72, 267-304 (1989) · Zbl 0692.73062
[2] Simo, J. C.; Fox, D. D.; Rifai, M. S., On a stress resultant geometrically exact shell model, Part II: The linear theory; Computational aspects, Comput. Methods Appl. Mech. Engrg., 73, 53-92 (1989) · Zbl 0724.73138
[3] Simo, J. C.; Fox, D. D.; Rifai, M. S., On a stress resultant geometrically exact shell model, Part III: Computational aspects of the nonlinear theory, Comput. Methods Appl. Mech. Engrg., 79, 21-70 (1990) · Zbl 0746.73015
[4] Hildebrand, F. B.; Reissner, E.; Thomas, G. B., Notes on the foundations of the theory of small displacements of Orthotropic shells, National Advisory Committee for Aerodynamics, Technical Note No. 1833 (1949)
[5] Green, A. E.; Zerna, W., The equilibrium of thin elastic shells, Quat. J. Mech. Appl. Math., III, 9-22 (1950) · Zbl 0041.10702
[6] Naghdi, P. M., The theory of shells, (Truesdell, C., Handbuch der Physik, Vol. Via/2, Mechanics of Solids II (1972), Springer: Springer Berlin) · Zbl 0154.22602
[7] Reissner, E., On finite axi-symmetrical deformations of thin elastic shells of revolution, J. Comput. Mech. (1989), to appear · Zbl 0687.73047
[8] Antman, S. S., Ordinary differential equations of nonlinear elasticity I: Foundations of the theories of non-linearly elastic rods and shell, Arch. Rat. Mech. Anal., 61, 4, 307-351 (1976) · Zbl 0354.73046
[9] Green, A. E.; Naghdi, P. M.; Wenner, M. L., Linear theory of Cosserat surface and elastic plates of variable thickness, (Proc. Camb. Phil. Soc., 69 (1971)), 227-254 · Zbl 0212.57501
[10] Antman, S. S., Existence and nonuniqueness of axisymmetric equilibrium states of nonlinearly elastic shells, Arch. Rat. Mech. Anal., 40, 5, 329-372 (1971) · Zbl 0254.73072
[11] Hughes, T. J.R.; Carnoy, E., Nonlinear finite element shell formulation accounting for large membrane strains, Comput. Methods. Appl. Mech. Engrg., 39, 69-82 (1983) · Zbl 0509.73083
[12] J.C. Simo and J.G. Kennedy, On a stress resultant geometrically exact shell model. Part V: Nonlinear plasticity. Formulation and integration algorithms, Comput. Methods Appl. Mech. Engrg., submitted.; J.C. Simo and J.G. Kennedy, On a stress resultant geometrically exact shell model. Part V: Nonlinear plasticity. Formulation and integration algorithms, Comput. Methods Appl. Mech. Engrg., submitted. · Zbl 0754.73042
[13] Hughes, T. J.R., The Finite Element Method (1987), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ
[14] Zienkiewicz, O. C.; Taylor, R. L., (The Finite Element Method. Vol. I: Linear Analysis (1989), McGraw-Hill: McGraw-Hill New York)
[15] Bertsekas, D. P., Constrained Optimization and Lagrange Multiplier Methods (1982), Academic Press: Academic Press New York · Zbl 0453.65045
[16] J.C. Simo and R.L. Taylor, Finite elasticity in principal stretches: Formulation and augmented Lagrangian algorithms.; J.C. Simo and R.L. Taylor, Finite elasticity in principal stretches: Formulation and augmented Lagrangian algorithms.
[17] Scordellis, A. C.; Lo, K. S., Computer analysis of cylindrical shells, J. Am. Concr. Inst., 61, 539-561 (1969)
[18] Morely, L. S.D., Skew plates and structures, (Internat. Series of Monographs in Aeronautics and Astronautics (1963), MacMillan: MacMillan New York) · Zbl 0124.17704
[19] Morely, L. S.D.; Morris, A. J., Conflict between finite elements and shell theory, Royal Aircraft Establishment Report (1978), London
[20] Reissner, E., A note on variational theorems in elasticity, Internat. J. Solids and Structures, 1, 93-95 (1965)
[21] Hughes, T. J.R.; Brezzi, F., On drilling degrees of freedom, Comput. Methods Appl. Mech. Engrg., 72, 105-121 (1989) · Zbl 0691.73015
[22] Reissner, E., A note on two-dimensional finite-deformation theories of shells, Internat. J. Non-Linear Mech., 17, 3, 217-221 (1982) · Zbl 0492.73034
[23] Simo, J. C.; Wong, K., Unconditionally stable algorithms for the orthogonal group that exactly preserve energy and momentum (1989), preprint
[24] Taber, L. A., Large deflection of a fluid-filled spherical shell under a point load, J. Appl. Mech., 49, 121-128 (1982) · Zbl 0487.73073
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