# Formulation and Computational Aspects of a Stress Resultant Geometrically Exact Shell Model

## Abstract

This paper considers the formulation and numerical implementation of a geometrically exact resultant based shell model for the analysis of large deformations of thin and moderately thick shells. The model is essentially a single extensible director Cosserat surface. Variable thickness and thickness stretch effects are properly modeled via the *extensibility* condition on the director field. A simple linear elastic constitutive model is given which possesses the correct asymptotic limits as the thickness tends to zero and recovers the plane stress constitutive relations in the thin shell limit. On the computational side, a configuration update procedure for the director field is presented which is *singularity free* and *exact* regardless of the magnitude of the director (rotation and thickness stretch) increment. The performance of the shell model is assessed through an extensive set of numerical examples.

## Keywords

Critical Load Director Field Momentum Balance Equation Multiplicative Decomposition Store Energy Function## Preview

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## References

- 1.Ahmad, S., B.M. Irons, and O.C. Zienkiewicz, [ 1970 ], “Analysis of thick and thin shell structures by curved finite elements,”
*Int. J. Num. Meth. Engng.*,**2**, pp. 419–451.CrossRefGoogle Scholar - 2.Antman, S.S., [ 1976a ], “Ordinary differential equations of nonlinear elasticity I: Foundations of the theories of non-linearly elastic rods and shell,”
*Arch. Rat. Mech. Anal*.**61**, Vol.4, p.307–351.MathSciNetMATHGoogle Scholar - 3.Antman, S.S., [ 1976b ], “Ordinary differential equations of nonlinear elasticity II: Existency and regularity theory for conservative boundary value problems,”
*Arch. Rat. Mech. Anal*.**61**, Vol.2, p.353–393.MathSciNetMATHGoogle Scholar - 4.Argyris, J.H., H. Balmer, J.St. Doltsinis, P.C. Dunne, M. Haase, M. Hasse, M. Kleiber, G.A. Malejannakis, J.P. Mlejenek, M. Muller, and D.W. Scharpf, [ 1979 ], “Finite Element Method — The Natural Approach”,
*Comput. Meths. Appl. Mech. Engng.*,**17/18**,1 106.Google Scholar - 5.Bathe, K.J., and E.N. Dvorkin, [ 1984 ], “A continuum mechanics based fournode shell element for general non-linear analysis,”
*Int. J. Computer-Aided Engng. Software*, Vol. 1.Google Scholar - 6.Belytschko, T., H. Stolarski, W.K. Liu, N. Carpenter, & J. S-J. Ong, [ 1985 ], “Stress Projection for Membrane and Shear Locking in Shell Finite elements,”
*Comp. Meth. Appl. Mech. Engng.*, Vol 51, pp 221–258.MathSciNetMATHCrossRefGoogle Scholar - 7.Bergan, P.G., Horrigmoe, G., Krakeland, B., and Soreide, T.H., [ 1978 ], “Solution techniques for nonlinear Finite Element problems,”
*Int. J. Num. Meth. Engng.*,**12**, pp. 1677–1696.MATHCrossRefGoogle Scholar - 8.Bushnell, D., [ 1985 ],
*Computerized Buckling Analysis of Shells*, Mechanics of Elastic Stability, Vol. 9, Martinus Nijoff Publishers, Boston.Google Scholar - 9.Ericksen, J.L., and Truesdell, C., [ 1958 ], “Exact theory of stress and strain in rods and shells,”
*Arch. Rat. Mech. Anal.*, Vol. 1, No. 4, pp. 295–323.MathSciNetMATHGoogle Scholar - 10.Hoff, N.J., and T.C. Soong [ 1965 ], “Buckling of Circular Cylindrical Shells in Axial Compression,”
*Int. Journ. Mech. Sci.*,**7**, pp. 489–520.CrossRefGoogle Scholar - 11.Horrigmoe, G., [ 1977 ], “Finite Element Instability Analysis of Free-Form Shells,” Report 77–2,
*Division of Structural Mechanics, Norwegian Institute of Technology, University of Trondheim, Norway*.Google Scholar - 12.Hughes, T.J.R., and W.K. Liu, [ 1981a ], “Nonlinear finite element analysis of shells: Part I-Three-dimensional shells,”
*Comp. Meth. Appl. Mech. Engng.*,**26**, 331–362.MATHCrossRefGoogle Scholar - 13.Hughes, T.J.R., and W.K. Liu, [ 1981b ], “Nonlinear finite element analysis of shells: Part II -Two-dimensional shells,”
*Comp. Meth. Appl. Mech. Engng.*,**27**, 167–182.MATHCrossRefGoogle Scholar - 14.Keller, H.B., [ 1977 ], “Numerical Solution of Bifurcation and Nonlinear eigenvalue problems,”
*Applications of Bifurcation Theory*, P. Rabinowitz, ed., Academic Press, New York, pp. 359–384.Google Scholar - 15.Naghdi, P.M., [ 1972 ], “The theory of shells,” in
*Handbuch der Physik*, Vol Via/2, Mechanics of Solids I I, C. Truesdell Ed., Springer-Verlag, Berlin.Google Scholar - 16.Parks, K.C., and G.M. Stanley, [ 1986 ], “A curved C° shell element based on assumed natural-coordinate strains,”
*J. Appl. Mech.*, Vol. 53, No. 2, pp. 278–290.CrossRefGoogle Scholar - 17.Reissner, E., [ 1964 ], “On the Form of Variationally Derived Shell Equations,”
*J. Appl. Mech.*, Vol. 31, pp. 233–238.MathSciNetMATHCrossRefGoogle Scholar - 18.Reissner, E., [ 1974 ], “Linear and Nonlinear Theories of Shells, in
*Sechler Anniversary Volume*, pp. 29–44, Prentice Hall, New York.Google Scholar - 19.Rheinboldt, W.C., [ 1974 ], “Methods for Solving Systems of Nonlinear Equations,”
*CBMS Regional Conference Series in Applied Mathematics*, 14, Society for Industrial and Applied Mathematics, Philadelphia.Google Scholar - 20.Rheinboldt, W.C., [ 1986 ],
*Numerical Analysis of Parametrized Nonlinear Equations*, Wiley Interscience, NewYork.MATHGoogle Scholar - 21.Schweizerhof, K.H., and P. Wriggers, [ 1986 ], “Consistent Linearization of Path Following Methods in Nonlinear FE Analysis,”
*Comp. Meth. Appl. Mech. Engng.*,**59**, 261–279MATHCrossRefGoogle Scholar - 22.Simo, J.C. and D.D. Fox, [ 1989 ], “On a Stress Resultant Geometrically Exact Shell Model. Part I: Formulation and Optimal Parametrization,”
*Comp. Meth. Appl. Mech. Engng*,**72**, 267–304.MathSciNetMATHCrossRefGoogle Scholar - 23.Simo, J.C., D.D. Fox and M.S. Rifai, [ 1989 ], “On a Stress Resultant Geometrically Exact Shell Model. Part II: The Linear Theory; Computational Aspects,”
*Comp. Meth. Appl. Mech. Engng*, to appear.Google Scholar - 24.Simo, J.C., D.D. Fox and M.S. Rifai, [ 1989 ], “On a Stress Resultant Geometrically Exact Shell Model. Part III: Computational Aspects of the Nonlinear Theory,”
*Comp. Meth. Appl. Mech. Engng*, to appear.Google Scholar - 25.Simo, J.C. and J.G. Kennedy, [ 1989 ], “On a Stress Resultant Geometrically Exact Shell Model. Part IV: Nonlinear Plasticity. Formulation and Integration Algorithms,”
*Comp. Meth. Appl. Mech. Engng*, to appear.Google Scholar - 26.Simo, J.C., M.S. Rifai and D.D. Fox, [ 1989 ], “On a Stress Resultant Geometrically Exact Shell Model. Part V: Variable Thickness Shells with Throughthe-Thickness Stretching,”
*Comp. Meth. Appl. Mech. Engng*, to appear.Google Scholar - 27.Simo, J.C. and L.V. Quoc, [ 1986 ], “A 3-Dimensional Finite Strain Rod Model. Part II: Geometric and Computational Aspects,”
*Comp. Meth. Appl. Mech. Engng.*,**58**, 79–116.MATHCrossRefGoogle Scholar - 28.Simo, J.C., and L. Vu-Quoc, [ 1987a ], “A beam model including shear and torsional warping distorsions based on an exact geometric description of nonlinear deformations,”
*Int. J. Solids Structures*, Submitted for publication.Google Scholar - 29.Simo, J.C. and L. Vu-Quoc, [ 1987b ], “On the dynamics in Space of rods undergoing large motions -A geometrically exact approach,”
*Comp. Meth. Appl. Mech. Engng.*, To appear.Google Scholar - 30.Simo, J.C., P. Wriggers, K.H. Schweizerhoff and R.L. Taylor, [ 1986 ], “Post-buckling Analysis Involving Inelasticity and Unilateral Constraints,”
*Int. J. Num. Meth. Engng*,**23**, 779–800.MATHCrossRefGoogle Scholar - 31.Taylor, R.L., [ 1987 ], “Finite Element Analysis of Linear Shell Problems,”
*Proceedings of the Mathematics of Finite Elements and Applications*, (MAFELAP 1987 ), S.R. Whitheman Editor.Google Scholar - 32.Timoshenko S.P., and J.M. Gere, [ 1961 ],
*Theory of Elastic Stability*, Mc-Graw Hill, New York.Google Scholar - 33.Wriggers, P., and J.C. Simo, [ 1989 ]. “A General Purpose Algorithm for Extended Systems in Continuation Methods,” Preprint.Google Scholar
- 34.Wriggers, P., P. Wagner, and C. Miehe. [ 1988 ], “A Quadratic Convergent Procedure for the Calculation of Stability Points in Finite Element Analysis,”
*Comp. Meth. Appl. Mech. Engng*, to appear.Google Scholar