×

zbMATH — the first resource for mathematics

F-bar-based linear triangles and tetrahedra for finite strain analysis of nearly incompressible solids. I: formulation and benchmarking. (English) Zbl 1179.74159
Summary: This paper proposes a new technique which allows the use of simplex finite elements (linear triangles in 2D and linear tetrahedra in 3D) in the large strain analysis of nearly incompressible solids. The new technique extends the F-bar method proposed by E. A. de Souza Neto et al. [Int. J. Solids Struct. 33, No. 20–22, 3277–3296 (1996; Zbl 0929.74102)] and is conceptually very simple: It relies on the enforcement of (near-) incompressibility over a patch of simplex elements (rather than the point-wise enforcement of conventional displacement-based finite elements). Within the framework of the F-bar method, this is achieved by assuming, for each element of a mesh, a modified (F-bar) deformation gradient whose volumetric component is defined as the volume change ratio of a pre-defined patch of elements. The resulting constraint relaxation effectively overcomes volumetric locking and allows the successful use of simplex elements under finite strain near-incompressibility. As the original F-bar procedure, the present methodology preserves the displacement-based structure of the finite element equations as well as the strain-driven format of standard algorithms for numerical integration of path-dependent constitutive equations and can be used regardless of the constitutive model adopted. The new elements are implemented within an implicit quasi-static environment. In this context, a closed form expression for the exact tangent stiffness of the new elements is derived. This allows the use of the full Newton-Raphson scheme for equilibrium iterations. The performance of the proposed elements is assessed by means of a comprehensive set of benchmarking two- and three-dimensional numerical examples.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74B20 Nonlinear elasticity
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Simo, Computer Methods in Applied Mechanics and Engineering 51 pp 177– (1985)
[2] Sussman, Computers and Structures 26 pp 357– (1987)
[3] Finite element formulation and solution of contact-impact problems in Continuum Mechanics-III. Report No. UC SESM 75-7, Department of Civil Engineering University of California, Berkeley, July 1975.
[4] Moran, International Journal for Numerical Methods in Engineering 29 pp 483– (1990)
[5] Simo, International Journal for Numerical Methods in Engineering 33 pp 1413– (1992)
[6] de Souza Neto, International Journal of Solids and Structures 33 pp 3277– (1996)
[7] Zienkiewicz, International Journal for Numerical Methods in Engineering 43 pp 565– (1998)
[8] Bonet, Communications in Numerical Methods in Engineering 14 pp 437– (1998)
[9] Bonet, Communications in Numerical Methods in Engineering 17 pp 551– (2001)
[10] Taylor, International Journal for Numerical Methods in Engineering 47 pp 205– (2000)
[11] Guo, International Journal for Numerical Methods in Engineering 47 pp 287– (2000)
[12] Thoutireddy, International Journal for Numerical Methods in Engineering 53 pp 1337– (2002)
[13] Mathematical Foundations of Elasticity. Prentice-Hall: Englewood Cliffs, NJ, 1983. · Zbl 0545.73031
[14] Simo, International Journal for Numerical Methods in Engineering 29 pp 1595– (1990)
[15] Korelc, Engineering Computations 13 pp 103– (1996)
[16] Glaser, Engineering Computations 14 pp 759– (1997)
[17] Enhanced lower-order element formulations for large strains. In Computational Plasticity: Fundamentals and Applications, Proceedings of the Fourth International Conference held in Barcelona, (eds). 3rd-6th April 1995. Pineridge Press: Swansea, 1995; 293-320. · Zbl 0840.73059
[18] Non-Linear Elastic Deformations. Ellis Horwood: Chichester, 1984.
[19] de Souza Neto, Communications in Numerical Methods in Engineering 11 pp 951– (1995)
[20] Tvergaard, Journal of Mechanics and Physics of Solids 29 pp 115– (1981)
[21] Simo, Computer Methods in Applied Mechanics and Engineering 85 pp 273– (1991)
[22] Peri?, International Journal for Numerical Methods in Engineering 35 pp 1289– (1992)
[23] Nagtegaal, Computer Methods in Applied Mechanics and Engineering 4 pp 153– (1974)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.