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An assessment of the average nodal volume formulation for the analysis of nearly incompressible solids under finite strains. (English) Zbl 1302.74173
Summary: This paper provides an assessment of the average nodal volume methodology originally proposed by Bonet and Burton (Commun. Numer. Meth. Engng. 1998; 14:437-449) for the analysis of finitely strained nearly incompressible solids. An implicit version of the average nodal pressure formulation is derived by re-casting the original concept in terms of average nodal volume change ratio within the framework of the F-bar method proposed by de Souza Neto et al. (Int. J. Solids Struct. 1996; 33: 3277-3296). In this context, a linear triangle for implicit plane strain and axisymmetric analysis of nearly incompressible solids under finite strains is obtained. An exact expression for the corresponding element stiffness matrix is presented. This allows the use of the full Newton-Raphson algorithm, ensuring quadratic rates of asymptotic convergence in the global equilibrium iterations. The performance of the procedure is thoroughly assessed by means of numerical examples. The results show that the nodal averaging technique substantially reduces the volumetric locking tendency of the linear triangle and allows an accurate prediction of deformed shapes and reaction forces in situations of practical interest. However, the formulation is found to produce considerable checkerboard-type hydrostatic pressure fluctuations which poses severe limitations on its range of applicability.

MSC:
74S10 Finite volume methods applied to problems in solid mechanics
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[1] Taylor, A non-conforming element for stress analysis, International Journal for Numerical Methods in Engineering 10 pp 1211– (1976) · Zbl 0338.73041
[2] Hughes, Generalization of selective integration procedures to anisotropic and nonlinear media, International Journal for Numerical Methods in Engineering 15 pp 1413– (1980) · Zbl 0437.73053
[3] Simo, Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes, International Journal for Numerical Methods in Engineering 33 pp 1413– (1992) · Zbl 0768.73082
[4] Moran, Formulation of implicit finite element methods for multiplicative finite deformation plasticity, International Journal for Numerical Methods in Engineering 29 pp 483– (1990) · Zbl 0724.73221
[5] Sussman, A finite element formulation for nonlinear incompressible elastic and inelastic analysis, Computers and Structures 26 pp 357– (1987) · Zbl 0609.73073
[6] de Souza Neto, Design of simple low order finite elements for large strain analysis of nearly incompressible solids, International Journal of Solids and Structures 33 pp 3277– (1996) · Zbl 0929.74102
[7] Zienkiewicz, Triangles and tetrahedra in explicit dynamic codes for solids, International Journal for Numerical Methods in Engineering 43 pp 565– (1998) · Zbl 0939.74073
[8] Bonet, A simple average nodal pressure tetrahedral element for incompressible and nearly incompressible dynamic explicit applications, Communications in Numerical Methods in Engineering 14 pp 437– · Zbl 0906.73060
[9] Bonet, An averaged nodal deformation gradient linear tetrahedral element for large strain explicit dynamic applications, Communications in Numerical Methods in Engineering 17 pp 551– · Zbl 1154.74307
[10] Taylor, A mixed-enhanced formulation for tetrahedral finite elements, International Journal for Numerical Methods in Engineering 47 pp 205– (2000) · Zbl 0985.74074
[11] Simo, Computational Inelasticity (1998)
[12] Marsden, Mathematical Foundations of Elasticity (1983) · Zbl 0545.73031
[13] Simo, A class of mixed assumed strain methods and the method of incompatible modes, International Journal for Numerical Methods in Engineering 29 pp 1595– (1990) · Zbl 0724.73222
[14] Korelc, Consistent gradient formulation for a stable enhanced strain method for large deformations, Engineering Computations 13 (1) pp 103– (1996)
[15] Glaser, On the formulation of enhanced strain finite element methods in finite deformations, Engineering Computations 14 (7) pp 759– (1997) · Zbl 1071.74699
[16] Crisfield, Computational Plasticity: Fundamentals and Applications, Proceedings of the Fourth International Conference held in Barcelona pp 293– (1995)
[17] Ogden, Non-Linear Elastic Deformations (1984) · Zbl 0541.73044
[18] Nagtegaal, On numerically accurate finite element solutions in the fully plastic range, Computer Methods in Applied Mechanics and Engineering 4 pp 153– (1974) · Zbl 0284.73048
[19] Tvergaard, Flow localization in the plane strain tensile test, Journal of the Mechanics and Physics of Solids 29 pp 115– (1981) · Zbl 0462.73082
[20] Lemaitre, How to use damage mechanics, Nuclear Engineering Design 80 pp 233– (1984)
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