Abstract
We compare some mixed methods based on different variational formulations, namely a displacement-pressure formulation employed by de Borst and coworkers, the three-field formulation investigated by Simo and Taylor and a two-field formulation which is directly based on an energy functional. It emerges that all these yield the same discrete results if the stored energy function contains a volumetric contribution 1/2k(J−1)2 whereJ is the volume dilatation, i.e., the Jacobian determinant of the deformation, andk is the bulk modulus. The equivalence holds for arbitrary 3D and plane strain elements. In the numerical examples the mixed formulations are discretized by the quadrilateral Q1/P0 and Q2/P1 elements and the triangular Crouzeix-Raviart P2+/P1 element. We also compare with standard displacement elements and the enhanced strain Q1/E4 element.
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References
Atluri, S. N.;Reissner, E. (1989): On the formulation of variational theorems involving volume constraints. Comput. Mech. 5: 337–344.
van den Bogert, P. A. J.;de Borst, R. (1994): On the behaviour of rubberlike materials in compression and shear. Arch. Appl. Mech. 64: 136–146
van den Bogert, P. A. J.;de Borst, R.;Luiten, G. T.;Zeilmaker, J. (1991): Robust finite elements for 3D-analysis of rubber-like materials. Engng. Comput. 8: 3–17
de Borst, R.;van den Bogert, P. A. J.;Zeilmaker, J. (1988): Modelling and analysis of rubberlike materials. HERON 33: 1–57
Brezzi, F. (1974): On the existence, uniqueness and approximation of saddlepoint problems arising from Lagrangian multipliers. RAIRO Anal. Numer. 8: 129–151
Brezzi, F.;Fortin, M. (1991): Mixed and Hybrid Finite Element Methods. Berlin, Heidelberg, New York: Springer
Chang, T. Y. P.;Saleeb, A. F.;Li, G. (1991): Large strain analysis of rubber-like materials based on a perturbed Lagrangian variational principle. Comput. Mech. 8: 221–233
Chapelle, D.;Bathe, K.-J. (1993): The inf-sup test. Comp. & Struct. 47: 537–545
Chen, J. S.;Satyamurthy, K.;Hirschfelt, L. R. (1994): Consistent finite element procedures for nonlinear rubber elasticity with a higher order strain energy function. Comp. & Struct. 50: 715–727
Crouzeix, M.;Raviart, P.-A. (1973): Conforming and non conforming finite element methods for solving the stationary Stokes equations. RAIRO R3: 33–76
Engelman, M. S.;Sani, R. L.;Gresho, P. M.;Bercovier, M. (1982): Consistent vs. reduced integration penalty methods for incompressible media using several old and new elements. Int. J. Numer. Meth. Fluids, 2: 25–42
Fortin, M. (1981): Old and new finite elements for incompressible flows. Int. J. Numer. Meth. Fluids. 1: 347–364
Girault, V.;Raviart, P.-A. (1986): Finite Element Methods for Navier-Stokes Equations. Berlin, Heidelberg, New York: Springer
Le Tallec, P. (1981): Compatibility condition and existence results in discrete finite incompressible elasticity. Comp. Meth. Appl. Mech. Engrg. 27: 239–259
Le Tallec, P. (1994): Numerical methods for nonlinear three-dimensional elasticity. In: Ciarlet, P. G.; Lions, J. L. (eds.): Handbook of Numerical Analysis, vol. 3, Amsterdam: Elsevier
Liu, C. H.;Hofstetter, G.;Mang, H. A. (1994): 3D finite element analysis of rubber-like materials at finite strains. Engng. Comput. 11: 111–128
Miehe, C. (1993): Computation of isotropic tensor functions. Comm. Numer. Meth. Engrg. 9: 889–896
Miehe, C. (1994): Aspects of the formulation and finite element implementation of large strain isotropic elasticity. Int. J. Numer. Meth. Engrg 37: 1981–2004
Ogden, R. W. (1984): Non-Linear Elastic Deformations. Chicester: Ellis Horwood-John Wiley
Reddy, B. D.;Simo, J. C. (1995): Stability and convergence of a class of enhanced strain methods. SIAM J. Numer. Anal. 32: 1705–1728
Seki, W.;Atluri, S. N. (1994): Analysis of strain localization in strainsoftening hyperelastic materials, using assumed stress hybrid elements. Comput. Mech. 14: 549–585
Seki, W.;Atluri, S. N. (1995): On newly developed assumed stress finite element formulations for geometrically and materially nonlinear problems. Finite Elements in Analysis and Design 21: 75–110
Simo, J.;Armero, F. (1992): Geometrically nonlinear enhanced strain mixed methods and the method of incompatible modes. Int. J. Numer. Meth. Engrg. 33: 1413–1449
Simo, J.;Armero, F.;Taylor, R. L. (1993): Improved versions of assumed enhanced strain tri-linear elements for 3D finite deformation problems. Comp. Meth. Appl. Mech. Engrg. 110: 359–386
Simo, J.;Taylor, R. L. (1991): Quasi-incompressible finite elasticity in principle stretches. Continuum basis and numerical algorithms. Comp. Meth. Appl. Mech. Engrg. 85: 273–310
Simo, J.;Taylor, R. L.;Pister, K. S. (1985): Variational and projection methods for the volume constraint in finite deformation elasto-plasticity. Comp. Meth. Appl. Mech. Engrg. 51: 177–208
Stein, E.;Müller-Hoeppe, N. (1987): Finite element analysis and algorithms for large elastic strains. In: Pande, G. N.; Middleton, J. (eds.): Numerical Techniques for Engineering Analysis and Design: Proceedings of NUMETA 87, D4, pp. 1–8 Dordrecht: Nijhoff Publ.
Sussman, T.;Bathe, K.-J. (1987): A finite element formulation for nonlinear incompressible elastic and inelastic analysis. Comp. & Struct. 26: 357–409.
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Communicated by S. N. Atluri, 2 August, 1996
This work was supported by the German Research Foundation (DFG) under Grant No. Ste 238/35-1.
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Brink, U., Stein, E. On some mixed finite element methods for incompressible and nearly incompressible finite elasticity. Computational Mechanics 19, 105–119 (1996). https://doi.org/10.1007/BF02824849
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DOI: https://doi.org/10.1007/BF02824849