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An extension of Key's principle to nonlinear elasticity

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Abstract

A variational principle for finite isothermal deformations of anisotropic compressible and nearly incompressible hyperelastic materials is presented. It is equivalent to the nonlinear elastic field (Lagrangian) equations expressed in terms of the displacement field and a scalar function associated with the hydrostatic mean stress. The formulation for incompressible materials is recovered from the compressible one simply as a limit. The principle is particularly useful in the development of finite element analysis of nearly incompressible and of incompressible materials and is general in the sense that it uses a general form of constitutive equation. It can be considered as an extension of Key's principle to nonlinear elasticity. Various numerical implementations are used to illustrate the efficiency of the proposed formulation and to show the convergence behaviour for different types of elements. These numerical tests suggest that the formulation gives results which change smoothly as the material varies from compressible to incompressible.

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References

  1. I. Fried, Influence of Poisson's ratio on the condition of the finite elements stiffness matrix. Int. J. Sol. Struct. 9 (1973) 323–329.

    Google Scholar 

  2. M. D. Gunzburger, Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory and Practice, and Algorithms, London: Academic Press (1989) 269 pp.

    Google Scholar 

  3. T. J. R. Hughes, The Finite Element Method-Linear Static and Dynamic Finite Element Analysis, Englewood Cliffs, NJ: Prentice-Hall (1987) 803 pp.

    Google Scholar 

  4. I. Fried, Finite element analysis of incompressible material by residual energy balancing. Int. J. Sol. Struct. 10 (1974) 993–1002.

    Google Scholar 

  5. T. Scharnorst and T. H. H. Pian, Finite element analysis of rubber-like materials by a mixed model. Int. J. Num. Meth. Engng. 12 (1978) 655–676.

    Google Scholar 

  6. D. S. Malkus and T. J. R. Hughes, Mixed finite element methods-Reduced and selective integration techniques: A unification of concepts. Comp. Meth. Appl. Mech. Engng. 15 (1978) 63–81.

    Google Scholar 

  7. S. Cescotto and G. Fonder, A finite element approach for large strains of nearly incompressible rubber-like materials. Int. J. Sol. Struct. 15 (1979) 589–605.

    Google Scholar 

  8. M. H. B. M. Shariff, A constrained element-by-element method for slightly compressible and incompressible materials and its parallel implementation on transputers. In: B. H. V. Topping and M. Papadrakakis (eds), Advances in Parallel and Vector Processing for Structural Mechanics. Edinburgh: Civil-Comp. Press (1994) pp. 53–57.

    Google Scholar 

  9. M. Bercovier, Y. Hasbani, Y. Gilon and K. J. Bathe, On a finite element procedure for non-linear incompressible elasticity. In: S. N. Atluri, R. H. Gallagher and O. C. Zienkiewicz (eds), Hybrid and Mixed Element Methods. New York: Wiley-Interscience (1983) pp. 497–518.

    Google Scholar 

  10. M. H. B. M. Shariff, An extension of Herrmann's principle to nonlinear elasticity. Appl. Math. Mod. 21 (1997) 97–107.

    Google Scholar 

  11. C. Miehe, Aspects of the formulation and finite element implementation of large strain isotropic elasticity. Int. J. Numer. Methods Eng. 37 (1994) 1981–2004.

    Google Scholar 

  12. A. G. K. Jinka and R. L. Lewis, Incompressibility and axisymmetry: A modified mixed and penalty formulation. Int. J. Num. Meth. Engng. 37 (1994) 1623–1649.

    Google Scholar 

  13. U. Hueck and H. L. Schreyer, The use of orthogonal projections to handle constraints with applications to incompressible four-node quadrilateral elements. Int. J. Num. Meth. Engng. 35 (1992) 1633–1661.

    Google Scholar 

  14. M. H. B. M. Shariff, Linear finite-element equations for nonlinear deformation of slightly compressible elastic material. Q. J. Mech. Appl. Math. 45 (1992) 169–181.

    Google Scholar 

  15. A. J. M. Spencer, Finite deformation of an almost incompressible elastic solid. In: M. Reiner and D. Abir (eds), Second-order Effects in Elasticity, Plasticity and Fluid Dynamics. Oxford: Pergamon Press (1964) pp. 200–216.

    Google Scholar 

  16. S. V. Tsinopoulos, D. Polyzos and D. E. Beskos, BEM formulation for (3D) elastic incompressible solids. In: B. H. V. Topping (ed.), Advances in Boundary Element Methods. Edinburgh: Civil-Comp. Press (1996) pp. 1–8.

    Google Scholar 

  17. R. T. Fenner and R. M. Remzi, A boundary integral equation of compression moduli of bonded rubber blocks. J. Strain Anal. 18 (1983) 217–223.

    Google Scholar 

  18. N. Phan-Tien, BEM for finite-elasticity. Comp. Mech. 6 (1990) 205–219.

    Google Scholar 

  19. I. Babuska and M. Suri, Locking effects in the finite element approximation of elasticity problems. Num. Math. 62 (1992) 439–463.

    Google Scholar 

  20. PERFINE-The Manual, eds Barrett et al. University of Coventry, UK (1989).

    Google Scholar 

  21. M. H. B. M. Shariff, A modified augmented Lagrangian method for nearly incompressible problems, In: B. H. V. Topping (ed.), Developments in Computational Mechanics with High Performance Computing. Edinburgh: Civil-Comp. Press (1999) pp. 101–106.

    Google Scholar 

  22. R. W. Ogden, Non-linear Elastic Deformations. Chichester: Ellis Horwood (1984) 532 pp.

    Google Scholar 

  23. S. W. Key, A variational principle for incompressible and nearly incompressible anisotropic elasticity. Int. J. Sol. Struct. 5 (1969) 951–964.

    Google Scholar 

  24. D. F. Parker, Finite-amplitude oscillations of a spherical cavity in a nearly-incompressible elastic material. Arch. of Mech. 30 (1978) 777–790.

    Google Scholar 

  25. N. H. Scott, The slowness surfaces of incompressible and nearly incompressible elastic materials. J. Elasticity 16 (1986) 239–250.

    Google Scholar 

  26. Q. Niu and M. Shepard, Superconvergent boundary stress extraction and some experiments with adaptive pointwise error control. Int. J. Num. Meth. Engng. 37 (1994) 877–891.

    Google Scholar 

  27. L. R. Herrmann and R. M. Toms, A reformulation of the elastic field equations, in terms of displacements, valid for all admissible values of Poisson's ratio. J. Appl. Mech. 31 (1964) 148–149.

    Google Scholar 

  28. L. R. Herrmann, Elasticity equations for incompressible and nearly incompressible materials by a variational theorem, AIAA J. 3 (1965) 1896–1900.

    Google Scholar 

  29. P. J. Blatz, Finite elastic deformation of a plane strain wedge-shaped radial crack in a compressible cylinder. In: M. Reiner and D. Abir (eds), Second-order Effects in Elasticity, Plasticity and Fluid Dynamics. Oxford: Pergamon Press (1964) pp. 145–161.

    Google Scholar 

  30. A. E. Green and W. Zerna, Theoretical Elasticity, 2nd edition. Oxford: Clarendon Press (1968) 442 pp.

    Google Scholar 

  31. J. T. Oden, Finite Elements of Nonlinear Continua. New York: McGraw-Hill (1972) 432 pp.

    Google Scholar 

  32. M. H. B. M. Shariff, Parallel finite element software for nonlinear stress problems. In: M. Valero et al. (eds), Parallel Computing and Transputer Applications. Amsterdam: IOS Press (1992) pp. 1261–1270.

    Google Scholar 

  33. M. H. B. M. Shariff, A general approach to axial deformation of bonded elastic mounts of various cross-sectional shapes. Appl. Math. Mod. 17 (1993) 430–436.

    Google Scholar 

  34. A. N. Gent and P. B. Lindley, Compression of bonded rubber blocks. Proc. Inst. Mech. Engrs. 173 (1959) 111–117.

    Google Scholar 

  35. R. W. Ogden, Volume changes associated with the deformation of rubber-like solids. J. Mech. Phys. Solids 24 (1976) 323–338.

    Article  Google Scholar 

  36. M. A. Crisfield, A fast incremental/iteration solution procedure that handles 'snap-through'. Comp. and Struc. 13 (1981) 55–62.

    Google Scholar 

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Authors and Affiliations

  1. School of Computing and Mathematics, University of Teesside, Middlesborough, Cleveland, TS1 3BA, U.K.

    M. H. B. M. Shariff

  2. Department of Mathematics and Statistics, University of Edinburgh, The King's Buildings, Edinburgh, EH9 3JZ, U.K.

    D. F. Parker

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  1. M. H. B. M. Shariff
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  2. D. F. Parker
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Shariff, M.H.B.M., Parker, D.F. An extension of Key's principle to nonlinear elasticity. Journal of Engineering Mathematics 37, 171–190 (2000). https://doi.org/10.1023/A:1004734311626

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