Invariant formulaiton of hyperelastic transverse isotropy based on polyconvex free energy functions

International Journal of Solids and Structures

Volumn 40, Issue 2, 2003, Pages 401-445

  Schröder, Jörg ^a   Neff, Patrizio ^b  

^a UNIVERSITY OF DUISBURG ESSEN   (Germany)

^b DARMSTADT UNIVERSITY OF TECHNOLOGY   (Germany)

Author keywords

Anisotropy; Existence of minimizers; Non linear elasticity; Polyconvexity

Indexed keywords

ANISOTROPY; ELASTICITY; MATHEMATICAL MODELS; STRESSES; TENSORS;

POLYCONVEX ANISOTROPIC HYPERELASTICITY;

FREE ENERGY;

ENERGY;

EID: 0037245463     PISSN: 00207683     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0020-7683(02)00458-4     Document Type: Article

References (47)

1. Ball, J.M., 1977a. Convexity conditions and existence theorems in non-linear elasticity. Archive of Rational Mechanics and Analysis 63, 337-403.
2. Ball, J.M., 1977b. Constitutive equalities and existence theorems in elasticity. In: Knops, R.J. (Ed.), Symposium on Non-Well Posed Problems and Logarithmic Convexity. Springer, Lecture Notes in Math. 316.
3. Betten, J., 1987. Formulation of anisotropic constitutive equations. In: Boehler J.P. (Ed.), Applications of Tensor Functions in Solid Mechanics, CISM Course no. 292. Springer-Verlag.
4. Boehler, J.P., 1979. A simple derivation of representations for non-polynomial constitutive equations in some cases of anisotropy. Zeitschrift für angewandte Mathematik und Mechanik 59, 157-167.
5. Boehler, J.P., 1987. Introduction to the invariant formulation of anisotropic constitutive equations. In: Boehler J.P. (Ed.), Applications of Tensor Functions in Solid Mechanics, CISM Course no. 292. Springer-Verlag.
6. Ciarlet, P.G., 1988. Three Dimensional Elasticity. In: Mathematical Elasticity, vol. 1. Elsevier Science Publishers BV, North-Holland.
7. Criscione, J.C., Douglas, A.S., Hunter, W.C., 2001. Physically based strain invariant set for materials exhibiting transversely isotropic behavior. Journal of the Mechanics and Physics of Solids 49, 871-897.
8. Dacorogna, B., 1989. Direct methods in the calculus of variations. Applied Mathematical Science 78. Springer-Verlag.
9. Hackl, K., 1999. On the representation of anisotropic elastic materials by symmetric irreducible tensors. Continuum Mechanics and Thermodynamics 11, 353-369.
10. Hartmann, S., Ne., P., 2002. Existence theory for a modiffied hyperelastic relation of generalized polynomial-type in the case of nearly incompressibility, Int J. Solids and Structures, submitted for publication.
11. Holzapfel, G.A., Gasser, Th.C., Ogden, R.W., 2000. A new constitutive framework for arterial wall mechanics and a comparative study of material models. Journal of Elasticity 61, 1-48.
12. Hughes, T.J.R., 1987. The Finite Element Method--Linear Static and Dynamic Finite Element Analysis. Prentice-Hall.
13. Itskov M. 2001. A generalized orthotropic hyperelastic material model with application to incompressible shells. International Journal for Numerical Methods in Engineering 50, 1777-1799.
14. Jones R.M. 1975. Mechanics of Composite Materials. McGraw-Hill Book Company.
15. Lürding, D., 2001. Modellierung grosser Deformationen in orthotropen, hyperelastischen Schalenstrukturen. Bericht Nr.: 125, Mitteilungen aus dem Institut für Mechanik, Ruhr-Universität Bochum.
16. Marsden, J.E., Hughes, J.R., 1983. Mathematical Foundations of Elasticity. Prentice-Hall.
17. Morrey, C.B., 1952. Quasi-convexity and the lower semicontinuity of multiple integrals. Pacific Journal of Mathematics 2, 25-53.
18. Miehe, C., 1994. Aspects of the formulation and finite element implementation of large strain isotropic elasticity. International Journal for Numerical Methods in Engineering 37, 1981-2004.
19. Ne., P., 2000. Mathematische Analyse multiplikativer Viskoplastizität. Dissertation im Fachbereich Mathematik, Technische Universität Darmstadt, D 17.
20. Raoult A. 1986. Non-polyconvexity of the stored energy function of a St.-Venant-Kirchho. material. Aplikace Matematiky 6, 417-419.
21. Rockafellar, T., 1970. Convex Analysis. Princeton University Press, Princeton.
22. Rüter, M., Stein, E., 2000. Analysis, finite element computation and error estimation in transversely isotropic nearly incompressible finite elasticity. Computer Methods in Applied Mechanics and Engineering 190, 519-541.
23. Rychlewski J. Zhang J.M. 1991. On representation of tensor functions: a review. Advances in Mechanics 14, 75-94.
24. Schröder, J., 1996. Theoretische und algorithmische Konzepte zur phänomenologischen Beschreibung anisotropen Materialverhaltens. Bericht Nr.: I-1, Institut für Mechanik (Bauwesen), Lehrstuhl I, Universität Stuttgart.
25. Schröder, J., Ne., P., 2001. On the construction of polyconvex transversely isotropic free energy functions. In: Miehe, C. (Ed.), Proceedings of the IUTAM Symposium on Computational Mechanics of Solid Materials at Large Strains. Kluwer Academic Publishers, submitted for publication.
26. Silhavý, M., 1977. The Mechanics and Thermodynamics of Continuous Media. Springer-Verlag.
27. Smith, G.F., 1965. On isotropic integrity bases. Archive for Rational Mechanics and Analysis 18, 282-292.
28. Smith, G.F., 1970. On a fundamental error in two papers of C.C. Wang. Archive for Rational Mechanics and Analysis 36, 161-165.
29. Smith, G.F., 1971. On isotropic functions of symmetric tensors, skew-symmetric tensors and vectors. International Journal of Engineering Science 19, 899-916.
30. Smith, G.F., Rivlin, R.S., 1957. The anisotropic tensors. Quarterly of Applied Mathematics 15, 309-314.
31. Smith, G.F., Rivlin, R.S., 1958. The strain-energy function for anisotropic elastic materials. Transactions of the American Mathematical Society 88, 175-193.
32. Smith, G.F., Smith, M.M., Rivlin, R.S., 1963. Integrity basis for a symmetric tensor and a vector. The crystal classes. Archive for Rational Mechanics and Analysis, 93-133.
33. Spencer, A.J.M., 1971. Theory of invariants. In: Eringen, A.C. (Ed.), Continuum Physics, vol. 1. Academic Press, New York, pp. 239-353.
34. Spencer, A.J.M., 1987. Isotropic polynomial invariants and tensor functions. In: Boehler, J.P. (Ed.), Applications of Tensor Functions in Solid Mechanics, CISM Course no. 292. Springer-Verlag.
35. Spencer, A.J.M., 1987b. Kinematic constraints, constitutive equations and failure rules for anisotropic materials. In: Boehler, J.P. (Ed.), Applications of Tensor Functions in Solid Mechanics, CISM Course no. 292. Springer-Verlag.
36. Ting, T.C.T., 1996. Anisotropic Elasticity-Theory and Applications. Oxford Science Publications.
37. Truesdell, C., Noll, W., 1965. The nonlinear .eld theories of mechanics. In: Flügge, S. (Ed.), Handbuch der Physik Bd.III/3. Springer.
38. Wang, C.C., 1969a. On representations for isotropic functions: Part I. Isotropic functions of symmetric tensors and vectors. Archive of Rational Mechanics and Analysis 33, 249-267.
39. Wang, C.C., 1969b. On representations for isotropic functions: Part II. Isotropic functions of skew-symmetric tensors, symmetric tensors and vectors. Archive of Rational Mechanics and Analysis 33, 268-287.
40. Wang, C.C., 1970. A new representation theorem for isotropic functions: an answer to professor G.F. Smith's criticism of my papers on representations for isotropic functions. Archive of Rational Mechanics and Analysis 33, 249-287.
41. Wang, C.C., 1971. Corrigendum to my recent papers on: representations for isotropic functions. Archive of Rational Mechanics and Analysis 43, 392-395.
42. Weiss, J.A., Maker, B.N., Govindjee, S., 1996. Finite element implementation of incompressible, transversely isotropic hyperelasticity. Computer Methods in Applied Mechanics and Engineering 135, 107-128.
43. Weyl, H., 1946. The Classical Groups, Their Invariants and Representation. Princeton Univ. Press, Princeton New Jersey.
44. Zheng, Q.S., Spencer, A.J.M., 1993a. On the canonical representations for Kronecker powers of orthogonal tensors with application to material symmetry problems. International Journal of Engineering Science 31 (4), 617-635.
45. Zheng, Q.S., Spencer, A.J.M., 1993b. Tensors which characterize anisotropies. International Journal of Engineering Science 31 (5), 679-693.
46. Zheng, Q.S., Boehler, J.P., 1994. The description, classi.cation, and reality of material and physical symmetries. Acta Mechanica 102, 73-89.
47. Zienkiewicz, O.C., Taylor, R.L., 2000. The Finite Element Method, vols. I, II. Butterworth-Heinemann.