Abstract
This paper is a critical review of the so-called polar method, a mathematical technique based upon tensor invariants introduced as early as 1979 by Professor G. Verchery to treat anisotropic plane problems. The paper contains a brief presentation of the method and a summary of some results obtained by the author and co-workers in some problems mainly concerning elasticity and the elastic design of composite laminates.
Similar content being viewed by others
References
S.W. Tsai N.J. Pagano (1968) ‘Invariant properties of composite materials’ S.W. Tsai J.C. Halpin N.J. Pagano (Eds) Composite Materials Workshop Technomic USA 233–253
Verchery, G., ‘Les invariants des tenseurs d’ordre 4 du type de l’élasticité’.Colloque Euromech 115, Villard-de-Lans, 1979. Published in 1982 by Editions du CNRS, Paris, pp. 93–104 (in French).
F. Klein (1896) The Mathematical Theory of the Top Princeton USA
J.H. Michell (1902) ArticleTitle‘The inversion of plane stress’ Proc. London Math. Soc. 34 134–142 Occurrence Handle33.0819
G.V. Kolosov (1909) ‘On an Application of Complex Function Theory to a Plane Problem of the Mathematical Theory of Elasticity’ Yuriev Russia
Muskhelishvili, N.J., Some Basic Problems of the Mathematical Theory of Elasticity. 1933. English translation by J.R.M. Radok, 1953. P. Noordhoff, Gröningen, NL, 1933.
A. Green W. Zerna (1954) Theoretical Elasticity Clarendon Press Oxford, UK Occurrence Handle0056.18205
P. Vannucci (2002) Thèse pour l‘Habilitation à Diriger les Recherches Université de Bourgogne Nevers
Wu, E.M., Jerina, K.L. and Lavengood, R.E., ‘Data averaging of anisotropic composite material constants’, in Analysis of Test Methods for High Modulus Fibers and Composites, ASTM SPT 521 229–252 (1973).
S.W. Tsai H.T. Hahn (1980) Introduction to Composite Materials Technomic USA
Verchery, G. and Vong, T.S., ‘Une méthode d’aide graphique à la conception des séquences d’empilement dans les stratifiés’, Comptes rendus de JNC5 (5èmes Journées Nationales sur les Composites), Paris, 1986 (in French), pp. 267–280.
Pedersen, P., ‘Lecture notes on optimal design using anisotropic, non-linear materials’, in: Cinquini, C. (ed), Workshop on Optimum Strucutral Design: Analytical Approach and Computer Aided Design, Università di Pavia, Italy, 1993.
P. Vannucci (2002) ArticleTitle‘A special planar orthotropic material’ J. Elasticity 67 81–96 Occurrence Handle10.1023/A:1023949729395 Occurrence Handle1089.74565 Occurrence Handle1985443
A. Vincenti G. Verchery P. Vannucci (2001) ArticleTitle‘Anisotropy and symmetries for elastic properties of laminates reinforced by balanced fabrics’ Composites Part A. 32 1525–1532 Occurrence Handle10.1016/S1359-835X(01)00052-5 Occurrence Handle000171098400016
R.M. Jones (1975) Mechanics of Composite Materials Mc Graw-Hill New York
P. Vannucci G. Verchery (2001) ArticleTitle‘Stiffness design of laminates using the polar method’ Int. J. Solids Str. 38 9281–9294 Occurrence Handle1016.74019
E. Valot P. Vannucci (2005) ArticleTitle‘Some exact solutions for fully orthotropic laminates’ Compos. Struct. 69 157–166 Occurrence Handle10.1016/j.compstruct.2004.06.007
P. Pedersen (1990) ArticleTitle‘Combining material and element rotation in one formula’ Comm. In Appl. Num. Meth. 6 549–555 Occurrence Handle0709.73077
P. Vannucci G. Verchery (2001) ArticleTitle‘A special class of uncoupled and quasi-homogeneous laminates’ Compos. Sci. Techn. 61 1465–1473 Occurrence Handle10.1016/S0266-3538(01)00039-2
P. Vannucci (2001) ArticleTitle‘On bending-tension coupling of laminates’ J. Elasticity. 64 13–28 Occurrence Handle10.1023/A:1014865428771 Occurrence Handle1022.74007
A. Vincenti P. Vannucci G. Verchery (2003) ArticleTitle‘Influence of orientation errors on quasi-homogeneity of composite laminates’ Compos. Sci. Technol. 63 739–749 Occurrence Handle10.1016/S0266-3538(02)00263-4
Vincenti, A., Vannucci, P. and Verchery, G., ‘Design of composite laminates as an optimization problem: a new genetic algorithm approach based upon tensor polar invariants’, Proc. of WCSMO 5 (Fifth World Congress on Structural and Multidisciplinary Optimization), Lido di Jesolo – Venice, Italy, 2003.
Vincenti, A., and Vannucci, P., ‘Tailoring expansion coefficients of laminates: a new general optimal approach based upon the polar-genetic method’, Proc. of WCSMO 6 (Sixth World Congress on Structural and Multidisciplinary Optimization), Rio de Janeiro, Brazil, 2005.
Ishikawa, T. and Fukunaga, H., ‘Taylored laminates with null or arbitrary coefficient of thermal expansion’, Proc. of Composites ‘86: Recent Advances in Japan and the USA, Tokyo, 1986.
R. Riche ParticleLe J. Gaudin (1998) ArticleTitle‘Design of of dimensionally stable composites by evolutionary optimization’ Composite Structures. 41 97–111
Fernandez, A., Vincenti, A., Vannucci, P. and Pouget, J., ‘Optimal design of composites including piezoelectric active layers’, Proc. of II ECCOMAS Thematic Conference on Smart Structures and Materials, Lisbon, Portugal, 2005.
Vincenti, A., `Conception et optimisation de composites par méthode polaire et algorithmes génétiques’, PhD Thesis, Université de Bourgogne, Nevers, 2002 (in French).
E. Valot P. Vannucci G. Verchery (2002) ArticleTitle‘Complete in-plane elastic characterisation under tensile tests of angle-ply laminates composed of polymer-matrix layers’ Polym. and Polym. Compos. 10 483–492
E. Valot P. Vannucci G. Verchery (2003) ArticleTitle‘A linear theory for laminates composed of coupled layers’ Compos.Struct. 60 413–429 Occurrence Handle10.1016/S0263-8223(03)00022-9
P. Vannucci G. Verchery (2002) ArticleTitle‘A new method for generating fully isotropic laminates’ Compos. Struct. 58 75–82 Occurrence Handle10.1016/S0263-8223(02)00038-7
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper is dedicated to Professor Georges Verchery, on the occasion of his 60th birthday.
Rights and permissions
About this article
Cite this article
Vannucci, P. Plane Anisotropy by the Polar Method*. Meccanica 40, 437–454 (2005). https://doi.org/10.1007/s11012-005-2132-z
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11012-005-2132-z