American Institute of Mathematical Sciences

Advanced
January  2009, 8(1): 435-456. doi: 10.3934/cpaa.2009.8.435

Symmetry groups in nonlinear elasticity: an exercise in vintage mathematics

Annie Raoult 1,

1. 

Laboratoire MAP5, Université Paris Descartes and CNRS, 45 rue des Saints Pères, 75006 Paris, France

Received  March 2008 Revised  August 2008 Published  October 2008

This manuscript aims at characterizing energy densities and constitutive laws of transversely isotropic materials, orthotropic elastic materials and materials with non orthogonal families of fibers. It makes explicit references to results that are scattered over the literature and, although said to be well-known, are not always easy to locate. Direct proofs that are thought to be new and simplified expressions of constitutive laws for materials with two preferred directions are given.
Keywords: fibers, symmetry groups, transverse isotropy, constitutive laws, orthotropy, anisotropy, Nonlinear elasticity.
Mathematics Subject Classification: 74A20, 74B20, 74E10.
Citation: Annie Raoult. Symmetry groups in nonlinear elasticity: an exercise in vintage mathematics. Communications on Pure & Applied Analysis, 2009, 8 (1) : 435-456. doi: 10.3934/cpaa.2009.8.435
[1]

Daniele Boffi, Franco Brezzi, Michel Fortin. Reduced symmetry elements in linear elasticity. Communications on Pure & Applied Analysis, 2009, 8 (1) : 95-121. doi: 10.3934/cpaa.2009.8.95
[2]

Yohei Yamazaki. Transverse instability for a system of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 565-588. doi: 10.3934/dcdsb.2014.19.565
[3]

Claude Vallée, Camelia Lerintiu, Danielle Fortuné, Kossi Atchonouglo, Jamal Chaoufi. Modelling of implicit standard materials. Application to linear coaxial non-associated constitutive laws. Discrete & Continuous Dynamical Systems - S, 2013, 6 (6) : 1641-1649. doi: 10.3934/dcdss.2013.6.1641
[4]

Dmitry V. Zenkov. Linear conservation laws of nonholonomic systems with symmetry. Conference Publications, 2003, 2003 (Special) : 967-976. doi: 10.3934/proc.2003.2003.967
[5]

Irena Lasiecka, W. Heyman. Asymptotic behavior of solutions in nonlinear dynamic elasticity. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 237-252. doi: 10.3934/dcds.1995.1.237
[6]

Dwayne Chambers, Erica Flapan, John D. O'Brien. Topological symmetry groups of $K_{4r+3}$. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1401-1411. doi: 10.3934/dcdss.2011.4.1401
[7]

Lorena Bociu, Jean-Paul Zolésio. Existence for the linearization of a steady state fluid/nonlinear elasticity interaction. Conference Publications, 2011, 2011 (Special) : 184-197. doi: 10.3934/proc.2011.2011.184
[8]

Hirofumi Notsu, Masato Kimura. Symmetry and positive definiteness of the tensor-valued spring constant derived from P1-FEM for the equations of linear elasticity. Networks & Heterogeneous Media, 2014, 9 (4) : 617-634. doi: 10.3934/nhm.2014.9.617
[9]

Meng Qu, Ping Li, Liu Yang. Symmetry and monotonicity of solutions for the fully nonlinear nonlocal equation. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1337-1349. doi: 10.3934/cpaa.2020065
[10]

Gui-Qiang Chen, Monica Torres. On the structure of solutions of nonlinear hyperbolic systems of conservation laws. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1011-1036. doi: 10.3934/cpaa.2011.10.1011
[11]

Maciek Korzec, Andreas Münch, Endre Süli, Barbara Wagner. Anisotropy in wavelet-based phase field models. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1167-1187. doi: 10.3934/dcdsb.2016.21.1167
[12]

Toshikazu Ito, Bruno Scárdua. Holomorphic foliations transverse to manifolds with corners. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 537-544. doi: 10.3934/dcds.2009.25.537
[13]

Xiangnan He, Wenlian Lu, Tianping Chen. On transverse stability of random dynamical system. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 701-721. doi: 10.3934/dcds.2013.33.701
[14]

Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065
[15]

Julian Braun, Bernd Schmidt. On the passage from atomistic systems to nonlinear elasticity theory for general multi-body potentials with p-growth. Networks & Heterogeneous Media, 2013, 8 (4) : 879-912. doi: 10.3934/nhm.2013.8.879
[16]

Rita Ferreira, Elvira Zappale. Bending-torsion moments in thin multi-structures in the context of nonlinear elasticity. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1747-1793. doi: 10.3934/cpaa.2020072
[17]

Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125
[18]

Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395
[19]

Shiren Zhu, Xiaoli Chen, Jianfu Yang. Regularity, symmetry and uniqueness of positive solutions to a nonlinear elliptic system. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2685-2696. doi: 10.3934/cpaa.2013.12.2685
[20]

Sven Jarohs, Tobias Weth. Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2581-2615. doi: 10.3934/dcds.2014.34.2581

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences