To investigate complex physical phenomena, bi-dimensional models are often an interesting option. It allows spatial couplings to be produced while keeping them as simple as possible. For linear physical laws, constitutive equations involve the use of tensor spaces. As a consequence the different types of anisotropy that can be described are encoded in tensor spaces involved in the model. In the present paper, we solve the general problem of computing symmetry classes of constitutive tensors in using mathematical tools coming from representation theory. The power of this method is illustrated through the tensor spaces of Mindlin strain-gradient elasticity.
JerphagnonJChemlaDBonnevilleR.The description of the physical properties of condensed matter using irreducible tensors. Adv Phys1978; 27: 609–650.
2.
DesmoratBVannucciP.An alternative to the Kelvin decomposition for plane anisotropic elasticity. Math Meth Appl Sci2015; 38(1): 164–175.
3.
De SaxcéGValléeC. Invariant measures of the lack of symmetry with respect to the symmetry groups of 2D elasticity tensors. J Elast2013; 111: 21–39.
4.
ForteSVianelloM.A unified approach to invariants of plane elasticity tensors. Mecc2014; 49(9): 2001–2012.
5.
VincentiAVannucciPAhmadianMR.Optimization of laminated composites by using genetic algorithm and the polar description of plane anisotropy. Mech Adv Mater Struc2013; 20: 242–255.
6.
FrançoisML.A damage model based on Kelvin eigentensors and Curie principle. Mech Mater2012; 44: 23–34.
7.
AuffrayNRoparsP.Invariant-based reconstruction of bidimensional elasticity tensors. Int J Solids Struct2016; 87: 183–193. DOI: 10.1016/j.ijsolstr.2016.02.013.
8.
BacigalupoAGambarottaL.Homogenization of periodic hexa- and tetrachiral cellular solids. Compos Struct2014; 116: 461–476.
9.
ChenYLiuXHuGet al. Micropolar continuum modelling of bi-dimensional tetrachiral lattices. Proc R Soc A2014; 470(2165): 20130734.
10.
AuffrayNDirrenbergerJRosiG.A complete description of bi-dimensional anisotropic strain-gradient elasticity. Int J Solids Struct2015; 69–70: 195–206.
11.
RosiGAuffrayN.Anisotropic and dispersive wave propagation within strain-gradient framework. Wav Mot2016. In press.
12.
BoehlerJPKirillovAAJrOnatET.On the polynomial invariants of the elasticity tensor. J Elast1994; 34(2): 97–110.
13.
AuffrayNKolevBPetitotM.On anisotropic polynomial relations for the elasticity tensor. J Elast2014; 115: 77–103.
14.
GeymonatGWellerT.Symmetry classes of piezoelectric solids. CR Math2002; 335(10): 847–852.
15.
VannucciP.The polar analysis of a third order piezoelectricity-like plane tensor. Int J Solids Struct2007; 44: 7803–7815.
16.
ForestS.Mechanics of generalized continua: Construction by homogenizaton. J Phys IV1998; 8: Pr4–39.
17.
TrinhDKJanickeRAuffrayNet al. Evaluation of generalized continuum substitution models for heterogeneous materials. Int J Multisc Com2012; 10(6): 527–549.
HeQCZhengQS.On the symmetries of 2D elastic and hyperelastic tensors. J Elast1996; 43: 203–225.
26.
AuffrayNBouchetRBréchetY.Derivation of anisotropic matrix for bi-dimensional strain-gradient elasticity behavior. Int J Solids Struct2009; 46(2): 440–454.
27.
ZhengQSBoehlerJP.The description, classification, and reality of material and physical symmetries. Acta Mech1994; 102(1–4): 73–89.
28.
BredonG.Introduction to compact transformation groups (Pure and Applied Mathematics, vol. 46). New York, NY: Academic Press, 1972.
29.
ArmstrongMA.Groups and symmetry. New York, NY: Springer, 1983.
30.
HermanB.Some theorems of the theory of anisotropic media. CR (Dokl) Acad Sci URSS1945; 48(2): 89–92.
31.
MindlinR.Micro-structure in linear elasticity. Arch Rat Mech Anal1964; 16(1): 51–78.
32.
MindlinREshelN.On first strain-gradient theories in linear elasticity. Int J Solids Struct1968; 4(1): 109–124.
33.
OlverP.Classical invariant theory (London Mathematical Society Student Texts, vol. 44). Cambridge: Cambridge University Press, 1999.
34.
AuffrayNKolevBPetitotM.A complete integrity basis for the elasticity tensor. To be submitted, 2016.
35.
GolubitskyMStewartISchaefferDG.Singularities and groups in bifurcation theory. Vol. II (Applied Mathematical Sciences, vol. 69). New York, NY: Springer-Verlag, 1988.
36.
AnsemilJMFloretK.The symmetric tensor product of a direct sum of locally convex spaces. Stud Math1998; 129(3): 285–295.
37.
SternbergS.Group theory and physics. Cambridge: Cambridge University Press, 1994.
38.
ToupinRA.Elastic materials with couple-stresses. Arch Rat Mech Anal1962; 11: 385–414.
