The classification of all fourth-order anisotropic tensor classes for classical linear elasticity is well known. In this article, we review the related problem of explicitly computing the dimension and the expressions of the elements belonging to these classes, and we extend this computation to fourth-order elasticity tensors acting on non-symmetric matrices. These tensors naturally appear in generalized continuum models. Based on tensor symmetrization, we provide the most general forms of these tensors for orthotropic, transversely isotropic, cubic, and isotropic materials. We present a self-contained discussion and provide detailed calculations for simple examples.
AuffrayNLe QuangHHeQC.Matrix representations for 3D strain-gradient elasticity. J Mech Phys Solids2013; 61(5): 1202–1223.
19.
AuffrayNHeQCLe QuangH.Complete symmetry classification and compact matrix representations for 3D strain gradient elasticity. Int J Solids Struct2019; 159: 197–210.
20.
OliveMAuffrayN.Symmetry classes for odd-order tensors. Z Angew Math Phys2014; 94(5): 421–447.
21.
AuffrayN.On the algebraic structure of isotropic generalized elasticity theories. Math Mech Solids2015; 20(5): 565–581.
22.
OliveMAuffrayN.Symmetry classes for even-order tensors. Math Mech Complex Syst2013; 1(2): 177–210.
23.
OliveMKolevBAuffrayN.A minimal integrity basis for the elasticity tensor. Arch Ration Mech Anal2017; 226(1): 1–31.
24.
DesmoratBAuffrayN.Space of 2D elastic materials: a geometric journey. Continuum Mech Thermodyn2019; 31: 1205–1229.
25.
AuffrayNKolevBOliveM.Handbook of bi-dimensional tensors: Part I: harmonic decomposition and symmetry classes. Math Mech Solids2017; 22(9): 1847–1865.
26.
OliveMKolevBDesmoratR, et al. Characterization of the symmetry class of an elasticity tensor using polynomial covariants. Math Mech Solids2022; 27(1): 144–190.
27.
ForteSVianelloM.Symmetry classes for elasticity tensors. J Elast1996; 43(2): 81–108.
28.
LazarMAgiasofitouE.Toupin–Mindlin first strain-gradient elasticity for cubic and isotropic materials at small scales. PAMM2023; 23(3): e202300121.
29.
LazarMAgiasofitouEBöhlkeT.Mathematical modeling of the elastic properties of cubic crystals at small scales based on the Toupin–Mindlin anisotropic first strain gradient elasticity. Continuum Mech Thermodyn2022; 34(1): 107–136.
30.
VossJRizziGNeffP, et al. Modeling a labyrinthine acoustic metamaterial through an inertia-augmented relaxed micromorphic approach. Math Mech Solids2023; 28: 2177–2201.
31.
NeffPEidelBd’AgostinoMV, et al. Identification of scale-independent material parameters in the relaxed micromorphic model through model-adapted first order homogenization. J Elast2020; 139(2): 269–298.
32.
d’AgostinoMVBarbagalloGGhibaID, et al. Effective description of anisotropic wave dispersion in mechanical band-gap metamaterials via the relaxed micromorphic model. Journal of Elasticity2020; 139: 299–329.
33.
d’AgostinoMVBarbagalloGGhibaID, et al. A panorama of dispersion curves for the weighted isotropic relaxed micromorphic model. Z Angew Math Mech2017; 97(11): 1436–1481.
34.
BarbagalloGMadeoAd’AgostinoMV, et al. Transparent anisotropy for the relaxed micromorphic model: macroscopic consistency conditions and long wave length asymptotics. Int J Solids Struct2017; 120: 7–30.
35.
RizziGColletMDemoreF, et al. Exploring metamaterials’ structures through the relaxed micromorphic model: switching an acoustic screen into an acoustic absorber. Front Mater2021; 7: 589701.
36.
RizziGd’AgostinoMVNeffP, et al. Boundary and interface conditions in the relaxed micromorphic model: exploring finite-size metastructures for elastic wave control. Math Mech Solids2022; 27(6): 1053–1068.
37.
RizziGNeffPMadeoA.Metamaterial shields for inner protection and outer tuning through a relaxed micromorphic approach. Philos Trans R Soc A2022; 380(2231): 20210400.
38.
RamirezLAPRizziGMadeoA. Multi-element metamaterial’s design through the relaxed micromorphic model. In: AltenbachHBerezovskiAdell’IsolaF, et al. (eds) Sixty shades of generalized continua: dedicated to the 60th birthday of Prof. Victor A. Eremeyev. Cham: Springer, 2023, pp. 579–600.
39.
DemoreFRizziGColletM, et al. Unfolding engineering metamaterials design: relaxed micromorphic modeling of large-scale acoustic meta-structures. J Mech Phys Solids2022; 168: 104995.
40.
RizziGTallaricoDNeffP, et al. Towards the conception of complex engineering meta-structures: relaxed-micromorphic modelling of low-frequency mechanical diodes/high-frequency screens. Wave Motion2022; 113: 102920.
41.
ChossatPLauterbachRMelbourneI.Steady-state bifurcation with -symmetry. Arch Ration Mech Anal1991; 113(4): 313–376.
42.
AzziPOliveM.Clips operation between type-II and type-III -subgroups with application to piezoelectricity. Math Mech Complex Syst2023; 11(2): 235–269.
43.
OliveM.Effective computation of and linear representation symmetry classes. Math Mech Complex Syst2019; 7(3): 203–237.
44.
DeitmarAEchterhoffS.Principles of harmonic analysis. Cham: Springer, 2014.
45.
HallBC.Lie groups, lie algebras, and representations. In: HallB (ed.) Quantum theory for mathematicians. Cham: Springer, 2013, pp. 333–366.
46.
GolubitskyMStewartISchaefferDG.Singularities and groups in bifurcation theory. Cham: Springer Science & Business Media, 2012.
BatraRC.On heat conduction and wave propagation in non-simple rigid solids. Lett Appl Eng Sci1975; 3: 97–107.
53.
HuangYNBatraRC.Energy-momentum tensors in nonsimple elastic dielectrics. J Elast1996; 42(3): 275–281.
54.
FerrettiMMadeoAdell’IsolaF, et al. Modeling the onset of shear boundary layers in fibrous composite reinforcements by second-gradient theory. Z Angew Math Phys2014; 65(3): 587–612, http://link.springer.com/10.1007/s00033-013-0347-8
55.
MadeoABarbagalloGd’AgostinoMV, et al. Continuum and discrete models for unbalanced woven fabrics. Int J Solids Struct2016; 94–95(2016): 263–284.
56.
ElżanowskiMEpsteinM.The symmetry group of second-grade materials. Int J Non Lin Mech1992; 27(4): 635–638.
57.
MurdochIA.Symmetry considerations for materials of second grade. J Elast1979; 9(1): 43–50.
58.
MünchINeffP.Rotational invariance conditions in elasticity, gradient elasticity and its connection to isotropy. Math Mech Solids2018; 23(1): 3–42.
59.
JohnF.Rotation and strain. Commun Pure Appl Math1961; 14(3): 391–413.
60.
NeffPMünchI.Curl bounds grad on SO(3). ESAIM: Control Optim Calc Var2008; 14(1): 148–159.
61.
NeffPJeongJRamézaniH.Subgrid interaction and micro-randomness - novel invariance requirements in infinitesimal gradient elasticity. Int J Solids Struct2009; 46(25–26): 4261–4276.
62.
BatraRC.Private communications, 2018.
63.
NeffPMünchIGhibaID, et al. On some fundamental misunderstandings in the indeterminate couple stress model. A comment on recent papers of A.R. Hadjesfandiari and G.F. Dargush. Int J Solids Struct2016; 81: 233–243.
64.
MünchINeffPMadeoA, et al. The modified indeterminate couple stress model: why Yang et al.’s arguments motivating a symmetric couple stress tensor contain a gap and why the couple stress tensor may be chosen symmetric nevertheless. Z Angew Math Phys2017; 97(12): 1524–1554.
65.
GhibaIDNeffPMadeoA, et al. A variant of the linear isotropic indeterminate couple-stress model with symmetric local force-stress, symmetric nonlocal force-stress, symmetric couple-stresses and orthogonal boundary conditions. Math Mech Solids2017; 22(6): 1221–1266.
66.
NeffPGhibaIDMadeoA, et al. Null-Lagrangians and the indeterminate couple stress model. Proc Appl Math Mech2016; 16(1): 379–380.
67.
SarhilMScheunemannLSchröderJ, et al. Size-effects of metamaterial beams subjected to pure bending: on boundary conditions and parameter identification in the relaxed micromorphic model. Comput Mech2023; 72: 1091–1113.
68.
DesmoratBOliveMAuffrayN, et al. Computation of minimal covariants bases for 2D coupled constitutive laws. Int J Eng Sci2023; 191: 103880.
69.
FischleANeffP.Grioli’s theorem with weights and the relaxed-polar mechanism of optimal Cosserat rotations. Rendiconti Lincei2017; 28(3): 573–600.
70.
NeffPLankeitJMadeoA.On Grioli’s minimum property and its relation to Cauchy’s polar decomposition. Int J Eng Sci2014; 80: 209–217.
71.
GrafarendEWKuhnelW.A minimal atlas for the rotation group SO(3). GEM - International Journal on Geomathematics2011; 2(1): 113–122.
72.
MarsdenJERatiuTS.Introduction to Mechanics and Symmetry. Phys Today1995; 48(12): 65.
73.
DiebelJ.Representing attitude: Euler angles, unit quaternions, and rotation vectors. Matrix2006; 58(15–16): 1–35.
74.
AbudMSartoriG.The geometry of spontaneous symmetry breaking. Ann Phys1983; 150(2): 307–372.
75.
IhrigEGolubitskyM.Pattern selection with O(3) symmetry. Physica D1984; 13(1–2): 1–33.
