We present a general, constructive procedure to find the basis for tensors of arbitrary order subject to linear constraints by transforming the problem to that of finding the nullspace of a linear operator. The proposed method derives from well-established representation-theoretic foundations and utilizes standard numerical linear algebra techniques that are highly optimized and well-behaved. Our primary applications are in mechanics where modulus tensors and so-called structure tensors can be used to characterize anisotropy of functional dependencies on other inputs such as strain. Like modulus tensors, structure tensors are defined by their invariance to transformations by symmetry group generators but have more general applicability. The fully automated method is an alternative to classical, more intuition-reliant methods such as the Pipkin–Rivlin polynomial integrity basis construction. We demonstrate the utility of the procedure by: (a) enumerating elastic modulus tensors for common symmetries and (b) finding the lowest-order structure tensors that can represent all common point groups/crystal classes. Furthermore, we employ these results in two calibration problems using neural network models following classical function representation theory: (a) learning the symmetry class and orientation of a hyperelastic material given stress–strain data and (b) representing strain-dependent anisotropy of the stress response of a soft matrix-stiff fiber composite in a sequence of uniaxial loadings. These two examples demonstrate the utility of the method in model selection and calibration by: (a) determining structural tensors of a selected order across multiple symmetry groups and (b) determining a basis for a given group that allows the characterization of all subgroups. Using a common order in both cases allows sparse regression to operate on a common function representation to select the best-fit symmetry group for the data.
BiedenharnLCLouckJDCarruthersPA. Angular momentum in quantum physics: theory and application. Cambridge: Cambridge University Press, 1984.
2.
PopeSB. A more general effective-viscosity hypothesis. J Fluid Mech1975; 72(2): 331–340.
3.
ItskovM. On the theory of fourth-order tensors and their applications in computational mechanics. Comput Methods Appl Mech Eng2000; 189(2): 419–438.
4.
ItinYHehlFW. The constitutive tensor of linear elasticity: its decompositions, Cauchy relations, null Lagrangians, and wave propagation. J Math Phys2013; 54(4): 042903.
5.
WallaceDC. Thermodynamics of crystals. Hoboken, NJ: Wiley, 1972.
6.
WeinerJH. Statistical mechanics of elasticity. Chelmsford, MA: Courier Corporation, 2012.
7.
BarrettTW. Topological foundations of electromagnetism. Singapore: World Scientific, 2022.
8.
TarnJQChangHH. A Hamiltonian state space approach to anisotropic elasticity and piezoelasticity. Acta Mech2013; 224(6): 1271–1284.
9.
MasonW. A phenomenological derivation of the first- and second-order magnetostriction and morphic effects for a nickel crystal. Phys Rev1951; 82(5): 715.
10.
WallaceDC. Thermoelastic theory of stressed crystals and higher-order elastic constants. Solid State Phys1970; 25: 301–404.
11.
HikiY. Higher order elastic constants of solids. Annu Rev Mater Sci1981; 11: 51–73.
12.
d’AgostinoMVMartinRJLewintanP, et al. On the representation of fourth- and higher-order anisotropic elasticity tensors in generalized continuum models. Math Mech Solids2025; 30(8): 1789–1849.
13.
ClaytonJ. On symmetries of higher-order elastic constants. Found Crystallogr2025; 81(1): 1–4.
14.
OgdenRW. Nonlinear elasticity, anisotropy, material stability and residual stresses in soft tissue. In: HolzapfelGAOgdenRW (eds) Biomechanics of soft tissue in cardiovascular systems. Vienna: Springer, 2003, pp. 65–108.
15.
CazacuO. New yield criteria for isotropic and textured metallic materials. Int J Solids Struct2018; 139: 200–210.
16.
ZhengQS. On the representations for isotropic vector-valued, symmetric tensor-valued and skew-symmetric tensor-valued functions. Int J Eng Sci1993; 31(7): 1013–1024.
17.
PipkinACRivlinR. The formulation of constitutive equations in continuum physics. I. In: BarenblattGIJosephDD (eds) Collected papers of R. S. Rivlin: volume I and II. New York: Springer, 1959, pp. 1111–1126.
18.
FuhgJNBouklasNJonesRE. Learning hyperelastic anisotropy from data via a tensor basis neural network. J Mech Phys Solids2022; 168: 105022.
GoodmanRWallachNR. Representations and invariants of the classical groups, vol. 68. Cambridge: Cambridge University Press, 1998.
21.
KarafillisABoyceM. A general anisotropic yield criterion using bounds and a transformation weighting tensor. J Mech Phys Solids1993; 41(12): 1859–1886.
22.
ItinY. Irreducible matrix resolution of the elasticity tensor for symmetry systems. arXiv2018; ArXiv:1812.03367.
23.
OliveMAuffrayN. Isotropic invariants of a completely symmetric third-order tensor. J Math Phys2014; 55(9): 092901.
24.
DesmoratRAuffrayNDesmoratB, et al. Minimal functional bases for elasticity tensor symmetry classes. J Elast2021; 147(1): 201–228.
25.
AguiarARda RochaGL. Construction of invariant relations of n symmetric second-order tensors. J Elast2023; 154(1): 45–60.
26.
KemperGSteelA. Some algorithms in invariant theory of finite groups. In: DräxlerPRingelCMMichlerGO (eds) Computational methods for representations of groups and algebras. Basel: Birkhäuser, 1999, pp. 267–285.
27.
SturmfelsB. Algorithms in invariant theory. Vienna: Springer, 2008.
28.
KingSA. Minimal generating sets of non-modular invariant rings of finite groups. J Symb Comput2013; 48: 101–109.
29.
DerksenHKemperG. Computational invariant theory. Berlin and Heidelberg: Springer, 2015.
30.
RivlinRS. Further remarks on the stress-deformation relations for isotropic materials. J Ration Mech Anal1955; 4: 681–702.
31.
SmithGRivlinRS. The anisotropic tensors. Q Appl Math1957; 15(3): 308–314.
SpencerAJMRivlinRS. The theory of matrix polynomials and its application to the mechanics of isotropic continua. Arch Ration Mech Anal1958; 2: 309–336.
34.
SpencerAJMRivlinRS. Finite integrity bases for five or fewer symmetric 3x3 matrices. Arch Ration Mech Anal1958; 2(1): 435–446.
35.
SpencerARivlinRS. Isotropic integrity bases for vectors and second-order tensors. Arch Ration Mech Anal1962; 9(1): 45–63.
36.
SmithGSmithMRivlinRS. Integrity bases for a symmetric tensor and a vector—the crystal classes. Arch Ration Mech Anal1963; 12(1): 93–133.
RivlinRSSmithG. Orthogonal integrity basis for n symmetric matrices. Contrib Mech1969; 1969: 121.
39.
KiralESmithG. On the constitutive relations for anisotropic materials—triclinic, monoclinic, rhombic, tetragonal and hexagonal crystal systems. Int J Eng Sci1974; 12(6): 471–490.
40.
SmithGBaoG. Irreducible constitutive expressions. In: CarrollMMHayesMA (eds) Nonlinear effects in fluids and solids. Boston, MA: Springer, 1996, pp. 321–337.
41.
SpencerAJMRivlinRS. Further results in the theory of matrix polynomials. Arch Ration Mech Anal1959; 4: 214–230.
42.
SpencerA. Isotropic integrity bases for vectors and second-order tensors. Arch Ration Mech Anal1965; 18(1): 51–82.
43.
SpencerA. A note on the decomposition of tensors into traceless symmetric tensors. Int J Eng Sci1970; 8(6): 475–481.
44.
SpencerA. Isotropic polynomial invariants and tensor functions. In: BoehlerJP (ed.) Applications of tensor functions in solid mechanics. Vienna: Springer, 1987, pp. 141–169.
45.
BoehlerJP. A simple derivation of representations for non-polynomial constitutive equations in some cases of anisotropy. ZAMM1979; 59(4): 157–167.
46.
BoehlerJP. Representations for isotropic and anisotropic non-polynomial tensor functions. In: BoehlerJP (ed.) Applications of tensor functions in solid mechanics. Vienna: Springer, 1987, pp. 31–53.
47.
BoehlerJ. On a rational formulation of isotropic and anisotropic hardening. In: BoehlerJP (ed.) Applications of tensor functions in solid mechanics. Vienna: Springer, 1987, pp. 99–122.
48.
ZhengQSBoehlerJ. The description, classification, and reality of material and physical symmetries. Acta Mech1994; 102(1–4): 73–89.
49.
WangCC. On a general representation theorem for constitutive relations. Arch Ration Mech Anal1969; 33(1): 1–25.
50.
WangCC. A new representation theorem for isotropic functions: part I. Scalar-valued isotropic functions. Arch Ration Mech Anal1970; 36: 166–197.
WinemanASPipkinA. Material symmetry restrictions on constitutive equations. Arch Ration Mech Anal1964; 17(3): 184–214.
53.
JemiołoSKwiecińskiM. On irreducible number of invariants and generators in the constitutive relationships. Eng Trans1991; 39(2): 241–253.
54.
ZhangJRychlewskiJ. Structural tensors for anisotropic solids. Arch Mech1990; 42(3): 267–277.
55.
SmithG. On a fundamental error in two papers of C.-C. Wang. Arch Ration Mech Anal1970; 36: 161–165.
56.
SmithG. On isotropic functions of symmetric tensors, skew-symmetric tensors and vectors. Int J Eng Sci1971; 9(10): 899–916.
57.
BoehlerJP. Introduction to the invariant formulation of anisotropic constitutive equations. In: Applications of tensor functions in solid mechanics. Springer, 1987, pp. 13–30.
58.
ZhengQS. Theory of representations for tensor functions—a unified invariant approach to constitutive equations. Appl Mech Rev1994; 47(11): 545–587.
59.
XuYSmithMSmithG. Computer aided generation of anisotropic constitutive expressions. Int J Eng Sci1987; 25(6): 711–722.
60.
MieheC. Comparison of two algorithms for the computation of fourth-order isotropic tensor functions. Comput Struct1998; 66(1): 37–43.
61.
DanescuA. The number of elastic moduli for anisotropic polynomial constitutive equations. Mech Res Commun1997; 24(3): 289–301.
62.
CoxDLittleJO’SheaD. Ideals, varieties, and algorithms, vol. 3. New York: Springer, 1997.
HalkoNMartinssonPGTroppJA. Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev2011; 53(2): 217–288.
69.
BradburyJFrostigRHawkinsP, et al. JAX: composable transformations of Python+NumPy programs. 2018.
70.
LandauLDPitaevskiiLKosevichAM, et al. Theory of elasticity, vol. 7. Amsterdam: Elsevier, 2012.
71.
HillR. The mathematical theory of plasticity, vol. 11. Oxford: Oxford University Press, 1950.
72.
BarlatFLegeDJBremJC. A six-component yield function for anisotropic materials. Int J Plast1991; 7(7): 693–712.
73.
DerksenHKemperG. Computational invariant theory. New York: Springer, 2001.
74.
FisherDKleinbockDSoiferG. Dynamics, geometry, number theory: the impact of Margulis on modern mathematics. Chicago, IL: University of Chicago Press, 2022.
75.
SmithMSmithG. Group-averaging methods for generating constitutive equations. In: Mechanics and physics of energy density: characterization of material/structure behaviour with and without damage. Dordrecht: Springer, 1992, pp. 167–178.
76.
AuffrayNKolevBPetitotM. Invariant-based approach to symmetry class detection. arXiv2011; ArXiv:1111.0861.
77.
JadoonAAKalinaKARauschMK, et al. Inverse design of anisotropic microstructures using physics-augmented neural networks. J Mech Phys Solids2025; 203: 106161.
78.
KalinaKABrummundJSunW, et al. Neural networks meet anisotropic hyperelasticity: a framework based on generalized structure tensors and isotropic tensor functions. Comput Methods Appl Mech Eng2025; 437: 117725.
79.
KambouchevNFernandezJRadovitzkyR. A polyconvex model for materials with cubic symmetry. Model Simul Mater Sci Eng2007; 15(5): 451.
80.
BonetJBurtonA. A simple orthotropic, transversely isotropic hyperelastic constitutive equation for large strain computations. Comput Methods Appl Mech Eng1998; 162(1–4): 151–164.
KleinDKRothFJValizadehI, et al. Parametrized polyconvex hyperelasticity with physics-augmented neural networks. Data-Centric Eng2023; 4: e25.
86.
FuhgJNJadoonAWeegerO, et al. Polyconvex neural network models of thermoelasticity. J Mech Phys Solids2024; 192: 105837.
87.
ArrudaEMBoyceMC. Evolution of plastic anisotropy in amorphous polymers during finite straining. Int J Plast1993; 9(6): 697–720.
88.
AhziSLeeBAsaroR. Plasticity and anisotropy evolution in crystalline polymers. Mater Sci Eng A1994; 189(1–2): 35–44.
89.
HiwatashiSVan BaelAVan HoutteP, et al. Modelling of plastic anisotropy based on texture and dislocation structure. Comput Mater Sci1997; 9(1–2): 274–284.
90.
SchiferlSKMaudlinPJ. Evolution of plastic anisotropy for high-strain-rate computations. Comput Methods Appl Mech Eng1997; 143(3–4): 249–270.
91.
LiSHoferlinEVan BaelA, et al. Finite element modeling of plastic anisotropy induced by texture and strain-path change. Int J Plast2003; 19(5): 647–674.
92.
KowalczykKGambinW. Model of plastic anisotropy evolution with texture-dependent yield surface. Int J Plast2004; 20(1): 19–54.
93.
HarryssonMRistinmaaM. Description of evolving anisotropy at large strains. Mech Mater2007; 39(3): 267–282.
94.
KweonSRajaDS. Comparison of anisotropy evolution in bcc and fcc metals using crystal plasticity and texture analysis. Eur J Mech A Solids2017; 62: 22–38.
95.
NegahbanMWinemanAS. The evolution of anisotropies in the elastic response of an elastic-plastic material. Int J Plast1992; 8(5): 519–542.
96.
RajagopalKRWinemanAS. A constitutive equation for nonlinear solids which undergo deformation induced microstructural changes. Int J Plast1992; 8(4): 385–395.
