**39**, 2, pp. 241-253, 1991

### On Irreducible Number of Invariants and Generators in the Constitutive Relationships

The Pipkin-Rivlin method for determining the generators of a polynomial representation of a symmetric isotropic second-order tensor-valued function is modified. Generators of an anisotropic and orthotropic symmetric second-order tensor-valued function are thus shown as dependent on a finite number of symmetric second-order tensors. The obtained results coincide with those arrived at by Boehler. Next, irreducible invariants of anisotropic scalar functions, depending on a single symmetric second-order tensor a.re found. Types of anisotropy are considered in which the material symmetry group is described by means of vectors and symmetric second-order tensors. The anisotropic scalar functions derived can be used to construct the constitutive equations for nonlinear elasticity of Green's material as well as potentials and yield conditions in plasticity. As an example, the equations are derived for a material reinforced with two orthogonal families of bars.

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#### References

J.P.BOEHLER, On irreducible representations for isotropic scalar functions, ZAMM, 57, pp.323-327, 1977.

J.P.BOEHLER, J.RACLIN, Representations irreductibles des fonctions tensorielles non-polynomiales de deux tenseurs symetriques dans quelques cas d'anisotropie, Arch. Mech., 29, 3, pp. 431-444, 1977.

J.P.BOEHLER, Lois de comportement anisotrope des milieux continus, J.Mec., 17, 2, pp.153-190, 1978.

J.P.BOEHLER, A simple derivation of representations for non-polynomial constitutive equations in some cases of anisotropy, ZAMM, 59, pp. 157-167, 1979.

J.P.BOEHLER, Representations for isotropic and anisotropic non-polynomial tensor functions, CISM Courses and Lectures, No. 292, pp.31-53, 1987.

I-SHIH LIU, On representations of anisotropic invariants, Int. J.Engng Sci., 20, 10, pp. 1099-1109, 1982.

W.W.LOKHIN, L.l.SEDOV, Nonlinear tensor Junctfon3 of certain tensorial arguments [in Russian], Prikladnaya Matematika i Mekhanika, 27, pp.393-417, 1963.

A.C.PIPKIN, A.S.WINEMAN, Material symmetry restrictions on non-polynomial constitutive equations, Arch. Rat. Mech. Anal., 12, pp.420-426, 1963.

A.C.PIPKIN, R.S.RIVLIN, The formulation of constitutive equations in continuum physics I, Arch. Rat. Mech. Anal. 4, pp.129-144, 1959.

J.RYCHLEWSKI, Symmetry of causes and effects, [in Polish], Wydawnictwo Naukowe PWN, Warszawa 1991.

L.I.SEDOV, W.W.LOKHIN, On descriptions of finite symmetry groups with the use of tensors [in Russian], Doklady Akademii Nauk SV, 149, pp. 796-799, 1963.

G.F.SMITH, On a fundamental error in two papers of C.C. Wang, Arch. Rat. Mech. Ana!., 36, pp.161-165, 1970.

G.F.SMITH, On isotropic functions of symmetric tensors, skew-symmetric tensors and vectors, Int. J.Engng Sci., 9, pp.899-916, 1971.

A.J.M.SPENCER, Constitutive theory for strongly anisotropic solids, CISM Courses and Lectures, No. 282, pp.1-32, 1984.

J.J.TELEGA, Some aspects of invariants theory in plasticity, Part I. New results relative to representation of isotropic and anisotropic tensor functions, Arch. Mech., 36, 2, pp.147-162, 1984.

C.C.WANG, On representations for isotropic functions, Part I and II, Arch. Rat. Mech. Anal., 33, pp.249-287, 1969.

C.C.WANG, A new representation theorem for isotropic functions, Part I and II, Arch. Rat. Mech. Anal., 36, pp.166-223, 1970.

C.C.WANG, Gorrigendum, Arch. Rat. Mech. Anal., 43, pp.392-395, 1971.

A.S.WINEMAN, A.C.PIPKIN, Material symmetry restrictions on constitutive equations, Arch