The Formulation of Constitutive Equation for Anisotropic Solids

The plan of this lecture is that I shall first review some results in the theory of algebraic invariants of vectors and tensors in three dimensions, under the orthogonal group of transformations. Then I will show how, in some cases of interest, it is possible to use these results for invariants under orthogonal transformations to determine invariants of vectors and tensors under groups of transformations which are sub–groups of the orthogonal group. As examples, I shall consider (a) the case in which the group of transformations is the group of rotations about an axis, which is the symmetry group for a transversely isotropic material, and (b) the case in which the transformation group is generated by the set of reflections in three orthogonal planes, which is the symmetry group for an orthotropic material. Next I will show how these results can be applied to the problem of determining mechanical constitutive equations for anisotropic materials with various types of stress response; in particular I shall consider materials whose stress response is that of a linear or a non–linear elastic solid, or that of a plastic solid, and illustrate the results by considering transversely isotropic and orthotropic materials. Many fibre–reinforced and laminated materials are, on the macroscopic scale, either transversely isotropic or orthotropic, and I shall give particular consideration to materials of this kind, and especially to the case in which the preferred directions are not uniform, but vary with position in a body. Finally, I will discuss the effect on the constitutive equations of the kinematic constraints of incompressibility and inextensibility in specified directions.

[1]  A.J.M. Spencer,et al.  Theory of invariants , 1971 .

[2]  Further results in the theory of matrix polynomials , 1959 .

[3]  J. Boehler.  On Irreducible Representations for Isotropic Scalar Functions , 1977 .

[4]  J. E. Adkins,et al.  Large Elastic Deformations , 1971 .

[5]  G. F. Smith.  On isotropic functions of symmetric tensors, skew-symmetric tensors and vectors , 1971 .

[6]  R. S. Rivlin,et al.  The formulation of constitutive equations in continuum physics. II , 1959 .

[7]  Ronald S. Rivlin,et al.  Further Remarks on the Stress-Deformation Relations for Isotropic Materials , 1955 .

[8]  G. F. Smith.  On isotropic integrity bases , 1965 .

[9]  A.J.M. Spencer,et al.  Finite integrity bases for five or fewer symmetric 3×3 matrices , 1958 .

[10]  A.J.M. Spencer,et al.  The theory of matrix polynomials and its application to the mechanics of isotropic continua , 1958 .

[11]  A. Spencer,et al.  Isotropic integrity bases for vectors and second-order tensors , 1962 .

[12]  Ronald S. Rivlin,et al.  Large Elastic Deformations of Homogeneous Anisotropic Materials , 1954 .

[13]  A. Spencer,et al.  Deformations of fibre-reinforced materials, , 1972 .

[14]  A. Spencer.  The invariants of six symmetric 3 × 3 matrices , 1961 .

[15]  J. Boehler,et al.  A Simple Derivation of Representations for Non‐Polynomial Constitutive Equations in Some Cases of Anisotropy , 1979 .

[16]  G. F. Smith,et al.  Integrity bases for vectors — The crystal classes , 1964 .

[17]  R. S. Rivlin,et al.  Isotropic integrity bases for vectors and second-order tensors , 1962 .

[18]  G. F. Smith,et al.  The anisotropic tensors , 1957 .