Representations for Isotropic and Anisotropic Non-Polynomial Tensor Functions
Material symmetries of a continuum impose definite restrictions on the form of constitutive relations. The restrictions are specified in the representations of isotropic and anisotropic tensor functions and indicate the type and the number of independent
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REPRESENTATIONS FOR ISOTROPie AND ANISOTROPie NON-POL YNOMIAL TENSOR FUNeTIONS
J.P. Boehler University of Grenoble, France
1. INTRODUCTION 1.1 Invariant scalar functions and form-invariant tensor functions Material symmetries of a continuuro impose definite restrictions on the form of constitutive relations. The restrictions are specified in the representations of isotropic and anisotropic tensor functions and indicate the type and the nurober of independent variables involved in a constitutive relation. Thus, in a properly written constitutive equation, the material symmetries are automatically verified. In this Chapter, "tensors" and "vectors" mean second order tensors and vectors in a three-dimensional space. We restriet ourselves to constitutive functions of the arguroents: A • A • . .. A _l _2 _a •
w •w • • .• Wb. _l _2 _
V • _l
V • _2
• •• V
_c
(1 )
where theA., W. and vk are respectively an arbitrary nurober of symmetric -1
-J
-
tensors, skew-symmetric tensors and vectors.
J. P. Boehler (ed.), Applications of Tensor Functions in Solid Mechanics © Springer-Verlag Wien 1987
32
J.P. Boehler Consider first a constitutive law specifying a scalar A =
A
(2)
f(A., W., vk) -1
-J
-
If S is the group of transformations characterizing the material symmetries of the medium, the function f is subjected to the conditions:
VQ
e:
s
(3)
The scalar A is called scalar invariant under the group s. Consider now a constitutive equation specifying a symmetric second order tensor T: (4)
T = F(A., W., ~k) - -1 -J
Similarly, the function F is subjected to the conditions:
The function F is then called form-invariant under the group S and T is a tensorial invariant. 1.2 Representations for polynomial scalar and tensor functions In this Section, we suppose that the function f and the components in a reference frame of the function F are polynomials in the components of the arguments (1). The values A and T are then polynomial invariants under the group
s.
The problern of the representation for the scalar-valued function f is to determine a basic set of polynomial scalar invariants (I , I , ... I ), 1 2
p
such that an arbitrary polynomial scalar invariant of the same arguments can be expressed as a polynomial in the basic invariants. Such a set is called "integrity basis" for the considered list of arguments (1) and the group of transformations s. An integritybasis is termed irreducible if none of its subsets constitutes a complete representation by itself, i.e.
33
Representations for Tensor Functions
if no element of it can be expressed as a polynomial in the remainder. Even if an integrity basis is irreducible, polynomial relations may exist between its elements; but these relations do not enable any one of it to be expressed as apolynomial of the others. Such relations are called syzygies. The problern of the representation for the tensor-valued function F is to determine a generating set of tensors G., thus symmetric secend -1
order tensors which are invariant under the group s, such that the ten
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