On the Representation of Symmetric Isotropic 4-Tensors (original) (raw)
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International Journal of Pure and Apllied Mathematics, 2014
In a 4-dimensional Euclidean space, representation theorems have been recently obtained for isotropic functions depending on an arbitrary number of scalars, skew-symmetric second order tensors and symmetric second order tensors; the cases has been treated where at least one of these last ones has an eigenvalue with multiplicity 1 or two distinct eigenvalues with multiplicity 2. The case with at least a non null vector, among the independent variables, was already treated in literature. There remain the case where every symmetric tensor has an eigenvalue with multiplicity 4; but, in this case, it plays a role only through its trace. Consequently, it remains the case where the independent variables, besides scalars, are skew-symmetric tensors. This case is treated in the present paper. As in the other cases, the result is a finite set of scalar valued isotropic functions such that every other scalar function of the same variables can be expressed as a function of the elements of this set. Similarly, a set of tensor valued isotropic functions is found such that every other tensor valued function of the same variables can be expressed as a linear combination, trough scalar coefficients, of the elements of this set. This result is achieved both for symmetric functions , and for skew-symmetric functions.
On the representation theorem for linear, isotropic tensor functions
Journal of Elasticity, 1995
The well-know representation theorem for the elasticity tensor C of an isotropic body shows that C[E] = 2#E + ~ tr(E)I (1) for all symmetric tensors E, where tr(E) denotes the trace of E and I is the identity tensor. This theorem is actually a special case of a classical result (cf. e.g. [Je 31, Chapter 7]) on linear, tensor-valued mappings that are isotropic, i.e. C[QHQ T] = QC[HjQ T for all tensors H in the domain of C and all orthogonal tensors Q, where Q~ denotes the transpose of Q.
Intrinsic characterization of space-time symmetric tensors
Journal of Mathematical Physics, 1992
This paper essentially deals with the classification of a symmetric tensor on a fourdimensional Lorentzian space. A method is given to find the algebraic type of such a tensor. A system of concomitants of the tensor is constructed, which allows one to know the causal character of the eigenspace corresponding to a given eigenvalue, and to obtain covariantly their eigenvectors. Some algebraic as well as differential applications are considered.
On the Construction of Linearly Independent Tensors
—We consider various methods for constructing linearly independent isotropic, gyrotropic, orthotropic, and transversally isotropic tensors. We state assertions and theorem that permit one to construct these tensors. We find linearly independent above-mentioned tensors up to and including rank six. The components of the tensor may have no symmetry or have symmetries of various types. It is known [1–5] that E ˜ E = r i r i = g ij r i r j is the only isotropic tensor of rank 2 that can be used to represent any other isotropic tensor a ˜ a of rank 2 in the form a ˜ a = aE˜E, where a is a scalar; i.e., an arbitrary isotropic tensor of rank 2 is a spherical tensor. The tensors C ˜ C ˜ C (1) = E ˜ EE˜E = r i r i r j r j , C ˜ C ˜ C (2) = r i r j r i r j , C ˜ C ˜ C (3) = r i E ˜ Er i = r i r j r j r i (1.1) are three linearly independent (irreducible to each other) tensors of rank 4. The general expression for an arbitrary isotropic tensor of rank 4 is their linear combination C ˜ C ˜ C = 3 k=1 a k C ˜ C ˜ C (k). If we pay attention to the structure of isotropic tensors of rank 2 and rank 4 in (1.1), then we easily see that they can be obtained from the corresponding multiplicative bases by pairwise convolution (contraction) of indices of the basis vectors and by exhausting all possible cases of such contraction. By way of example, let us also construct all linearly independent isotropic tensors of rank 6. The multiplicative basis of a tensor of rank 6 is r i r j r k r l r m r n. By contracting the indices pairwise arbitrarily, we obtain some isotropic tensor of rank 6. For example, r i r i r k r k r m r m = E ˜ EE˜EE˜E. (1.2) All other isotropic tensors of rank 6 can be obtained from (1.2) by permutations of basis vectors. Obviously, by rearranging the basis vectors in (1.2), we obtain 6! = 720 permutations (isotropic tensors of rank 6) in the general case. Of these tensors, only fifteen are linearly independent (irreducible to each other) [3, 5]. To obtain these linearly independent tensors of rank 6, it suffices, for example, to consider the following tensors: r i r i r k r k r m r m , r i r k r i r k r m r m , r i r k r k r i r m r m , r i r k r k r m r i r m , r i r k r k r m r m r i. (1.3) It is clear that the basis vector r i occupies all possible positions in (1.3). Now by keeping the vectors r i and r i at their positions in (1.3) and by permuting the other vectors, we obtain two additional tensors
On Symmetric irreducible tensors in d -dimensions
ARI - An International Journal for Physical and Engineering Sciences, 1998
Aim of the paper is to establish the notion of symmetric irreducible tensors in arbitrary dimensions d ≥ 2. These tensors are generalisations of a symmetric traceless, second order tensor and their significance stems from their close connection to spherical harmonics. We introduce the general concepts and derive some fundamental relations with respect to these tensors. Special considerations are given to proofs, because those are hardly (or not at all) to find in the literature. The relation to spherical harmonics mentioned above is used to obtain a theorem which allows L 2 (S d )-functions to be expressed as tensor series with respect to symmetric irreducible tensors. Such representations can be (and have been) used in the theory of liquid crystals to deal with orientation distribution functions and to introduce suitable order parameters.
Mechanics Research Communications, 2011
In the present paper we propose new results concerning linear tensorial algebra for third-order and non symmetric isotropic sixth-order tensors in the most general case (i.e. having not the major and minor symmetries). Such tensors are used, for instance, in the theory of microstructured elastic media. A formalism based on an irreducible basis for isotropic sixth-order tensors is introduced, which is useful for performing the classical tensorial operations. Specially, a condensed expression for the product between two isotropic sixth-order tensors is provided, which allows the obtaining of a condition on these tensors for being invertible and a closed form expression of the inverse of such a tensor. Finally, the condition of positiveness of third-order tensor-valued quadratic functions is derived. For instance, such conditions are required for computing the elastic energy of microstructured media.
A note on the nonabelian tensor square
Indian Journal of Science and …, 2012
In this paper, we determine the nonabelian tensor square G ⊗G for special orthogonal groups SO n (F q ) and spin groups Spin n (F q ), where F q is a field with q elements.