Construction of Invariant Relations of n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n$\end{document} Symmetric Second-Order Tensors

A methodology is presented to find either implicit or explicit relations, called syzygies, between invariants in a minimal integrity basis for n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n$\end{document} symmetric second-order tensors defined on a three-dimensional euclidean space. The methodology i) yields explicit non-polynomial expressions for certain invariants in terms of the remaining invariants in the integrity basis and ii) allows the construction of the implicit relations. The results of this investigation are important in modeling biological structures, which, in general, are non-homogeneous and made of anisotropic viscoelastic materials that are subjected to large deformations and are modeled through constitutive relations that depend on symmetric tensors.

Recent mathematical developments of the theory are reported by Kemper [9] and [10], who have worked on optimal homogeneous systems of parameters and separating sets, Olive and Auffray [13] and Chen et al. [4], on isotropic invariants of third-order tensors, Olive et al. [15], Desmorat et al. [6], Desmorat et al. [7], on minimal integrity basis and separating set for the fourth-order elasticity tensor, and Olive and Desmorat [14], on effectivce rationality of second-order symmetric tensors.Our approach is based on the invariant theory found in classical texts of mechanics, such as Spencer [29], Smith [28], and Zheng [32].
The application of invariance principles in continuum mechanics leads to the proposition of constitutive relations that depend on a list of invariants of physical variables, such as vectors and second-order tensors.Given a group of transformations acting on these variables, the central problem of the associated theory of invariants is to find a list of invariants from which all the other invariants can be generated without having redundant members.In this work, the invariants are real-valued polynomial functions of their arguments and this list is called an integrity basis if any invariant can be expressed as a polynomial of the members in the list.The integrity basis is minimal if it contains the smallest possible number of members.
The construction of minimal integrity bases in continuum mechanics has been the subject of intense investigation since the 1950s, an account of which can be found in Spencer [29], and is, by now, well established.Concerning symmetric second-order tensors defined on the three-dimensional euclidean space, Spencer [29] presents a thorough analysis about the construction of the minimum integrity basis for a finite number n of these tensors.As noted in Sect.2, the number of elements in this basis increases rapidly with n (see also Olive and Desmorat [14]), which represents a serious limitation for the application of the corresponding results in practice.
The members of a minimal integrity basis may satisfy polynomial relations between invariants which do not permit any one invariant to be expressed as a polynomial in the remainder.These relations may be explicit, in which case an invariant may be expressed as a rational function of the remainder, or, implicit.If the invariants do not depend on the other invariants through any type of relation, implicit or explicit, they are called independent.The number of independent variables in a minimal integrity basis is 6 n − 3 (see, for instance, da Rocha and Aguiar [5], Aguiar and da Rocha [1], Shariff [25,26], and Shariff et al. [27]).
The determination of the number of syzygies in a minimal integrity basis together with the construction of these syzygies continues to be an active area of research.In terms of applications in continuum mechanics, syzygies provide additional expressions which the invariants of the constitutive relations must satisfy.Thus, in case the values of the invariants are obtained experimentally, the syzygies can be used to verify the accuracy between the experimental values and the corresponding theoretical values of the invariants.
In this work, we propose a methodology, based on the works of Smith [28] and Zheng [32], that i) yields explicit non-polynomial expressions for certain invariants in terms of the remaining invariants in the integrity basis and ii) allows the construction of implicit relations between the invariants.To obtain these results, the first step in the methodology is to construct the set of 6 n − 3 independent invariants.In Sect. 2 we present some preliminary results, which include the total number of invariants in a minimal integrity basis for n symmetric tensors.In Sect. 3 we investigate the cases n = 1, . . ., 5, and then generalize the results for n > 5.In Sect. 4 we present some concluding remarks.

Preliminaries
We are concerned with symmetric second-order tensors in three-dimensional euclidean space.The summation convention is not used and indices of tensor components take the values 1, 2, 3.
Let us consider that all the symmetric tensors in the set A (n) def = {A (1) , A (2) , . . ., A (n) }, where n ≥ 1, have three distinct eigenvalues and that no two tensors in this set do not have parallel eigenvectors.If {e 1 , e 2 , e 3 } is the set of eigenvectors of A (1) with associated eigenvalues (λ 1 , λ 2 , λ 3 ), we write where are the six components of the tensor A (r) in the basis {e 1 , e 2 , e 3 }.It is then clear from (1) that the maximum number of distinct components of all the tensors in the set Also, the trace of A (r) is given by tr Now, let Q be a second-order orthogonal tensor, such that Q Q T = 1, where Q T is the transpose of Q and 1 is the second-order identity tensor.We say that a real-valued function f : A (n) → R is an invariant of the tensors in A (n) under the group of second-order orthogonal tensors if f Ā(1) , Ā(2) , . . ., Ā(n) = f A (1) , A (2) , . . ., A (n) , Ā(r for every Q of the orthogonal group.From classical invariant theory,1 all the invariants of second-order tensors only can be expressed in terms of traces of products of the tensors in A (n) .We consider invariants that are real-valued polynomial functions in their arguments.
The central problem of the theory of invariants, as it applies to this work, is to determine a set of polynomial invariants from which all the other polynomial invariants can be generated and which contains the smallest possible number of members.In our work, this set is the minimal integrity basis for the set A (n) under orthogonal transformations and is completely characterized in Spencer [1971].For completeness, in Table 2 of Appendix A we present the matrix products whose traces generate the minimal integrity basis for A (n) .Observe from this table that the total number of invariants of A (n) , n > 0, is given by where β i is an integer given in the first column of Table 2 and 26 n 5 + 10 n 6 , which shows that N(n) increases rapidly with n.This expression is presented in Table 1 of Olive and Desmorat [14]. 2  Based on the works of Smith [28] and Zheng [32], we present below a methodology to find either implicit or explicit relations between invariants in the minimal integrity basis for A (n) .The methodology i) yields explicit non-polynomial expressions for certain invariants in terms of the remaining invariants in the integrity basis, and ii) allows the construction of implicit relations between the invariants.To obtain these results, the first step is to construct the set of 6 n − 3 independent invariants, which is included here for completeness of presentation.

Methodology
We investigate the cases n = 1, . . ., 5, and then generalize for n > 5. Similarly to the works of Smith [28] and Zheng [32], the basic idea is to construct a bijection between subsets of invariants and sets of combinations of products of the components α (r)  ij defined in (2).Based on this bijection, we clearly identify the independent invariants and syzygies between elements of the subsets of invariants.

The Case n = 1
Claim All the 3 classical invariants are independent.

Proof The three invariants
depend on the three eigenvalues λ i , i = 1, 2, 3, which are independent variables.In addition, these variables are roots of the characteristic equation λ 3 − J 1 λ 2 + J 2 λ − J 3 = 0, where It is well known that this characteristic equation yields three real-valued expressions for λ in terms of the invariants J i , i = 1, 2, 3, and, in view of (6), in terms of the invariants I 1 i , i = 1, 2, 3. Thus, the three invariants in (6) are independent.
In this way, we have shown that there is a bijection between the sets Ψ and Ω.Since α (2)  12 α (2)  23 α ( 2) and since the integrity basis is minimal and contains the 10 invariants given by both ( 6) and (7), it follows from steps a)-d) above that relation (8) yields a syzygy between I 2 3 and the other invariants.Since the elements of the set Ω I def = Ω\{α (2)  12 α (2)  23 α (2)  13 } are independent, the claim is proved.

The Case n = 3
Claim From N(3) = 28 classical invariants, 15 invariants are independent, 11 invariants satisfy, at least, 19 syzygies, and the remaining 2 invariants are rational functions of the other invariants.
In summary, observe from steps a) -d) that only 21 elements in the set Ω are independent.They are the elements in the set Ω I def = Ω a \{α (r)  12 α (r) 23 α (r) 13 }, r = 2, 3, 4. Since, by Step a), there is a bijection between Ω I and the set Ψ I def = Ψ a \{I r 3 }, r = 2, 3, 4, we see that Ψ I contains 21 independent invariants.All the other invariants depend on these invariants through both explicit expressions discussed in Step e) and implicit relations obtained from (8), ( 10) thru (12), and (15) thru (18).Thus, the claim is proved.
Let the set of 251 invariants be given by Ψ . ., 26, and, analogous to the previous sections, let Ω be a set of products between the components of the symmetric tensors in A (5) , which will be determined from these invariants by following the steps below.13 }, k, l = 1, 2, 3, l > k, p = 2, . . ., 5. We also have that the last 4 terms of Ω a satisfy 4 relations, given by ( 8), ( 10), (15), and a relation that can be obtained from (15)  23 } for p, q = 2, . . ., 5, q > p, and, therefore, there is a bijection between the sets Ψ b and Ω b .The terms in Ω b are not independent.They must satisfy (11), (16), and 12 relations that can be obtained from ( 16) by replacing the superscript (4) with the superscript (5), where, here, i = 2, 3, 4, and the terms (α   2) (n − 3) elements.We also find that the elements of Ω d are not independent and satisfy (n − 1) (n − 2)(n − 3) relations having the forms given in (20) with the superscript (5) replaced by (n).It follows from above that the set 1.These elements determine all the possible combinations between the components of the tensors and yield all the N(n) invariants for A (n) .Since there is a bijection between the sets Ω and the difference between N(n) and the number of invariants in Ψ ∪ yields the number of invariants that can be obtained explicity in terms of the invariants in the set Ψ ∪ , which is given by N R (n) in Table 1.It also follows from above that (n − 1)(n 2 − n/2 − 2) invariants in Ψ ∪ satisfy, at least, (n − 1)(n 2 + 7n/2 − 10) implicit relations, and these numbers correspond to, respectively, N D (n) and N S (n) in Table 1.

Conclusions
The minimal integrity basis for n symmetric second-order tensors in the set A (n) has N(n) invariants given by ( 5), from which N I (n) = 6 n − 3 invariants are independent, N D (n) = (n − 1)(n 2 − n/2 − 2) invariants satisfy, at least, N S (n) = (n − 1)(n 2 + 7n/2 − 10) implicit relations between the invariants, and the remaining N R (n) = N(n) − (N I (n) + N D (n)) invariants are rational functions of the other invariants and are, therefore, determined explicitly.

Appendix A: Minimal Integrity Basis for A (n)
Let A (r) , r = 1, . . ., n, be given by (1).For n ≤ 6, the minimal integrity basis is formed by the traces of the tensor products in Table 2, where β i , i = 1, . . ., 6, is the number of these products in the i-th row and we recall from Sect.3.4 that ( †) means cyclic permutation of the superscripts, together with the traces of tensor products of all the subsets of tensors that can be chosen from the given set.For example, the minimal integrity basis for the case n = 3 has β 1 n + β 2 n + β 3 = 28 elements and its elements are given by the expressions in ( 6), (7), and (9).For n > 6, the integrity bases are obtained by forming the sum of the integrity bases taken six at a time in all the possible combinations.
The case n = 2: In addition to the list of invariants given by (22), we have the list of invariants presented below.

a)
Following the steps a) thru d) in Sect.3.2, we find that there is a bijection between the set of 17 invariants given by Ψ a def i = 1, 2, 3, j = 1, . . ., 4, p = 2, 3, and the set of 17 terms given by Ω a def The 3 invariants in the set Ψ b def = {I 23 1 , I 123 1 , I 123 2 } depend linearly on the 3 elements in the set Ω b def = {α

a)
Following the steps a) thru d) in Sect.3.3, we have that there is a bijection between the set of 24 invariants given by Ψ a def i = 1, 2, 3, j = 1, . . ., 4, p = 2, 3, 4, and the set of 24 terms given by Ω a def = {λ k , α p, q = 2, 3, 4, q > p, depend linearly on the 9 elements in the set Ω b def = {α 13 , j = 2, 3, 4, were obtained in Step a).The bijection between Ψ b and Ω b then implies that the elements of Ψ b are not independent and must satisfy syzygies obtained from the expressions in(16).c) Analogously, the 18 invariants in the set Ψ c p, q = 2, 3, 4, q > p, depend linearly on the 18 elements in the set Ω c def = {α

6 }
13 , j = 2, 3, 4, 5, were obtained in Step a).The bijection between the sets Ψ b and Ω b then implies that the elements of Ψ b are not independent and must satisfy syzygies obtained from expressions analogous to(16).c) The 36 invariants in the set Ψ c def , p, q = 2, . . ., 5, q > p, depend linearly on the 36 elements in the set Ω c def

Table 1
Number of invariants and syzygies n