Representations of m -linear functions on tensor spaces: Duals and transpositions with applications in continuum mechanics

Abstract

The main goal of the present paper is to introduce certain duals and transpositions of a high-order tensor playing an important role in continuum mechanics. Emphasis is also placed on the comparison of the duals and the transpositions. In contrast to the duals, the transpositions depend on a metric of an underlying metric space. A high-order tensor is itself a representation of a multilinear function on a tensor space, obtained by means of the multilinear extension. The duals and the transpositions of a high-order tensor are identified as multilinear maps defined by the generalized scalar and the inner product, respectively. Consequently, the duals and the transpositions distinguish and define symmetries and symmetry-preserving transformation rules of a co-, contra- and mixed-variant tensor, respectively. As an application in continuum mechanics, the duals and the transpositions of a usual fourth-order tensor are defined and are employed to the determination of symmetries involved.

Keywords

Transpositions duals contractions symmetry transformation rules

1. Introduction

The duals and the transpositions of a tensor have numerous applications in continuum mechanics. They are needed for calculation of a tensor norm, for the definition of pull-backs and push-forwards and in connection with tensor symmetrizations. In the literature, the transpose of a bilinear map or a second-order tensor is well defined whereas the transpositions of higher-order tensors have remained abstract. On the basis of physical considerations [1], the scalar and the inner product are distinguished and are used to define the duals and the transpositions, respectively. Depending on the order of the tensor under consideration, various numbers of duals and transpositions can be defined. In the present paper, the major transposition, also termed the transpose, is defined separately from the minor transpositions. A tensor is itself a representation of a multilinear ( $m$ -linear) function which, after $m$ -linear extension, is given by the tensor products on the underlying tensor space. High-order tensors are considered here in terms of a theory involving algebraic equations expressed in the absolute notation and the construction of decomposable tensors from second-order tensors. In its organization, this paper follows the basics of differential geometry, essential for an in-depth mathematical analysis of variational initial-boundary value problems in continuum mechanics [2 –7]. In addition, $m$ -linear algebra is applied [8, 9]. Differential geometry and $m$ -linear algebra have also been used to provide an understanding of kinematics and internal geometry of multibody systems [10, 11] and of certain terms such as co-variance and contra-variance [5, 6, 12]. Examples of this are to be found in nonlinear continuum mechanics [13 –15], in finite elasto-plasticity [16, 17] and in micro-mechanics [18 –21].

In contrast to classical tensor algebra which is employed on inner product spaces, the classification of vector and tensor spaces in this study is based on tensor analysis on manifolds [6]. The derivations of these concepts differ since in the classical approach the identification of dual and primary vector spaces is performed a priori in contrast to tensor analysis on manifolds in which the different forms of tensors are clearly identified by the explicit use of a metric. This results in the transpositions as well as certain operations, such as pull-back, push-forward and tensor symmetrization, being correctly defined. For instance, the members involved in the symmetrization lie on one and the same space when their summation makes sense. Within this treatment, a fourth-order tangent stiffness tensor is of major importance. It is desired in numerical solution methods and enters the linearization of the stationary condition of a variational principle with respect to tensor-valued arguments. The existence of an associated quadratic potential in the variational principle results in a tangent stiffness tensor which is self-adjoint, i.e. it has the major symmetry property. If the tensor arguments are symmetric, the tangent stiffness tensor has corresponding sub or minor symmetries. In order to avoid the difficulties referred to above, the tangent stiffness tensor as well as its duals and transpositions for symmetrization should first be developed on suitable metric spaces. Thereafter, the component matrices for a numerical solution method can be evaluated on inner product spaces (e.g. in a Cartesian or in a curvilinear coordinate system) using explicit expressions of metrics [22]. Since all of the results developed on a one tensor space are available on its sub-spaces but not vice versa, the proposed concept is readily validated.

The paper is organized as follows. First, a brief survey of $m$ -linear algebra is given. $m$ -linear functions are represented by tensors based on the existence of a decomposable tensor which forms a basis for an underlying tensor space and enables the definition of a composition map on an underlying tensor space. The dual tensor basis is also introduced since it allows one to define the dual tensor as well as the mixed tensor on a dual and on a coupled primary–dual tensor space, respectively. Apart from the dual tensor basis, the dual tensor is defined in terms of the generalized scalar product for high-order tensors. Similarly, the transposed tensor is defined on the basis of the generalized inner product. In addition to the major dual and transpose, two minor duals and transpositions are introduced with the aid of the scalar and the inner product, respectively. Attention will also be directed at presentation of the commutative rules applying to the duals, the transpositions and the inverse of a tensor. It will be shown that the representation of a $m$ -linear function by a tensor depends on a specific identification of tensor spaces and, correspondingly, the duals and the transpositions obtain different explicit representations. An identification being appropriate for the requirements of continuum mechanics enables the duals and transpositions to be specified and employed for the determination of symmetries. It will be pointed out that the transpositions in connection with the inner products are required for defining symmetries of mixed-variant forms, whereas the duals related to the scalar products define symmetries of co- and contra-variant forms. As an application of the scalar and the inner product, the double contraction is defined consistently with the chain rule and the product derivative rule for tensor-valued functions [23]. Based on the calculation of the double contraction by the scalar products some frequently used results, in contrast to prior approaches, are solely given in terms of the dual tensors rather than the transposes [16, 24]. Within the framework adopted, symmetries are determined in a systematic manner involving their closed-form representations for different forms of a fourth-order tensor using either the duals or the transpositions. The results can be applied e.g. to the development of an algorithmically consistent tangent tensor for numerical solution procedures. An important result of this work is also the symmetry-preserving transformation rules for high-order tensors which are determined by pull-backs and push-forwards, respectively. These transformation rules, as will be shown, need a correction in the definition present in prior approaches. The correction results in the pull backs and push-forwards of different forms of tensors being consistent with each other.

2. Tensor representations on linear spaces

2.1. Introduction to m-linear functions

The present theory of tensor algebra involves the extension of classical linear algebra to $m$ -linear algebra. Since $m$ -linear algebra is only rarely applied to continuum mechanics, a brief survey, relevant for our purposes, will be provided. Roughly speaking, whereas in linear algebra one deals with vectors on vector spaces, $m$ -linear algebra concerns $m$ -linear functions on product vector spaces or tensors on tensor spaces. Tensor spaces are defined as combinations of two or more vector spaces which are all either the scalar or the inner product spaces. The existence and uniqueness of a tensor space are verified based on the universal factorization property (UFP) [9]. A tensor itself is defined on a tensor space where the connections between vector spaces are identified as usual tensor products, “ $⊠$ ”. Attention is also focused on the construction of a decomposable high order tensor from second order tensors.

Definition 2.1 Let $V_{1}, V_{2}, \dots, V_{m}$ be vector spaces and $T$ a tensor space. A function $Ф : V_{1} \times V_{2} \times \dots \times V_{m} \to T$ is multilinear ( $m$ -linear) if $Ф$ is linear separately in each component of the product vector space $V_{1} \times V_{2} \times \dots \times V_{m} = {(v_{1}, v_{2}, \dots, v_{i}, \dots, v_{m}) : v_{i} \in V_{i}, 1 \leq i \leq m}$ , i.e.

\begin{array}{l} Ф (v_{1}, v_{2}, \dots, c u_{i} + d w_{i}, \dots, v_{m})= c Ф (v_{1}, v_{2}, \dots, u_{i}, \dots, v_{m}) + d Ф (v_{1}, v_{2}, \dots, w_{i}, \dots, v_{m}), \\ v_{i} = c u_{i} + d w_{i} (1 \leq i \leq m), c, d \in ℝ . \end{array}

Definition 2.2 Let $V_{1}, V_{2}, \dots, V_{m}$ be vector spaces. A tensor space $T$ and a $m$ -linear function $Φ : V_{1} \times V_{2} \times \dots \times V_{m} \to T$ are said to satisfy the UFP if, for every tensor space $S$ and every $m$ -linear function $f : V_{1} \times V_{2} \times \dots \times V_{m} \to S$ , there exists a linear map $h : T \to S$ such that $f = h \circ Ф$ . The pair $(T, Ф)$ which satisfies UFP is called a universal pair.

Remark 2.3 To verify the existence of a universal pair, one needs to show that there exist $T, Ф$ and $h$ such that the diagram in Figure 1 commutes for each $f$ , i.e. $φ = h^{- 1}$ exists.

Figure 1.

Commutative diagram.

Definition 2.4 Suppose that the pairs $(S, f)$ and $(T, Ф)$ satisfy UFP for $V_{1}, V_{2}, \dots, V_{m}$ . The pairs are isomorphic if there exists an invertible linear transformation $φ$ from $S$ onto $T$ such that $Ф = φ \circ f$ .

It follows from Remark 2.3 and Definition 2.4 that the universal pairs $(S, f)$ and $(T, Ф)$ are isomorphic. The representation of a $m$ -linear function by a tensor is based on the existence of a decomposable tensor, one that constitutes a tensor space and enables a composition map $h$ to be developed on the underlying tensor space. A comprehensive discussion with regards to the representation of tensors on tensor spaces can be found in [8] and [9].

Definition 2.5 Let $Ф : V_{1} \times V_{2} \times \dots \times V_{m} \to T$ be a $m$ -linear function. If the pair $(T, Ф)$ satisfies UFP for $V_{1}, V_{2}, \dots, V_{m}$ , then a decomposable tensor

Ф (v_{1}, v_{2}, \dots, v_{m}) = v_{1} ⊠ v_{2} ⊠ \dots ⊠ v_{m} \in T,

exists and

T

is a tensor product space or a tensor space of

V_{1}, V_{2}, \dots, V_{m}

, written as

T : = V_{1} ⊠ V_{2} ⊠ \dots ⊠ V_{m}

Example 2.6 Let there be $x, y, u, v \in V$ . The tensor-valued map $Ф_{1} (x, y, u, v) = x ⊠ y + u ⊠ v \in V ⊠ V$ and the symmetrization $Ф_{2} (x, y) = s y m_{1} (x ⊠ y) : = 1 / 2 (x ⊠ y + y ⊠ x) \in V ⊠ V$ are not decomposable. The dyadic $Ф_{3} (x, y, v) = x ⊠ v + y ⊠ v = (x + y) ⊠ v \in V ⊠ V$ and the symmetrization $Ф_{4} (x, y) = s y m_{2} (x ⊠ y) : = 1 / 2 (x + y) ⊠ 1 / 2 (y + x) \in V ⊠ V$ are both decomposable since they are linear in their each vector, cf. Definition 2.1.

Definition 2.7 If $(S, f)$ satisfies UFP, then $S$ is said to be a model for a tensor product $T$ , cf. Figure 1.

Theorem 2.8 (Associativity) We say that $V_{1} ⊠ V_{2} ⊠ \dots ⊠ V_{m}$ is a model for $V_{1} ⊠ \dots ⊠ (V_{k} ⊠ \dots ⊠ V_{l}) ⊠ \dots ⊠ V_{m}$ , $1 \leq k < l \leq m$ where

v_{1} ⊠ v_{2} ⊠ \dots ⊠ (v_{k} ⊠ \dots ⊠ v_{l}) ⊠ \dots ⊠ v_{m} = v_{1} ⊠ v_{2} ⊠ \dots ⊠ v_{k} ⊠ \dots ⊠ v_{l} ⊠ \dots ⊠ v_{m} .

Proof. Let us define $T : = V_{1} ⊠ \dots ⊠ (V_{k} ⊠ \dots ⊠ V_{l}) ⊠ \dots ⊠ V_{m}$ , $v_{1} ⊠ \dots ⊠ (v_{k} ⊠ \dots ⊠ v_{l}) ⊠ \dots ⊠ v_{m} \in T$ and $S : = V_{1} ⊠ V_{2} ⊠ \dots ⊠ V_{m}$ . Since ${v_{1} ⊠ v_{2} ⊠ \dots ⊠ v_{m} :$ $v_{i} \in V_{i}, 1 \leq i \leq m}$ spans $V_{1} ⊠ V_{2} ⊠ \dots ⊠ V_{m}$ , the transformation

\begin{array}{l} h : T \to S such that \\ h (v_{1} ⊠ \dots ⊠ (v_{k} ⊠ \dots ⊠ v_{l}) ⊠ \dots ⊠ v_{m}) = v_{1} ⊠ v_{2} ⊠ \dots ⊠ v_{m} \in S \end{array}

is onto and, by a dimension argument, isomorphic. Within the framework of the theory, let define a unique function $f$ having the extension

f (v_{1} ⊠ \dots ⊠ v_{k - 1}, (v_{k} ⊠ \dots ⊠ v_{l}), v_{l + 1} ⊠ \dots ⊠ v_{m}) = v_{1} ⊠ v_{2} ⊠ \dots ⊠ v_{m} \in S,

for all $v_{i} \in V_{i}, 1 \leq i \leq m$ . Then the existence of $h$ is established by UFP, cf. Definition 2.2. $_{□}$

According to the associativity of tensor products, cf. Theorem 2.8, $Ф_{1} : = v_{1} ⊠ v_{2} ⊠ \dots ⊠ v_{m - 1} ⊠ v_{m}$ and $Ф_{2} : = (v_{1} ⊠ v_{2}) ⊠ \dots ⊠ (v_{m - 1} ⊠ v_{m})$ are equal and, through the sum and the multiplication with a scalar, constitute the same tensor space. Suppose that $T_{i} : V_{i} \to W_{i}$ , $1 \leq i \leq m$ (not necessarily decomposable) and define $S : = W_{1} ⊠ W_{2} ⊠ \dots ⊠ W_{m}$ . Owing to a unique composition ‘o’ (e.g. the scalar or the inner product) and the associativity of tensor products, there exists a unique (invertible) linear transformation

\begin{array}{l} h & : = T_{1} ⊠ T_{2} ⊠ \dots ⊠ T_{m} : T \to S such that \\ f & = h \circ Ф = T_{1} v_{1} ⊠ T_{2} v_{2} ⊠ \dots ⊠ T_{m} v_{m} \in S \end{array}

(1)

is unique, cf. Figure 1.

Definition 2.9 The unique linear transformation

\begin{array}{l} T : = T_{1} ⊠ T_{2} ⊠ \dots ⊠ T_{m} \end{array}

is defined to be a tensor composed of linear maps $T_{i} : V_{i} \to W_{i}$ , $1 \leq i \leq m$ .

2.2. Dual and reciprocal bases associated with the scalar and the inner product

In order to define the dual and the transposed tensor, the definition for the dual basis and the dual tensor space are needed. The dual tensor basis, which is also a dual tensor, is defined as a linear map of the primary tensor basis on the dual tensor space being an extension of vector space isomorphism. Based on the associativity property of the tensor products, cf. Theorem 2.8, one can uniquely define a mixed-form tensor basis which forms a mixed primary-dual tensor space. A comprehensive discussion related to the tensor presentations on mixed tensor spaces is given in the textbooks [5] and [8]. Since the dual and the transposed tensors are defined by means of the scalar and the inner product, respectively, we first start by briefly reviewing the fundamental ingredients for metric scalar and inner product spaces. A Riemannian metric is considered, the metric thus being symmetric and positive-definite (spd) [25].

The inner product (or the dot product induced by the metric $G$ ) between a vector $a$ and $b$ and the scalar product between the associated co-vector $a^{b} : = G a$ and a vector $b$ are related according to

∴ V \times V \to ℝ, a \cdot b : = {(G a, b)}_{V} .

(2)

If the metric space is obvious, $(\cdot, \cdot)$ can be used instead of ${(\cdot, \cdot)}_{V}$ . The inverse of the metric or the contra-variant metric $G^{- 1} = \det {(G)}^{- 1} cof (G) = : G^{♯}$ allows one to define the inner product on the dual space as well, i.e.

∴ V^{*} \times V^{*} \to ℝ, a \cdot b : = {(a, G^{- 1} b)}_{V},

(3)

where $a$ is a co-vector and $b^{♯} : = G^{- 1} b$ is called the vector associated to $b$ . Let ${G_{A}}$ and ${g_{A}}$ be ordered bases of the vector spaces $V$ and $W$ , respectively. On the vector spaces $(V, G)$ and $(W, g)$ , equipped with the metric $G$ and $g$ , the so-called reciprocal bases ${G^{B}}$ and ${g^{b}}$ to ${G_{A}}$ and ${g_{a}}$ , respectively, are defined by the inner product as

\begin{array}{l} G_{A} \cdot G^{B} & = δ_{A}^{B}, & a n d g_{a} \cdot g^{b} & = δ_{a}^{b}, \end{array}

(4)

where $δ_{A}^{B}$ and $δ_{a}^{b}$ are Kronecker’s delta symbols. The inner product is defined between objects existing on the one and the same space. With the notation $G_{A B} : = G_{A} \cdot G_{B}$ and $G^{A B} : = G^{A} \cdot G^{B}$ , and analogously for $g_{ab}$ and $g^{ab}$ , one can deduce the linear relations between the reciprocal base vectors and the base vectors as being

G^{A} = G^{A B} G_{B} and g^{a} = g^{a b} g_{b} .

(5)

Moreover,

δ_{B}^{A} = G^{A C} G_{C B} a n d δ_{b}^{a} = g^{a c} g_{c b} \Rightarrow [G^{A B}] = {[G_{A B}]}^{- 1} a n d [g^{a b}] = {[g_{a b}]}^{- 1},

(6)

where $[\cdot] \in ℝ^{3 \times 3}$ denotes the component matrix of the metric tensor. It then follows that

G_{A} = G_{A B} G^{B} a n d g_{a} = g_{a b} g^{b}

(7)

and, thereby, a vector can be written as $a = a^{A} G_{A} = a_{A} G^{A}$ . The vector’s component $a^{A}$ is related to the reciprocal vector’s component $a_{B}$ through $a^{A} = G^{A B} a_{B}$ .

The corresponding dual bases ${G^{A}}$ and ${g^{a}}$ on the dual spaces $V^{*}$ and $W^{*}$ , respectively, are defined as

\begin{array}{l} {(G^{A}, G_{B})}_{V} & = δ_{B}^{A}, & {(g^{a}, g_{b})}_{W} & = δ_{b}^{a}, \end{array}

(8)

where the parentheses, $(,)$ , express the dual pairing also termed the scalar product. The relations between the dual and the reciprocal base vectors are given by

G^{A} = G G^{A} = : {(G^{A})}^{♯} \in V^{*}, g^{a} = g g^{a} = : {(g^{a})}^{♯} \in W^{*},

(9)

or the dual vectors $G^{A}$ and $g^{a}$ equal to the co-vectors associated to $G^{A}$ and $g^{a}$ , respectively. This allows the metric tensors to be defined as the co-variant form, i.e. $G = G_{A B} G^{A} ⊠ G^{B} \in V^{*} ⊠ V^{*}$ and $g = g_{a b} g^{a} ⊠ g^{b} \in W^{*} ⊠ W^{*}$ . Use of Equations (5), (6) and (9) in (8) makes it possible to write the scalar product as an inner product of two vectors on $V$ , naturally involving the reciprocal base vectors as

(G^{A}, G_{B}) = (G G^{A}, G_{B}) = G^{A} \cdot G_{B} = δ_{B}^{A} .

(10)

From Equations (5), (6) and (10) it follows that

G_{A} \cdot G^{- 1} = G_{A}, G^{A} \cdot G^{- 1} = G^{A B} G_{B} = G^{A},

(11)

i.e. the vectors and the reciprocal vectors are immutable for the contra-variant metric.

Let $F : V \to W$ be a linear map represented by a tensor product between a vector $u \in W$ and a co-vector $b \in V^{*}$ as $F = u ⊠ b \in W ⊠ V^{*}$ . The linear mapping $F a, a \in V$ is carried out by the scalar product of $b$ and $a$ as

F a = (u ⊠ b) a = {(b, a)}_{V} u \in W .

(12)

2.3. Representations of linear maps on mixed tensor spaces: the generalized scalar and the inner product

In addition to a decomposable tensor $T$ by Definition 2.9, there exists a unique (invertible) linear transformation (not necessarily decomposable) $C : T \to S$ such that the diagram in Figure 1 commutes [5, 8].

Definition 2.10 Let $V_{i}$ , $1 \leq i \leq m$ , be $m_{i}$ -dimensional (real) vector spaces and ${G_{1 i_{1}}$ , $G_{2 i_{2}}$ , …, $G_{m i_{m}}}$ their ordered bases. The co-vectors ${G^{1 i_{1}}, G^{2 i_{2}}, \dots, G^{m i_{m}}}$ , for which $(G^{j i_{j}}, G_{j k_{j}})$ = $δ_{k_{j}}^{i_{j}}$ (Kronecker’s delta), form the bases of $V_{i}^{*}$ .Similarly, let $W_{j}$ , $1 \leq j \leq n$ ,be $n_{j}$ -dimensional (real) vector spaces, ${g_{1 j_{1}}, g_{2 j_{2}}, \dots, g_{n j_{n}}}$ their ordered bases, and ${g^{1 j_{1}}, g^{2 j_{2}}, \dots, g^{n j_{n}}}$ the co-vector bases of $W_{j}^{*}$ .Then a $(r, s; t, u)$ mixed-variant tensor space is defined as

\begin{array}{l} T (r, s; t, u) & = \underset{r}{\underset{︸}{V_{1} ⊠ \dots ⊠ V_{r}}} ⊠ \underset{s}{\underset{︸}{V_{1}^{*} ⊠ \dots ⊠ V_{s}^{*}}} ⊠ \underset{t}{\underset{︸}{W_{1} ⊠ \dots ⊠ W_{t}}} ⊠ \underset{u}{\underset{︸}{W_{1}^{*} ⊠ \dots ⊠ W_{u}^{*}}} \end{array}

or it is a space of $r$ -fold on the vector spaces $V_{i}$ , $s$ -fold on the co-vector (or dual) spaces $V_{j}^{*}$ , $t$ -fold on the vector spaces $W_{k}$ and $u$ -fold on the co-vector space $W_{l}^{*}$ , $1 \leq i \leq r \leq m$ , $1 \leq j \leq s \leq m$ , $1 \leq k \leq t \leq n$ , $1 \leq l \leq u \leq n$ .

Specifically, if $d i m (V_{i}) = m_{i}, 1 \leq i \leq r$ , and $d i m (V_{j}^{*}) = n_{j}$ , 1 ≤ j ≥ s, the dimension of a $(r, s)$ -mixed variant tensor space is $\prod_{i} \prod_{j} m_{i} n_{j}$ . If $m_{i} = n_{j} = 3$ , $1 \leq i \leq 2$ , $1 \leq j \leq 2$ , for instance a (2,2)-mixed variant tensor space consists of three-dimensional vector spaces and the dimension is $3^{2 + 2} = 81$ .

Definition 2.11 Let $C \in T (r, s; t, u)$ . Then $C$ is termed a $(r, s; t, u)$ -mixed variant tensor.

Note here that e.g. the tensor basis $G_{1 i_{1}} ⊠ \dots ⊠ G_{k i_{k}} ⊠ G^{1 j_{1}} ⊠ \dots ⊠ G^{l j_{l}} \in T (2,2; 0,0)$ is itself a mixed-variant tensor (decomposable). Let it be agreed on that a first-order contra-variant tensor is a vector, a first order co-variant tensor is a co-vector, and that both 0-contra-variant and 0-co-variant tensors are considered as real numbers.

The following proposition results from Definition 2.11, cf. also [5, p. 67, Proposition 4.4]

Proposition 2.12 Let ${G_{i a_{i}} : 1 \leq a_{i} \leq n_{i}, 1 \leq i \leq r}$ , ${G^{j b_{j}} : 1 \leq b_{j} \leq n_{j}, 1 \leq j \leq s}$ , ${g_{k α_{k}} : 1 \leq α_{k} \leq n_{k}, 1 \leq k \leq t}$ and ${g^{l β_{l}} : 1 \leq β_{l} \leq n_{l}, 1 \leq l \leq u}$ be ordered bases of $V^{i}$ , $V^{* j}$ , $W^{k}$ and $W^{* l}$ , respectively. Often, $n_{i} = n_{j} = n_{k} = n_{l} = n = 3$ . The representation of a tensor $C \in T (r, s; t, u)$ becomes

\begin{array}{l} C = {C^{1 a_{1}, 2 a_{2}, \dots, r a_{r}}}_{1 b_{1}, 2 b_{2}, \dots, s b_{s}}^{1 α_{1}, 2 α_{2}, \dots, t α_{t}}_{1 β_{1}, 2 β_{2}, \dots, u β_{u}} G_{1 a_{1}} ⊠ \dots ⊠ G_{r a_{r}} ⊠ G^{1 b_{1}} ⊠ \dots \\ ⊠ G^{s b_{s}} ⊠ g_{1 α_{1}} ⊠ \dots ⊠ g_{t α_{t}} ⊠ g^{1 β_{1}} ⊠ \dots ⊠ g^{u β_{u}} . \end{array}

As an illustration, a special class of tensors, called one- and two-point tensors, will be introduced being essential in continuum mechanics [26]. A tensor is called a two-point tensor if it is defined on two different vector spaces $V, W$ , whereas a one-point tensor is solely defined either on $V$ or on $W$ .

Example 2.13 A second order one-point tensor can be represented in different bases by Equations (5),(7), and (9) as

\begin{array}{l} C^{b} = C_{i j} G^{i} ⊠ G^{j} = C_{i j} G G^{i} ⊠ G G^{j} = C_{i j} G^{i k} G^{j l} G G_{k} ⊠ G_{l} G = C^{i j} G G_{i} ⊠ G_{j} G = \\ G C^{♯} G = \dots = C_{i j} G^{j l} G^{i} ⊠ G_{l} G = C_{i .}^{j} G^{i} ⊠ G_{j} G = \dots = C_{. j}^{i} G G_{i} ⊠ G^{j} = : G C^{♮}, w h e r e \\ C^{b} = C_{i j} G^{i} ⊠ G^{j} \in T (0,2; 0,0), C^{♯} = C^{i j} G_{i} ⊠ G_{j} \in T (2,0; 0,0) a n d \\ C^{♮} = C_{. j}^{i} G_{i} ⊠ G^{j} \in T (1,1; 0,0) \end{array}

are the co-, contra-, and mixed-variant form, respectively. Above the relation, $a ⊠ G b = (a ⊠ b) G$ was made use of. As shown, the identities $C^{b} = G C^{♯} G = G C^{♮}$ hold. The components are given by

\begin{array}{l} C^{i j} = C_{k l} G^{i k} G^{j l}, \dots, C_{. j}^{i} = C_{k j} G^{k i} . \end{array}

If some of the base (co-)vectors are replaced by base (co-)vectors of a different (dual-)vector space, the tensor is called a second-order two-point tensor, cf. [5, 26].

Example 2.14 The co-, contra-, and mixed-variant forms of a fourth-order one-point tensor are given by

\begin{array}{l} C^{b} = C_{i j k l} G^{i} ⊠ G^{j} ⊠ G^{k} ⊠ G^{l} = C_{i j k l} G^{i m} G G_{m} ⊠ G^{j n} G G_{n} ⊠ G^{k q} G G_{q} ⊠ G^{l r} G G_{r} \\ = \dots = {C^{i}}_{. j}^{k}_{. l} G G_{i} ⊠ G^{j} ⊠ G G_{k} ⊠ G^{l} w h e r e \\ C^{b} = C_{i j k l} G^{i} ⊠ G^{j} ⊠ G^{k} ⊠ G^{l} \in T (0,4; 0,0), \\ C^{♯} = C^{i j k l} G_{i} ⊠ G_{j} ⊠ G_{k} ⊠ G_{l} \in T (4,0; 0,0), \dots, \\ C^{♮} = {C^{i}}_{. j}^{k}_{. l} G_{i} ⊠ G^{j} ⊠ G_{k} ⊠ G^{l} \in T (2,2; 0,0) \end{array}

having the components

\begin{array}{l} C^{i j k l} : = C_{m n q r} G^{i m} G^{j n} G^{k q} G^{l r} e t c . \end{array}

If some of the base-(co-)vectors are replaced by the base-(co-)vectors of a different (dual-)vector space, the tensor in question is called a fourth-order two-point tensor.

As a result of Equation (9), one cannot equalize the different forms as one can in classical tensor algebra, i.e. $C^{b} \neq C^{♮}$ etc., cf. Examples 2.13 and 2.14. Owing to the universal nature of tensors, the usual notations ‘ $b$ ’, ‘ $♮$ ’, ‘ $♯$ ’ for co-, mixed-, and contra-variant forms, respectively, will be only used if necessary, such as when the different forms are compared.

Let $Ф_{1}$ and $Ф_{2}$ be decomposable tensors that constitute $T_{1}$ and $T_{2}$ , respectively. Assuming there exist $h_{1} \in T_{1}$ and $h_{2} \in T_{2}$ such that the diagram in Figure 1 commutes for both $h_{1} \in T_{1}$ and $h_{2} \in T_{2}$ . Let define a tensor product ‘ $\land$ ’ (e.g. ‘ $⊠$ ’) and a unique composition ‘ $\circ$ ’ (e.g. the scalar or the inner product). Owing to a unique composition $\circ \circ$ and the associativity, cf. Theorem 2.8, there exists a unique linear transformation such that the diagram in Figure 1 commutes, i.e.

h_{1} \circ Ф_{1} \land h_{2} \circ Ф_{2} = (h_{1}) \land (h_{2}) \circ \circ (Ф_{1}) \land (Ф_{2}) = (h_{1} \land h_{2}) \circ \circ (Ф_{1} \land Ф_{2}) = : h \circ Ф .

(13)

On the basis of Equation (13), $(h_{1}) \land (h_{2}) = h_{1} \land h_{2} = h$ , since $Ф_{1} \land Ф_{2} = Ф$ is a model for $(F_{1}) \land (F_{2})$ . The representations of the linear maps $h_{1}$ and $h_{2}$ by the tensors $C_{1}$ and $C_{2}$ depend on a specific identification of vector and tensor spaces [6]. Since the notions $T (2,2; 0,0)$ , $T (1,1,1,1)$ , etc., are ambiguous, use is made of an eligible identification in which any bilinear map is represented by a second-order tensor, whereas any $2 m$ -linear map is represented by a $2 m$ th-order tensor.

Definition 2.15 Let

\begin{array}{l} T \in L i n (V_{1} ⊠ V_{2} ⊠ \dots ⊠ V_{m}; W_{1} ⊠ W_{2} ⊠ \dots ⊠ W_{m}) \\ : = {V : = V_{1} ⊠ V_{2} ⊠ \dots ⊠ V_{m} \to W : = W_{1} ⊠ W_{2} ⊠ \dots ⊠ W_{m}} \\ ≅ W_{1} ⊠ V_{1}^{*} ⊠ W_{2} ⊠ V_{2}^{*} ⊠ \dots ⊠ W_{m} ⊠ V_{m}^{*} \end{array}

be a $2 m$ -linear map ¹ such that

\begin{array}{l} T (V) : = T \circ V = (T_{1} ⊠ T_{2} ⊠ \dots ⊠ T_{m}) \circ (v_{1} ⊠ v_{2} ⊠ \dots ⊠ v_{m}) \\ : = T_{1} v_{1} ⊠ T_{2} v_{2} ⊠ \dots ⊠ T_{m} v_{m} \in W, \\ T_{i} \in L i n (V_{i}; W_{i}) : = {T_{i} : V_{i} \to W_{i}), 1 \leq i \leq m} ≅ W_{i} ⊠ V_{i}^{*}, v_{i} \in V_{i}, 1 \leq i \leq m . \end{array}

Moreover, the identifications $V_{i} = V$ and $W_{i} = W, 1 \leq i \leq m$ are frequently employed in continuum mechanics.

Next, the definitions of the scalar and the inner product, given through Equations (3)–(13), are extended to concern higher-order tensors. Using the identification in Definition 2.15 subsequently two times yields

\begin{array}{l} ((v_{1} ⊠ v_{2} ⊠ \dots ⊠ v_{m}) \circ (w_{2} ⊠ w_{4} ⊠ \dots ⊠ w_{m})) \circ (w_{1} ⊠ w_{3} ⊠ \dots ⊠ w_{m - 1}) = \prod_{i = 1}^{m} (v_{i} w_{i}) \in ℝ . \end{array}

Identifying a composition ‘ $\circ$ ’ as the generalized scalar or the generalized inner product allows for the following definitions.

Definition 2.16 Let $v : = v_{1} ⊠ v_{2} ⊠ \dots ⊠ v_{m}$ and $w : = w_{1} ⊠ w_{2} ⊠ \dots ⊠ w_{m}$ be decomposable tensors on $V_{1} ⊠ V_{2} ⊠ \dots ⊠ V_{m}$ and $V_{1}^{*} ⊠ V_{2}^{*} ⊠ \dots ⊠ V_{m}^{*}$ , respectively. The scalar product is defined as

\begin{array}{l} (v, w) : = (v_{1} ⊠ v_{2} ⊠ \dots ⊠ v_{m}, w_{1} ⊠ w_{2} ⊠ \dots ⊠ w_{m}) = \prod_{i = 1}^{m} (v_{i}, w_{i}) \in ℝ, \end{array}

where $v_{i} \in V_{i}, w_{i} \in V_{i}^{*}, 1 ⩽ i ⩽ m$ . Let $w : = w_{1} ⊠ w_{2} ⊠ \dots ⊠ w_{m} \in V_{1} ⊠ V_{2} ⊠ \dots ⊠ V_{m}$ be a decomposable tensor. The inner product is defined as

\begin{array}{l} v • w : = (v_{1} ⊠ v_{2} ⊠ \dots ⊠ v_{m}) • (w_{1} ⊠ w_{2} ⊠ \dots ⊠ w_{m}) = \prod_{i = 1}^{m} (v_{i} \cdot w_{i}) \in ℝ, \end{array}

$w_{i} \in V_{i}, 1 \leq i \leq m$ .

The generalized inner product is formed with objects existing on one and the same space, whereas the generalized scalar product is formed between the objects that exist on the primary and the associated dual space. Similar to the inner product which is defined between two vectors by Equation (2), the generalized inner product for tensors is given in terms of the generalized scalar product as

∴ V_{1} \times V_{2} \times \dots \times V_{2 m} \to ℝ, v • w : = {(\tilde{G} v, w)}_{V},

(14)

where $\tilde{G} : = G_{1} ⊠ G_{2} ⊠ \dots ⊠ G_{m}$ , $G_{i} \in V_{i}^{*} ⊠ V_{i}^{*}$ , $1 \leq i \leq m$ , and $\tilde{G} v = : v^{b}$ is termed the co-tensor associated with $v$ .

The components of a high-order tensor given on a primary tensor space are evaluated by taking the generalized scalar product between a primary and a dual tensor basis, cf. Definition 2.16.

Proposition 2.17 Let ${G^{i a_{i}} : 1 \leq a_{i} \leq n_{i}, 1 \leq i \leq r}$ , ${G_{j b_{j}} : 1 \leq b_{j} \leq n_{j}, 1 \leq j \leq s}$ , ${g^{k α_{k}} : 1 \leq α_{k} \leq n_{k}, 1 \leq k \leq t}$ , and ${g_{l β_{l}} : 1 \leq β_{l} \leq n_{l}, 1 \leq l \leq u}$ be ordered bases of $V^{* i}$ , $V^{j}$ , $W^{* k}$ , and $W^{l}$ , respectively. The components of the tensor $C \in T (r, s; t, u)$ in Proposition 2.12 are evaluated as

\begin{array}{l} {C^{1 a_{1}, 2 a_{2}, \dots, r a_{r}}}_{1 b_{1}, 2 b_{2}, \dots, s b_{s}}^{1 α_{1}, 2 α_{2}, \dots, t α_{t}}_{1 β_{1}, 2 β_{2}, \dots, u β_{u}} = C ( & G^{1 a_{1}}, \dots, G^{r a_{r}}, G_{1 b_{1}}, \dots, G_{s b_{s}}, \\ g^{1 α_{1}}, \dots, g^{t α_{t}}, g_{1 β_{1}}, \dots, g_{u β_{u}}) = (C, \tilde{G}), \end{array}

where $\tilde{G} : = G^{1 a_{1}} ⊠ \dots ⊠ G^{r a_{r}} ⊠ G_{1 b_{1}} ⊠ \dots ⊠ G_{s b_{s}} ⊠ g^{1 α_{1}} ⊠ \dots ⊠ g^{t α_{t}} ⊠ g_{1 β_{1}} ⊠ \dots ⊠ g_{u β_{u}}$ .

3. The dual, transposed, and inverse map

3.1. Introduction of second-order tensors on tangent spaces

In continuum mechanics, vector and tensor spaces are often induced by manifolds since a manifold as a basic notion of a continuum provides essential clarifications for the variational initial boundary value problems. Consequently, the representation of a linear map by a tensor can naturally be constructed on points of a manifold or on tangent spaces of the same dimension. In addition, the duals, transpositions and, for example, the push-forward and the pull-back operations can be clearly defined by the explicit use of a metric [5, 6].

Let $P_{0}$ and $P$ be the placements of a manifold and let $T_{X} P_{0}$ and $T_{x} P$ be tangent spaces at the points $X \in P_{0}$ and $x \in P$ respectively. In order to apply the results of the preceding sections, use is made of the identification of the vector spaces $V, W$ and the tangent spaces $T_{X} P_{0}, T_{x} P$ . The natural bases on $T_{X} P_{0}$ and $T_{x} P$ are ${G_{A}, A = 1,2,3}$ and ${g_{a}, a = 1,2,3}$ respectively. Similarly, the corresponding dual bases of the cotangent spaces $T_{X}^{*} P_{0}$ and $T_{x}^{*} P$ are ${G^{A}, A = 1,2,3}$ and ${g^{a}, a = 1,2,3}$ , respectively. The bilinear tangent map² is defined locally as a two-point tensor [6],

F = F_{. A}^{a} g_{a} ⊠ G^{A} \in L i n (T_{X} P_{0}; T_{x} P) .

(15)

Definition 3.1 The adjoint or the dual $F^{*} \in Lin (T_{x}^{*} P; T_{X}^{*} P_{0})$ $F$ is defined by means of the dual pairing, Equation (12), as

{(F^{*} f, a)}_{T_{x} P_{0}} = {(f, Fa)}_{T_{x} P} \in ℝ \forall a \in T_{X} P_{0}, f \in T_{x}^{*} P .

Since the inner product is defined between the objects existing on the same space, the transposed map instead of the dual is provided.

Definition 3.2 The transpose of a two-point tensor $F \in Lin ((T_{X} P_{0}, G); (T_{x} P, g))$ is defined by means of the inner product, Equation (2), i.e.

\begin{array}{l} F^{T} a \cdot b = a \cdot Fb, \forall a \in T_{x} P, b \in T_{X} P_{0} . \end{array}

Corollary 3.3 The relation between the transposed and the dual map is given by

F^{T} = G^{- 1} F^{*} g \in L i n ((T_{x} P, g); (T_{X} P_{0}, G)), F \in Lin ((T_{X} P_{0}, G); (T_{x}, P g)) .

Proof. According to Definition 3.2, the transpose is evaluated on the basis of the inner product, Equation (2), and of the dual according to Definition 3.1, i.e.

F^{T} a \cdot b : = {(G F^{T} a, b)}_{T_{X} P_{0}} = {(g a, F b)}_{T_{x} P} = {(F^{*} g a, b)}_{T_{X} P_{0}} ._{□}

Using Equations (8) and (15) in Definition 3.1 and in Corollary 3.3 for the evaluation of the dual and transpose, respectively, one obtains

F^{*} = F_{. A}^{a} G^{A} ⊠ g_{a} and Ф^{T} = F_{. B}^{b} G^{A B} g_{ab} G_{A} ⊠ g^{a} .

(16)

Remark 3.4 Similar to a two-point tensor, the transpose of a one-point co-, mixed-, or contra-variant tensor, $A^{b} \in Lin (T_{X} P_{0}; T_{X}^{*} P_{0})$ $B^{♮} \in Lin (T_{X} P_{0}; T_{X} P_{0})$ and $C^{♯} \in L i n (T_{X}^{*} P_{0}, T_{X} P_{0})$ , respectively, are given by [1]

\begin{array}{l} A^{b, T} = G^{- 1} (A_{B A} G^{A} ⊠ G^{B}) G^{- 1} = A_{B A} G^{A} ⊠ G^{B} = A_{B A} G^{A C} G^{B D} G_{C} ⊠ G_{D} \\ = : {(A^{b, T})}^{C D} G_{C} ⊠ G_{D}, \\ B^{♮, T} = G^{- 1} (B_{. B}^{A} G^{B} ⊠ G_{A}) G = B_{. B}^{A} G_{A C} G^{B} ⊠ G^{C} = (B_{. B}^{A} G_{A C} G^{B D}) G_{D} ⊠ G^{C} \\ = : {(B^{♮, T})}_{. C}^{D} G_{D} ⊠ G^{C}, \\ C^{♯, T} = G (C^{B A} G_{A} ⊠ G_{B}) G = C^{B A} G_{A C} G_{B D} G^{C} ⊠ G^{D} \\ = : {(C^{♯, T})}_{C D} G^{C} ⊠ G^{D} . \end{array}

The representations in Remark 3.4 show that the transpose depends on the metric (tensor) of the metric vector or tangent space involved, whereas the dual map is defined independently of the metric.

Remark 3.5 Since the co-variant metric tensor is spd, $G = G^{*}$ and thus $G^{T} = G^{- 1} G^{*} G^{- 1} = G^{- 1} = : G^{♯}$ . Instead, for the mixed form $G^{♮ T} = G^{♮} = I$ (identity) because, by Example 2.13, $G^{♮} = G^{- 1} G = I$ .

Assuming $F \in Lin (T_{X} P_{0}; T_{x} P)$ is isomorphic (invertible). The left and the right inverses are defined by the compositions that lead to the identities, $I, i$ , respectively. In detail, depending on the sequence of the composition,

F^{- 1} F = I \in Lin (T_{X} P_{0}; T_{X} P_{0}) and F F^{- 1} = i \in L i n (T_{x} P; T_{x} P) .

(17)

Similarly, the left and right dual inverse are defined as

F^{- *} F^{*} = i^{*} \in Lin (T_{x}^{*} P; T_{x}^{*} P) a n d F^{*} F^{- *} = I^{*} \in L i n (T_{X}^{*} P_{0}; T_{X}^{*} P_{0}),

(18)

where $Ф^{- *} : = {(Ф^{*})}^{- 1} = {(Ф^{- 1})}^{*}$ was employed as a result of isomorphism. Using Equation (15) in (17) and Equation (16)₁ in (18) gives³

F^{- 1} = F_{. a}^{A} G_{A} ⊠ g^{a}, F^{- *} = F_{a .}^{A} g^{a} ⊠ G_{A},

(19)

in which use was made of the notions, $F_{. a}^{A} : = {(F^{- 1})}_{. a}^{A}$ and $F_{a .}^{A} = {(F^{- *})}_{a .}^{A}$ . The metrics as well as the transposed, the dual, and the inverse maps are given in Figure 3.1.

Figure 2.

Domain and range of the linear maps and the commutative diagram for dual, transpose, and metrics.

3.2. Extension to higher-order tensors

The dual and the transposed map of a high order tensor will be defined here using the identification in Definition 2.15 two times in succession. Moreover, the concept will be concretized using the identification in Definition A.5 for fourth-order tensors. In accordance with the bilinear maps, the tensor representations of transposed $2 m$ -linear maps are given by the inner products, whereas the duals are given by the scalar products. As was pointed out in Section 2.1, a unique map $h$ (e.g. the dual and transposed map) exists for the decomposable tensors $V \in T$ and $W \in S$ . However, one can define the dual and the transposed map for all $A \in T$ and $B \in S$ (not necessarily decomposable) such that the diagram in Figure 1 commutes.

Note: In the following, the set of linear maps between tangent or vector spaces is denoted by $Lin$ or more precisely by ${Lin}_{k}, k \in {2, (3),4, \dots, 2 m | order of a tensor}$ if no need for specification exists. Similarly, the set of invertible and symmetric maps is denoted by $In v_{k} \subset {Lin}_{k}$ and $Sy m_{k} \subset {Lin}_{k}$ respectively.

Definition 3.6 Let $V, W \in {Lin}_{m}$ (e.g. $V : = v_{1} ⊠ v_{2} ⊠ \dots ⊠ v_{m}$ and $W : = w_{1} ⊠ w_{2} ⊠\dots⊠ w_{m})$ be decomposable tensors. The dual map $C^{*}$ of $C \in {Lin}_{2 m}$ is defined as

\begin{array}{l} W C V : = (C (V), W)= (W, C (V))= (C^{*} (W), V)= V C^{*} W \in ℝ . \end{array}

Definition 3.7 Let $V, W \in {Lin}_{m}$ be decomposable tensors. The transposed map $C^{T}$ of $C \in {Lin}_{2 m}$ is defined as

\begin{array}{l} W • C (V):= C (V) • W = C^{T} (W) • V = V • C^{T} (W) \in ℝ . \end{array}

Corollary 3.8 The transpose of a high-order tensor $C \in {Lin}_{2 m}$ is given by

C^{T} = G_{1}^{- 1} C^{*} G_{2},

(20)

where $G_{1}, G_{2} \in In v_{2 m}$ are decomposable tensors being composed of the tensor products of metric tensors.

Proof. Using Definition 3.6 for the dual and Equation (14) for the generalized inner product, the transpose is evaluated by Definition 3.7 as

C^{T} (W) • V = (G_{1} C^{T} (W), V)= (G_{2} W, C (V))= (C^{*} G_{2} (W), V) ._{□}

The representations of $G_{1}$ and $G_{2}$ by tensors depend on the identification employed, cf. Definition 2.15. Explicit expressions of $G_{1}$ and $G_{2}$ are given below, cf. Example 5.8.

Theorem 3.9 Suppose $T_{i} \in {Lin}_{2}, 1 \leq i \leq m$ . The dual of a decomposable tensor $T \in {Lin}_{2 m}$ (with respect to the scalar product) is defined as

T^{*} : = {(T_{1} ⊠ T_{2} ⊠\dots⊠ T_{m})}^{*} = T_{1}^{*} ⊠ T_{2}^{*} ⊠\dots⊠ T_{m}^{*} .

Proof. Without loss of generality, let there be $T_{i} \in L i n (T_{X} P_{0}; T_{x} P), 1 \leq i \leq m$ and the decomposable tensors $V, W \in {Lin}_{m}$ . Employing the identification 2.15 in Definition 3.6 yields

\begin{array}{l} (T (V), W) = ((T_{1} ⊠ T_{2} ⊠\dots⊠ T_{m}) (v_{1} ⊠ v_{2} ⊠\dots⊠ v_{m}), w_{1} ⊠ w_{2} ⊠\dots⊠ w_{m}) \\ = (w_{1}, T_{1} v_{1}) (w_{2}, T_{2} v_{2}) \dots (w_{m}, T_{m} v_{m})= (v_{1}, T_{1}^{*} w_{1}) (v_{2}, T_{2}^{*} w_{2}) \dots (v_{m}, T_{m}^{*} w_{m}) \\ = ((T_{1}^{*} ⊠ T_{2}^{*} ⊠\dots⊠ T_{m}^{*}) (w_{1} ⊠ w_{2} ⊠\dots⊠ w_{m}), v_{1} ⊠ v_{2} ⊠\dots⊠ v_{m}) ∥ byDefinition \\ =: ({(T_{1} ⊠ T_{2} ⊠\dots⊠ T_{m})}^{*} (w_{1} ⊠ w_{2} ⊠\dots⊠ w_{m}), v_{1} ⊠ v_{2} ⊠\dots⊠ v_{m}) . \end{array}

Since the duals $T_{i}^{*}$ , $1 \leq i \leq m$ are unique and the decomposable tensors $V$ and $W$ constitute $T_{X} P_{0,1} ⊠ T_{X} P_{0,2} ⊠ \dots ⊠ T_{X} P_{0, m}$ and $T_{x}^{*} P_{1} ⊠ T_{x}^{*} P_{2} ⊠ \dots ⊠ T_{x}^{*} P_{m}$ , respectively, $T^{*}$ is unique. $_{□}$

Theorem 3.10 Suppose there is $T_{i} \in {Lin}_{2}, 1 \leq i \leq m$ . The transpose of a decomposable tensor $T \in {Lin}_{2 m}$ (with respect to the inner product) is defined as

T^{T} : = {(T_{1} ⊠\dots⊠ T_{m})}^{T} = T_{1}^{T} ⊠\dots⊠ T_{m}^{T} .

Proof. According to Definition 3.2, $w_{i} \cdot T_{i} v_{i} = v_{i} \cdot T_{i}^{T} w_{i}$ , $1 \leq i \leq m$ in Definition 3.7, cf. the proof of Theorem 3.9. According to Corollary 3.3, the transposes are given by $T_{i}^{T} = G_{i}^{- 1} T_{i}^{*} g_{i}$ , $1 \leq i \leq m$ . Since the dual $T_{i}^{*}$ is unique, the transpose $T_{i}^{T}$ as an isomorphism is likewise unique. Since the decomposable tensors $V$ and $W$ constitute $T_{X} P_{0,1} ⊠ T_{X} P_{0,2} ⊠ \dots ⊠ T_{X} P_{0, m}$ and $T_{x} P_{1} ⊠ T_{x} P_{2} ⊠ \dots ⊠ T_{x} P_{m}$ , respectively, $T^{T}$ is unique. $_{□}$

For later purposes, some important operations, widely used for second-order tensors, will be introduced subsequently.

Lemma 3.11 The dual and the transpose of the composition satisfy ${(C_{1} C_{2})}^{*} = C_{2}^{*} C_{1}^{*}$ and ${(C_{1} C_{2})}^{T} = C_{2}^{T} C_{1}^{T}$ , $C_{1}, C_{2} \in Lin$ .

Proof. According to Definition 3.6,

V {(C_{1} C_{2})}^{*} W = W C_{1} C_{2} V = (W C_{1}) C_{2} V = V C_{2}^{*} (W C_{1})= V C_{2}^{*} C_{1}^{*} W .

By analogy, using Definition 3.7 ${(C_{1} C_{2})}^{T} = C_{2}^{T} C_{1}^{T} ._{□}$

Lemma 3.12 The transpose and the dual commute, i.e. ${(C^{*})}^{T} = {(C^{T})}^{*}$ , $C \in Lin$ .

Proof. In accordance with Corollary 3.8, ${(C^{*})}^{T} = G_{2} C G_{1}^{- 1}$ , $C \in Lin$ . Using Lemma 3.11 in Corollary 3.8 and Theorem 3.9 for $G_{1}$ and $G_{2}$ yield

{(C^{T})}^{*} = G_{2}^{*} C G_{1}^{- *} = G_{2} C G_{1}^{- 1}, G_{1}, G_{2} \in Sym ._{□}

According to Corollary 3.3, ${(T_{i}^{*})}^{T} = g T_{i} G^{- 1} = {(T_{i}^{T})}^{*}, T_{i} \in {Lin}_{2}$ , $1 \leq i \leq m$ . Substituting these results into the transposed tensor product, Theorem 3.10, and taking note of Theorem 3.9 yields ${(T^{*})}^{T} = {(T^{T})}^{*}$ .

Lemma 3.13 We have ${(C^{T})}^{T} = C$ , and ${(C^{*})}^{*} = C \in Lin$ .

Proof. Owing to the natural identification, $T_{X} P_{0, i}^{* *} ≅ T_{X} P_{0, i}$ and $T_{x} P_{i}^{* *} ≅ T_{x} P_{i}$ , ${(C^{*})}^{*} = C$ $G_{1}$ $i = 1,2, \dots$ (finite dimensional). Thus, ${(C^{*})}^{*} = C$ . First using Corollary 3.8 and Lemma 3.11 and thereafter Theorem 3.10 and Remark 3.5 for and $G_{2}$ yield

{(C^{T})}^{T} = {(G_{1}^{- 1} C^{*} G_{2})}^{T} = G_{2}^{T} {(C^{*})}^{T} G_{1}^{- T} = G_{2}^{- 1} G_{2} C G_{1}^{- 1} G_{1} = C ._{□}

In accordance with the bilinear maps, the left and the right inverse of $C \in Inv$ are defined by the identities, $ℐ$ and $i$ , respectively, i.e.

C^{- 1} C = ℐ and C C^{- 1} = i .

(21)

The left and the right dual inverse are defined as

C^{- *} C^{*} = i^{*} and C^{*} C^{- *} = ℐ^{*},

(22)

respectively. Owing to the isomorphism, ${(C^{*})}^{- 1} = {(C^{- 1})}^{*} =: C^{- *}$ , which is the identification used in Equation (22).

Lemma 3.14 We have ${(C_{1} C_{2})}^{- 1} = C_{2}^{- 1} C_{1}^{- 1}$ , $C_{1}, C_{2} \in Inv$ .

Proof. Let $F_{1} = C_{1} C_{2} F_{2}$ be a map in which $F_{1}, F_{2} \in Lin$ , $C_{1}, C_{2} \in Inv$ . Taking the compositions in terms of $C_{1}^{- 1}$ and $C_{2}^{- 1}$ from the left gives $F_{2} = C_{2}^{- 1} C_{1}^{- 1} F_{1}$ . On the other side, $F_{2} = {(C_{1} C_{2})}^{- 1} F_{1}$ holds. $_{□}$

Lemma 3.15 The inverse and the transpose commute, i.e. ${(C^{T})}^{- 1} = {(C^{- 1})}^{T}$ .

Proof. Using Corollary 3.8 and Lemma 3.14 yields

{(C^{T})}^{- 1} = {(G_{1}^{- 1} C^{*} G_{2})}^{- 1} = G_{2}^{- 1} {(C^{*})}^{- 1} G_{1} = G_{2}^{- 1} {(C^{- 1})}^{*} G_{1} = {(C^{- 1})}^{T} ._{□}

Theorem 3.16 Let $T \in Inv$ be an invertible decomposable tensor. Then,

\begin{array}{l} T^{- 1} : = T_{1}^{- 1} ⊠\dots⊠ T_{m}^{- 1} = {(T_{1} ⊠\dots⊠ T_{m})}^{- 1}, \\ T^{- *} : = T_{1}^{- *} ⊠\dots⊠ T_{m}^{- *} = {(T_{1} ⊠\dots⊠ T_{m})}^{- *} \\ T^{- T} : = T_{1}^{- T} ⊠\dots⊠ T_{m}^{- T} = {(T_{1} ⊠\dots⊠ T_{m})}^{- T}, forall T_{i} \in In v_{2}, 1 \leq i \leq m . \end{array}

Proof. Employing Corollary 3.3 in Lemma 3.14, one obtains

\begin{array}{l} {(T_{i}^{T})}^{- 1} = {(G_{i}^{- 1} T_{i}^{*} g_{i})}^{- 1} = g_{i}^{- 1} {(T_{i}^{*})}^{- 1} G_{i} = g_{i}^{- 1} {(T_{i}^{- 1})}^{*} G_{i} = {(T_{i}^{- 1})}^{T}, T_{i} \in In v_{2}, 1 \leq i \leq m . \end{array}

Substitution of these results into the transposed decomposable tensor, given by Theorem 3.10, yields

\begin{array}{l} {(T_{1}^{T})}^{- 1} ⊠\dots⊠ {(T_{m}^{T})}^{- 1} = {(T_{1}^{- 1} ⊠\dots⊠ T_{m}^{- 1})}^{T} . \end{array}

Similarly for the dual

\begin{array}{l} {(T_{1}^{*})}^{- 1} ⊠\dots⊠ {(T_{m}^{*})}^{- 1} = {(T_{1}^{- 1} ⊠\dots⊠ T_{m}^{- 1})}^{*} . \end{array}

Then, according to UFP, there exist inverse maps

\begin{array}{l} (T_{1} ⊠ \dots ⊠ T_{m}) \circ (T_{1}^{- 1} ⊠ \dots ⊠ T_{m}^{- 1}) = T_{1} T_{1}^{- 1} ⊠ \dots ⊠ T_{m} T_{m}^{- 1} = i ⊠ \dots ⊠ i o r \\ (T_{1}^{- 1} ⊠ \dots ⊠ T_{m}^{- 1}) \circ (T_{1} ⊠ \dots ⊠ T_{m}) = T_{1}^{- 1} T_{1} ⊠ \dots ⊠ T_{m}^{- 1} T_{m} = I ⊠ \dots ⊠ I, \end{array}

where $I, i$ are the second order identity tensors. This gives $T_{1}^{- 1} ⊠\dots⊠ T_{m}^{- 1} = {(T_{1} ⊠\dots⊠ T_{m})}^{- 1}$ . $_{□}$

4. The minor duals and transpositions of high-order tensors

The dual and the transpose of a high-order tensor were defined by Definitions 3.6 and 3.7, respectively. Analogously, use of the identification present in Definition 2.15 allows one to define the minor dual and the minor transpose of a high-order tensor. In addition, the commutative rules for the duals, the transpositions and the tensor inverse will be introduced. Since tensors are understood as being invariant quantities, absolute notation will be employed.

Definition 4.1 Let there be $V, W \in {Lin}_{m}$ and $C \in {Lin}_{2 m}$ . The duals of $C$ , denoted by $C^{*}$ , $C^{⋆}$ and $⋆ C$ , are defined as follows:

\begin{array}{l} W C V = V C^{*} W \in ℝ (m a j o r) d u a l o f C, \\ W C V = W^{*}^{⋆} C V = W C^{⋆} V^{*} \in ℝ l e f t a n d r i g h t m i n o r d u a l o f C, \\ W C V = V^{*}^{⋆} (C^{*}) W = V {(C^{*})}^{⋆} W^{*} \in ℝ l e f t a n d r i g h t m i n o r d u a l o f C^{*} . \end{array}

Definition 4.2 Let there be $V, W \in {Lin}_{m}$ and $C \in {Lin}_{2 m}$ . The transpositions of $C$ , denoted by $C^{T}$ , $C^{t}$ and ${}^{t}C$ , are defined as follows:

\begin{array}{l} W • C V = V • C^{T} W \in ℝ major transposition or the transpose of C, \\ W • C V = W^{T} •^{t} C V = W • C^{t} V^{T} \in ℝ left and right minor transposition of C, \\ W • C V = V^{T} •^{t} (C^{T}) W = V • {(C^{T})}^{t} W^{T} \in ℝ left and right minor transposition of C^{T} . \end{array}

It follows that $C V = V C^{*}$ , whereas $C V \neq V • C^{T}$ since the right- and the left-hand side lie on different spaces.

Example 4.3 The following identities are derived on the basis of Definitions 4.1 and 4.2:

W • C V = V • C^{T} W = V C^{*} • W,

(23a)

W • C V = W^{T} • {{}^{t}C}^{t} V^{T} = W^{*} • {{}^{⋆}C}^{⋆} V^{*},

(23b)

W • C V = V^{T} {}^{t}{(C^{*})}^{t} • W^{T} = V^{*} {}^{⋆}{(C^{*})}^{⋆} • W^{*},

(23c)

W • C V = V^{*} {}^{t}{(C^{T})}^{t} • W^{*} = V^{T} {}^{⋆}{(C^{T})}^{⋆} • W^{T} .

(23d)

Corollary 4.4 The relation between the duals and transpositions are given by ${}^{t}{(C^{T})}^{t} = {}^{⋆}{(C^{T})}^{⋆}, C \in {Lin}_{2 m}$ .

Proof. A comparison of Equations (23c)₃ and (23d)₂ reveals that ${}^{t}{(C^{T})}^{t} = {}^{⋆}{(C^{T})}^{⋆}, C \in {Lin}_{2 m}$

Next, attention will be directed at the commutative rules for the duals, the transpositions and the tensor inverses.

Theorem 4.5 Let there be $C \in {Lin}_{2 m}$ . The following identities hold separately for the duals and the transpositions:

\begin{array}{l} {(C^{*})}^{*} = C, {(C^{T})}^{T} = C, \\ {(^{⋆} C)}^{⋆} =^{⋆} (C^{⋆})=^{⋆} C^{⋆},^{⋆} (C^{*})= {(C^{⋆})}^{*}, {(^{⋆} C)}^{*} = {(C^{*})}^{⋆}, \\ {(^{⋆} C^{⋆})}^{*} =^{⋆} {(C^{*})}^{⋆}, {[{(^{⋆} C)}^{*}]}^{⋆} =^{⋆} [{(C^{⋆})}^{*}]= C^{*}, \\ {(^{t} C)}^{t} =^{t} (C^{t})=^{t} C^{t},^{t} (C^{T})= {(C^{t})}^{T}, {(^{t} C)}^{T} = {(C^{T})}^{t}, \\ {(^{t} C^{t})}^{T} =^{t} {(C^{T})}^{t}, {[{(^{t} C)}^{T}]}^{t} =^{t} [{(C^{t})}^{T}] = C^{T} . \end{array}

Proof. The first two results for the dual and the transpose emanate directly from Lemma 3.13. It follows then from Definition 4.1 that

\begin{array}{l} V {(C^{*})}^{⋆} W^{*} = V C^{*} W = W C V = W^{*} {}^{⋆}C V = V {({}^{⋆}C)}^{*} W^{*}, \end{array}

$\forall V, W \in L i n_{m}, C \in L i n_{2 m}$ . Analogously, ${}^{⋆}{(C^{*})} = {(C^{⋆})}^{*}$ . It follows then from Definition 4.2 that

\begin{array}{l} V • {(C^{T})}^{t} W^{T} = V • C^{T} W = W • C V = W^{T} • {}^{⋆}C V = V • {({}^{⋆}C)}^{T} W^{T}, \end{array}

$\forall V, W \in L i n_{m}, C \in L i n_{2 m}$ Similarly, ${}^{t}{(C^{T})} = {(C^{t})}^{T}$ , etc. $□$

Theorem 4.6 Let there be $C \in L i n_{2 m}$ . The following identities apply to the duals and the transpositions:

\begin{array}{l} {(C^{*})}^{T} = {(C^{T})}^{*}, {(C^{⋆})}^{T} =^{⋆} (C^{T}), {(^{⋆} C)}^{T} = {(C^{T})}^{⋆}, {(C^{*})}^{t} = {(^{t} C)}^{*}, \\ {}^{t}{(C^{*})} = {(C^{t})}^{*}, {(C^{⋆})}^{t} = {(C^{t})}^{⋆},^{⋆} (C^{t}) = {(^{⋆} C)}^{t}, {(^{t} C)}^{⋆} =^{t} (C^{⋆}),^{⋆} {({{}^{t}C}^{t})}^{⋆} =^{t} {(^{⋆} C^{⋆})}^{t} . \end{array}

Proof. The first result has already been given in Lemma 3.12. According to Definitions 4.1 and 4.2 and Example 4.3,

\begin{array}{l} V • {(C^{T})}^{⋆} W^{*} = V • C^{T} W = W • C V = W^{*} •^{⋆} C V = V • {(^{⋆} C)}^{T} W^{*}, \end{array}

$\forall V, W \in L i n_{m}, C \in L i n_{2 m}$ and similarly, ${(C^{⋆})}^{T} =^{⋆} (C^{T})$ . Then,

\begin{array}{l} V {(C^{*})}^{t} • W^{T} = V (C^{*}) • W = W^{T} •^{t} C V = V {(^{t} C)}^{*} • W^{T}, \forall V, W \in L i n_{m} \end{array}

and, similarly, ${}^{t}{(C^{*})} = {(C^{t})}^{*}$ . Then, by analogy

\begin{array}{l} C^{⋆} V^{*} = {(C^{⋆})}^{t} • V = C^{⋆} • V^{T} = C • {(V^{T})}^{*} = C • {(V^{*})}^{T} = {(C^{t})}^{⋆} • V, \forall V \in L i n_{m} \end{array}

and, similarly, ${}^{⋆}{(C^{t})} = {(^{⋆} C)}^{t}$ , etc. $_{□}$

Theorem 4.7 Let there be $C \in L i n_{2 m}$ . The following identities apply to the minor duals, the minor transpositions and the tensor inverse:

{(C^{⋆})}^{- 1} =^{⋆} (C^{- 1}), {(^{⋆} C)}^{- 1} = {(C^{- 1})}^{⋆}, {(C^{t})}^{- 1} =^{t} (C^{- 1}), {(^{t} C)}^{- 1} = {(C^{- 1})}^{t} .

Proof. Let there be $W, V \in L i n_{m}$ and $C \in I n v_{2 m}$ such that $W : = C V$ . By Definition 4.1,

W = C V = C^{⋆} V^{*} o r V^{*} = {(C^{⋆})}^{- 1} W .

(24)

On the other hand,

V^{*} =^{⋆} (C^{- 1}) W .

(25)

A comparison of Equations (24) and (25) reveals that $^{⋆} (C^{- 1}) = {(C^{⋆})}^{- 1}$ , etc. $□$

5. The duals and the transpositions of fourth-order tensors

Based on Definitions 4.1 and 4.2 the duals and the transpositions of important fourth-order tensor will be determined and further applied to specific forms taking the metric of an underlying tensor space into account. To obtain explicit representations that are suitable for purposes in continuum mechanics, any fourth-order tensor is identified as a four-linear map and the composition therein is defined to be the double contraction, cf. Definition A.5. Then the duals and the transpositions of the two different tensor products are defined to be consistent with the contractions being employed. If the double contraction is calculated on the basis of the inner products, it is termed the double-dot product, denoted by ‘:’.

Theorem 5.1 Let $A, B \in {Lin}_{2}$ be second-order tensors. The duals of the decomposable tensors (with respect to the double contractions) $A ⊠ B \in {Lin}_{4}$ and $A ⊡ B \in {Lin}_{4}$ are given by ⁴

\begin{array}{l} {(A ⊠ B)}^{*} = A^{*} ⊠ B^{*},^{⋆} {(A ⊠ B)}^{⋆} = B^{*} ⊠ A^{*}, {(A ⊡ B)}^{*} = B ⊡ A, \\ {(A ⊡ B)}^{⋆} = A ⊡ B^{*},^{⋆} (A ⊡ B) =^{*} A ⊡ B,^{⋆} {(A ⊡ B)}^{⋆} = A^{*} ⊡ B^{*} . \end{array}

Proof. Without loss of generality, let there be $A, B \in L i n (T_{X} P_{0}; T_{x} P)$ and $v_{1}, v_{2} \in T_{X} P_{0}$ , $w_{1}, w_{2} \in T_{x}^{*} P$ . Application of Definition 4.1 for the duals results in

\begin{array}{l} (A ⊠ B : v_{1} ⊠ w_{1}) : w_{2} ⊠ v_{2} = (w_{2}, A v_{1}) (w_{1}, B v_{2}) = (v_{1}, A^{*} w_{2}) (v_{2}, B^{*} w_{1}) = \\ (A^{*} ⊠ B^{*} : w_{2} ⊠ v_{2}) : v_{1} ⊠ w_{1} = : ({(A ⊠ B)}^{*} : w_{2} ⊠ v_{2}) : v_{1} ⊠ w_{1}, \\ (A ⊡ B : w_{1} ⊠ v_{2}) : w_{2} ⊠ v_{1} = (w_{2}, A v_{1}) (w_{1}, B v_{2}) = (w_{1}, B v_{2}) (w_{2}, A v_{1}) = \\ (B ⊡ A : w_{2} ⊠ v_{1}) : w_{1} ⊠ v_{2} = : ({(A ⊡ B)}^{*} : w_{2} ⊠ v_{1}) : w_{1} ⊠ v_{2}, \\ (A ⊠ B : v_{1} ⊠ w_{1}) : w_{2} ⊠ v_{2} = (w_{2}, A v_{1}) (w_{1}, B v_{2}) = (v_{2}, B^{*} w_{1}) (v_{1}, A^{*} w_{2}) = \\ (B^{*} ⊠ A^{*} : w_{1} ⊠ v_{1}) : v_{2} ⊠ w_{2} = : (^{⋆} {(A ⊠ B)}^{⋆} : w_{1} ⊠ v_{1}) : v_{2} ⊠ w_{2} etc ._{□} \end{array}

Theorem 5.2 Let $A, B \in {Lin}_{2}$ be second-order tensors. The transpositions of the decomposable tensors (with respect to the double-dot products) $A ⊠ B \in L i n_{4}$ and $A ⊡ B \in L i n_{4}$ are given by

\begin{array}{l} {(A ⊠ B)}^{T} = A^{T} ⊠ B^{T},^{t} {(A ⊠ B)}^{t} = B^{T} ⊠ A^{T}, {(A ⊡ B)}^{T} = B^{* T} ⊡ A^{* T}, \\ {(A ⊡ B)}^{t} = A ⊡ B^{T},^{t} (A ⊡ B) = A^{T} ⊡ B,^{t} {(A ⊡ B)}^{t} = A^{T} ⊡ B^{T} . \end{array}

Proof. Let there be $A, B \in Lin (T_{X} P_{0}; T_{x} P)$ and $v_{1} \in T_{X} P_{0}$ , $v_{2} \in T_{X}^{*} P_{0}$ , $w_{1} \in T_{x}^{*} P$ , and $w_{2} \in T_{x} P$ . Based on Definition 4.2 the transpose ${(A ⊡ B)}^{T}$ is obtained by

\begin{array}{l} (A ⊡ B : w_{1} ⊠ v_{1}) : w_{2} ⊠ v_{2} = (w_{2} \cdot A \cdot v_{2}) (w_{1}, B v_{1}) = (w_{1} \cdot B^{* T} \cdot v_{1}) (w_{2}, A^{* T} v_{2}) = \\ (B^{* T} ⊡ A^{* T} : w_{2} ⊠ v_{2}) : w_{1} ⊠ v_{1} = : ({(A ⊡ B)}^{T} : w_{2} ⊠ v_{2}) : w_{1} ⊠ v_{1} etc ._{□} \end{array}

Remark 5.3 The transposition ${(\cdot)}^{R}$ , given by Equation (41), does not commute with the dual ${(\cdot)}^{*}$ and the transpose ${(\cdot)}^{T}$ :

\begin{array}{l} {({({(T_{1} ⊡ T_{2})}^{R})}^{*})}^{R} = {({(T_{1} ⊠ T_{2})}^{*})}^{R} = T_{1}^{*} ⊡ T_{2}^{*} =^{⋆} {(T_{1} ⊡ T_{2})}^{⋆} \neq {({({(T_{1} ⊡ T_{2})}^{R})}^{R})}^{*} \\ = {(T_{1} ⊡ T_{2})}^{*} = T_{2} ⊡ T_{1} . \end{array}

By analogy, ${({(T_{1} ⊡ T_{2})}^{R})}^{T} \neq {({(T_{1} ⊡ T_{2})}^{T})}^{R}$ .

Let $A, B, C, D, E \in {Lin}_{2}$ be second order tensors. Using Definitions A.1, A.3 and A.5 for the contractions, Definition 4.1 and Theorem 5.1 for the duals and Definition 4.2 and Theorem 5.2 for the transpositions, the following identities can be obtained:

C : (A ⊠ B) : D = D : {(A ⊠ B)}^{T} : C = D : (A^{T} C B^{T}) = C : A D B,

(26a)

C : (A ⊡ B) : D = D : {(A ⊡ B)}^{T} : C = (A^{* T} : C) (B^{* T} : D) = (A : C) (B : D),

(26b)

{(A ⊠ B)}^{⋆} : C = A C^{*} B = A \cdot C^{T} \cdot B = {(A ⊠ B)}^{t} : C,

(26c)

{(A ⊡ B)}^{⋆} : C = (B : C^{*}) A = (B : C^{T}) A = {(A ⊡ B)}^{t} : C,

(26d)

C :^{t} (A ⊠ B) = A^{*} \cdot C^{T} \cdot B^{*} = A^{*} C^{*} B^{*},

(26e)

C :^{t} (A ⊡ B) = (A : C^{T}) B = (A : C^{*}) B .

(26f)

Using Definitions A.1, A.3 and A.5 for the contractions and exploiting Definition 4.1 for the duals, also the following results can be obtained.

Lemma 5.4 Let there be $A, B, C, D, E \in L i n_{m}$ , $A, ℬ, C$ and $D \in L i n_{2 m}$ . Then,

\begin{array}{l} {(C : ℬ)}^{*} = ℬ^{*} : C^{*}, {(C : ℬ)}^{⋆} = C : ℬ^{⋆},^{⋆} (C : ℬ) =^{⋆} C : ℬ, \\ {(C : C)}^{⋆} =^{⋆} C : C =^{⋆} (C : C), {(D : C)}^{⋆} = D : C^{⋆} =^{⋆} (D : C),^{⋆} (^{⋆} A A) = A^{*} A, \\ ^{⋆} (B^{⋆} A A) = A^{*} A B^{*},^{⋆} A A :^{⋆} D =^{⋆} A^{⋆} A : D =^{⋆} (A^{*} A^{⋆}) : D =^{⋆} ((A^{*} A^{⋆}) : D). \end{array}

Similar results can be obtained for the corresponding transpositions, i.e. replacing $*$ by $T$ and $⋆$ by $t$ .

Lemma 5.5 Let there be $A, B \in I n v_{2}$ . The inverse of the decomposable tensor $A ⊠ B \in {Inv}_{4}$ is given by

\begin{array}{l} {(A ⊠ B)}^{- 1} = A^{- 1} ⊠ B^{- 1} . \end{array}

Proof. Using Definition A.5 in Equation (17) yields

\begin{array}{l} A^{- 1} ⊠ B^{- 1} : A ⊠ B = I ⊠ i a n d A ⊠ B : A^{- 1} ⊠ B^{- 1} = i ⊠ I . \end{array}

Then, $i ⊠ I : A = A$ . Use of Lemma 3.14 results in

\begin{array}{l} {(i ⊠ I : A)}^{- 1} = {(i A I)}^{- 1} = I A^{- 1} i = I ⊠ i : A^{- 1} = A^{- 1}, i . e . \\ {(A ⊠ B)}^{- 1} = A^{- 1} ⊠ B^{- 1} ._{□} \end{array}

Example 5.6 Employing Theorem 5.2 and Lemma 5.5 and noting Remark 3.5 enable the following relations to be derived for the tensor product of the co-variant metric tensors:

\begin{array}{l} {(g^{- 1} ⊠ G)}^{- 1} = {(g^{- 1} ⊠ G)}^{T} =^{t} {(G ⊠ g^{- 1})}^{t} = g ⊠ G^{- 1} . \end{array}

5.1. Representations of the transpositions based on the explicit use of a metric

Since the duals and the transpositions are both tensorial operations, they were defined using absolute notation, cf. Definitions 4.1 and 4.2. Evaluation of the transpositions, however, depends on the metric of the metric tensor spaces being involved. Although the explicit representations of the transpositions will be given for only a restricted set of tensors, all of the results can analogously be obtained for tensors of any form and order using the tensor products and contractions which are consistent with the identification employed.

Figure 3.

Domain and range of the four-linear map $C \in Lin (T_{x} P_{0} ⊠ T_{X}^{*} P; T_{x} P ⊠ T_{X}^{*} P_{0}$ and the commutative diagram for the dual, transpose, and metrics.

Corollary 5.7 The transpose of the eight-liner map $C \in Lin (T_{x}^{*} P ⊠ T_{X} P_{0} ⊠ T_{x}^{*} P ⊠ T_{X} P_{0}; ℝ) \subseteq {Lin}_{4}$ is given by

C^{T} = G^{- 1} ⊠ g : C^{*} : g ⊠ G^{- 1} \in L i n (T_{X}^{*} P_{0} ⊠ T_{x} P ⊠ T_{X}^{*} P_{0} ⊠ T_{x} P; ℝ).

Proof. Without loss of generality, let there be $C_{1} = F_{1} ⊠ F_{2} \in L i n (T_{x}^{*} P ⊠ T_{X} P_{0} ⊠ T_{x}^{*} P ⊠ T_{X} P_{0}; ℝ)$ . At first, using Corollary 3.3 for the transpose of a bilinear map and Theorem 5.1 for the dual of a fourth-order tensor yield

\begin{array}{l} C_{1}^{*} : = {(F_{1} ⊠ F_{2})}^{*} = F_{1}^{*} ⊠ F_{2}^{*} = G F_{1}^{T} g^{- 1} ⊠ G F_{2}^{T} g^{- 1} \\ = G ⊠ g^{- 1} : F_{1}^{T} ⊠ F_{2}^{T} : g^{- 1} ⊠ G ∥ by Definition A.5 \\ = G ⊠ g^{- 1} : {(F_{1} ⊠ F_{2})}^{T} : g^{- 1} ⊠ G ∥ by Theorem 5.2, \\ \Leftrightarrow C_{1}^{T} : = {(F_{1} ⊠ F_{2})}^{T} = F_{1}^{T} ⊠ F_{2}^{T} = G^{- 1} ⊠ g : {(F_{1} ⊠ F_{2})}^{*} : g ⊠ G^{- 1} ∥ by Theorem 5.5 \\ = G^{- 1} ⊠ g : C_{1}^{*} : g ⊠ G^{- 1}, \forall C_{1} \in L i n (T_{x}^{*} P ⊠ T_{X} P_{0} ⊠ T_{x}^{*} P ⊠ T_{X} P_{0}; ℝ)._{□} \end{array}

It can be concluded that the transposed map operates from the actual primary tensor space into the original primary tensor space and the dual transposed map operates from the original dual tensor space into the actual dual tensor space whereas the original tensor and its dual operates vice versa, cf. Figures 3.1 and 5.1.

Example 5.8 According to Corollaries 3.8 and 5.7, the transpose of $C \in L i n (T_{x}^{*} P ⊠ T_{X} P_{0} ⊠ T_{x}^{*} P ⊠ T_{X} P_{0}; ℝ)$ can be given by $C^{T} = G_{1}^{- 1} : C^{*} : G_{2}$ where $G_{1} = {(G^{- 1} ⊠ g)}^{- 1} = G ⊠ g^{- 1}$ and $G_{2} = g ⊠ G^{- 1}$ .

Example 5.9 Let there be $C : = {C^{I}}_{. J}^{K}_{. L} G_{I} ⊠ G^{J} ⊠ G_{K} ⊠ G^{L} \in L i n (T_{X}^{*} P_{0} ⊠ T_{X} P_{0} ⊠ T_{X}^{*} P_{0} ⊠ T_{X} P_{0}; ℝ)$ . In accordance with Corollary 5.7, the transpose takes the form

C^{T} = G^{- 1} ⊠ G : C^{*} : G ⊠ G^{- 1} \in L i n (T_{X}^{*} P_{0} ⊠ T_{X} P_{0} ⊠ T_{X}^{*} P_{0} ⊠ T_{X} P_{0}; ℝ).

The components are calculated according to Proposition 2.17 as

{(C^{T})}^{M}_{. N}^{Q}_{. R} = {C^{I}}_{. J}^{K}_{. L} G^{J M} G_{I N} G^{L Q} G_{K R} .

Let there be $C_{1} \in L i n (T_{X} P_{0}; T_{X} P_{0})$ and $C_{2} \in L i n (T_{X}^{*} P_{0}; T_{X}^{*} P_{0})$ . Based on the $(2 : 4 : 2)$ contraction, one obtains

\begin{array}{l} C_{1} : C : C_{2} = C_{2} : C^{*} : C_{1} = \\ {(C_{2})}_{. N}^{M} G_{M} ⊠ G^{N} : {C^{I}}_{. J}^{K}_{. L} G^{J} ⊠ G_{I} ⊠ G^{L} ⊠ G_{K} : {(C_{1})}_{Q .}^{R} G^{Q} ⊠ G_{R} = \\ {(C_{2})}_{. N}^{M} {(C_{1})}_{Q .}^{R} {C^{Q}}_{. M}^{N}_{. R}, \end{array}

i.e. the dual $C^{*}$ takes the form $C^{*} = {C^{I}}_{. J}^{K}_{. L} G^{J} ⊠ G_{I} ⊠ G^{L} ⊠ G_{K}$ . Use of the transpose $C^{T}$ and calculation of the $(2 : 4 : 2)$ contraction by the double-dot products must yield the same result:

\begin{array}{l} C_{1} : C : C_{2} = C_{2} : C^{T} : C_{1} = \\ {(C_{2})}_{. N}^{M} G_{M} ⊠ G^{N} : {C^{A}}_{. B}^{C}_{. D} G^{B I} G_{A J} G^{D K} G_{C L} G_{I} ⊠ G^{J} ⊠ G_{K} ⊠ G^{L} : {(C_{1})}_{Q .}^{R} G^{Q} ⊠ G_{R} = \\ {(C_{2})}_{. N}^{M} {(C_{1})}_{Q .}^{R} ({C^{A}}_{. B}^{C}_{. D} G^{B I} G_{A J} G^{D K} G_{C L} G^{N L} G_{M I} G^{Q J} G_{R K}) = {(C_{2})}_{. N}^{M} {(C_{1})}_{Q .}^{R} {C^{Q}}_{. M}^{N}_{. R} . \end{array}

Corollary 5.10 Suppose that $C \in L i n (T_{X}^{*} P_{0} ⊠ T_{X} P_{0} ⊠ T_{X}^{*} P_{0} ⊠ T_{X} P_{0}; ℝ)$ . The transpositions of $C$ are given by

\begin{array}{l} C^{T} = G^{- 1} ⊠ G : C^{*} : G ⊠ G^{- 1}, \\ C^{t} = C : {(G^{- 1} ⊠ G)}^{⋆}, \\ {}^{t}C =^{⋆} (G ⊠ G^{- 1}) : C, \\ {{}^{t}C}^{t} =^{⋆} (G ⊠ G^{- 1}) : C : {(G^{- 1} ⊠ G)}^{⋆}, \\ {}^{t}{(C^{T})}^{t} = {(C^{t})}^{T} =^{⋆} (C^{*}) : G ⊠ G^{- 1}, \\ {(C^{T})}^{t} = {(^{t} C)}^{T} = (G^{- 1} ⊠ G) : {(C^{*})}^{⋆}, \end{array}

i.e. the transpositions lie on the same space $Lin (T_{X}^{*} P_{0} ⊠ T_{X} P_{0} ⊠ T_{X}^{*} P_{0} ⊠ T_{X} P_{0}; ℝ)$ .

Proof. The first result is obtained analogously by use of Corollary 5.7.

Let there be $B \in L i n (T_{X} P_{0}; T_{X} P_{0})$ . Using first Definition 4.1, then Corollary 3.3 and, finally, Definition A.5 results in

\begin{array}{l} C : B & = C^{t} : B^{T} = C^{t} : (G^{- 1} B^{*} G) = C^{t} : {(G^{- 1} ⊠ G)}^{⋆} : B \\ \Leftrightarrow & C^{t} : = C : {({(G^{- 1} ⊠ G)}^{⋆})}^{- 1} = C : {(G^{- 1} ⊠ G)}^{⋆}, \end{array}

(27)

where Theorems 4.7 and 5.1 and Lemma 5.5 were employed in the last phase. Let it be that $B_{1} : = C : B$ Applying first Lemma 5.4, then Corollary 3.3 and finally using Theorem 5.1 in Definition 4.1 yields

\begin{array}{l} {}^{t}C : B = B_{1}^{T} = B_{1} :^{⋆} ({(G^{- 1} ⊠ G)}^{*}) = B_{1} : {(G ⊠ G^{- 1})}^{⋆} \\ \Leftrightarrow B_{1} = (^{t} C : B) : {({(G ⊠ G^{- 1})}^{⋆})}^{- 1} = {(G^{- 1} ⊠ G)}^{⋆} : (^{t} C : B) ‖ by Theorem 4.7 \\ = ({(G^{- 1} ⊠ G)}^{⋆} :^{t} C) : B = C : B \\ \Rightarrow^{t} C = {({(G^{- 1} ⊠ G)}^{⋆})}^{- 1} : C = {(G^{- 1} ⊠ G)}^{⋆} : C, ∥ by Theorems 4.7 and 5.1 . \end{array}

Taking Lemma 5.4 and Theorems 4.6 and 5.1 into account, the transpose of ${}^{t}C$ is given by

\begin{array}{l} {(^{t} C)}^{T} = C^{T} : {({(G^{- 1} ⊠ G)}^{⋆})}^{T} = G^{- 1} ⊠ G : C^{*} : G ⊠ G^{- 1} : {(G^{- 1} ⊠ G)}^{⋆} = (G^{- 1} ⊠ G) : {(C^{*})}^{⋆} . \end{array}

Taking the transpose of $C^{t}$ and employing Theorem 4.6 and Lemma 5.4 and, finally, Theorem 4.6, one obtains

\begin{array}{l} {(C^{t})}^{T} =^{t} (C^{T}) = {(^{⋆} (G ⊠ G^{- 1}))}^{T} : C^{T} = {(G^{- 1} ⊠ G)}^{⋆} : C^{T} = ℐ^{⋆} : C^{*} : G ⊠ G^{- 1} \\ =^{⋆} (C^{*}) : G ⊠ G^{- 1} . \end{array}

Finally, on the basis of the results obtained,

\begin{array}{l} {{}^{t}C}^{t} = {(^{t} C)}^{t} =^{⋆} (G ⊠ G^{- 1}) : C^{t} =^{⋆} (G ⊠ G^{- 1}) : C : {(G^{- 1} ⊠ G)}^{⋆} ._{□} \end{array}

Example 5.11 According to Definition 4.1, the duals of the contra-variant one-point fourth-order tensor in Example 2.14 are given by

\begin{array}{l} C : = C^{I J K L} G_{I} ⊠ G_{J} ⊠ G_{K} ⊠ G_{L} \\ C^{*} = C^{I J K L} G_{J} ⊠ G_{I} ⊠ G_{L} ⊠ G_{K} = C^{J I L K} G_{I} ⊠ G_{J} ⊠ G_{K} ⊠ G_{L}, (c f . E x a m p l e) \\ C^{⋆} = C^{I J K L} G_{I} ⊠ G_{K} ⊠ G_{J} ⊠ G_{L} = C^{I K J L} G_{I} ⊠ G_{J} ⊠ G_{K} ⊠ G_{L}, \\ {}^{⋆}C = C^{I J K L} G_{L} ⊠ G_{J} ⊠ G_{K} ⊠ G_{I} = C^{L J K I} G_{I} ⊠ G_{J} ⊠ G_{K} ⊠ G_{L} . \end{array}

That is, $C, C^{*}, C^{⋆},^{⋆} C \in L i n (T_{X}^{*} P_{0} ⊠ T_{X}^{*} P_{0} ⊠ T_{X}^{*} P_{0} ⊠ T_{X}^{*} P_{0}; ℝ)$

Example 5.12 Let $C^{♯} : = C^{I J K L} G_{I} ⊠ G_{J} ⊠ G_{K} ⊠ G_{L}, C_{2}^{♯} : = A ⊠ A \in L i n (T_{X}^{*} P_{0} ⊠ T_{X}^{*} P_{0} ⊠ T_{X}^{*} P_{0} ⊠ T_{X}^{*} P_{0}; ℝ)$ be contravariant fourth-order one-point tensors. The duals were already given in Example 5.11 and the transpositions are obtained in accordance with Corollary 5.10:

C_{2}^{♯ T} : = A^{T} ⊠ A^{T} = G A^{*} G ⊠ G A^{*} G = G ⊠ G : C_{2}^{♯ *} : G ⊠ G,

i.e.

\begin{array}{l} C^{♯ T} = G ⊠ G : C^{♯ *} : G ⊠ G = C^{I J K L} G G_{J} ⊠ G G_{I} ⊠ G_{L} G ⊠ G_{K} G \\ = (C^{I J K L} G_{A J} G_{B I} G_{C L} G_{D K}) G^{A} ⊠ G^{B} ⊠ G^{C} ⊠ G^{D} = : {\tilde{C}}_{B A D C} G^{A} ⊠ G^{B} ⊠ G^{C} ⊠ G^{D} \\ ⇝ {\tilde{C}}_{B A D C} G^{A} ⊠ G^{B} ⊠ G^{C} ⊠ G^{D} ⇝ C^{I J K L} G_{J} ⊠ G_{I} ⊠ G_{L} ⊠ G_{K}, \\ C^{♯ t} = C^{♯} :^{⋆} (G ⊠ G) = C^{I J K L} G_{I} ⊠ G_{A B} (G_{J}, G^{D}) G^{B} ⊠ G_{C D} G^{C} ⊠ (G^{A}, G_{K}) G_{L} \\ = (C^{I J K L} G_{K B} G_{C J}) G_{I} ⊠ G^{B} ⊠ G^{C} ⊠ G_{L} = : {\tilde{C}}_{. C B}^{I . L} G_{I} ⊠ G^{B} ⊠ G^{C} ⊠ G_{L} \\ ⇝ {\tilde{C}}_{. C B}^{I . L} G_{I} ⊠ G^{B} ⊠ G^{C} ⊠ G_{L} ⇝ C^{I J K L} G_{I} ⊠ G_{K} ⊠ G_{J} ⊠ G_{L}, \\ {{}^{t}C}^{♯} = {(G ⊠ G)}^{⋆} : C^{♯} = C^{I J K L} G_{A B} G^{A} ⊠ (G^{C}, G_{I}) G_{J} ⊠ G_{K} ⊠ G_{C D} (G_{L}, G^{B}) ⊠ G^{D} \\ = (C^{I J K L} G_{A L} G_{I D}) G^{A} ⊠ G_{J} ⊠ G_{K} ⊠ G^{D} = : {\tilde{C}}_{D . A}^{. J K} G^{A} ⊠ G_{J} ⊠ G_{K} ⊠ G^{D} \\ ⇝ {\tilde{C}}_{D . A}^{. J K} G^{A} ⊠ G_{J} ⊠ G_{K} ⊠ G^{D} ⇝ C^{I J K L} G_{L} ⊠ G_{J} ⊠ G_{K} ⊠ G_{I}, \\ {}^{t}{(C^{♯})}^{t} = {(G ⊠ G)}^{⋆} : C^{♯} :^{⋆} (G ⊠ G) \\ = C^{I J K L} G_{E F} G^{E} ⊠ (G^{G}, G_{I}) G_{A B} (G_{J}, G^{D}) G^{B} ⊠ G_{C D} G^{C} ⊠ (G^{A}, G_{K}) G_{G H} (G_{L}, G^{F}) G^{H} \\ = (C^{I J K L} G_{K B} G_{C J} G_{E L} G_{I H}) G^{E} ⊠ G^{B} ⊠ G^{C} ⊠ G^{H} = : {\tilde{C}}_{H C B E} G^{E} ⊠ G^{B} ⊠ G^{C} ⊠ G^{H} \\ ⇝ {\tilde{C}}_{H C B E} G^{E} ⊠ G^{B} ⊠ G^{C} ⊠ G^{H} ⇝ C^{I J K L} G_{L} ⊠ G_{K} ⊠ G_{J} ⊠ G_{I}, \end{array}

where $G_{I}, G^{I}$ and $G^{I}, I = 1,2,3$ are the base vectors, the reciprocal base vectors and the dual base vectors, respectively. In the first place, the symbol $⇝$ denotes the transition from tensor algebra on manifolds to tensor algebra on inner-product spaces (TAIS) and in the second places it refers to the transition from TAIS to classical tensor algebra on inner-product spaces (CTAIS) [6]. According to TAIS, the dual and the reciprocal vector are considered equal, cf. Equation (9), whereas according to CTAIS, the transpositions and the duals are equal, i.e. $C^{T} = C^{*}$ and $G = I$ . Comparing the first result for $C^{♯ T}$ with Example 2.14 and assuming dual-symmetry property it can be concluded that $C^{♯ T} = C^{b}$ .

6. Symmetry and transformation rules of high-order tensors

In this section two different approaches for tensor symmetries will be introduced. An example in the first approach are symmetric co- and contra-variant tensor fields (on topological manifold $ℳ$ ) of which every value at every point $p \in ℳ$ is a symmetric tensor. An example in the second approach are symmetric mixed-variant tensor fields and tensors. The most important symmetric tensor field on $ℳ$ is the Riemannian metric of which every value at every point $p \in ℳ$ is a spd metric tensor $G \in S y m^{+} (T_{p} ℳ; T_{p}^{*} ℳ) \subset L i n (T_{p} ℳ; T_{p}^{*} ℳ)$ . Here, $T_{p} ℳ$ denotes the tangent space of $ℳ$ at $p$ .Another important example in continuum mechanics is a symmetric fourth-order tangent stiffness tensor, a matter of that will be discussed subsequently.

There are two approaches for symmetry:

symmetry based on the duals or self-duality (S-D);

symmetry based on the transpositions or transposition-symmetry (T-S).

Definition 6.1 A $k$ -contra-variant tensor $T^{♯}$ is completely symmetric if ⁵

T^{♯} (G^{1}, \dots, G^{I}, \dots, G^{J}, \dots, G^{K}) = T^{♯} (G^{1}, \dots, G^{J}, \dots, G^{I}, \dots, G^{K}) (S - D)

(28)

for all $1 ≼ I ≺ J ≼ K$ Similarly, symmetry is defined for a k-co-variant tensor $T^{b}$ . A $(k, l)$ -mixed-variant tensor $T^{♮}$ is completely symmetric if

\begin{array}{l} T^{♮} (G_{1}, \dots, G_{I}, \dots, G_{J}, \dots, G_{K}, G^{L}, \dots, G^{M}, \dots, G^{N}, \dots, G^{Q}) \\ = T^{♮} (G_{1}, \dots, G_{I}, \dots, G_{N}, \dots, G_{K}, G^{L}, \dots, G^{M}, \dots, G^{J}, \dots, G^{Q}) (T - S) \\ = T^{♮} (G_{1}, \dots, G_{J}, \dots, G_{I}, \dots, G_{K}, G^{L}, \dots, G^{N}, \dots, G^{M}, \dots, G^{Q}) (S - D) \end{array}

(29)

for all $1 ≼ I ≺ J ≼ K ≺ L ≼ M ≺ N ≼ Q$

One can also define the corresponding operation, called symmetrization consisting of permutations in Equations (28) and (29). According to Definition 6.1, the duals define symmetries and symmetrizations of co- and contra-variant forms (self-duality), whereas the transpositions enable symmetrization of mixed-variant forms on mixed primary and dual tensor spaces. In contrast to complete symmetry, a special class of symmetries is of major importance in continuum mechanics.

Definition 6.2 A tensor, $C \in Lin$ is (major-)symmetric if $C^{b *} = C^{b}$ , $C^{♯ *} = C^{♯}$ or $C^{♮ T} = C^{♮}$ .

As a result of Definitions 3.6 and 6.2, S-D is defined on the basis of the scalar products whereas T-S is defined on the basis of the inner products, cf. Definition 3.7. Symmetry defined as

C^{T} = C^{*} = C, C \in S y m

(30)

holds if $C^{T} = G_{1}^{- 1} C^{*} G_{2} = C$ or $G_{1} C G_{2}^{- 1} = C^{*} = C$ cf. Corollary 3.8. Consequently, $G_{1} = G_{2} = ℐ$ , i.e. they are both identified as identities.⁶ Since tensors are invariant and the transposition is a homomorphism or the structural preserving map (or even isomorphism), symmetry of a tensor remains under change of the basis, i.e. $C^{♮ T} = C^{♮} \Leftrightarrow C^{b *} = C^{b} \Leftrightarrow C^{♯ *} = C^{♯}$ . As an illustration, one can compare the relations present in Example 2.13 and Remark 3.4. It can be concluded that $C^{♯ T} = C^{b} = G C^{♮} \in S y m$ if $C^{b} = C^{b *} \in Sym$ , i.e. $C^{♯ T}$ and $C^{♯}$ lie on different spaces. Since ${(C^{b})}^{*} = {(G C^{♯} G)}^{*} = G C^{♯ *} G = C^{b}$ , $G \in S y m^{+}$ , it follows that $C^{♯ *} = C^{♯}$ . Then, from Lemma 3.11 and Remark 3.5 it follows that

\begin{array}{l} G^{♯} = {(G C^{♮})}^{T} = C^{♮ T} G^{T} = G^{- 1} C^{♮ *} G G^{- 1} = G^{- 1} C^{♮ *} = G^{♯ *} = C^{♮} G^{- 1} | | \cdot G \\ ∴ C^{♮ T} = C^{♮} . \end{array}

As shown, $C^{b}, C^{♯}$ , and $C^{♮} \in Sym$ are symmetric only in the sense of the dual and the transpose, respectively. Consequently, the two-point tensor in Equation (15) cannot be symmetric, only its component matrix can be symmetric.

The corresponding operation for self-duality is dual-symmetrization, defined as $sym (C^{b}) : = 1 / 2 (C^{b} + C^{b *})$ , and similarly for the contra-variant form. Based on the discussion above, the symmetrization using the transpose $sym (C^{♮}) : = 1 / 2 (C^{♮} + C^{♮ T})$ is defined only for mixed-variant forms when the members in summation lie on one and the same space.

An example in continuum mechanics is the linearization of the stationary condition of a variational principle with respect to symmetric tensor-valued arguments. The linearization results in the quadratic form for the energy power given by the (2:4:2) double contractions between the tangent stiffness tensor and the second-order tensor arguments. As a result, the fourth-order tangent stiffness tensor has both minor and sub-symmetries, i.e. ${}^{⋆}{(C)}^{b ⋆} = C^{b}$ ${}^{⋆}{(C)}^{♯ ⋆} = C^{♯}$ or ${}^{⋆}{(C)}^{♮ ⋆} = C^{♮}$ and it is called a sub-symmetric tensor [27]. The major symmetry property follows from the existence of a quadratic potential present in a variational principle. Having the major symmetry as well as both minor symmetries, i.e. $C^{b} = C^{b *} =^{⋆} C^{b} = C^{b ⋆}$ $C^{♯} = C^{♯ *} =^{⋆} C^{♯} = C^{♯ ⋆}$ or $C^{♮} = C^{♮ T} =^{t} C^{♮} = C^{♮ t}$ , a tensor is called super-symmetric tensor [23]. The corresponding operation, super-symmetrization, is defined for the mixed-variant form as

s s y m m : = \frac{1}{8} (C^{♮} + C^{♮ t} +^{t} C^{♮} +^{t} C^{♮ t} + C^{♮ T} + {(C^{♮ T})}^{t} +^{t} (C^{♮ T}) +^{t} {(C^{♮ T})}^{t}).

(31)

The terms in Equation (31) can be found from Corollary 5.10. Similarly, using the duals super-symmetrization for co- and contra-variant forms are defined by the terms given in Example 5.11.

6.1. Transformation rules

What is still missing are the generalized transformation rules for high-order tensors. The formulae for the computation of push-forwards $φ_{*}$ of co-, contra-, and mixed-variant tensors $C^{b}$ , $C^{♯}$ , and $C^{♮} \in L i n_{2 m}$ and pullbacks $φ^{*}$ of co-, contra-, and mixed-variant tensors $c^{b}$ , $c^{♯}$ , and $c^{♮} \in L i n_{2 m}$ are extended on the basis of the isomorphisms $ℱ_{1} \in L i n_{2 m}$ and $ℱ_{2} \in L i n_{2 m}$ . Exploiting the generalized scalar and the inner product, it follows from Definition 3.7 that

φ_{*} C^{♮} (ℱ_{1} (V)) • ℱ_{2} (W) = ℱ_{2}^{T} φ_{*} C^{♮} ℱ_{1} (V) • W, V, W \in L i n_{m} .

The invariance $ℱ_{2}^{T} φ_{*} C^{♮} ℱ_{1} (V) • W = C^{♮} (V) • W$ requires that $φ_{*} C^{♮} = ℱ_{2}^{- T} C^{♮} ℱ_{1}^{- 1}$ . Similarly, applying Definition 3.6 one obtains $(ℱ_{2}^{*} φ_{*} C^{b} ℱ_{1} (V), W) = (C^{b} (V), W)$ and the invariance requires that $φ_{*} C^{b} = ℱ_{2}^{- *} C^{b} ℱ_{1}^{- 1}$ . By analogy, the push-forward of a contra-variant form is derived, i.e.

\begin{array}{l} (φ_{*} C^{b} (ℱ_{1} (V)), ℱ_{2} (W)) & = (ℱ_{2}^{*} φ_{*} (G C^{♯} G) ℱ_{1} (V), W) = (ℱ_{2}^{*} g φ_{*} C^{♯} g ℱ_{1} (V), W) \\ = (G C^{♯} G (V), W), \end{array}

where Example 2.13 was exploited for $C^{b}$ . Using Cor. 3.8 for $ℱ_{1}$ and $ℱ_{2}$ , the invariance results in $φ_{*} C^{♯} = g^{- 1} ℱ_{2}^{- *} G C^{♯} G ℱ_{1}^{- 1} g^{- 1} = ℱ_{2}^{- T} C^{♯} ℱ_{1}^{- * T}$ . The pull-back is an inverse transformation from the actual placement to the reference placement, i.e. $φ^{*} c^{♮} = ℱ_{2}^{T} c^{♮} ℱ_{1}$ , etc. Specifically, if $ℱ_{2} = ℱ_{1}$ , the push-forward and the pull-back operations preserve symmetry of tensors. The expressions for push-forwards and pull-backs are summarized in Table 1. In contrast to prior approaches [26], the symmetry-preserving transformation rules of a contra-variant tensor require a correction in the definition. The present approach results in push-forwards and pull-backs being unambiguous, e.g. the transformation rules for a mixed-variant tensor are immediately derived using the left isomorphism $ℱ_{2}$ for a contra-variant form and the right isomorphism $ℱ_{1}$ for a co-variant form. Moreover, according to Example 2.13,

\begin{array}{l} {(φ_{*} C^{b})}^{*} & = {(ℱ^{- *} C^{b} ℱ^{- 1})}^{*} = ℱ^{- *} C^{#, *} ℱ^{- 1} = ℱ^{- *} C^{b} ℱ^{- 1} = φ_{*} C^{b}, C^{b} \in S y m, \\ {(φ_{*} C^{♯})}^{*} & = {(ℱ^{- T} C^{♯} ℱ^{- * T})}^{*} = ℱ^{- T} C^{#, *} ℱ^{- * T} = ℱ^{- T} C^{♯} ℱ^{- * T} = φ_{*} C^{♯}, C^{♯} \in S y m, \\ {(φ_{*} C^{♮})}^{T} & = {(ℱ^{- T} C^{♮} ℱ^{- 1})}^{T} = ℱ^{- T} C^{♮, T} ℱ^{- 1} = ℱ^{- T} C^{♮} ℱ^{- 1} = φ_{*} C^{♮}, C^{♮} \in S y m, \end{array}

Table 1.

Symmetry-preserving transformation rules of high-order tensors.

A high-order tensor	Push-forward, $φ_{*}$	Pull-back, $φ^{*}$
$C^{b}$ , $c^{b}$	$φ_{} C^{b} = ℱ^{- } C^{b} ℱ^{- 1}$	$φ^{} c^{b} = ℱ^{} c^{b} ℱ$
$C^{♯}$ , $c^{♯}$	$φ_{} C^{♯} = ℱ^{- T} C^{♯} ℱ^{- T}$	$φ^{} c^{♯} = ℱ^{T} c^{♯} ℱ^{ T}$
$C^{♮}$ , $c^{♮}$	$φ_{*} C^{♮} = ℱ^{- T} C^{♮} ℱ^{- 1}$	$φ^{*} c^{♮} = ℱ^{T} c^{♮} ℱ$

i.e. symmetry is preserved in the transformations.

Example 6.3 Push-forwards by $F \in Lin (T_{X} P_{0}; T_{x} P)$ that preserve symmetry of a co-, contra- and mixed-variant tensor $A^{b}$ , $B^{♯}$ and $C^{♮}$ are given by $F^{- *} A^{b} F^{- 1}$ , $F^{- T} B^{♯} F^{- * T}$ and $F^{- T} C^{♮} F^{- 1}$ , respectively.⁷ Pull-backs are $F^{*} a^{b} F$ , $F^{T} b^{♯} F^{* T}$ and $F^{T} c^{♮} F$ .

Remark 6.4 The symmetry-preserving transformation rules for the transposed inverse can be readily obtained by taking the inverse and transpose of the rules presented in Table 6.4, i.e.

{(φ_{*} C^{♯})}^{- T} = ℱ {(C^{♯})}^{- T} ℱ^{*} = φ_{*} {(C^{♯})}^{- T} etc .

Since a tensor of any form $C$ and its transposed inverse $C^{- T}$ lie on one and the same space, one can formally write ⁸

φ_{*} C^{♯} = ℱ C^{♯} ℱ^{*}, φ^{*} c^{♯} = ℱ^{- 1} c^{♯} ℱ^{- *} .

(32)

Assuming that the metric $g$ and $G$ are identities, it follows from Corollary 3.3 and 5.7 that $ℱ^{T} = ℱ^{*}$ . Using this assumption, the transformation rules of the contra-variant form in (32) are equal to those given in literature, cf., e.g., [16, 23, 28].

Example 6.5 In an implicit finite-element solution process, the stress–strain relation needs to be linearized. In state-of-the-art models for polymeric materials, the Kirchhoff stress $τ^{♮} \in Sym$ in terms of the elastic stretch tensor $v^{e, ♮} \in Sym$ is defined as

τ^{♮} : = 2 μ {(\ln v^{e, ♮})}^{dev} + κ trace (\ln v^{e, ♮}) i = 2 μ (i ⊠ i + \frac{3 κ + 2 μ}{6 μ} i ⊡ i): \ln v^{e, ♮} =: L^{e, ♮} : \ln v^{e, ♮}

(33)

where ${(\ln v^{e, ♮})}^{dev} : = \ln v^{e, ♮} - \frac{1}{3} trace (\ln v^{e, ♮}) i$ denotes the deviatoric component and $μ$ and $κ$ are the elastic constitutive parameters, cf. [21, 29]. Again, ‘:’ denotes the double contraction defined by A.5. In the following, use is made of the usual assumption that the kinetic, stress-like quantities are given in the contra-variant form, whereas the kinematic quantities are considered as the co-variant form. For the sake of simplicity, the notation $♯$ and $♯$ will be neglected. Based on Example 2.13, the contra-variant form of the Kirchhoff stress becomes

\begin{array}{l} τ = τ^{♮} g^{- 1} : = 2 μ g^{- 1} {(\ln v^{e})}^{dev} g^{- 1} + κtrace (g^{- 1} \ln v^{e}) g^{- 1} \\ = 2 μ (g^{- 1} ⊠ g^{- 1} + \frac{3 κ + 2 μ}{6 μ} g^{- 1} ⊡ g^{- 1}): \ln v^{e} =: L^{e} : \ln v^{e} \in Sym . \end{array}

(34)

In contrast to the elastic tangent stiffness tensor $L^{e, ♮}$ in Equation (33), the representation of the contra-variant form $L^{e}$ depends on the contra-variant metric $g^{- 1}$ under consideration. Using the transformations in (32), the algorithmic tangent stiffness (ATS) tensor takes in terms of the second Piola–Kirchhoff stress $S : = F^{- 1} τ F^{- *}$ and the Cauchy–Green deformation tensor $C : = F^{*} gF$ the form

\begin{array}{l} C : = \frac{d S}{d C} =' {(F^{- 1} ⊠ F^{- *})}^{'} : \frac{d τ}{d F} : {(g^{- 1} F^{- *} ⊠ I)}^{'}' {(S ⊠ C^{- 1})}^{'} \\ \in L i n (T_{X}^{*} P_{0} ⊠ T_{X}^{*} P_{0} ⊠ T_{X}^{*} P_{0} ⊠ T_{X}^{*} P_{0}; ℝ) \end{array}

(35)

where $I$ is the identity, defined in the material configuration. In the above the left, right, and their coupled sub-symmetrization, denoted by

' (•) = \frac{1}{2} ((•) +^{⋆} (•)), {(•)}^{'} = \frac{1}{2} ((•) + {(•)}^{⋆}),' {(•)}^{'} = \frac{1}{4} ((•) + {(•)}^{⋆} +^{⋆} (•) +^{⋆} {(•)}^{⋆}),

(36)

were introduced. The left sub-symmetry results from the symmetry of the stress $S$ , whereas the strain $C$ and the stress $t$ result in the right sub-symmetries in Equation (35). The minor transpositions needed in Equation (36) are given in Example 5.11. As an example, the symmetrized components of $L^{e}$ are given by

{(L^{e})}^{ijkl} : = 2 μ [\frac{1}{2} (g^{ij} g^{kl} + g^{jl} g^{ik}) + \frac{3 κ + 2 μ}{6 μ} g^{il} g^{jk}]

(37)

where the pairs $il$ and $jk$ form the row and the column index of the component matrix, respectively. It should be noted that the ATS tensor $C$ possesses sub-symmetries but not the major symmetry property. Using Equation (32), the Lagrangian form (35) can be pushed forward to the spatial configuration which yields

\begin{array}{l} c : = \frac{d τ}{d F} : {(g^{- 1} ⊠ F)}^{'} - 2^{'} {(t ⊠ g^{- 1})}^{'} \in Lin (T_{x}^{*} P ⊠ T_{x}^{*} P ⊠ T_{x}^{*} P ⊠ T_{x}^{*} P; ℝ) . \end{array}

(38)

The linearization of the Kirchhoff stress $τ$ results in

\begin{array}{l} \frac{d t}{d F} & = L^{e} : \frac{\partial (\ln v^{e})}{\partial v^{e}} : \frac{d v^{e}}{d F} . \end{array}

(39)

The derivative of the tensor logarithm in Equation (39) can be calculated following, e.g., [30]. The details how to obtain the derivative $d v^{e} / d F$ which is consistent with the integration algorithm employed can be found, e.g., in [31]. If the metrics are considered as identities, Equations (35) and (38) take the form

\begin{array}{l} C =' {(F^{- 1} ⊠ - F^{- T})}^{'} : \frac{d τ}{d F} : {(F^{- T} ⊠ I)}^{'} - 2' {(S ⊠ C^{- 1})}^{'}, \\ c = \frac{d τ}{d F} : {(i ⊠ F)}^{'} - 2^{'} {(τ ⊠ i)}^{'} . \end{array}

(40)

A comparison of Equations (35) and (38) with Equation (40) reveals a difference of the two frameworks. An advantage of tensor calculus on inner-product spaces is its applicability over the all sub-spaces, whereas the classical approach can be applied only on the sub-space under consideration. In other words, it is not clear how Equations (35) and (38) can be obtained if the classical approach and the results in Equation (40) are employed. Moreover, different tensor forms cannot be identified in the classical approach and as a consequence, pull-back and push-forward operations become ambiguous.

Footnotes

A. Appendix: Special tensor products and contractions in continuum mechanics

There are defined several tensor products and contractions for different purposes in continuum mechanics. Some of them will be introduced being suitable for the second- and the fourth-order tensors, widely used in continuum mechanics. As it was pointed out in Section 2.1, cf. Remark 2.3 and Definition 2.5, decomposable tensors allow us to uniquely define a map $h : T \to S$ . However, a map $h$ exists and is unique for all $A \in T$ and $B \in S$ (not necessarily decomposable) assuming that the diagram in Figure 1 commutes.

Obviously, the simple contraction can be extended to concern tensors of any order. The simple contraction between tensors $C_{1}, C_{2}$ , and $C_{3} \in Lin$ satisfies both the associative and the distributive rule, i.e.

C_{1} (C_{2} C_{3}) = (C_{1} C_{2}) C_{3}, C_{1} (C_{2} + C_{3}) = C_{1} C_{2} + C_{1} C_{3} .

Obviously, the simple contraction does not obey the commutative rule, i.e. $C_{1} C_{2} \neq C_{2} C_{1}$ .

The double contraction between two second-order tensors is commutative: $F_{1} : F_{2} = F_{2} : F_{1}$ . It also satisfies the associative and the distributive rule:

A : ((B + C) : D) = (A : (B + C)) : D = A : B : D + A : C : D .

The double contraction between a fourth-order and a second-order tensor, denoted by (4:2), is defined to be consistent with the chain rule and consequently with the essential rate relation $\dot{A} (B) = A_{, B} : \dot{B}$ as well as with the product derivative rule, cf. [23].

where

\begin{array}{l} C_{1} : = {(C_{1})}^{i}_{. j}^{k}_{. l} g_{i} ⊠ G^{j} ⊠ g_{k} ⊠ G^{l} \in Lin (T_{X} P_{0} ⊠ T_{x}^{*} P; T_{x} P ⊠ T_{X}^{*} P_{0}), \\ F_{2} : = {(F_{2})}_{. n}^{m} G_{m} ⊠ g^{n} \in L i n (T_{x} P; T_{X} P_{0}) . \end{array}

The double contraction between two fourth order tensors is a four-linear map $Li n_{4} \times L i n_{4} \to L i n_{4}$ defined as

\begin{array}{l} ∴ Lin (T_{X} P_{0} ⊠ T_{x}^{*} P; T_{x} P ⊠ T_{X} P_{0}) \times Lin (T_{x} P ⊠ T_{X}^{*} P_{0}; T_{X} P_{0} ⊠ T_{x}^{*} P) \to \\ Lin (T_{x} P ⊠ T_{X}^{*} P_{0}; T_{x} P ⊠ T_{X}^{*} P_{0}), \\ (a ⊠ b ⊠ c ⊠ d, e ⊠ f ⊠ g ⊠ h) \mapsto (a ⊠ b ⊠ c ⊠ d) : (e ⊠ f ⊠ g ⊠ h) : = \\ {(b, e)}_{T_{X} P_{0}} {(c, h)}_{T_{x}} a ⊠ f ⊠ g ⊠ d = (b_{j} e^{j}) (c^{l} h_{l}) a ⊠ f ⊠ g ⊠ d a n d \\ (C_{1}, C_{2}) \mapsto C_{1} : C_{2} : = {(C_{1})}^{i}_{. j}^{k}_{. l} {(C_{2})}^{m}_{. n}^{q}_{. r} g_{i} (G^{j}, G_{m}) g^{n} ⊠ G_{q} (g^{r}, g_{k}) G^{l} = \\ {(C_{1})}^{i}_{. j}^{k}_{. l} {(C_{2})}^{j}_{. n}^{q}_{. k} g_{i} ⊠ g^{n} ⊠ G_{q} ⊠ G^{l} \in L i n (T_{x} P ⊠ T_{X}^{*} P_{0}; T_{x} P ⊠ T_{X}^{*} P_{0}), \end{array}

where $C_{2} : = {(C_{2})}^{m}_{. n}^{q}_{. r} G_{m} ⊠ g^{n} ⊠ G_{q} ⊠ g^{r} \in Lin (T_{x} P ⊠ T_{X}^{*} P_{0}; T_{X} P_{0} ⊠ T_{x}^{*} P)$ .

The double contractions (4:2) and (4:4) fulfil both the associative and distributive rule, i.e.

(A : (ℬ + C)) : D = A : ((ℬ + C) : D) = A : ℬ : D + A : C : D, D \in L i n_{2}, A, ℬ, C \in L i n_{4} .

However, they do not obey the commutative rule $A : ℬ \neq ℬ : A, A : ℬ \neq ℬ : A$ .

Based on the identification of a four-linear map in Definition A.5, another important tensor product ‘ $⊡$ ’ is now introduced. Since the tensor representations are based on the standard tensor product ‘ $⊠$ ’ (also termed the Kronecker product), a transformation ${(\cdot)}^{R}$ which enables a representation of this new tensor product by means of the standard tensor product needs to be introduced.

The tensor products ‘ $\land$ ’ (‘ $⊠$ ’ or ‘ $⊡$ ’) satisfy the distributive rule: $F \land (H + K) = F \land H + F \land K, \forall F, H, K \in Lin$ . According to Theorem 2.8 the tensor products are associative: $F \land H \land K = (F \land H) \land K = F \land (H \land K)$ . Noting that $A : B = B : A$ and $α A = A α$ but that, in general, $A B \neq B A$ , make use of the rate relation results in the following derivative rules:

\begin{array}{l} {(α A)}_{, B} & = A ⊡ α_{, B} + α A_{, B}, {(A : B)}_{, C} = A : B_{, C} + B : A_{, C}, {(A B)}_{, C} = A_{, C} B + A B_{, C} . \end{array}

Note here that the usual operation

{(C_{1})}^{i}_{. j}^{k}_{. l} g_{i} ⊠ G^{j} ⊠ g_{k} ⊠ G^{l} : {(F_{2})}_{m .}^{n} g^{m} ⊠ G_{n} = ({(C_{1})}^{i}_{. j}^{k}_{. l} {(F_{2})}_{k .}^{l} g_{i} ⊠ G^{j}

does not satisfy the rate relation

{(A (C) B)}_{, C} : \dot{C} = (A_{, C} : \dot{C}) B = A_{, C} B : \dot{C} .

The relation between the standard tensor product $A ⊠ B$ and the tensor product in Example A.6 is obtained by means of the double contraction, i.e. $C : {(A ⊠ B)}^{R} : D = C : (A ⊡ B) : D$ . It then follows that

(41)

{(A ⊠ B)}^{R} : = A ⊡ B or {(A_{. j}^{i} B_{. l}^{k} G_{i} ⊠ G^{j} ⊠ G_{k} ⊠ G^{l})}^{R} : = A_{. j}^{i} B_{. l}^{k} G_{i} ⊠ G^{l} ⊠ G^{j} ⊠ G_{k} .

Based on Definitions A.1 and A.5 for the simple and double contraction, respectively, the following identities are derived for all $A \in {Lin}_{4}$ and $A, B, C, D \in {Lin}_{2}$ :

(42a)

(A ⊠ B) : C = ACB (A ⊡ B) : C = (B : C) A

(42b)

C : (A ⊡ B) = (C : A) B (A ⊠ B) : (C ⊠ D) = A C ⊠ D B

(42c)

(A ⊡ B) : (C ⊡ D) = (B : C) A ⊡ D A (A : C) B = (A A B) : C

Acknowledgements

The author acknowledges numerous discussions with professor Tapio Salmi and Dr Jari Mäkinen on various aspects of this research. The author would also like to thank professors Matti Ristinmaa and Niels Saabye Ottosen in Lund University in Sweden for their valuable comments and criticism that have contributed to improvements in the present paper.

Notes

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Conflict of interest

None declared.

References

van der Giessen

Kollmann

. On mathematical aspects of dual variables in continuum mechanics. part 1: Mathematical principles. Z Angew Math Mech 1996; 76: 447–462.

Abraham

Marsden

Ratiu

. Manifolds, Tensor Analysis and Applications. Reading, MA: Addison-Wesley, 1983.

Haupt

. On the mathematical modelling of material behavior in continuum mechanics. Acta Mechanica 1993; 100: 129–154.

Lee

. Introduction to Smooth Manifolds. New York: Springer, 2003.

Marsden

Hughes

TJR

. Mathematical Foundations of Elasticity. New York: Dover Publications, Inc., 1993.

Stumpf

Hoppe

. The application of tensor algebra on manifolds to nonlinear continuum mechanics-invited survey artile. Z Angew Math Mech 1997; 77: 327–339.

Truesdell

Noll

. The Nonlinear Field Theories of Mechanics, Handbuch der Physik, vol. III/3. Berlin: Springer-Verlag, 1965.

Greub

. Multilinear Algebra. Berlin: Springer-Verlag, 1967.

Merris

. Multilinear Algebra. Hayward, CA: Gordon and Breach Science/California State University, 1997.

10.

Cardona

Geradin

Doan

. Rigid and flexible joint modelling in multibody dynamics using finite elements. Comput Meth Appl Mech Engng 1991; 89: 395–418.

11.

Mäkinen

. Total Lagrangian Reissner"s geometrically exact beam element without singularities. Int J Numer Meth Engng 2007; 70: 1009–1048.

12.

Yavari

Marsden

Ortiz

. On spatial and material covariant balance laws in elasticity. J Math Phys 2006; 47: 1–53.

13.

Ball

. Convexity conditions and existence theorems in nonlinear elasticity. Arch Rat Mech Anal 1977; 63: 337–403.

14.

Haupt

Tsakmakis

. On the application of duals variables in continuum mechanics. Continuum Mech Thermodyn 1989; 1: 165–196.

15.

Sansour

Bednarczyk

. A study on rate-type constitutive equations and the existence of a free energy function. Acta Mechanica 1993; 100: 205–221.

16.

Kintzel

. Modeling of Elasto-plastic Material Behavior and Ductile Micropore Damage of Metallic Materials at Large Deformations. Aachen: Shaker-Verlag, 2007.

17.

Miehe

Stein

Wagner

. Associative multiplicative elasto-plasticity: Formulation and aspects of the numerical implementation including stability analysis. Comp Struct 1994; 52: 969–978.

18.

Beatty

. Constitutive equations for the backstress in amorphous glassy polymers. Math Mech Solids 2005; 10: 167–181.

19.

Coleman

Hodgdon

. On shear bands in ductile materials. Arch Rat Mech Anal 1985; 90: 219–247.

20.

Nemat-Nasser

Hori

. Micromechanics - Overall Properties of Heterogenous Solids. Amsterdam: North-Holland, 1993.

21.

van der Giessen

. On improved network models for rubber elasticity and their applications to orientation hardening in glassy polymers. J Mech Phys Solids 1993; 41: 427–456.

22.

Park

Youn

S-K

. Finite element analysis and constitutive modelling of anisotropic nonlinear hyperelastic bodies with convected frames. Comput Meth Appl Mech Engng 1998; 151: 605–618.

23.

Itskov

. Tensor Algebra and Tensor Analysis for Engineers with Applications to Continuum Mechanics. Berlin: Springer-Verlag, 2007.

24.

Basar

Weichert

. Nonlinear Continuum Mechanics of Solids. Berlin: Springer-Verlag, 2000.

25.

Carmo

MPD

. Riemannian Geometry. Cambridge, MA: Harvard University Press, 1988.

26.

Kollman

Hackenberg

. On the algebra of two-point tensors on manifolds with applications in nonlinear solid mechanics. Z Angew Math Mech 1993; 73: 307–314.

27.

Bertram

. Elasticity and Plasticity of Large Deformations. Berlin: Springer-Verlag, 2005.

28.

Belytschko

Liu

Moran

. Nonlinear Finite Elements for Continua and Structures. Chichester: Wiley, 2000

29.

Arruda

Boyce

. Evolution of plastic anisotropy in amorphous polymers during finite straining. Int J Plasticity 1993; 9: 697–720.

30.

Miehe

. Comparison of two algorithms for the computation of fourth-order isotropic tensor functions. Comp Struct 1998; 66: 37–43.

31.

Ekh

Runesson

. Modeling and numerical issues in hyperelasto-plasticity with anisotropy. Int J Solids Structures 2001; 52: 1487–1500.