Vector and Tensor Calculus an Introduction E1 E2 E3 Α11 Α21 Α22

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Universit¨at Stuttgart Institut f¨ur Mechanik Prof. Dr.-Ing. W. Ehlers www. mechbau.uni-stuttgart.de

Vector and Tensor Calculus An Introduction

e3 ∗ e3 e∗2

α22 α21 e 2 α11 e1 e∗1

Last Change: 10 April 2018

Chair of Continuum Mechanics, Pfaﬀenwaldring 7, D - 70 569 Stuttgart, Tel.: (0711) 685-66346

Contents

1 Mathematical Prerequisites 1 1.1 Basicsofvectorcalculus ...... 1

2 Fundamentals of tensor calculus 9 2.1 Introductionofthetensorconcept...... 9 2.2 Basicrulesoftensoralgebra ...... 10 2.3 Speciﬁctensorsandoperations...... 13 2.4 Changeofthebasis...... 16 2.5 Higherordertensors ...... 25 2.6 Fundamental tensor of 3rd order (Ricci permutationtensor) ...... 28 2.7 Theaxialvector...... 28 2.8 Theoutertensorproductoftensors ...... 33 2.9 The eigenvalue problem and the invariants of tensors ...... 34

3 Fundamentals of vector and tensor analysis 36 3.1 Introductionoffunctions ...... 36 3.2 Functions of scalar variables ...... 36 3.3 Functionsofvectorandtensorvariables...... 37 3.4 Integraltheorems ...... 42 3.5 Transformations between actual and reference conﬁgurations...... 47

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1 Mathematical Prerequisites

1.1 Basics of vector calculus

(a) Symbols, summation convention, Kronecker δ

Single- or multiple subscripts

ui u1, u2, u3, ... u v −→ u v , u v , u v , ... i k −→ 1 1 1 2 1 3 u2 v1, u2 v2, ...... tik t11, t12, ... −→ ...

Summation convention of Einstein

Deﬁnition: Whenever the same subscript occurs twice in a term, a summation over that “double” subscript has to be carried out.

Example: uj vj = u1 v1, + u2 v2, + ... + un vn, n

= uj vj j=1 X Kronecker symbol

Deﬁnition: It exists a symbol δik with the following properties

0 if i = k δik = 6  1 if i = k   Example: ui δik = u1 δ1k + u2 δ2k + ... + un δnk

u1 δ11 = u1 u δ = 0  1 12 with u δ = 1 1k   ·  · u1 δ1n = 0  ui δik = uk −→  If the Kronecker symbol is multiplied with another quantity and if there is a double subscript in this term, the Kronecker symbol disappears, the “double” subscript can be dropped and the free subscript remains.

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Rem.: Subscripts occuring two times in a term can be renamed arbitrarily. (b) Terms and definitions of vector algebra

Rem.: The following statements are related to the standard three-dimensional (3-d) physical space, i. e. the Euclidean vector space 3. V Generally, SPACE is a mathematical concept of a set and does not directly refer to the 3-d point space 3 and the 3-d vector space 3. E V A: Vector addition Requirement: u, v, w, ... 3 { }∈ V The following relations hold:

u + v = v + u : commutative law u +(v + w) =(u + v)+ w : associative law u + 0 = u : 0 : identity element of vector addition u +( u) = 0 : u : inverse element of vector addition − −

Examples to the commutative and the associative law:

u + v v u + v v v + w u u w u v v + u u + v + w

B: Multiplication of a vector with a scalar quantity Requirement: u, v, w, ... 3; α, β, ... R { } ∈V { } ∈

1 v = v : 1: identity element α (β v) =(α β) v : associative law (α + β) v = α v + β v : distributive law (addition of scalars) α (v + w) = α v + α w : distributive law (addition of vectors) α v = v α : commutative law

Rem.: In the general vector calculus, the deﬁnitions A and B constitute the “aﬃne vector space”.

Linear dependency of vectors Rem.: In 3, 3 non-coplanar vectors are linearly independent; i. e. each further vector V can be expressed as an multiple of these vectors.

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Theorem: The vectors vi (i =1, 2, 3, ..., n) are linearly dependent, if real numbers αi exist which are not all equal to zero, such that

αi vi = 0 or α1 v1 + α2 v2 + ... + αn vn = 0

Example (plane case):

v1 + v2 + v3 = 0 v2 α2 v2 6 but: α1 v1 + α2 v2 + α3 v3 = 0 v3 α v 1 1 α3 v3 v1, v2, v3 : linearly dependent v1 −→ { } v , v : linearly independent −→ { 1 2}

Rem.: The αi can be multiplied by any factor λ.

Basis vectors in 3 V ex. : v , v , v : linearly independent { 1 2 3} then : v , v , v , v : linearly dependent { 1 2 3 } Thus, it follows that

α1 v1 + α2 v2 + α3 v3 + λ v = 0 λ v = α v −→ − i i α or v = − i v =: β v λ i i i α β = − i : coeﬃcients (of the vector components) with i λ ( vi : basis vectors of v

Choice of a speciﬁc basis

Rem.: In 3, each system of 3 linearly independent vectors can be selected as a basis; V e. g.

vi : general basis

ei : speciﬁc, orthonormal basis (Cartesian, right-handed)

v e3 v v3

v2 e2 v1 e1

Basissystem vi Basissystem ei

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Representation of the vector v:

βi vi v = ( γi ei here: Speciﬁc choice of the Cartesian basis system ei

Notations

v = vi ei = v1 e1 + v2 e2 + v3 e3

vi ei : vector components with ( vi : coeﬃcients of the vector components

C: Scalar product of vectors The following relations hold:

u v = v u : commutative law · · u (v + w) = u v + u w : distributive law · · · α (u v) = u (α v)= (α u) v : associative law · · · u v = 0 u, if v 0 · ∀ ≡ u u = 0 , if u = 0 −→ · 6 6

Rem.: The deﬁnitions A, B and C constitute the “Euclidean vector space”. If instead of u u = 0 especially · 6 u u > 0 , if u = 0, · 6 holds, then A, B and C deﬁne the “proper Euclidean vector space 3” (physical space). V

Square and norm of a vector

v2 := v v , v = v = √v2 · | |

Rem.: The norm is the value or the positive square root of the vector.

Angle between two vectors

v u v − <)(u ; v) =: α α u

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Law of cosines

u v 2 = u 2 + v 2 2 u v cos α | − | | | | | − | || | u2 + v2 (u v)2 cos α = − − −→ 2 u v u v | || | = · u v | || | or u v = u v cos α · | || | Scalar products (inner products) in an orthonormal basis

Scalar product of the basis vectors ei: 90◦ if i = k : cos90◦ =0 6 <)(ei ; ek) ◦ ◦ ( 0 if i = k : cos0 =1 thus

e e = e e cos <)(e ; e ) i · k | i|| k| i k = cos <)(ei ; ek)

It follows with the Kronecker δ

1 if i = k ei ek = δik = · ( 0 if i = k 6

Scalar product of two vectors: u v =(u e ) (v e ) · i i · k k = u v (e e ) i k i · k = ui vk δik

= ui vi = u1 v1 + u2 v2 + u3 v3

D: Vector or cross product (outer product) of vectors One deﬁnes the following vector product u v = u v sin <)(u ; v) n × | || | with n: unit vector u , v (corkscrew rule or right-hand rule, see page 7) ⊥ From the above deﬁniton, the following relations can be derived u v = v u : no commutative law × − × u (v + w) = u v + u w : distributive law × × × α (u v) =(α u) v = u (α v) : associative law × × ×

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Scalar triple product (parallelepidial product): u (v w)= v (w u)= w (u v) · × · × · × Arithmetic laws for the vector product (without proof)

u u = 0 × (u + v) w = u w + v w × × × u (u v) = v (u u)=0 · × · × Expansion theorem: u (v w)=(u w) v (u v) w × × · − · Lagrangean identity (Jean Louis Lagrange: 1736-1813): (u v) (w z)=(u w)(v z) (u z)(v w) × · × · · − · · Norm of the vector product: u v = u v sin <)(u ; v) | × | | || | Vector product in an orthonormal basis here: simpliﬁed representation in matrix notation Calculation of

e1 e2 e3

u = v w = v1 v2 v3 ×

w1 w2 w3

= (v w v w ) e (v w v w ) e +(v w v w ) e 2 3 − 3 2 1 − 1 3 − 3 1 2 1 2 − 2 1 3 Rem.: u v, w ; i. e. u v = u w = 0 holds ⊥ · · Example: u v = u v = (v w v w ) v (v w v w ) v +(v w v w ) v = 0 q. e. d. · i i 2 3 − 3 2 1 − 1 3 − 3 1 2 1 2 − 2 1 3 Remarks on the products between vectors on the scalar product • Decomposition of a vector (example: in 2-d):

u2 u = u1 + u2

with u1 = u1 e1 and u2 = u2 e2 u2 β u e2 α u1 , u2 : vector components u1 e1 u1 , u2 : coeﬃcients of the vector components u1

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Projection of u on the directions of ei: u = u e i · i Veriﬁcation of the projection law: u e = (u e ) e · i k k · i = uk δki = ui q. e. d. Calculation of the projections: u = u e cos α 1 | || 1| = u cos α = u cos α | | with u = u | | u2 = u cos β = u cos (90◦ α) = u sin α −

Note: For the values of the vector components, the following relations hold

u = u cos α u2 u 1 α u2 = u sin α u1

on the vector product • Orientation of the vector u = v w: × u

(1) u v , w w ⊥ (2) corkscrew rule (right-hand rule) α v It is obvious that w

α z = w v × v v w = w v −→ × − × z Value of the vector product: w w sin α v w = v w sin α | × | | || | α = v (w sin α) v

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Note: The vector v w is perpendicular to v and w (corkscrew orientation); its × value corresponds to the area spanned by v and w.

Scalar triple product (parallelepidial product):

u (v w) =: [ uvw ] z · × with z = v w × u follows u z = u z cos γ · | || | = z (u cos γ) w with (u cos γ) : projection of u on the direction of z v γ Rem.: The parallelepidial product yields the volume of the parallelepiped spanned by u, v and w.

Remark: The preceding and the following relations are valid with respect to an arbitrary basis system. For simplicity, the following material is restricted to the orthonormal basis, whenever a basis notation occurs. Concerning a more general basis representation, cf., e. g., de Boer, R.: Vektor- und Tensorrechnung f¨ur Ingenieure. Springer-Verlag, Berlin 1982.

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2 Fundamentals of tensor calculus

Rem.: The following statements are related to the proper Euklidian vector space 3 and the corresponding dyadic product space 3 3 3 (n times)V of V ⊗V ⊗ ··· ⊗V n-th order.

2.1 Introduction of the tensor concept

(a) Tensor concept and linear mapping

Deﬁnition: A 2nd order (2nd rank) tensor T is a linear mapping which transforms a vector u uniquely in a vector w:

w = T u

u, w 3 ; T ( 3, 3) ∈V ∈ L V V therein: set of all 2nd order tensors or linear  ( 3, 3):  L V V mappings of vectors, respectively  (b) Tensor concept and dyadic product space

Deﬁnition: There is a “simple tensor” (a b) with the property ⊗ (a b) c =: (b c) a ⊗ ·

a b 3 3 (dyadic product space) therein: ⊗ ∈ V ⊗V ( : dyadic product (binary operator of 3 3) ⊗ V ⊗V It follows directly that a b ( 3, 3) 3 3 ( 3, 3) ⊗ ∈ L V V −→ V ⊗V ⊂L V V Rem.: (a b) maps a vector c onto a vector d = (b c) a . ⊗ ·

Basis notation of a simple tensor: A := a b =(a e ) (b e )= a b (e e ) ⊗ i i ⊗ k k i k i ⊗ k

ai bk : coeﬃcients of the tensor components with e e : tensor basis i ⊗ k Tensors A 3 3 have 9 independent components (and directions); e. g. a b (e∈V e⊗V) etc. 1 3 1 ⊗ 3

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Introduction of arbitrary tensors T 3 3 : ∈V ⊗V T = t (e e ) ik i ⊗ k t t t 11 12 13 matrix of coeﬃcients of T with with t = t t t : ik  21 22 23  9 independent quantities t31 t32 t33   2.2 Basic rules of tensor algebra

Requirement: A, B, C, ... 3 3 . { }∈V ⊗V

(a) Tensor addition

A + B = B + A : commutative law A +(B + C) =(A + B)+ C : associative law A + 0 = A : 0 : identical element A +( A) = 0 : A : inverse element − − Tensor addition with respect to an orthonormal tensor basis: A = a (e e ), B = b (e e ) ik i ⊗ k ik i ⊗ k C = A + B =(a + b )(e e ) −→ ik ik i ⊗ k cik | {z } Rem.: A tensor addition carried out as an addition of the tensor coeﬃcients requires that both tensors have the same tensor basis.

(b) Multiplication of tensors by a scalar

1 A = A : 1 : identical element α (β A) =(α β) A : associative law (α + β) A = α A + β A : distributive law (with respect to the addition of scalars) α (A + B) = α A + α B : distributive law (with respect to the addition of tensors) α A = A α : commutative law

The following deﬁnitions make use of the linear mapping (cf. 2.1) w = T u

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Rem.: In the literature, the multiplication of a vector by a tensor is also called “contraction”. The following relations hold: A (u + v) = A u + A v : distributive law A (α u) = α(A u) : associative law (A + B) u = A u + B u : distributive law (α A) u = α (A u) : associative law 0 u = 0 : 0 : zero element of the linear mapping I u = u : I : identity element of the linear mapping

Linear mapping in basis notation: A = a (e e ) , u = u e ik i ⊗ k i i A u = (a e e )(u e ) = a u (e e ) e ik i ⊗ k j j ik j i ⊗ k j One obtains i : free index (basis index) w = A u = aik uj δkj ei = aik uk ei mit k : silent index (double index of wi) wi

| {z } e3 Rem.: In general, a linear mapping A causes both u a rotation and a stretch of a vector u. 0 e 2 A u e Identity tensor I 3 3 : 1 ∈V ⊗V I = δ e e = e e ik i ⊗ k i ⊗ i Proof of the deﬁning property: u = I u =(e e ) u e = u (e e ) e = u δ e = u e q. e. d. i ⊗ i j j j i ⊗ i j j ij i i i Rem.: Tensors built from basis vectors are called fundamental tensors, i. e.

I 3 3 is the fundamental tensor of 2nd order. ∈V ⊗V

(d) Scalar product of tensors (inner product) The following relations hold: A B = B A : commutative law · · A (B + C) = A B + A C : distributive law · · · (α A) B = A (α B)= α (A B) : associative law · · · A B = 0 A , if B 0 · ∀ ≡ A A > 0 for A = 0 −→ · 6

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Scalar product of A with a simple tensor a b 3 3 : ⊗ ∈ V ⊗V A (a b)= a A b · ⊗ · Scalar product of A and B in basis notation: A = a (e e ), B = b (e e ) ik i ⊗ k ik i ⊗ k α = A B = a (e e ) b (e e )= a b (e e ) (e e ) · ik i ⊗ k · st s ⊗ t ik st i ⊗ k · s ⊗ t One obtains α = aik bst δis δkt = aik bik

Rem.: The result of the scalar product is a scalar.

(e) Tensor product of tensors

Deﬁnition: The tensor product of tensors yields

(AB) v = A (B v)

Rem.: With this deﬁnition, the tensor product of tensors is directly linked to the linear mapping (cf. 2.1 (a)).

The following relations hold: (AB) C = A (BC) : associative law A (B + C) = AB + AC : distributive law (A + B) C = AC + BC : distributive law α (AB) =(α A) B = A (α B) : associative law IT = TI = T : I : identity element 0T = T0 = 0 : 0 : zero element

Rem.: In general, the commutative law is not valid, i. e. AB = BA. 6 Tensor product of simple tensors: A = a b , B = c d ⊗ ⊗ It follows with the above deﬁnition (AB) v = A (B v) [(a b)(c d)] v = (a b)[(c d) v ] −→ ⊗ ⊗ ⊗ ⊗ = (a b)(d v) c ⊗ · = (b c)(d v) a · · = [(b c)(a d)] v · ⊗

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Consequence: (a b)(c d) =(b c) a d ⊗ ⊗ · ⊗ Tensor product in basis notation:

AB = a (e e ) b (e e ) ik i ⊗ k st s ⊗ t = a b (e e )(e e ) ik st i ⊗ k s ⊗ t = a b δ (e e ) ik st ks i ⊗ t = a b (e e ) ik kt i ⊗ t Rem.: The result of a tensor product is a tensor.

2.3 Speciﬁc tensors and operations

(a) Transposed tensor

Deﬁnition: The transposed tensor AT belonging to A exhibits the property

w (A u) =(AT w) u · ·

The following relations hold:

(A + B)T = AT + BT (α A)T = α AT (AB)T = BT AT

Transposition of a simple tensor a b: ⊗ It follows with the above deﬁnition w (a b) u = w (b u) a · ⊗ · · = (w a)(b u) · · = (b a) w u ⊗ · (a b)T = b a −→ ⊗ ⊗

Transposed tensor in basis notation: A = a (e e ) ik i ⊗ k AT = a (e e ) −→ ik k ⊗ i = a (e e ) : renaming the indices ki i ⊗ k

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Note: The transposition of a tensor A 3 3 can be carried out by an ∈ V ⊗ V exchange of the tensor basis or by an exchange of the subscripts of the tensor coeﬃcients.

(b) Symmetric and skew-symmetric tensor

Deﬁnition: A tensor A 3 3 is symmetric, if ∈V ⊗V A = AT

and skew-symmetric (antimetric), if

A = AT −

Symmetric and skew-symmetric parts of an arbitrary tensor A 3 3 : ∈V ⊗V 1 T sym A = 2 (A + A )

skw A = 1 (A AT ) 2 − A = sym A + skw A −→ Properties of symmetric and skew-symmetric tensors:

w (sym A) v = (sym A) w v · · v (skw A) v = (skw A) v v = 0 · − · Positive deﬁnite symmetric tensors:

sym A is positive deﬁnite, if sym A (v v)= v (sym A) v > 0 • · ⊗ · sym A is positive semi-deﬁnite, if sym A (v v)= v (sym A) v 0 • · ⊗ · ≥ (c) Inverse tensor

Deﬁnition: If A−1 inverse to A exists, it exhibits the property

v = Aw w = A−1 v ←→

The following relations hold:

AA−1 = A−1 A = I (A−1)T = (AT )−1 =: AT −1 (AB)−1 = B−1 A−1

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Rem.: The computation of the inverse tensor in basis notation is carried out by intro- ducing the “double cross product” (outer tensor product of tensors), cf. 2.8.

(d) Orthogonal tensor

Deﬁnition: An orthogonal tensor Q 3 3 exhibits the property ∈V ⊗V Q−1 = QT Q QT = I ←→ (det Q)2 = 1 : orthogonality Additionally det Q = 1 : proper orthogonality

Rem.: The computation of the determinant of 2nd order tensors is deﬁned with the aid of the double cross product, cf. 2.8. Properties of orthogonal tensors: Q v Qw = QT Q v w = v w · · · Q u Q u = u u −→ · · Rem.: Linear mapping with Q preserves the norm of the respective vector. Illustration: u in general: linear mapping with A 3 3 ∈V ⊗V causes a rotation and a stretch

in special: linear mapping with Q 3 3 Q u A u ∈V ⊗V causes only a rotation

(e) Trace of a tensor

Deﬁnition: The trace tr A of a tensor A 3 3 is the scalar product ∈V ⊗V tr A = A I ·

The following relations hold: tr(α A) = α tr A tr(a b) = a b ⊗ · tr AT = tr A tr(AB) = tr(BA) (AB) I = B AT = BT A −→ · · · tr(ABC) = tr(BCA) = tr(CAB)

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2.4 Change of the basis

Rem.: The goal is to ﬁnd a relation between vectors and tensors which belong to dif- ferent basis systems. here: Restriction to orthonormal basis systems which are rotated against each other.

(A) Rotation of the basis system

Illustration:

e3 ∗ ∗ e3 e2

0, e : basis system { i} α22 ∗ α21 e 2 0, ei : rotated basis system ∗ α11 { } e1 e1 αik : angle between the basis vectors { } ∗ ei and ek

Development of the transformation tensor: The following relations hold: ∗ ∗ e = I e and I = e e i i j ⊗ j Thus, ∗ ∗ ∗ ei =(ej ej) ei =(ej ei) ej ∗ ∗ ⊗ · using ei = δik ek leads to ∗ ∗ ∗ e =(e δ e ) e =(e e )(e e ) e i j · ik k j j · k i · k j one obtains ∗ ∗ ∗ e =(e e )(e e )e =: R e with R =(e e ) e e i j · k j ⊗ k i i j · k j ⊗ k

Rem.: R is the transformation tensor which transforms the basis vectors ei into the ∗ basis vectors ei.

Coeﬃcient matrix Rjk: ∗ ∗ ∗ ∗ R = e e = e e cos<)(e ; e ) = cos α with e = e =1 jk j · k | j|| k| j k jk | j| | k|

Rem.: Rjk contains the 9 cosines of the angles between the directions of the basis ∗ vectors ej and ek.

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Orthogonality of the transformation tensor: ∗ Rem.: By R, the basis vectors ei are only rotated towards ei, thus, R is an orthogonal tensor. Orthogonality condition:

RRT =! I = R (e e ) R (e e )= R R δ e e jk j ⊗ k pn n ⊗ p jk pn kn j ⊗ p = R R (e e ) jk pk j ⊗ p It follows with I = δ (e e ) by comparison of coeﬃcients jp j ⊗ p

R R = δ ( ) jk pk jp ∗

Rem.: ( ) contains 6 constraints for the 9 cosines (RRT = sym (RRT )), i. e. only 3 of∗ 9 trigonometrical functions are independent. Thus, the rotation of the basis system is deﬁned by 3 angles.

(B) Introduction of “CARDANO angles”

Idea: Rotation around 3 axes which are given by the basis directions ei. This procedure was ﬁrstly investigated by Girolamo Cardano (1501-1576).

Procedure: The rotation of the basis system is carried out by 3 independent rotations around the axes e1, e2, e3. Each rotation is expressed by a transformation tensor Ri (i = 1, 2, 3).

Rotation of ei around e3, e2, e1:

∗ ∗ ∗ e = R [R (R e )] = R e with R = R R R i { 1 2 3 i } i 1 2 3

Rotation of ei around e1, e2, e3:

¯e = R [R (R e )] = R¯ e with R¯ = R R R i { 3 2 1 i } i 3 2 1

Obviously, ∗ ∗ R = R¯ e = ¯e 6 −→ 6 i

Rem.: The result of the orthogonal transformation depends on the sequence of the rotations.

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Illustration:

◦ (a) Rotation around e3, e2, e1 (e. g. each about 90 )

(e3) ◦ 90 3 e3 2

e2 ◦ 1 90 (e2) ∗ e1 2 e1

∗ 1 e2 3 ∗ e3 90◦ (e1)

◦ (b) Rotation around e1, e2, e3 (e. g. each about 90 ) with e3 2

e2 3 ◦ 90 (e2) e1 ◦ 1 90 ∗ (e1) e1

(e3) ◦ 90 ∗ 3 ¯e2 e2 ¯e3 ∗ 2 e3 180◦ 1 ¯e1

Deﬁnition of the orthogonal rotation tensors Ri

(a) Rotation around the e3-axis

e2 ◦ The following relations hold: e2 ◦ ◦ e1 e1 = cos ϕ3 e1 + sin ϕ3 e2 ◦ ϕ3 e = sin ϕ e + cos ϕ e 2 − 3 1 3 2 ϕ ◦ 3 e3 = e3 e1 e3

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In general, ◦ e = R e = R (e e ) e = R δ e = R e i 3 i 3jk j ⊗ k i 3jk ki j 3ji j Thus, by comparison of coeﬃcients

cos ϕ sin ϕ 0 3 − 3 R = R (e e ) with R = sin ϕ cos ϕ 0 3 3ji j i 3ji  3 3  ⊗ 0 0 1  

(b) Rotation around the e2- and e1-axis Analogously,

cos ϕ2 0 sin ϕ2 R = R (e e ) with R = 0 1 0 2 2ji j i 2ji   ⊗ sin ϕ 0 cos ϕ − 2 2   10 0 R = R (e e ) with R = 0 cos ϕ sin ϕ 1 1ji j ⊗ i 1ji  1 − 1  0 sin ϕ1 cos ϕ1   Rem.: The rotation tensor R can be composed of single rotations under consideration of the rotation sequence.

(c1) it follows from rotation of ei around e3, e2, e1 that

∗ R R = R R R −→ 1 2 3 = R (e e ) R (e e ) R (e e ) 1ij i ⊗ j 2no n ⊗ o 3pq p ⊗ q = R R R δ δ (e e ) 1ij 2no 3pq jn op i ⊗ q = R R R (e e ) 1ij 2jo 3oq i ⊗ q ∗ Riq with | {z }

cos ϕ2 cos ϕ3 cos ϕ2 sin ϕ3 sin ϕ2 ∗ − = sin ϕ1 sin ϕ2 cos ϕ3 + cos ϕ1 sin ϕ3 sin ϕ1 sin ϕ2 sin ϕ3 + cos ϕ1 cos ϕ3 sin ϕ1 cos ϕ2 Riq  − −  cos ϕ1 sin ϕ2 cos ϕ3 + sin ϕ1 sin ϕ3 cos ϕ1 sin ϕ2 sin ϕ3 + sin ϕ1 cos ϕ3 cos ϕ1 cos ϕ2  −   

(c2) it follows from rotation of ei around e1, e2, e3 that

R R¯ = R R R −→ 3 2 1 = R R R (e e ) 3ij 2jo 1oq i ⊗ q R¯iq | {z }

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cos ϕ2 cos ϕ3 sin ϕ1 sin ϕ2 cos ϕ3 cos ϕ1 sin ϕ3 cos ϕ1 sin ϕ2 cos ϕ3 + sin ϕ1 sin ϕ3 − R¯ = cos ϕ2 sin ϕ3 sin ϕ1 sin ϕ2 sin ϕ3 + cos ϕ1 cos ϕ3 cos ϕ1 sin ϕ2 sin ϕ3 sin ϕ1 cos ϕ3 iq  −  sin ϕ2 sin ϕ1 cos ϕ2 cos ϕ1 cos ϕ2  −   

Orthogonality of “Cardano rotation tensors”: ∗ For all R R , R , R , R, R¯ , the following relations hold ∈{ 1 2 3 } R−1 = RT , i. e. RRT = I and (det R)2 =1 orthogonality −→ Furthermore, all rotation tensors hold the following relation det R = 1 : “proper” orthogonality

Rem.: A basis transformation with “non-proper” orthogonal transformations (det R = 1) transforms a “right-handed” into a “left-handed” basis system. −

Example: here: Investigation of the orthogonality properties of R = R (e e ) 3 3ij i ⊗ j cos ϕ sin ϕ 0 3 − 3 with R = sin ϕ cos ϕ 0 3ij  3 3  0 0 1 One looks at   R RT = R (e e ) R (e e ) 3 3 3ij i ⊗ j 3on n ⊗ o = R R δ (e e ) = R R (e e ) 3ij 3on jn i ⊗ o 3in 3on i ⊗ o where 2 2 sin ϕ3 + cos ϕ3 0 0 R R = 0 sin2 ϕ + cos2 ϕ 0 = δ 3in 3on  3 3  io 0 01 and one obtains   R RT = δ (e e )= I q. e. d. 3 3 io i ⊗ o Furthermore, det R := det (R )=1 R is proper orthogonal 3 3ij −→ 3

Description of rotation tensors: ◦ In general, the transformation between basis systems ¯ei and basis systems ei satisﬁes the following relation: ◦ e = R¯ ¯e with R¯ =R¯ ¯e ¯e i i ik i ⊗ k T ◦ − T ¯e = R¯ e with R¯ 1 R¯ −→ i i ≡

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Otherwise, ◦ ◦ ◦ ◦ ◦ ◦ ¯e = R e with R = R e e i i ik i ⊗ k Consequence: By comparing both relations, it follows that ◦ ◦ ◦ T ◦ ◦ R = R¯ , i. e., R e e =(R¯ )T ¯e ¯e R =R¯ ik i ⊗ k ik i ⊗ k −→ ik ki In particular,

◦ ◦ ◦ ◦ ◦ R = Rik (ei ek)= Rik (R¯ e¯i R¯ e¯k) ◦ ⊗ ⊗ ◦ ! T = Rik R¯ni e¯n R¯pk e¯p =(R¯ni Rik R¯pk) e¯n e¯p = R¯pn e¯n e¯p = R¯ ⊗ ⊗ ⊗

◦ ◦ ! R¯ni Rik R¯pk = R¯pn R¯ni Rik = δnk −→ ←→

◦ Rem.: The coeﬃcient matrices R¯ni and Rik are inverse to each other, i. e., in general, ◦ ◦ R¯ni Rik = δnk implies 6 equations for the 9 unknown coeﬃcients Rik. Due to ¯ −1 ¯ T ¯−1 ¯ T ¯ R = R , one has Rni =(Rni) = Rin, i. e.

◦ T Rik=(R¯ik) = R¯ki

Rem.: Rotation of a basis system ei around three speciﬁc axes. ∗ Introduction of 3 speciﬁc angles around e3, e¯1, e˜3 =e3 Illustration:

∗ Idea: Given are 2 planes and with e3 ∗ ∗ ∗ ∗ F F e e2 e in-plane vectors e1, e2 and e1, e2 3 δ ˜2 δ ∗ ψ ¯e2 e2 ∗ and surface normals e3 and e3. ϕ e ∗ 1 The basis systems ei and ei are c ψ c ¯e1 related to each other by the Eu- ϕ lerian rotation tensor R: e1 ∗ ei := R ei F ∗ F

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Rotation of ei in plane around e with the angle ϕ, 1st step: F 3 such that e¯i is directed towards c – c. This yields the rotation tensor cos ϕ sin ϕ 0 e¯3 = e3 − R = sin ϕ cos ϕ 0 e e . 3   j ⊗ k ϕ 0 0 1 e¯2 ϕ   Then, the new system e¯ is computed as follows e2 i c c e¯i = R3 ei = R3jk (ej ek) ei = R3ji ej . e¯1 ⊗ ϕ Thus,

e1 e¯1 = R3j1 ej = cos ϕ e1 + sin ϕ e2 e¯ = R e = sin ϕ e + cos ϕ e 2 3j2 j − 1 2 e¯3 = R3j3 ej = e3 .

2nd step: Rotation of e¯i around e¯1 with the angle δ, such that ∗ e˜2 lies in the plane , and e˜3 is directed normal to the ∗ F plane . This yields the rotation tensor F e¯3 δ 10 0 e˜ e˜ 3 2 R¯ = 0 cos δ sin δ e¯ e¯ . δ 1  −  j ⊗ k e¯2 0 sin δ cos δ   δ Then, the new system e˜ is computed as follows c c i

e˜ = e¯ e˜i = R¯ 1 e¯i = R¯1jk (e¯j e¯k) e¯i = R¯1ji e¯j . 1 1 ⊗ Thus,

e˜1 = R¯1j1 e¯j = e¯1

e˜2 = R¯1j2 e¯j = cos δ e¯2 + sin δ e¯3 e˜ = R¯ e¯ = sin δ e¯ + cos δ e¯ . 3 1j3 j − 2 3

3rd step: ∗ Rotation of e˜i in plane around e˜3 with the angle ψ. This yields the rotationF tensor

∗ ∗ e3 = e˜3 e ψ cos ψ sin ψ 0 2 e˜ − 2 R˜ = sin ψ cos ψ 0 e˜ e˜ . 3   j ⊗ k ∗ 0 0 1 ψ e1   ψ ∗ c c Then, the new system ei is computed as follows e˜ 1 ∗ e = R˜ e˜ = R˜ (e˜ e˜ ) e˜ = R˜ e˜ . i 3 i 3jk j ⊗ k i 3ji j

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Thus, ∗ e1 = R˜3j1 e˜j = cos ψ e˜1 + sin ψ e˜2 ∗ e2 = R˜3j2 e˜j = sin ψ e˜1 + cos ψ e˜2 ∗ − e3 = R˜3j3 e˜j = e˜3 .

Summary:

(a) Inserting e˜i = R¯ 1 e¯i ∗ e1 = cos ψ e¯1 + sin ψ (cos δ e¯2 + sin δ e¯3) ∗ e2 = sin ψ e¯1 + cos ψ (cos δ e¯2 + sin δ e˜3) ∗ − e = e˜ = sin δ e¯ + cos δ e¯ 3 3 − 2 3 Result: ∗ e1 = cos ψ e¯1 + sin ψ cos δ e¯2 + sin ψ sin δ e¯3 ∗ e2 = sin ψ e¯1 + cos ψ cos δ e¯2 + cos ψ sin δ e˜3 ∗ − e3 = sin δ e¯2 + cos δ e¯3 ∗ − e = R˜ (R¯ e¯ ) =: R¯ e¯ with R¯ = R˜ R¯ −→ i 3 1 i i 3 1 e˜i | {z } (b) Inserting e¯i = R3 ei ∗ e1 = cos ψ (cos ϕ e1 + sin ϕ e2) + sin ψ cos δ ( sin ϕ e1 + cos ϕ e2) + sin ψ sin δ e3 ∗ − e2 = sin ψ (cos ϕ e1 + sin ϕ e2) + cos ψ cos δ ( sin ϕ e1 + cos ϕ e2) + cos ψ sin δ e3 ∗ − − e = sin δ ( sin ϕ e + cos ϕ e ) + cos δ e 3 − − 1 2 3 Result: ∗ e = (cos ψ cos ϕ sin ψ cos δ sin ϕ) e + 1 − 1 +(cos ψ sin ϕ + sin ψ cos δ cos ϕ) e2 + sin ψ sin δ e3 ∗ e = ( sin ψ cos ϕ cos ψ cos δ sin ϕ) e + 2 − − 1 +( sin ψ sin ϕ + cos ψ cos δ cos ϕ) e2 + cos ψ sin δ e3 ∗ − e = sin δ sin ϕ e sin δ cos ϕ e + cos δ e 3 1 − 2 3 ∗ e = R¯ (R e ) =: R e with R = RR¯ = R˜ R¯ R −→ i 3 i i 3 3 1 3 e¯i

∗ | {z } Rotation tensors R and R: For the total rotation the following relation holds: ∗ ei = (R˜ 3 R¯ 1 R3) ei =: R ei

= (R˜ 3 R¯ 1)(R3 ei)= R˜ 3 (R¯ 1 e¯i)= R˜ 3 e˜i ∗ e¯i e˜i ei | {z } | {z } | {z } Institute of Applied Mechanics, Chair of Continuum Mechanics 24 Supplement to Continuum Mechanics Research

Furthermore,

∗ ∗ ∗ ∗ ∗ e = R e e = RT e =: R e R = RT i i −→ i i i −→

∗ T Analogously to the previous considerations Rik=(Rik) = Rki −→ Description:

cos ψ cos ϕ sin ψ cos δ sin ϕ sin ψ cos ϕ cos ψ cos δ sin ϕ sin δ sin ϕ − − − R = cos ψ sin ϕ + sin ψ cos δ cos ϕ sin ψ sin ϕ + cos ψ cos δ cos ϕ sin δ cos ϕ e e  − −  i ⊗ k  sin ψ sin δ cos ψ sin δ cos δ    Combining rotation tensors with diﬀerent basis systems:

Example: R¯ := R˜ 3 R¯ 1 ∗ ei = R˜ 3 e˜i =(R˜ 3 R¯ 1) e¯i R¯ = R˜ (e˜ e˜ ) R¯ (e¯ e¯ ) −→ 3ik i ⊗ k 1no n ⊗ o = R˜ ( R¯ e¯ R¯ e¯ ) R¯ (e¯ e¯ ) 3ik 1 i ⊗ 1 k 1no n ⊗ o R¯ e¯ R¯ e¯ 1si s ⊗ 1tk t | {z } R¯ = R¯ R˜ R¯ (e¯ e¯ ) R¯ (e¯ e¯ ) −→ 1si 3ik 1tk s ⊗ t 1no n ⊗ o = R¯ R˜ R¯ R¯ δ (e¯ e¯ ) 1si 3ik 1tk 1no tn s ⊗ o = R¯ R˜ R¯ R¯ (e¯ e¯ ) 1si 3ik 1tk 1to s ⊗ o R¯so | {z } Thus, the rotation tensor R¯ is given by cos ψ sin ψ 0 − R¯ = sin ψ cos δ cos ψ cos δ sin δ e¯ e¯  −  i ⊗ k sin ψ sin δ cos ψ sin δ cos δ     Rem.: Concerning Cardano angles, all partial rotations (e. g. R = R3 R2 R1 with ∗ ei = R ei) are carried out with respect to the same basis ei, i. e. the combination of the partial rotations is much easier.

Rotation around a ﬁxed axis: Rem.: A rotation around 3 independent axes can also be described by a rotation around the resulting axis of rotation: Euler-Rodrigues representation of the spatial rotation −→ The Euler-Rodrigues representation of the rotation is discussed later (see section 2.7).

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2.5 Higher order tensors

Deﬁnition: An arbitrary n-th order tensor is given by

n A 3 3 3 (n times) ∈ V ⊗V ⊗···⊗V with 3 3 3 : n-th order dyadic product space V ⊗V ⊗···⊗V

Rem.: Usually, n 2. However, there exist special cases for n = 1 (vector) and n =0 (scalar). ≥

General description of the linear mapping

Deﬁnition: A linear mapping is a “contracting product” (contraction) given by

n s n−s A B = C with n s ≥

Descriptive example on simple tensors:

(a b c d) (e f) =(c e)(d f) a b ⊗ ⊗ ⊗ ⊗ · · ⊗ 4 A B C | {z } | {z } | {z } Fundamental 4-th order tensors

Rem.: 4-th order fundamental tensors are built by a dyadic product of 2nd order identity tensors and the corresponding independent transpositions.

One introduces:

I I = (ei ei) (ej ej) ⊗ 23 ⊗ ⊗ ⊗ T (I I) = ei ej ei ej ⊗ 24 ⊗ ⊗ ⊗ (I I)T = e e e e ⊗ i ⊗ j ⊗ j ⊗ i

ik with ( )T : transposition, deﬁned by the exchange of the i-th and the k-th basis system ·

Rem.: Further transpositions of I I do not lead to further independent tensors. The fundamental tensors from above⊗ exhibit the property

4 4 4 4 13 24 A = AT with AT =(AT )T

Consequence: The 4-th order fundamental tensors are symmetric (concerning an exchange of the ﬁrst two and the second two basis systems).

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Properties of 4-th order fundamental tensors

(a) identical map

23 (I I)T A =(e e e e ) a (e e ) ⊗ i ⊗ j ⊗ i ⊗ j st s ⊗ t = a δ δ (e e )= a (e e )= A st is jt i ⊗ j ij i ⊗ j 4 23 I := (I I)T is 4-th order identity tensor −→ ⊗ (b) “transposing” map

24 (I I)T A =(e e e e ) a (e e ) ⊗ i ⊗ j ⊗ j ⊗ i st s ⊗ t = a δ δ (e e )= a (e e )= AT st js it i ⊗ j ji i ⊗ j

(I I) A =(e e e e ) a (e e ) ⊗ i ⊗ i ⊗ j ⊗ j st s ⊗ t = a δ δ (e e )= a (e e ) st js jt i ⊗ i jj i ⊗ i =(A I) I = (tr A) I · with A I = a (e e ) (e e )= a δ δ = a · st s ⊗ t · j ⊗ j st sj tj jj Speciﬁc 4-th order tensors 4 Let A, B, C, D be arbitrary 2nd order tensors. Then, a 4-th order tensor A can be deﬁned exhibiting the following properties:

4 23 14 A = (A B)T = (BT AT )T ( ) 4 ⊗ 23 ⊗ 23 ∗ A T = [(A B)T ]T = (AT BT )T ⊗ 23 ⊗ 23 4 − A 1 = [(A B)T ]−1 = (A−1 B−1)T ⊗ ⊗ Furthermore, following relation holds:

4 4 13 24 ( ) T = [( ) T ]T · · From ( ), the following relations can be derived: ∗ 23 23 23 (A B)T (C D)T = (AC BD)T ⊗ 23 ⊗ ⊗ (A B)T (C D) = (ACBT D) ⊗ ⊗ 23 ⊗ (A B)(C D)T = (A CT BD) ⊗ ⊗ ⊗ and 23 (A B)T C = ACBT ⊗ 23 23 (A B)T v = [A (B v)]T ⊗ ⊗

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4 Deﬁning a 4-th order tensor B with the properties

4 24 13 B = (A B)T = [(A B)T ]T 4 ⊗ 24 ⊗ 24 B T = [(A B)T ]T = (B A)T ⊗ 24 ⊗ 24 4 − B 1 = [(A B)T ]−1 =(BT −1 AT −1)T ⊗ ⊗ it can be shown that

24 24 23 (A B)T (C D)T = (ADT BT C)T ⊗ 23 ⊗ 24 ⊗ 24 (A B)T (C D)T = (AC DBT )T ⊗ 24 ⊗ 23 ⊗ 24 (A B)T (C D)T = (AD CT B)T ⊗ 24 ⊗ ⊗ (A B)T (C D) = (ACT B D) ⊗ ⊗ 24 ⊗ (A B)(C D)T = (A DBT C) ⊗ ⊗ ⊗ and 24 (A B)T C = ACT B ⊗ Furthermore, the following relation holds:

4 4 4 4 (CD)T = D T C T

4 4 where C and D are arbitrary 4-th order tensors.

High order tensors and incomplete mappings If higher order tensors are applied to other tensors in the sense of incomplete mappings, one has to know how many of the basis vectors have to be linked by scalar products. Therefore, a underlined supercript ( )i indicates the order of the desired result after the · tensor operation has been carried out. Examples in basis notation:

4 3 3 3 (A B) = [aijkl (ei ej ek el) bmno (em en eo)] ⊗ ⊗ ⊗ ⊗ ⊗ = a b δ δ (e e e ) ijkl mno km ln i ⊗ j ⊗ o 3 1 1 (A B) = [aij (ei ej) bmno (em en eo)] ⊗ ⊗ ⊗

= aij bmno δim δjn eo

Note: Note in passing that the incomplete mapping is governed by scalar products of a suﬃcient number of inner basis systems.

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2.6 Fundamental tensor of 3rd order (Ricci permutation tensor)

Rem.: The fundamental tensor of 3rd order is introduced in the context of the “outer product” (e. g. vector product between vectors).

3 Deﬁnition: The fundamental tensor E satisﬁes the rule

3 u v = E (u v) × ⊗

3 Introduction of E in basis notation:

There is 3 E = e (e e e ) ijk i ⊗ j ⊗ k with the “permutation symbol” eijk

1 : even permutation e123 = e231 = e312 =1 e = 1 : odd permutation e = e = e = 1 ijk  − −→  321 213 132 −  0 : double indexing  all remaining eijk vanish

3  Application of E to the vector product of vectors: From the above deﬁnition, 3 u v = E (u v) × ⊗ = e (e e e )(u e v e ) ijk i ⊗ j ⊗ k s s ⊗ t t = eijk us vt δjs δkt ei = eijk uj vk ei =(u v u v ) e +(u v u v ) e +(u v u v ) e 2 3 − 3 2 1 3 1 − 1 3 2 1 2 − 2 1 3 Comparison with the computation by use of the matrix notation, cf. page 5

e1 e2 e3 u v = u1 u2 u3 = q. e. d. × ··· v1 v2 v3

3 An identity for E: Incomplete mapping of two Ricci-tensors yielding a 2nd or 4th order object

3 3 3 3 23 24 (E E)2 =2 I , (E E)4 =( I I )T ( I I )T ⊗ − ⊗ 2.7 The axial vector

Rem.: The axial vector (pseudo vector) can be used for the description of rotations (rotation vector).

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A Deﬁnition: The axial vector t is associated with the skew-symmetric part skw T of an arbitrary tensor T 3 3 via ∈V ⊗V A 3 1 T t := 2 E T

One calculates,

A t = 1 e (e e e ) t (e e ) 2 ijk i ⊗ j ⊗ k st t ⊗ s 1 1 = 2 eijk tst δjt δks ei = 2 eijk tkj ei

= 1 [(t t ) e +(t t ) e +(t t ) e ] 2 32 − 23 1 13 − 31 2 21 − 12 3 It follows from 2.3 (b) T = sym T + skw T Thus, the axial vector of T is given by

A 3 1 T t = 2 E (sym T + skw T)

3 3 = 1 E (skw TT )= 1 E (skw T) 2 − 2 Rem.: A symmetric tensor has no axial vector.

Axial vector and linear mapping: The following relation holds:

A (skw T) v = t v v 3 × ∀ ∈V Axial vector and the vector product of tensors:

Deﬁnition: The vector product of 2 tensors T, S 3 3 satisﬁes { }∈V ⊗V 3 S T = E (STT ) ×

Rem.: The vector product (cross product) of 2 tensors yields a vector.

In comparison with the deﬁnition of the axial vector follows

3 A I T = E TT =2 t × Furthermore, the vector product of 2 tensors yields

S T = T S × − ×

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Axial vector and outer tensor product of vector and tensor:

Deﬁnition: The outer tensor product of a vector u 3 and a tensor T 3 3 satisﬁes ∈ V ∈ V ⊗V (u T) v = u (T v); v 3 × × ∈V

Rem.: The outer tensor product of vector and tensor yields a tensor.

The following relations hold:

u T = (u T)T = T u × − × − × i. e. u T is skew-symmetric −→ × 3 u T = [ E (u T)]2 × ⊗ with ( )2 : “incomplete” linear mapping (association) · resulting in a 2nd order tensor.

Evaluation in basis notation leads to u T = [(e e e e )(u e t e e )]2 × ijk i ⊗ j ⊗ k r r ⊗ st s ⊗ t = e u t δ δ (e e ) ijk r st jr ks i ⊗ t = e u t (e e ) ijk j kt i ⊗ t In particular, if T I, the following relation holds: ≡ 3 u I = [ E (u I)]2 = e u δ (e e )= e u (e e ) × ⊗ ijk j kt i ⊗ t ijt j i ⊗ t Furthermore, for the special tensor u I follows × 3 E (u I)= 2 u × − 3 3 u = 1 E (u I)= 1 E (u I)T −→ − 2 × 2 × Consequence: In the tensor u I, the vector u is already the corresponding axial × vector. Finally, the following relation holds:

3 u I = E u × − 3 3 3 3 3 E (u I)= E (E u)= (E E)2 u =! 2 u −→ × − − − 3 3 i. e. (E E)2 =2 I

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Some additional rules: (a b) c = a (b c) × ⊗ × ⊗ (I T) w = T Ω with Ω = w I × · · × Application to the tensor product of vector and tensor

Rotation around a ﬁxed spatial axis

Rotation of x around axis e ϕ ∗ ∗ ∗ u u x = a + u = a + C1 u + b b a =(x e) e · ∗ x a x with  u = x a −   b = C (e x) 2 × e  and ϕ= ϕ e ; e =1 | | O Determination of the constants C1 and C2: ∗ (a) For the angle between u and u, the following relation holds

∗ u u ∗ cos ϕ = · ∗ with u = u u u | | | | | || | Furthermore, the following relation holds ∗ u u = u (C u + b)= C u u + u b = C u 2 · · 1 1 · · 1 | | =0, da u b ⊥ Thus, | {z } C u 2 cos ϕ = 1 | | = C C = cos ϕ u 2 1 −→ 1 ∗| | (b) For the angle between b and u, the following relation holds

∗ ◦ b u cos(90 ϕ) = sin ϕ = · ∗ − b u | || | Furthermore, the following relation holds ∗ b u = b (C u + b)= C b u + b b = b 2 · · 1 1 · · | | =0, da u b ⊥ and | {z } b = C e x = C e x sin <)(e ; x) = C u | | 2 | × | 2 | | | | 2 | | 1 u | | |{z} | {z } Institute of Applied Mechanics, Chair of Continuum Mechanics 32 Supplement to Continuum Mechanics Research

Thus, leading to

b 2 b C u sin ϕ = | | = | | = 2 | | = C C = sin ϕ b u u u 2 −→ 2 | || | | | | | ∗ Thus, x is given by

∗ x =(x e) e + cos ϕ [x (x e) e] + sin ϕ (e x) · − · ×

Determination of the rotation tensor R: For the tensor product of vector and tensor, the following relation holds:

(e I) x = e (I x)= e x × × × Thus, ∗ x=(e e) x + cos ϕ (I e e) x + sin ϕ (e I) x =! R x ⊗ − ⊗ ×

R = e e + cos ϕ (I e e) + sin ϕ (e I) ( ) −→ ⊗ − ⊗ × ∗

Rem.: ( ) is the Euler-Rodrigues form of the spatial rotation. ∗ Example: Rotation with ϕ3 around the e3 axis

R = R = e e + cos ϕ (I e e ) + sin ϕ (e I) 3 3 ⊗ 3 3 − 3 ⊗ 3 3 3 × The following relation holds:

3 e I = [ E (e I)]2 3 × 3 ⊗ = [e (e e e )(e e e )]2 ijk i ⊗ j ⊗ k 3 ⊗ l ⊗ l = e δ δ (e e )= e (e e ) ijk j3 kl i ⊗ l i3l i ⊗ l = e e e e 2 ⊗ 1 − 1 ⊗ 2 Thus, leading to

R = e e + cos ϕ (e e + e e ) + sin ϕ (e e e e ) 3 3 ⊗ 3 3 1 ⊗ 1 2 ⊗ 2 3 2 ⊗ 1 − 1 ⊗ 2 = R (e e ) 3ij i ⊗ j cos ϕ sin ϕ 0 3 − 3 with R3ij =  sin ϕ3 cos ϕ3 0  q. e. d. 0 0 1    

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2.8 The outer tensor product of tensors

Deﬁnition: The outer tensor product of tensors (double cross product) is deﬁned via

@ @ (A B)(u u ) := Au Bu Au Bu 1 × 2 1 × 2 − 2 × 1

As a direct consequence, one ﬁnds

@ @

@ @ A B = B A

Furthermore, the following relations hold:

T T T

@ @

(A B) = A B

@ @ @ @

(A B)(C D)= (AC BD)+(AD BC)

@ @

(I I)= 2 I

@ @ (a b) (c d)= (a c) (b d)

⊗ ⊗ × ⊗ ×

@ @ @

@ @ @ (A B) C = (B C) A =(C A) B · · ·

From the above deﬁnition, it is easily proved that

@ @ [(A B) C][(u u ) u ]= e (Au Bu ) Cu · 1 × 2 · 3 ijk i × j · k

The outer tensor product in basis notation

@ @

@ @ A B = a (e e ) b (e e ) ik i ⊗ k no n ⊗ o = a b (e e ) (e e ) ik no i × n ⊗ k × o 3 ei en = E (ei en) = einj ej with × ⊗  3  ek eo = E (ek eo) = ekop ep

× ⊗

@  @ A B = aik bno einj ekop (ej ep) −→ ⊗

Furthermore, it follows that

@ @ A I = (A I) I A · −

T T

@ @ A B = (A I)(B I) I (A B) I (A I) B · · − · − · − (B I) AT + AT BT + BT AT − ·

T T

@ @ (A B) C = (A I)(B I)(C I) (A I)(B C) (B I)(A C) · · · · − · · − · · − (C I)(AT B)+(AT BT ) C +(BT AT ) C − · · · · The cofactor, the adjoint tensor and the determinant:

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The following relations hold: +

1 T

@ @ cof A = 2 A A =: A , adj A = (cof A)

1 (Au1 Au2) Au3

@ @ det A = (A A) A = det a = × · 6 · | ik| (u u ) u 1 × 2 · 3 In basis notation the following relation holds:

+ + A = 1 (a a e e )(e e )= a (e e ) 2 ik no inj kop j ⊗ p jp j ⊗ p

+ Rem.: The coeﬃcient matrix a of the cofactor cof A contains at each position ( ) jp · jp the corresponding subdeterminant of A

+ a = a a a a etc. 11 22 33 − 23 32 The inverse tensor: The following relation holds: A−1 = (det A)−1 adj A ; A−1 exists if det A =0 6 Rules for the cofactor, the determinant and the inverse tensor:

det (AB) = det A det B det (α A)= α3 det A det I = 1 det AT = det A

+ detA = (det A)2 det A−1 = (det A)−1

+ + det (A + B) = det A + A B + A B + det B · · + + + (AB) = A B

+ + (A)T = (AT )

2.9 The eigenvalue problem and the invariants of tensors

Deﬁnition: The eigenvalue problem of an arbitrary 2nd order tensor A is given by

γA : eigenvalue (A γA I) a = 0 , where − ( a : eigenvector

Institute of Applied Mechanics, Chair of Continuum Mechanics Supplement to Continuum Mechanics Research 35

Formal solution for a yields 0 −1 a =(A γA I) 0 = adj(A γA I) − − det (A γA I) − Consequence: Non-trivial solution for a only if the characteristic equation is fulﬁlled, i. e. det (A γA I)=0 − With the determinant rule

@ @ det (A + B) = [(A + B) (A + B)] (A + B) 6 ·

1 1 1

@ @ @

@ @ @ = (A A) A + (A A) B + (A B) A + 6 · 6 · 3 ·

1 1 1

@ @ @

@ @ @ + 3 (A B) B + 6 (B B) A + 6 (B B) B + · + · · = det A +A B + A B+ det B · · follows + + det (A γA I) = det A +A ( γA I)+ A ( γA I) + det ( γA I) − · − · − −

1 2 1 3

@ @

@ @ = det A γA (A A) I + γA A (I I) γA det I =0 − 2 · 2 · − With the abreviations

@ @ IA = (A I) I 2 ·

@ @ IIA = (A A) I 2 ·

@ @ IIIA = (A A) A 6 · the characteristic equation can be simpliﬁed to 2 3 det (A γA I)= IIIA γA IIA + γA IA γA =0 − − − Rem.: The abreviations IA, IIA and IIIA are the three scalar principal invariants of a tensor A which play an important role in the ﬁeld of continuum mechanics.

Alternative representations of the principal inavriants Scalar product representation:

IA = A I = tr A · 1 2 1 2 IIA = (IA AA I) = [(tr A) tr(AA)] 2 − · 2 − 1 3 1 2 1 T T IIIA = IA IA (AA I)+ A A A = 6 − 2 · 3 · = 1 [(tr A)3 3tr A tr(AA)+2tr(AAA)] = det A 6 − Eigenvalue representation:

IA = γA(1) + γA(2) + γA(3)

IIA = γA(1) γA(2) + γA(2) γA(3) + γA(3) γA(1)

IIIA = γA(1) γA(2) γA(3) Caley-Hamilton-Theorem:

AAA IA AA + IIA A IIIA I = 0 − −

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3 Fundamentals of vector and tensor analysis

3.1 Introduction of functions

Notation:

φ( ) : scalar-valued function scalar variables · exists v( ) : vector-valued function of ( ) vector variables  ·  ·   T( ) : tensor-valued function   tensor variables ·    Example: φ(A) : scalar-valued tensor function

Notions:

Domain of a function: set of all possible values of the independent variable quantities • (variables); usually contiguous

Range of a function: set of all possible values of the dependent variable quantities: • φ( ); v( ); T( ) · · ·

3.2 Functions of scalar variables here: Vector- and tensor-valued functions of real scalar variables

(a) Vector-valued functions of a single variable

It exists: u : unique vector-valued function, u = u(α) with range in the open domain 3  V  α : real scalar variable  Derivative of u(α) with the diﬀerential quotient:

du(α) w(α) := u′(α) := dα Diﬀerential of u(α): du = u′(α) dα

Introduction of higher derivatives and diﬀerentials:

d2u(α) d2u = d(du)= u′′(α) dα2 = dα2 etc. dα2

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(b) Vector-valued functions of several variables

It exists:

u = u(α, β, γ, ...) with α, β, γ, ... : real scalar variable { } Partial derivative of u(α, β, γ, ...):

∂u( ) w (α, β, γ, ...) := · =: u, α ∂α α Total diﬀerential of u(α, β, γ, ...):

du = u, dα + u, dβ + u, dγ + α β γ ··· Higher partial derivative (examples):

∂2u( ) ∂2u( ) u, = · ; u, = · αα ∂α2 γβ ∂γ ∂β

Rem.: The order of partial derivatives is permutable.

Rem.: Tensor-valued functions are treated analogously to the above procedure. (d) Derivative of products of functions

Some rules: (a b)′ = a′ b + a b′ ⊗ ⊗ ⊗ (AB)′ = A′ B + AB′ (A−1)′ = A−1 A′ A−1 −

3.3 Functions of vector and tensor variables

(a) The gradient operator

Rem.: Functions of the position (placement) vector are called ﬁeld functions. Deriva- tives with respect to the position vector are called “gradient of a function”. Scalar-valued functions φ(x)

dφ(x) grad φ(x) := =: w(x) result is a vector ﬁeld dx −→ or in basis notation ∂φ(x) grad φ(x) := ei =: φ,i ei ∂xi

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Vector-valued functions v(x)

dv(x) grad v(x) := =: S(x) result is a tensor ﬁeld dx −→ or in basis notation

∂vi(x) grad v(x) := ei ej =: vi,j ei ej ∂xj ⊗ ⊗

Tensor-valued functions T(x)

dT(x) 3 grad T(x) := =: U (x) result is a tensor ﬁeld of 3-rd order dx −→ or in basis notation

∂tik(x) grad T(x) := ei ek ej =: tik,j ei ek ej ∂xj ⊗ ⊗ ⊗ ⊗

Rem.: The gradient operator grad( )= ( ) (with : Nabla operator) increases the order of the respective function· by∇ one.· ∇ (b) Derivative of functions of arbitrary vectorial and tensorial variables

Rem.: Derivatives concerning the respective variables are built analogously to the preceding procedures, e. g.

∂R(T, v) ∂Rij(T, v) = ei ej es et ∂T ∂tst ⊗ ⊗ ⊗ Some speciﬁc rules for the derivative of tensor functions with respect to tensors For arbitrary 2-nd order tensors A, B, C, the following rules hold:

∂(AB) 23 = (A I)T ∂B ⊗ ∂(AB) 23 = (I BT )T ∂A ⊗ ∂(AA) 23 23 = (A I)T +(I AT )T ∂A ⊗ ⊗ ∂(AT A) 23 24 = (AT I)T +(I A)T ∂A ⊗ ⊗ ∂(AAT ) 24 23 = (A I)T +(I A)T ∂A ⊗ ⊗ ∂(AT AT ) 24 24 = (I AT )T +(AT I)T ∂A ⊗ ⊗

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∂(ABC) 23 = (A CT )T ∂B ⊗ ∂AT 24 = (I I)T ∂A ⊗ ∂A−1 23 = (A−1 AT −1)T ∂A − ⊗ ∂AT −1 24 = (AT −1 AT −1)T ∂A − ⊗ + ∂A 24 = det A [(AT −1 AT −1) (AT −1 AT −1)T ] ∂A ⊗ − ⊗ ∂(α β) ∂β ∂α = α + β ∂C ∂C ∂C ∂(α v) ∂α ∂v = v + α ∂C ⊗ ∂C ∂C ∂(α A) ∂α ∂A = A + α ∂C ⊗ ∂C ∂C ∂(A v) ∂A 24 23 ∂v = T T v + A 3 ∂C " ∂C ! # " ∂C# ∂(u v) ∂u 13 ∂v 13 · = T v T + T u T ∂C " ∂C! # " ∂C! # ∂(A B) ∂A ∂B · = T B + T A ∂C ∂C ! ∂C! ∂(AB) ∂A 24 24 ∂B 14 14 = T B 4 T + T AT 4 T ∂C " ∂C ! # ! " ∂C! # ! Furthermore, ∂A 23 4 = (I I)T =: I ∂A ⊗ ∂AT 24 = (I I)T ∂A ⊗ ∂(A I) I · = (I I) ∂A ⊗ A ∂ t(A) 3 = 1 E ∂A − 2

Principal invariants and their derivatives (see also section 2.9)

∂IA = I with IA = A I ∂A ·

∂IIA 1 2

@ @ = A I with IIA = (IA AA I) ∂A 2 − · ∂IIIA + = A with IIIA = det A ∂A

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(c) Specific operators here: Introduction of the further diﬀerential operators div ( ) and rot( ). · · Divergence of a vector ﬁeld v(x)

div v(x) := grad v(x) I =: φ(x) result is a scalar ﬁeld · −→ or in basis notation div v(x) = v , (e e ) (e e ) i j i ⊗ j · n ⊗ n = vi,j δin δjn = vn,n ∂v ∂v ∂v = 1 + 2 + 3 ∂x1 ∂x2 ∂x3 Divergence of a tensor ﬁeld T(x)

div T(x) = [grad T(x)] I =: v(x) result is a vector ﬁeld −→ or in basis notation

div T(x) = t , (e e e )(e e ) ik j i ⊗ k ⊗ j n ⊗ n = tik,j δkn δjn ei = tin,n ei

Rem.: The divergence operator div ( )= ( ) decreases the order of the respective · ∇· · function by one.

Rotation of a vector ﬁeld v(x)

3 rot v(x) := E [grad v(x)]T =: r(x) result is a vector ﬁeld −→ or in basis notation

rot v(x) = e (e e e ) v , (e e ) ijn i ⊗ j ⊗ n o p p ⊗ o = eijn vo,p δjp δno ei = eijn vn,j ei

Consequence: rot v(x) yields twice the axial vector corresponding to the skew- symmetric part of grad v(x). Rem.: The rotation operator rot( ) = curl( ) = ( ) preserves the order of the · · ∇ × · respective function.

Laplace operator

∆( ) := div grad ( ) analogical to the precedings · · −→ Rem.: The Laplace operator ∆( )= ( ) preserves the order of the diﬀerentiated function. · ∇·∇ ·

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Rules for the operators grad( ), div ( ), and rot( ) · · ·

grad(φψ) = φ grad ψ + ψ grad φ

grad(φv) = v grad φ + φ grad v ⊗ grad(φT) = T grad φ + φ grad T ⊗ grad(u v) = (grad u)T v + (grad v)T u · grad(u v) = u grad v + grad u v × × × 23 grad(a b) = [grad a b + a (grad b)T ]T ⊗ ⊗ ⊗ 23 grad(Tv) = (grad T)T v + T grad v

23 23 grad(TS) = [(grad T)T S]3 T +(T grad S)3

13 13 grad(T S) = (grad T)T ST + (grad S)T TT · grad x = I

div (u v) = u div v + (grad u) v ⊗ div (φ v) = v grad φ + φ div v · div (Tv) = (div TT ) v + TT grad v · · div (grad v)T = grad div v

div (u v) = (grad u v) I (grad v u) I × × · − × · = v rot u u rot v · − · div (φ T) = T grad φ + φ div T

div (T S) = (grad T) S + T div S

div (v T) = v div T + grad v T × × × div (v T) = v div T + (grad v) TT ⊗ ⊗ 3 3 3 13 23 div (v T) = v div T + [(grad v)( TT )T ]3 ⊗ ⊗ div (grad v)+ = 0

div [grad v (grad v)T ] = div grad v grad div v ± ± div rot v = 0

rotrot v = grad div v div grad v − rot grad φ = 0

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rot grad v = 0

rot (grad v)T = gradrot v

rot(φ v) = φ rot v + grad φ v × rot(u v) = div (u v v u) × ⊗ − ⊗ = u div v + (grad u)v v div u (grad v)u − − Grassmann evolution:

v rot v = 1 grad(v v) (grad v) v = (grad v)T v (grad v) v × 2 · − −

3.4 Integral theorems

Rem.: In what follows, some integral theorems for the transformation of surface integrals into volume integrals are presented. Requirement: u = u(x) is a steady and sufficiently often steadily differentiable vector field. The domain of u is in 3. V

(a) Proof of the integral theorem

u(x) da = grad u(x)dv with da = n da ⊗ ZS VZ da : surface element and ( n : outward oriented unit surface normal vector

u(x) da2 u¯2

u¯4 dx2

e2 da4 da1 X dx1

x u¯1 dx e1 3 0 da3 u¯3 e3 da5 u¯5

Basis: Consideration of an inﬁnitesimal volume element dv spanned in the point X by the position vector x, and u¯i, i. e. the values of u(x) in the centroid of the partial surfaces 1 - 6.

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Determination of the surface element vectors dai: da = dx dx = dx dx (e e ) 1 2 × 3 2 3 2 × 3 = dx dx e = da e = n = n 2 3 1 − 4 −→ 1 1 − 4 Furthermore, one obtains da = dx dx e = da e = n = n 2 3 1 2 − 5 −→ 2 2 − 5 da = dx dx e = da e = n = n 3 1 2 3 − 6 −→ 3 3 − 6

Rem.: The surface vectors hold the condition dai = 0 . i=1 X Determination of the volume elements dv: dv = (dx dx ) dx = dx dx dx 1 × 2 · 3 1 2 3

Values of u(x) in the centroids of the partial surfaces:

Rem.: The increments of u(x) in the directions of dx1, dx2, dx3 are approximated by the ﬁrst term of a Taylor series. 1 ∂u 1 ∂u u¯4 = u(x)+ dx2 + dx3 2 ∂x2 2 ∂x3 ∂u u¯1 = u¯4 + dx1 ∂x1 Furthermore, one obtains ∂u ∂u u¯2 = u¯5 + dx2 , u¯3 = u¯6 + dx3 ∂x2 ∂x3

Computation of the surface integral yields

6 u(x) da u¯ da = u¯ da + u¯ da + ⊗ −→ i ⊗ i 1 ⊗ 1 4 ⊗ 4 ··· Z i=1 S(dv) X ∂u (u¯ 1− dx1) ( da1) ∂x|1 {z ⊗} − Thus 6 ∂u ∂u ∂u u¯ da = dx da + dx da + dx da i ⊗ i ∂x 1 ⊗ 1 ∂x 2 ⊗ 2 ∂x 3 ⊗ 3 i=1 1 2 3 X with da1 = dx2 dx3 e1 , da2 = dx1 dx3 e2 , da3 = dx1 dx2 e3 yields 6 ∂u ∂u ∂u u¯ da = e + e + e dx dx dx i ⊗ i ∂x ⊗ 1 ∂x ⊗ 2 ∂x ⊗ 3 1 2 3 i=1 1 2 3 X ∂ui dv ei ej = grad u | ∂xj ⊗ {z } | {z }

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Thus 6 u¯ da = grad u dv i ⊗ i i=1 X Integration over an arbitrary volume V yields

u(x) da = grad u(x) dv q. e. d. ( ) ⊗ ∗ ZS VZ

(b) Proof of the GAUSSian integral theorem

u(x) da = div u(x) dv · ZS VZ

Basis: Integral theorem ( ) after scalar multiplication with the identity tensor ∗

I u(x) da = I grad u(x) dv · ⊗ · ZS VZ

I [u(x) da] = I grad u(x) dv −→ · ⊗ · ZS u(x) da ZV div u(x) · | {z } | {z } Thus, leading to u(x) da = div u(x) dv ( ) · ∗∗ ZS VZ

T(x) da = div T(x) dv ZS VZ

Basis: Scalar multiplication of the surface integral with a constant vector b 3 ∈V

b T(x) da = b T(x) da = [TT (x) b] da =: u(x) da · · · · ZS ZS ZS ZS

with u(x) := TT (x) b

It follows with the integral theorem ( ) ∗∗

b T(x) da = div [TT (x) b] dv · ZS VZ

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In particular, with b = const. and a divergence rule follows

div [TT (x) b] = div T(x) b · leading to

b T(x) da = div T(x) b dv · · ZS VZ

Thus

T(x) da = div T(x) dv q. e. d. ZS VZ

Rem.: At this point, no further proofs are carried out.

(d) Summary of some integral theorems For the transformation of surface integrals into volume integrals, the following relations hold:

u da = grad u dv ⊗ ZS VZ

φ da = grad φ dv ZS VZ

u da = div u dv · ZS VZ

u da = rot u dv × − ZS VZ

T da = div T dv ZS VZ

u T da = div (u T) dv × × ZS VZ

u T da = div (u T) dv ⊗ ⊗ ZS VZ

For the transformation of line into surface integrals the following relations hold:

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u dx = grad u da ⊗ − × IL ZS

φ dx = grad φ da − × IL ZS

u dx = (rot u) da · · IL ZS

u dx = (I div u grad T u) da × − IL ZS

T dx = (rot T)T da IL ZS with da = n da

Rem.: If required, further relations of the vector and tensor calculus will be presented in the respective context. The description of non-orthogonal and non-unit basis systems was not discussed in this contribution.

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3.5 Transformations between actual and reference conﬁgurations

Given are the deformation gradient F = ∂x/∂X and arbitrary vectorial and tensorial field functions v and A. Then, with ∂ Grad( ) = ( ) · ∂X · reference configuration   Div ( ) = [Grad ( )] I or [Grad ( )] I  · · · ·  ∂  grad( ) = ( ) · ∂x · actual configuration   div ( ) = [grad ( )] I or [grad ( )] I  · · · ·  the following relations hold:

Grad v = (grad v) F Grad A = [(grad A) F]3 grad v = (Grad v) F−1 grad A = [(Grad A) F−1]3

Div v = (grad v) FT Div A = (grad A) FT · div v = (Grad v) FT −1 div A = (Grad A) FT −1 ·

Furthermore, it can be shown that

Div FT −1 = FT −1 (FT −1 Grad F)1 = (det F)−1 FT −1 [Grad (det F)] − − div FT = FT (FT grad F−1)1 = (det F) FT [grad (det F)−1] − −

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