Appendix A Basic Operations of Tensor Algebra

The tensor calculus is a powerful tool for the description of the fundamentals in continuum mechanics and the derivation of the governing equations for applied problems. In general, there are two possibilities for the representation of the tensors and the tensorial equations: • the direct (symbolic, coordinate-free) notation and • the index (component) notation The direct notation operates with scalars, vectors and tensors as physical objects defined in the three-dimensional space (in this book we are limit ourselves to this case). A vector (first rank tensor)a is considered as a directed line segment rather than a triple of numbers (coordinates). A second rank tensor A is any finite sum ofordered vector pairs A = a ⊗ b +···+c ⊗ d. The scalars, vectors and tensors are handled as invariant (independent from the choice of the coordinate system) quantities. This is the reason for the use of the direct notation in the modern literature of mechanics and rheology, e.g. Altenbach (2012), Antman (1995), Besseling and van der Giessen (1994), Giesekus (1994), Haupt (2002), Lurie (1990), Palmov (1998), Truesdell and Noll (1992), Zhilin (2001) among others. The basics of the direct tensor calculus are given in the classical textbooks of Wilson (1901) (founded upon the lecture notes of Gibbs) and Lagally (1962). The index notation deals with components or coordinates of vectors and tensors. For a selected basis, e.g. gi , i = 1, 2, 3 one can write   i i j i j a = a gi , A = a b +···+c d gi ⊗ g j

Here the Einstein’s summation convention is used: in one expression the twice repeated indices are summed up from 1 to 3, e.g.

3 3 k k ik ik a gk ≡ a gk , A bk ≡ A bk k=1 k=1

© Springer International Publishing Switzerland 2016 323 K. Naumenko and H. Altenbach, Modeling High Temperature Materials Behavior for Structural Analysis, Advanced Structured Materials 28, DOI 10.1007/978-3-319-31629-1 324 Appendix A: Basic Operations of Tensor Algebra

In the above examples k is a so-called dummy index. Within the index notation the basic operations with tensors are defined with respect to their coordinates, e.g. the sum of two vectors is computed as the sum of their coordinates ci = ai + bi .The introduced basis remains in the background. It must be noted that a change of the coordinate system leads to the change of the components of tensors. In this book we prefer the direct tensor notation over the index one. When solv- ing applied problems the tensor equations can be “translated into the language” of matrices for a specified coordinate system. The purpose of this Appendix is to give a brief guide to notations and rules of the tensor calculus applied throughout this book. For more comprehensive overviews on tensor calculus we recommend Betten (1987), de Boer (1982), Giesekus (1994), Lebedev et al. (2010), Lippmann (1993), Lurie (1990), Trostel (1993), Zhilin (2001). The calculus of matrices is presented in Bellmann (1970), Faddejew and Faddejewa (1964), Zurmühl and Falk (1992), for example. AppendixA provides a summary of basic algebraic operations with vectors and second rank tensors. Several rules from tensor analysis are given in Appendix B. Basic sets of invariants for different groups of symmetry transformation are presented in Sect. B.5, where a novel approach to find the functional basis is discussed.

A.1 Polar and Axial Vectors

A vector in the three-dimensional Euclidean space is defined as a directed line segment with specified scalar-valued magnitude and direction. The magnitude (the length) of a vector a is denoted by |a|. Two vectors a and b are equal if they have the same direction and the same magnitude. The zero vector 0 has a magnitude equal to zero. In mechanics two types of vectors can be introduced. The vectors of the first type are directed line segments. These vectors are asso- ciated with translations in the three-dimensional space. Examples for polar vectors include the force, the displacement, the velocity, the acceleration, the momentum,

(a) (b) (c) a

a∗ a ∗ a ∗

a

Fig. A.1 Spin vector and its representation by an axial vector. a Spin vector, b axial vector in the right-screw oriented reference frame, c axial vector in the left-screw oriented reference frame Appendix A: Basic Operations of Tensor Algebra 325 etc. The second type is used to characterize spinor motions and related quantities, i.e. the moment, the angular velocity, the angular momentum, etc., Fig.A.1ashowsthe so-called spin vector a ∗ which represents a rotation about the given axis. The direc- tion of rotation is specified by the circular arrow and the “magnitude” of rotation is the corresponding length. For the given spin vector a ∗ the directed line segment a is introduced according to the following rules (Zhilin 2001): 1. the vector a is placed on the axis of the spin vector, 2. the magnitude of a is equal to the magnitude of a ∗, 3. the vector a is directed according to the right-handed screw, Fig.A.1b, or the left-handed screw, Fig. A.1c The selection of one of the two cases in of the third item corresponds to the convention of orientation of the reference frame (Zhilin 2001). The directed line segment is called a polar vector if it does not change by changing the orientation of the reference frame. The vector is called to be axial if it changes the sign by changing the orientation of the reference frame. The above definitions are valid for scalars and tensors of any rank too. The axial vectors (and tensors) are widely used in the rigid body dynamics, e.g. Altenbach et al. (2007), Zhilin (1996), in the theories of rods, plates and shells, e.g. Altenbach and Zhilin (1988), Naumenko and Eremeyev (2014), in the asymmetric theory of elasticity, e.g. Nowacki (1986), as well as in dynamics of micro-polar media, e.g. Altenbach et al. (2003), Eringen (1999). When dealing with polar and axial vectors it should be remembered that they have different physical meanings. Therefore, a sum of a polar and an axial vector has no sense.

A.2 Operations with Vectors

A.2.1 Addition

For a given pair of vectors a and b of the same type the sum c = a + b is defined according to one of the rules in Fig. A.2. The sum has the following properties • a + b = b + a (commutativity), • (a + b) + c = a + (b + c) (associativity), • a + 0 = a

(a) (b)

b c c b

a a

Fig. A.2 Addition of two vectors. a Parallelogram rule, b triangle rule 326 Appendix A: Basic Operations of Tensor Algebra

A.2.2 Multiplication by a Scalar

For any vector a and for any scalar α a vector b = αa is defined in such a way that •|b|=|α||a|, • for α>0 the direction of b coincides with that of a, • for α<0 the direction of b is opposite to that of a. For α = 0 the product yields the zero vector, i.e. 0 = 0a. It is easy to verify that

α(a + b) = αa + αb, (α + β)a = αa + βa

A.2.3 Scalar (Dot) Product of Two Vectors

For any pair of vectors a and b a scalar α is defined by

α = a · b =|a||b| cos ϕ, where ϕ is the angle between the vectors a and b.Asϕ one can use any of the two angles between the vectors, Fig. A.3a. The properties of the scalar product are

• a · b = b · a (commutativity), • a · (b + c) = a · b + a · c (distributivity)

Two nonzero vectors are said to be orthogonal if their scalar product is zero. The unit vector directed along the vector a is defined by (see Fig. A.3b) a n = a |a|

(a) (b)

b b

a a n b a na a a a

Fig. A.3 Scalar product of two vectors. a Angles between two vectors, b unit vector and projection Appendix A: Basic Operations of Tensor Algebra 327

(a) (b) (c) c c

a b a b a b

Fig. A.4 Vector product of two vectors. a Plane spanned on two vectors, b spin vector, c axial vector in the right-screw oriented reference frame

The projection of the vector b onto the vector a is the vector (b · a)na ,Fig.A.3b. The length of the projection is |b|| cos ϕ|.

A.2.4 Vector (Cross) Product of Two Vectors

For the ordered pair of vectorsa andb the vectorc = a ×b is defined in two following steps (Zhilin 2001):

• the spin vector c∗ is defined in such a way that – the axis is orthogonal to the plane spanned on a and b,Fig.A.4a, – the circular arrow shows the direction of the “shortest” rotation from a to b, Fig. A.4b, – the length is |a||b| sin ϕ, where ϕ is the angle of the “shortest” rotation from a to b, • from the resulting spin vector the directed line segment c is constructed according to one of the rules listed in Sect.A.1. The properties of the vector product are

a × b =−b × a, a × (b + c) = a × b + a × c

The type of the vector c = a ×b can be established for the known types of the vectors a and b (Zhilin 2001). If a and b are polar vectors the result of the cross product will be the axial vector. An example is the moment of momentum for a mass point m defined by r × (mvv)˙ , where r is the position of the mass point and v is the velocity of the mass point. The next example is the formula for the distribution of velocities in a rigid body v = ω × r . Here the cross product of the axial vector ω (angular velocity) with the polar vector r (position vector) results in the polar vector v. 328 Appendix A: Basic Operations of Tensor Algebra

The mixed product of three vectors a, b and c is defined by (a ×b) · c. The result is a scalar. For the mixed product the following identities are valid

a · (b × c) = b · (c × a) = c · (a × b) (A.2.1)

If the cross product is applied twice, the first operation must be set in parentheses, e.g., a × (b × c). The result of this operation is a vector. The following relation can be applied a × (b × c) = b(a · c) − c(a · b) (A.2.2)

By use of (A.2.1) and (A.2.2) one can calculate

(a × b) · (c × d) = a ··[b × (c × d)] = a · (ccbb · d − ddbb · c) (A.2.3) = a · ccbb · d − a · ddbb · c

A.3 Bases

Any triple of linear independent vectors e1, e2, e3 is called basis. A triple of vectors ei is linear independent if and only if e1 · (e2 × e3) = 0. For a given basis ei any vector a can be represented as follows

1 2 3 i a = a e1 + a e2 + a e3 ≡ a ei

i The numbers a are called the coordinates of the vector a for the basis ei . In order to compute the coordinates ai the dual (reciprocal) basis ek is introduced in such a way that  1, k = i, ek · e = δk = i i 0, k = i

δk i i is the Kronecker symbol. The coordinates a can be found by

i · = · i = m · i = m δi = i e a a e a em e a m a

For the selected basis ei the dual basis can be found from

e × e e × e e × e e1 = 2 3 , e2 = 3 1 , e3 = 1 2 (A.3.4) (e1 × e2) · e3 (e1 × e2) · e3 (e1 × e2) · e3 Appendix A: Basic Operations of Tensor Algebra 329

By use of the dual basis a vector a can be represented as follows

1 2 3 i m a = a1e + a2e + a3e ≡ aie , am = a · em , a = am

In the special case of the orthonormal vectors ei , i.e. |ei |=1 and ei · ek = 0for k k i = k, from (A.3.4) follows that e = ek and consequently ak = a .

A.4 Operations with Second Rank Tensors

A second rank tensor is a finite sum of ordered vector pairs A = a ⊗b +···+c ⊗d. One ordered pair of vectors is called the dyad (Wilson 1901). The symbol ⊗ is called the dyadic (tensor) product of two vectors. A single dyad or a sum of two dyads are special cases of the second rank tensor. Any finite sum of more than three dyads can be reduced to a sum of three dyads. For example, let

n A = a (i) ⊗ b(i) i=1 be a second rank tensor. Introducing a basis ek the vectors a (i) can be represented by = k k a (i) a(i)ek , where a(i) are coordinates of the vectors a (i). Now we may write

n n n = k ⊗ = ⊗ k = ⊗ k , k ≡ k A a(i)ek b(i) ek a(i)b(i) ek d d a(i)b(i) i=1 i=1 i=1

A.4.1 Addition

The sum of two tensors is defined as the sum of the corresponding dyads. The sum has the properties of associativity and commutativity. In addition, the following operation can be introduced

a ⊗ (b + c) = a ⊗ b + a ⊗ c,(a + b) ⊗ c = a ⊗ c + b ⊗ c

A.4.2 Multiplication by a Scalar

This operation is introduced first for one dyad. For any scalar α and any dyad a ⊗ b

α(a ⊗ b) = (αa) ⊗ b = a ⊗ (αb), (A.4.5) (α + β)a ⊗ b = αa ⊗ b + βa ⊗ b 330 Appendix A: Basic Operations of Tensor Algebra

By setting α = 0 in the first equation of (A.4.5) the zero dyad can be defined, i.e. 0(a ⊗ b) = 0 ⊗ b = a ⊗ 00. The above operations can be generalized for any finite sum of dyads, i.e. for second rank tensors.

A.4.3 Inner Dot Product

For any two second rank tensors A and B the inner dot product is specified by A · B. The rule and the result of this operation can be explained in the special case of two dyads, i.e. by setting A = a ⊗ b and B = c ⊗ d

A · B = a ⊗ b · c ⊗ d = (b · c)a ⊗ d = αa ⊗ d,α≡ b · c

Note that in general A · B = B · A. This can be again verified for two dyads. The operation can be generalized for two second rank tensors as follows

3 3 A · B = a (i) ⊗ b(i) · c(k) ⊗ d (k) i=1 k=1 3 3 3 3 = (b(i) · c(k))a (i) ⊗ d (k) = α(ik)a (i) ⊗ d (k) i=1 k=1 i=1 k=1 with α(ik) ≡ b(i) · c(k). The result of this operation is a second rank tensor.

A.4.4 Transpose of a Second Rank Tensor

The transpose of a second rank tensor A is defined as follows   3 T 3 T A = a (i) ⊗ b(i) = b(i) ⊗ a (i) i=1 i=1

A.4.5 Double Inner Dot Product

For any two second rank tensors A and B the double inner dot product is specified by A ·· B. The result of this operation is a scalar. This operation can be explained for two dyads A = a ⊗ b and B = c ⊗ d as follows

A ·· B = a ⊗ b ·· c ⊗ d = (b · c)(a · d) Appendix A: Basic Operations of Tensor Algebra 331

By analogy to the inner dot product one can generalize this operation for two second rank tensors. It can be verified that A·· B = B ·· A for arbitrary second rank tensors A and B. For a second rank tensor A and for a dyad a ⊗b the double inner dot product yields A ·· a ⊗ b = b · A · a (A.4.6)

A scalar product of two second rank tensors A and B is defined by

α = A ·· B T

One can verify that A ·· B T = B T ·· A = B ·· AT

A.4.6 Dot Products of a Second Rank Tensor and a Vector

The right dot product of a second rank tensor A and a vector c is defined by   3 3 3 A · c = a (i) ⊗ b(i) · c = (b(i) · c)a (i) = α(i)a (i) i=1 i=1 i=1 with α(i) ≡ b(i) · c. The left dot product is defined by   3 3 3 c · A = c · a (i) ⊗ b(i) = (c · a (i))b(i) = β(i)b(i) i=1 i=1 i=1 with β(i) ≡ c · a (i). The results of these operations are vectors. One can verify that

A · c = c · A, A · c = c · AT

A.4.7 Cross Products of a Second Rank Tensor and a Vector

The right cross product of a second rank tensor A and a vector c is defined by   3 3 3 A × c = a (i) ⊗ b(i) × c = a (i) ⊗ (b(i) × c) = a (i) ⊗ d (i) i=1 i=1 i=1 with d (i) ≡ b(i) × c. The left cross product is defined by 332 Appendix A: Basic Operations of Tensor Algebra   3 3 3 c × A = c × a (i) ⊗ b(i) = (c × a (i)) ⊗ b(i) = e(i) ⊗ b(i) i=1 i=1 i=1 with e(i) ≡ b(i) × c. The results of these operations are second rank tensors. It can be shown that A × c =−[c × AT]T

A.4.8 Trace

The trace of a second rank tensor is defined by   3 3 tr A = tr a (i) ⊗ b(i) = a (i) · b(i) i=1 i=1

By taking the trace of a second rank tensor the dyadic product is replaced by the dot product. It can be shown that

tr A = tr AT, tr (A · B) = tr (B · A) = tr (AT · B T) = A ·· B

A.4.9 Symmetric Tensors

A second rank tensor is said to be symmetric if it satisfies the following equality

A = AT

An alternative definition of the symmetric tensor can be given as follows. A second rank tensor is said to be symmetric if for any vector c = 0 the following equality is valid c · A = A · c

An important example of a symmetric tensor is the unit or identity tensor I , which is defined by such a way that for any vector c

c · I = I · c = c

The representations of the identity tensor are

k k I = ek ⊗ e = e ⊗ ek Appendix A: Basic Operations of Tensor Algebra 333

k · m = δm for any basis ek and e , ek e k . For three orthonormal vectors m, n and p the identity tensor has the form

I = n ⊗ n + m ⊗ m + p ⊗ p

A symmetric second rank tensor P satisfying the condition P · P = P is called projector. Examples of projectors are

m ⊗ m, n ⊗ n + p ⊗ p = I − m ⊗ m, where m, n and p are orthonormal vectors. The result of the dot product of the tensor m ⊗ m with any vector a is the projection of the vector a onto the line spanned on the vector m, i.e. m ⊗ m · a = (a · m)m. The result of (n ⊗ n + p ⊗ p) · a is the projection of the vector a onto the plane spanned on the vectors n and p.

A.4.10 Skew-Symmetric Tensors

A second rank tensor is said to be skew-symmetric if it satisfies the following equality

A =−AT or if for any vector c = 0

c · A =−A · c

Any skew-symmetric tensor A can be represented by

A = a × I = I × a

The vector a is called the associated vector. Any second rank tensor can be uniquely decomposed into the symmetric and skew-symmetric parts

1   1   A = A + AT + A − AT = A + A , 2 2 1 2 1   A = A + AT , A = AT, 1 2 1 1 1   A = A − AT , A =−AT 2 2 2 2 334 Appendix A: Basic Operations of Tensor Algebra

A.4.11 Linear Transformations of Vectors

A vector valued function of a vector argument f (a) is called to be linear if f (α1a 1 + α2a 2) = α1 f (a 1) + α2 f (a 2) for any two vectors a 1 and a 2 and any two scalars α1 and α2. It can be shown that any linear vector valued function can be represented by f (a) = A · a, where A is a second rank tensor. In many textbooks, e.g. Antman (1995), Smith (1993), a second rank tensor A is defined to be the linear transformation of the vector space into itself.

A.4.12 Determinant and Inverse of a Second Rank Tensor

Let a, b and c be arbitrary linearly-independent vectors. The determinant of a second rank tensor A is defined by

(A · a) ··[(A · b) × (A · c)] det A = (A.4.7) a · (b × c)

The following identities can be verified

det(AT) = det(A), det(A · B) = det(A) det(B)

The inverse of a second rank tensor A−1 is introduced as the solution of the following equation A−1 · A = A · A−1 = I

A is invertible if and only if det A = 0. A tensor A with det A = 0 is called singular. Examples for singular tensors are projectors. Applying (A.4.7) one may derive the following identities valid for a non-singular tensor A and vectors a and b (Zhilin 2001)

det AAAA−T · (a × b) = (A · a) × (A · b), (A.4.8) det AAAA−T · (a × I ) · A−1 = (A · a) × I

A.4.13 Principal Values and Directions of Symmetric Second Rank Tensors

Consider a dot product of a second rank tensor A and a unit vector n. The resulting vector a = A · n differs in general from n both by the length and the direction. However, one can find those unit vectors n,forwhichA · n is collinear with n, i.e. Appendix A: Basic Operations of Tensor Algebra 335 only the length of n is changed. Such vectors can be found from the equation

A · n = λn or (A − λI ) · n = 0 (A.4.9)

The unit vector n is called the principal vector (principal direction) and the scalar λ the principal value of the tensor A. The problem to find the principal values and principal directions of Eq. (A.4.9) is the eigen-value problem for A. The principal values are the eigen-values, the principal directions are the eigen-directions. Let A be a symmetric tensor. In this case the principal values are real numbers and there exist at least three mutually orthogonal principal vectors. The principal values can be found as roots of the characteristic polynomial

3 2 det(A − λI ) =−λ + J1(A)λ − J2(A)λ + J3(A) = 0 (A.4.10)

Here Ji (A)(i = 1, 2, 3) are the principal invariants of the tensor A

J1(A) = tr A, 1 J (A) = [(tr A)2 − tr A2], 2 2 (A.4.11) 1 1 1 J (A) = det A = (tr A)3 − tr Atr A2 + tr A3 3 6 2 3

The principal values are specified by λI ,λII,λIII. The following three cases can be introduced

• three distinct values λi , i = I, II, III, i.e. λI = λII = λIII,or • one single value and one double solution, e.g. λI = λII = λIII,or • one triple solution λI = λII = λIII

For a fixed solution λi , i = I, II, III the eigen-directions can be found from

(A − λi I ) · n(i) = 0 (A.4.12)

The eigen-direction is defined with respect to an arbitrary scalar multiplier. For known principal values and principal directions the second rank tensor can be represented as follows (spectral representation)

A = λI n I ⊗ n I + λIIn II ⊗ n II + λIIIn III ⊗ n III for λI = λII = λIII, A = λI (I − n III ⊗ n III) + λIIIn III ⊗ n III for λI = λII = λIII, A = λI for λI = λII = λIII = λ

A.4.14 Cayley-Hamilton Theorem

Any second rank tensor satisfies the following equation 336 Appendix A: Basic Operations of Tensor Algebra

3 2 A − J1(A)A + J2(A)A − J3(A)I = 0, (A.4.13) where A2 = A · A, A3 = A · A · A. The Cayley-Hamilton theorem can be applied to compute the powers of a tensor higher than two or negative powers of a non-singular tensor. For example, the fourth power of A can be computed by multiplying (A.4.13)byA

4 3 2 A − J1(A)A + J2(A)A − J3(A)A = 0

After eliminating the third power we get    4 = 2( ) − ( ) 2 + ( ) − ( ) ( ) + ( ) ( ) A J1 A J2 A A J3 A J1 A J2 A A J1 A J3 A I

For a non-singular tensor A (det A = 0) the inverse can be computed by multiplying (A.4.13)byA−1. As a result one obtains

−1 2 1 A = A − J1(A)A + J2(A)I J3(A)

A.4.15 Vector Invariant

The vector invariant or “Gibbsian Cross” of a second rank tensor A is defined by   3 3 A× = a (i) ⊗ b(i) = a (i) × b(i) i=1 × i=1

The result of this operation is a vector. The vector invariant of a symmetric tensor is the zero vector. The following identities can be verified

(a × I )× =−2a, (A.4.14) a × I × b = b ⊗ a − (a · b)I

For any vector a and any second rank tensor B the following identity is valid

T (a × B)× = B · a − (tr B)a,(B × a)× = B · a − (tr B)a (A.4.15)

For any vector a and symmetric tensors A, B the following identity can be established Appendix A: Basic Operations of Tensor Algebra 337

[(A × a) · B]× =[A · (a × B)]× = C · a, 3 3 = λA A × × A = λB B × × B C i ni B ni i ni A ni (A.4.16) i=1 i=1 =[tr (A · B) − tr Atr B]I + (tr B)A + (tr A)B − A · B − B · A,

λA λB where i and i are principal values of the tensors A and B, respectively. The A B othonormal vectors ni and ni are corresponding principal directions.

A.4.16 Coordinates of Second Rank Tensors

k Let ei be a basis and e the dual basis. Any two vectors a and b can be represented as follows i j l m a = a ei = a je , b = b el = bme

A dyad a ⊗ b has the following representations

i j i j i j j i a ⊗ b = a b ei ⊗ e j = a b jei ⊗ e = ai b je ⊗ e = ai b e ⊗ e j

For the representation of a second rank tensor A one of the following four bases can be used i j i j ei ⊗ e j , e ⊗ e , e ⊗ e j , ei ⊗ e

With these bases one can write

= ij ⊗ = i ⊗ j = i∗ ⊗ j = ∗ j i ⊗ A A ei e j Aije e A∗ jei e Ai∗ e e j

For a selected basis the coordinates of a second rank tensor can be computed as follows

ij i j A = ei · A · e j , Aij = e · A · e , i∗ = · · j , ∗ j = i · · A∗ j ei A e Ai∗ e A e j

A.4.17 Orthogonal Tensors

A second rank tensor Q is said to be orthogonal if it satisfies the equation

Q T · Q = I

If Q operates on a vector, its length remains unchanged, i.e. let b = Q · a, then 338 Appendix A: Basic Operations of Tensor Algebra

|b|2 = b · b = a · Q T · Q · a = a · a =|a|2

Furthermore, the orthogonal tensor does not change the scalar product of two arbitrary vectors. For two vectors a and b as well as a  = Q · a and b = Q · b one can calculate a  · b = a · Q T · Q · b = a · b

From the definition of the orthogonal tensor follows

Q T = Q −1, Q T · Q = Q · Q T = I , det(Q · Q T) = (det Q)2 = det I = 1 ⇒ det Q =±1

Orthogonal tensors with det Q = 1 are called proper orthogonal or rotation tensors. The rotation tensors are widely used in the rigid body dynamics, e.g. Zhilin (1996), and in the theories of rods, plates and shells, e.g. Altenbach and Zhilin (1988), Antman (1995). Any orthogonal tensor is either the rotation tensor or the composition of the rotation and the tensor −I .LetP be a rotation tensor, det P = 1, then an orthogonal tensor Q with det Q =−1 can be composed by

Q = (−I ) · P = P · (−I ), det Q = det(−I ) det P =−1

For any two orthogonal tensors Q 1 and Q 2 the composition Q 3 = Q 1 · Q 2 is the orthogonal tensor, too. This property is used in the theory of symmetry and symmetry groups, e.g. Nye (1992), Zheng and Boehler (1994). Two important examples for orthogonal tensors are the • rotation tensor about a fixed axis

Q(ϕm) = m ⊗ m + cos ϕ(I − m ⊗ m) + sin ϕm × I , det Q = 1, (A.4.17)

where the unit vector m represents the axis and ϕ is the angle of rotation, • reflection tensor Q = I − 2n ⊗ n, det Q =−1, (A.4.18)

where the unit vector n represents the normal to the mirror plane. One can prove the following identities (Zhilin 2001)

(Q · a) × (Q · b) = det QQQQ · (a × b), (A.4.19) Q · (a × Q T) = Q · (a × I ) · Q T = det Q [(Q · a) × I ] (A.4.20) Appendix A: Basic Operations of Tensor Algebra 339

A.4.18 Polar Decomposition

Any regular second rank tensor A with det A > 0 can be represented in a unique manner as following polar decompositions of the positive-semidefinite symmetric tensors U or V and a rotation tensor R

A = R · U = V · R (A.4.21)

R · U is the right and V · R the left polar decomposition. The following statements are valid • R is the proper orthogonal tensor: R · R T = I , det R =+1. • U and V are symmetric, positive-semidefinite tensors:

U = U T, V = V T,(U · a) · a > 0,(V · b) · b > 0

a,b are arbitrary non-zero vectors. • U , V can be uniquely computed from F . • The eigen-values of the tensors U and V are the same. If η is an eigen-vector of U , then R · η is an eigen-vector of V . The proof of the statements is elementary. For any tensor A and vector a = 00we can write   (A · a) · (A · a) = a · AT · A · a > 0

Hence AT · A is a symmetric positive-semidefinite tensor. Therefore the tensors U =   / AT · A 1 2 andU −1 are also symmetric and positive-semidefinite. The orthogonality of R follows from

   T  −1 R · R T = A · U −1 · A · U −1 = A · U −2 · AT = A · U 2 · AT

 −1  −1 = A · AT · A · AT = A · A−1 · AT · AT

   −1 = A · A−1 · AT · AT = I · I = I

−1 −1 With det F > 0 we obtain det U > 0 and it follows det R = det Adet U > 0. The orthogonality condition R · R T = I yields det R · R T = (det R)2 =+1 and det R =+1. The proof of the uniqueness can be given as follows. From A = T T R · U = R 1 · U 1 it follows (R · U ) = (R 1 · U 1) . Hence

= T, = T, · T = · T U U U 1 U 1 U R U 1 R 1 340 Appendix A: Basic Operations of Tensor Algebra

Finally,     U 2 = U · R T · R · U = U · R T · (R · U )     = · T · ( · ) = · T · · = 2, U 1 R 1 R 1 U 1 U 1 R 1 R 1 U 1 U 1 that meansU = U 1. The proof of the second decomposition is similar. For F = V · R   / with V = F · F T 1 2 we obtain

V 2 = F · F T = (R · U ) · (R · U )T = R · U 2 · RT = (R · U ) · (RT · R) · (U · RT)      2 = R · U · RT · R · U · RT = R · U · RT ,   V = R · U · RT =⇒ V · R = R · U · RT · R = R · U = F

If η and λ are eigen-vector and eigen-value of U , then ληη = U · η and

λ(R · ηη) = (R · U ) · η = (V · R) · η = V · (R · ηη)

U and V have the same eigen-value λ and η or R · η are the eigen-vectors of U and V , respectively.

References

Altenbach H (2015) Kontinuumsmechanik - Einführung in die materialunabhängigen und materi- alabhängigen Gleichungen. Springer Vieweg, Berlin, Heidelberg Altenbach H, Zhilin PA (1988) Osnovnye uravneniya neklassicheskoi teorii uprugikh obolochek (Basic equations of a non-classical theory of elastic shells, in Russ.). Adv Mech 11:107–148 Altenbach H, Naumenko K, Zhilin P (2003) A micro-polar theory for binary media with application to phase-transitional flow of fiber suspensions. Continuum Mech Thermodyn 15:539–570 Altenbach H, Naumenko K, Pylypenko S, Renner B (2007) Influence of rotary inertia on the fiber dynamics in homogeneous creeping flows. ZAMM-J Appl Math Mech/Zeitschrift für Angewandte Mathematik und Mechanik 87(2):81–93 Antman S (1995) Nonlinear problems of elasticity. Springer, Berlin Bellmann R (1970) Introduction to matrix analysis. McGraw Hill, New York Besseling JF, van der Giessen E (1994) Mathematical modelling of inelastic deformation. Chapman and Hall, London Betten J (1987) Tensorrechnung für Ingenieure. Springer, Berlin de Boer R (1982) Vektor und Tensorrechnung für Ingenieure. Springer, Berlin Eringen AC (1999) Microcontinuum field theories, vol I: Foundations and Solids. Springer, New York Faddejew DK, Faddejewa WN (1964) Numerische Methoden der linearen Algebra. Deutscher Verlag der Wissenschaften, Berlin Giesekus H (1994) Phänomenologische Rheologie. Springer, Berlin Haupt P (2002) Continuum mechanics and theory of materials. Springer, Berlin Lagally M (1962) Vorlesungen über Vektorrechnung. Geest and Portig, Leipzig Lebedev LP,Cloud MJ, Eremeyev VA(2010) Tensor analysis with applications in mechanics. World Scientific Appendix A: Basic Operations of Tensor Algebra 341

Lippmann H (1993) Angewandte Tensorrechnung. Springer, Berlin Lurie AI (1990) Nonlinear theory of elasticity. North-Holland, Dordrecht Naumenko K, Eremeyev VA(2014) A layer-wise theory for laminated glass and photovoltaic panels. Compos Struct 112:283–291 Nowacki W (1986) Theory of asymmetric elasticity. Pergamon Press, Oxford Nye JF (1992) Physical properties of crystals. Oxford Science Publications, Oxford Palmov V (1998) Vibrations in elasto-plastic bodies. Springer, Berlin Smith DR (1993) An introduction to continuum mechanics. Kluwer, Dordrecht Trostel R (1993) Mathematische Grundlagen der Technischen Mechanik, vol I: Vektor- und Ten- soralgebra. Vieweg, Braunschweig Truesdell C, Noll W (1992) The non-linear field theories of mechanics, 2nd edn. Springer, Berlin Wilson EB (1901) Vector analysis, founded upon the lectures of G. W. Gibbs. Yale University Press, New Haven Zheng QS, Boehler JP (1994) The description, classification and reality of material and physical symmetries. Acta Mech 102:73–89 Zhilin PA (1996) A new approach to the analysis of free rotations of rigid bodies. ZAMM-J Appl Math Mech/Zeitschrift für Angewandte Mathematik und Mechanik 76(4):187–204 Zhilin PA (2001) Vektory i tensory vtorogo ranga v trekhmernom prostranstve (Vectors and second rank tensors in three-dimensional space, in Russ.). Nestor, St. Petersburg Zurmühl R, Falk S (1992) Matrizen und ihre Anwendungen. Springer, Berlin Appendix B Elements of Tensor Analysis

B.1 Coordinate Systems

The vector r characterizing the position of a point P can be represented by use of the Cartesian coordinates xi as follows, Fig. B.1,

1 2 3 1 2 3 i r (x , x , x ) = x e1 + x e2 + x e3 = x ei

Instead of coordinates xi one can introduce any triple of curvilinear coordinates q1, q2, q3 by means of one-to-one transformations

xk = xk (q1, q2, q3) ⇔ qk = qk (x1, x2, x3)

It is assumed that the above transformations are continuous and continuous differ- entiable as many times as necessary and for the Jacobians     ∂xk ∂qi det = 0, det = 0 ∂qi ∂xk must be valid. With these assumptions the position vector can be considered as a function of curvilinear coordinates qi , i.e. r = r (q1, q2, q3). Surfaces q1 = const, q2 = const, and q3 = const, Fig. B.1, are called coordinate surfaces. For given fixed 2 2 3 3 1 values q = q∗ and q = q∗ a curve can be obtained along which only q varies. This curve is called the q1-coordinate line, Fig. B.1. Analogously, one can obtain the q2- and q3-coordinate lines. The partial derivatives of the position vector with respect the to selected coordi- nates ∂r ∂r ∂r r = , r = , r = , r · (r × r ) = 0 1 ∂q1 2 ∂q2 3 ∂q3 1 2 3

© Springer International Publishing Switzerland 2016 343 K. Naumenko and H. Altenbach, Modeling High Temperature Materials Behavior for Structural Analysis, Advanced Structured Materials 28, DOI 10.1007/978-3-319-31629-1 344 Appendix B: Elements of Tensor Analysis

r

r

r r

e e e

Fig. B.1 Cartesian and curvilinear coordinates define the tangential vectors to the coordinate lines in a point P,Fig.B.1. The vectors k r i are used as the local basis in the point P.Byuseof(A.3.4) the dual basis r can be introduced. The vector drr connecting the point P with a point P in the differential neighborhood of P is defined by

∂r ∂r ∂r dr = dq1 + dq2 + dq3 = r dqk ∂q1 ∂q2 ∂q3 k

The square of the arc length of the line element in the differential neighborhood of P is calculated by

2 i k i k ds = dr · dr = (r i dq ) · (r k dq ) = gikdq dq , where gik ≡ r i · r k are the so-called contravariant components of the metric tensor. k With gik one can represent the basis vectors r i by the dual basis vectors r as follows

k k r i = (r i · r k )r = gikr

Similarly i i k ik ik i k r = (r · r )r k = g r k , g ≡ r · r , where gik are termed covariant components of the metric tensor. For the selected k bases r i and r the second rank unit tensor has the following representations

i ik ik i k i I = r i ⊗ r = r i ⊗ g r k = g r i ⊗ r k = gikr ⊗ r = r ⊗ r i Appendix B: Elements of Tensor Analysis 345

B.2 Hamilton (Nabla) Operator

A scalar field is a function which assigns a scalar to each spatial point P for the domain of definition. Let us consider a scalar field ϕ(r ) = ϕ(q1, q2, q3). The total differential of ϕ by moving from a point P to a point P in the differential neighborhood is ∂ϕ ∂ϕ ∂ϕ ∂ϕ dϕ = dq1 + dq2 + dq3 = dqk ∂q1 ∂q2 ∂q3 ∂qk

Taking into account that dqk = dr · r k ∂ϕ dϕ = dr · r k = dr ··∇∇ϕ ∂qk

The vector ∇ϕ is called the gradient of the scalar field ϕ and the invariant operator ∇ (the Hamilton or nabla operator) is defined by ∂ ∇∇=r k ∂qk

For a vector field a(r ) one may write

∂a ∂a da = (dr · r k ) = dr · r k ⊗ = dr ··∇∇∇⊗a = (∇∇⊗a)T · dr , ∂qk ∂qk ∂a ∇∇⊗a = r k ⊗ ∂qk

The gradient of a vector field is a second rank tensor. The operation ∇ can be applied to tensors of any rank. For vectors the following additional operations are defined

∂a diva ≡∇∇∇·· a = r k · , ∂qk ∂a rota ≡∇∇∇×a = r k × ∂qk

The divergence and the rotation (curl) of tensor fields of any rank higher 1 can be calculated in a similar manner. The following identities can be verified

∂r ∇∇⊗r = r k ⊗ = r k ⊗ r = I , ∇∇·· r = 3 ∂qk k

For a scalar α, a vector a and for a second rank tensor A the following identities are valid 346 Appendix B: Elements of Tensor Analysis   ∂(αa) ∂α ∂a ∇(αa) = r k ⊗ = r k ⊗ a+αr k ⊗ = (∇α)⊗ a+α∇∇⊗a, (B.2.1) ∂qk ∂qk ∂qk

∂(A · a) ∂ A ∂a ∇∇·· (A · a) = r k · = r k · · a + r k · A · (B.2.2) ∂ k ∂ k ∂ k q  q  q ∂a = (∇∇·· A) · a + A ·· ⊗ r k ∂qk = (∇∇·· A) · a + A ·· (∇∇⊗a)T

For a second rank tensor A and a position vector r one can prove the following identity

∂(A × r ) ∂ A ∂r ∇∇·· (A × r ) = r k · = r k · × r + r k · A × (B.2.3) ∂qk ∂qk ∂qk k = (∇∇·· A) × r + r · A × r k = (∇∇·· A) × r − A×

Here we used the definition of the vector invariant as follows   = ⊗ k · = × ( k · ) =− k · × A× r k r A × r k r A r A r k

B.3 Integral Theorems

Let ϕ(r ), a(r ) and A(r ) be continuously differentiable scalar, vector and second rank tensor fields. Let V be the volume of a bounded domain with a regular surface A(V ) and n be the outer unit normal to the surface at r . The integral theorems can be summarized as follows • Gradient Theorems   ∇ϕ dV = nϕ dA,  V A(V ) ∇∇⊗a dV = n ⊗ a dA, (B.3.4) V A(V ) ∇∇⊗A dV = n ⊗ A dA

V A(V )

• Divergence Theorems Appendix B: Elements of Tensor Analysis 347   ∇∇·· a dV = n · a dA, (B.3.5) V A(V ) ∇∇·· A dV = n · A dA

V A(V )

• Curl Theorems   ∇∇×a dV = n × a dA, (B.3.6) V A(V ) ∇∇×A dV = n × A dA

V A(V )

Based on the first equation in (B.3.5) and Eq. (B.2.2) the following formula can be derived    n · A · a dA = ∇∇·· (A · a)dV = (∇∇·· A) · a+A··(∇∇⊗a)T dV (B.3.7)

A(V ) V V

With the second equation in (B.3.5) and Eq. (B.2.3) the following relation can be obtained   r × (n · A)dA =− n · A × r dA A(V ) A(V ) =− ∇∇·· (A × r )dV (B.3.8)

 V = r × (∇∇·· A) + A× dV

V

B.4 Scalar-Valued Functions of Vectors and Second Rank Tensors

Let ψ be a scalar valued function of a vector a and a second rank tensor A, i.e. ψ = ψ(a, A). Introducing a basis ei the function ψ can be represented as follows

i ij i ij ψ(a, A) = ψ(a ei , A ei ⊗ e j ) = ψ(a , A ) 348 Appendix B: Elements of Tensor Analysis

The partial derivatives of ψ with respect to a and A are defined according to the following rule ∂ψ ∂ψ dψ = dai + dAij (B.4.9) ∂ai ∂ Aij ∂ψ ∂ψ = da · ei + dA ·· e j ⊗ ei ∂ai ∂ Aij In the coordinate-free form the above rule can be rewritten as follows   ∂ψ ∂ψ T T dψ = da · + dA ·· = da · ψ, + dA ·· (ψ, ) (B.4.10) ∂a ∂ A a A with ∂ψ ∂ψ ∂ψ ∂ψ i i j ψ, ≡ = e ,ψ, ≡ = e ⊗ e a ∂a ∂ai A ∂ A ∂ Aij

It can be verified that ψ,a and ψ,A are independent from the choice of the basis. As an example let us calculate the partial derivatives of the function

ψ(a,b, A) = a · A · b with respect to a, b and A. With

dψ = da · A · b + a · dA · b + a · A · db = da · A · b + dA ·· (b ⊗ a) + db · (a · A) T = da · ψ,a + dA ·· (ψ,A ) + db · ψ,b we obtain ψ,a = A · b,ψ,b = a · A,ψ,A = a ⊗ b

k k Let us calculate the derivatives of the functions J1(A ) = tr A , k = 1, 2, 3 with respect to A. With

2 3 J1(A) = A ·· I , J1(A ) = A ·· A, J1(A ) = A ·· (A · A) we can write

2 3 dJ1(A) = dA ·· I , dJ1(A ) = 2dA ·· A, dJ1(A ) = 3dA ·· (A · A)

Consequently

2 T 3 2T J1(A),A = I , J1(A ),A = 2A , J1(A ),A = 3A (B.4.11) Appendix B: Elements of Tensor Analysis 349

With (B.4.11) the derivatives of principal invariants of a second rank tensor A can be calculated as follows

J1(A),A = I , T J2(A),A = J1(A)I − A , (B.4.12) 2T T T −1 J3(A),A = A − J1(A)A + J2(A)I = J3(A)(A )

To find the derivative of the function ψ(A) = ψ(J1(A), J2(A), J3(A)) with respect to A we may write      ∂ψ T ∂ψ T ∂ψ T dψ = dA ·· J1(A),A + J2(A),A + J3(A),A ∂ J1 ∂ J2 ∂ J3

Taking into account (B.4.12) we obtain    ∂ψ ∂ψ ∂ψ ψ J1(A), J2(A), J3(A) = + J1 + J2 I (B.4.13) , ∂ ∂ ∂ A  J1 J2 J3 ∂ψ ∂ψ ∂ψ T T2 − + J1 A + A ∂ J2 ∂ J3 ∂ J3

B.5 Orthogonal Transformations and Orthogonal Invariants

An application of the theory of tensor functions is to find a basic set of scalar invariants for a given group of symmetry transformations, such that each invariant relative to the same group is expressible as a single-valued function of the basic set. The basic set of invariants is called functional basis. To obtain a compact representation for invariants, it is required that the functional basis is irreducible in the sense that removing any one invariant from the basis will imply that a complete representation for all the invariants is no longer possible. Such a problem arises in the formulation of constitutive equations for a given group of material symmetries. For example, the strain energy density of an elastic non-polar material is a scalar valued function of the second rank symmetric strain tensor. In the theory of the Cosserat continuum two strain measures are introduced, where the first strain measure is a polar tensor while the second one is an axial tensor, e.g. Eringen (1999). The strain energy density of a thin elastic shell is a function of two second rank tensors and one vector, e.g. Altenbach and Zhilin (1988), Altenbach et al. (2005). In all cases the problem is to find a minimum set of functionally independent invariants for the considered tensorial arguments. For the theory of tensor functions we refer to Boehler (1987). Representations of tensor functions are reviewed in Rychlewski and Zhang (1991), Zheng (1994). An orthogonal transformation of a scalar α, a vector a and a second rank tensor A is 350 Appendix B: Elements of Tensor Analysis defined by Altenbach and Zhilin (1988), Zhilin (1982)

α ≡ (det Q)ζ α, a  ≡ (det Q)ζ Q · a, A ≡ (det Q)ζ Q · A · Q T, (B.5.14) where Q is an orthogonal tensor, ζ = 0 for absolute (polar) scalars, vectors and tensors and ζ = 1 for axial ones. An example of the axial scalar is the mixed product of three polar vectors, i.e. α = a · (b ×c). A typical example of the axial vector is the cross product of two polar vectors, i.e.c = a ×b. An example of the second rank axial tensor is the skew-symmetric tensor W = a ×I , where a is a polar vector. Consider a group of orthogonal transformations S (e.g., the material symmetry transformations) characterized by a set of orthogonal tensors Q. A scalar-valued function of a second rank tensor f = f (A) is called to be an orthogonal invariant under the group S if

∀Q ∈ S : f (A) = (det Q)η f (A), (B.5.15) where η = 0 if values of f are absolute scalars and η = 1 if values of f are axial scalars. Any second rank tensor B can be decomposed into a symmetric and a skew- symmetric part, i.e. B = A + a × I , where A is a symmetric tensor and a is an associated vector. Therefore f (B) = f (A,a).IfB is a polar (axial) tensor, then a is an axial (polar) vector. For the set of symmetric second rank tensors and vectors the definition of an orthogonal invariant (B.5.15) can be generalized as follows

∀ ∈ : (  ,  ,...,  ,  ,  ,...,  ) Q S f A1 A2 An a 1 a 2 a k η (B.5.16) = (det Q) f (A1, A2,...An,a 1,a 2,...,a k )

B.6 Invariants for the Full Orthogonal Group

In Zhilin (2003) orthogonal invariants for different sets of second rank tensors and vectors with respect to the full orthogonal group are presented. It is shown that orthogonal invariants are integrals of a generic partial differential equation (basic equations for invariants). Let us present two following examples • Orthogonal invariants of a symmetric second rank tensor A are

k Ik = tr A , k = 1, 2, 3

Instead of Ik it is possible to use the principal invariants Jk defined by (A.4.11). • Orthogonal invariants of a symmetric second rank tensor A and a vector a are

k Ik = tr A , k = 1, 2, 3, I4 = a · a, I5 = a · A · a, 2 2 (B.6.17) I6 = a · A · a, I7 = a · A · (a × A · a) Appendix B: Elements of Tensor Analysis 351

In the above set of invariants only 6 are functionally independent. The relation between the invariants (so-called syzygy) can be formulated as follows      I4 I5 I6  2 =  · 3 ·  , I7  I5 I6 a A a  (B.6.18)  3 4  I6 a · A · aaaa · A · a

3 4 where a · A · a and a · A · a can be expressed by Il , l = 1,...6 applying the Cayley-Hamilton theorem (A.4.13). The set of invariants for a symmetric second rank tensor A and a vector a can be applied for a non-symmetric second rank tensor B since it can be represented by B = A + a × I , A = AT.

B.7 Invariants for the Transverse Isotropy Group

Transverse isotropy is an important type of the symmetry transformation due to a variety of applications. Transverse isotropy is usually assumed in constitutive mod- eling of fiber reinforced materials Altenbach et al. (2003a), Kröner et al. (2009), fiber suspensions, Altenbach et al. (2003b, 2007), directionally solidified alloys, Mücke and Bernhardi (2003), deep drawing sheets, Betten (1976, 2001), forgings Naumenko and Gariboldi (2014), Gariboldi et al. (2016) and piezoelectric materials Schröder and Gross (2004). The invariants and generating sets for tensor-valued functions with respect to different cases of transverse isotropy are discussed in Bruhns et al. (1999), Xiao et al. (2000) (see also relevant references therein). In what follows we analyze the problem of a functional basis within the theory of linear first order partial differ- ential equations rather than the algebra of polynomials. Let us discuss the approach proposed in Altenbach et al. (2006) for the invariants with respect to the group of transverse isotropy. The invariants will be found as integrals of the generic partial differential equations. Although a functional basis formed by these invariants does not include any redundant element, functional relations between them may exist. It may be therefore useful to find out simple forms of such relations. We show that the proposed approach may supply results in a direct, natural manner.

B.7.1 Invariants for a Single Second Rank Symmetric Tensor

Consider the proper orthogonal tensor which represents a rotation about a fixed axis, i.e.

Q(ϕm) = m ⊗ m + cos ϕ(I − m ⊗ m) + sin ϕm × I , det Q(ϕm) = 1, (B.7.19) 352 Appendix B: Elements of Tensor Analysis where m is assumed to be a constant unit vector (axis of rotation) and ϕ denotes the angle of rotation about m. The symmetry transformation defined by this tensor corresponds to the transverse isotropy, where five different cases are possible, e.g. Spencer (1987), Zheng and Boehler (1994). Let us find scalar-valued functions of a second rank symmetric tensor A satisfying the condition   f A(ϕ) = f Q(ϕm) · A · Q T(ϕm) = f (A), A(ϕ) ≡ Q(ϕm) · A · Q T(ϕm) (B.7.20) Equation (B.7.20) must be valid for any angle of rotation ϕ.In(B.7.20) only the left-hand side depends on ϕ. Therefore its derivative with respect to ϕ can be set to zero, i.e.   d f dA ∂ f T = ·· = 0 (B.7.21) dϕ dϕ ∂ A

The derivative of A with respect to ϕ can be calculated by the following rules

dA(ϕ) = dQ(ϕm) · A · Q T(ϕm) + Q(ϕm) · A · dQ T(ϕm), (B.7.22) dQ(ϕm) = m × Q(ϕm)dϕ ⇒ dQ T(ϕm) =−Q T(ϕm) × m dϕ

By inserting the above equations into (B.7.21) we obtain   ∂ f T (m × A − A × m) ·· = 0 (B.7.23) ∂ A

Equation (B.7.23) is classified in Courant and Hilbert (1989) to be the linear homoge- neous first order partial differential equation. The characteristic system of (B.7.23)is

dA = (m × A − A × m) (B.7.24) ds Any system of n linear ordinary differential equations has not more than n − 1 functionally independent integrals (Courant and Hilbert 1989). By introducing a ij basis ei the tensor A can be written down in the form A = A ei ⊗ e j and (B.7.24) is a system of six ordinary differential equations with respect to the coordinates Aij. The five integrals of (B.7.24) may be written down as follows

gi (A) = ci , i = 1, 2,...,5, where ci are integration constants. Any function of the five integrals gi is the solution of the partial differential equation (B.7.23). Therefore the five integrals gi represent the invariants of the symmetric tensor A with respect to the symmetry transformation (B.7.19). The solutions of (B.7.24)are

k ( ) = ( ) · k · T( ), = , , , A s Q smm A0 Q smm k 1 2 3 (B.7.25) Appendix B: Elements of Tensor Analysis 353 where A0 plays the role of the initial condition. In order to find the integrals, the variable s must be eliminated from (B.7.25). Taking into account the following identities

tr (Q · Ak · Q T) = tr (Q T · Q · Ak ) = tr Ak , m · Q(smm) = m, (B.7.26) (Q · a) × (Q · b) = (det Q)Q · (a × b) and using the notation Q m ≡ Q(smm) the integrals can be found as follows

( k ) = ( k ), = , , , tr A tr A0 k 1 2 3 · l · = · · l · T · m A m m Q m A0 Q m m = · l · , = , , m A0 m l 1 2 · 2 · ( × ·· ) = · · 2 · T · ( × · · T · ) m A m A m m Q m A0 Q m m Q m A0 Q m m = · 2 · T · ( · ) × ( · · ) m A0 Q m Q m m Q m A0 m = · 2 · ( × · ) m A0 m A0 m (B.7.27) As a result we can formulate the six invariants of the tensor A with respect to the symmetry transformation (B.7.19)asfollows

k Ik = tr (A ), k = 1, 2, 3, I4 = m · A · m, 2 2 · (B.7.28) I5 = m · A · m, I6 = m · A · (m × A ·· m)

The invariants with respect to various symmetry transformations are discussed in Bruhns et al. (1999). For the case of the transverse isotropy six invariants are derived in Bruhns et al. (1999) by the use of another approach. In this sense our result coincides with the result given in Bruhns et al. (1999). However, from the derivations presented here it follows that only five invariants listed in (B.7.28) are functionally independent. Taking into account that I6 is the mixed product of vectors m, A · m and A2 · m the relation between the invariants can be written down as follows ⎡ ⎤ 1I4 I5 2 = ⎣ · 3 · ⎦ I6 det I4 I5 m A m (B.7.29) 3 4 I5 m · A · mmmm · A · m

One can verify thatm · A3 · m andm · A4 · m are transversely isotropic invariants, too. However, applying the the Cayley-Hamilton theorem (A.4.13) they can be uniquely expressed by I1, I2,...I5 in the following way (Betten 1987)

3 m · A · m = J1 I5 + J2 I4 + J3, · 4 · = ( 2 + ) + ( + ) + , m A m J1 J2 I5 J1 J2 J3 I4 J1 J3 where J1, J2 and J3 are the principal invariants of A defined by (A.4.11). Let us note that the invariant I6 cannot be dropped. In order to verify this, it is enough to consider two different tensors = · · T, A and B Q n A Q n 354 Appendix B: Elements of Tensor Analysis where

Q n ≡ Q(πn) = 2n ⊗ n − I , n · n = 1, n · m = 0, det Q n = 1

One can prove that the tensor A and the tensor B have the same invariants I1, I2,...,I5. Taking into account that m · Q n =−m and applying the last identity in (B.7.26) we may write

( ) = · 2 · ( × · ) = · 2 · T · ( × · · ) I6 B m B m B m m A Q n m Q n A m 2 =−m · A · (m × A · m) =−I6(A)

We observe that the only difference between the two considered tensors is the sign 2 2 of I6. Therefore, the triples of vectors m, A · m, A · m and m, B · m, B · m have different orientations and cannot be combined by a rotation. It should be noted that the functional relation (B.7.29) would in no way imply that the invariant I6 should be “dependent” and hence “redundant”, namely should be removed from the basis (B.7.28). In fact, the relation (B.7.29) determines the magnitude but not the sign of I6. To describe yielding and failure of oriented solids a dyad M = v ⊗ v has been used in Betten (1985), Boehler and Sawczuk (1977), where the vector v specifies a privileged direction. A plastic potential is assumed to be an isotropic function of the symmetric Cauchy stress tensor and the tensor generator M . Applying the represen- tation of isotropic functions the integrity basis including ten invariants was found. In the special case v = m the number of invariants reduces to the five I1, I2,...I5 defined by (B.7.28). Further details of this approach and applications in continuum mechanics are given in Betten (2008), Boehler (1987). However, the problem state- ment to find an integrity basis of a symmetric tensor A and a dyad M , i.e. to find scalar valued functions f (A, M ) satisfying the condition

f (Q · A · Q T, Q · M · Q T) = (det Q)η f (A, M ), (B.7.30) ∀Q, Q · Q T = I , det Q =±1 essentially differs from the problem statement (B.7.20). In order to show this we take into account that the symmetry group of a dyad M, i.e. the set of orthogonal solutions of the equation Q · M · Q T = M includes the following elements

Q 1,2 =±I , v Q = Q(ϕm), m = , 3 |v| (B.7.31)

Q 4 = Q(πn) = 2n ⊗ n − I , n · n = 1, n · v = 0, where Q(ϕm) is defined by (B.7.19). The solutions of the problem (B.7.30)areat the same time the solutions of the following problem Appendix B: Elements of Tensor Analysis 355

( · · T, ) = ( )η ( , ), = , , , , f Q i A Q i M det Q i f A M i 1 2 3 4 i.e. the problem to find the invariants of A relative to the symmetry group (B.7.31). However, (B.7.31) includes much more symmetry elements if compared to the prob- lem statement (B.7.20). An alternative set of transversely isotropic invariants can be formulated by the use of the following decomposition

A = αm ⊗ m + β(I − m ⊗ m) + A pD + t ⊗ m + m ⊗ t , (B.7.32) where α, β, A pD and t are projections of A. With the projectors P 1 = m ⊗ m and P 2 = I − m ⊗ m we may write

α = m · A · m = tr (A · P 1), 1 1 β = (tr A − m · A · m) = tr (A · P ), (B.7.33) 2 2 2 A pD = P 2 · A · P 2 − βP 2,

t = m · A · P 2

The decomposition (B.7.32) is the analogue to the following representation of a vector a a = I · a = m ⊗m · a + (I −m ⊗m) · a = ψm +ττ, ψ = a · m,ττ = P 2 · a (B.7.34) Decompositions of the type (B.7.32) are applied in Bischoff-Beiermann and Bruhns (1994), Bruhns et al. (1999). The projections introduced in (B.7.33) have the follow- ing properties

tr (A pD) = 0, A pD · m = m · A pD = 0, t · m = 0 (B.7.35)

With (B.7.32) and (B.7.35) the tensor equation (B.7.24) can be transformed to the following system of equations ⎧ α ⎪ d ⎪ = 0, ⎪ ds ⎪ ⎪ dβ ⎪ = 0, ⎨ ds A (B.7.36) ⎪ dA pD = × − × , ⎪ m A pD A pD m ⎪ ds ⎪ ⎪ dt ⎪ = m × t ⎩ ds 356 Appendix B: Elements of Tensor Analysis

From the first two equations we observe that α and β are transversely isotropic invari- ants. The third equation can be transformed to one scalar and one vector equation as follows dA d(A ·· A ) db pD ·· A = 0 ⇒ pD pD = 0, = m × b ds pD ds ds

≡ · ( 2 ) = ·· with b A pD t . We observe that tr A pD A pD A pD is a transversely isotropic invariant, too. Finally, we have to find the integrals of the following system ⎧ dt ⎨⎪ = t × m, ds (B.7.37) ⎩⎪ db = b × m ds The solutions of (B.7.37)are

t (s) = Q(smm) · t 0, b(s) = Q(smm) · b0, where t 0 and b0 play the role of initial conditions. The vectors t and b belong to the plane of isotropy, i.e. t · m = 0 and b · m = 0. Therefore, one can verify the following integrals t · t = t 0 · t 0, b · b = b0 · b0, t · b = t 0 · b0,(t × b) · m = (t 0 × b0) · m (B.7.38) We found seven integrals, but only five of them are functionally independent. In order to formulate the relation between the integrals we compute

· = · 2 · , · = · · b b t A pD t t b t A pD t

For any plane tensor A p satisfying the equations A p · m = m · A p = 0 the Cayley- Hamilton theorem can be formulated as follows, see e.g. Boehler (1987)

1 A2 − (tr A )A + (tr A )2 − tr (A2 ) (I − m ⊗ m) = 0 p p p 2 p p

Since tr A pD = 0wehave

1 2A2 = tr (A2 )(I − m ⊗ m), t · A2 · t = tr (A2 )(t · t ) pD pD pD 2 pD

( 2 ) · · Because tr A pD and t t are already defined, the invariant b b can be omitted. The vector t × b is spanned on the axis m. Therefore

t × b = γm,γ= (t × b) · m, γ 2 = (t × b) · (t × b) = (t · t )(b · b) − (t · b)2 Appendix B: Elements of Tensor Analysis 357

Now we can summarize six invariants and one relation between them as follows 1 ¯ = α, ¯ = β, ¯ = ( 2 ), ¯ = · = · · , I1 I2 I3 tr A pD I4 t t t A m ¯ ¯ 2 (B.7.39) I5 = t · A pD · t , I6 = (t × A pD · t ) · m, ¯2 = ¯2 ¯ − ¯2 I6 I4 I3 I5

Let us assume that the symmetry transformation Q n ≡ Q(πn) belongs to the symmetry group of the transverse isotropy, as it was made in Betten (2008), Boehler ( ) = ( · · T) = ( ) (1987). In this case f A f Q n A Q n f A must be valid. Taking into account that Q n · m =−m we can write

α = α, β = β,  = ,  =− · A pD A pD t Q n t

¯ = ¯ = , ,..., Therefore in (B.7.39) Ik Ik , k 1 2 5 and   ¯ = (  ×  · ) · = ( · ) × · · · I6 t A pD t m Q n t Q n A pD t m ¯ = (t × A pD · t ) · Q n · m =−(t × A pD · t ) · m =−I6

Consequently

( ) = ( ¯ , ¯ ,..., ¯ , ¯ ) = ( ¯ , ¯ ,..., ¯ , − ¯ ) f A f I1 I2 I5 I6 f I1 I2 I5 I6 ⇒ ( ) = ( ¯ , ¯ ,..., ¯ , ¯2) f A f I1 I2 I5 I6

¯2 and I6 can be omitted due to the last relation in (B.7.39).

B.7.2 Invariants for a Set of Vectors and Second Rank Tensors

By setting Q = Q(ϕm) in (B.5.16) and taking the derivative of (B.5.16) with respect to ϕ results in the following generic partial differential equation   n ∂ f T k ∂ f ·· (m × A − A × m) + · (m × a ) = 0 (B.7.40) ∂ A i i ∂a j i=1 i j=1 j

The characteristic system of (B.7.40)is ⎧ ⎪ dA ⎨⎪ i = (m × A − A × m), i = 1, 2,...,n, s i i d (B.7.41) ⎪ da ⎩ j = m × a , j = 1, 2,...,k ds j 358 Appendix B: Elements of Tensor Analysis

The above system is a system of N ordinary differential equations, where N = 6n+3k is the total number of coordinates of Ai anda j for a selected basis. The system (B.7.41) has not more then N − 1 functionally independent integrals. Therefore we can formulate: Theorem B.1 A set of n symmetric second rank tensors and k vectors with N = 6n+ 3k independent coordinates for a given basis has not more than N − 1 functionally independent invariants for N > 1 and one invariant for N = 1 with respect to the symmetry transformation Q(ϕm). In essence, the proof of this theorem is given within the theory of linear first order partial differential equations (Courant and Hilbert 1989). As an example let us consider the set of a symmetric second rank tensor A and a vector a. This set has eight independent invariants. For a visual perception it is useful to keep in mind that the considered set is equivalent to

A, a, A · a, A2 · a

Therefore it is necessary to find the list of invariants, whose fixation determines this set as a rigid whole. The generic equation (B.7.40) takes the form   ∂ f T ∂ f ·· (m × A − A × m) + · (m × a) = 0 (B.7.42) ∂ A ∂a

The characteristic system of (B.7.42)is dA da = m × A − A × m, = m × a (B.7.43) ds ds This system of ninth order has eight independent integrals. Six of them are invariants of A and a with respect to the full orthogonal group. They fix the considered set as a rigid whole. The orthogonal invariants are defined by Eqs. (B.6.17) and (B.6.18). Let us note that the invariant I7 in (B.6.17) cannot be ignored. To verify this it is enough to consider two different sets , = · · T, , A a and B Q p A Q p a where Q p = I − 2p ⊗ p, p · p = 1, p · a = 0. One can prove that the invariants I1, I2,...,I6 are the same for these two sets. The only difference is the invariant I7, i.e. a · B 2 · (a × B · a) =−a · A2 · (a × A · a). Therefore the triples of vectors a, A · a, A2 · a and a, B · a, B 2 · a have different orientations and cannot be combined by a rotation. In order to fix the considered set with respect to the unit vector m it is enough to fix the next two invariants

I8 = m · A · m, I9 = m · a (B.7.44)

The eight independent transversely isotropic invariants are (B.6.17), (B.6.18) and (B.7.44). Appendix B: Elements of Tensor Analysis 359

B.8 Invariants for the Orthotropic Symmetry Group

The orthogonal tensors

Q 1 = 2n1 ⊗ n1 − I , Q 2 ≡ n2 ⊗ n2 − I , det Q 1 = det Q 2 = 1 represent the rotations on the angle π about the axes n1 and n2. These tensors are the symmetry elements of the orthotropic (orthorhombic) symmetry group. Let us find the scalar-valued functions of a symmetric tensor A satisfying the following conditions ( · · T) = ( · · T) = ( ) f Q 1 A Q 1 f Q 2 A Q 2 f A (B.8.45)

· · T Replacing the tensor A by the tensor Q 2 A Q 2 we find that

( · · · T · T) = ( · · T) = ( ) f Q 1 Q 2 A Q 2 Q 1 f Q 2 A Q 2 f A (B.8.46)

Consequently the tensor Q 3 = Q 1 · Q 2 = 2n3 ⊗ n3 − I = Q(πn3) belongs to the symmetry group, where the unit vector n3 is orthogonal to n1 and n2. Consider  three tensors Ai formed from the tensor A by three symmetry transformations i.e.,  ≡ · · T · = Ai Q i A Q i . Taking into account that Q i ni ni (no summation over i) and Q i · n j =−n j , i = j we can write

( k ) = ( k ), = , , , = , , , tr Ai tr A k 1 2 3 i 1 2 3 ·  · = · · · T · ni Ai ni ni Q i A Q i ni = ni · A · ni , i = 1, 2, 3, (B.8.47) · 2 · = · · 2 · T · ni Ai ni ni Q i A Q i ni 2 = ni · A · ni , i = 1, 2, 3

The above set of includes 9 scalars. The number can be reduced to 7 due to the obvious relations

k k k k tr (A ) = n1 · A · n1 + n2 · A · n2 + n3 · A · n3, k = 1, 2

Therefore the orthotropic scalar-valued function of the symmetric second rank tensor can be represented as a function of the following seven arguments

I1 = n1 · A · n1, I2 = n2 · A · n2, I3 = n3 · A · n3, 2 2 2 3 (B.8.48) I4 = n1 · A · n1, I5 = n2 · A · n2, I6 = n3 · A · n3, I7 = tr A

Instead of I4, I5, I6 and I7 in (B.8.48) one may use the following list of arguments (Lurie 1990) 360 Appendix B: Elements of Tensor Analysis

2 2 2 J1 = (n1 · A · n2) , J2 = (n2 · A · n3) , J3 = (n1 · A · n3) , (B.8.49)

J4 = (n1 · A · n2)(n1 · A · n3)(n2 · A · n3)

The invariants J1, J2, J3, J4 can be uniquely expressed through I1,...,I7 by use of the following relations

= 2 + + , = 2 + + , = 2 + + , I4 I1 J1 J3 I5 I2 J1 J2 I6 I3 J2 J3 = ( − 2) + ( − 2) + ( − 2) + (B.8.50) I7 2I1 I4 I1 2I2 I5 I2 2I3 I6 I3 J4

Let us note that if A is the polar tensor, then the lists of invariants (B.8.48) and (B.8.49) are also applicable to the class of the orthotropic symmetry characterized by the following eight symmetry elements

Q =±n1 ⊗ n1 ± n2 ⊗ n2 ± n3 ⊗ n3 (B.8.51)

In Sect. B.7 we derived the generic partial differential equation for the case of the transverse isotropy. Applying this approach one may find the list of functionally independent invariants among all possible invariants. Let us formulate the generic partial differential equation for the case of orthotropic symmetry. To this end let us find the scalar valued arguments of the tensor A from the following condition ( ,  ⊗  ,  ⊗  ,  ⊗  ) = ( , ⊗ , ⊗ , ⊗ ), f A n1 n1 n2 n2 n3 n3 f A n1 n1 n2 n2 n3 n3 (B.8.52)

 = · · T  = · ∀ , = where A Q A Q , ni Q ni , Q det Q 1. The symmetry group of a single dyad is given by Eqs. (B.7.31). It can be shown that the symmetry group of three dyads ni ⊗ ni includes eight elements (B.8.51). Among all rotation tensors Q the three rotations Q 1, Q 2 and Q 3 belong to the symmetry group ofni ⊗ni . Therefore Eq. (B.8.52) is equivalent to the following three equations

( · · T, ⊗ , ⊗ , ⊗ ) = ( , ⊗ , ⊗ , ⊗ ), f Q 1 A Q 1 n1 n1 n2 n2 n3 n3 f A n1 n1 n2 n2 n3 n3 ( · · T, ⊗ , ⊗ , ⊗ ) = ( , ⊗ , ⊗ , ⊗ ), f Q 2 A Q 2 n1 n1 n2 n2 n3 n3 f A n1 n1 n2 n2 n3 n3 ( · · T, ⊗ , ⊗ , ⊗ ) = ( , ⊗ , ⊗ , ⊗ ) f Q 3 A Q 3 n1 n1 n2 n2 n3 n3 f A n1 n1 n2 n2 n3 n3

Consequently, the scalar-valued arguments of A found from (B.8.52) satisfy three Eqs. (B.8.45) and (B.8.46). To derive the generic partial differential equation for invariants we follow the approach presented in Zhilin (2003, 2006). Let Q(τ) be a continuous set of rotations depending on the real parameter τ. In this case d d Q(τ) = ωω(τ) × Q(τ) ⇒ Q T(τ) =−Q T(τ) × ωω(τ), dτ dτ Q(0) = I ,ωωω(0) = ω0, where the axial vector ω has the sense of the angular velocity of rotation. Taking the derivative of Eq. (B.8.52) with respect to τ we obtain the following partial differential equation Appendix B: Elements of Tensor Analysis 361   ∂ f T (ωω × A − A × ωω) ·· ∂ A 3   (B.8.53)  ∂ f T + (ωω × n ⊗ n − n ⊗ n × ωω) ·· = 0, i i i i ∂n ⊗ n i=1 i i

(τ) = (τ) · · T(τ),  (τ) = (τ) · τ = where A Q A Q ni Q ni .For 0Eq.(B.8.53) takes the form   ∂ f T (ωω × A − A × ω ) ·· 0 0 ∂ A 3   (B.8.54)  ∂ f T + (ωω × n ⊗ n − n ⊗ n × ω ) ·· = 0 0 i i i i 0 ∂n ⊗ n i=1 i i

Taking into account the following identities

(a × A) ·· B = a · (A · B)×, B ·· (A × a) = a · (B · A)×,

Eq. (B.8.54) can be transformed to      ∂ f T ∂ f T ω · A · − · A 0 ∂ A ∂ A      3 ∂ f T 3 ∂ f T + n ⊗ n · − · n ⊗ n = 0 i i ∂n ⊗ n ∂n ⊗ n i i i=1 i i i=1 i i ×

Because ω0 is the arbitrary vector we obtain      ∂ f T ∂ f T A · − · A ∂ A ∂ A      (B.8.55) 3 ∂ f T 3 ∂ f T + n ⊗ n · − · n ⊗ n = 0 i i ∂n ⊗ n ∂n ⊗ n i i i=1 i i i=1 i i ×

The vector partial differential equation (B.8.55) corresponds to three scalar differen- tial equations. The total number of scalar arguments of the function f is 9 including 6 components of the symmetric tensor A and three parameters (e.g. three Euler angles) characterizing three dyads ni ⊗ni . Each of the scalar partial differential equations in (B.8.55) reduces the number of independent arguments by one. Therefore, the total number of independent arguments is 6. It can be shown that all seven arguments presented by Eqs. (B.8.48)or(B.8.49) satisfies (B.8.55). Because only six of them are independent, one functional relation must exist. In the case of the list (B.8.49) the functional relation is obvious. Indeed, we can write 362 Appendix B: Elements of Tensor Analysis

2 = J4 J1 J2 J3 (B.8.56)

To derive the functional relation for the list (B.8.48) one may apply Eqs. (B.8.50)to express J1,...,J4 through I1,...I7. The result should be inserted into Eq. (B.8.56).

References

Altenbach H, Zhilin PA (1988) Osnovnye uravneniya neklassicheskoi teorii uprugikh obolochek (Basic equations of a non-classical theory of elastic shells, in Russ.). Adv Mech 11:107–148 Altenbach H, Naumenko K, L’vov GI, Pylypenko S (2003a) Numerical estimation of the elastic properties of thin-walled structures manufactured from short-fiber reinforced thermoplastics. Mech Compos Mater 39:221–234 Altenbach H, Naumenko K, Zhilin P (2003b) A micro-polar theory for binary media with application to phase-transitional flow of fiber suspensions. Continuum Mech Thermodyn 15:539–570 Altenbach H, Naumenko K, Zhilin PA (2005) A direct approach to the formulation of constitutive equations for rods and shells. In: Pietraszkiewicz W, Szymczak C (eds) Shell structures: theory and applications, pp 87–90. Taylor and Francis, Leiden Altenbach H, Naumenko K, Zhilin P (2006) A note on transversely isotropic invariants. ZAMM-J Appl Math Mech/Zeitschrift für Angewandte Mathematik und Mechanik 86:162–168 Altenbach H, Naumenko K, Pylypenko S, Renner B (2007) Influence of rotary inertia on the fiber dynamics in homogeneous creeping flows. ZAMM-J Appl Math Mech/Zeitschrift für Angewandte Mathematik und Mechanik 87(2):81–93 Betten J (1976) Plastic anisotropy and Bauschinger-effect - general formulation and comparison with experimental yield curves. Acta Mech 25(1–2):79–94 Betten J (1985) On the representation of the plastic potential of anisotropic solids. In: Boehler J (ed) Plastic behavior of anisotropic solids, CNRS, Paris, pp 213–228 Betten J (1987) Tensorrechnung für Ingenieure. Springer, Berlin Betten J (2001) Kontinuumsmechanik. Springer, Berlin Betten J (2008) Creep mechanics, 3rd edn. Springer, Berlin Bischoff-Beiermann B, Bruhns O (1994) A physically motivated set of invariants and tensor gen- erators in the case of transverse isotropy. Int J Eng Sci 32:1531–1552 Boehler JP (ed) (1987) Application of tensor functions in solid mechanics. CISM Lecture Notes No. 292, Springer, Wien Boehler JP, Sawczuk A (1977) On yielding of oriented solids. Acta Mech 27:185–206 Bruhns O, Xiao H, Meyers A (1999) On representation of yield functions for crystals, quasicrystals and transversely isotropic solids. Eur J Mech A/Solids 18:47–67 Courant R, Hilbert D (1989) Methods of mathematical physics, vol 2: Partial differential equations. Wiley Interscience Publication, New York Eringen AC (1999) Microcontinuum field theories, vol I: Foundations and Solids. Springer, New York Gariboldi E, Naumenko K, Ozhoga-Maslovskaja O, Zappa E (2016) Analysis of anisotropic damage in forged Al-Cu-Mg-Si alloy based on creep tests, micrographs of fractured specimen and digital image correlations. Mater Sci Eng: A 652:175–185 Kröner C, Altenbach H, Naumenko K (2009) Coupling of a structural analysis and flow simulation for short-fiber-reinforced polymers: property prediction and transfer of results. Mech Compos Mater 45(3):249–256 Lurie AI (1990) Nonlinear theory of elasticity. North-Holland, Dordrecht Mücke R, Bernhardi O (2003) A constitutive model for anisotropic materials based on Neuber’s rule. Comput Methods Appl Mech Eng 192:4237–4255 Appendix B: Elements of Tensor Analysis 363

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A Clausius-Duhem inequality, 88, 169 Ageing, 62, 242, 264, 300 Clausius-Planck inequality, 87, 168 Aluminium alloy BS 1472, 299 Climb-plus-glide deformation mechanism, Angular velocity vector, 152, 153, 191, 232 226 Anisotropy, 26, 249 Coarse-grained region, 58, 308 damage induced, 27, 206, 254, 266, 271 Coarsening, 263 deformation induced, 27, 206 carbide precipitates, 9 induced, 27 subgrain microstructure, 9 initial, 26, 206, 271 subgrains, 4 Annihilation, 8 Coble creep, 227 Annihilation of dislocations, 4 Coffin-Manson equation, 21 Arrhenius function of the temperature, 97, Columnar region, 58, 308 304 Compressibility, 203 Arrhenius temperature dependence, 126, Conjugate variable, 91, 174 225, 243, 264 Connection in parallel, 311 Connection in series, 313 Conservation of mass, 83, 159 B Constituent Backstress, 111, 112, 243, 289, 296 inelastic-hard, 251 Bailey-Orowan recovery hypothesis, 244 inelastic-soft, 251 Balance of angular momentum, 165 Continuity, 124 Balance of momentum, 84, 85, 159, 161, Continuum mechanics modeling, 62 164, 167 Continuum micromechanics, 63 Base metal, 57, 308 Convexity condition, 260 Bauschinger effect, 114 Crack driving force, 134 Biaxial tests, 258 Creep Bulk modulus, 237 accelerated, 6 anisotropic, 13, 26, 266, 303, 315 cyclic, 13 C dynamic, 15 Cauchy formula, 163 fracture, 256 Cauchy-Green tensor, 147 grain boundary, 300 Cavitated area fraction, 263 high cyclic, 15 Cavitation, 264 McVetty-type, 269 Cavities, 123, 124, 266 multi-axial rupture data, 265 Cayley-Hamilton theorem, 154, 233 power law, 124, 260 © Springer International Publishing Switzerland 2016 365 K. Naumenko and H. Altenbach, Modeling High Temperature Materials Behavior for Structural Analysis, Advanced Structured Materials 28, DOI 10.1007/978-3-319-31629-1 366 Index

primary, 6, 240, 264, 289, 299, 300, 309 D ratcheting, 59 Damage, 62, 123, 124, 254, 255, 287, 296, recovery, 10, 247 315, 318 reduced, 6 activation, 271 rupture, 301 deactivation, 271 secondary, 6, 125, 126, 256, 269, 300, elastic-brittle, 268 309, 318 Damage driving force, 134 stationary, 6 Damage effect tensor, 269 Damage equivalent stress, 285, 286 steady-state, 196, 201, 248, 312 Damage evolution, 286, 308, 309 tertiary, 6, 105, 129, 255, 288, 292, 300, Damage evolution equation, 127, 256 309, 314 Damage mechanism, 132 transient, 13, 242 Damage parameter, 292 transversely isotropic, 315 scalar-valued, 254, 269, 285 volumetric, 203, 262 tensor-valued, 268 Creep condition, 195 Damage process Creep damage, 21, 61, 129 evolution equation, 124 Kachanov-Rabotnov model, 124, 255 Damage rate, 124 mechanim-based model, 262 Damage state variable, 124 micromechanically-consistent model, Damage tensor, 268 260 higher order, 271 model based on dissipation, 265 second rank, 269 Creep damage evolution, 18 Damage variable, 124 Creep equation, 203 tensor-valued, 254 classical, 201, 204 Darboux problem, 181 Decomposition non-classical, 201 polar, 339 non-quadratic, 216 Deformation gradient, 81, 145, 150, 191, 206 tensorial non-linear, 200 decomposition, 228 Creep failure, 257, 308 multiplicative decomposition, 229 Creep-fatigue, 18 Deformation rate tensor, 151 Creep fracture, 129, 131 Degradation function, 133, 135 Creep history, 267 Degradation of stiffness, 132 Creep in structures, 30 Diffusion creep, 226 Creep potential, 195, 196, 255, 260, 315, 316 Diffusion equation, 93, 175 classical, 199 Diffusion of vacancies, 66 Norton-Bailey-Odqvist, 196, 255, 316 Dilatation, 203, 262 Creep rate Discrete dislocation dynamics, 65 volumetric, 203 Dislocation creep, 226 Creep strain tensor, 285 Dislocation density hardening, 121 Creep strength, 127 Displacement, 80, 143 Dissipation inequality, 88, 97, 131, 169, 170, Creep test, 57, 308, 315 179, 190, 229 biaxial, 254 Dissipation power, 197, 266 multi-axial, 256, 301 Drag stress, 62, 107, 121 short-term uni-axial, 60 Ductile creep rupture, 105, 293 triaxial, 254 Dyad, 329 uni-axial, 5, 254, 262, 265, 266 Dyadic product, 332 Creep theory von Mises-Odqvist, 241 Cross section shrinkage, 104, 105 E Cross-weld specimens, 57, 308 Effective stress, 125 Cubic symmetry, 215 Effective stress concept, 125, 271 Index 367

Effective stress tensor, 255, 269, 271 Fracture, 124 Eigen-direction, 335 Frederick-Armstrong model, 114, 253 Eigen-value, 335 Free-surface effect, 64 Eigen-value problem, 335 Friction welding, 99 Elasticity tensor, 188, 190 Function of temperature, 285 Elastic range, 2 Functions of stress Elastic regime, 3 exponential law, 224 Elastic springback, 3 hyperbolic sine, 301 Elastoplasticity, 100 hyperbolic sine law, 224 Empirical modeling, 60 power, 301, 316 Energy power law, 223 free, 92, 176, 179, 184, 186, 189, 190 power law breakdown equation, 224 internal, 85, 166, 175 souble power law, 224 kinetic, 85, 166 total, 85, 166 Energy balance, 87, 93, 95, 96, 167, 168, G 175, 189 Grain boundary cavitation, 64 Energy supply, 167 Grain boundary sliding, 53, 64, 227 Engineering stress, 103 Grain boundary strengthening, 52 Entropy inequality, 88, 169 Grain size, 309 Equivalent creep rate von Mises, 24 Equivalent creep strain, 263 H von Mises, 24 Hall-Petch effect, 52 Equivalent deformation rate Hamilton operator, 144 Hardening, 5, 13, 53, 62, 129, 226, 250, 264, von Mises, 194 288, 289, 315, 318 Equivalent stress, 200, 202, 204, 214, 257, anisotropic, 249 263, 315 cyclic, 16, 59, 111 damage, 256, 258 initial, 249 generalized, 258 isotropic, 242 von Mises, 22, 194, 200, 204, 243, 258, kinematic, 242, 248 288, 302 Hardening parameter, 107 Hardening rate, 109 Hardening regime, 2 F Hardening rule, 242 Fatigue damage, 16, 21, 59, 61 Hardening variable, 122, 248, 300 surface induced, 52 Harmonic stress variation, 14 Fatigue failure, 16 Heat-affected zone, 56, 57, 265, 308, 309 Fatigue life, 18, 19 Heat capacity, 93, 175, 189 Fatigue strength, 16 Heat equation, 93, 175, 189 Fine-grained region, 58, 308 Heat flow, 89, 167 First law of dynamics, 84 Heat flow vector, 168 First principal direction, 255 Heat flux, 97, 175 First principal stress, 286, 295 Heat flux vector, 191 Flow potential, 195, 243 Heat transfer equation, 96, 189 Flow rule, 195, 196, 202 Helmholtz free energy, 89, 170 Odqvist, 199 Heterogeneous structure, 309 Fluid High-temperature inelasticity, 64 incompressible, 193 linear viscous, 98 non-linear viscous, 98, 192 I non-Newtonian, 193 Identity tensor, 333 Fourier law of heat conduction, 92, 175 Independent variable, 91, 174 368 Index

Inelastic-hard constituent, 116 poly-crystalline, 227 Inelastic regime, 3 transversely-isotropic, 179, 185 with hardening, 3 Material processing, 12 with necking, 3 Materials science modeling, 61 Inelastic strain rate, 61 Material symmetry, 67, 176, 177, 315 Internal state variable, 242, 300, 315 Material symmetry group, 188, 315 Invariant Maximum tensile stress, 255, 286 cubic, 202 Maxwell model, 100 linear, 202 Mean dislocation density, 121 principal, 155, 183, 185 Mean dislocation density model, 109 quadratic, 202 Mechanical action, 160 Isochronous rupture surface, 258 Mechanical power, 86, 99, 108, 110, 131, Iso-strain approach, 252 166, 175, 190, 227 Iso-strain concept, 311 Mechanism-based model, 262, 299 Iso-stress approach, 99, 108, 131, 231, 313 Method of characteristics, 180 Micromechanically-consistent model, 260 Micromechanical modeling, 63 K Microstructural analysis, 67 Kachanov-Rabotnov model, 310 Minimum creep rate, 126, 304 Kachanov’s continuity, 124 Mirror reflection, 176, 177 Kinematic hardening model, 242 Mixture model, 288 Mixture rule, 116, 252 Momentum, 83, 159 L angular, 165 Lamé constants, 190, 237 Monkman-Grant relationship, 60, 129 LCF test, 114 Motion, 80, 143 Least squares method, 284 Multi-pass weld, 56, 309 Linear damage summation rule, 61 Linear hardening rule, 112 Linear stress-strain relationship, 2 N Loading Nabarro-Herring creep, 227 combined, 26 Nabla operator, 144 complex, 242 Necking, 3, 5 cyclic, 13 Net-stress, 125 multi-axial, 254 Net-stress tensor, 269 multi-axial non-proportional, 271 Neumann principle, 177 non-isothermal, 51 Newton law of cooling, 93, 175 non-proportional, 26, 27, 248, 254 Non-mechanical energy supply, 86, 166 proportional, 26 Norton-Bailey law, 248 simple, 248 Norton-Bailey type potential, 201 thermo-mechanical cyclic, 39 Notation Low cycle fatigue, 16 component, 323 coordinate-free, 323 direct, 173, 323 M index, 323 Martensitic transformation, 4 symbolic, 323 Mass, 83 Notch stress effect, 64 Material aelotropic, 177 anisotropic, 177, 206 O crystalline, 198 Offset yield point, 3, 5 cubic symmetry, 215 Orientation averaging, 271 Hencky, 231 Orientation distribution function, 271 isotropic, 179, 182, 190 Orientation tensor, 271 Index 369

Orowan hardening, 121 S Orowan stress, 121 Saturation, 4 Orthotropic symmetry, 211 Saturation stress, 117 Overstress, 117 Sdobyrev criterion, 257 Shear flow, 193 Shear modulus, 190, 298 P Softening, 4, 5, 13, 250, 253, 287, 290, 296 Parent material, 309 cyclic, 16, 59 Particle hardening, 121 Specific dissipation work, 265 Peach-Köhler force, 65 Specimen Permanent plastic strain, 3 circumferentially notched, 22 Phase mixture model, 250 cruciform, 22 Physical symmetry, 176, 177 cylindrical, 5 Piping system, 308 standard, 2 Plasticity thin-walled tube, 22 rate-independent, 98, 195 Spectral form, 178 rigid, 98 Spectral representation, 148, 153 Plastic pre-strain, 249 Spin tensor, 152, 153 Poisson’s ratio, 190 left, 152 Polar decomposition, 152, 231 State variables Position vector, 142 hidden, 62 Power-law function of stress, 97 internal, 62 Poynting-Swift effect, 194, 200 Steady-state flow regime, 4 Prandtl model, 100 Steel 13CrMo4-5, 286 Pre-damage, 267 Strain, 81 Pre-loading, 267 creep, 7 Pressure vessel, 308 cyclic, 18 Pressurized cylindrical shell, 302 elastic, 7 Pre-straining, 267 Hencky, 94 Principal direction, 335 infinitesimal, 150 Principal value, 335 local, 81 Principal vector, 335 local normal, 146 Projector, 333 Strain deviator, 237 Strain equivalence principle, 125, 256, 269, 271 R Strain hardening, 240, 242 Ratchetting, 15 Strain hardening model, 240 Rate of plastic strain, 111 Strain local shear, 147 Recovery, 4, 8, 13, 62, 226, 288 Strain tensor Redistribution, 302 additive split, 236 Relaxation, 10, 302 Almansi, 148 Relaxation-fatigue, 18 Biot, 148 Relaxation test, 12 Cauchy-Green, 148 Representative volume element, 64 left, 148 Requirements for the modeling, 66 Green-Lagrange, 149 Resistance function, 133, 135 Hencky, 148, 156, 158, 230 Rigid body rotation, 186 right Cauchy-Green, 181 Rotation, 315 Strain trajectory, 26, 256 infinitesimal, 150 Strengthening Rotation about a fixed axis, 176 mechanism, 54 Rotation tensor, 148, 152, 153, 181, 191, 232 Stress linearized, 150 effective, 125 Rupture time, 258 engineering, 89 370 Index

hydrostatic, 193 Thermal conductivity, 92, 132, 179 Kirchhoff, 95 tensor, 175, 177 mean, 193 Thermal expansion Stress concentration, 53 coefficient, 190 Stress deviator, 193 tensor, 188, 190 active, 288, 297 Thermal expansion coefficient, 95 Stress function Thermally-activated dislocation climb, 66 hyperbolic sine, 264 Thermodynamic force, 91, 174 power law, 243 Thermoelasticity, 101 Stress redistribution, 308, 309, 312, 314 Thermo-elasto-plasticity, 236 Stress relaxation, 10, 12 Thermo-mechanical fatigue, 19 Stress state effect, 25, 27, 254, 266 Time hardening, 240, 242, 299 Stress state index, 265, 300 Time hardening function, 299 Stress tensor Time hardening model, 240 active part, 243 Time to fracture, 127 additional part, 243 Transient heat transfer, 45 Cauchy, 163, 180, 183, 185, 190–192, Transverse isotropy, 207 230, 232 Transversely loaded rectangular plate, 302 deviatoric part, 22, 227, 243 Type 316 stainless steel, 285 dissipative, 227, 230, 231 effective part, 243 Piola-Kirchhoff, 164, 168, 182 U second, 182 Uni-axial tension, 316 quasi-conservative, 227, 229 Unified model, 44 spherical part, 243 Uniform shear, 316 translation part, 243 Stress vector, 160 V Stretch, 81 Variable local, 81, 146 hidden, 107 principal, 148, 183 internal state, 107, 118, 122, 132 Stretch tensor, 153 Vector left, 148 axial, 324 right, 148 polar, 324 St. Venant model, 98 spin, 325 Symmetries of microstructure, 67 Velocity field, 81, 144 Symmetry group, 176, 187 Velocity gradient, 151, 152, 190, 191, 206 physical, 230 Viscoelasticity, 100 Symmetry transformation, 176, 178 Viscoplasticity, 98 Void, 123, 124, 254, 260, 270 Void growth, 64 T Volume constancy, 23 Taylor hardening, 121 Volume fraction, 252 Tensile test Von Mises-Odqvist type inelastic flow, 237 uni-axial, 2 Von Mises type potential, 200 Tensor Voronoi tessellation, 64 orthogonal, 337 Vorticity vector, 151 second rank, 329 singular, 334 skew-symmetric, 333 W symmetric, 332 Weakening, 53 Test Weissenberg effect, 193 multi-axial, 22 Welded joint, 308, 309 stress controlled, 23 Weld metal, 57, 265, 283, 308, 309 Index 371

Weld metal 9CrMoNbV, 308, 316 Yield limit, 3 Work hardening, 264 Yield point, 2 lower, 3 Y upper, 3 Yield condition, 195 Young’s modulus, 190, 298