tensor notation[1]

8/10/2019 Tensor Notation[1]

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2 Tensor Notation

It will be convenient in this monograph to use the compact notation oftenreferred to as indicial or index notation . It allows a strong reduction in thenumber of terms in an equation and is commonly used in the current litera-ture when stress, strain, and constitutive equations are discussed. Therefore,a basic knowledge of the index notation is helpful in studying continuummechanics, especially constitutive modelling of materials. With such a no-tation, the various stress-strain relationships for materials under multi-axialstates of stress can be expressed in a compact form. Thus, greater attentioncan be paid to physical principles rather than to the equations themselves.A short outline of this notation should therefore be given in the following.In comparison, some expressions or equations shall also be written in sym-bolic or matrix notation , employing whichever is more convenient for thederivation or analysis at hand, but taking care to establish the interrelation-ship between the two distinct notations.

2.1 Cartesian Tensors

We consider vectors and tensors in three-dimensional E UCLID ean space.

For simplicity, rectangular Cartesian coordinates xi , i = 1 , 2, 3, are usedthroughout. Results may, if desired, be expressed in terms of curvlinear coor-dinate systems by standard techniques of tensor analysis (B ETTEN , 1987c),as has been pointed out in Section 2.2 and used in Chapter 5.

In a rectangular Cartesian coordinate system, a vector V can be decom-posed in the following three components

V = ( V 1, V 2, V 3) = V 1e 1 + V 2e 2 + V 3e 3 , (2.1)

where e 1, e 2, e 3 are unit base vectors:

e i e j = ij . (2.2)


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10 2 Tensor Notation

In 2.2 the Symbol ij is known as K RONECKER s delta. Thus, the set of unitvectors, {e i}, constitutes an orthonormal basis .

A very useful notational device in the manipulation of matrix, vector,

and tensor expressions is the summation convention introduced by E INSTEIN(1916):

Whenever an index occurs twice in the same term, summation overthe values 1, 2, and 3 of that index is automatically assumed, and thesummation sign is omitted.

Thus, the decomposition (2.1) can be written in the more compact form

V = V i e i V k e k . (2.1*)The repeated index i or k in (2.1*) is often called summation index or dummyindex because the choice of the letter for this index is immaterial. However,we have to notice that an index must not appear more than twice in the sameterm of an expression or equation. Otherwise, there is a mistake. An expres-sion such as A ijk Bkk would be meaningless.

Consider a sum in which one of the repeated indices is on the K RO -NECKER

delta, for example, ik Aij = 1k A1 j + 2k A2 j + 3k A3 j .

Only one term in this sum does not vanish, namely the term in whichi = k = 1 , 2, 3. Consequently, the sum reduces to

ik Aij = Akj .

A similar example is: ij V j = V i . Notice that the summation, involving oneindex of the K RONECKER delta and one of another factor, has the effect of substituting the free index of the delta for the repeated index of the otherfactor. For this reason the K RONECKER delta could be called substitutiontensor (B ETTEN , 1987c).

Another example of the summation convention is the scalar product (dotproduct, inner product) of two vectors U and V . It can be written as

U

V = U 1V 1 + U 2V 2 + U 3V 3

U k V k . (2.3)

Note that the expression

(Aii )2 (A11 + A22 + A33)2


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2.1 Cartesian Tensors 11

is different from the sum

A2ii A211 + A222 + A233 ,where the rst one is the square of the sum A ii , while the second one is thesum of the squares.

The vector product (cross product) has the following form:

C = A B =e 1 e2 e3

A1 A2 A3

B1 B2 B3ijk e i A j Bk . (2.4a)

Its components are given by

C i = ijk A j Bk . (2.4b)

In (2.4a,b) the alternating symbol ijk (also known as permutation symbolor third-order alternating tensor) is used. It is dened as:

ijk= +1 if ijk represents an even permutation of 123 ;= 0 if any two of ijk indices are equal ;= 1 if ijk represents an odd permutation of 123 .

(2.5)

It follows from this denition that ijk has the symmetry properties

ijk = jki = kij = ikj = jik = kji . (2.6)The triple scalar product can also be calculated by using the alternating

symbol:

(A B ) C =A1 A2 A3

B1 B2 B3

C 1 C 2 C 3

= ijk Ai B j C k . (2.7)

The following relation between the K RONECKER delta and the alternat-ing tensor is very important and useful (B ETTEN , 1987c):

ijk pqr = ip iq ir

jp jq jr

kp kq kr3! i[ p] j [q] k [r ] , (2.8)


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where on the right-hand side the operation of alternation is used. This pro-cess is indicated by placing square brackets around those indices to whichit applies, that is, the three indices pqr are permutated in all possible ways.

Thus, we obtain 3! terms. The terms corresponding to even permutations aregiven a plus sign, those which correspond to odd permutations a minus sign,and they are then added and divided by 3!.

From (2.8) we immediately obtain the contraction

ijk pqk = ip jq iq jp , (2.9a)for instance, i.e., the tensor of rank six in (2.8) is reduced to the fourth-ordertensor (2.9a). Other contractions are

pqi pqj = 2 ij and pqr pqr = 6 , (2.9b,c)

for instance.Now let us consider a coordinate transformation, i.e., we introduce a new

rectangular right-handed Cartesian coordinate system and new base vectorse i , i = 1 , 2, 3. The new system may be regarded as having been derivedfrom the old by a rigid rotation of the triad of coordinate axes about the same

origin. Let a vector V have components V i in the original coordinate systemand components V i in the new system. Thus, one can write:

V = V i e i = V

i e

i . (2.10)

We denote by a ij the cosine of the angle between e

i and e j , so that

a ij cos(e

i , e j ) = e

i e j , (2.11)i.e., a ij are the direction cosines of e i relative to the rst coordinate system,or, equivalently, aij are the components of the new base vectors e

i , in therst system. Thus

e i = a ij e j . (2.12)

It is geometrically evident that the nine quantities a ij are not independent.Since e i are mutually perpendicular unit vectors,

e i

e j = ij , (2.13)

we arrive at

e i e

j = a ip e p a jr e r = a ip a jr e p e r = a ip a jr pr = a ip a jp


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by considering (2.2) and (2.12). Hence

a ip a jp = ij or in matrix notation aa t = , (2.14a)

where a t is the transpose of the matrix a . Because of the symmetry ij = ji ,the result (2.14a) represents a set of six relations between the nine quantitiesa ij . Similarly to (2.14a), we nd:

a pi a pj = ij or in matrix notation a t a = . (2.14b)

It follows immediately from (2.14a,b) that |a ij | = 1 and, furthermore,that the transpose a t is identical to the inverse a 1. Thus, the matrix a is

orthogonal , and the reciprocal relation to (2.12) is

e i = a ji e

j . (2.15)

Inserting (2.15) or (2.12) into (2.10), we arrive at

V i = a ij V j or V i = a ji V

j , (2.16a,b)

respectively. In particular, if V is the position vector x of the point P relative

to the origin, then

xi = a ij x j and xi = a ji x

j , (2.17a,b)

where xi and xi are the coordinates of the point P in the new and originalcoordinate systerns, respectively.

Now, let us consider a vector function

Y = f (X ) or in index notation: Y i = f i (X p) , (2.18a,b)

where X is the argument vector which is transformed to another vector Y .The simplest form is a linear transformation

Y i = T ij X j , (2.19)

where T ij are the cartesian components of a second-order tensor T , alsocalled second-rank tensor , i.e., a second-order tensor can be interpreted as alinear operator which transforms a vector X into an image vector Y .

In extension of the law (2.16a,b), the components of a second-order ten-sor T transform according to the rule

T ij = a ip a jq T pq or T ij = a pi aqj T

pq , (2.20a,b)


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which can be expressed in matrix notation:

T = aT a t or T = a t T a . (2.21a,b)

Second-rank tensors play a central role in continuum mechanics, for in-stance, strain and stress tensors are second-order tensors. It is sometimesuseful in continuum mechanics, especially in the theory of plasticity or increep mechanics, to decompose a tensor into the sum of its deviator and aspherical tensor as follows:

T ij = T ij + T kk ij / 3 . (2.22)

For instance, the stress deviator

ij := ij kk ij / 3 (2.23)is responsible for the change of shape ( distortion ), while the hydrostaticstress kk ij / 3 produces volume change without change of shape in anisotropic continuum, i.e., in a material with the same material properties inall directions. Clearly, a uniform all-around pressure should merely decreasethe volume of a sphere of material with the same strength in all directions.However, if the sphere were weaker in one direction, that diameter would bechanged more than others. Thus, hydrostatic pressure can produce a changeof shape in anisotropic materials.

The deviator (2.23) is often called a traceless tensor , since its tracetr kk is identical to zero.

A second-order tensor has three irreducible invariants

J 1

ij T ji = T jj

T kk , (2.24a)

J 2 T i [i]T j [ j ] = ( T ij T ji T ii T jj )/ 2 , (2.24b)J 3 T i [i]T j [ j ]T k [k ] = det( T ij ) |T ij | , (2.24c)

which are scalar quantities appearing in the characteristic equation

det( ij T ij ) = 3 J 12 J 2 J 3 = 0 . (2.25)In (2.24b,c) the operation of alternation is used and indicated by placingsquare brackets around those indices to which it applies. This process is al-ready illustrated in the context with (2.8).

We read from (2.24a,b,c): The rst (linear) invariant J 1 is the trace of T ,the second (quadratic) invariant J 2 is dened as the negative sum of the three


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principal minors of order 2, while the third (cubic) invariant is given by thedeterminant of the tensor. A deviator has only two non-vanishing invariants:

J 2 = T ij T ji / 2 , J 3 = det( T ij ) . (2.26a,b)Remark : Because of the denition (2.24b) the second invariant (2.26a) of the deviator is always positive . Therefore, J 2 is dened as the negative sumof the principal minors in this text.

The invariants (2.24a,b,c) can be expressed through the principal valuesT I , T II , T III of the Tensor T , i.e., the elementary symmetric functions of the three arguments T I , . . . , T III are related to the irreducible invariants(2.24a,b,c) as follows:

T I + T II + T III = J 1 , (2.27a)T I T II + T II T III + T III T I = J 2 , (2.27b)

T I T II T III = J 3 . (2.27c)

After some manipulation one can arrive from (2.22), (2.24), and (2.26) atthe relations

J 2 = J 2 + 13J 21 , J 3 = J 3 + 13J 1J 2 + 227J

31 . (2.28a,b)

In the theory of invariants the H AMILTON -C AYLEY theorem plays animportant role. It states that

T (3)ij J 1T (2)ij J 2T ij J 3 ij = 0 ij , (2.29)

where T (3)ij

T ip T pr T rj and T

(2)ij

T ip T pj are, respectively, the third and

the second power of the tensor T . Thus, every second-order tensor ( linear operator ) satises its own characteristic equation (2.25). B ETTEN (1987c;2001c) has proposed extended characteristic polynomials in order to ndirreducible invariants for fourth-order tensors (Section 4.3.2).

By analogy with (2.19), a fourth-order tensor A , having 81 componentsAijkl , can be interpreted as a linear operator:

Y ij = A ijkl X kl (2.30)

where X kl and Y ij are the cartesian components of the second-rank tensorsX and Y . For example, the constitutive equation

ij = E ijkl kl (2.31)


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describes the mechanical behavior of an anisotropic linear-elastic material,where ij are the components of C AUCHY s stress tensor, ij are the compo-nents of the innitesimal strain tensor, and E ijkl are the components (elatic

constants) of the fourth-order material tensor characterising the anisotropyof the material.

2.2 General Bases

In the foregoing Section we have introduced an orthonormal basis {e i}char-acterized by (2.2), i.e., we have restricted ourselves to rectangular cartesiancoordinates. This is the simplest way to formulate the basic equations of continuum mechanics and the constitutive or evolutional equations of var-ious materials. However, solving particular problems, it may be preferableto work in terms of more suitable coordinate systems and their associatedbases.

In particular, cylindrical polar coordinates are useful for congurationswhich are symmetric about an axis, e.g., thick-walled tubes in Chapter 5. An-other example is the system of spherical polar coordinates, which should bepreferred when there is some symmetry about a point. Thus, it is useful to ex-press the basic equations of continuum mechanics and the constitutive lawsof several materials in terms of general (most curvlinear) coordinates. Thus,in the following some fundamentals of curvlinear tensor calculus should bediscussed.

Letx i = x i ( p) i = i (x p) (2.32)

be an admissible transformation of coordinates with J ACOBI ans

J |x i / j | and K | i /x j | ,which does not vanish at any point of the considered region, then J K = 1 .Further important properties of admissible coordinate transformations arediscussed, for instance, by S OKOLNIKOFF (1964) and B ETTEN (1974) inmore detail.

The coordinates xi in (2.32) referred to a right-handed orthogonal carte-sian system of axes dene a three-dimensional E UCLID ean space , whilethe i are curvlinear coordinates . Because of the admissible transforma-tion (2.32), each set of values of xi corresponds a unique set of values of i , and vice versa. The values i therefore determine points in the denedthree-dimensional E UCLID ean space. Hence we may represent our space by


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2.2 General Bases 17

the variables i instead by the cartesian system xi , but the space remains, of course, E UCLID ean.

In this Section, we consider symbols characterized by one or several in-

dices which may be either subscripts or superscripts , such as Ai , Ai, Bij ,

B ij , B i j , etc., where indices as superscripts are not taken as powers. Some-times it is necessary to indicate the order of the indices when subscripts andsuperscripts occur together. In that case, for example, we write Ai j wherethe dot before j indicates that j is the second index while i is the rst one.

As explained later, the values Ai and Ai can be considered as the co-variant and the contravariant components , respectively, of the vector A .However, the position of the indices on the kernel letters x and in the

transformation of coordinates (2.32) has nothing to do with covariance orcontravariance and is therefore immaterial. In this context we refer to thefollowing remarks of other authors:

FUN G (1965, p38): The differential di is a contravariant vec-tor, the set of variables i itself does not transform like a vector.Hence, in this instance, the position of the index of i must beregarded as without signicance.

GREEN /Z ERNA (1968, pp5/6): The differentials di transformaccording to the law for contravariant tensors, so that the posi-tion of the upper index is justied. The variables i themselvesare in general neither contravariant nor covariant and the positionof their index must be recognized as an exception. In future the in-dex in non-tensors will be placed either above or below accordingto convenience. For example, we shall use either i or i .

GREEN /A DKINS (1970, p1): The position of the index on coordi-nates x i , yi and i is immaterial and it is convenient to use either

upper or lower indices. The differential involving general curvi-linear coordinates will always be denoted by di since di has adifferent meaning and is not a differential. For rectangular coor-dinates, however, we use either dxi , dxi for differentials, sincedx i = dx i .

MALVERN (1969, p603): Warning : Although the differentialsdxm are tensor components, the curvilinear coordinates xm arenot, since the coordinate transformations are general functional

transformations and not the linear homogeneous transformationsrequired for tensor components.

According to the above remarks, it is immaterial, if we write i or i . Since the differentials d i transform corresponding to the law for con-


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travariant tensors, the position of the upper index, i , is justied. Thus, thedifferential dT of a stationary scalar eld T = T ( i ) should be written asdT = ( T/ i )d i .

The position of a point P can be determined by xi or, alternatively, by i

as illustrated in Fig.2.1.

P

3 x

2 x

1 x

e1

e2

e3 R g1

g2

g3

Fig. 2.1 Orthonormal and covariant base vectors

The position vector R of any point P (x i ) can be decomposed in the form

R = xk e k . (2.33)

Since the orthonormal base vectors e i are independent of the position of thepoint P (x i ), we deduce from (2.33) that

R /x i = e k (x k /x i ) = e k ki = e i , (2.34)

i.e., the orthonormal base vectors can be expressed by partial derivatives of the position vector R with respect to the rectangular cartesian coordinatesx i .

Analogous to (2.34), we nd

R i = ( R / x p ) x p i = e p x p i = g i , (2.35)

i.e., the geometrical meaning of the vector R / i is simple: it is a base vec-tor directed tangentially to the i -coordinate curve. From (2.35) we observe


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that the covariant base vectors g i are no longer independent of the coordi-nates i , in contrast to the orthonormal base vectors. Furthermore, they neednot be mutually perpendicular or of unit length.

The inverse form to (2.35) is given by

e i R / x i = ( R / p) ( p/ x i ) = g p ( p/ x i ) . (2.36)Besides the covariant base vectors dened in (2.35), (2.36), a set of con-

travariant base vectors g i is obtained from the constant unit vectors e i e ias follows

g i = i x p e p ei = ( x i / p) g p . (2.37)

This set of contravariant base vectors, g i , are often called the dual or recip-rocal basis of the covariant basis g i , and they are denoted by superscripts.In the special case of rectangular cartesian coordinates, the covariant andcontravariant base vectors are identical ( e i e i ).

From (2.35), (2.37) and considering the orthonormal condition we arriveat the relation

g i g j = ij ji . (2.38)

between the two bases. For example, the contravariant base vector g 1 is or-thogonal to the two covariant vectors g 2 and g 3. Since these vectors directedtangentially to the 2- and 3-curves, the contravariant base vector g 1 is per-pendicular to the 1-surface (Fig.2.2).

g1

g2g3

-surface

-curve

-surface

e1e2

e3

-curve

-surface-curve

x i= x i ( )

covariant basisorthonormal basise

i e

j=

ij g

i g

j = g

ij

p

Fig. 2.2 Coordinate surfaces and base vectors


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In addition to the relation (2.38) one can form the following scalar prod-ucts:

g i g j gij , g i g j gij . (2.39a,b)These quantities, gij and gij , are called the covariant and contravariant met-ric tensors , respectively, and, because of (2.38), the K RONECKER tensor i j can be interpreted as a mixed metric tensor . The metric tensors (2.38),(2.39a,b) are symmetric since the scalar products of two vectors are commu-tative.

Inserting the base vectors (2.35) or (2.37) into (2.39a,b), respectively, wecan express the covariant or contravariant metric tensors as

gij = x k / i x k / j gij = i /x k j /x k ,

(2.40a,b)from which we immediately arrive at the reciprocal relation

gik g jk = ji . (2.41)

This result represents a system of linear equations from which the con-travariant metric tensor can be calculated accordingly

gij = Gij /g with Gij (1)i+ j U (gij ) , (2.42)when the covariant metric tensor is given, where Gij is the cofactor of theelement gij in the determinant g |gij |. From the reciprocal relation (2.41)we deduce the determinant of the contravariant metric tensor as

|gij

| = 1 /g.

The magnitudes of the covariant and contravariant base vectors followdirectly from (2.39a,b):

|g i | = g i g ( i ) = gi ( i) , g i = g i g ( i) = gi( i ) , (2.43a,b)where the index is not summed, as indicated by parentheses.An increment dR of the position vector R in Fig.2.1 can be decomposed

in the following ways

dR = e i dx i = g i d i = g i d i . (2.44)

Forming the scalar product


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g j e i dx i = g j g i d i = ji d

i = d j ,

and then inserting (2.37), we nd the transformation

d j = j / x p e p e i dx i = j / x p dx por

d i = i / x p dx p , (2.45a)

i.e., the d i in (2.44) transforms contravariant (matrix i /x p) and can beidentied with the usual total differential of the variable i , so that the use of upper index is justied.

In a similar way one can nd the covariant transformationd i = x p i dx p , (2.45b)

which essentially differs from (2.45a) and cannot be interpreted as the totaldifferential.

Using (2.44) with (2.39a,b) the square of the line element ds can be writ-ten in the form

ds2

= dR dR = dxk dxk = gij d id

j

= gij

d i d j . (2.46)

Hence, the reason for the term metric tensor gij is account for. In additionto (2.46), the mixed form ds2 = d k d k is also possible. This form followsimmediately from (2.46) because of the rule of raising (gij A j = Ai ) andlowering (gij A j = A i ) the indices.

Considering the decompositions of two vectors

A = Ak g k = Ak g k , B = B k g k = Bk g k , (2.47a,b)

we then can represent the scalar product in the following forms

A B = gij Ai B j = Ak Bk = Ak B k = gij Ai B j , (2.48)from which we can determine the length of a vector B A as

|A | A =

gij Ai A j =

Ak Ak =

Ak Ak =

gij Ai A j . (2.49)

Alternatively, the scalar product (2.48) can be expressed by AB cos , so thatthe angle between two vectors can be calculated from the following formula:

cos = gij Ai B j /( AB ) . (2.50)


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In particular, the angle 12 between the 1-curve and 2-curve, i.e., betweenthe covariant base vectors g1 and g2 in Fig.2.2 can be determined in thefollowing way

g 1 g 2

g12

|g1 ||g 2|cos 12 cos 12 =

g12 g11 g12 , (2.51a)

where the relations (2.39a) and (2.43a) have been used. Cyclic permutationsyield

cos 23 = g23 g22g33 and cos 31 =

g31 g33g11 . (2.51b,c)

From the result (2.51a,b,c) we deduce the following theorem:A necessary and sufcient condition that a given curvlinear co-ordinate system be orthogonal is that the gij vanish for i = j atevery point in a region considered, i.e., the matrix (gij ) has thediagonal form.

According to (2.47a), there are two decompositions of a vector A : Thecontravariant components Ak are the components of A in the directions of the covariant base vectors gk , while the covariant components Ak are thecomponents of A corresponding with the contravariant base vectors g k , asillustrated in Fig.2.3.

g gA1

g1

g1A

A

g g2

g2

A

gA2

11

1

2

2

2

Fig. 2.3 Decomposition of a vector in covariant and contravariant components

A relation between the covariant and contravariant components, Ai andAi , can be achieved by forming the scalar products of (2.47a), respectively,with the base vectors g i and g i according to


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Ai = gik Ak and Ai = gik Ak . (2.52a,b)

These results express the rule of lowering and raising the indices , respec-tively. This operation can also be applied to the base vectors in order to ndrelations between the covariant and contravariant bases :

g i = gik g k and g i = gik g k . (2.53a,b)

Comparing the decompositions (2.47a) with the decompositions

A = Ak e k

Ak e k (2.54)

with respect to the orthonormal basis e k e k , and considering (2.36), (2.37),one arrives at the folllowing relations between the covariant or contravariantcomponents, A i or A i , and the cartesian components Ak Ak of the vectorA :

Ai = x p i A p , Ai = i / x p A p . (2.55a,b)

We see in (2.55a,b) the same transformation matrices (x p/ i ) and( i /x p), respectively, as in (2.35) and (2.37).

In the tensor analysis the Nabla operator , sometimes called del op-erator , plays a fundamental role. It is a differential operator, and can bedecomposed with respect to the orthonormal basis:

= e ii e i

x i. (2.56a)

Substituting e i e i in (2.56a) by (2.37), and utilizing the chain rule

x i=

p

x i

p ,

we nd the decomposition of the Nabla operator with respect to the con-travariant basis:

= g k

k . (2.56b)

The divergence of a vector eld A is dened as the scalar product of theNabla operator (2.56b) and the vector (2.47a):


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div A = A = Ai / i + Ai j.ij Ai |i . (2.57)

The functions T k.ij in (2.57) are called the C HRISTOFFEL symbols of the sec-ond kind . They are the coefcients in the following decompositions

g i / j k.ij g k kij

g k = ijk g k , (2.58a)

g i / j i.kj g k ikj

g k . (2.58b)

Because of the denition (2.58a), and considering (2.35), (2.36), the C HRISTOF -FE L symbols of the second kind can be expressed in the form

k.ij = k

x p 2x p

i j . (2.59a)

They are symmetric with respect to the lower indices ( i, j ) and can be relatedto the metric tensors in the following way

k.ij

kij =

12g

klgil

j+ g jl

i

gij l

. (2.59b)

In addition, the C HRISTOFFEL symbols of the rst kind are given by

kij [ij,k ] = [ ji, k ] = 12

gik j + g jk i gij k .(2.60)

Comparing the two sets (2.59b) and (2.60), we read gkl lij k.ij in agree-ment with the rule of raising the indices.

Evidently there are n = 3 distinct C HRISTOFFEL symbols of each kindfor each independent gij , and, since the number of independent gij s is

n (n + 1) / 2 ,

the number of independent C HRISTOFFEL symbols is

n2(n + 1) / 2 .

Note that the C HRISTOFFEL symbols, in general, are not tensors. This isvalid also for the partial derivatives Ai / j and Ai / j with respect tocurvlinear coordinates.


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However, the derivative

A / j = A i | j g i = A i | j g i (2.61)is a tensor, where the expressions

Ai | j A i / j + Ak i.kj and Ai | j A i / j Ak k.ij (2.62a,b)are the covariant derivatives of the contravariant and covariant vector com-ponents, respectively. These derivatives transform like the components of asecond-rank-tensor. The trace of (2.62a) immediately yields the divergence(2.57).

Let the vector A in (2.61) be the gradient of a scalar eld. We then have

A j = () j = T ij g i , (2.63)

where

T ij = T ji = 2 i j

k.ij

k ,i | j (2.64)

is a symmetric second-rank covariant tensor.

The L APLACE operator is dened as =div grad . Using (2.56b),(2.57), (2.58), (2.39b), and the abbreviation (2.64), the following relation isobtained:

= = g i i g

k k

= gij T ij gij ,i | j . (2.65)The gradient of a vector A is formed by the dyadic product of the Nabla

operator (2.56b) and the eld vector (2.47a) in connection with (2.62a,b):

T A = g j j

Ai g i = Ai j g jg i T i j g j g i(2.66a)

T A = g j j

Ai g i = Ai | j g j g i T ij g jg i .(2.66b)

In contrast to (2.64), the tensor T ij Ai | j in (2.66b) is not symmetric.From the representations (2.66a,b) we read that the mixed components T i jand the covariant components T ij of dyadic T A are identical withthe covariant derivatives (2.62a,b), respectively.


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In curvlinear coordinates, the constitutive equation (2.31) of the lineartheory of elasticity should be expressed in the form

ij

= E ijkl

kl , (2.67)where the innitesimal strain tensor is formed by covariant derivativesfrom the displacement vector w according to

ij = ( wi | j + w j | i ) /2 . (2.68)In the absence of body forces the divergence of the stress tensor must be

equal to zero. Thus, the equations of equilibrium are then given by

ij |i = 0 j . (2.69)Note that for applications in solid mechanics the physical components of

the tensors ij , ij and wi , used in (2.67), (2.68), (2.69), have to be calculatedaccordingly to

ij = ij

g( ii )g( jj ) , ij = ij g(ii )g( jj ) , ui = wi g( ii ) , (2.70a,b,c)

where the bracketed indices should not be summed.Considering a vector (2.47a) the components of which, A1g 1, A2g 2 ,

A3g 3 , form the edges of the parallelepiped whose diagonal is A . Since theg i are not unit vectors in general, we see that the lengths of edges of thisparallelepiped, or the physical components of A , are determined by the ex-pressions

A1 g11 , A2 g22 , A3 g33 ,since g11 = g 1 g 1, . . . , g33 = g 3 g 3 .

As an example, let us introduce cylindrical coordinates

x1 = 1 cos 2 , x2 = 1 sin 2 , x3 = 3 , (2.71)

where 1 r , 2 and 3 z. The covariant base vectors (2.35) arethen given by

g 1 = e 1 cos + e 2 sin ,

g 2 = e 1r sin + e 2r cos ,g 3 = e 3 , (2.72a)while the contravariant base vectors (2.37) are immediately found accordingto


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g 1 = g 1 , g2 = g 2 r 2 , g3 = g 3 . (2.72b)

Since the cylindrical coordinates (2.71) are orthogonal, the metric tensors

(2.39a,b), (2.40a,b) have the diagonal forms

gij =

1 0 0

0 r2 0

0 0 1

, gij =

1 0 0

0 1/r 2 0

0 0 1

, (2.73a,b)

while the nonvanishing C HRISTOFFEL symbols (2.59a,b), (2.60) are givenby

122 [12, 2] = 1.22 = r, 221 [21, 2] = [12, 2] = r, 2.12 = 1 /r.(2.74)

Taking the covariant derivative (2.62b) into account, the tensor (2.68) canalso be represented in the following way

ij = wi j + w j i /2 wk k.ij . (2.75)The physical components (2.70c) of the displacement vector are in cylindri-cal coordinates because of (2.42b) very simple:

u r = w1 , u = w2/r , u z = w3 , (2.76)

and, likewise, we calculate from (2.70b) the physical components

r = 11 , = 22 /r 2 , z = 33 ,

r = 12 /r , rz = 13 , z = 32 /r ,

(2.77)

of the innitesimal strain tensor, so that we nally arrive at the followingcomponents

r = ur / r , = ( u / + u r ) /r , z = uz / z ,r = [(u r / ) / r + u / r u / r ] /2 ,rz = ( u r /z + u z /r )/ 2 ,

z = [(u z / ) / r + u / z ] /2 ,

(2.78)

by considering (2.74), (2.75), and (2.76).Finally, the physical components of the stress tensor are deduced from

(2.70a):


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r = 11 , = r 2 22 , z = 33 ,

r = r 12 , z = r 23 , zr = 31 , (2.79)

so that we arrive from (2.69) by considering

Aij |k = Aij k + i.kp A pj + j.kp A

ip (2.80)

and (2.74) at the equations of equilibrium

r / r + ( r / ) / r + zr / z + ( r ) / r = 0 ,r / r + ( / ) / r + z / z + 2 r / r = 0 ,

zr / r + ( z / ) / r + z / z + zr / r = 0 .

(2.81)

For isotropic materials the fourth-order elasticity tensor in (2.31) has theform

E ijkl = ij kl + ( ik jl + il jk ) , (2.82a)

while the material tensor in (2.67) is represented by

E ijkl = g ij gkl + gik g jl + gil g jk , (2.82b)

so that we nd from (2.67) in connection with (2.77) and (2.79) the followingconstitutive equations:

r = 2 r + (r + + z ) , = 2 + (r + + z ) ,z = 2z + (r + + z ) ,

r = 2 r , z = 2z , zr = 2zr .

(2.83)

According (2.65) together with (2.64), the L APLACE operator is denedin general. In the special case of cylindrical coordinates with (2.73a,b), (2.74)this operator takes the form

= 2

r 2 +

1r 2

2

2 +

1r

r

+ 2

z2 , (2.84)

which occurs as a differential, for example, in the L APLACE or POISSON

equation. These partial differential equations are fundamental for many ap-plications in continuum mechanic.

In the representation theory of tensor functions the irreducible basic in-variants


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S 1 = ij A ji , S 2 = Aij A ji , S 3 = Aij A jk Aki (2,85a,b,c)

or, alternatively, the irreducible main invariants

J 1 S 1 , J 2 = S 2 S 21 / 2 , (2.86a,b)

J 3 = 2S 3 3S 2S 1 + S 31

6 (2.86c)

play a central role, since they form an integrity basis . In (2,85a,b,c) the Aijare the components of the tensor A with respect to the orthonormal basise i . The invariants (2,85a,b,c), e.g., can be expressed in terms of covariant orcontravariant components of the tensor A . To do this we need the followingtransformations

Aij = x p i

x q j

A pq Aij = p

x i q

x jA pq , (2.87)

and

Aij = i

x p j

x qA pq Aij =

xi p

x j q

A pq , (2.88)

which are extended forms of (2.35), (2.36), and (2.37). Inserting the trans-formation (2.87) in (2,85a,b,c) and considering (2.40a,b), we nally nd theirreducible basic invariants in the following forms:

S 1 = g pq A pq = g pq A pq = Akk , (2.89a)

S 2 = gip g jq A ji A pq = gip g jq A ji A pq = A ik Aki , (2.89b)

S 3 = gip g jq gkr Aij Aqk Arp = = A i j A jk A

ki , (2.89c)

and then the main invariants (2.86a,b,c) also in terms of covariant , con-travariant or mixed tensor components .

A lot of tensor operations are included in the MAPLE tensor package .Examples are illustrated in the following MAPLE program, where the metrictensors and the C HRISTOFFEL symbols have been calculated for cylindricaland spherical coordinates

2 1.mws> with(tensor):

> cylindrical_coord:=[r,phi,z]:

covariant metric tensor:> g_compts:=array(symmetric,sparse,1..3,1..3):


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> g_compts[1,1]:=1: g_compts[2,2]:=r2:

> g_compts[3,3]:=1:

> g:=create([-1,-1], eval(g_compts));

g := table([ compts =1 0 00 r 2 00 0 1

, index char = [1, 1]])> D1g:=d1metric(g,cylindrical_coord):> Gamma[kij]:=[ijk];

> CHRISTOFFEL[first_kind][cylindrical]:=

> Gamma[kij]=Christoffel1(D1g): kij := [ijk ]

The results are printed as a list on the CD-ROM. They are identicalto those values in (2.74).> spherical_coord:=[r,phi,theta]:

covariant metric tensor:> g_compts:=array(symmetric,sparse,1..3,1..3):

> g_compts[1,1]:=1: g_compts[2,2]:=r2:

> g_compts[3,3]:=(r * sin(phi))2:

> g:=create([-1,-1], eval(g_compts));

g := table([ compts =1 0 00 r 2 00 0 r2 sin()2

, index char = [1, 1]])> D1g:=d1metric(g,spherical_coord):> Gamma[kij]:=[ijk];

> CHRISTOFFEL[first_kind][spherical]:=

> Gamma[kij]=Christoffel1(D1g):

kij := [ijk ]The results are printed as a list on the CD-ROM. In a similar way one cannd the C HRISTOFFEL symbols of the second kind.


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http://www.springer.com/978-3-540-85050-2

tensor notation[1]

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