plate and shell

7/31/2019 Plate and Shell

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PLATE AND SHELL THEORY

Sren R. K. Nielsen, Jesper W. Strdahl and Lars Andersen

1

2

x1

x2

x3

aa

a) b)

f

1

1

Aalborg UniversityDepartment of Civil Engineering

November 2007


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Contents

1 Preliminaries 71.1 Tensor Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.1 Vectors, curvilinear coordinates, covariant and contravariant bases . . . 71.1.2 Tensors, dyads and polyads . . . . . . . . . . . . . . . . . . . . . . . . 121.1.3 Gradient, covariant and contravariant derivatives . . . . . . . . . . . . 191.1.4 Riemann-Christoffel tensor . . . . . . . . . . . . . . . . . . . . . . . . 23

1.2 Differential Theory of Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 281.2.1 Differential geometry of surfaces, rst fundamental form . . . . . . . . 281.2.2 Principal curvatures, second fundamental form . . . . . . . . . . . . . 351.2.3 Codazzis equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 441.2.4 Surface area elements . . . . . . . . . . . . . . . . . . . . . . . . . . 47

1.3 Continuum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491.3.1 Three-dimensional continuum mechanics . . . . . . . . . . . . . . . . 491.3.2 Fiber-reinforced composite materials . . . . . . . . . . . . . . . . . . 59

1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

2 Shell and Plate Theories 692.1 Plates and Shells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 712.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

7 Index 74

8 Bibliography 76

3


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4 Contents


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Preface

This book has been written as a part of ......

Aalborg, November 2007Sren R.K. Nielsen, Jesper W. Strdahl and Lars Andersen

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6 Contents


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C HAPTER 1Preliminaries

Mathematical and mechanical preliminaries.

Sections 1.1 and 1.2: basic results of tensor calculus and differential theory of surfaces havebeen outlined in below. Partly based on (Spain, 1965), (Malvern, 1969)

Section 1.3: Three-dimensional continuum mechanics. Composite materials. Based on (Reddy,2004).

1.1 Tensor Calculus

1.1.1 Vectors, curvilinear coordinates, covariant and contravari-ant bases

Fig. 11 Spherical coordinate system.

Fig. 1-1 shows a Cartesian (x1, x2, x3)-coordinate system, as well as a spherical (1, 2, 3)-coordinate system. 1 is the zenith angle , 2 is the azimuth angle , and 3 is the radial distance .

7


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8 Chapter 1 Preliminaries

Notice that upper indices are use for the identication of the coordinates, which should not beconfused with power raising. Instead, this will be indicated by parentheses, so if x1 speciesthe rst Cartesian coordinate, (x1)2 indicates the corresponding coordinate raised to the secondpower. With the restrictions 1 [0, ],

2]0, 2] and

3 0 a one-to-one correspondencebetween the coordinates of the two systems exists except for points at the line x1 = x2 = 0 .These represent the singular points of the mapping. For regular points the relations become,see e.g. (Zill and Cullen, 2005)

x1

x2

x3=

3 sin 1 cos 2

3 sin 1 sin 2

3 cos 1

1

2

3=

cos 1x3

(x1)2 + ( x2)2 + ( x3)2tan 1

x2

x1

(x1)2 + ( x2)2 + ( x3)2(11)

We shall refer to l , l = 1 , 2, 3, as curvilinear coordinates . Generally, the relation between theCartesian and curvilinear coordinates are given by relations of the type

x j = f j (l) (12)

The Jacobian of the mapping (1-2) is dened as

J = detf j

k(13)

Points, where J = 0 , represent singular points of the mapping. In any regular point, whereJ = 0 the inverse mapping exists locally, given as

j = h j (xl) (14)

For 1 = constant (1-3) denes a surface in space, dened by the parametric description

x j = f j (c1, 2, 3) (15)

Similar parametric description of surfaces arise, when 2 or 3 are kept constant, and the remain-ing two coordinates are varied independently. The indicated three surfaces intersect pairwise

along three curves s j

, j = 1 , 2, 3, at which two of the curvilinear coordinates are constant, e.g.the intersection curve s1 is determined from the parametric description x j = f j (1, c2, c3). Allthree surfaces intersect at the point P with the Cartesian coordinates x j = f j (c1, c2, c3). Lo-cally, at this point an additional curvilinear (s1, s2, s3)-coordinate system may be dened withaxes made up of the said intersection curves as shown on Fig. 1-1. We shall refer to thesecoordinates as the arc length coordinates .


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1.1 Tensor Calculus 9

The position vector x from the origin of the Cartesian coordinate system to the point P has thevector representation

x = x1i1 + x2i2 + x3i3 = x j i j (16)

where i j , j = 1 , 2, 3, signify the orthonormal base vectors of the Cartesian coordinate sys-tem. In the last statement the summation convention over dummy indices has been used. Thisconvention will extensively be used in what follows. The rules is that dummy latin indices in-volves summation over the range j = 1 , 2, 3, whereas dummy greek indices merely involvessummation over the range = 1 , 2. As an example a j b j = a1b1 + a2b2 + a3b3, whereasa b = a1b1 + a2b2. The summation convention is abolished, if the dummy indices are sur-rounded by parentheses, i.e. a( j )b( j ) merely means the product of the j th components a j and b j .

Let dx j denote an innitesimal increment of the j th coordinate, whereas the other coordinatesare kept constant. From (1-6) follows that this induces a change of the position vector given asdx = i( j )dx( j ) . Hence,

i j = xx j

(17)

A corresponding independent innitisimal increment d j of the j th curvilinear coordinate in-duce a change of the position vector given as dx = g ( j )d( j ) , so

g j = x j

(18)

g j is tangential to the curve s j at the point P , see Fig. 1-1. In any regular point the vectors g j , j = 1 , 2, 3, may be used as base vector for the arc length coordinate system at P . The indicatedvector base vectors is referred to as the covariant vector base , and g j are denoted the covariant base vectors . Especially, if the motion is described in arc length coordinates, g j becomes equalto the unit tangent vectors t j , i.e.

t j = xs j

(19)

Use of the chain rule of partial differentiation provides

g j = x j

= x

x kx k j

= ck j ik (110)

i j = xx j

= xk

k

x j= dk j g k (111)

where


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ck j =x k

j=

f k

j

dk j =k

x j=

h k

x j

(112)

ck j specify the Cartesian components of the covariant base vector g j . Similarly, d

k j speciesthe components in the covariant base of the Cartesian base vector i j . Obviously, the following

relation prevails

ckl dl j =

x k

l l

x j= k j (113)

k j denotes Kroneckers delta in the applied notation, dened as

k j =0 , j = k

1 , j = k(114)

Fig. 12 Covariant and contravariant base vectors.

Generally, the covariant base vectors g j are neither orthogonal nor normalized to the length1, as is the case for the Cartesian base vectors i j . In order to perform similar operations oncomponents of vectors and tensorz as for an orthonormal vector base, a so-called contravariant vector base or dual vector base is introduced. The corresponding contravariant base vectors g j

are dened from

g j g k = k j (115)where indicates a scalar product. (1-15) implies that g 3 is orthogonal to g 1 and g 2 . Further,the angle between g 3 and g 3 is acute in order that g 3 g 3 = +1 , see Fig. 1-2. Generally, thecontravariant base vectors can be determined from the covariant base vectors by means of thevector products


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v j = g jk vk

v j = g jk vk(121)

where

g jk = g j g k = gkjg jk = g j g k = gkj

(122)

The indicated symmetry property of the quantities g jk and g jk follows from the commutativityof the involved scalar products. From (1-21) follows

v j = g jl vl = g jl glk vk g jl glk = k j (123)

By the use of (1-20) and (1-21) the following relations between the covariant and the contravari-

ant base vectors may be derived

v j g j = v j g j = g jk vkg j = v j g jk g k

v j g j = v j g j = g jk vkg j = v j g jk g k g j = g jk g k

g j = g jk g k(124)

As seen g jk represent the components of g j in the contravariant vector base. Similarly, g jksignify the components of g j in the covariant vector base. Use of (1-10) and (1-11) providesthe following relation between the Cartesian and the covariant vector components

v j i j = v j g j = v j ck j ik = c jkv

k i j

v j g j = v j i j = v j dk j g k = d jk vkg j

v j = c jkvk

v j = d jk vk(125)

Finally, using (1-21) the scalar product of two vectors u and v can be evaluated in the followingalternative ways

u v =u j v j = u j v j = g jk u j vk

u j v j = u j v j = g jk u j vk(126)

where it is noticed that u j = u j .

1.1.2 Tensors, dyads and polyads

A second order tensor T is dened as a linear mapping of a vector v onto a vector u by meansof a scalar product, i.e.

u = T v (127)


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Since the vectors u and v are coordinate independent quantities, the 2nd order tensor T as wellmust be independent of any selected coordinate system chosen for the specication of the rela-tion (1-27). Equations in solid mechanics are independent of the chosen coordinate system forwhich reason these are basically formulated as tensor equations.

A dyad (or outer product or tensor product ) of two vectors a and b is denoted as ab . The polyad abc formed by three vectors a , b and c is denoted a triad , and the polyad abcd formed bythree vectors a , b , c and is denoted a tetrad .

For dyads the following associative rules apply

m (ab ) = ( ma ) b = a (mb ) = ( ab ) m

abc = ( ab ) c = a (bc )

(ma ) (nb ) = mn ab

(128)

where m and n are arbitrary constants. Further, the following distributive rules are valid

a (b + c) = ab + ac

(a + b ) c = ab + bc(129)

No commutative rule is valid for dyads formed by two vectors a and b . Hence, in general

ab = ba (130)

If the outer product of two vectors entering a polyad is replaced by a scalar product of the samevectors a polyad is obtained of an order less by two. This operation is known as contraction .Contraction of a triad is possible in the following two ways

a bc = ( a b ) cab c = ( b c) a

(131)

The scalar product of two dyads, the so-called double contraction , can be dened in two ways

ab cd = ( a c)(b d )ab cd = ( a d )(b c)

(132)

As seen the symbol

denes scalar multiplication between the two rst and the two last

vectors in the two duads, whereas denes scalar multiplication between the rst and thelast vector, and the last and the rst vectors in the two dyads. Because of the commutativity of the scalar product of two vectors follows

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

(133)


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Use of (1-32) and (1-33) provides the following identities for double contraction of two dyads

ab cd = ba dc = cd ab = dc baab cd = ba dc = cd ab = dc ba

(134)

From the second relation of (1-31) follows that the dyad ab / (b c ) is mapping the vectorc onto the vector a via a scalar multiplication. From the denition (1-27) follows that thisquantity is a second order tensor. Now, it can be proved that any second order tensor can berepresented as a linear combination of nine dyads, formed as outer product of three arbitrarylinearly independent base vectors. Hence, we have following alternative representations of atensor T in the Cartesian, the covariant and the contravariant bases

T = T jk i j ik = T jk g j g k = T jk g j g k (135)

The dyads i j ik , g j g k and g j g k form so-called tensor bases . Obviously, (1-35) represents thegeneralization to second order tensors of (1-20) for the decomposition of a vector in the cor-responding vector bases. T jk , T jk and T jk denotes the Cartesian components , the covariant components , and the contravariant components of the second order tensor T . Using (1-10), (1-11), (1-13) and the associate rules (1-28) the following relations between the dyads and tensorcomponents related to the considered three tensor bases may be derived

i j ik = dl j dmk g lg m

g j g k = cl j cmk i lim = g jl gkm g

lg m

g j g k = g jl gkm g lg m

(136)

T jk = c jl ckm T

lm , T jk = d jl dkm T

lm

T jk = g jl gkm T lm , T jk = g jl gkm T lm (137)

As seen, cl j cmk and g jl gkm specify the Cartesian and contravariant tensor components of the dyadg j g k , whereas g jl gkm denotes the covariant tensor components of g j g k . (1-36) and (1-37) rep-resent the equivalent to the relations (1-23) and (1-24) for base vectors and vector components.In some outlines of tensor calculus the transformation rules in (1-37) between Cartesian andcurvilinear tensor components are used as a dening property of the tensorial character of adoubled indexed quantity, (Spain, 1965), (Synge and Schild, 1966).

Alternatively, T may be decomposed after a tensor base with dyads of mixed covariant and

contravariant base vectors, i.e.

T = T jk g j gk = T jk g

kg j (138)

T jk and T j

k represent the mixed covariant and contravariant tensor components . In general,T jk = T

jk as a consequence of g j g

k = g kg j , i.e. the relative horizontal position of the upperand lower indices of the tensor components is important, and specify the sequence of covariant


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and contravariant base vector in the dyads of the tensor base. Tensor bases with dyads of mixedCartesian and curvilinear base vectors may also be introduced. However, in what follows onlythe mixed curvilinear dyads in (1-38) will be considered. The following identities may bederived from (1-35) and (1-38) by the use of (1-24)

T = T jk g j g k = T jl g j gl = T jlg

lk g j g k

T = T jk

g j g k = T k

l gl

g k = T k

l glj

g j g kT = T jk g j g k = T lk g lg

k = T lkglj g j g k

T = T jk g j g k = T l j g j g l = T l j glk g

j g k

T jk = glk T jl

T jk

= g jl

T k

l

T jk = g jl T lkT jk = glk T l j

(139)

Using (1-23) and (1-39) the following representations of the mixed components in terms of covariant and contravariant tensor components may be derived

T k j = g jl T kl

T k j = g jl T lk

T k j = gkl T lj

T k j = gkl T jl

(140)

It follows from (1-40) that, if T jk = T kj then T k j = T j

k and T jk = T kj . A second order tensorfor which the covariant components fulll the indicated symmetry property is denoted a sym-metric tensor .

Let T jk denote the covariant components of a second order tensor T . The related so-calledtransposed tensor T T is dened as the tensor with the covariant components T kj , i.e.

T T = T kj g j g k (141)

For a symmetric tensor T jk = T kj for which reason T T = T .

The identity tensor is dened as the tensor with the covariant and contravariant components g jkand g jk , i.e.

g = g jk g j g k = g jk g j g k = k j g j g k (142)

The mixed components follows from (1-23) and (1-40)

gk j = g jl gkl = k j

g k j = g jl glk = k j

(143)

Hence, the mixed components are equal to the Kroneckers delta, which explains the last state-ment for the mixed representation in (1-42). The name identity tensor stems from the fact thatg maps any vector v onto itself. Actually,


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g v = g jk g j g k vl g l = g jk vl g j lk = g jk vk g j = v j g j = v (144)The length of a vector v is determined from

|v |2

= v v = v g v = g jk

v j vk = g jk v jv

k(145)

Because of its relationship to the length of a vector the identity tensor is also designated themetric tensor or the fundamental tensor .

The increment dx of the the position vector x as given by (1-6), due to independent incrementdx j and d j of the Cartesian or curvilinear coordinates becomes

dx = dx j i j = d j g j (146)

Then, the length ds of the incremental position vector becomes, cf. (1-45)

ds2 = dx j dx j = g jk d j dk (147)

The inverse tensor T 1 of the tensor T is dened from the identity

T 1 T = T T 1 = g (148)

Let T 1 jl denote the contravariant components of T 1, and T mk the covariant components of T .

Then,

g = k j g j g k = T 1 T = T

1 jl g

j g l T mk g m g k = T 1

jl T mk lm g j g k = T

1 jl T

lk g j g k T 1 jl T

lk = k j (149)

In matrix notation this means that the three dimensional matrix, which stores the componentsT 1 jl , is the inverse of the matrix, which stores the components T lk . Similarly, the covariantcomponents (T 1) jl of T 1 and the contravariant components T lk of T are stored in inversematrices. In contrast, the matrices which store the contravariant components T 1 jl and T lk will

not be mutual inverse. From (1-15) follows that

g j T g k = g j T lm g lg m g k = T lm jl

km ) = T

jk

T jk = g j T g k (150)Similarly,


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T jk = g j T g kT k j = g j T g kT jk = g

j T g kT jk = i j

T

ik

(151)

Next, the second Piola-Kirchhoff stress tensor S and its virtual work conjugated strain measurethe Green strain tensor E are considered. In the Cartesian, covariant, contravariant and mixedtensor bases these are represented as follows

S = S jk i j ik = S jk g j g k = S jk g j g k = S k j g j g k = S j k g j g

k

E = E jk i j ik = E jk g j g k = E jk g j g k = E k j g j g k = E j k g j g

k(152)

Because the Piola-Kirchhoff stress tensor and the Green strain tensor are both symmetric tensorstheir mixed tensor components fulll S k j = S

jk and E k j = E

jk . For this reason we shall

simply use the notaions S k j and E k j for these quantities in what follows. The indicated Cartesian,covariant, contravariant and mixed tensor components for the stress tensor are related as follows,cf. (1-37), (1-39)

S jk = g jl gkm S lm , S jk = g jl S klS jk = g jl gkm S lm , S jk = g jl S lkS k j = g

kl S lj , S k j = g jl S lk

S jk = d jl dkm S

lm , S jk = c jl ckm S

lm

(153)

Completely analog relations prevail for the components of the Green tensor. For a physicallinear material a linear mapping of the Green tensor onto the Piola-Kirchhoff stress tensor viaa double contraction with the elasticity tensor C , i.e.

S = C E (154)C is a fourth order tensor , which can be specied in any tensor base with tetrads made up of either the base vectors of the Cartesian, the covariant or the contravariant vector bases as follows

C = C jk lm g j g kg lg m = C klm j g j g kg lg m = C j lmk g j g kg lg m = C jk ml g j g kg lg m= C jk l m g j g kg lg

m = C lm jk g j g kg lg m = C k m j l g

j g kg lg m = C kl j m g j g kg lg m

= C j mkl g j gkg lg m = C j lk m g j g

kg lg m = C jk lm g j g kglg m = C jklm g j g

kg lg m

= C k j lm g j g kg lg m = C l jk m g

j g kg lg m = C m jkl g j g kg lg m = C jk lm g j g kg lg m

= C jk lm i j ik ilim

(155)


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Formally, the relations between the various mixed tensor components can be derived by raisingand lowering indices by means of the covariant components g jk and the contravariant compo-nents g jk of the identity tensor g , cf. (1-53). As an example

C jklm = glr gms C jkrs (156)

Normally, the components of the elasticity tensor is specied in the Cartesian coordinate sys-tem. The relation between the Cartesian components C rstu and the covariant components C jk lmbecomes, cf. (1-53)

C jklm = d jr dks d

lt d

mu C

rstu (157)

The following relation between the covariant components of S and E may be derived, cf. (1-15),(1-32)

S = S jk

g j g k = C E = C jk

lm g j g kgl

gm

E rs

g r g s =C jk lm E

rs g j g k (g lg m g r g s ) = C jk

lm E rs g j g k(g l g r )(g m g s ) =

C jk lm E rs lr

ms g j g k = C

jklm E

lm g j g k

S jk = C jk lm E lm (158)

If the alternative denition of the double contraction in (1-32) is used, the following result isobtained for the tensor representation of S

S = C E = C jk lm E ml g j g k (159)Since, E lm = E ml , the indicated denitions provide the same result for the stress components.Expressed in the mixed components of E and S the constitutive equations on components formmay be given in the following ways

S k j = C k l

j m E ml = C

kl j m E

ml = C

j lk m E

ml = C

j lkm E

ml (160)

Clearly, the symmetry of the stress and strain tensor implies the following symmetry proper-ties of the mixed elasticity tensor components C jk lm = C

kjlm = C

jkml = C

kjml and C k l j m =

C kl j m = C j lk m = C j lkm .

Finally, the double contraction of S and E may be evaluated in the following alternative ways

S E = S E = S jk E jk = S jk E jk = S k j E jk (161)


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1.1.3 Gradient, covariant and contravariant derivatives

Consider a scalar function a = a(xl) = a(l) of the Cartesian or curvilinear coordinates.The gradient of a, denoted as a, is a vector with the following Cartesian and contravariantrepresentations

a =a

x jik =

a

jg j (162)

Let k denote the increments of the curvilinear coordinated, and dene the incremental vector = k g k . The corresponding increment a of the scalar is determined by

a = a =a j

g j k g k =a j

j (163)

The gradient of a vector function v = v (xl) = v (l), denoted as v , is a second order tensor,which in complete analogy to (1-63) associates to any incremental vector = k g k acorresponding increment v = v = v j j of the vector v . The gradient tensor hasthe following representations

v =

vx k

ik = (v j i j )

x kik =

v j

x ki j ik

vk

g k = (v j g j )

kg k =

v j

kg j + v j

g jk

g k

vk

g k = (v j g j )

kg k =

v jk

g j + v j g j

kg k

(164)

At the derivation of the Cartesian representation it has been used that the base vector i j is

constant as a function of xl. In contrast, the curvilinear base vectors depend on the curvilinearcoordinates, which accounts for the second term within the parentheses in (1-64). Clearly, g j k

and gj

k are vectors, which may hence be decomposed in the covariant and contravariant vectorbases as follows

g jk

= l j k

g l

g j

k=

jl k

g l(165)

l

j k signify the covariant components of g j k , and

j

l k is the contravariant components of g j k .

l

j k is denoted the Christoffel symbol . This is related to the co- and contravariant componentsof the identity tensor as follows

l j k

=12

glmg jmk

+gkm j

g jkm

(166)


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(1-65) and (1-66) have been proved in Box 1.1. From (1-66) follows that the Christoffel symbolsfulll the symmetry condition

l j k

= lk j

(167)

From (1-8) follows that

g jk

= 2x

j k=

2xk j

= g k j

(168)

Alternatively, the symmetry property (1-67) follows from (1-65) and (1-66).

Box 1.1: Proof of (1-65) and (1-66)

We consider the rst relation in (1-65) as a denition of the Christoffel symbol, and provethat this implies the second. From (1-15) and the rst relation in (1-65) follows

k

jm =

kg m g j =

g mk g

j + g m g j

k= 0

g j

k g m = l

m kg l g j =

lm k

jl = j

m k=

jl k

lm =

j

l kg l g m = 0

g j

k+ j

l kg l g m = 0 (169)

Since, (1-69) is valid for any of the three covariant base vectors the term within thebracket must be equal to 0 . This proves the validity of the second relation (1-65).

From (1-22) and the rst relation (1-65) follows

g jmk

= (g j g m )

k=

g jk g m +

g mk g j =

n j k

gnm +n

m kgnj (170)


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From (1-70) follows

g jmk

+gkm j

g jkm

=

n j k

gnm +n

m kgnj +

nk j

gnm +n

m jgnk

n j m

gnk n

k mgnj =

2 n j k

gnm (171)

where the symmetry property (1-67) has been used. Next, (1-66) follows from (1-71)upon multiplication on both sides with glm and use of (1-23).

Use of (1-65) in (1-66) provides the following representations of v

v = v j;k g j g k = v j ;k g j g k (172)

where

v j;k =v j

k+ j

k lvl

v j ;k =v jk

l j k

vl

(173)

Hence, v j;k species the mixed co-and contravariant components of v , and v j ;k species thecontravariant components. For the partial derivative of the vector function v (l) the followingrepresentations may be derived by the use of (1-64)

vk

=

(v j g j )k

=v j

kg j +

g jk

v j = v j;k g j

(v j g j )k

=v jk

g j + g j

kv j = v j ;k g j

(174)

Hence, v j

;k and v j ;k may alternatively be interpreted as the covariant components and the con-travariant components of the vector v k . For this reason v j

;k is referred to as the covariant derivative , and v j ;k as the contravariate derivative of the components v j and v j , respective. Inthe applied notation these derivatives will always be indicated by a semicolon.

Further, by the use of (1-65) the following results for the derivatives of the dyads entering thecovariant, mixed and contravariant tensor bases become


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l

(g j g k = g j l

g k + g j g k l

= m j l

g m g k +m

k lg j g m

l

(g j g k) = g j l

g k + g j g k

l= m

j lg m g k

km l

g j g m

l

(g j g k) = g j

lg k + g j

g k l

= j

m lg m g k +

mk l

g j g m

l

(g j g k) = g j

lg k + g j

g k

l=

jm l

g m g kk

m lg j g m

(175)

Finally, the covariant and covariant representations of the increment v of the vector v due tothe incremental vector = K g k of the curvilinear coordinates becomes

v = v = v j;k g j g k l g l = v j ;k g j g k l g l v = v

j;k

kg j = v j ;k

kg

j(176)

The gradient of a second order tensor function T = T (l), denoted as T , is a third order tensor with the following representations

T = T l

g l =

(T jk g j g k) l

g l =T jk

lg j g k + T jk

g j l

g k + T jk g j g k l

g l

(T jk g j g k) l

g l =T jk l

g j g k + T jk g j l

g k + T jk g j g k

lg l

(T k

j g j

g k) l

g l = T k

j l

g j g k + T k j g j

lg k + T k j g j g k l

g l

(T jk g j g k) l

g l =T jk l

g j g k + T jk g j

lg k + T jk g j

g k

lg l

T =

T jk

l+ T mk j

m l+ T jm k

m lg j g kg l = T jk ;l g j g kg

l

T jk l

+ T mkj

m l T jm

mk l

g j g kg l = T jk;l g j gkg l

T k j l T

km

m j l

+ T m jk

m lg j g kg l = T k j ;l g

j g kg l

T jk l T km

m j l T jm

mk l

g j g kg l = T jk ;l g j g kg l

(177)


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where

T jk ;l =T jk

l+ T mk j

m l+ T jm k

m l

T jk;l =T jk l

+ T mkj

m l T jm

mk l

T k j ;l = T k

j l T km m j l + T m j km l

T jk ;l =T jk l T km

m j l T jm

mk l

(178)

where (1-65) has been used in the last statement. Then, the partial derivative of the tensorfunction T (m ) may be written in any of the following alternative second order tensor repre-sentations, cf. (1-74)

T l

= T jk ;l g j g k = T jk;l g j g

k = T jk ;l g j g k (179)

Partial differentiation of both sides of (1-44) provides

vk

= gk v + g

vk

= gk v +

vk

gk v = 0 (180)

Since v is arbitrary (1-80) implies that

gk

= 0 (181)

In turn this means that g = 0 . Hence, the identity tensor is constant under covariant differen-tiation.

1.1.4 Riemann-Christoffel tensor

The gradient of a vector v with decomposition into Cartesian and curvilinear tensor bases havebeen indicated by (1-64). The gradient of this second order tensor is given as, cf. (1-73), (1-74)

( v ) =

x l

vx k

ik i l =

x l (v j i j )

x kik il =

2v j

x k x li j ik il

l

vk

g k g l =

lv j

k+ j

k mvm g j g k g l = v j ;kl g j g

kg l

l

vk

g k g l =

lv jk

m j k

vm g j g k g l = v j ;kl g j g kg l

(182)


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where the following tensor components have been introduced

v j ;kl = ( v j

;k);l = 2v j

k l+ j

k mvm

l+ j

l mvm

k m

k lv j

m j

n mn

k lvm +

l

j

k m

vm + j

l n

n

k m

vm (183)

v j ;kl = ( v j ;k);l = 2v j

k l m

j kvm l

m j l

vmk

mk l

v jm

+ nk l

mn j

vm

lm

j kvm +

n j l

mn k

vm (184)

From (1-83) and (1-84) follow that the indices k and l can be interchanged in the rst ve termson the right hand sides without changing the value of this part of the expressions, whereasthis is not the case for the last two terms. This implies that the sequence in which the covariant

differentiations is performed is signicant, i.e. in general v j ;kl = v j ;lk . In order to investigate thisfurther the so-called Riemann-Christoffel tensor R is introduced, which is a fourth order tensorwith the mixed representation R = Rm jkl g m g j g kg l , where the mixed curvilinear componentsare dened as

Rm jkl =n

j lm

n k n

j km

n l+

k

m j l

l

m j k

(185)

Obviously,

Rm jkl = Rm jlk (186)Further, the following relations are valid

Rm jkl + Rm

klj + Rm

ljk = 0

R j jkl = 0(187)

The relations (1-87) follow by insertion of (1-85) and use of the symmetry property (1-67) of the Christoffel symbol. From (1-84) and (1-85) follows that

Rm jkl vm = v j ;kl v j ;lk (188)The Cartesian components of R follow from (1-82)

R jklm vm = 2v j

x k x l 2v j

x lx k= 0

R jklm = 0 (189)

Hence, it can be concluded that R = 0 in an Euclidean three-dimensional space, i.e. a spacespanned by a Cartesian vector basis. In turn this means that also the curvilinear components


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(1-85) must vanish in this space. A space, where everywhere R = 0 is called at . Reversely,a non-vanishing Riemann-Cristoffel tensor indicates a curved space. In a at space Rm jkl = 0 ,with the implication that v j ;kl = v j ;lk . The three-dimensional Eucledian space is at, and anyplane in this space forms a at two-dimensional subspace. In contrast, a curved surface in theEucledian space is not a at subspace. An example of a curved four-dimensional space is thetime-space continuum used at the formulation of the general theory of relativity, where the in-dices correspondingly range over j = 1 , 2, 3, 4.

Because of the identities (1-86), (1-87) it can be shown that only 112 N 2(N 2 1) of the ten-

sor components Rm jkl are independent and non-trivial, where N denotes the dimension of thespace, (Spain, 1965). Hence, for a two-dimensional space only one independent componentexists, which can be chosen as R1 212 . In the three-dimensional case six independent and non-trivial components exist, which are chosen as R1212 , R1 213 , R1 223 , R1 313 , R1 323 and R2 323 .

Example 1.1: Covariant base vectors, identity tensor and Riemann-Christoffel tensor inspherical coordinates

By the use of (1-10) the rst equation (1-65) becomes

(cmj im ) k

=c mj k

im =l

j kcml im

c mj k

=l

j kcml (190)

The Cartesian components cmj of the covariant base vector g j is stored in the column matrix g j = [cmj ]. Then,

(1-88) may be written in the following matrix form

gj

k=

l j k

gl =

1 j k

g 1 +2

j kg 2 +

3 j k

g 3 (191)

The spherical coordinate system dened by (1-1) is considered. In this case the column matrices attain the form,cf. (1-12)

g 1 =

3 cos 1 cos 2

3 cos 1 sin 2

3 sin 1

, g 2 =

3 sin 1 sin 2

3 sin 1 cos 2

0, g 3 =

sin 1 cos 2

sin 1 sin 2

cos 1(192)

The covariant components of the identity tensor is given by (1-22) as gjk = g j g k = gT j g k , where the laststatement is obtained by evaluating the scalar product in Cartesian coordinates. The covariant and contravariantcomponents of the identity tensor are conveniently stored in matrices. Using (1-92) these becomes

[gjk ] =

(3 )2 0 0

0 (3 )2 sin2 1 0

0 0 1

, gjk =

1

(3

)2 0 0

01

(3 )2 sin2 10

0 0 1

(193)

The result for the contravariant components follows from (1-23). Next, (1-91) will be used to determine theChristoffel symbols by direct specication of the decomposition on the right hand side. Using (1-92) the followingresults are obtained


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The non-trivial components of the Riemann-Christoffel tensor follow from (1-85) and (1-94)

R 1 212 =n

2 21

n 1 n

2 11

n 2+

1

12 2

2

12 1

=

12

sin(21 ) 0 + 0 0 3 sin2 1

13

+ 0 0 +1

tan 1 12

sin(21 ) 0 0 cos(21 ) 0 = 0

R 1 213 =n

2 31

n 1 n

2 11

n 3+

1

12 3

3

12 1

=

0 0 +13 0 + 0

13 0

13

1tan 1 0 0 0 + 0 0 = 0

R 1 223 =n

2 31

n 2 n

2 21

n 3+

2

12 3

3

12 2

=

0 0 13

12

sin(21 ) + 0 0 +12

sin(21 ) 13 0 0 +

3 sin2 1 0 + 0 0 = 0

R 1 313 =n

3 3

1

n 1 n

3 1

1

n 3+

1

1

3 3

3

1

3 1=

0 0 + 0 0 + 0 13

13

13 0 0 0 0 +

1(3 )2

= 0

R 1 323 =n

3 31

n 2 n

3 21

n 3+

2

13 3

3

13 2

=

0 0 0 12

sin(21 ) + 0 0 0 13 0 0 0 0 + 0 0 = 0

R 2 323 =n

3 32

n 2 n

3 22

n 3+

2

23 3

3

23 2

=

0 1tan 1 + 0 0 + 0 13 0 0 13 13 0 0 + 0 + 1(3 )2 = 0

(195)

As expected all components of the Riemann-Christoffel tensor vanish as a consequence of the atness of the three-dimensional Euclidean space.


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1.2 Differential Theory of Surfaces

1.2.1 Differential geometry of surfaces, rst fundamental form

Fig. 13 a) Parameter space. b) Surface space.

Let the spherical 3 coordinate be xed at the value 3 = r . Then, the mapping (1-1) of thespherical coordinates onto the Cartesian coordinates takes the form

x1

x2

x3=

r 1

r 2

r 3=

r sin 1 cos 2

r sin 1 sin 2

r cos 1(196)

(r 1, r 2, r 3) denotes the Cartesian components of the position vector r to a given point of thesurface. Obviously, (r 1)2 + ( r 2)2 + ( r 3)2 = r 2. Hence, with the zenith angle varied in theinterval 1 [0, ], and the azimuth angle varied in the interval

2]0, 2], (1-96) represents

the parametric description of a sphere with the radius r and the center at (r 1, r 2, r 3) = (0 , 0, 0).In what follows it is assumed that the parametric description of all considered surfaces is denedby a constant value of the curvilinear coordinate 3 = c3 in the mapping (1-2). Then, a givensurface is given as, cf. (1-5)

r j = f j (1, 2, c3) = f j ( ) (197)

In the last statement of (1-97) the explicit dependence on the constant c3 is ignored, as willalso be the case in the following. Let the mapping (1-97) be dened within a domain in the

parameter space. For each point pdetermined by the parameters (

1,

2), a given point P

is dened on with Cartesian coordinates given by (1-97). Assume that a curve through p isspecied by the parametric description 1(t), 2(t) , where t is the free parameter. Then, thiscurve is mapped onto a curve s = s(t) through P on as shown on Fig. 1-3. Especially, if thecurvilinear coordinate 1 is xed, whereas 2 is varied independently a curve s1 through P isdened on . Similarly, a curve s2 on the surface through P is obtained, if 1 is xed and 2 isvaried. The positive direction of s1 and s2 are dened, so positive increments of correspond


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1.2 Differential Theory of Surfaces 29

to positive increments of s . Then, these curves dene a local two-dimensional arch length co-ordinate system (s1, s2) through P . Assume that is a rectangular domain [a1, a2][b1, b2] withsides parallel to the axes as shown on Fig. 1-3a. As an example this is the case for the map-ping (1-96), where = [0, ]]0, 2]. In such cases the surface will be bounded by the arclength coordinate curves given by the parameter descriptions r j = f j (a1, 2), r j = f j (a1, 2),r j = f j (1, b1) and r j = f j (1, b2), see Fig. 1-3b.

Since d3 = 0 , when merely 1 and 2 are varied in , the increment dr of the position vector rto the point P on the surface is given as by the following Cartesian and curvilinear representa-tions, cf. (1-6)

dr = dr 1 i1 + dr 2 i2 + dr 3 i3 = d1 a 1 + d2 a 2 dr = dr j i j = d a (198)

where the covariant base vectors becomes, cf. (1-8)

a = r

(199)

Fig. 14 Covariant and contravariant base vectors in the tangent plane.

a are tangential to the arch length curves at P , see Fig. 1-3b. a may then be interpreted as alocal two-dimensional covariant vector bases, which spans the tangent plane A at the point P ,see Fig. 1-4. The indicated covariant vectors may be represented in the Cartesian vector basisas, cf. (1-10)

a = c j i j (1100)

where the Cartesian components c j are given by (1-12).


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At the point P a unit normal vector a 3 to the surface and the tangent plane is dened as

a 3 =a 1 a 2|a 1 a 2|

=1A

a 1 a 2 (1101)A denotes the area of the parallelogram spanned by a 1 and a 2, and given as

A = |a 1 a 2| = |a 1||a 2|sin (1102)where denotes the angle between a 1 and a 2, see Fig. 1-4. Using (1-16) with V = 1 A = A,a contravariant basis at P may be dened by the relations

a 1 =1A

a 2 a 3 =1

(A)2a 2 (a 1 a 2) =

1(A)2 |a 2|

2 a 1 (a 1 a 2) a 2a 2 =

1A

a 3 a 1 =1

(A)2(a 1 a 2) a 1 =

1(A)2 |a 1|

2 a 2 (a 1 a 2) a 1a 3 = 1

Aa 1 a 2 = a 3

(1103)

The last statements of the results for a 1 and a 2 follow from the vector identities a (b c) =(a c) b (b a ) c and (a b ) c = ( a c) b (b c) a . The basic relation between thecovariant and contravariant base vectors in the tangent plane becomes, cf. (1-15)

a a = (1104)(1-104) may alternatively be proved from the last statements in (1-103), using (1-102) anda 1

a 2 =

|a 1

||a 2

|cos.

The unit normal vector a 3 = a 3 is orthogonal to both the covariant and the contravariant basevectors in the tangent plane, so

a 3 a = a 3 a = 0 (1105)Then, the covariant and contravariant components g jk and g jk of the three-dimensional identitytensor can be stored in the following matrices

[g jk ] =a11 a12 0a21 a22 00 0 1

, g jk =a

11

a12

0a21 a22 00 0 1

(1106)

A surface vector function v = v (1, 2) is a vector eld, which everywhere (i.e. for anyparameters (1, 2) ) is tangential to the surface. Then, v may be represented by the followingCartesian, covariant and contravariant representations


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v = v j i j = v a = v , a (1107)

where v j , v and v denote the Cartesian, the covariant and the contravariant components of v .v3 = v3 = 0 for a surface vector, for which reason only two components enter the curvilinearrepresentations. The Cartesian and the covariant components are related as, cf. (1-25)

v j = c j v

v = d j v j

(1108)

Similarly, the covariant and contravariant components are related by the following relationsanalog to (1-22)

v = a v

v

= a

v (1109)

where

a = a a a = a a

(1110)

As a consequence of the structure (1-106) of the covariant and contravariant components of theidentity tensor it follows that, cf. (1-23)

a a = (1111)

where denotes the Kroneckers delta in two dimensions. Both a and a are surface vectors.Then, these can be expanded in two-dimensional contravariant and covariant vector bases asfollows, cf. (1-24)

a = a a

a = a a (1112)

In analogy to (1-27), a surface second order tensor is dened as a second order tensor, whicheverywhere maps a surface vector v onto another surface vector u by means of a scalar product.A second order surface tensor T admits the following Cartesian, covariant, contravariant andmixed representations, cf. (1-35)

T = T jk i j ik = T a a = T a a = T a a = T a

a (1113)


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T , T , T and T denotes the covariant, the contravariant and the mixed covafriant andcontravariant components of the surface tensor. These are related as, cf. (1-39), (1-40)

T = a a T , T = a T

T = a a T , T = a T

T = a T , T

= a

T

(1114)

The surface identity tensor a is dened as a second order tensor with the covariant componentsa , the contravariant components a , and the mixed components , corresponding to thefollowing representations, cf. (1-42)

a = a a a = g a a = a a (1115)

a will map any surface vector onto itself, i.e.

a v = v (1116)(1-116) is proved in the same way as (1-44), using the representation (1-115). Let dr be a differ-ential increment of the position vector r due to an increment d of the parameters. Obviously,dr is a surface vector with the representation (1-98). The length ds of the incremental vectorbecomes

ds2 = dr dr = dr a dr = d a a d = a d d (1117)Obviously, the tensor a determines the length of any surface vector, for which reason the alter-native naming surface metric tensor is used. The quadratic form in the last statement of (1-117)is called the rst fundamental form of the surface.

The partial derivatives of the covariant and contravariant base vectors are unchanged given by(1-65)

a j

= l j

a l =

j a +

3 j

a 3

a j

=

jl

a l = j

a

j3

a 3(1118)

a 3 = a 3 is orthogonal to both a and a , so a 3 a = 0 and a 3 a = 0 . Then, scalarmultiplication of the rst equation in (1-118) with a 3 and scalar multiplication of the second

equation with a 3 provides

a 3 a j

= 3 j

a 3 a j

=

j3

(1119)


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where it has been used that a 3 a 3 = 1 . Because a 3 = a 3, the left hand sides of (1-119) areidentical for j = 3 . However, the right hand sides have opposite signs in this case. Both can thenonly be truth if the right hand sides vanish, leading to the following result for the Christoffelsymbol

3

3 = 0 (1120)

(1-120) implies that a 3 is orthogonal to the surface normal a 3 = a3 , and hence must be a

surface vector.

Obviously, the increment vector = a with the covariant components is a surfacevector. The gradient v of a surface vector function v = v ( ) is a surface second order tensor,which to the increment vector associates a surface increment vector v = v = v

of the vector v . The gradient tensor has the following curvilinear representations,which merely follow by replacing the latin indices with greek indices in (1-72), (1-73)

v = v; a a = v ; a a (1121)

where

v; =v

+

v

v ; =v

v

(1122)

v; and v ; will be referred to as the surface contravariant derivative and surface covariant derivative of the components v and v .

Correspondingly, the gradient T of a surface second order tensor function T = T ( ) is asurface third order tensor with the following curvilinear representations, cf. (1-77), (1-78)

T = T ; a a a = T ; a a

a = T ; a a a = T ; a a a (1123)

where

T ; =T

+ T

+ T

T ; =T

+ T

T T ; =

T T

+ T

T ; =T T

T

(1124)


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The gradient of the gradient of a surface vector is a third order surface tensor with the followingrepresentations, cf. (1-82), (1-83), (1-84)

( v ) = v; a a a = v ; a a a (1125)

where

v; = ( v; ); =

2v

+

v

+

v

v

v +

v +

v (1126)

v ; = ( v ; ); = 2v

v

v

v

+

v

v +

v (1127)

The surface Riemann-Christoffel tensor is denoted B = B (1, 2) to distinguish it from theequivalent tensor R in the three dimensional space. The mixed covariant and contravariantrepresentation becomes, cf. (1-85)

B = B a a a a (1128)

B =

+

(1129)

In analogy to (1-86) - (1-88) the tensor components B fulll the following identities

B = B B + B

+ B

= 0

B = 0

(1130)

B v = v ;

v ; (1131)

Due to the identities (1-130) only one independent and non-trivial component exists, which istaken as B 1212 .

From (1-85) and (1-129) follow that the components in the tangent plane R of the Riemann-Christoffel tensor in the three-dimensional space is related to the corresponding componentsB of the surface Riemann-Christoffel tensor as follows


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R = B +

3

3

3

3

(1132)

Since, the three-dimensional Eucledian space is at, we have everywhere R = 0 . Then, thefollowing simplied representation for tensor components B is obtained from (1-132)

B =3

3 3

3 (1133)

1.2.2 Principal curvatures, second fundamental form

Fig. 15 a) Unit tangent vector of a curve. b) Radius of curvature of a curve.

An arbitrary curve s(t) on the surface is dened by the parametric description r = r (t) =r 1(t), 2(t) , where t is a free parameter as explained subsequent to (1-97). P and Q denotetwo neighboring points on the curve dened by the position vectors r and r + dr , correspondingto the parameter values t and t + dt , respectively. The increment dr with the length ds is asurface vector placed in the tangent plane at P , and specied by the covariant representationdr = d a . Hence, the unit tangent vector to the curve at P is given as

t =drds

=d

dsa (1134)

The unit tangent vector in Q deviates innitesimally from t , and may be written as t + dt .Given, that the length unchanged is equal to 1, the increment dt must fulll

1 = ( t + dt ) (t + dt ) = t t + 2 dt t + dt dt = 1 + 2 dt t dt t = 0 (1135)


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where the second order term dt dt is ignored. (1-135) shows that the increment dt is orthogonalto t . Let n denote the unit normal vector n co-directional to dt . Then, the following identitymay be dened

dtds

= 0 n (1136)

(1-136) is referred to as Frenets formula . dtds and n are called the curvature vector and the principal normal vector of the curve, respectively, and the proportionality factor 0 is referredto as the curvature . Generally, this is always assumed to be positive, so (1-136) is dening theorientation of n . In Fig. 1-5b lines orthogonal to s have been drawn through P and Q in theplane spanned by t and n , which intersect each other at the curvature center O under the angled as shown on Fig. 1-5b. The position of O relative to s is dened by the orientation of n .Then, the following geometric relation prevails, see Fig. 1-5b

|dt ds| = |t ds| d |dt | = | t | d = 1 d =

dsR (1137)

where R = OP = OQ denotes the radius of curvature at P . Since, |n | = 1 it follows from(1-136) and (1-137) that the curvature has the following geometric interpretation

0 =1R

(1138)

Fig. 16 Normal plane to unit tangent vector.The unit tangent vector t is normal to a plane , which contains both to the unit normal vectora 3 = a 3 and to the principal normal vector n , see Fig. 1-6. A third linear independent unitvector a g in is given by the vector

a g = a 3 t (1139)


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We may then represent the curvature vector dtds = 0 n as a linear combination of a 3 and a g asfollows

0 n = a 3 + g a g (1140)

The expansion components and are known as the normal curvature and the geodesic cur-

vature of the curve s . Using that a 3 a g = 0 , and both are unit vectors, these may be expressedin terms of the curvature 0 of the curve as follows

= a 3 dtds

= ( a 3 n ) 0g = a 0

dtds

= ( a 0 n ) 0(1141)

Using the chain rule of partial differentiation it follows from (1-134) that the curvature vectordtds has the following representation

dtds

=d2

ds2a +

a

d

dsd

ds(1142)

Since a 3 = a 3 is orthogonal to a , it follows from (1-119), (1-141) and (1-142) that

= a 3 dtds

= a 3 a

d

dsd

ds= 3

d

dsd

ds= b

d

dsd

ds(1143)

where the following indexed quantities have been introduced

b = a 3 a

= 3

= b (1144)

The neighbouring points P and Q on the curve s are given by the position vectors r (s) andr (s + ds) = r (s) + dr . By the use of (1-134) the increment dr is given by the Taylor expansion

dr = r (s + ds) r (s) =drds

ds +12

d2rds2

ds2 + O ds3 = t ds +12

dtds

ds2 + O (ds3) (1145)

The distance dz of Q from the tangent plane at P is given by the projection of dr on the surfaceunit normal vector a 3, see Fig. 1-5. Hence,

dz = a 3 dr =12

a 3 dtds

ds2 =12

ds 2 =12

b d d (1146)

where (1-143) and the orthogonality of a 3 and t has been used. The larger the distance dz, thelarger is the curvature of the curve at P . In (1-146) this quantity is partly determined by the


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components b and partly by the parameter increments d . As seen from (1-144) the com-ponents b are entirely determined from properties related to the surface, and independent of the specic direction of the curve s . In contrast this direction is determined by the incrementsd of the curvilinear coordinates. The quadratic form b d d entering (1-146) is calledthe second fundamental form of the surface. In what follows b is considered the contravariantcomponents of a surface second order tensor b = b a a , which is called the curvature tensor of the surface.

Using a = a a , it follows from (1-119), (1-144) that

b = a 3 (a a )

= a 3 a

a

+ a a 3 a

= a a 3

a

= a

3

3

= a b = b (1147)

where the orthogonality of a 3 and a is used. Insertion of (1-144) and (1-147) into (1-133)

provides the following alternative representation of the components of the surface Riemann-Christoffel tensor

B = b b + b b B = b b + b b (1148)

where the transformation rules (1-114) of covariant and contravariant tensor components havebeen used in the nal statement. (1-148) provides the following form for the selected indepen-dent tensor component B1212

B1212 = b21b12 + b22b11 = det b = b (1149)where b denotes the determinant formed by the contravariant components of the curvature ten-sor b . (1-149) is called Gausss equation . If B1212 = 0 , all other contravariant components of the tensor will vanish as well, and it may be concluded that B = 0. Then, a surface is at, if b = 0 everywhere. Noticed that if the determinant of the contravariant components vanish, thedeterminant of the components of any other representation of b vanishes as well.

From (1-117) and (1-143) follows that the normal curvature of a curve on the surface with adirection determined by the increment d is given as

=b d d

a d d (1150)

(1-150) may be interpreted as a Rayleigh quotient related to the following generalized eigen-value problem, (Nielsen, 2006)


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b a t = 0 (1151)t = d

ds represent the covariant components of a unit tangent vector t = t a to the surface,

which species the a direction for which (1-151) is fullled. With a = a a a and b =b a a , (1-151) is seen to represent the curvilinear component relation of the following tensor

relation

b a t = 0 (1152)(1-152) is fullled for two unit eigenvectors t 1 and t 2 related to the eigenvalues 1 and 2 of (1-151). The directions specied on the surface by the eigenvectors t are denoted the principal curvature directions , and the related eigenvalues are called principal curvatures . The uniteigenvectors fulll the orthogonality conditions, (Nielsen, 2006)

t a t =0 , =

1 , = (1153)

t b t =0 , =

, = (1154)

Let the principal curvatures be ordered, so 1 denotes the smallest and 2 the largest curvature.As follows from the well-known bounds on the Rayleigh quotient the normal curvature of an arbitrary curve passing through P will be bounded by the principal curvatures as, (Nielsen,2006)

1 2 (1155)Using the mixed representations a = a a and b = b a a the component form of (1-152)attains the form

b t = 0 (1156)Formally, (1-156) may be obtained from (1-151) by pre-multiplication of a and use of (1-111), (1-114). Solutions t = 0 of the homogeneous system of linear equations (1-156) are

obtained, if the determinant of the coefcient matrix vanishes. This provides the followingcharacteristic equation for the determination of the principal curvatures

det b = 2 b11 + b22 + b11b22 b21b12 = 0 2 b +

ba

= 0 (1157)


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In the last statement it has been used that det b =ba , where a = det a denotes the

determinant of the matrix formed by the contravariant components of the surface identity tensor.This relation follows from the identities

b = a b

det b = det a det b

det b =ba

(1158)

The solutions to (1-157) may be written as

12

= H H 2 K (1159)

where

H =12

(1 + 2) =12

b (1160)

K = 12 =ba

= b11b22 b21b12 (1161)

H is called the mean curvature of the surface, and K is the Gaussian curvature . As seen K is equal to the determinant of the matrix formed by the mixed components b of the curvaturetensor, and 2H is equal to the trace of this matrix. It follows from that K = 0 , if either 1 = 0or 2 = 0 . A surface, where the Gaussian curvature vanishes everywhere is developable, i.e. the

surface can be constructed by transforming a plane without distortion. Conical and cylindricalsyrfaces are developable, whereas a spherical surface in not. From (1-149) and (1-161) follows

B1212 = aK (1162)

Since, the eigenvectors t are surface vectors, it follows from (1-116) and (1-153) that t 1a t 2 =t 1 t 2 = 0 . This means that the principal curvature directions through P are orthogonal to eachother . It is then possible to construct an (s1, s2) arc-length coordinate system at each point onthe surface, which everywhere has the principal curvature eigenvectors t 1 and t 2 as tangents.This so-called principal curvature coordinate system is specic in the sense that both [a ] and[b

] become diagonal, i.e a

12= b

12= 0 . This is analog to the formulation in modal coordinates

of the equations of motion of linear structural dynamics, where the modal mass and stiffnessmatrices become diagonal, (Nielsen, 2006). In this specic coordinate system the principalcurvatures are given as

=b( )( )a( )( )

(1163)


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Next, various relations are indicated, which involve the partial derivative of the surface unitnormal vector a 3 = a 3 . Partial differentiation of the equation a 3 a = 0 and use of (1-144)provide

a 3 a + a 3

a

= 0

a a 3

= a 3 a

= 3

= b (1164)

a 3 is a unit vector, so a 3 a 3 = 1 . Then, partial differentiation provides

a 3 a 3

= 0 (1165)

Hence, a 3 is orthogonal to a 3 , and accordingly can be decomposed in the covariant surfacevector base a as follows

a 3

= l a (1166)

Scalar multiplication of both sides with a and use of (1-110) and (1-164) provides the followingsolution for the covariant components l

a a 3

= b = l a a = a l l = a b = b (1167)

Then, insertion into (1-166) provides the following alternative representations

a 3

= a 3

= b a = b a a = b a =

3

a (1168)

where (1-112) and (1-144) have been used. Similarly, (1-118) and (1-144) provide the followingrelation for the partial derivatives of the surface covariant base vectors

a

=

a + b a 3 (1169)

(1-169) is called Gausss formula . Next, e and e are introduced as alternative two-dimensionalpermutation symbols dened as


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e =0 1

1 0

e = 0 11 0

(1170)

As seen [e ] = [e ] 1 = [e ]T = [e ]. From (1-110) follows that

a1 = |a 1| = ( a 1 a 1)1/ 2 = a11a2 = |a 2| = ( a 2 a 2)1/ 2 = a22

cos =a 1 a 2|a 1||a 2|

=a12

a11a22 sin = 1 a212a11a22(1171)

Then, (1-102) may be written as

A = a11a22 sin = a11a22 (a12)2 = a (1172)where a = det[ a ]. By the use of (1-170) and (1-172) the vector products in (1-103) can bewritten in the following compact forms

a 3 a = a e a a

a = a e a 3 (1173)

Further, the following component identities may be derived

a = a e a e

a =1a

e a e

e =1a

a e a

e = a a e a

(1174)

The rst relation in (1-174) follows from the following matrix identities

a11 a12a21 a22

= a11 a12

a21 a22

1

= a a22 a12

a21 a11= a 0 1

1 0 a11 a12

a21 a22 0 11 0T

(1175)


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where it has been used that det[a ] = 1/ det[a ] = 1a . The proof of the remaining relations in(1-174), which is left an exercise, may be carried out by simple matrix operations on (1-175).

Next, a three-dimensional vector eld v = v (1, 2) is considered, dened on the parameterdomain . Hence, a vector is connected to each point of the surface . Since, v3 = v3 = 0 , v isnot a surface vector. Physically, v may represent a force vector, a moment vector, a displacementvector or a rotation vector related to a material point on the surface. In this case the followingrepresentations prevail

v = v a + v3 a 3 = v a + v3 a 3 (1176)

Since, v 3 = 0 , the gradient becomes

v = v

a = v j ; a j a = v j ; a j a (1177)

Similarly, (1-74) becomes

v = v j ; a j = v; a + v3; a 3

v

= v j ; a j = v ; a + v3; a 3(1178)

where, cf. (1-73)

v; =v

+

v +

3v3

v3; =v3

+ 3

v + 3

3v3

v ; =v

v 3

v3

v3; =v3

3

v 3

3 v3

(1179)

The right hand sides of (1-178) both consist of a surface vector and a vector in the normaldirection. The surface and normal components of these representations must pair-wise be equalcorresponding to

v; a = v ; a

v3; a 3 = v3; a 3

v; = a v ;

v ; = a v ; v3; = v3;

(1180)


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The nal relations in (1-180) follow from the initial vector identities using (1-112) and a 3 = a 3 .Further, v3 = v3, v3; = v3; ,

3 3 =

33 = 0 and

3 = b , cf. (1-120), (1-144). Then,

withdrawal of the second equation in (1-179) from the fourth equation provides the followingrelations for the components of the curvature tensor

3 v =

3

v = b v

3 a v = b v

3 v = b a v

b =

3 a

b = 3 (1181)

1.2.3 Codazzis equations

From (1-164), (1-168) and (1-169) follows

b

=

a

a 3

= a

a 3 a

2a 3

=

a + b a 3 b a a 2a 3

= b

a

2a 3

(1182)

Interchange of the indices and in (1-182) provides

b

= b

a 2a 3

(1183)

Withdrawal of (1-182) from (1-183) gives the relation

b b = b b (1184)

(1-184) is trivially fullled for = . Further, the index combinations (, ) = (1 , 2) and(, ) = (2 , 1) lead to the same relation. Hence, merely two non-trivial and independent rela-tions exist, which are taken corresponding to the index combinations (, , ) = (1 , 1, 2) and(, , ) = (2 , 2, 1), leading to the following relations


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b112 b1

1 2

=b121 b2

1 1

b22

1 b2

2 1=

b21

2 b1

2 2

(1185)

The relations (1-185) are called Codazzis equations . It can be shown that arbitrary compo-nent functions a = a (1, 2) and b = b (1, 2) of the curvilinear parameters 1 and2 uniquely determine a surface (saved an arbitrary rigid body translation and rotation), whichhas a d d and b d d as its rst and second fundamental form, if and only if thesecomponents fulll the Codazzs equations (1-185) and the Gausss equation (1-165), (Spain,1965).

In a principal curvature coordinate system we have a12 = a21 = 0 and b12 = b21 = 0 . In thiscase the Codazzis equations can be reduced to

2

a1R1

=1

R2a 12

1

a2R2

=1

R1a 21

(1186)

where a1 and a2 are given by (1-171), and R1 and R2 are the principal curvature radii . A proof of (1-186) has been given in Box 1.2.

Box 1.2: Proof of (1-186)

Since b12 = b21 = 0 in the principal curvature coordinate system, (1-185) can be reducedto

b112

= b11 11 2 b22 21 1b221

= b222

2 1 b111

2 2

(1187)


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[a ] =r 2 00 r 2 sin2 1

(1191)

The contravariant components of the curvature tensor follow from (1-94) and (1-144)

[b ] = r 00 r sin

2

1 (1192)

Since both [a and [b ] are diagonal, it is concluded that the spherical coordinate system is a principal curvaturecoordinate system. The principal curvatures and related principal curvature radii follow from (1-163)

1 = 2 = 1r

R 1 = R 2 = r (1193) 1 and 2 are negative, because unit surface normal vector is directed away from the curvature center, which inthis case is identical to the origin of the Cartesian coordinate system.The selected independent component of the Riemann-Christoffel tensor becomes, cf. (1-149)

B 1212 = r 2 sin2 1 (1194)

Left and right hand sides of the Codazzi equations (1-186) vanish in this case. Hence, these equations are triviallyfullled.

1.2.4 Surface area elements

Fig. 17 Area elements in parallel surfaces.

Fig. 1.7 shows two surfaces and 0 , described by the same parameters 1 and 2. The pointsP 0 and P denotes the mapping points for the same parameters, specied by the position vectors


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r = r (1, 2) and x = x (1, 2), respectively. Further, the unit normal vectors related to the twosurfaces at the said points are assumed to be identical and equal to a 3 = a 3 . Then, the distancebetween any such corresponding points P 0 and P on the surfaces are everywhere constant andequal to s3, where s3 is measured from the surface 0. On the surfaces arc length coordinatecoordinate systems (s1, s2) and (S 1, S 2) are introduced with origins P 0 and P . The relatedsurface covariant base vectors are denoted a and g , respectively. Then, the position vectorsare related as follows

x = r + s3 a 3 (1195)

dA0 denotes a differential area element of shape as a parallelogram with sides of the length ds1and ds2 parallel to a 1 and a 2 , see Fig. 1.7. The length of the sides follows from (1-117)

ds1 = a11 d1ds2 = a22 d2 (1196)

Let denote the angle between a 1 and a 2 . Then, cf. (1-102), (1-172)

dA0 = ds1ds2 sin = a11a22 sin d1d2 =

a11a22 (a12)2 d1d2 = a d1d2 (1197)The surface covariant base vectors g follows from (1-99), (1-168), (1-195)g =

x

= r

+ s3

a 3

= a s3b a = s3b a (1198)Then, the contravariant components g of the surface identity tensor of becomes

g = g g = s3b s3b

a a = s3b

s3b

a =

a a s3b a s3b = a a s3b a s3b (1199)Then, the differential area element dA on , corresponding to dA0 on 0, becomes

dA = g d1d2 (1200)where g = det[ g ] follows from (1-199)

g = det a det a s3b det a s3b = 1a det a s3b

2 =

1a

det a det s3b

2=

det a 2

adet s3b

2=

a 1 b s3 +ba

s3 22

= a 1 2H s 3 + K s32 2

(1201)


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1.3 Continuum Mechanics 49

where (1-157) and (1-158) have been used. Further, the mean curvature H and the Gaussiancurvature as given by (1-160) and (1-161) have been introduced. Then the ratio between thedifferential area elements become

dAdA0

=ga = 1 2Hs

3 + K (s3)2 (1202)

1.3 Continuum Mechanics

1.3.1 Three-dimensional continuum mechanics

x1

x2

x3

i1

i2

i3

O

X

dX0dV 0

A0

V 0

u (X k , t)

x (X k , t )

dV

A

V dx

Fig. 18 Referential and current conguration of a exible body.

Fig. 1-8 shows a exible body, i.e. a set of connected material points, which at the time t = 0occupies the volume V 0 with the surface A0. The indicated conguration of the body is referredto as the referential conguration . Material points (mass particles) are identied by their po-sition vector X relative to the origin of a given frame of reference. Although any curvilinearcoordinate system may be used, vectors and tensors will in this and the next subsection exclu-sively be described in the Cartesian (x1, x2, x3)-coordinate system with the origin O and thebase unit vectors (i1, i2, i3) shown on Fig. 1-8, with the exception that the dual Cartesian vectorbase (i1, i2, i3) will be used as well. Because the unit base vectors in the indicated vector basesare identical, i.e. i j = i j , covariant, contravariant and mixed Cartesian components will be

identical. Since no misapprehension with covariant and contravariant curvilinear coordinates ispossible, the bar atop the Cartesian components of vectors and tensors will be skipped through-out this subsection. The Cartesian components X j of X = X j i j are denoted the material or Lagrangian coordinates of the particle. At the time t the body occupies a new volume V withthe surface A due to translation, rotation and mutual distortion of the particles. The indicatedconguration is referred to as the current conguration of the body. The position in the cur-rent conguration of a particle with the referential coordinates X j is dened by the position


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vector x (X k , t ) = x j (X k , t ) i j . The current coordinates x j are known as the Eulerian cordi-nates of the particle. Traditionally, the governing equations of uid mechanics are formulatedusing Eulerian coordinates. This permits that large distortions in the continuum motion can behandle with relative ease. By contrast formulations in Lagrangian coordinates is preferred forthe kinematical description in solid mechanics, which allows an easy tracking of the materialhistory. Its weakness lies in its inability to follow large distortions of the computational domain.

The displacement of the particle is dened as the difference between the position vector in thecurrent and referential congurations, see Fig. 1-8

u (X k , t ) = x (xk , t ) X (1203)Consider a neighbor particle with the referential position vector X + dX . The indicated particleis displaced to the position x (X k + dX k , t ) = x (X k , t ) + dx (X k , t ), where the incrementalvectors are related as

dx (X k , t ) = F (X k , t ) dX (1204)F = x is the so-called material deformation gradient , cf. (1-76). This second order tensor, aswell as the transposed tensor F T , the inverse tensor F 1 and the transpose of the inverse tensorF T have the following dual Cartesian representations, cf. (1-41), (1-47)

F = F jk i j ik =x jX k

i j ik , F T = F kj i j ik =x kX j

i j jk

F 1 = F 1 jk i j ik =

X jx k

i j ik , F T = F 1kj i j ik =

X kx j

i j ik(1205)

The inverse material deformation gradient presumes the existence of the inverse mapping X =

X (xk , t ). This requires that F is non-singular, i.e. that the Jacobian det[F jk ] > 0. Let dV anddV 0 denote the volume of a particle in the current and the referential conguration. Further, let and 0 be the mass density of the particle in these congurations. Since the mass of the particleis unchanged we must have dV = 0dV 0. The Jacobian species the quotient between thedifferential volumes of the particle in the current and referential congurations correspondingto

det[F jk ] =dV dV 0

=0

(1206)

The distance between the neighboring particles in the referential and the current conguration

is given by the lengths ds and dS of the incremental position vectors dx and dX . Clearly, thedifference between these lengths is a measure of the local distortion of the material. Amongseveral strain measures suitable for description of large strains we shall adopt the so-calledGreen strain tensor E , which is introduced in the following way

ds2 dS 2 = dx dx dX dX = dX F T F g dX = 2 dX E dX (1207)


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where (1-44) and (1-204) have been used. g denotes the identity tensor, which has the follow-ing Cartesian representation g = jk i j ik = k j i j ik = jk i j ik . jk = k j = jk specify theKroneckers delta (1-14) in various tensor bases. Hence, the Green strain tensor is related to thematerial deformation gradient as follows

E =1

2F T

F

g (1208)

The dual Cartesian components of E may be expressed in terms of the components of thedisplacement eld by (1-203), (1-205) and (1-208)

E = E jk i j ik =12

F lj i j il F mk im ik jk i j ik =12

F lj F mk lm jk i j ik =12

F lj F lk jk i j ik =12

x lX j

x lX k jk i

j ik =

12 lj +

u lX j lk +

u lX k jk i

j

ik

=12

u jX k +

u kX j +

u lX j

u lX k i

j

ik

E jk =12

u jX k

+u kX j

+u lX j

u lX k

(1209)

For deformation gradients fullling |u jX k | 1 the quadratic term in (1-209) may be ignored.The corresponding Green small strain tensor e has the components

e jk =12

u jX k

+u kX j

(1210)

The Eulerian small strain tensor represents the corresponding strain tensor, where the defor-mation gradient is taken with the respect to the Eulerian coordinates. This tensor has the dualCartesian components

jk =12

u jx k

+u kx j

(1211)

The Cauchy equations of motion of a mass particle is formulated in Eulerian coordinates. Intensor notation these read, (Malvern, 1969)

a = + b (1212)a (xk , t ) denotes the acceleration vector of the particle, and b (xk , t ) is the external load per unit mass . (xk , t ) represents the Cauchy stress tensor , which is a symmetric tensor, = T .The divergence operation species a vector with the components x l ( jk i j ik ) il = jkx l

i j ik i l = jk

x ki j , cf. (1-77). Then, the dual Cartesian component form of (1-212) becomes


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dA

A1

A2

u

Fig. 19 Denition of boundary conditions.

a j =

x k jk + b j (1213)

The surface traction (xk , t ) at the current surface A is related to Cauchys stress tensor (xk , t )by Cauchys boundary condition , (Malvern, 1969)

(xk , t ) = n (1214)where n = n (xk , t ) signies the outward directed unit vector to the surface A. A is dividedinto two disjoint sub-surfaces A1 and A2, at which either mechanical or kinematical boundaryconditions are prescribed, see Fig. 1-9. Then, (1-212) is solved with the following boundaryconditions

(xk , t ) = (xk , t ) , xkA1u (xk , t ) = u (xk , t ) , xk

A2(1215)

(xk , t ) denotes the prescribed surface traction on the sub-surface A1, and u (xk , t ) is the pre-scribed displacement on the sub-surface A2.

As a basis for any nite element method the equations of dynamic equilibrium and relatedboundary conditions are reformulated in a weak form. This is a variational principle for somefunctional of the unknown elds of the problem. At stationarity the varied functional deter-mines the discretized unknown elds, so these full the equilibrium equation and the boundaryconditions at best or in average throughout V and A, when the elds are varied withina predened functional sub-space. For so-called displacement based nite element formula-

tions, where the only unknown is the displacement eld, the relevant variational principle is the principle of virtual displacements , which may be formulated as

V dV = V u b a dV + A1 u dA (1216)with the component form


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V jk jk dV = V u j b j a j dV + A1 u j j dA (1217)u (xk) = u j (xk) i j denotes an arbitrary variational displacement eld with the restrictions thatu (xk) = 0 on A2. Then, (1-217) can be derived by scalar multiplication of both sides (1-213)

with u j (xk), followed by an integration over V . Next, the identity is obtained by the use of thedivergence theorem, the mechanical boundary condition on A1, and the kinematical constrainthat allowable variations fulll u j (xk) = 0 on A2 . Save that certain continuity requirementsneed to fullled to make the indicated mathematical operations meaningful, the variational eldu (xk) may else be prescribed with an arbitrary magnitude. Especially, the variations do notnecessarily have any relation to the actual displacement eld.

= jk i j ik in (1-216) denotes the Eulerian small strain tensor for the variational displace-ment eld u (xk), i.e. the components are given as, cf. (1-211)

jk = 12 u j

x k+ ukx j

(1218)

Because u j (xk) may be of nite magnitude, this is not identical to the variation of the com-ponents of the Eulerian small strain tensor of the actual displacement eld as given by (1-211).This is so because x j depends on u j , for which reason the Eulerian gradient x k will be variedas well. Actually

u jx k

= u jX l

X lx k

=u jX l

X lx k

+u jX l

X lx k

=u jx k

+u jX l

F 1lk (1219)

where it has been used that the Lagrangian coordinates X k are not subjected to any varia-tion, i.e. the variation concerns the displacement of a specic mass particle, for which reason( u jX k ) =

u jX k

. Only for small displacement variations will ( u jx k ) =u jx k

.

Since, and are symmetric tensors, it follows that = , cf. (1-59), (1-61).Hence, the same internal virtual work is obtained in (1-216) independent of which denition of the double contraction of the involved tensors is used.

Finally, the acceleration vector a enters the volume integral on the right hand side of (1-216) viathe combination b a . Formally, b a may be interpreted as an equivalent quasi-static loadper unit mass, which takes the dynamic effects into consideration via the the so-called inertialload a dV on the particle. This is the essence of the dAlemberts principle . Then, theright hand side of (1-216) represents the external virtual work W e performed by the equivalentload per unit mass b a throughout V , and the surface traction on A1. The left hand siderepresents the internal virtual work W i performed by the internal forces (stress components)on the particles. Then, the principle of virtual displacements postulates that the internal andexternal virtual works are equal, i.e.


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W i = W e (1220)

The idea is to formulate nite element theories based on the Lagrangian formulation, for whichreason (1-216) at rst is reformulated into the following equivalent form based on quantitiesdened in the referential conguration

V 0 E S dV 0 = V 0 u 0 b a dV 0 + A01 u 0 dA0 (1221)with the component form

V 0 E jk S jk dV 0 = V 0 u j 0 b j a j dV 0 + A01 u j j0 dA0 (1222)(b a ) dV = ( b a ) 0dV 0 specify the equivalent static load on a given particle, so the rstintegrals on the right hand side of (1-216) and (1-221) denote the summation of these loadsfrom all particles within the structure. Hence, these integrals are equal.

The surface A0 in the referential conguration is divided into the sub-surfaces A01 and A02 ,which are mapped onto the A1 and A2 in the current conguration. Correspondingly, a surfacearea element dA0 on A01 is mapped onto the surface area element dA on A1. 0 entering thesecond integral on the right hand side of (1-221) denotes a ctitious surface traction denedin the referential conguration, which is related to the actual surface traction in the currentconguration as

0dA0 = dA (1223)

Then, the second integrals on the right hand side of (1-216) and (1-221) represent the summa-tion of identical loads from corresponding area elements in the two conguration. It followsthat also these integrals are equal. 0 will be referred to as a pseudo surface traction , sincethe referential conguration is actually free of any stresses, save for a possible known residualstress conguration with no relation to the indicated external loads and boundary conditions.

As mentioned, denotes the Eulerian small strain tensor of the variational displacement eld

u (xk), and must not be confused with the variation of the Eulerian small strain tensor of theactual displacement eld. For this reason it could be anticipated that E entering the integral onthe left hand side of (1-221) in the same way represents the Green strain tensor of the variationaldisplacement eld. This is not so. Indeed, E denotes the variation of the nite Green straintensor due to the variation the displacement eld, and depends on both u and u . In order toderive the relation between E and the variation of the material gradient tensor is at rstdetermined. Use of (1-205) provides with x j = (X j + u j ) = u j


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F = x j

X mi j im =

x jX m

i j im =u jx k

x kX m

i j im =u jx k

x lX m

i j (kl im ) =

u jx k

x lX m

i j (ik il im ) =u jx k

i j ik x l

X milim =

u jx k

i j ik F (1224)

Then, the variation of the Green strain tensor follows from (1-208) and (1-222)

E =12

F T F + F T F g = F T 12

ukx j

+u jx k

i j ik F = F T F (1225)where it has been used that g = g u = 0 , cf. (1-81). The related component formbecomes, cf. (1-205)

E jk = E jk =x lX j

lmx mX k

(1226)

The tensor and component form of the inverse relation read

= F T E F 1 (1227)

jk = jk =X lx j

E lmX mx k

(1228)

The tensor S entering the integral on the left hand side of (1-221) is another pseudo-stress

quantity denoted the second Piola-Kirchhoff stress tensor. This is the virtual work conjugatedstress measure to E to insure that the left hand sides of (1-216) and (1-221) become identical.This necessitates that jk jk dV = E lm S lm dV 0. Then, use of (1-206) and (1-226) providesthe following relation between the components of the Cauchy stress tensor and the componentsof the second Piola-Kirchhoff stress tensor

jk jk dV = E lm S lm dV 0 =x jX l

jkx kX m

S lm dV 0

jk = jk =dV 0dV

x jX l

x kX m

S lm =0

x jX l

S lmx k

X m(1229)

The related tensor form reads

=0

F S F T (1230)The tensor and component form of the inverse relation read


56/77


S =0

F 1 F T (1231)

S jk = S jk

=0

X jx l

lmX kx m

(1232)

For hyperelastic materials the existence of an elastic potential W (E ) is postulated, dependingentirely on the Green strain tensor, from which the components of the second Piola-Kirchhoff tensor may be obtained from the gradient relation

S jk =W (E )

E jk(1233)

For a St.Venant-Kirchhoff type of material the following linear relation between S and E ispostulated

S = S 0 + C E (1234)S 0 signies a known residual stress conguration in the referential conguration, which has nodependence on the external loads, the surface traction on A01 , or on the prescribed displace-ments on A02 . The fourth order tensor C is known as the elasticity tensor. In the rst place thishas 34 = 81 independent components C jk lm . However, because S = S T and E = E T are bothsymmetric tensors, the following symmetry relations prevail C jk lm = C kjlm = C jkml , whichreduces the number of independent components to 66 = 36 . Additionally, if the linear materialis hyperelastic, it follows from (1-233) and (1-234) that

C jklm = 2W (E )

E jk E lm=

2W (E )E lm E jk

= C lmjk (1235)

The symmetry condition (1-235) further reduces the number of independent elastic constants to21. In this case the constitutive relation between the 6 non-trivial stress components (due to thesymmetry of S ), and the 6 non-trivial components of E can be given in the following matrixform

S = S 0 + C E (1236)

S =

S 11S 22

S 33

S 12

S 13

S 23

, S 0 =

S 110S 220S 330S 120S 130S 230

, E =

E 11E 22E 33E 12E 13E 23

(1237)


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C =

C 1111 C 1122 C 1133 C 1112 C 1113 C 1123

C 1122 C 2222 C 2233 C 2212 C 2213 C 2223

C 1133 C 2233 C 3333 C 3312 C 3313 C 3323

C 1112 C 2212 C 3312 C 1212 C 1213 C 1223

C 1113

C 2213

C 3313

C 1213

C 1313

C 1323

C 1123 C 2223 C 3323 C 1223 C 1323 C 1313

(1238)

For an orthotropic material the number of material parameters further reduces to 9, correspond-ing to the constitutive matrix

C =

C 1111 C 1122 C 1133 0 0 0C 1122 C 2222 C 2233 0 0 0C 1133 C 2233 C 3333 0 0 0

0 0 0 2G12 0 00 0 0 0 2G13 00 0 0 0 0 2G23

(1239)

G12 , G13 and G23 denotes the shear moduli of the material. Finally, in case of an isotropicmaterial merely two independent parameters remains. In this case the constitutive matrix maybe written, (Malvern, 1969)

C =

+ 2 0 0 0 + 2 0 0 0

+ 2 0 0 00 0 0 2 0 00 0 0 0 2 00 0 0 0 0 2

(1240)

where and are known as the Lam e parameters . these may be related to Youngs modulus E ,and the Poissons ratio as follows, (Malvern, 1969)

= E

(1 + v)(1

2 )

, = G =E

2(1 + )(1241)

where G is the shear modulus. It should be noticed that although the relation (1-236) is physicallinear, the relation becomes geometrical non-linear, when the Green tensor is expressed in thedisplacement components.

The use of the variational formulation (1-221) as a basis for a nite element formulation re-quires that the variational eld u (X k) an

plate and shell

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