PLATE AND SHELL THEORY

Transcription

1 PLATE AND SHELL THEORY Søren R. K. Nielsen, Jesper W. Stærdahl and Lars Andersen a) b) 1 θ 2 x 3 f ω a x 2 1 θ 1 a x 1 Aalborg University Department of Civil Engineering November 2007

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3 Contents 1 Preliminaries Tensor Calculus Vectors, curvilinear coordinates, covariant and contravariant bases Tensors, dyads and polyads Gradient, covariant and contravariant derivatives Riemann-Christoffel tensor Differential Theory of Surfaces Differential geometry of surfaces, first fundamental form Principal curvatures, second fundamental form Codazzi s equations Surface area elements Continuum Mechanics Three-dimensional continuum mechanics Fiber-reinforced composite materials Exercises Shell and Plate Theories Plates and Shells Exercises Index 74 8 Bibliography 76 3

4 4 Contents

5 Preface This book has been written as a part of... Aalborg, November 2007 Søren R.K. Nielsen, Jesper W. Stærdahl and Lars Andersen 5

6 6 Contents

7 CHAPTER 1 Preliminaries Mathematical and mechanical preliminaries. Sections 1.1 and 1.2: basic results of tensor calculus and differential theory of surfaces have been 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 Vectors, curvilinear coordinates, covariant and contravariant bases Fig. 1 1 Spherical coordinate system. Fig. 1-1 shows a Cartesian (x 1, x 2, x 3 )-coordinate system, as well as a spherical (θ 1, θ 2, θ 3 )- coordinate system. θ 1 is the zenith angle, θ 2 is the azimuth angle, and θ 3 is the radial distance. 7

8 8 Chapter 1 Preliminaries Notice that upper indices are use for the identification of the coordinates, which should not be confused with power raising. Instead, this will be indicated by parentheses, so if x 1 specifies the first Cartesian coordinate, (x 1 ) 2 indicates the corresponding coordinate raised to the second power. With the restrictions θ 1 [0, π], θ 2 ]0, 2π] and θ 3 0 a one-to-one correspondence between the coordinates of the two systems exists except for points at the line x 1 = x 2 = 0. These represent the singular points of the mapping. For regular points the relations become, see e.g. (Zill and Cullen, 2005) x 1 x 2 x 3 θ 3 sin θ 1 cos θ 2 = θ 3 sin θ 1 sin θ 2 θ 3 cosθ 1 θ 1 θ 2 θ 3 cos 1 x 3 (x1 ) 2 + (x 2 ) 2 + (x 3 ) 2 = 1 x2 tan x 1 (x1 ) 2 + (x 2 ) 2 + (x 3 ) 2 (1 1) We shall refer to θ l, l = 1, 2, 3, as curvilinear coordinates. Generally, the relation between the Cartesian and curvilinear coordinates are given by relations of the type x j = f j (θ l ) (1 2) The Jacobian of the mapping (1-2) is defined as [ ] f j J = det θ k (1 3) Points, where J = 0, represent singular points of the mapping. In any regular point, where J 0 the inverse mapping exists locally, given as θ j = h j (x l ) (1 4) For θ 1 =constant (1-3) defines a surface in space, defined by the parametric description x j = f j (c 1, θ 2, θ 3 ) (1 5) Similar parametric description of surfaces arise, when θ 2 or θ 3 are kept constant, and the remaining 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 s 1 is determined from the parametric description x j = f j (θ 1, c 2, c 3 ). All three surfaces intersect at the point P with the Cartesian coordinates x j = f j (c 1, c 2, c 3 ). Locally, at this point an additional curvilinear (s 1, s 2, s 3 )-coordinate system may be defined with axes made up of the said intersection curves as shown on Fig We shall refer to these coordinates as the arc length coordinates.

9 1.1 Tensor Calculus 9 The position vector x from the origin of the Cartesian coordinate system to the point P has the vector representation x = x 1 i 1 + x 2 i 2 + x 3 i 3 = x j i j (1 6) where i j, j = 1, 2, 3, signify the orthonormal base vectors of the Cartesian coordinate system. In the last statement the summation convention over dummy indices has been used. This convention will extensively be used in what follows. The rules is that dummy latin indices involves summation over the range j = 1, 2, 3, whereas dummy greek indices merely involves summation over the range α = 1, 2. As an example a j b j = a 1 b 1 + a 2 b 2 + a 3 b 3, whereas a α b α = a 1 b 1 + a 2 b 2. The summation convention is abolished, if the dummy indices are surrounded by parentheses, i.e. a (j) b (j) merely means the product of the jth components a j and b j. Let dx j denote an infinitesimal increment of the jth coordinate, whereas the other coordinates are kept constant. From (1-6) follows that this induces a change of the position vector given as dx = i (j) dx (j). Hence, i j = x x j (1 7) A corresponding independent infinitisimal increment dθ j of the jth curvilinear coordinate induce a change of the position vector given as dx = g (j) dθ (j), so g j = x θ j (1 8) g j is tangential to the curve s j at the point P, see Fig 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 indicated vector 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 equal to the unit tangent vectors t j, i.e. t j = x s j Use of the chain rule of partial differentiation provides (1 9) g j = x θ j = x x k x k θ j = ck j i k (1 10) i j = x x j = x θ k θ k x j = dk j g k (1 11) where

10 10 Chapter 1 Preliminaries c k j = xk θ = fk j θ j d k j = θk x = hk j x j (1 12) c k j specify the Cartesian components of the covariant base vector g j. Similarly, d k j specifies the components in the covariant base of the Cartesian base vector i j. Obviously, the following relation prevails c k l dl j = xk θ l θ l x j = δk j (1 13) δ k j denotes Kronecker s delta in the applied notation, defined as δ k j = { 0, j k 1, j = k (1 14) Fig. 1 2 Covariant and contravariant base vectors. Generally, the covariant base vectors g j are neither orthogonal nor normalized to the length 1, as is the case for the Cartesian base vectors i j. In order to perform similar operations on components 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 defined from g j g k = δ k j (1 15) 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 Generally, the contravariant base vectors can be determined from the covariant base vectors by means of the vector products

11 1.1 Tensor Calculus 11 g 1 = 1 V g 2 g 3 g 2 = 1 V g 3 g 1 g i = 1 V eijk g j g k (1 16) g 3 = 1 V g 1 g 2 where V denotes the volume of the parallelepiped spanned by the covariant base vectors g j, see Fig This is given as V = g 1 (g 2 g 3 ) = g 2 (g 3 g 1 ) = g 3 (g 1 g 2 ) (1 17) e ijk is the permutation symbol defined as 1, (i, j, k) = (1, 2, 3), (2, 3, 1), (3, 1, 2) e ijk = 1, (i, j, k) = (1, 3, 2), (3, 2, 1), (2, 1, 3) 0, else (1 18) The permutation symbol does not indicate the components of a 3rd order tensor in any coordinate system, and should merely be considered as an indexed sequence of numbers. For this reason we shall not make distinction between upper and lover indices, so we may write e ijk = e ijk. The permutation symbol and the Kronecker delta are related by the following so-called e δ relation e ijk e klm = δ i l δj m δi m δj l (1 19) In the Cartesian coordinate system we have i j = i j, i.e. the dual vector basis is identical to original. A vector is envisioned as a geometric quantity in space (an arrow with a given length and orientation). From this interpretation it follows that a vector is independent of any coordinate system used for its specification. Actually, infinitely many coordinate systems can be used for the representation (or decomposition) of one and the same vector. In the Cartesian, the covariant and the contravariant bases a given vector v can be represented in the following ways v = v j i j = v j g j = v j g j (1 20) where v j, v j and v j denotes the Cartesian vector components, the covariant vector components, and the contravariant vector components of the vector v. Generally, Cartesian components of vectors and tensors will be indicated by a bar. Use of (1-15), and scalar multiplication of the two last relations in (1-20) with g k and g k, respectively, provides the following relations between the covariant and contravariant vector components

12 12 Chapter 1 Preliminaries v j = g jk v k v j = g jk v k (1 21) where g jk = g j g k = g kj g jk = g j g k = g kj (1 22) The indicated symmetry property of the quantities g jk and g jk follows from the commutativity of the involved scalar products. From (1-21) follows v j = g jl v l = g jl g lk v k g jl g lk = δ k j (1 23) By the use of (1-20) and (1-21) the following relations between the covariant and the contravariant base vectors may be derived v j g j = v j g j = g jk v k g j = v j g jk g k v j g j = v j g j = g jk v k g j = v j g jk g k { gj = g jk g k g j = g jk g k (1 24) As seen g jk represent the components of g j in the contravariant vector base. Similarly, g jk signify the components of g j in the covariant vector base. Use of (1-10) and (1-11) provides the following relation between the Cartesian and the covariant vector components v j i j = v j g j = v j c k ji k = c j k vk i j v j g j = v j i j = v j d k j g k = d j k vk g j { v j = c j k vk v j = d j k vk (1 25) Finally, using (1-21) the scalar product of two vectors u and v can be evaluated in the following alternative ways { ūj v j = u j v j = g jk u j v k u v = (1 26) ū j v j = u j v j = g jk u j v k where it is noticed that ū j = ū j Tensors, dyads and polyads A second order tensor T is defined as a linear mapping of a vector v onto a vector u by means of a scalar product, i.e. u = T v (1 27)

13 1.1 Tensor Calculus 13 Since the vectors u and v are coordinate independent quantities, the 2nd order tensor T as well must be independent of any selected coordinate system chosen for the specification of the relation (1-27). Equations in solid mechanics are independent of the chosen coordinate system for which 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 by three 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) = mnab (1 28) where m and n are arbitrary constants. Further, the following distributive rules are valid a (b + c) = ab + ac (1 29) (a + b)c = ab + bc No commutative rule is valid for dyads formed by two vectors a and b. Hence, in general ab ba (1 30) If the outer product of two vectors entering a polyad is replaced by a scalar product of the same vectors 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)c ab c = (b c)a (1 31) The scalar product of two dyads, the so-called double contraction, can be defined in two ways ab cd = (a c)(b d) ab cd = (a d)(b c) (1 32) As seen the symbol defines scalar multiplication between the two first and the two last vectors in the two duads, whereas defines scalar multiplication between the first and the last vector, and the last and the first 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) (1 33)

14 14 Chapter 1 Preliminaries Use of (1-32) and (1-33) provides the following identities for double contraction of two dyads ab cd = ba dc = cd ab = dc ba ab cd = ba dc = cd ab = dc ba (1 34) From the second relation of (1-31) follows that the dyad ab/(b c) is mapping the vector c onto the vector a via a scalar multiplication. From the definition (1-27) follows that this quantity is a second order tensor. Now, it can be proved that any second order tensor can be represented as a linear combination of nine dyads, formed as outer product of three arbitrary linearly independent base vectors. Hence, we have following alternative representations of a tensor T in the Cartesian, the covariant and the contravariant bases T = T jk i j i k = T jk g j g k = T jk g j g k (1 35) The dyads i j i k, g j g k and g j g k form so-called tensor bases. Obviously, (1-35) represents the generalization to second order tensors of (1-20) for the decomposition of a vector in the corresponding 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 tensor components related to the considered three tensor bases may be derived i j i k = d l j dm k g lg m g j g k = c l j cm k i li m = g jl g km g l g m g j g k = g jl g km g l g m (1 36) T jk = c j l ck m T lm, T jk = d j l dk m T lm T jk = g jl g km T lm, T jk = g jl g km T lm (1 37) As seen, c l jc m k and g jlg km specify the Cartesian and contravariant tensor components of the dyad g j g k, whereas g jl g km denotes the covariant tensor components of g j g k. (1-36) and (1-37) represent 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 and curvilinear tensor components are used as a defining property of the tensorial character of a doubled 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 j k g jg k = T j k gk g j (1 38) T j k and T j k represent the mixed covariant and contravariant tensor components. In general, T j k T j k as a consequence of g jg k g k g j, i.e. the relative horizontal position of the upper and lower indices of the tensor components is important, and specify the sequence of covariant

15 1.1 Tensor Calculus 15 and contravariant base vector in the dyads of the tensor base. Tensor bases with dyads of mixed Cartesian and curvilinear base vectors may also be introduced. However, in what follows only the mixed curvilinear dyads in (1-38) will be considered. The following identities may be derived from (1-35) and (1-38) by the use of (1-24) T = T jk g j g k = T j l g jg l = T j l glk g j g k T jk = g lk T j l T = T jk g j g k = T k l g l g k = T k l g lj g j g k T = T jk g j g k = T l k g l g k = T l kg lj g j g k T = T jk g j g k = Tj l gj g l = Tj l g lk g j g k T jk = g jl T k l T jk = g jl T l k T jk = g lk Tj l (1 39) 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 = g jl T lk T k j T k j = gkl T lj T k j = g kl T jl (1 40) 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 tensor for which the covariant components fulfill the indicated symmetry property is denoted a symmetric tensor. Let T jk denote the covariant components of a second order tensor T. The related so-called transposed tensor T T is defined as the tensor with the covariant components T kj, i.e. T T = T kj g j g k (1 41) For a symmetric tensor T jk = T kj for which reason T T = T. The identity tensor is defined as the tensor with the covariant and contravariant components g jk and g jk, i.e. g = g jk g j g k = g jk g j g k = δ k j g j g k (1 42) The mixed components follows from (1-23) and (1-40) g k j = g jl g kl = δ k j g k j = g jl g lk = δ k j (1 43) Hence, the mixed components are equal to the Kronecker s delta, which explains the last statement for the mixed representation in (1-42). The name identity tensor stems from the fact that g maps any vector v onto itself. Actually,

16 16 Chapter 1 Preliminaries g v = g jk g j g k v l g l = g jk v l g j δ l k = gjk v k g j = v j g j = v (1 44) The length of a vector v is determined from v 2 = v v = v g v = g jk v j v k = g jk v j v k (1 45) Because of its relationship to the length of a vector the identity tensor is also designated the metric tensor or the fundamental tensor. The increment dx of the the position vector x as given by (1-6), due to independent increment dx j and dθ j of the Cartesian or curvilinear coordinates becomes dx = dx j i j = dθ j g j (1 46) Then, the length ds of the incremental position vector becomes, cf. (1-45) ds 2 = dx j dx j = g jk dθ j dθ k (1 47) The inverse tensor T 1 of the tensor T is defined from the identity T 1 T = T T 1 = g (1 48) Let T 1 jl denote the contravariant components of T 1, and T mk the covariant components of T. Then, g = δj k 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 δm l g j g k = T 1 jl T lk g j g k T 1 jl T lk = δ k j (1 49) In matrix notation this means that the three dimensional matrix, which stores the components T 1 jl, is the inverse of the matrix, which stores the components T lk. Similarly, the covariant components (T 1 ) jl of T 1 and the contravariant components T lk of T are stored in inverse matrices. 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 l g m ) g k = T lm δ j l δk m ) = T jk T jk = g j T g k Similarly, (1 50)

17 1.1 Tensor Calculus 17 T jk = g j T g k = g j T g k T k j T j k = gj T g k T jk = i j T i k (1 51) Next, the second Piola-Kirchhoff stress tensor S and its virtual work conjugated strain measure the Green strain tensor E are considered. In the Cartesian, covariant, contravariant and mixed tensor bases these are represented as follows S = S jk i j i k = S jk g j g k = S jk g j g k = S k j g j g k = S j k g jg k E = Ējk i j i k = E jk g j g k = E jk g j g k = E k j g j g k = E j k g jg k (1 52) Because the Piola-Kirchhoff stress tensor and the Green strain tensor are both symmetric tensors their mixed tensor components fulfill Sj k = S j k and E j k = E j k. For this reason we shall simply use the notaions Sj k and Ek 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 g km S lm, S jk = g jl Sl k S jk = g jl g km S lm, S jk = g jl Sk l Sj k = g kl S lj, Sj k = g jl S lk S jk = d j l dk S m lm, Sjk = c j l ck m Slm (1 53) Completely analog relations prevail for the components of the Green tensor. For a physical linear material a linear mapping of the Green tensor onto the Piola-Kirchhoff stress tensor via a double contraction with the elasticity tensor C, i.e. S = C E (1 54) C is a fourth order tensor, which can be specified 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 jklm g j g k g l g m = C klm j = C jkl m g jg k g l g m = C lm jk g j g k g l g m = C j lm k g j g k g l g m = C k m j l g j g k g l g m = C jk m l g j g k g l g m g j g k g l g m = Cj kl m gj g k g l g m = C j m kl g j g k g l g m = C j l k m g jg k g l g m = C jk lm g jg k g l g m = C j klm g jg k g l g m = Cj k lm gj g k g l g m = Cjk l m gj g k g l g m = Cjkl m g j g k g l g m = C jklm g j g k g l g m = C jklm i j i k i l i m (1 55)

18 18 Chapter 1 Preliminaries Formally, the relations between the various mixed tensor components can be derived by raising and lowering indices by means of the covariant components g jk and the contravariant components g jk of the identity tensor g, cf. (1-53). As an example C jk lm = g lrg ms C jkrs (1 56) Normally, the components of the elasticity tensor is specified in the Cartesian coordinate system. The relation between the Cartesian components C rstu and the covariant components C jklm becomes, cf. (1-53) C jklm = d j r dk s dl t dm u C rstu (1 57) 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 jg k g l g m E rs g r g s = C jk lm Ers g j g k (g l g m g r g s ) = C jk lm Ers g j g k (g l g r )(g m g s ) = C jk lm Ers δ l r δm s g jg k = C jk lm Elm g j g k S jk = C jk lm Elm (1 58) If the alternative definition of the double contraction in (1-32) is used, the following result is obtained for the tensor representation of S S = C E = C jk lm Eml g j g k (1 59) Since, E lm = E ml, the indicated definitions provide the same result for the stress components. Expressed in the mixed components of E and S the constitutive equations on components form may be given in the following ways S k j = C k l j m Em l = C kl j m Em l = C j l k m Em l = C j l km Em l (1 60) Clearly, the symmetry of the stress and strain tensor implies the following symmetry properties of the mixed elasticity tensor components C jk lm = Ckj lm = Cjk ml = Ckj ml and C j k m l = Cj kl m = Cj l l k m = Cj km. 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 Ej k (1 61)

19 1.1 Tensor Calculus Gradient, covariant and contravariant derivatives Consider a scalar function a = a(x l ) = a(θ l ) of the Cartesian or curvilinear coordinates. The gradient of a, denoted as a, is a vector with the following Cartesian and contravariant representations a = a x j i k = a θ j gj (1 62) Let θ k denote the increments of the curvilinear coordinated, and define the incremental vector θ = θ k g k. The corresponding increment a of the scalar is determined by a = a θ = a θ j gj θ k g k = a θ j θj (1 63) The gradient of a vector function v = v(x l ) = v(θ l ), denoted as v, is a second order tensor, which in complete analogy to (1-63) associates to any incremental vector θ = θ k g k a corresponding increment v = v θ = v θ j of the vector v. The gradient tensor has θ j the following representations v = v x i k k = ( vj i j ) i x k k = vj x i ji k k ( v θ k gk = (vj g j ) v g k j = θ k v θ k gk = (v j g j ) g k = θ k θ g k j + v j g j θ k ( vj g j θ k gj + v j θ k ) g k ) g k (1 64) At the derivation of the Cartesian representation it has been used that the base vector i j is constant as a function of x l. In contrast, the curvilinear base vectors depend on the curvilinear coordinates, which accounts for the second term within the parentheses in (1-64). Clearly, g j θ k and gj are vectors, which may hence be decomposed in the covariant and contravariant vector θ k bases as follows g j θ k = g j θ k = { l j k { j l k g l g l (1 65) g signify the covariant components of j, and { j θ k l k is the contravariant components of g j. θ k j k is denoted the Christoffel symbol. This is related to the co- and contravariant components { l j k { l of the identity tensor as follows { l j k = 1 ( gjm 2 glm θ + g km g jk k θ j θ m ) (1 66)

20 20 Chapter 1 Preliminaries (1-65) and (1-66) have been proved in Box 1.1. From (1-66) follows that the Christoffel symbols fulfill the symmetry condition { { l l = (1 67) j k k j From (1-8) follows that g j θ k = 2 x θ j θ k = 2 x θ k θ j = g k θ j (1 68) 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 first relation in (1-65) as a definition of the Christoffel symbol, and prove that this implies the second. From (1-15) and the first relation in (1-65) follows θ kδj m = ( gm g j) = g m θ k { g j l θ g k m = m k { j g l g m = 0 l k θ k gj + g m gj θ = 0 k { g l g j l = δ j l = m k { { j j = δm l = m k l k ( { ) g j j θ + g l g k m = 0 (1 69) l k Since, (1-69) is valid for any of the three covariant base vectors the term within the bracket must be equal to 0. This proves the validity of the second relation (1-65). From (1-22) and the first relation (1-65) follows g jm θ = (g j g m ) = g j k θ k θ g k m + g m θ g k j = { { n n g nm + g nj (1 70) j k m k

21 1.1 Tensor Calculus 21 From (1-70) follows g jm θ + g km g jk k θ j θ = m { { n n g nm + g nj + j k m k 2 { n g nm j k { { { { n n n n g nm + g nk g nk g nj = k j m j j m k m (1 71) where the symmetry property (1-67) has been used. Next, (1-66) follows from (1-71) upon multiplication on both sides with g lm and use of (1-23). Use of (1-65) in (1-66) provides the following representations of v v = v j ;k g jg k = v j;k g j g k (1 72) where { v j ;k = vj j θ + k k l v j;k = v { j l θ k j k v l v l (1 73) Hence, v j ;k specifies the mixed co-and contravariant components of v, and v j;k specifies the contravariant components. For the partial derivative of the vector function v(θ l ) the following representations may be derived by the use of (1-64) v θ k = (v j g j ) θ k (v j g j ) θ k = vj θ kg j + g j θ k vj = v j ;k g j = v j θ kgj + gj θ k v j = v j;k g j (1 74) Hence, v j ;k and v j;k may alternatively be interpreted as the covariant components and the contravariant components of the vector v. For this reason v j θ k ;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. In the 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 the covariant, mixed and contravariant tensor bases become

22 22 Chapter 1 Preliminaries θ l(g jg k = g j θ l g k + g j g k θ l = θ l(g jg k ) = g j θ l gk + g j g k θ l = θ l(gj g k ) = gj θ l g k + g j g k θ l = θ l(gj g k ) = gj θ l gk + g j gk θ l = { m j l { m j l { j m l { j m l { m g m g k + k l { g m g k k { g m m g k + { g m g k k m l g j g m g j g m m l g j g m k l g j g m (1 75) Finally, the covariant and covariant representations of the increment v of the vector v due to the incremental vector θ = K g k of the curvilinear coordinates becomes v = v θ = v j ;k g jg k θ l g l = v j;k g j g k θ l g l v = v j ;k θk g j = v j;k θ k g j (1 76) The gradient of a second order tensor function T = T(θ l ), denoted as T, is a third order tensor with the following representations T = T θ l gl = T = ( (T jk g j g k ) T g l jk = θ l θ l (T j k g jg k ) θ l g l = (T k j g j g k ) θ l g l = (T jk g j g k ) θ l g l = ( T jk + T mk θ l ( T j k + T m θ l k ( T k j θ l ( Tjk θ l T k m T km ( T j k g j g k + T jk g j θ g l k + T jk g k g j θ l θ l g j g k + T j k ( T k j g j θ l gk + T j k g g k j θ l g j θ g l k + Tj k g j g k θ l g j g θ l k + Tj k ( Tjk g j g k g j + T θ l jk θ l gk + T jk g j gk θ l { j m l { m k l ) g l ) ) g l g l ) g l { { ) j + T jm k g j g k g l = T jk ;l m l m l g jg k g l ) T j m g j g k g l = T j k;l g jg k g l { { ) m + Tj m k g j g k g l = Tj k ;l j l m l gj g k g l ) { m j l T jm { m k l g j g k g l = T jk;l g j g k g l (1 77)

23 1.1 Tensor Calculus 23 where T jk ;l = T jk θ l T j k;l = T j k θ l j ;l = T j k θ l T k T jk;l = T jk θ l { { + T mk j + T jm k m l m l { { + T m j k T j m m m l k l { { Tm k m + Tj m k T km j l { m j l m l T jm { m k l (1 78) where (1-65) has been used in the last statement. Then, the partial derivative of the tensor function T(θ m ) may be written in any of the following alternative second order tensor representations, cf. (1-74) T θ l = T jk ;l g jg k = T j k;l g jg k = T jk;l g j g k (1 79) Partial differentiation of both sides of (1-44) provides v θ = g k θ v + g v k θ = g k θ v + v k θ k g θ v = 0 k Since v is arbitrary (1-80) implies that g θ k = 0 (1 80) (1 81) In turn this means that g = 0. Hence, the identity tensor is constant under covariant differentiation Riemann-Christoffel tensor The gradient of a vector v with decomposition into Cartesian and curvilinear tensor bases have been indicated by (1-64). The gradient of this second order tensor is given as, cf. (1-73), (1-74) ( ) v x l x i k k i l = ( ) ( v j i j ) i x l x k k i l = 2 v j x k x i ji l k i l ( ) ( ( v v ( v) = θ l θ k gk g l = { ) j j v )g θ l θ + m k j g k g l = v j ;kl k m g jg k g l ( ) ( v θ l θ k gk g l = ( vj { ) m v θ l θ k m )g j g k g l = v j;kl g j g k g l j k (1 82)

24 24 Chapter 1 Preliminaries where the following tensor components have been introduced { { { { { v j ;kl = (vj ;k ) ;l = 2 v j j v m j v m m v j j n θ k θ + + l k m θ l l m θ k k l θ v m + m n m k l { { { j v m j n + v m (1 83) θ l k m l n k m { { { { { v j;kl = (v j;k ) ;l = 2 v j m θ k θ vm m vm m l j k θ l j l θ vj n m k k l θ + v m m k l n j { { { m n m v θ l m + v m (1 84) j k j l n k From (1-83) and (1-84) follow that the indices k and l can be interchanged in the first five terms on the right hand sides without changing the value of this part of the expressions, whereas this is not the case for the last two terms. This implies that the sequence in which the covariant differentiations is performed is significant, i.e. in general v j;kl v j;lk. In order to investigate this further the so-called Riemann-Christoffel tensor R is introduced, which is a fourth order tensor with the mixed representation R = R m jkl g mg j g k g l, where the mixed curvilinear components are defined as R m jkl = Obviously, { n j l { m n k { { n m + { m { m j k n l θ k j l θ l j k (1 85) R m jkl = Rm jlk Further, the following relations are valid R m jkl + Rm klj + Rm ljk = 0 R j jkl = 0 (1 86) (1 87) 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 R m jkl v m = v j;kl v j;lk (1 88) The Cartesian components of R follow from (1-82) R jklm v m = 2 v j x k x 2 v j l x l x = 0 k R jklm = 0 (1 89) Hence, it can be concluded that R = 0 in an Euclidean three-dimensional space, i.e. a space spanned by a Cartesian vector basis. In turn this means that also the curvilinear components

25 1.1 Tensor Calculus 25 (1-85) must vanish in this space. A space, where everywhere R = 0 is called flat. Reversely, a non-vanishing Riemann-Cristoffel tensor indicates a curved space. In a flat space R m jkl = 0, with the implication that v j;kl = v j;lk. The three-dimensional Eucledian space is flat, and any plane in this space forms a flat two-dimensional subspace. In contrast, a curved surface in the Eucledian space is not a flat subspace. An example of a curved four-dimensional space is the time-space continuum used at the formulation of the general theory of relativity, where the indices correspondingly range over j = 1, 2, 3, 4. Because of the identities (1-86), (1-87) it can be shown that only 1 12 N2 (N 2 1) of the tensor components R m jkl are independent and non-trivial, where N denotes the dimension of the space, (Spain, 1965). Hence, for a two-dimensional space only one independent component exists, which can be chosen as R In the three-dimensional case six independent and nontrivial components exist, which are chosen as R 1 212, R1 213, R1 223, R1 313, R1 323 and R Example 1.1: Covariant base vectors, identity tensor and Riemann-Christoffel tensor in spherical coordinates By the use of (1-10) the first equation (1-65) becomes (c m j i { m) θ k = cm j l θ k i m = c m l i m j k { c m j l θ k = c m l j k (1 90) The Cartesian components c m j of the covariant base vector g j is stored in the column matrix g j = [c m j ]. Then, (1-88) may be written in the following matrix form { { { { g j l θ k = g j k l = g j k 1 + g j k 2 + g j k 3 (1 91) The spherical coordinate system defined by (1-1) is considered. In this case the column matrices attain the form, cf. (1-12) g 1 = θ 3 cosθ 1 cosθ 2 θ 3 cosθ 1 sin θ 2, g 2 = θ 3 sin θ 1 θ 3 sin θ 1 sin θ 2 θ 3 sin θ 1 cosθ 2, g 3 = 0 sinθ 1 cosθ 2 sinθ 1 sin θ 2 (1 92) cosθ 1 The covariant components of the identity tensor is given by (1-22) as g jk = g j g k = g Tg, where the last j k statement is obtained by evaluating the scalar product in Cartesian coordinates. The covariant and contravariant components of the identity tensor are conveniently stored in matrices. Using (1-92) these becomes (θ 3 ) [ g jk ] = 0 (θ 3 ) 2 sin 2 θ , 1 [ g jk ] (θ 3 ) = 1 0 (θ 3 ) 2 sin 2 0 θ (1 93) The result for the contravariant components follows from (1-23). Next, (1-91) will be used to determine the Christoffel symbols by direct specification of the decomposition on the right hand side. Using (1-92) the following results are obtained

26 26 Chapter 1 Preliminaries g 1 θ 1 = θ 3 sin θ 1 cosθ 2 θ 3 sin θ 1 sinθ 2 = 0 g g 2 θ 3 g 3 θ 3 cosθ 1 θ g 3 cosθ 1 sin θ 2 1 θ 2 = θ 3 cosθ 1 cosθ 2 = 0 g tan θ 1 g + 0 g cosθ g 1 cosθ 2 1 θ 3 = cosθ 1 sin θ 2 = 1 θ 3 g + 0 g + 0 g sin θ 1 θ g 3 cosθ 1 sin θ 2 2 θ 1 = θ 3 cosθ 1 cosθ 2 = 0 g tan θ 1 g + 0 g g 2 θ 2 = θ 3 sin θ 1 cosθ 2 θ 3 sin θ 1 sinθ 2 = 1 2 sin(2θ1 ) g g 2 θ 3 sin 2 θ 1 g 3 0 sin θ g 1 sin θ 2 2 θ 3 = sin θ 1 cosθ 2 = 0 g θ 3 g + 0 g cosθ g 1 cosθ 2 3 θ 1 = cosθ 1 sin θ 2 = 1 θ 3 g + 0 g + 0 g sin θ 1 sin θ g 1 sin θ 2 3 θ 2 = sin θ 1 cosθ 2 = 0 g θ 3 g + 0 g g 3 θ 3 = 0 0 = 0 g g g 3 0 { { { { { { { { { { { { { { { { { { { { { { { { { { { = 0 = 0 = θ 3 = 0 1 = tan θ 1 = 0 = 1 = 0 = 0 θ 3 = 0 1 = tan θ 1 = 0 1 = 2 sin(2θ1 ) = 0 = θ 3 sin 2 θ 1 = 0 1 = = 0 θ 3 = 1 = 0 = 0 θ 3 = 0 1 = = 0 = 0 = 0 = 0 θ 3 (1 94)

27 1.1 Tensor Calculus 27 The non-trivial components of the Riemann-Christoffel tensor follow from (1-85) and (1-94) R = { { { { n 1 n n n sin(2θ1 ) θ 3 sin 2 θ 1 R = { { { n 1 n 2 3 n { { + 1 θ θ θ tan θ sin(2θ1 ) 0 0 cos(2θ 1 ) 0 = 0 { 1 n 3 { + 1 θ θ θ θ 3 1 tan θ = 0 { { n 1 R = 2 3 n θ 3 1 R = { n { { n n 3 { 1 θ { { + 1 θ θ sin(2θ1 ) sin(2θ1 ) θ θ3 sin 2 θ = 0 { { { { { 1 n n n 3 θ θ 3 = θ 3 1 θ 3 1 θ (θ 3 ) 2 = 0 { { { { n 1 n 1 R = 3 3 n n sin(2θ1 ) { { n 2 R = 3 3 n 2 0 { { + 1 θ θ θ = 0 { { n n 3 { { + 2 θ θ tan θ θ θ 3 1 θ (θ 3 ) 2 = 0 = = = = = (1 95) As expected all components of the Riemann-Christoffel tensor vanish as a consequence of the flatness of the threedimensional Euclidean space.

28 28 Chapter 1 Preliminaries 1.2 Differential Theory of Surfaces Differential geometry of surfaces, first fundamental form Fig. 1 3 a) Parameter space. b) Surface space. Let the spherical θ 3 coordinate be fixed at the value θ 3 = r. Then, the mapping (1-1) of the spherical coordinates onto the Cartesian coordinates takes the form x 1 x 2 x 3 = r 1 r 2 r 3 r sin θ 1 cosθ 2 = r sin θ 1 sin θ 2 (1 96) r cosθ 1 (r 1, r 2, r 3 ) denotes the Cartesian components of the position vector r to a given point of the surface. Obviously, (r 1 ) 2 + (r 2 ) 2 + (r 3 ) 2 = r 2. Hence, with the zenith angle varied in the interval θ 1 [0, π], and the azimuth angle varied in the interval θ 2 ]0, 2π], (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 defined by a constant value of the curvilinear coordinate θ 3 = c 3 in the mapping (1-2). Then, a given surface Ω is given as, cf. (1-5) r j = f j (θ 1, θ 2, c 3 ) = f j (θ α ) (1 97) In the last statement of (1-97) the explicit dependence on the constant c 3 is ignored, as will also be the case in the following. Let the mapping (1-97) be defined within a domain ω in the parameter space. For each point p ω determined by the parameters (θ 1, θ 2 ), a given point P is defined on Ω with Cartesian coordinates given by (1-97). Assume that a curve through p is specified by the parametric description ( θ 1 (t), θ 2 (t) ), where t is the free parameter. Then, this curve is mapped onto a curve s = s(t) through P on Ω as shown on Fig Especially, if the curvilinear coordinate θ 1 is fixed, whereas θ 2 is varied independently a curve s 1 through P is defined on Ω. Similarly, a curve s 2 on the surface through P is obtained, if θ 1 is fixed and θ 2 is varied. The positive direction of s 1 and s 2 are defined, so positive increments of θ α correspond

29 1.2 Differential Theory of Surfaces 29 to positive increments of s α. Then, these curves define a local two-dimensional arch length coordinate system (s 1, s 2 ) through P. Assume that ω is a rectangular domain [a 1, a 2 ] [b 1, b 2 ] with sides parallel to the θ α axes as shown on Fig. 1-3a. As an example this is the case for the mapping (1-96), where ω = [0, π] ]0, 2π]. In such cases the surface Ω will be bounded by the arc length coordinate curves given by the parameter descriptions r j = f j (a 1, θ 2 ), r j = f j (a 1, θ 2 ), r j = f j (θ 1, b 1 ) and r j = f j (θ 1, b 2 ), see Fig. 1-3b. Since dθ 3 = 0, when merely θ 1 and θ 2 are varied in ω, the increment dr of the position vector r to the point P on the surface is given as by the following Cartesian and curvilinear representations, cf. (1-6) dr = dr 1 i 1 + dr 2 i 2 + dr 3 i 3 = dθ 1 a 1 + dθ 2 a 2 dr = dr j i j = dθ α a α where the covariant base vectors becomes, cf. (1-8) (1 98) a α = r θ α (1 99) Fig. 1 4 Covariant and contravariant base vectors in the tangent plane. a α are tangential to the arch length curves at P, see Fig. 1-3b. a α may then be interpreted as a local two-dimensional covariant vector bases, which spans the tangent plane A at the point P, see Fig The indicated covariant vectors may be represented in the Cartesian vector basis as, cf. (1-10) a α = c j α i j (1 100) where the Cartesian components c j α are given by (1-12).

30 30 Chapter 1 Preliminaries At the point P a unit normal vector a 3 to the surface and the tangent plane is defined as a 3 = a 1 a 2 a 1 a 2 = 1 A a 1 a 2 (1 101) 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 ϕ (1 102) where ϕ denotes the angle between a 1 and a 2, see Fig Using (1-16) with V = 1 A = A, a contravariant basis at P may be defined by the relations a 1 = 1 A a 2 a 3 = 1 (A) 2 a 2 (a 1 a 2 ) = 1 (A) 2 ( a 2 2 a 1 (a 1 a 2 )a 2 ) a 2 = 1 A 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 1 ) (1 103) a 3 = 1 A a 1 a 2 = a 3 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 the covariant and contravariant base vectors in the tangent plane becomes, cf. (1-15) a α a β = δ β α (1 104) (1-104) may alternatively be proved from the last statements in (1-103), using (1-102) and a 1 a 2 = a 1 a 2 cosϕ. The unit normal vector a 3 = a 3 is orthogonal to both the covariant and the contravariant base vectors in the tangent plane, so a 3 a α = a 3 a α = 0 (1 105) Then, the covariant and contravariant components g jk and g jk of the three-dimensional identity tensor can be stored in the following matrices a 11 a 12 0 [ g jk ] = a 21 a 22 0, [ ] a 11 a 12 0 g jk = a 21 a 22 0 (1 106) A surface vector function v = v(θ 1, θ 2 ) is a vector field, which everywhere (i.e. for any parameters (θ 1, θ 2 ) ) is tangential to the surface. Then, v may be represented by the following Cartesian, covariant and contravariant representations

31 1.2 Differential Theory of Surfaces 31 v = v j i j = v α a α = v α,a α (1 107) where v j, v α and v α denote the Cartesian, the covariant and the contravariant components of v. v 3 = v 3 = 0 for a surface vector, for which reason only two components enter the curvilinear representations. The Cartesian and the covariant components are related as, cf. (1-25) v j = c j α vα v α = d α j v j (1 108) Similarly, the covariant and contravariant components are related by the following relations analog to (1-22) v α = a αβ v β v α = a αβ v β (1 109) where a αβ = a α a β a αβ = a α a β (1 110) As a consequence of the structure (1-106) of the covariant and contravariant components of the identity tensor it follows that, cf. (1-23) a αγ a γβ = δ β α (1 111) where δ β α denotes the Kronecker s delta in two dimensions. Both a α and a α are surface vectors. Then, these can be expanded in two-dimensional contravariant and covariant vector bases as follows, cf. (1-24) a α = a αβ a β a α = a αβ a β (1 112) In analogy to (1-27), a surface second order tensor is defined as a second order tensor, which everywhere 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 and mixed representations, cf. (1-35) T = T jk i j i k = T αβ a α a β = T αβ a α a β = T α β a α a β = T β α a α a β (1 113)

32 32 Chapter 1 Preliminaries T αβ, T αβ, T α β and T α β denotes the covariant, the contravariant and the mixed covafriant and contravariant components of the surface tensor. These are related as, cf. (1-39), (1-40) T αβ = a αγ a βδ T γδ, T α β = aαγ T γβ T αβ = a αγ a βδ T γδ, T β α = a αγ T γβ T αβ = a αγ T γ β, T αβ = a αγ T β γ (1 114) The surface identity tensor a is defined as a second order tensor with the covariant components a αβ, the contravariant components a αβ, and the mixed components δα β, corresponding to the following representations, cf. (1-42) a = a αβ a α a β = g αβ a α a β = δ β α aα a β (1 115) a will map any surface vector onto itself, i.e. a v = v (1 116) (1-116) is proved in the same way as (1-44), using the representation (1-115). Let dr be a differential increment of the position vector r due to an increment dθ α of the parameters. Obviously, dr is a surface vector with the representation (1-98). The length ds of the incremental vector becomes ds 2 = dr dr = dr a dr = dθ α a α a β dθ β = a αβ dθ α dθ β (1 117) Obviously, the tensor a determines the length of any surface vector, for which reason the alternative naming surface metric tensor is used. The quadratic form in the last statement of (1-117) is called the first fundamental form of the surface. The partial derivatives of the covariant and contravariant base vectors are unchanged given by (1-65) a j θ α = a j θ α = { l a l = j α { j a l l α { { β 3 a β + j α j α { a β j 3 α { j = β α a 3 a 3 (1 118) a 3 = a 3 is orthogonal to both a β and a β, so a 3 a β = 0 and a 3 a β = 0. Then, scalar multiplication of the first equation in (1-118) with a 3 and scalar multiplication of the second equation with a 3 provides a 3 a j θ α = a 3 aj θ α = { 3 j α { j 3 α (1 119)

33 1.2 Differential Theory of Surfaces 33 where it has been used that a 3 a 3 = 1. Because a 3 = a 3, the left hand sides of (1-119) are identical for j = 3. However, the right hand sides have opposite signs in this case. Both can then only be truth if the right hand sides vanish, leading to the following result for the Christoffel symbol { 3 = 0 3 α (1-120) implies that a 3 θ α surface vector. (1 120) is orthogonal to the surface normal a 3 = a 3, and hence must be a Obviously, the increment vector θ = θ β a β with the covariant components θ β is a surface vector. The gradient v of a surface vector function v = v(θ γ ) is a surface second order tensor, which to the increment vector θ associates a surface increment vector v = v θ = v θ α θ α 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 α ;β a αa β = v α;β a α a β (1 121) where { v α ;β = vα θ + α β β γ v α;β = v α θ β { γ α β v γ v γ (1 122) v α ;β and v α;β will be referred to as the surface contravariant derivative and surface covariant derivative of the components v α and v α. Correspondingly, the gradient T of a surface second order tensor function T = T(θ γ ) is a surface third order tensor with the following curvilinear representations, cf. (1-77), (1-78) T = T αβ ;γ a αa β a γ = T α ;βγ a αa β a γ = T β α ;γ aα a β a γ = T αβ;γ a α a β a γ (1 123) where { { T αβ αβ T ;γ = + T δβ α + T αδ β θ γ δ γ δ γ T α β;γ = T α { { β + T δ α θ γ β T α δ δ δ γ β γ Tα β ;γ = T { { α β T β δ θ γ δ + Tα δ β α γ δ γ T αβ;γ = T { { αβ δ δ T θ γ βδ T αδ α γ β γ (1 124)

34 34 Chapter 1 Preliminaries The gradient of the gradient of a surface vector is a third order surface tensor with the following representations, cf. (1-82), (1-83), (1-84) ( v) = v α ;βγ a αa β a γ = v α;βγ a α a β a γ (1 125) where { { { { { v α ;βγ = (vα ;β ) ;γ = 2 v α α v µ α v µ µ v α α ν θ β θ + γ β µ θ + γ γ µ θ β β γ θ v µ + µ ν µ β γ { { { α v µ α ν + v µ (1 126) θ γ β µ γ ν β µ { v α;βγ = (v α;β ) ;γ = 2 v α µ θ β θ γ α β { { { µ ν µ v θ γ µ + α β α γ ν β { { { { vµ µ θ vµ µ γ α γ θ vα ν µ β β γ θ + v µ µ β γ ν α v µ (1 127) The surface Riemann-Christoffel tensor is denoted B = B(θ 1, θ 2 ) to distinguish it from the equivalent tensor R in the three dimensional space. The mixed covariant and contravariant representation becomes, cf. (1-85) B = B δ αβγ a δ a α a β a γ (1 128) { { { { B δ ν δ ν δ αβγ = + { δ { δ α γ ν β α β ν γ θ β α γ θ γ α β (1 129) In analogy to (1-86) - (1-88) the tensor components B δ αβγ fulfill the following identities B δ αβγ = Bδ αγβ B δ αβγ + Bδ βγα + Bδ γαβ = 0 B α αβγ = 0 (1 130) B δ αβγ v δ = v α;βγ v α;γβ (1 131) Due to the identities (1-130) only one independent and non-trivial component exists, which is taken as B From (1-85) and (1-129) follow that the components in the tangent plane R δ αβγ of the Riemann- Christoffel tensor in the three-dimensional space is related to the corresponding components B δ αβγ of the surface Riemann-Christoffel tensor as follows

35 1.2 Differential Theory of Surfaces 35 { { { { R δ 3 δ 3 δ αβγ = Bδ αβγ + α γ 3 β α β 3 γ (1 132) Since, the three-dimensional Eucledian space is flat, we have everywhere R δ αβγ = 0. Then, the following simplified representation for tensor components B δ αβγ is obtained from (1-132) { { { { B δ 3 δ 3 δ αβγ = α β 3 γ α γ 3 β (1 133) Principal curvatures, second fundamental form Fig. 1 5 a) Unit tangent vector of a curve. b) Radius of curvature of a curve. An arbitrary curve s(t) on the surface Ω is defined by the parametric description r = r(t) = r ( θ 1 (t), θ 2 (t) ), where t is a free parameter as explained subsequent to (1-97). P and Q denote two neighboring points on the curve defined by the position vectors r and r+dr, corresponding to the parameter values t and t + dt, respectively. The increment dr with the length ds is a surface vector placed in the tangent plane at P, and specified by the covariant representation dr = dθ α a α. Hence, the unit tangent vector to the curve at P is given as t = dr ds = dθα ds a α (1 134) The unit tangent vector in Q deviates infinitesimally from t, and may be written as t + dt. Given, that the length unchanged is equal to 1, the increment dt must fulfill 1 = (t + dt) (t + dt) = t t + 2 dt t + dt dt = dt t dt t = 0 (1 135)

36 36 Chapter 1 Preliminaries where the second order term dt dt is ignored. (1-135) shows that the increment dt is orthogonal to t. Let n denote the unit normal vector n co-directional to dt. Then, the following identity may be defined dt ds = κ 0 n (1 136) dt (1-136) is referred to as Frenet s formula. and n are called the curvature vector and the ds principal normal vector of the curve, respectively, and the proportionality factor κ 0 is referred to as the curvature. Generally, this is always assumed to be positive, so (1-136) is defining the orientation of n. In Fig. 1-5b lines orthogonal to s have been drawn through P and Q in the plane spanned by t and n, which intersect each other at the curvature center O under the angle dα as shown on Fig. 1-5b. The position of O relative to s is defined by the orientation of n. Then, the following geometric relation prevails, see Fig. 1-5b dt ds = t ds dα dt = t dα = 1 dα = ds R (1 137) 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 κ 0 = 1 R (1 138) Fig. 1 6 Normal plane to unit tangent vector. The unit tangent vector t is normal to a plane π, which contains both to the unit normal vector a 3 = a 3 and to the principal normal vector n, see Fig A third linear independent unit vector a g in π is given by the vector a g = a 3 t (1 139)

37 1.2 Differential Theory of Surfaces 37 We may then represent the curvature vector dt ds = κ 0 n as a linear combination of a 3 and a g as follows κ 0 n = κa 3 + κ g a g (1 140) The expansion components κ and κ are known as the normal curvature and the geodesic curvature of the curve s. Using that a 3 a g = 0, and both are unit vectors, these may be expressed in terms of the curvature κ 0 of the curve as follows κ = a 3 dt ds = (a 3 n) κ 0 κ g = a 0 dt ds = (a 0 n) κ 0 (1 141) Using the chain rule of partial differentiation it follows from (1-134) that the curvature vector has the following representation dt ds dt ds = d2 θ α ds a 2 α + a α dθ α dθ β θ β ds ds Since a 3 = a 3 is orthogonal to a α, it follows from (1-119), (1-141) and (1-142) that (1 142) κ = a 3 dt ds = a3 a { α dθ α dθ β 3 dθ α θ β ds ds = dθ β α β ds ds = b dθ α dθ β αβ ds ds (1 143) where the following indexed quantities have been introduced b αβ = a 3 a { α 3 θ = = b β βα (1 144) α β The neighbouring points P and Q on the curve s are given by the position vectors r(s) and r(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) = dr ds ds+ 1 d 2 r 2 ds 2 ds2 +O ( ds 3) = t ds+ 1 dt 2 ds ds2 +O(ds 3 ) (1 145) The distance dz of Q from the tangent plane at P is given by the projection of dr on the surface unit normal vector a 3, see Fig Hence, dz = a 3 dr = 1 2 a 3 dt ds ds2 = 1 2 κ ds2 = 1 2 b αβ dθ α dθ β (1 146) where (1-143) and the orthogonality of a 3 and t has been used. The larger the distance dz, the larger is the curvature of the curve at P. In (1-146) this quantity is partly determined by the

38 38 Chapter 1 Preliminaries components b αβ and partly by the parameter increments dθ α. As seen from (1-144) the components b αβ are entirely determined from properties related to the surface, and independent of the specific direction of the curve s. In contrast this direction is determined by the increments dθ α of the curvilinear coordinates. The quadratic form b αβ dθ α dθ β entering (1-146) is called the second fundamental form of the surface. In what follows b αβ is considered the contravariant components of a surface second order tensor b = b αβ a α a β, which is called the curvature tensor of the surface. Using a α = a αγ a γ, it follows from (1-119), (1-144) that b αβ = a 3 (a αγ a γ ) = a θ β 3 a γ a { αγ θ + a β αγ a 3 aγ θ = a β αγ a 3 aγ θ = a γ β αγ 3 β { α = a αγ b γβ = b α β (1 147) 3 β where the orthogonality of a 3 and a γ 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 δ αβγ = b αβ b δ γ + b αγ b δ β B δαβγ = b αβ b δγ + b αγ b δβ (1 148) where the transformation rules (1-114) of covariant and contravariant tensor components have been used in the final statement. (1-148) provides the following form for the selected independent tensor component B 1212 B 1212 = b 21 b 12 + b 22 b 11 = det [ b αβ ] = b (1 149) where b denotes the determinant formed by the contravariant components of the curvature tensor b. (1-149) is called Gauss s equation. If B 1212 = 0, all other contravariant components of the tensor will vanish as well, and it may be concluded that B = 0. Then, a surface is flat, if b = 0 everywhere. Noticed that if the determinant of the contravariant components vanish, the determinant 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 a direction determined by the increment dθ α is given as κ = b αβ dθ α dθ β a αβ dθ α dθ β (1 150) (1-150) may be interpreted as a Rayleigh quotient related to the following generalized eigenvalue problem, (Nielsen, 2006)

39 1.2 Differential Theory of Surfaces 39 ( bαβ κ a αβ ) t β = 0 (1 151) t β = dθβ ds represent the covariant components of a unit tangent vector t = tβ a β to the surface, which specifies the a direction for which (1-151) is fulfilled. With a = a αβ a α a β and b = b αβ a α a β, (1-151) is seen to represent the curvilinear component relation of the following tensor relation ( b κa ) t = 0 (1 152) (1-152) is fulfilled for two unit eigenvectors t 1 and t 2 related to the eigenvalues κ 1 and κ 2 of (1-151). The directions specified on the surface by the eigenvectors t γ are denoted the principal curvature directions, and the related eigenvalues are called principal curvatures. The unit eigenvectors fulfill the orthogonality conditions, (Nielsen, 2006) t γ a t δ = { 0, γ δ 1, γ = δ (1 153) { 0, γ δ t γ b t δ = κ γ, γ = δ (1 154) Let the principal curvatures be ordered, so κ 1 denotes the smallest and κ 2 the largest curvature. As follows from the well-known bounds on the Rayleigh quotient the normal curvature κ of an arbitrary curve passing through P will be bounded by the principal curvatures as, (Nielsen, 2006) κ 1 κ κ 2 (1 155) Using the mixed representations a = δ α β a αa β and b = b α β a αa β the component form of (1-152) attains the form ( ) b α β κ δβ α t β = 0 (1 156) Formally, (1-156) may be obtained from (1-151) by pre-multiplication of a γα and use of (1-111), (1-114). Solutions t β 0 of the homogeneous system of linear equations (1-156) are obtained, if the determinant of the coefficient matrix vanishes. This provides the following characteristic equation for the determination of the principal curvatures det [ b α β κ δα β] = κ 2 ( b b2 2) κ + ( b 1 1 b 2 2 b2 1 b1 2) = 0 κ 2 b α α κ + b a = 0 (1 157)

40 40 Chapter 1 Preliminaries In the last statement it has been used that det [ b γ β] = b a, where a = det [ a αγ ] denotes the determinant of the matrix formed by the contravariant components of the surface identity tensor. This relation follows from the identities b αβ = a αγ b γ β det [ ] [ ] [ b αβ = det aαγ det b γ ] β det [ ] b γ b β = a The solutions to (1-157) may be written as (1 158) κ 1 κ 2 where = H H 2 K (1 159) H = 1 2 (κ 1 + κ 2 ) = 1 2 bα α (1 160) K = κ 1 κ 2 = b a = b1 1b 2 2 b 2 1b 1 2 (1 161) 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 γ β of the curvature tensor, and 2H is equal to the trace of this matrix. It follows from that K = 0, if either κ 1 = 0 or κ 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 cylindrical syrfaces are developable, whereas a spherical surface in not. From (1-149) and (1-161) follows B 1212 = ak (1 162) Since, the eigenvectors t α are surface vectors, it follows from (1-116) and (1-153) that t 1 a t 2 = t 1 t 2 = 0. This means that the principal curvature directions through P are orthogonal to each other. It is then possible to construct an (s 1, s 2 ) arc-length coordinate system at each point on the surface, which everywhere has the principal curvature eigenvectors t 1 and t 2 as tangents. This so-called principal curvature coordinate system is specific in the sense that both [a αβ ] and [b αβ ] 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 stiffness matrices become diagonal, (Nielsen, 2006). In this specific coordinate system the principal curvatures are given as κ α = b (α)(α) a (α)(α) (1 163)

41 1.2 Differential Theory of Surfaces 41 Next, various relations are indicated, which involve the partial derivative of the surface unit normal vector a 3 = a 3. Partial differentiation of the equation a 3 a α = 0 and use of (1-144) provide a 3 θ β a α + a 3 a α θ β = 0 a α a 3 θ β = a 3 a α θ β = { 3 = b αβ (1 164) α β a 3 is a unit vector, so a 3 a 3 = 1. Then, partial differentiation provides a 3 a 3 θ α = 0 (1 165) Hence, a 3 θ α is orthogonal to a 3, and accordingly can be decomposed in the covariant surface vector base a β as follows a 3 θ α = lβ α a β (1 166) Scalar multiplication of both sides witha γ and use of (1-110) and (1-164) provides the following solution for the covariant components l β α a γ a 3 θ α = b γα = l β α a γ a β = a γβ l β α l β α = a βγ b γα = b β α (1 167) Then, insertion into (1-166) provides the following alternative representations { a 3 θ = a3 α θ = α bβ α a β = b αγ a γβ a β = b αβ a β 3 = a β (1 168) α β where (1-112) and (1-144) have been used. Similarly, (1-118) and (1-144) provide the following relation for the partial derivatives of the surface covariant base vectors { a α γ θ = a β γ + b αβ a 3 (1 169) α β (1-169) is called Gauss s formula. Next, e αβ and e αβ are introduced as alternative two-dimensional permutation symbols defined as

42 42 Chapter 1 Preliminaries [ ] [ ] 0 1 eαβ = 1 0 [ ] [ ] e αβ 0 1 = 1 0 (1 170) As seen [e αβ ] = [e αβ ] 1 = [e αβ ] T = [e αβ ]. From (1-110) follows that a 1 = a 1 = (a 1 a 1 ) 1/2 = a 11 a 2 = a 2 = (a 2 a 2 ) 1/2 = a 22 cosϕ = a 1 a 2 a 1 a 2 = Then, (1-102) may be written as a 12 a11 a22 sin ϕ = 1 a2 12 a 11 a 22 (1 171) A = a 11 a22 sin ϕ = a 11 a 22 (a 12 ) 2 = a (1 172) where a = det[a αβ ]. By the use of (1-170) and (1-172) the vector products in (1-103) can be written in the following compact forms a 3 a α = a e αβ a β a α a β = a e αβ a 3 (1 173) Further, the following component identities may be derived a αβ = a e αγ a γδ e βδ a αβ = 1 a eγα a γδ e δβ e αβ = 1 a a αγ e δγ a δβ e αβ = a a αγ e δγ a δβ (1 174) The first relation in (1-174) follows from the following matrix identities [ ] [ ] 1 a 11 a 12 a = 11 a 12 = a a 21 a 22 a 21 a 22 [ ] a 22 a 12 a 21 a 11 [ ] [ 0 1 = a 1 0 a 11 a 12 a 21 a 22 ] [ ] T (1 175)

43 1.2 Differential Theory of Surfaces 43 where it has been used that det[a αβ ] = 1/ det[a αβ ] = 1. The proof of the remaining relations in a (1-174), which is left an exercise, may be carried out by simple matrix operations on (1-175). Next, a three-dimensional vector field v = v(θ 1, θ 2 ) is considered, defined on the parameter domain ω. Hence, a vector is connected to each point of the surface Ω. Since, v 3 = v 3 0, v is not a surface vector. Physically, v may represent a force vector, a moment vector, a displacement vector or a rotation vector related to a material point on the surface. In this case the following representations prevail v = v α a α + v 3 a 3 = v α a α + v 3 a 3 (1 176) Since, v θ 3 = 0, the gradient becomes v = v θ β aβ = v j ;β a ja β = v j;β a j a β (1 177) Similarly, (1-74) becomes v θ = β vj ;β a j = v α ;β a α + v 3 ;β a 3 v θ = v β j;β a j = v α;β a α + v 3;β a 3 where, cf. (1-73) { v α ;β = vα θ + α β β γ { v 3 ;β = v3 θ + 3 β β γ v α;β = v { α γ θ β α β v 3;β = v { 3 γ θ β 3 β { v γ α + β 3 v γ + v γ v γ { 3 β 3 { 3 α β { 3 3 β v 3 v 3 v 3 v 3 (1 178) (1 179) The right hand sides of (1-178) both consist of a surface vector and a vector in the normal direction. The surface and normal components of these representations must pair-wise be equal corresponding to v α ;β a α = v α;β a α v 3 ;β a 3 = v 3;β a 3 v α ;β = aαγ v γ;β v α;β = a αγ v γ ;β v 3;β = v 3 ;β (1 180)

44 44 Chapter 1 Preliminaries The final relations in (1-180) follow from the initial vector identities using (1-112) and a 3 = a 3. Further, v 3 = v 3, v 3 ;β = v 3;β, { { 3 β 3 = 3 { 3 β = 0 and 3 β γ = bβγ, cf. (1-120), (1-144). Then, withdrawal of the second equation in (1-179) from the fourth equation provides the following relations for the components of the curvature tensor { { γ 3 v γ = v γ = b βγ v γ 3 β β γ { α a αγ v γ = b βγ v γ 3 β { γ v γ = b βα a αγ v γ 3 β { α b βγ = a αγ 3 β b γ β = { γ 3 β (1 181) Codazzi s equations From (1-164), (1-168) and (1-169) follows ( a α a ) 3 = a α θ β θ a 3 γ θ a β α b αβ θ = γ θ γ ({ ) δ a δ + b αγ a 3 b βε a ε a α α γ 2 a 3 θ γ θ β = b βδ Interchange of the indices β and γ in (1-182) provides 2 a 3 θ γ θ = β { δ α γ a α 2 a 3 θ γ θ β (1 182) { b αγ δ θ = b 2 a 3 β γδ a α (1 183) α β θ β θ γ Withdrawal of (1-182) from (1-183) gives the relation { b αβ δ θ b γ βδ = b { αγ δ α γ θ b β γδ α β (1 184) (1-184) is trivially fulfilled for β = γ. Further, the index combinations (β, γ) = (1, 2) and (β, γ) = (2, 1) lead to the same relation. Hence, merely two non-trivial and independent relations exist, which are taken corresponding to the index combinations (α, β, γ) = (1, 1, 2) and (α, β, γ) = (2, 2, 1), leading to the following relations

45 1.2 Differential Theory of Surfaces 45 { b 11 δ θ b 2 1δ b 22 θ 1 b 2δ 1 2 { δ 2 1 = b { 12 δ θ b 1 2δ = b 21 θ 2 b 1δ 1 1 { δ 2 2 (1 185) The relations (1-185) are called Codazzi s equations. It can be shown that arbitrary component functions a αβ = a αβ (θ 1, θ 2 ) and b αβ = b αβ (θ 1, θ 2 ) of the curvilinear parameters θ 1 and θ 2 uniquely determine a surface (saved an arbitrary rigid body translation and rotation), which has a αβ dθ α dθ β and b αβ dθ α dθ β as its first and second fundamental form, if and only if these components fulfill the Codazz s equations (1-185) and the Gauss s equation (1-165), (Spain, 1965). In a principal curvature coordinate system we have a 12 = a 21 = 0 and b 12 = b 21 = 0. In this case the Codazzi s equations can be reduced to ( a1 θ 2 θ 1 R 1 ( a2 R 2 ) ) = 1 R 2 a 1 θ 2 = 1 R 1 a 2 θ 1 (1 186) where a 1 and a 2 are given by (1-171), and R 1 and R 2 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 b 12 = b 21 = 0 in the principal curvature coordinate system, (1-185) can be reduced to { { b θ = b 2 11 b { { (1 187) b θ = b 1 22 b

46 46 Chapter 1 Preliminaries Next, the Christoffel symbol in (1-187) are expressed in terms of the covariant and contravariant components of the identity tensor as given by the defining equation (1-66). Due to the principal curvature coordinates and the special structure (1-106) these components become g jk = g jk = 0 for j k, g (α)(α) = a (α)(α) and g (α)(α) = a (α)(α) = 1 a (α)(α). Then { 1 = 1 ( ) a a11 θ = { 2 = 12 ( 1 1 a a ) 11 = 1 θ 2 2 { 2 = 1 ( ) a a22 θ = { 1 = 12 ( 2 2 a a ) 22 = 1 θ a 11 a 11 θ 2 1 a 11 a 22 θ 2 1 a 22 a 22 θ 1 1 a 22 a 11 θ 1 (1 188) Further, from (1-163) follows that b 11 = (1/R 1 )a 11 and b 22 = (1/R 2 )a 22. Insertion of (1-188) in (1-187) then provides ( ) a11 = 1 ( ) a11 θ 2 R 1 2 R 1 R 2 θ 2 ( ) a22 = 1 ( ) (1 189) a22 θ 1 R 2 2 R 1 R 2 θ 1 Introduction of (a 1 ) 2 = a 11 and (a 2 ) 2 = a 22 provides the identities ( ) a11 = ( ) a 1 a θ 2 R 1 θ 2 1 = a ( ) 1 a 1 R 1 R 1 θ + a a1 2 1 θ 2 R 1 ( ) a22 = ( ) a 2 a θ 1 R 2 θ 1 2 = a ( ) 2 a 2 R 2 R 2 θ + a a2 1 2 θ 1 R 2 a 11 a 1 = 2a θ 2 1 θ 2 a 22 a 2 = 2a θ 1 2 θ 1 (1 190) Insertion of the results (1-190) into (1-189) immediately provides (1-186). Example 1.2: Identity tensor, curvature tensor, Riemann-Christoffel tensor and principal curvatures in spherical coordinates As seen from (1-93) the contravariant components of the surface identity tensor becomes

47 1.2 Differential Theory of Surfaces 47 [a αβ ] = [ ] r r 2 sin 2 θ 1 The contravariant components of the curvature tensor follow from (1-94) and (1-144) [ ] r 0 [b αβ ] = 0 r sin 2 θ 1 (1 191) (1 192) Since both [a αβ and [b αβ ] are diagonal, it is concluded that the spherical coordinate system is a principal curvature coordinate system. The principal curvatures and related principal curvature radii follow from (1-163) κ 1 = κ 2 = 1 r R 1 = R 2 = r (1 193) κ 1 and κ 2 are negative, because unit surface normal vector is directed away from the curvature center, which in this 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 sin 2 θ 1 (1 194) Left and right hand sides of the Codazzi equations (1-186) vanish in this case. Hence, these equations are trivially fulfilled Surface area elements Fig. 1 7 Area elements in parallel surfaces. Fig. 1.7 shows two surfaces Ω and Ω 0, described by the same parameters θ 1 and θ 2. The points P 0 and P denotes the mapping points for the same parameters, specified by the position vectors

48 48 Chapter 1 Preliminaries r = r(θ 1, θ 2 ) and x = x(θ 1, θ 2 ), respectively. Further, the unit normal vectors related to the two surfaces at the said points are assumed to be identical and equal to a 3 = a 3. Then, the distance between any such corresponding points P 0 and P on the surfaces are everywhere constant and equal to s 3, where s 3 is measured from the surface Ω 0. On the surfaces arc length coordinate coordinate systems (s 1, s 2 ) and (S 1, S 2 ) are introduced with origins P 0 and P. The related surface covariant base vectors are denoted a α and g α, respectively. Then, the position vectors are related as follows x = r + s 3 a 3 (1 195) da 0 denotes a differential area element of shape as a parallelogram with sides of the length ds 1 and ds 2 parallel to a 1 and a 2, see Fig The length of the sides follows from (1-117) ds 1 = a 11 dθ 1 ds 2 = a 22 dθ 2 Let ϕ denote the angle between a 1 and a 2. Then, cf. (1-102), (1-172) (1 196) da 0 = ds 1 ds 2 sin ϕ = a 11 a22 sin ϕ dθ 1 dθ 2 = a11 a 22 (a 12 ) 2 dθ 1 dθ 2 = a dθ 1 dθ 2 (1 197) The surface covariant base vectors g α follows from (1-99), (1-168), (1-195) g α = x θ = r α θ + a α s3 3 θ = a α α s 3 b δ α a δ = ( δα δ ) s3 b δ α aδ (1 198) Then, the contravariant components g αβ of the surface identity tensor of Ω becomes g αβ = g α g β = ( δα δ )( s3 b δ α δ γ β s3 b γ β) aδ a γ = ( δα δ ) ( s3 b δ α δ γ β s3 b γ β) aδγ = a ( )( ) γδ a γα s 3 b γα aδβ s 3 b ( ) ( ) δβ = a γδ a αγ s 3 b αγ aβδ s 3 b βδ (1 199) Then, the differential area element da on Ω, corresponding to da 0 on Ω 0, becomes da = g dθ 1 dθ 2 where g = det[g αβ ] follows from (1-199) g = det [ a γδ] det [ ] [ ] a αγ s 3 b αγ det aβδ s 3 1 b βδ = a 1 ( det [ ] [ a αγ det δ γ β a s3 b γ ] ) ( [ ]) 2 2 det aαγ β = a ( a 1 b α α s3 + b ( ) ) s = a (1 2H s 3 + K ( s 3) ) 2 2 a ( det [ a αβ s 3 b αβ ] ) 2 = ( det [ δ γ β s3 b γ β] ) 2 = (1 200) (1 201)

49 1.3 Continuum Mechanics 49 where (1-157) and (1-158) have been used. Further, the mean curvature H and the Gaussian curvature as given by (1-160) and (1-161) have been introduced. Then the ratio between the differential area elements become da da 0 = g a = 1 2Hs 3 + K(s 3 ) 2 (1 202) 1.3 Continuum Mechanics Three-dimensional continuum mechanics V 0 V dx dx ρ 0 dv 0 u(x k, t) ρdv A 0 x 2 X x(x k, t) A i 2 i 3 O i 1 x 1 x 3 Fig. 1 8 Referential and current configuration of a flexible body. Fig. 1-8 shows a flexible body, i.e. a set of connected material points, which at the time t = 0 occupies the volume V 0 with the surface A 0. The indicated configuration of the body is referred to as the referential configuration. Material points (mass particles) are identified by their position vector X relative to the origin of a given frame of reference. Although any curvilinear coordinate system may be used, vectors and tensors will in this and the next subsection exclusively be described in the Cartesian (x 1, x 2, x 3 )-coordinate system with the origin O and the base unit vectors (i 1,i 2,i 3 ) shown on Fig. 1-8, with the exception that the dual Cartesian vector base (i 1,i 2,i 3 ) will be used as well. Because the unit base vectors in the indicated vector bases are 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 is possible, the bar atop the Cartesian components of vectors and tensors will be skipped throughout 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 with the surface A due to translation, rotation and mutual distortion of the particles. The indicated configuration is referred to as the current configuration of the body. The position in the current configuration of a particle with the referential coordinates X j is defined by the position

50 50 Chapter 1 Preliminaries vector x(x k, t) = x j (X k, t)i j. The current coordinates x j are known as the Eulerian cordinates of the particle. Traditionally, the governing equations of fluid mechanics are formulated using Eulerian coordinates. This permits that large distortions in the continuum motion can be handle with relative ease. By contrast formulations in Lagrangian coordinates is preferred for the kinematical description in solid mechanics, which allows an easy tracking of the material history. Its weakness lies in its inability to follow large distortions of the computational domain. The displacement of the particle is defined as the difference between the position vector in the current and referential configurations, see Fig. 1-8 u(x k, t) = x(x k, t) X (1 203) Consider a neighbor particle with the referential position vector X + dx. The indicated particle is displaced to the position x(x k + dx k, t) = x(x k, t) + dx(x k, t), where the incremental vectors are related as dx(x k, t) = F(X k, t) dx (1 204) F = x is the so-called material deformation gradient, cf. (1-76). This second order tensor, as well as the transposed tensor F T, the inverse tensor F 1 and the transpose of the inverse tensor F T have the following dual Cartesian representations, cf. (1-41), (1-47) F = F jk i j i k = x j X k i j i k, F T = F kj i j i k = x k X j i j j k F 1 = F 1 jk ij i k = X j x k ij i k, F T = F 1 kj i j i k = X k x j i j i k (1 205) The inverse material deformation gradient presumes the existence of the inverse mapping X = X(x k, t). This requires that F is non-singular, i.e. that the Jacobian det[f jk ] > 0. Let dv and dv 0 denote the volume of a particle in the current and the referential configuration. Further, let ρ and ρ 0 be the mass density of the particle in these configurations. Since the mass of the particle is unchanged we must have ρdv = ρ 0 dv 0. The Jacobian specifies the quotient between the differential volumes of the particle in the current and referential configurations corresponding to det[f jk ] = dv = ρ 0 dv 0 ρ (1 206) The distance between the neighboring particles in the referential and the current configuration is given by the lengths ds and ds of the incremental position vectors dx and dx. Clearly, the difference between these lengths is a measure of the local distortion of the material. Among several strain measures suitable for description of large strains we shall adopt the so-called Green strain tensor E, which is introduced in the following way ds 2 ds 2 = dx dx dx dx = dx ( ) F T F g dx = 2 dx E dx (1 207)

51 1.3 Continuum Mechanics 51 where (1-44) and (1-204) have been used. g denotes the identity tensor, which has the following Cartesian representation g = δ jk i j i k = δ k j i j i k = δ jk i j i k. δ jk = δ k j = δ jk specify the Kronecker s delta (1-14) in various tensor bases. Hence, the Green strain tensor is related to the material deformation gradient as follows E = 1 2 ( ) F T F g (1 208) The dual Cartesian components of E may be expressed in terms of the components of the displacement field by (1-203), (1-205) and (1-208) E = E jk i j i k = 1 ) (F lj i j i l F mk i m i k δ jk i j i k = ) i j i k = 1 ) (F lj F lk δ jk i j i k = 1 ( xl 2 2 ( ( δ lj + u l 1 2 E jk = 1 2 X j x l X k δ jk X j )( δ lk + u l X k ) δ jk ) i j i k = 1 2 ( uj + u k + u ) l u l X k X j X j X k (F lj F mk δ lm δ jk ) i j i k = ( uj + u k + u ) l u l i j i k X k X j X j X k (1 209) For deformation gradients fulfilling u j X k 1 the quadratic term in (1-209) may be ignored. The corresponding Green small strain tensor e has the components e jk = 1 2 ( uj + u ) k X k X j (1 210) The Eulerian small strain tensor ε represents the corresponding strain tensor, where the deformation gradient is taken with the respect to the Eulerian coordinates. This tensor has the dual Cartesian components ε jk = 1 2 ( uj + u ) k x k x j (1 211) The Cauchy equations of motion of a mass particle is formulated in Eulerian coordinates. In tensor notation these read, (Malvern, 1969) ρa = σ + ρb (1 212) a(x k, t) denotes the acceleration vector of the particle, and b(x k, t) is the external load per unit mass. σ(x k, t) represents the Cauchy stress tensor, which is a symmetric tensor, σ = σ T. The divergence operation σ specifies a vector with the components x l (σ jk i j i k ) i l = i j i k i l = σjk x k i j, cf. (1-77). Then, the dual Cartesian component form of (1-212) becomes σ jk x l

52 52 Chapter 1 Preliminaries τ A 1 da ū A 2 Fig. 1 9 Definition of boundary conditions. ρ a j = x k σ jk + ρ b j (1 213) The surface traction τ(x k, t) at the current surface A is related to Cauchy s stress tensor σ(x k, t) by Cauchy s boundary condition, (Malvern, 1969) τ(x k, t) = σ n (1 214) where n = n(x k, t) signifies the outward directed unit vector to the surface A. A is divided into two disjoint sub-surfaces A 1 and A 2, at which either mechanical or kinematical boundary conditions are prescribed, see Fig Then, (1-212) is solved with the following boundary conditions τ(x k, t) = τ(x k, t), x k A 1 u(x k, t) = ū(x k, t), x k A 2 (1 215) τ(x k, t) denotes the prescribed surface traction on the sub-surface A 1, and ū(x k, t) is the prescribed displacement on the sub-surface A 2. As a basis for any finite element method the equations of dynamic equilibrium and related boundary conditions are reformulated in a weak form. This is a variational principle for some functional of the unknown fields of the problem. At stationarity the varied functional determines the discretized unknown fields, so these fulfil the equilibrium equation and the boundary conditions at best or in average throughout V and A, when the fields are varied within a predefined functional sub-space. For so-called displacement based finite element formulations, where the only unknown is the displacement field, the relevant variational principle is the principle of virtual displacements, which may be formulated as V δε σ dv = with the component form V δu ρ ( b a ) dv + δu τ da (1 216) A 1

53 1.3 Continuum Mechanics 53 V δε jk σ jk dv = V δu j ρ ( b j a j) dv + δu j τ j da (1 217) A 1 δu(x k ) = δu j (x k )i j denotes an arbitrary variational displacement field with the restrictions that δu(x k ) = 0 on A 2. Then, (1-217) can be derived by scalar multiplication of both sides (1-213) with δu j (x k ), followed by an integration over V. Next, the identity is obtained by the use of the divergence theorem, the mechanical boundary condition on A 1, and the kinematical constrain that allowable variations fulfill δu j (x k ) = 0 on A 2. Save that certain continuity requirements need to fulfilled to make the indicated mathematical operations meaningful, the variational field δu(x k ) may else be prescribed with an arbitrary magnitude. Especially, the variations do not necessarily have any relation to the actual displacement field. δε = δε jk i j i k in (1-216) denotes the Eulerian small strain tensor for the variational displacement field δu(x k ), i.e. the components are given as, cf. (1-211) δε jk = 1 2 ( δuj + δu ) k x k x j (1 218) Because δu j (x k ) may be of finite magnitude, this is not identical to the variation of the components of the Eulerian small strain tensor of the actual displacement field 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 varied as well. Actually ( ) ( ) uj uj X l δ = δ = δu j X l + u j δ x k X l x k X l x k X l ( ) Xl x k = δu j x k + u j X l δf 1 lk (1 219) where it has been used that the Lagrangian coordinates X k are not subjected to any variation, i.e. the variation concerns the displacement of a specific mass particle, for which reason δ( u j X k ) = δu j X k. Only for small displacement variations will δ( u j x k ) = δu j x k. Since, δε and σ are symmetric tensors, it follows that δε σ = δε σ, cf. (1-59), (1-61). Hence, the same internal virtual work is obtained in (1-216) independent of which definition 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) via the combination b a. Formally, b a may be interpreted as an equivalent quasi-static load per unit mass, which takes the dynamic effects into consideration via the the so-called inertial load a ρdv on the particle. This is the essence of the d Alembert s principle. Then, the right hand side of (1-216) represents the external virtual work δw e performed by the equivalent load per unit mass b a throughout V, and the surface traction τ on A 1. The left hand side represents the internal virtual work δw i performed by the internal forces (stress components) on the particles. Then, the principle of virtual displacements postulates that the internal and external virtual works are equal, i.e.

54 54 Chapter 1 Preliminaries δw i = δw e (1 220) The idea is to formulate finite element theories based on the Lagrangian formulation, for which reason (1-216) at first is reformulated into the following equivalent form based on quantities defined in the referential configuration ( ) δe SdV 0 = δu ρ 0 b a dv0 + δu τ 0 da 0 (1 221) V 0 V 0 A 01 with the component form ( δe jk S jk dv 0 = δu j ρ 0 b j a j) dv 0 + δu j τ j 0 da 0 (1 222) V 0 V 0 A 01 (b a) ρdv = (b a) ρ 0 dv 0 specify the equivalent static load on a given particle, so the first integrals on the right hand side of (1-216) and (1-221) denote the summation of these loads from all particles within the structure. Hence, these integrals are equal. The surface A 0 in the referential configuration is divided into the sub-surfaces A 01 and A 02, which are mapped onto the A 1 and A 2 in the current configuration. Correspondingly, a surface area element da 0 on A 01 is mapped onto the surface area element da on A 1. τ 0 entering the second integral on the right hand side of (1-221) denotes a fictitious surface traction defined in the referential configuration, which is related to the actual surface traction τ in the current configuration as τ 0 da 0 = τda (1 223) Then, the second integrals on the right hand side of (1-216) and (1-221) represent the summation of identical loads from corresponding area elements in the two configuration. It follows that also these integrals are equal. τ 0 will be referred to as a pseudo surface traction, since the referential configuration is actually free of any stresses, save for a possible known residual stress configuration with no relation to the indicated external loads and boundary conditions. As mentioned, δε denotes the Eulerian small strain tensor of the variational displacement field δu(x k ), and must not be confused with the variation of the Eulerian small strain tensor of the actual displacement field. For this reason it could be anticipated that δe entering the integral on the left hand side of (1-221) in the same way represents the Green strain tensor of the variational displacement field. This is not so. Indeed, δe denotes the variation of the finite Green strain tensor due to the variation the displacement field, and depends on both u and δu. In order to derive the relation between δe and δε the variation of the material gradient tensor is at first determined. Use of (1-205) provides with δx j = δ(x j + u j ) = δu j

55 1.3 Continuum Mechanics 55 ( ) xj δf = δ i j i m = δx j i j i m = δu j X m X m x k δu j x k x l X m i j (i k i l i m ) = δu j x k i j i k x k X m i j i m = δu j x k x l X m i j (δ kl i m ) = x l X m i l i m = δu j x k i j i k F (1 224) Then, the variation of the Green strain tensor follows from (1-208) and (1-222) δe = 1 2 ( ) δf T F+F T δf δg = F T 1 ( δuk + δu ) j i j i k F = F T δε F (1 225) 2 x j x k where it has been used that δg = g δu = 0, cf. (1-81). The related component form becomes, cf. (1-205) δe jk = δe jk = x l X j δε lm x m X k (1 226) The tensor and component form of the inverse relation read δε = F T δe F 1 (1 227) δε jk = δε jk = X l x j δe lm X m x k (1 228) 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 conjugated stress measure to δe to insure that the left hand sides of (1-216) and (1-221) become identical. This necessitates that δε jk σ jk dv = δe lm S lm dv 0. Then, use of (1-206) and (1-226) provides the following relation between the components of the Cauchy stress tensor and the components of the second Piola-Kirchhoff stress tensor δε jk σ jk dv = δe lm S lm dv 0 = x j X l δε jk x k X m S lm dv 0 σ jk = σ jk = dv 0 dv x j X l The related tensor form reads x k X m S lm = ρ ρ 0 x j X l S lm x k X m (1 229) σ = ρ ρ 0 F S F T (1 230) The tensor and component form of the inverse relation read

56 56 Chapter 1 Preliminaries S = ρ 0 ρ F 1 σ F T (1 231) S jk = S jk = ρ 0 ρ X j x l σ lm X k x m (1 232) For hyperelastic materials the existence of an elastic potential W(E) is postulated, depending entirely 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 (1 233) For a St.Venant-Kirchhoff type of material the following linear relation between S and E is postulated S = S 0 + C E (1 234) S 0 signifies a known residual stress configuration in the referential configuration, which has no dependence on the external loads, the surface traction on A 01, or on the prescribed displacements on A 02. The fourth order tensor C is known as the elasticity tensor. In the first place this has 3 4 = 81 independent components C jklm. However, because S = S T and E = E T are both symmetric tensors, the following symmetry relations prevail C jklm = C kjlm = C jkml, which reduces the number of independent components to 6 6 = 36. Additionally, if the linear material is hyperelastic, it follows from (1-233) and (1-234) that C jklm = 2 W(E) E jk E lm = 2 W(E) E lm E jk = C lmjk (1 235) The symmetry condition (1-235) further reduces the number of independent elastic constants to 21. In this case the constitutive relation between the 6 non-trivial stress components (due to the symmetry of S), and the 6 non-trivial components of E can be given in the following matrix form S = S 0 + C E S = S 11 S 22 S 33 S 12 S 13 S 23, S 0 = S0 11 S0 22 S0 33 S0 12 S0 13 S0 23, E = E 11 E 22 E 33 E 12 E 13 E 23 (1 236) (1 237)

57 1.3 Continuum Mechanics 57 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 = 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 (1 238) For an orthotropic material the number of material parameters further reduces to 9, corresponding to the constitutive matrix C 1111 C 1122 C C 1122 C 2222 C C = C 1133 C 2233 C G G G 23 (1 239) G 12, G 13 and G 23 denotes the shear moduli of the material. Finally, in case of an isotropic material merely two independent parameters remains. In this case the constitutive matrix may be written, (Malvern, 1969) λ + 2µ λ λ λ λ + 2µ λ C = λ λ λ + 2µ µ µ µ (1 240) where λ and µ are known as the Lamé parameters. these may be related to Young s modulus E, and the Poisson s ratio as follows, (Malvern, 1969) λ = ν E (1 + v)(1 2ν), µ = G = E 2(1 + ν) (1 241) where G is the shear modulus. It should be noticed that although the relation (1-236) is physical linear, the relation becomes geometrical non-linear, when the Green tensor is expressed in the displacement components. The use of the variational formulation (1-221) as a basis for a finite element formulation requires that the variational field δu(x k ) and displacement field u(x k, t) need to be interpolated

58 58 Chapter 1 Preliminaries between nodal values by means of suitable selected shape function. The shape function for the variational field must vanish, and the shape function for the displacement field must comply with the kinematical boundary conditions on A 2. Fig Shell like structure in referential configuration. In the present context the body V 0 in the referential state is either a plate or a shell. Shell structures have a surface like appearance, since the thickness is considerable smaller than the two other dimensions. The geometry of a shell structure is determined by a parameterized surface and a thickness function. In this case the surface A 0 is conveniently divided into three sub-surfaces Ω 0u, Ω 0 l and Γ 0. Ω 0 u and Ω 0l form the upper and lover surfaces of the body, whereas Γ 0 makes up the remaining cylindrical part of the boundary. Γ 0 is further sub-divided into the subsets Γ 01 and Γ 02. Kinematical boundary conditions are only prescribed on Γ 02. Hence, Γ 02 is identical to the sub-surface A 02 on Fig. 1-9, whereas A 01 consists of Ω 0 u, Ω 0 l and Γ 01. Further, the so-called midsurface Ω 0 has been shown in fig. 1-10, which is an artificial plane in the shell defined as the midst between Ω 0 u, Ω 0 l. The thickness t is the distance between the upper and lower surface in the direction of the normal of the midsurface. Fig Plate and shell surfaces in referential and current state. The referential position of a material point within the shell is specified by the position vector X(θ 1, θ 2, θ 3 ), indicating a vectorial parametric description in terms of the curvilinear coordi-

59 1.3 Continuum Mechanics 59 nates θ 1, θ 2, θ 3. The parametrization in the θ 3 -direction is centralized in a way that the midsurface is obtained for θ 3 = 0, i.e. the position vector to a particle or material point P 0 of the midsurface is given as R(θ 1, θ 2 ) = X(θ 1, θ 2, 0) (1 242) The thickness t = t(θ 1, θ 2 ) related to a point of the midsurface may vary over the surface. For a given value θ = θ 3 a surface Ω is defined in the referential state by the position vector X(θ 1, θ 2, θ) to a particle P. Especially, the upper and lower surfaces are described by the parametric descriptions R u (θ 1, θ 2 ) = X ( θ 1, θ 2, t(θ 1, θ 2 /2) ) R l (θ 1, θ 2 ) = X ( θ 1, θ 2, t(θ 1, θ 2 /2) ) (1 243) At the material points P 0 and P local arc-length coordinate systems have been defined in both the referential and the current states. The related covariant and contravariant base vectors in the referential state are denoted A j and A j, and G j and G j, respectively. The corresponding base vectors in the current state are denoted a j and a j, and g j and g j. The surface covariant base vectors become, cf. (1-99) A α = A α (θ 1, θ 2 ) = R θ α G α = G α (θ 1, θ 2, θ) = X θ α a α = a α (θ 1, θ 2 ) = r θ α g α = g α (θ 1, θ 2, θ) = x θ α (1 244) The displacement vector of the particles P 0 and P from the referential state to the current state are given as, see Fig v(θ 1, θ 2 ) = r(θ 1, θ 2 ) R(θ 1, θ 2 ) u(θ 1, θ 2 ) = x(θ 1, θ 2 ) X(θ 1, θ 2 ) (1 245) The idea in shell theory is to relate all vectors and tensors related to a particle P in the current state to the covariant or contravariant base vectors A j and A j of the related particle on the midsurface in the referential state. This will be the subject of section Fiber-reinforced composite materials Composite materials consist of two or more ingredients with specific mechanical properties. Typically, a composite contains a reinforcement material (fibres, steel bars), which are embedded into a matrix material (epoxy, cement). The strength and stiffness of the reinforcement

60 60 Chapter 1 Preliminaries Fig Transfer of axial stress in fiber to matrix material. material are much larger than those of the matrix material, as well as those of the combined material. The reinforcement material is primarily carrying axial stresses, which are transferred to the matrix material as shear stresses at the interface between the two materials. The basic mechanism can be explained with reference to Fig. 1-12, which shows two cross-sections separated by a distance l a of a circular fiber with the radius r embedded in a surrounding matrix material. At the left section a tensile force F = σ f A f has been applied to the fiber, where σ f is the axial stress, and A f is the area of the fiber. Over the length l a, which is denoted the anchor length, the axial stress drops from σ f to zero as a shear stress τ f develops at the interface between the materials. From static equilibrium follows that τ f = r 2 dσ f/dx. At the right section both σ f and τ f are zero, and a tensile axial stress σ m has developed in the matrix material in static equilibrium with F. Since, σ m is distributed to a significantly larger area than A f, this stress is correspondingly smaller than the fiber stress at the left cross-section. Fig Buckling of fiber in axial compression. During axial compression the fiber may tend to buckle. Whether this happens depends on the capability of the matrix material in providing lateral elastic support to the fiber as illustrated by the Winckler foundation model shown on Fig A lamina or ply represents the fundamental building block in a laminate. The fiber reinforcement may consist of discontinuous or continuous fibers. These may be unidirectional, bidirectional or randomly distributed, or consist of woven fiber nets as shown on Fig A unidirectional fiber arrangement only provides extra strength and stiffness in the direction of

61 1.3 Continuum Mechanics 61 Fig Fiber arrangements. a) Discontinuous randomly distributed fibres. b) Unidirectional fibers. c) Bidirectional fibers. d) Woven fiber net. (Reddy, 2004). the alignment. The strength and stiffness of continuously unidirectional fibers are larger than that of a corresponding discontinuous fiber arrangement. x 1 θ = 0 θ > 0 θ = 90 x 2 θ < 0 x 3 Fig Laminate built up of unidirectional laminae with different fiber orientations. (Reddy, 2004). A laminate consists of several stacked laminae to achieve the desired stiffness, strength and thickness. In Fig is illustrated how unidirectional fiber reinforced laminae may be stacked to achieved this. The layers are usually bonded together with the same matrix material as that of the adjacent laminae. Mismatch of the material properties between the layers may produce significant shear stresses at the interfaces, which in turn may cause delamination. Similarly, mismatch of the fiber and matrix materials may cause debonding of the fibers. The idea in what follows is to model the lamina as an equivalent smeared out homogeneous orthotropic material, with averaged mechanical properties determined by the constituent materials. In order to derive the material law a so-called material (x 1, x 2, x 3 )-coordinate system is introduced, with the (x 1, x 2 )-plane placed in the mid-plane of the lamina, and the x 1 - and

62 62 Chapter 1 Preliminaries x 2 x 3 x 1 Fig Orthotropic lamina. Definition of material coordinate system. (Reddy, 2004). x 2 -axes in the direction of the fibers, see Fig The Piola-Kirchhoff stress components S 33 and S 3α in the said coordinate system vanish at the upper and lower surface of the lamina. Then, continuity suggests that these stress components can be ignored throughout the lamina. From (1-239) follows that the shear strain components E 3α of the Green tensor vanish as well. However, the normal strain components E 33 is non-zero as seen from the third equation in (1-239)). As an example the isotropic case provides with (1-240) S 33 = 0 = λ (E 11 + E 22) + (λ + 2µ) E 33 E 33 = λ λ + 2µ (E 11 + E 22 ) = ν 1 ν (E 11 + E 22 ) (1 246) where E 33 in general is non-vanishing. The axial strain in the x 1 -direction is given as E 11 = S 11 /E 1 ν 12 S 22 /E 2, where E 1 and E 2 denote the Young s moduli for normal stresses in the x 1 - and x 2 -directions, and ν 12 denotes the Poisson s ratio for the contraction in the x 1 -direction due to an axial strain in the x 2 -direction. Similarly, the axial strain in the x 2-direction is given as E 22 = S 22/E 2 ν 21S 11/E 1, where ν 21 denotes the Poisson s ratio for the contraction in the x 2 -direction due to an axial strain in the x 3α 1-direction. For plates and shells shear stresses S are present in bending deformation, and cannot be disregarded. For this reason these will also be included in the constitutive relation for the laminae. Then, the material law for the retained second Piola-Kirchhoff stress and Green strain components can be presented by the following flexibility relation on matrix form E = D S (1 247) where

63 1.3 Continuum Mechanics 63 E = E 11 E 22 γ 12 γ 13 γ 23 S 11 S 22, S = S 12 S 13 S 23 (1 248) 1 E 1 ν 21 E 1 D = ν 12 E 2 1 E G G G 23 (1 249) As seen in (1-248) the angle strains γ 12 = 2E 12, γ 13 = 2E 13, γ 23 = 2E 23 have been introduced as shear strain measures. The symmetry of the flexibility matrix requires that the Poisson ratios fulfill ν 12 E 2 = ν 21 E 1 (1 250) Hence, at most six independent material parameters enter the constitutive law (1-248), (1-249). The inverse stiffness relation reads S = C E (1 251) E 1 ν 12 E ν 12ν 21 1 ν 12ν 21 ν 21 E 2 E C = 1 ν 12ν 21 1 ν 12ν G G G 23 (1 252) In finite element formulations of plate and shell theories the kinematical constraint E 33 = 0 is often presumed, with the consequence that the element thickness is preserved during deformation. This constraint can only be fulfilled if a normal stress S 33 0 is applied to prevent the

64 64 Chapter 1 Preliminaries Poisson contraction as given by (1-245), thus stiffening the configuration. The resulting stiffening effect has been termed thickness locking or Poisson thickness locking, which is manifesting in an overestimation of the bending stiffness. Despite the obvious contradiction this problem is usually eliminated in finite element analysis by applying the plane stress constitutive equations (1-251), (1-252). x 2 x 1 x 2 x 3, x 3 θ x 1 Fig Orthotropic lamina. Material and global coordinate systems. (Reddy, 2004). Next, the constitutive relations are formulated in a referential (x 1, x 2, x 3 )-coordinate system with the (x 1, x 2 )-plane coinciding with the (x 1, x 2 )-plane, and with the x 3 -axis co-directional to the x 3 -axis. Further, the x 1 -axis is rotated the angle θ from the x 1 -axis in the positive x 3 - direction, see Fig The base unit vectors i j of the material coordinate system are related to the base unit vectors i j of the referential coordinate system as i j = l k j i k (1 253) The relation for the corresponding dual base unit vectors is written i j = m j k ik (1 254) l k j and mj k specify the referential coordinates of the base unit vectors i j and i j.with j indicating a row index, and k a column index, these may be stored in the matrix cosθ sin θ 0 [lj k ] = [m j k ] = sin θ cosθ 0 (1 255) From the tensor representations S = S jk i j i k = S jk i ji k it follows from (1-253) that the referential and material components of the second Piola-Kirchhoff stress tensor are related as S jk = l j l lk lm m S (1 256) Evaluation of (1-256) provides the following explicit relations

65 1.3 Continuum Mechanics 65 S 11 = cos 2 θ S 11 + sin 2 θ S 22 2 sin θ cosθ S 12 S 22 = sin 2 θ S 11 + cos 2 θ S sinθ cosθ S 12 S 12 = sin θ cosθ S 11 sin θ cosθ S 22 + (cos 2 θ sin 2 θ) S 12 S 13 = cos θ S 13 sin θ S 23 S 23 = sin θ S 13 + cosθ S 23 (1 257) (2-257) may be written in the following matrix form S = T S (θ) S (1 258) where S and S are column matrices storing the non-trivial stress components as indicated in (1-248), and the transformation matrix T S (θ) is given as T S (θ) = cos 2 θ sin 2 θ 2 sin θ cosθ 0 0 sin 2 θ cos 2 θ 2 sinθ cosθ 0 0 sin θ cosθ sin θ cos θ cos 2 θ sin 2 θ cos θ sin θ sin θ cos θ (1 259) Similarly, the dual tensor representations E = E jk i j i k = E jk i j i k provides the following analog relation between the referential and material components of the Green tensor E jk = m l j mm k E lm (1 260) Explicit evaluation of (1-260) provides the following relation between the non-trivial strain measures on matrix form E = T E (θ) E (1 261) where E and E are column matrices storing the non-trivial stress components a T E (θ) = cos 2 θ sin 2 θ sin θ cosθ 0 0 sin 2 θ cos 2 θ sin θ cosθ sin θ cosθ 2 sin θ cos θ cos 2 θ sin 2 θ cos θ sin θ sin θ cos θ (1 262) It can easily be shown that T S (θ) and T E (θ) are related as

66 66 Chapter 1 Preliminaries T 1 E (θ) = T T S (θ) (1 263) Then, S = T S (θ) S = T S (θ) C E = T S (θ) C T 1 (θ) E = T (θ) E S C T T (θ) E (1 264) S From (1-264) is concluded that the components of the elasticity tensor in the two coordinate systems are related by the following relation on matrix form C(θ) = T S (θ) C T T (θ) (1 265) S C(θ) has the structure C 1111 C 1122 C C 1122 C 2222 C C(θ) = C 1112 C 2212 C C 1313 C C 1323 C 2323 (1 266) Although not indicated, the components in (1-266) can easily be explicitly calculated from (1-252) and (1-259).

67 1.4 Exercises Exercises 1.1 Given the following vectors a, b, c and d. Prove the following vector identities (a.) a (b c) = (a c)b (a b)c (b.) (a b) (c d) = (a c)(b d) (a d)(b c) (c.) (a b) c = (b c) a = (c a) b 1.2 Prove the e δ relation (1-19). 1.3 Given the vectors a, b, c and d with the Cartesian components [āj ] 1 = 2, [ bj ] 3 j = 2, [ c ] 1 [ = 2, ] 3 dj = Calculate the dyadic scalar products ab cd and ab cd. 1.4 Prove the relations (1-37) and (1-40). 1.5 Given the Cartesian components C rstu of the elasticity tensor C. Calculate the mixed curvilinear components C k m j l. 1.6 Prove (1-81) by applying (1-77) to the mixed representation g = δ k j gj g k of the identity tensor. a) b) 1 θ 2 x 3 f ω a x 2 1 θ 1 a x 1 Fig a) Parameter domain. b) Shell surface. 1.7 The mid-surface of a shell structure is defined by the parametric description x 1 a θ 1 x 2 = a θ 2 x 3 2f θ 1 θ 2 where the parameter domain ω has been indicated on Fig. 1-18a. a and f are parameters defining the length and the height of the shell structure as shown on Fig. 1-18b.

68 68 Chapter 1 Preliminaries (a.) Calculate the covariant and contravariant base vectors g j and g j in terms of the Cartesian base vectors. (b.) Calculate the contravariant components of the surface identity tensor and the curvature tensor, and determine principal curvature, the mean curvature, the Gaussian curvature and the principal curvature directions. (c.) Formulate Codazzi s equations for the surface in terms of the selected curvilinear coordinates. 1.8 Prove the last three relations in (1-74). 1.9 Derive (1-217) from (1-213).

69 CHAPTER 2 Shell and Plate Theories The history of plate theories started with Jacob II Bernoulli, who formulated a grid model of overlapping beams, where the torsional mode was neglected. The differential equations of a plate were derived independently by Lagrange, Poisson and Navier. However, their theories all had flaws with respect to the elastic constants and the boundary conditions. These problems were finally solved by Kirchhoff, (Kirchhoff, 1850), for which reason he is considered the founder of modern plate theory. It took almost a century before E. Reissner, (Reissner, 1944) and (Reissner, 1945), and Mindlin (Mindlin, 1951), extended the model to include transverse shear deformations, in much the same way as Timoshenko included shear deformation in the Bernoulli-Euler beam theory. Dealing with the eigenvibrations of a bell, Euler in 1764 developed the first mathematical model to shell theory by analyzing the axisymmetric bell structure as independent ring beams, ignoring the torsional and meridional bending effects. Again, a century passed before a consistent theory was derived by Love, (Love, 1888). It is based on Kirchhoff s method, and thus are referred to as the Kirchhoff-Love model. The inclusion of transverse shear deformations, although proposed by Naghdi, (Naghdi, 1972), is today mostly referred to as the Reisnner-Mindlin model, in recognition of their analog contributions to plate theory. The list of names contributing to plate and shell theory in the twentieth century is very long. Hardly any other area of structural mechanics has produced so many publications, which is partly an indication of the complexity of the involved mechanics, and partly of the practical relevance of the subject. While the initial interest in plate and shell theory grew from the need of solutions to problems in civil and architectural engineering, the focus has gradually changed to a wider scale of applications in mechanical engineering, aerospace and vehicle industry and biomechanics, which has been the driving force for further developments. Subjects as large displacements (e.g. v. Karman plate theory), finite rotations, buckling, composites, anisotropic materials, material nonlinearities and higher order plate and shell theories have been investigated. Central in these developments has been the concurrent development of numerical methods based on the finite element methods. In turn the numerical modeling has also changed the standard mechanical model for plate and shell elements from a Kirchhoff-Love to a Reissner- Mindlin kinematical formulation, motivated by the reduced compatibility requirements of the element shape functions. 69

70 70 Chapter 2 Shell and Plate Theories When applying a finite element method to plates and shell it is important to realize that errors on the numerical predictions stem from two sources, namely modeling errors intrinsic to the applied plate or shell theory, and discretization errors caused by the numerical analysis of this theory by a finite-degrees-of-freedom model. Modeling errors are caused by certain approximations applied to the three-dimensional continuum in terms of specific kinematic constraint on the motion of the mass particles, resulting in Love-Kirchhoff, Reissner-Mindlin or even higher order theories. Additionally, decisions on the inclusion or exclusion of material and geometrical non-linearities, of anisotropy and nonhomogeneity will influence the qualities of the theory. While the discretization errors can be removed by mesh refinement, the modeling errors always persist. When developing and applying a finite element formulation the aim is to achieve a certain accuracy of the numerical prediction relative to the theoretical model, using a minimum number of degrees of freedom. This rises a number of questions, which have to be answered in advance. Should one start from a genuine shell theory, or should the plate and shell elements be derived as degenerated continuum elements. Should the two-dimensional elements be derived from a variational formulation of the a genuine plate or shell theory, or should the dimensional reduction be achieved via degenerated three-dimensional continuum elements. Further, which simplifications in the formulation are admissible, and in which applications. The aim of the following outline is to provide an overview of the mathematical formulation of the theory for thin-walled based on the finite element formulation. It is not the intention to go in into depth with these theories, which consequently will be given in a somewhat descriptive manner. However, we do need to define some fundamental concepts precisely. Since, non-linear plates turn into shells after the first load increment has been applied the focus is from the start on shells. The formulations aim at the largest possible generality. Both thin and thick shells are covered. The shell structure may be layered (sandwich or laminar structures), anisotropic or nonhomogeneous. Only static problems will be covered. Further, the non-linear behavior is assumed to be of geometric nature. The extension to dynamic problems or material non-linearities are possible without fundamental difficulties. In the first case a mass and damping matrix need to be included. In the last case the incremental elastic stiffness matrix need to be modified.

71 2.1 Plates and Shells Plates and Shells Fig. 2 1 Bending and membrane deformation modes of axis-symmetric shells. The structural behavior of the shell is characterized by two principal different modes of deformations, namely membrane and bending deformations. Fig. 2-1 shows some illustrative examples for axis-symmetric shells, which have been chosen merely because the actions are easily illustrated in this case. The bending deformation typically causes an ovalization of the shell. Correspondingly, the cross-section is exposed to a linearly varying normal stress over the section with the zero line placed at the midsurface. In contrast, membrane dominated deformations induce normal stresses in the section, which are approximately constant over the thickness. Obviously, the membrane state is the desirable mode of deformation. If this can be established, the shell may carry very large external loads compared to its self-weight. If the external load varies significantly, or the supports are inappropriately designed, bending deformations may arise. These deformations and related stresses are considerably larger than those caused by the same loads in case of membrane action. if a membrane state