Chemnitz Scientific Computing Preprints

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1 Arnd Meyer The Koiter shell equation in a coordinate free description CSC/1-0 Chemnitz Scientific Computing Preprints ISSN Chemnitz Scientific Computing Preprints

2 Impressum: Chemnitz Scientific Computing Preprints ISSN ( : Preprintreihe des Chemnitzer SFB393) Herausgeber: Professuren für Numerische und Angewandte Mathematik an der Fakultät für Mathematik der Technischen Universität Chemnitz Postanschrift: TU Chemnitz, Fakultät für Mathematik Chemnitz Sitz: Reichenhainer Str. 41, 0916 Chemnitz

3 Chemnitz Scientific Computing Preprints Arnd Meyer The Koiter shell equation in a coordinate free description CSC/1-0 Abstract We give an alternate description of Koiter s shell equation that does not depend on the special mid surface coordinates, but uses differential operators defined on the mid surface. CSC/1-0 ISSN Febr. 01

4 Contents 1 Introduction 1 [4] J.P.Boehler, Applications of tensor functions in solid mechanics, CISM Courses no 9, Springer,(1987). Basic differential geometry 1.1 The initial mid surface The initial shell Special case: the plate The deformed shell The strain tensor and its simplifications 5 4 Linearization of E () to small strain and coordinate free description Change of metric tensor Change of curvature tensor The resulting Koiter energy 10 Author s addresses: Arnd Meyer TU Chemnitz Fakultät für Mathematik D Chemnitz 13

5 (Note that the volume element was simplified to hdτds). The third (original) Koiter-energy is obtained from a special ansatz for the unknown vector U as an expansion w.r.t. the contravariant surface basis: U = U i A i + U 3 A 3 with the 3 unknown functions (U 1, U, U 3 ) depending on (η 1, η ). Now, the derivatives of U lead to longer expressions, such as U,j = U i,j A i + U i A i,j + U 3,i A 3 + U 3 A 3,i = (U k,j U i Γ i jk)a k + (U 3,i U i B i j)a 3 U 3 B ik A k and the coefficients γ ij (U) and ϱ ij (U) are changed into the well-known complicate functions on derivatives of (U 1, U, U 3 ), compare [1]. If a more general material law is considered (still constant w.r.t. τ), we have a change in the expressions of the c ijkl only. For instance, a transversely isotropic material (compare [3, 4]) uses a spatial dependent normalized direction vector a(η 1, η ) of given fiber directions and 5 material parameters (µ, µ a, λ, α, β) to define the material tensor as C = µi + λ(i I) + α(aai + Iaa) + (µ a µ)ĉ + β(aaaa) (Here, Ĉ : X = (aa) X + X (aa) for each second order tensor X.) Obviously the fiber direction a is in the tangential plane of S 0 with a = a 1 A 1 + a A then the material coefficients are generalized to c ijkl = (A i A j ) : C : (A k A l ) = µa il A jk + λa ij A kl + α(a i a j A kl + A ij a k a l ) + (µ a µ)(a j a k A il + A jk a l a i ) + βa i a j a k a l without any change in the differential operators γ ij (U) and ϱ ij (U). References [1] P.G.Ciarlet,The Finite Element Method for Elliptic Problems, 530p., North-Holland, Amsterdam, (1978). [] P.G.Ciarlet, L.Gratie, A new approach to linear shell theory, Preprint Ser.No.17, Liu Bie Ju Centre for Math.Sc.(005). [3] A.F.Bower, Applied mechanics of solids, CRC(009). 1 Introduction We consider the deformation of a thin shell of constant thickness h under mechanical loads. If a usual linear elastic material behavior is proposed, then consequently a linearized strain tensor has to be considered. In this case, the well established Koiter shell equation is obtained after some additional simplifications. We consider these simplifications from the initial large strain equation to the Koiter shell equation in an easier form due to Ciarlet []. Based on this, we are able to find a coordinate free description, which means that differential operators (defined on the mid surface of the shell) are used instead of derivatives with respect to the surface parameters (coordinates) (η 1, η ). Basic differential geometry.1 The initial mid surface We start with the description of the basic differential geometry on both the undeformed shell (initial domain) and the shell after deformation. All vectors and matrices belonging to the initial configuration (mainly the co- and contravariant basis vectors and the matrices of first and second fundamental forms) are written as capital letters. All these quantities belonging to the deformed structure are the same lower case letters. Let S 0 = { Y ( η 1, η ) : ( η 1, η ) Ω R } be the mid surface of the undeformed shell, where Y denote the points of the surface in the 3-dimensional space and (η 1, η ) run through a parameter domain Ω. Then we have A i = Y the tangential vectors i = 1, ηi A 3 = A 3 = (A 1 A ) A 1 A surface normal vector. This defines the first metrical fundamental forms A ij = A i A j written as the ( )-matrix A = (A ij ) ij=1. The surface element is ds = A 1 A dη 1 dη = (deta) 1/ dη 1 dη 1 1

6 and the contravariant basis is A j = A jk A k with A j A k = δ j k and Ajk the entries of A 1. The second fundamental forms are ( ) B ij = η i η Y A j 3 = A i,j A 3 = A i A 3,j forming the matrix B = (B ij ) ij=1. We recall the Gauss- and Weingarten-equations A 3,i = B ij A jk A k = B ij A j A i,j =Γ k ija k + B ij A 3 with the Christoffel symbols Γ k ij = A kl (A il,j + A jl,i A ij,l ). Throughout this paper we use Einstein s summation convention, where consequently all indices run from 1 to only. Later on, we will need the two second order tensors A = A ij A i A j and B = B ij A i A j often referred as metric tensor and curvature tensor of the surface S 0. Throughout this paper a pair of vectors (first order tensors) as A 1 A (or A 1 A or similar) is understood as second order tensor. A second order tensor in general is any linear combination of such pairs. The main meaning of a second order tensor is its action as a map of the (3-dimensional) vector functions onto itself via the dot product: (A 1 A ) U = A 1 (A U) U (A 1 A ) = A (U A 1 ) consequently the second order tensor A 1 A has a trace tr(a 1 A ) = A 1 A and the transposed tensor of A 1 A is (A 1 A ) T = A A 1. The double dot product between two second order tensors such as (A 1 A ) : (A 3 A 4 ) = (A A 3 )(A 1 A 4 ) is a scalar function on (η 1, η ). Later on, we use 4th order tensors in the same manner, as a 4-tuple of vectors (A 1 A A 3 A 4 and an arbitrary linear combination of those) or as a pair of second order tensors. Here, the main operation is the double dot product as a map of second order tensors onto second order tensors. From this definition both tensors A and B are rank--tensors mapping each vector into the tangential space span(a 1, A ) = span(a 1, A ). Especially A is the orthogonal projector onto this -dimensional space, due to: A = A ij A i A j = A j A j = I A 3 A 3. with the Lamé constants µ = E 1 + ν and λ = µ ν 1 ν (for the plane stress assumption). Here, I is the 4th order identity map (I : X = X for each nd order tensor X ) and (I I) : X = I (I : X ) = I trx. Now, we end up with three different representations of the Koiter shell energy, depending on which strain formulation is inserted into (18). If the material tensor C is constant over the thickness (independent on τ), we integrate over τ [ 1/, +1/] and end up with the parts: with W (U) = h W a (U) + h3 1 W b (U) W a (U) = 1 E a : C : E a ds S 0 W b (U) = 1 E b : C : E b ds. S 0 With the formulas (13) and (17) this is a coordinate free form of Koiter s shell energy. The second equivalent formula (comparable to []) is obtained from the coordinate dependent strain equation E a = γ ij (U)A i A j and E b = ϱ ij (U)A i A j inserted into the energy functional. Here, we end up with and c ijkl = (A i A j ) : C : (A k A l ) = µa il A jk + λa ij A kl = E 1 + ν (Ail A jk + ν 1 ν Aij A kl ) W a (U) = 1 γ ij (U) c ijkl γ kl (U)dS S 0 W b (U) = 1 ϱ ij(u) c ijkl ϱ kl(u)ds S 0 11

7 The last term will be considered later, the first two terms are [ ] ( ) A i A k U η i,k A 3 [ ] =A i η Grad SU A i 3 = [Grad S Grad S U] A 3 This is a 3rd order tensor applied to A 3. It remains to investigate the last term from above, namely We have hence we can replace it by A i [ B k i A 3 (U,k A 3 ) ]. B k i = B ij A jk = B ij A j A k, = A i (B ij A j A k )A 3 (U,k A 3 ) = (B ij A i A j ) A k A 3 (U,k A 3 ) = B A k (U,k A 3 ) A 3 = B Grad S U (A 3 A 3 ), which is a product of 3 second order tensors. Therefore, the end result is E b = [Grad S Grad S U] A 3 B Grad S U (A 3 A 3 ) (17) 5 The resulting Koiter energy We complete the resulting deformation energy of the shell by inserting E (3) into the energy functional. Due to the desired small strain assumption in E (3), we use a linear material law, such as W (U) = 1 H 0 E : C : E dv (18) with a (possibly space dependent) 4th order material tensor C. The most simple case, the St.Vernant-Kirchhoff material, is considered to be C = µi + λ(i I) (Here, I denotes the identity tensor mapping each vector U onto itself). It should be stressed that the two vectors A 1 and A are dependent on the parametrization (η 1, η ) chosen to define S 0 but A 3 not, hence A and B are independent on the special coordinates (η 1, η ) but functions on the given point Y of S 0 only. So, (Y A 3 ) is called the Gaussian map and B the Weingarten map. Furthermore the surface gradient as gradient operator on the tangential space also is independent on the special parametrization (η 1, η ), obviously Grad S = A i η. i The matrix A 1 B has two eigenvalues λ 1 and λ as main curvatures at Y (η 1, η ), as well as the tensor B has these eigenvalues (together with a 0 as rank- tensor), so H = (λ 1 + λ )/ = trb = tr(a 1 B) is the mean curvature and K = λ 1 λ = det(a 1 B) the Gaussian curvature at Y.. The initial shell The initial shell is the 3-dimensional manifold { H 0 = X ( η 1, η, τ = η 3) = Y ( η 1, η ) + τha 3, ( η 1, η ) Ω, τ 1 } with the constant thickness h and A 3 from.1. For an easy description of the following let τ = η 3 be a synonym for the (dimensionless) thickness coordinate. We may use η 1 and η dimensionless as well, then A i have length dimension (in m) and A i in 1/m while A 3 = A 3 is dimensionless in any case. In 3D we have to consider the covariant basis G i = η i X = A i + τha 3,i, i = 1, and G 3 = ha 3 as well as the contravariant tensor basis G i (i = 1, ) and G 3 = h 1 A 3. The volume element of the shell is dv = [G 1, G, G 3 ] dη 1 dη dτ = h det(g) 1/ dη 1 dη dτ with the ( )-matrix G = (G ij ) i,j=1, G ij = G i G j, which is simply calculated as G = A ( I τha 1 B ) = (A τhb) A 1 (A τhb). () (1) 10 3

8 From this, the volume element is well-known as dv = h (1 τhh + (τh) K) dτ ds (but later approximately used as h dτ ds)..3 Special case: the plate Here we have a simplification on S 0 such as Y = L 1 e 1 η 1 + L e η, yielding A 3 = e 3 independent on (η 1, η ). From this a lot of simplifications arise:.4 The deformed shell G i = A i, B = O, B = 0 The basic assumption of the simple shell models consists in keeping a straight line of the points { Y (η 1, η ) + τha 3 (η 1, η ) : τ 1 } after the deformation also, i. e. the mid surface is deformed as S t = { y(η 1, η ) = Y (η 1, η ) + U(η 1, η ) : (η 1, η ) Ω } with an unknown displacement vector U (a function of (η 1, η ) as well as of Y ). The weaker assumption defines the deformed shell as H t = { x(η 1, η, τ) = y(η 1, η ) + τh d(η 1, η ) } (3) with an additional vector field d(η 1, η ) (the so called director vector). Here, we concentrate on the stronger Kirchhoff assumption, where d is the new surface normal vector a 3 of the deformed surface S t following its differential geometry: Let a i = η y = A i i + U,i the tangential vectors after deformation, then analogously we have a 3 = (a 1 a ) a 1 a The first term ϱ ij (U) is exactly found in the Koiter s shell equation (compare []) and will be transformed into a coordinate free description at the end of this chapter. The other part ζ () ij has a simple coordinate free meaning and will vanish after linearizing β. So, The term is so This leads to and 1 α (A A 3 ) U,1 + 1 α (A 3 A 1 ) U, =A 1 U,1 + A U, =Div S U = Grad S U ζ () ij = (Div S U) B ij. β 1 = a 1 a α A α (U,1 A ) + 1 α (A 1 U, ) + h.o.t., β = α (U,1 A ) A α (A 1 U, ) A 3 + h.o.t. = 1 + Div S U + h.o.t. β = 1 (Div S U) + h.o.t. b ij =ζ ij β = (B ij + ϱ ij + B ij (Div S U))(1 (Div S U)) + h.o.t. (15) =B ij + ϱ ij + h.o.t. (16) Hence, the linearization of leads to with (b ij B ij )A i A j E b = ϱ ij A i A j ϱ ij = A 3 (U,ij Γ k iju,k ). This tensor can be written in a coordinate free manner due to the following manipulations. ϱ ij A i A j = A i [ A j (U,ij A 3 ) Γ k ija j (U,k A 3 ) ] = A i [ A k (U,ik A 3 ) + (A k,i B k i A 3 ) (U,k A 3 ) ] 4 9

9 Hence, we have E a = A (Grad S U) T + (Grad S U) A (13) with the orthogonal projector A onto the tangential plane at Y. 4. Change of curvature tensor Here, the linearization of (b ij B ij )A i A j to E b = ϱ ij (U)A i A j is a longer calculation. We start with abbreviations for B ij = A 3 A i,j = 1 α [A 1, A, A i,j ] with the spatial inner product [a, b, c] = (a b) c and α = A 1 A. as surface normal vector of S t. Again we have a = (a ij ) i,j=1 with a ij = a i a j b = (b ij ) i,j=1 with b ij = a i,j a 3 as new first and second fundamental forms. With the 3D covariant basis is now g i = d = a 3, (4) η i x = a i + τha 3,i = A i + U,i + τha 3,i and g 3 = ha 3. Hence, we can define the ( )-matrix g = (g ij ) from g ij = g i g j, which is analogously to G: g = a ( I τh a 1 b ) = (a τh b) a 1 (a τh b). (5) In the same way we have b ij = ζ ij β with 3 The strain tensor and its simplifications ζ ij = 1 α [a 1, a, a i,j ] α β = a 1 a and we linearize both parts ζ ij and β separately. In ζ ij we delete all quadratic terms in U to: ζ ij = B ij + 1 α [U,1, A, A i,j ] + 1 α [A 1, U,, A i,j ] + 1 α [A 1, A, U,ij ]. (14) (The last term is A 3 U ij ). Now, let us start with the first two terms. Using Gauss Weingarten equations we get and 1 α [A, A i,j, U,1 ] = 1 [ ] A, Γ k α ija k + B ij A 3, U,1 = From the definition of the deformed shell as (3,4), we may deduce the 3Ddeformation gradient F = g i G i + g 3 G 3 = g i G i + a 3 A 3, the right Cauchy Green tensor C = F T F and the strain tensor E = 1 (C I) = 1 (g ij G ij ) G i G j = 1 (g ijg i G j A), (6) which is a rank- tensor only without components of G i G 3 or G 3 G 3. In the following we expand this tensor with respect to the surface basis vectors A i A j instead of G i G j which is possible due to span(g 1, G ) = span(a 1, A ): G i = A i + τha 3,i So, = Γ 1 ij(a 3 U,1 ) + 1 α B ij [A, A 3, U,1 ]. ζ ij = B ij + ϱ ij + ζ () ij ϱ ij = A 3 (U,ij Γ k iju,k ) ζ () ij = 1 α B ij ([A, A 3, U,1 ] + [A 3, A 1, U, ]). yields = A i τhb ij A jk A k = (A ij τhb ij )A j G k = G ki G i = G ki (A ij τhb ij )A j. (7) Hence, g ij G i G j is written as [ (Aks τhb ks )G si] g ij [ G jm (A ml τhb ml ) ] A k A l 8 5

10 which means that E = ɛ ij A i A j (8) and the matrix of the coefficients ɛ ij is the following matrix product (ɛ ij ) i,j=1 = A(A τhb) 1 g(a τhb) 1 A A (9) = (I τhba 1 ) 1 g(i τha 1 B) 1 A. (10) Note, that the expression (8) describes the correct strain tensor without any simplifications belonging to the Kirchhoff-assumption (4). In contrast to the formula E = (g ij G ij ) G i G j all dependencies on the thickness coordinate τ are contained in ɛ ij only, the tensor basis is constant w.r.t. τ. From this formula the simplifications towards Koiter s shell equation can be deduced. This is done in 3 steps: 1. We expand E w.r.t. powers of (τh) and neglect all quadratic and higher order terms, yielding E (1).. Then E (1) still contains nonlinear expressions similar to large strain, which are products of derivatives of U, such as (U,i U,j ). As usually for small strain equations we neglect all such products, yielding E (3) being a linear differential operator applied to U. 3. Before this last linearization can be done, we neglect a term which is the product of τh (small) with the derivatives of U. This is: τha 1 a is simplified to τhi, because a = A+(U,i A j +A i U,j ) i,j=1+ h.o.t.. These 3 steps result in an approximate strain tensor, which is typically a linear differential operator acting on U (as small strain) and is a sum of change of metric and change of curvature as E (3) = E a τhe b E a = γ ij (U)A i A j E b = ϱ ij (U)A i A j The calculations within these 3 steps are easily done: 1. First we expand the matrix (10) w.r.t. (τh) powers and neglect all quadratic and higher terms, yielding (ɛ (1) ij ) i,j=1 = a A τh(ba 1 a + aa 1 B b) (11) and Now, the remark of step 3 leads to E (1) = ɛ (1) ij Ai A j. E () = [(a ij A ij ) τh(b ij B ij )]A i A j, which is still nonlinearly dependent on U.. We linearize E () to E (3) = E a τhe b which can be seen as Koiter-strain-tensor because the coefficients γ ij (U) and ϱ ij (U) are linear differential operators applied to U and are the same expressions as in Koiter s shell model. 4 Linearization of E () to small strain and coordinate free description We recall and linearize both parts separately. 4.1 Change of metric tensor From simply follows E () = [ 1 (a ij A ij ) τh(b ij B ij )]A i A j a ij = A ij + A i U,j + A j U,i + U,i U,j as found in the Koiter shell equation as well. The tensor γ ij = 1 (A i U,j + A j U,i ) (1) E a = γ ij (U)A i A j has a simple coordinate free representation from: E a = E 1 + E T 1 E 1 = (A j U,i )A i A j = A i U,i A j A j = Grad S U A. 6 7