Appendix A Rudiments of Tensor Calculus

This section is devoted to the fundamentals of the Tensor calculus. Through its mathematical structure, it forms the framework for the rational presentation in present treatise. Nevertheless, only the most necessary tools are to be explained at this point. For an exhaustive discussion of this topic, please refer to the literature, such as [2, 4, 6–8] or [1], among others. The mechanical considerations refer to the Euclidean space. This space is selected as it has a metric, so provides measures for lengths and angles. This space is given a canonical orientation by a positively oriented base {ei}. This linear independent system of vectors is called orthonormal if the following applies.  1ifi = j ei · ej = (A.1) 0ifi = j

First, second, and fourth-order tensors are introduced, giving priority to symbolic notation. In the following, these tensors are represented in component representation with respect to an orthonormal basis {ei}. The Einstein sum convention [3] applies here. Duplicate (dummy) indices are summed up.

3 a = aiei = aiei (A.2) i=1 3 3 A = Aijei ⊗ej = Aijei ⊗ej (A.3) i=1 j=1 3 3 3 3 A = Aijkl ei ⊗ej ⊗ek ⊗el = Aijkl ei ⊗ej ⊗ek ⊗el (A.4) i=1 j=1 k=1 l=1

Usually the indices i, j, k, l ∈{1, 2, 3} are used in present explanations. The use of α, β, γ, δ ∈{1, 2} is analogous. Already introduced indirectly in Eq. (A.1), the Kronecker symbol (also Kronecker delta) is defined as follows.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2019 105 M. Aßmus, Structural Mechanics of Anti-Sandwiches, SpringerBriefs in Continuum Mechanics, https://doi.org/10.1007/978-3-030-04354-4 106 Appendix A: Rudiments of Tensor Calculus  1ifi = j δij = (A.5) 0ifi = j

It proves to be useful in the presentation of second- 1 = δijei ⊗ej = ei ⊗ei or fourth- order metric tensors I = δilδjk ei ⊗ej ⊗ek ⊗el = ei ⊗ej ⊗ej ⊗ei. In addition, the Levi- Civita or permutation symbol is also required. When considering three dimensions, it is triple indexed. ⎧ ⎨⎪+1, if (i, j, k) is an even permutation of (1, 2, 3)

ijk = −1, if (i, j, k) is an odd permutation of (1, 2, 3) (A.6) ⎩⎪ 0, if (i, j, k) is no permutation of (1, 2, 3)

The scalar product between first-, first- and second- and second-order tensors is defined as follows.

a · b = ai bi (A.7)

A · b = Aijbj ei (A.8)

A · B = AijBklei ⊗ei · ek ⊗el = AijBjlei ⊗el (A.9)

The scalar product between first-order tensors is commutative, but not between first- and second-order tensors, and that between fourth-order tensors. With the scalar product, e.g. the components of a tensor are determined.

Aij = ei · A · ej (A.10)

Two vectors a = o and b = o are mutually orthogonal if their scalar product yields zero (a · b = 0). The double-scalar product between second-order tensors is defined as follows.

A : B = Aij Bkl ei ⊗ej : ek ⊗el

= Aij Bji (A.11)

Between a tensor fourth and a second-order tensors it can be determined as follows.

A : B = Aijkl Bmn ei ⊗ej ⊗ek ⊗el : em ⊗en

= Aijkl Blk ei ⊗ej =F (A.12)

The operation between two fourth-order tensors is determined as follows.

A : B = Aijkl Bmnop ei ⊗ej ⊗ek ⊗el : em ⊗en ⊗eo ⊗ep

= Aijkl Blkopei ⊗ej ⊗eo ⊗ep =G (A.13) Appendix A: Rudiments of Tensor Calculus 107

The dyadic product between first-order tensors is as follows.

a⊗b = aibjei ⊗ej = Aijei ⊗ej Aij = Aji (A.14)

b⊗a = bjaiej ⊗ei = Ajiei ⊗ej aibj = bjai (A.15)

It is not commutative (a⊗b = b⊗a). The dyadic product between second-order ten- sors can be determined as follows.

A⊗B = AijBklei ⊗ej ⊗ek ⊗el = Cijklei ⊗ej ⊗ek ⊗el (A.16)

B⊗A = BklAijek ⊗el ⊗ei ⊗ej = Dijklei ⊗ej ⊗ek ⊗el (A.17)

Also, this link is not commutative. (A⊗B = B⊗A). The transposition of a second-order tensor A is defined by

  a · A · b = b · A · aA= Ajiei ⊗ej (A.18) and the transposition of a tensor of fourth-order A by

  A : A : B = B : A : A A = Aklijei ⊗ej ⊗ek ⊗el . (A.19)

The cross product between two first-order tensors is defined as follows.

c = a × b = aibjei × ej = a b  e i j ijk k √ = ab sin ϕ ec a = a · a (A.20)

The result is a vector c which is orthogonal to the plane spanned by a and b. ϕ is the smaller angle between a and b. The cross product is anti-commutative.

a × b =−b × a (A.21)

Also of interest is the difference between polar and axial vectors. Due to the special consideration of rotations, axial vectors are used in the present work. Physically, this describes a vector whose length corresponds to the length of a circular line segment and whose direction describes the direction of rotation. The polar vector, on the other hand, is characterized by defining the magnitude and direction of a straight line segment [11]. The cross product between a second-order and a first-order tensor is defined by the following expression.

A × c = (a⊗b) × c = a⊗(b × c) = a⊗d (A.22)

This cross product has the following property. 108 Appendix A: Rudiments of Tensor Calculus

  A × c =− c × A (A.23)

The vectorial invariant of a second-order tensor can be determined by the scalar cross product with the unit tensor 1 = ei ⊗ei. This is often designated with an subscript cross as A×, cf. [9, 10].

A× = 1 · ×A

= (ei ⊗ei) · × (Akl ek ⊗el)

= Akl δik ei × el

= Aij ei × ej (A.24)

For a compact notation, the box product is introduced, which corresponds to the scalar cross product and has the following characteristics.

1 · ×A = 1  A = A  1 =−1  A =−A  1 (A.25)

If A ∈ Sym holds true, the vectorial invariant is vanishing. Every can tensor can be assigned its symmetric Asym (A = AT resp. b · A = A · b) and anti(sym)metric (or skew) part Askw (A =−A resp. b · A =−A · b).  Asym = 1 A + A A = Asym + Askw 2  (A.26) skw = 1 −  A 2 A A

Skew tensors can also be represented as follows.

Askw = a × 1 = 1 × a (A.27)

For fourth-order tensors, the symmetric portion of the unit tensor is of interest. For this purpose, the transposer T should be introduced first [2].

T = ei ⊗ej ⊗ei ⊗ej (A.28)

It maps all second-order tensors into their transpose. For application, the tensor A ∈ Lin is used here.

T : A = ei ⊗ej ⊗ei ⊗ej : Aklek ⊗el

= δjk δilAklei ⊗ej

= Ajiei ⊗ej = A

Thus, the symmetric part of the fourth-order unit tensor can be determined. Appendix A: Rudiments of Tensor Calculus 109   sym 1 1 I = I + T = e ⊗e ⊗e ⊗e + e ⊗e ⊗e ⊗e (A.29) 2 2 i j j i i j i j This tensor, also called symmetrizer, maps all second-order tensors A ∈ Lin into their symmetric part.  sym 1 I : A = e ⊗e ⊗e ⊗e + e ⊗e ⊗e ⊗e : A e ⊗e 2 i j j i i j i j kl k l 1  = δ δ A e ⊗e + δ δ A e ⊗e 2 ik jl kl i j jk il kl i j 1  = A e ⊗e + A e ⊗e 2 ij i j ji i j 1  = A + A 2 = Asym

The symmetrizer of the planar surface continuum Psym can be derived analogously by replacing the indices from ijkl to αβγ δ. Furthermore, each tensor can be decomposed additively into its dilatoric Adil and deviatoric part Adev. For three dimensional problems, this is done as follows.

Adil = 1 [A : 1] 1 A = Adil + Adev 3 (A.30) Adev = A − Adil

For two dimensional problems, however, the following relationships apply by using the unit tensor of the surface continuum P = eα ⊗eα.

Bdil = 1 [B : P] P B = Bdil + Bdev 2 (A.31) Bdev = B − Bdil

The inverse of a tensor is defined by the following relationships.

 −1 A−1 · A = A · A−1 = 1 A−1 = A (A.32)

− − − −1 A 1 : A = A : A 1 = I sym A 1 = A (A.33)

The inverse of the second-order metric tensor coincides with the metric tensor itself (1−1 = 1). For the fourth-order metric tensor this only applies to its symmetric part ([Isym]−1 = Isym). The determinant of a tensor is considered as criterion for its in- vertibility. The invertibility of any tensor A is guaranteed if det A = 0 holds. The determinants of second- and fourth-order tensors can be defined by their eigenval- ues λi. 110 Appendix A: Rudiments of Tensor Calculus

ND 3ND detA = λi detA = λi (A.34) i=1 i=1

The respective eigenvalues can be determined by the associated eigenproblem.

A · ci = λici A : Ci = λiCi (A.35)

Herein, ci and Ci are first- and second-order eigentensors. The characteristic poly- nomials for the non-trivial solutions can be represented as follows.

λ3−I λ2+II λ−III=0 (A.36) λ9 − Iλ8 + II λ7 − III λ6 + IV λ5 − V λ4 + VI λ3−VII λ2+VIII λ−IX =0 (A.37)

Therein I, II , III , ..., IX are principal invariants of A and A, respectively. The above transition allows the solution with standard methods of linear algebra. Further hints for the solution of the eigenvalue problem of fourth-order tensors are presented in [5], while the solution of second-order tensors is described in detail in the textbooks mentioned at the outset of this annex. A tensor A ∈ Lin is called positive-definite if

b · A · b > 0 ∀ b = o (A.38) holds. Also, if A ∈ Sym, then all eigenvalues are positive. The Nabla operator is defined by the following relationship. With the aid of this, the divergence, gradient, and rotation of a tensor can be formed. ⎧ ⎪∇· ∂ ⎨ (divergence) ∇ = ∇ ⊗ ei ⎪ (gradient) (A.39) ∂Xi ⎩ ∇× (curl)

In the present treatise the gradient is written as ∇ = ∇ ⊗ for the sake of brevity. The scalar product of the Nabla operator with itself ∇·∇gives the scalar-valued Laplace operator ∇2. When applied to a twice differentiable scalar field, this operator assigns the divergence of its gradient to the scalar.

(∇·∇)w = ∇2w = ∇·(∇w) (A.40)

The application in two dimensional Cartesian coordinates provides the following relation.

∂2w ∂2w ∇2w = + (A.41) ∂ 2 ∂ 2 X1 X2 Appendix A: Rudiments of Tensor Calculus 111

For reasons of a consistent notation, the Laplace operator will not be denoted as here, as commonly found in the literature. For the conversion of surface into volume integrals and vice versa, this treatise uses the Gauss integral theorem (representative for the scientists Gauss, Ostrogradski, Stokes and Green) used. For explanation, V is a volume in E3 with its surface A and a surface normal n. At this point, a generalized integral theorem is introduced to represent gradient, divergence and rotation theorems. The transformation of the integrals can thus be described as follows . ⎧ ⎨⎪⊗ ∇ ◦  dV = n ◦  dA ∀  ≡{R ≥ 1}∧◦= · (A.42) ⎩⎪ V A ×

The symbol  stands here for a tensor field of order R ≥ 1.

References

1. Altenbach J, Altenbach H (1994) Einführung in die Kontinuumsmechanik. Teubner, Stuttgart 2. Bertram A (2012) Elasticity and plasticity of large deformations: an introduction, 2nd edn. Springer, Berlin. http://dx.doi.org/10.1007/978-3-642-24615-9 3. Einstein A (1916) Die Grundlage der allgemeinen Relativitätstheorie. Annalen der Physik 354(7):769–822. http://dx.doi.org/10.1002/andp.19163540702 4. Haupt P (2002) Continuum mechanics and theory of materials, 2nd edn. Springer, Berlin. http://dx.doi.org/10.1007/978-3-662-04775-0 5. Itskov M (2000) On the theory of fourth-order tensors and their applications in computational mechanics. Comput Methods Appl Mech Eng 189(2):419–438. http://dx.doi.org/10.1016/ S0045-7825(99)00472-7 6. Itskov M (2015) Tensor algebra and tensor analysis for engineers, 4th edn. Springer, Cham. http://dx.doi.org/10.1007/978-3-319-16342-0 7. Lai W, Rubin D, Krempl E (2010) Introduction to continuum mechanics, 4th edn. Butterworth- Heinemann, Oxford 8. Lebedev LP, Cloud MJ, Eremeyev VA (2010) Tensor analysis with applications in mechanics. World Scientific, New Jersey. http://dx.doi.org/10.1142/9789814313995 9. Legally M (1962) Vorlesungen über Vektorrechnung. Geest & Portig, Leipzig 10. Wilson EB (1901) Vector analysis, founded upon the lectures of G.W. Gibbs. Yale University Press, New Haven 11. Zhilin PA (2001) Vectors and second-rank tensors in three-dimensional space (in Russian). Nestor, St. Petersburg Appendix B Elastic Potential of Simple Materials

In the modeling of reversible, time-independent processes, the stress resultants are solely determined by the state of deformation [6]. When restricting to decoupled deformation states of a surface continuum, this leads to a unambiguous assignment.

N = N(G) q = q (g) L = L(K) (B.1)

The requirement of path independence is given when the kinetic measures can be derived by differentiating an elastic potential function W with respect to the conjugate kinematic quantities.

∂W ∂W ∂W N = q = L = (B.2) ∂ G ∂ g ∂K

The strain energy function W depends only on the deformations [5]. It is a homoge- neous quadratic function of G, g, and K [4].

1  W = G : A : G + K : D : K + g · Z · g (B.3) 2 Therein, A and D are constitutive tensors of fourth-order, and Z is a constitutive tensor of second order. A discussion of the possible coupling of membrane strains, curvature changes, and more specifically with transverse shear strains, can be re- viewed in e.g. [1, 2]. The strain energy should be positive definite.

W(G, K, g)>0 ∀{G, K} = 0 ∧ g = o (B.4)

Thus, the constitutive tensors must also be positive-definite. The following individual portions can be identified.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2019 113 M. Aßmus, Structural Mechanics of Anti-Sandwiches, SpringerBriefs in Continuum Mechanics, https://doi.org/10.1007/978-3-030-04354-4 114 Appendix B: Elastic Potential of Simple Materials

1  W = W(G) = G : A : G M 2 Yh Yhν = G : G + [G : P]2 2(1 + ν) 2(1 − ν2) 1 = GhG : G + (B−G)h [G : P]2 (B.5) 2 1  W = W(K) = K : D : K B 2 Yh3 Yh3ν = K : K + [K : P]2 24(1 + ν) 24(1 − ν2) h3 1 h3 = G K : K + (B−G) [K : P]2 (B.6) 12 2 12 1  W = W(g) = g · Z · g S 2 κYh = g · g 4(1 + ν) 1 = κ Ghg · g (B.7) 2 For the forces and moments we obtain the following constitutive equations.

N = A : G = 2Gh G + (B−G)h [G : P] P (B.8) h3 h3 L = D : K = 2G K + (B−G) [K : P] P (B.9) 12 12 q = Z · g = κGh g (B.10)

Equations (B.8)–(B.10) are valid only in the simplest case of material symmetries. Terms for the coupling of membrane and bending state as well as transverse shear state, as they are required in the generally anisotropic case as well as in the consid- eration of initially curved surfaces are not considered here. The stiffness tensors A, D, and Z can be given as follows.

sym A = 2Gh P + (B−G) h P ⊗P (B.11) 3 3 h sym h D = 2G P + (B−G) P ⊗P (B.12) 12 12 Z = κ GhP (B.13)

Correlations to the engineering parameters Y and ν are as follows.

4BG B − G Y = ν = (B.14) B + G B + G Appendix B: Elastic Potential of Simple Materials 115

In the classical theories of thin-walled structural elements, representations of mem- brane stiffness (index M), bending stiffness (index B), and transverse shear stiff- ness (index S) have been manifested [8]. These are as follows.

Yh Yh3 h3 DM = = (B + G) hDB =   = (B + G) DS = κ Gh 1 − ν2 12 1 − ν2 12 (B.15)

This leads to an alternative representation, as it is maintained e.g. in [7].

sym A = DM (1−ν) P + DM ν P ⊗P (B.16) sym D = DB (1−ν) P + DB ν P ⊗P (B.17)

Z = DS P (B.18)

It is trivial to show that Eqs. (B.11)–(B.13)or(B.16)–(B.18) can be transformed into Eq. (2.52). Altenbach [3] introduces a further way of representation by grouping the bases while restricting to second-order tensors. This strategy follows differential-geometric thoughts. Due to the equivalence of these representations, however, this possibility should not be explicitly presented here. The representation in the Eqs. (B.16)–(B.18) competes with the representations in (2.52). While the projector representation is useful in terms of mathematical operations, the representation presented above char- acterized by engineering interpretations (DM, DB, DS) is advantageous in the context of the numerical treatment as presented in subsequent sections.

References

1. Altenbach H (1984) Analytische Modelle zur Beschreibung von in Dickenrichtung homogenen und inhomogenen dünnen Platten und Schalen. Zeitschrift für Angewandte Mathematik und Mechanik 64(10):M430–M431. http://dx.doi.org/10.1002/zamm.19840641003 2. Altenbach H (1985) Zur Theorie der inhomogenen Cosserat-Platten. Zeitschrift für Angewandte Mathematik und Mechanik 65(12):638–641. http://dx.doi.org/10.1002/zamm. 19850651219 3. Altenbach H (1987) The direct approach in the theory of viscoelastic shells (in Russian). Ha- bilitation thesis, Leningrad Polytechnic Institute 4. Altenbach H, Zhilin P (1988) A general theory of elastic simple shells (in Russian). Adv Mech (Uspekhi Mekhaniki) 11(4):107–148 5. Backhaus G (1983) Deformationsgesetze. Akademie Verlag, Berlin 6. Bertram A, Glüge R (2015) Solid mechanics: theory, modeling, and problems. Springer, Cham. http://dx.doi.org/10.1007/978-3-319-19566-7 7. Naumenko K, Eremeyev VA (2014) A layer-wise theory for laminated glass and photovoltaic panels. Compos Struct 112:283–291. http://dx.doi.org/10.1016/j.compstruct.2014.02.009 8. Timoshenko S, Woinowsky-Krieger S (1987) Theory of plates and shells, 2nd edn. McGraw- Hill, New York, (1st edn. 1959) Appendix C Vector-Matrix Formulation

To handle the boundary value problem numerically efficient, the basic equations must be transformed into a form readable by a computer algebra system. At first it seems beneficial to introduce a vector-matrix notation for the constitutive law. For this purpose, the schematic of the original Voigt notation [5] is used here. When ap- plied to the three dimensional Cauchy continuum, the symmetry of stress and strain tensors as well as the main, left and right sub symmetry of the elasticity tensor are exploited. The reduction by a semi-circulating mnemonic rule, taking into account the orthonormal system, leads to a 6×1 vector representation for second-order ten- sors and a 6×6 matrix representation for fourth-order tensors. When restricting to orientation-independent material behavior, the following representation applies to linear constitutive relation T = C : E known as Hookes law [2].

t = Ce ⎡ ⎤ ⎡ ⎤⎡ ⎤ T11 C1111 C1122 C1122 000 E11 ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢T22⎥ ⎢ C1111 C1122 000⎥⎢ E22 ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢T33⎥ ⎢ C1111 000⎥⎢ E33 ⎥ ⎢ ⎥ = ⎢ ⎥⎢ ⎥ (C.1) ⎢T23⎥ ⎢ C2323 00⎥⎢2E23⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦ T13 C2323 0 2E13 T12 sym C2323 2E12

Herein, Tij are stresses related to the Cauchy stress tensor, Ekl are linearized strains, and Cijkl are material parameters of the constitutive tensor. This representation is mathematically inconsistent due to the following problems, cf. [4].

tt = T : T ee = E : E te = T : E

However, it has found wide application especially in dealing with numerical solution techniques. The factor 2 in the representation of the strain vector is used for the identical representation of the strain energy function in tensor and vector-matrix notation. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2019 117 M. Aßmus, Structural Mechanics of Anti-Sandwiches, SpringerBriefs in Continuum Mechanics, https://doi.org/10.1007/978-3-030-04354-4 118 Appendix C: Vector-Matrix Formulation

2W = te = T : E

By exploiting the symmetries of the membrane force tensor N and the moment tensor L, the Voigt notation can also be applied to the structural mechanics problem presented in present work as introduced in [3]. The first-order transverse shear force tensor q is also converted into the vector notation. For the single layer, the kinetic measures can be represented as follows, using s as the global kinetic quantity for the sake of simplicity and indexed according to the loading case.

  sM = N11 N22 N12 (C.2)   sS = Q1 Q2 (C.3)   sB = M11 M22 M12 (C.4)

Analogous approach is used for the kinematic measures G, K, and g with e being used as a global kinematic variable for simplification.

  eM = G11 G22 2G12 (C.5)   eS = g1 g2 (C.6)   eB = K11 K22 2K12 (C.7)

Based on the constitutive tensors (B.16)–(B.18), a matrix notation can be introduced in an analogous manner as done in Hooke’s law as shown in Eq. (C.1). Membrane, plate, and transversal shear stiffness are then represented as follows, with isotropy being assumed. ⎡ ⎤ ⎡ ⎤   A1111 A1122 0 D1111 D1122 0 ⎣ ⎦ ⎣ ⎦ Z11 0 A = A2222 0 D = D2222 0 Z = (C.8) sym Z22 sym A1212 sym D1212

Values of the coefficients Aαβγ δ , Dαβγ δ and Zαβ can be derived directly from the Eqs. (2.53)–(2.55) in conjunction with Eq. (2.52). Based on this, the following rep- resentation is mostly to be found in the literature, cf. [1]. ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ sM A00 eM ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ sB = D0 eB (C.9) sS sym Z eS

Consequently, the matrices required in the FEM are to be given in terms of global variables (Indices ◦, , c) for the three layered composite in vector-matrix form. The constitutive tensors of the global quantities can be introduced in matrix notation as follows, introducing C as a global stiffness quantity for the sake of simplicity. Appendix C: Vector-Matrix Formulation 119 ⎡ ⎤ K + K K aM 2bM bM 0 ˆ K = ⎣ K K + K ⎦ ∀ ∈ {◦, , } CM aM aM 2bM 0 K c (C.10) 00bK ⎡ M⎤ K + K K aB 2bB bB 0 ˆ K = ⎣ K K + K ⎦ ∀ ∈ {◦, , } CB aB aB 2bB 0 K c (C.11) K 00bB

  10 Cˆ K = aK ∀ K ∈ {◦, ,c} (C.12) S S 01

Here, the following abbreviations have been introduced based on the engineering interpretations for membrane stiffness DM, bending stiffness DB, and transverse shear stiffness DS. ⎧ ⎪ t νt + bνb =◦ ⎨DL DL if K aK = Dt νt − Dbνb if K = ∀ L ∈ {M, B} (C.13) L ⎩⎪ L L Dc νc if K = c ⎧ L ⎪ 1−νt t + 1−νt b =◦ ⎨ 2 DL 2 DL if K t t bK = 1−ν Dt − 1−ν Db K = ∀ L ∈ {M, B} (C.14) L ⎪ 2 L 2 L if ⎩ t 1−ν Dc if K = c ⎧ 2 L ⎪ t + b =◦ ⎨DS DS if K aK = Dt − Db if K = (C.15) S ⎩⎪ S S c = DS if K c

With the above representation, the generalized stiffness matrices can be specified. ⎡ ⎤ ˆ ◦ CM 000 ⎢ ˆ ◦ ⎥ ◦ ⎢ 0 C 00⎥ C = ⎢ M ⎥ MB ⎣ ˆ ◦ ⎦ (C.16) 00CB 0 000Cˆ ◦ ⎡ ⎤ B 0 Cˆ 00 ⎢ M ⎥ = ⎢0000⎥ CMB ⎣ ˆ ⎦ (C.17) 000CB 0000 ˆ ◦ ◦ C 0 C = S (C.18) S 0 Cˆ ◦  S ˆ 0 C C = S (C.19) S 00 120 Appendix C: Vector-Matrix Formulation

The zero matrices in the Eqs. (C.16) and (C.17) each possess three columns and rows, while the null matrices in Eqs. (C.18) and (C.19) have only two columns and rows each. The B matrices for combining the approximation of local continuous kinematic measures with the discrete degrees of freedom of the element are given as follows.

  ˆ ◦ ˆ ˆ ◦ ˆ B = BMB BMB ... BMB B = B B B B (C.20) MB 1 2 N MBi Mi Mi Bi Bi   ˆ ◦ ˆ B = BS BS ... BS B = B B (C.21) S 1 2 N Si Si Si

The sub measures introduced herein are given in the following matrices. ⎡ ⎤ i N,1 00000000 Bˆ ◦ = ⎣ N i ⎦ (C.22) Mi 0 ,2 0000000 i i N, N, 0000000 ⎡ 2 1 ⎤ i 00N,1 000000 Bˆ = ⎣ N i ⎦ (C.23) Mi 000 ,2 00000 i i 00N,2 N,1 00000 ⎡ ⎤ i 000000N,1 00 ˆ ◦ ⎣ i ⎦ B = 00000−N, 000 (C.24) Bi 2 i i 00000−N, N, 00 ⎡ 1 2 ⎤ i 00000000N,1 ˆ ⎣ i ⎦ B = 0000000−N, 0 (C.25) Bi 2 − i i 0000000N,1 N,2   i ◦ 0000N, 0 i 00 Bˆ = 1 N (C.26) Si i i 0000N, − 000  2 N  i 00000000 Bˆ = N (C.27) Si 0000000−N i 0

The differential operators for membrane, bending, and transverse shear state as well as their auxiliary matrices are structured as follows.

  = ◦ ◦ DMB DM DM DB DB (C.28)   = ◦ DS DS DS (C.29)

The sub measures are structured as follows. Appendix C: Vector-Matrix Formulation 121 ⎡ ⎤ ∂ ∂ 0 0000000 ⎢ X1 ⎥ ◦ = ⎣ ∂ ⎦ DM 0 ∂X 0000000 (C.30) ∂ ∂ 2 ∂ ∂ 0000000 ⎡ X2 X1 ⎤ ∂ 00 ∂ 0 00000 ⎢ X1 ⎥ = ⎣ ∂ ⎦ DM 00 0 ∂X 00000 (C.31) ∂ ∂ 2 00 ∂X ∂X 00000 ⎡ 2 1 ⎤ ∂ 00000 ∂ 000 ⎢ X1 ⎥ ◦ = ⎣ ∂ ⎦ DB 00000 0 ∂X 00 (C.32) ∂ ∂ 2 00000 ∂ ∂ 00 ⎡ X2 X1 ⎤ ∂ 0000000 ∂ 0 ⎢ X1 ⎥ = ⎣ ∂ ⎦ DB 0000000 0 ∂X (C.33) ∂ ∂ 2 0000000 ∂X ∂X  2 1 ∂ ◦ 0000 ∂ 1000 = X1 DS ∂ (C.34) 0000 ∂ 0100  X2  000000010 D = (C.35) S 000000001

The auxiliary matrices Ai ∀ i ∈ {1,...,5} for transforming the terms of virtual work into the vector-matrix notation are defined as follows.   1 00−200− (h◦ + hc) 0 −h 0 A = (C.36) 1 c 00 0 −20 0 − (h◦ + hc) 0 −h h  1000 A2 = (C.37)  0100 = 1 1 ◦ A3 I0 2 h I 2 h I (C.38) 1  A = 0 2I h◦I h I 4 c (C.39) ⎡h ⎤ 10 0 0 00000000 0 ⎢ ⎥ ⎢ 01 0 0 00000000 0⎥ ⎢ − 2 ⎥ ⎢ 00 1 0 0000000 hc 0 ⎥ ⎢ − 2 ⎥ ⎢ 00 0 1 00000000 hc ⎥ = ⎢ 00 0 0 00100000 0⎥ A5 ⎢ ◦ ⎥ ⎢ 1 1 ◦ + c − − h +hc ⎥ h 0 (h h ) 0000010 0 c 0 ⎢ 2 2 h ◦ ⎥ ⎢ 1 1 ( ◦ + c) − h +hc ⎥ ⎢ 0 2 h 0 2 h h 00010000 hc ⎥ ⎣ 1 ◦ − − h ⎦ 00 0 0 2 h 00000 1 hc 0 1 ◦ − h 00 0 0 02 h 00 0 1 0 0 hc (C.40) 122 Appendix C: Vector-Matrix Formulation

The unit I and the zero matrices 0 in Eqs. (C.38)–(C.39) each have three columns and rows. The auxiliary matrix for generating the mass matrix has nine columns and nine rows. ⎡ ⎤ H11 H12 H13 H14 H15 H16 H17 H18 H19 ⎢ ⎥ ⎢ H22 H23 H24 H25 H26 H27 H28 H29⎥ ⎢ ⎥ ⎢ H33 H34 H35 H36 H37 H38 H39⎥ ⎢ ⎥ ⎢ H44 H45 H46 H47 H48 H49⎥ ⎢ ⎥ H = ⎢ H55 H56 H57 H58 H59⎥ (C.41) ⎢ ⎥ ⎢ H66 H67 H68 H69⎥ ⎢ ⎥ ⎢ H77 H78 H79⎥ ⎣ ⎦ H88 H89 sym H99

The parameters Hij ∀ i, j ∈{1,...,9} in H ∈ Sym are shorthand for the following components.

◦ H11 = H22 = H55 = ρ 2 H = ρ◦ − ρc 33 3     ◦ c ◦ 1 c ◦ 2 2 H = H = ρ − ρ β + ρ β + ρ (h ) + 3 h 66 77 12     ◦ c ◦ 1 c 2 ◦ 2 H = H = ρ − ρ β + ρ β + ρ h + 6 (h ) 88 99 12   1 H = H = ρ◦ − ρc α◦ + ρ α + ρch 16 27 2   1 H = H = ρ◦ − ρc α + ρ α◦ + ρch◦ 18 29 2   1 H = H = ρ◦ − ρc α + ρ α◦ + ρch◦ 36 47 6   1 H = H = ρ◦ − ρc α◦ + ρ α + ρch 38 49 6   1 H = H = ρ◦ − ρc β◦ + ρ β◦ + ρch◦h 68 79 3 All other expressions in matrix (C.41) are identical to zero.

References

1. Altenbach H, Altenbach J, Kissing W (2004) Mechanics of composite structural elements. Springer, Berlin. http://dx.doi.org/10.1007/978-3-662-08589-9 2. Altenbach J, Altenbach H (1994) Einführung in die Kontinuumsmechanik. Teubner, Stuttgart 3. Aßmus M, Bergmann S, Eisenträger J, Naumenko K, Altenbach H (2017) Consideration of non-uniform and non-orthogonal mechanical loads for structural analysis of photovoltaic composite structures. In: Altenbach H, Goldstein RV, Murashkin E (eds) Mechanics for Appendix C: Vector-Matrix Formulation 123

materials and technologies, advanced structured materials, vol 46, Springer, Singapore, pp 73–122. http://dx.doi.org/10.1007/978-3-319-56050-2_4 4. Nordmann J, Aßmus M, Altenbach H (2018) Visualising elastic anisotropy: theoretical back- ground and computational implementation. Contin Mech Thermodyn 30(4):689–708. http://dx.doi.org/10.1007/s00161-018-0635-9 5. Voigt W (1966) Lehrbuch der Kristallphysik (mit Ausschluss der Kristalloptik). Springer, Wiesbaden. http://dx.doi.org/10.1007/978-3-663-15884-4, Reproduktion des 1928 erschiene- nen Nachdrucks der ersten Auflage von 1910 Appendix D Numerical Integration

The numerical integration is explicitly presented here, since artificial stiffening ef- fects can be a problem of the finite element solution with the presented element. Such stiffening effects are characterized by the fact that the size sought with the numerical solution of the problem is smaller than that of the closed-form solution. The solu- tion of structure mechanical problems is therewith characterized that the resulting values of the degrees of freedom are too small due to the too rigid mapping of the structure. Although convergence to the exact solution occurs with increasing mesh refinement, it does so much slower than with locking-free elements [7]. A distinction must be made between geometric and material locking effects. Geometric locking effects include plane shear locking, transverse shear locking, membrane locking, and trapezoidal locking, or curvature-thickness locking [4]. There is also a material locking effect, volumetric locking (also known as Poisson locking), which occurs predominantly in materials with a Poisson ratio near 0.5. As is known from the literature, the element used in the present context tends to transverse shear locking [3, 6]. This stiffening effect becomes particularly relevant when bending states are studied in slender structures [1]. In this case, the shear stiff- ness is parasitic. To counter such problems, there are alternatives to full integration. Doherty et al. [2] introduced a procedure with a reduced order of integration (too low for an exact integration). In doing so, they limited themselves to defined stiffness terms, which from today’s perspective is understood as selective integration. Reduced integration thus means the sub-integration of all stiffness terms. Selective integration is thus limited to the sub-integration of the terms associated to the transverse shear stiffness.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2019 125 M. Aßmus, Structural Mechanics of Anti-Sandwiches, SpringerBriefs in Continuum Mechanics, https://doi.org/10.1007/978-3-030-04354-4 126 Appendix D: Numerical Integration

Table D.1 Gauss points and weights of Gauss–Legendre-quadrature [5] ξ i ξ j αi,αj 1  2  ξ 1 =− 3 ξ 1 =− 3 α1 = 5 Full 1 5 2 5 9 ξ 2 = ξ 2 = α2 = 8 1 0 2 0 9   ξ 3 =+ 3 ξ 3 =+ 3 α3 = 5 1 5 2 5 9 Reduced ξ 1 =−√1 ξ 1 =−√1 α1 = 1 1 3 2 3 ξ 2 =+√1 ξ 2 =+√1 α2 = 1 1 3 2 3

In order to determine stiffness matrices, mass matrices and load vectors, it is necessary to integrate over the element surface. In the context of this work the Gauss- Legendre quadrature is used [7]. The analytic integration I of a function f (ξ) over the two dimensional element d is defined as follows.

1 1 e I = f (ξ) d = f (ξ) |J(ξ)| dξ1 dξ2 (D.1) e −1 −1

For this problem, the integral can be defined as a weighted summation of the function values [5].

NG 1 NG 2 ≈ αi αj (ξ i ,ξj ) I 1 2 f 1 2 (D.2) i=1 j=1

The integration is performed in the interval ξi ∈[−1, 1]. The function to be integrated ξ j αj is evaluated at the Gauss points i and multiplied by the weighting factors i .Here ξ i NG i is the number of Gauss points in the direction considered. The coordinates i of αj the Gauss points as well as their weighting factors i for the planar SERENDIPITY element with quadratic shape functions are summarized in Table D.1 for complete and reduced integration. In addition, the different integration modes are visualized in Fig. D.1. However, this distinction only takes place in the integration of the stiffness terms. Appendix D: Numerical Integration 127

Fig. D.1 Usage of different Gauss points in the variation of integration types

References

1. Babuška I, Suri M (1992) On Locking and robustness in the finite element method. SIAM J Numer Anal 29(5):1261–1293. http://dx.doi.org/10.1137/0729075 2. Doherty WP, Wilson EL, Taylor RL (1969) Stress analysis of axisymmetric solids utilizing higher-order quadrilateral finite elements. Report sesm 69-3, Struct Eng Lab 3. Hughes TJR (1987) The finite element method. Linear static and dynamic finite element analysis. Prentice-Hall, Inc., Englewood Cliffs 4. Koschnick F (2004) Geometrische Locking-Effekte bei Finiten Elementen und ein allge- meines Konzept zu ihrer Vermeidung. Dissertation, Technische Universität München. http:// nbn-resolving.de/urn:nbn:de:bvb:91-diss2004100700624 5. Schwarz HR, Köckler N (2004) Numerische Mathematik, 5th edn. B.G. Teubner, Stuttgart. http://dx.doi.org/10.1007/978-3-322-96814-2 6. Szabó B, Babuška I (1991) Finite element analysis. Wiley, New York 7. Zienkiewicz OC, Taylor RL (2005) The finite element method for solid and structural mechanics, 6th edn. Elsevier Butterworth-Heinemann, Oxford