On Linear Plate Models

Submitted by Ludwig Mitter

Submitted at Institute of Computational Mathematics

Supervisor A.Univ.-Prof. Dipl.-Ing. Dr. Walter Zulehner

November, 2017 On Linear Plate Models

Master Thesis to obtain the academic degree of Diplom-Ingenieur in the Master’s Program Industrial Mathematics

JOHANNES KEPLER UNIVERSITY LINZ Altenbergerstraße 69 4040 Linz, Österreich www.jku.at DVR 0093696

Abstract

In this master thesis we study the consistent semi-discretization of a plane linear elastic body. First we consider the theory of a three-dimensional linear elastic body by means of a primal formulation and a mixed formulation (Hellinger-Reissner- Principle). Assuming, that such a body is plane and its thickness is small compared to the planar dimensions, we separate the general three-dimensional problem into a so-called bending and a membrane problem. For both of these problems we perform a semi-discretization of the displacement by virtue of a series expansion. The obtained auxiliary problems are then solved on a two- dimensional computational domain. On this basis we analyse the consistency of the semi-discretized bending problem to the original three-dimensional problem for two diﬀerent boundary conditions.

Zusammenfassung

In dieser vorliegenden Masterarbeit befassen wir uns mit der konsistenten Semidiskretisierung eines ﬂachen linear elastischen Körpers. Zuerst erörtern wir hierzu die Theorie dreidimensionaler linear elastischer Körper anhand eines Verschiebungsansatzes und einer gemischten Formu- lierung (Hellinger-Reissner-Prinzip). Unter der Annahme, dass ein solcher Körper ﬂach und dessen Dicke im Vergleich zur planaren Ausdehnung klein ist, separieren wir das allgemeine dreidimensionale Belastungsproblem in ein sogenanntes Biegungs- und ein Membran-Problem. Für beide Probleme füh- ren wir eine Semidiskretisierung der Verschiebung über eine Reihenentwick- lung durch. Die dadurch gewonnenen Ersatzprobleme werden dann auf ei- nem zweidimensionalen Rechengebiet gelöst. Darauf aufbauend analysieren wir die Konsistenz des semidiskretisierten Biegungs-Problems mit dem ur- sprünglichen dreidimensionalen Belastungsproblem für zwei unterschiedliche Randbedinungen.

iii

Contents

1. Introduction 1 1.1. Motivation ...... 1 1.2. Linearized 3D Elasticity ...... 5 1.2.1. The Pure Displacement Formulation ...... 9 1.2.2. The Hellinger-Reissner-Principle ...... 16

2. Models of Linearized Plate Theory 33 2.1. The 3D Plate Model ...... 33 2.2. Splitting into Plate Bending and Stretching ...... 37 2.3. Hierarchic Plate Models ...... 40 2.4. The Clamped Plate ...... 46 2.4.1. The Plate Hypotheses ...... 47 2.4.2. Notations Regarding Plate Models ...... 48 2.4.3. Model Calculations ...... 51 2.4.4. Remarks on the (1,1,0)- and the (1,1,2)-Models . . . . 56 2.4.5. The Perturbed (1,1,2)-Kirchhoff-Love Plate Model . . . 59 2.4.6. Justification of the (1,1,2)-Kirchhoff-Love Plate Model 62 2.4.7. Justification of the (1,1,2)-Reissner-Mindlin Plate Model 67 2.5. The Simply Supported Plate ...... 68 2.5.1. Simple Support ...... 68 2.6. Lower Order Plate Bending Models ...... 74 2.6.1. The Reissner-Mindlin Plate Model ...... 77 2.6.2. An Analogon of the Hellinger-Reissner Principle for the Reissner-Mindlin Plate Model ...... 79 2.6.3. The Kirchhoff-Love Plate Model ...... 80 2.6.4. An Analogon of the Hellinger-Reissner Principle for the Kirchhoff-Love Plate Model ...... 81 2.6.5. Babuška’s Plate Paradox ...... 85 2.7. On Modified Lower Order Plate Bending Models ...... 87 2.7.1. Refined Consistency Results ...... 91

v Contents

A. Additional Information on Dimensional Reduction 95 A.1. Discs ...... 95 A.1.1. Plain Stress State ...... 95 A.1.2. Plain Strain State ...... 96

B. Notations 97

vi 1. Introduction

1.1. Motivation

In the area of computational elastodynamics and elastostatics, plate theory has become a well established and widely used tool for efficient as well as reliable numerical simulations of “thin” three-dimensional entities by means of dimensional reduction. Historically, much attention has been given to the derivation of various plate theories and models, which lead to different solutions depending on the model used [7]. In the field of linearized elasticity, the so-called Kirchhoff-Love plate theory as well as the more sophisticated Reissner-Mindlin plate theory are among the oldest and possibly most widely spread approaches. These models have been more amenable to an analytical solution, for example by Fourier series, than the fully three-dimensional equations. However, considering basic benchmark calculations on the fully three- dimensional model, the Reissner-Mindlin as well as the Kirchhoff-Love plate models one is confronted with an unphysical behaviour, for example, while dealing with so called simple support boundary conditions, limits of polygonal domains or boundary layers [3–5, 8]. This observation raises the question of the all-over applicability of Kirchhoff-Love plates and their related model assumptions, as well as their relation to the generic three-dimensional equations of elasticity. Hence we intend to review tools to identify applications for which the model assumptions of the different plate models are legitimate (or at least plausible). Since it is the intention of this thesis to examine linearized structural plate models in Chapter 2 based on a mixed formulation approach we – in addition to a brief introduction to linearized three-dimensional elasticity – prepend the theoretical basis for mixed problems to familiarize the reader with our notations (outlined in Chapter B) and formulations. It is henceforth assumed that the reader is acquainted with basic results on structural continuum

1 1. Introduction mechanics[15, 31], as well as the related theory behind conforming finite element methods (FEM) for elliptic problems[11, 12, 17, 23]. In the absence of the adjective “thin”, plate theory would be no more and no less than the analysis of a class of boundary value problems in three- dimensional elasticity. Interest in a theory of “plates” is governed by the expectation that the quality of thinness makes it possible to reach significant conclusions on the basis of two-dimensional rather than three-dimensional considerations [42, 47]. According to [16, Preface to Volume II] plate models combine two inher- ent advantages over a full three-dimensional model approach as the thickness becomes very small: First and foremost, plate models transfer the very general three-dimensional elasticity problem to a lower dimensional problem with a much simpler mathematical structure. Secondly, the resulting lower dimensional problem can more efficiently be handled by numerical computations as soon as model assumptions can be applied. Consequently it is a good idea to strive for replacing the full three-dimensional problem with a “fitting” lower-dimensional model. At this stage, [16] raises two questions regarding the general permissibility of treating “thin” elastic entities with a lower-dimensional model.

Given a “lower-dimensional” elastic body, together with speciﬁc loadings and boundary conditions, how to choose between the manifold lower-dimensional models that are available? This question is of paramount practical importance, for it makes no sense to devise accurate methods for approximating the solution of an inappropriate model. Consequently, before approximating the exact solution of a given lower-dimensional model, we should ﬁrst know whether it is “close enough” to the exact solution of the three-dimensional model it is intended to approximate. This observation leads to a second question: How to mathematically justify in a rational fashion a lower dimensional model from the three-dimensional model?

Both of these questions can extensively be answered in the realm of three very diﬀerent ideas on the linkage of the lower- to the higher dimensional model. For this purpose, one ﬁrst aspires towards inferring expedient lower- dimensional models, which can either be coherently embedded into physical

2 1.1. Motivation considerations, such as certain kinematic hypotheses, or be derived by formally expanding the original three-dimensional displacement with respect to the transverse direction and truncating the emergent formal series to acquire various models. An ansatz due to the ﬁrst and second approach will be referred to as being a constrained and hierarchic model, respectively. The most prominent models of Kirchhoﬀ-Love and Reissner-Mindlin can be derived by either approach. In a second step, one wants to justify the resultant models by analysing the asymptotic behaviour compared to the displacement of the three-dimensional model with respect to a feasible norm.

Constrained Method The first and oldest approach consists in using the constraint method, whose governing principle is an a priori assumption that, due to physical reasoning, the admissible displacement fields are restricted to a specific form, leading to a modified constitutive law going back to the basal work of Kirchhoff in 1850 [42]. However, lacking a self-contained mathematical foundation, the constraint method cannot serve as a viable answer to the question of asymptotic correctness.

Asymptotic Analysis The only direct derivation and (partial) justification is achieved by harnessing the previously mentioned formal asymptotic method, for which the three-dimensional solution (the displacement field and, in some cases, the stress field) is first scaled in an appropriate manner so as to be defined on a fixed domain, then expanded as a formal series expansion in terms of a small parameter > 0, which is the “dimensionless” half-thickness of a plate. See Figure 1.1. The convergence results emerging from this kind of analysis is given with respect to the elastic energy norm. Refer to [16, 18, 47].

Estimate A third approach consists in directly estimating the difference between the three-dimensional solution and the solution of a given, i.e. “known in advance”, lower-dimensional model. For linearized elastic plates, the first such estimate goes back to [36], who cleverly utilized the two-energies principle (originating from the celebrated Theorem of Prager and Synge, see Theorem 2.35) of the linearized theory. See Fig- ure 1.2. Apart from [2, 13], who further clarified several missing details in the context of arbitrarily given applied forces, this approach constitutes the only rigorously worked out proof of the relationship between

3 1. Introduction

the three-dimensional solution and the plate model, with respect to a “stronger” norm than the energy norm. Prima facie, an asymptotic analysis seems to be the natural choice [4, 5, 7, 16, 18], since the approaches one and three do not yield self-contained derivations of plate models, for they require some a-priori assumptions on the approximate behaviour of “thin” geometric elastic entities. While the constrained method a-priori neglects any kind of a more general class of deformations per assumption, the estimate approach can be easily extended to cope with any difficulties arising from actual applications and computations by virtue of hierarchic plate modelling. However, even if one is not content with this approach it nonetheless leads to sound statements on the error between the theory on three-dimensional elasticity and the ansatz by virtue of Kirchhoff-Love plate theory. Furthermore, since the latter is more restrictive and therefore disregards more general geometric deformations it automatically gives us statements on more complex and general plate models such as the Reissner-Mindlin plate model we are going to discuss in Chapter 2 [2–4, 8, 13]. Remark 1.1 (The evolution of plate modelling). The original idea behind plate modelling – namely to reduce the dimensional dependence of the displacement – was first performed due to kinematic hypotheses about the trans- formation of normal lines to the plate’s mid-surface (see Definition 2.4.1 and Definition 2.4.1), motivating the constrained method. Nowadays, the asymptotic approach, which consists of the asymptotic expansion of the displacement solution of the three-dimensional linearized elasticity model is fully known and no longer requires heuristic arguments to motivate two-dimensional models. Apart from the lowest order models we are going to discuss in Section 2.4, the asymptotic expansion also gives us arbitrarily many higher order models, for which the model error converges to zero, as the plate’s thickness vanishes. These two-dimensional models are called hierarchical plate models. However, aside from these very consistent hierarchical models, an even simpler class of plate models (Definition 2.18, Definition 2.19) is frequently used, which do not directly provide us with the desired convergence to the three-dimensional solution (Section 2.6). These models, however, can be easily modified to “mimique” the behaviour of the related hierarchical model, for which one can indeed show the desired convergence to the three-dimensional solution. This

4 1.2. Linearized 3D Elasticity procedure is referred to as shear correction (see Section 2.7), which historically has not emerged from the asymptotic approach, but from the estimate approach, which discusses the singularly perturbed character of hierarchical models, see Section 2.4.6. [42] Apart from linearized plate models, many linear and non-linear plate theories have been proposed to provide different degrees of sophistication when it comes to material laws or different hypotheses on the behaviour of the desired material [16, and the related chapters therein]. In the context of the lowest order Kirchhoff-Love plate model and a discussion on their different mixed formulations, a new approach has recently been proposed, to replace the fourth order elliptic (biharmonic) problem of Kirchhoff-Love by three sequentially solvable second order elliptic problems which avoids the relatively cumbersome introduction of more complex finite element function spaces [33, 40, 49]. On account of these revelations, this thesis is targeted at scrutinising the details regarding the more general Reissner-Mindlin plate theory and its susceptibility to the new theory on mixed methods. The outline of this thesis is as follows: The subsequent Section 1.2 introduces notations and presents the three-dimensional boundary value problem of linearized elasticity, for which we consider the pure displacement, as well as a mixed formulation (Hellinger-Reissner principle). In Section 2.1 we adapt our notations to the treatment of plates. In Section 2.2 we separate the problem of a three-dimensional linearized elastic plate into a plate bending and a plate stretching problem. Section 2.3 introduces a hierarchy of semi-discretizations of the displacement, which we call plate models. In the Sections 2.4, 2.5 we give consistency results of higher-order plate bending models to the original model of three-dimensional linearized elasticity for clamped and simply supported plates. In Section 2.6 we elaborate the use of lower-order plate bending models and their consistency. In the final Section 2.7 we present modified lower-order models of a clamped plate and refined consistency results.

1.2. Three-Dimensional Linearized Elasticity

This preparatory section is intended to brieﬂy introduce the reader to the notion of linearized elasticity [12, 15, 21, 27] upon which all our considerations

5 1. Introduction

Ωˆ =ω ˆ × [−/2, /2] ωˆ × {0} Ωˆ 1 =ω ˆ × [−1, 1]

Figure 1.1.: The “Asymptotic Analysis” approach proposed by [16] directly derives the Kirchhoﬀ-Love plate model by discussing convergence results as the plate thickness tends to zero.

Ωˆ =ω ˆ × [−/2, /2] ωˆ × {0}

(1,1,2) (1,1,2) E , Σ E , Σ K K

(1,1,2) 2 2 2 kE(u ) − E(u )k + kΣ − Σ k −1 = O()kΣ k −1 K C eq C C

Figure 1.2.: The “Estimate” approach starts off with the separated treatment of the three-dimensional and the Kirchhoff-Love plate model. On the basis of the celebrated two-energies-principle by Prager and Synge (Theorem 2.35) one gains an estimate for the error in the resulting stresses and strains while comparing the three- dimensional to the Kirchhoff-Love plate model solution. If ∈ (0, 1] specifies the thickness of the plate, then the error is of order O(1/2) .

6 1.2. Linearized 3D Elasticity of plate theory will be based on in Chapter 2. 3 We assume that an origin 0 ∈ R and an orthonormal basis e1, e2, e3 ∈ R3 have been chosen in the three-dimensional Euclidean space, which will therefore be identified with the space R3. Let λ, µ be the Lamé constants, ν the Poisson ratio and E Young’s modulus of a St. Venant-Kirchhoff material of an elastic body “at rest” filling a reference configuration Ωˆ ⊂ R3, which is defined as a domain1 with a Lipschitz continuous boundary parametrization. The second Piola-Kirchhoff stress tensor is given by the response function (constitutive equation)

E ν Σ(u) := E(u) := E + (tr E)I = 2µE + λ(tr E)I C 1 + ν 1 − 2ν 3 3 = CDu with C : S → S .

Here Mn constitutes the set of all real square matrices of order n, Sn := > {B ∈ Mn : B = B } and the linearized strain tensor E(u) is given by a linear mapping called the symmetric gradient D (kinematic equation)

1 E(u) := u := (∇u + ∇u>), D 2 where the unknown function u : Ωˆ → R3 designates the linearized strain tensor and the displacement of our body Ωˆ to which one relates the deformation ϕ : Ωˆ → R3 by ϕ(xˆ) := (1 + u)(xˆ) := x, where 1(.) constitutes the identity mapping. One calls xˆ and x the Lagrangian respectively Eule- rian coordinates. As Γˆ is Lipschitz continuous, a unit outer normal vector n ∈ R3, |n| ≡ 1 can be deﬁned a.e. on Γˆ [15, Sect. 1.6.]. Remark 1.2 (Translation of elasticity constants). The diﬀerent elasticity constants can be easily converted by setting

λ µ(3λ + 2µ) ν = E = ν ∈ [0, 1/2),E ∈ + 2(λ + µ) λ + µ R Eν E λ = µ = λ, µ ∈ +. (1 + ν)(1 − 2ν) 2(1 + ν) R 1open, connected and bounded subset of Rn, n ∈ N whose interior is locally inside of its boundary

7 1. Introduction

Remark 1.3 (Properties of the elasticity tensor and notations). We note that C is a one-to-one mapping, E ν E = E + (tr E)I = Σ = (σ )3 , (1.1) C 1 + ν 1 − 2ν ij i,j=1 whose inverse (compliance tensor) is given by

1 + ν ν −1Σ = Σ − (tr Σ)I = E = ( )3 . (1.2) C E E ij i,j=1

One calls σii, i = 1, 2, 3 the normal stress components, whereas σij, i, j = 1, 2, 3, i 6= j are most commonly addressed as shear stress components. Proof of (1.2). The application of the trace operator to (1.1) leads to E tr Σ = tr E. 1 − 2ν Therefore one obtains from (1.1)

1 + ν ν E = Σ − (tr Σ)I. E E

Remark 1.4 (Further properties of the elasticity tensor). One can write (1.1), (1.2) componentwise (matrix representation of C, C−1)       Σ11 λ + 2µ λ λ 0 0 0 E11 Σ22  λ λ + 2µ λ 0 0 0  E22       Σ33  λ λ λ + 2µ 0 0 0  E33   =     (1.3) Σ12  0 0 0 2µ 0 0  E12       Σ13  0 0 0 0 2µ 0  E13 Σ23 0 0 0 0 0 2µ E23       E11 1 −ν −ν 0 0 0 Σ11 E22 −ν 1 −ν 0 0 0   Σ22        E33 1 −ν −ν 1 0 0 0   Σ33    =     (1.4) E12 E  0 0 0 1 + ν 0 0   Σ12        E13  0 0 0 0 1 + ν 0   Σ13  E23 0 0 0 0 0 1 + ν Σ23.

8 1.2. Linearized 3D Elasticity

Through (1.1) one can immediately infer, that for ν ∈ [0, 1/2) the elasticity tensor C has two diﬀerent eigenvalues λmin(C) = 2µ, λmax(C) = 2µ + 3λ, hence C is positive deﬁnite. Furthermore, from the symmetry of C one concludes that the elasticity tensor C is self-adjoint in (Sn, :), n ∈ N.

1.2.1. The Pure Displacement Formulation In this section we are going to derive the weak form of the linearized displacement-traction problem, whose classical formulation [15] is given as follows. Problem 1.5 (Classical formulation of the linearized displacement-traction problem). Find a solution u : cl{Ωˆ} → R3 (cl{Ωˆ} constitutes the closure of Ωˆ) of the linear boundary value problem

− div (CDu) = f in Ωˆ (1.5) ˆ u ≡ 0 on Γ0 (1.6) ˆ (CDu) n = g on Γ1. (1.7) To cast the boundary value problem (1.5)-(1.7) into variational form, we multiply (1.5) componentwise by a test function v : cl{Ωˆ} → R3 and integrate over the computational domain Ωˆ and adding, one obtains Z Z − (div CDu) · vdxˆ = f · vdxˆ. (1.8) Ωˆ Ωˆ ˆ Green’s formula can be stated for a symmetric tensor ﬁeld S ∈ H(div, Ω)S and vector ﬁeld v ∈ H1(Ω)ˆ , Z Z Z div S · vdxˆ = − S : ∇vdxˆ + Sn · vdsxˆ , Ωˆ Ωˆ Γˆ

n n where the matrix inner product for arbitrary A = (aij)i,j=1, B = (bij)i,j=1 ∈ n Pn M is given by A : B := i,j=1 aijbij, the vector product for arbitrary n n n Pn a = (ai)i=1, b = (bi)i=1R is given by a · b := i=1 aibi. Furthermore we 1 1 3 1 2 introduce the spaces H (Ω) := (H (Ω)) ,H (Ω) := {v ∈ L (Ω) : ∂iv ∈ 2 2 R 1/2 L (Ω), 1 ≤ i ≤ 3},L (Ω) := {v :Ω → R such that kvk0,Ω := Ω vdx < 2 3 2 ∞}, L (Ω)S := {Σ = (σij)i,j=1 : σij = σji ∈ L (Ω), 1 ≤ i, j ≤ 3} and ˆ 2 ˆ 2 ˆ H(div, Ω)S := {T ∈ L (Ω)S : div T ∈ L (Ω)}. In this regard we have

9 1. Introduction

1 ˆ 2 ˆ D : H (Ω) → L (Ω)S. By the virtue of the symmetry of the matrix S, this formula can be restated as Z Z Z 1 > div S · vdxˆ = − S : (∇v + ∇v )dxˆ + Sn · vdsxˆ Ωˆ Ωˆ 2 Γˆ Z Z = − S :(Dv)dxˆ + Sn · vdsxˆ . Ωˆ Γˆ We now apply Green’s formula to the left side of (1.8), Z Z Z − (div CDu) · vdxˆ = (CDu):(Dv)dxˆ − (CDu)n · vdsxˆ Ωˆ Ωˆ ∂Ωˆ Z = f · vdxˆ for ∀u, v ∈ H1(Ω)ˆ . Ωˆ Assume g ∈ L2(Γ)ˆ , f ∈ L2(Ω)ˆ . Let Z Z Z hF, vi := f · vdxˆ + g · vdsxˆ a(u, v) := (Du):(CDv)dxˆ Ωˆ Γˆ1 Ωˆ then we get Z 1 ˆ a(u, v) = (CDu)n · vdsxˆ + hF, vi for ∀v ∈ H (Ω). Γˆ0 ˆ This naturally leads to assuming that v ≡ 0 on Γ0 [15, 50], therefore we arrive at the following problem. Problem 1.6. Find a solution u ∈ V of the equation

a(u, v) = hF, vi for ∀v ∈ V (1.9) where

3 Z X Z a(u, v) := (Du):(CDv)dxˆ = (Du)ij(CDv)ijdxˆ ˆ ˆ Ω i,j=1 Ω 3 3 Z Z X Z X Z hF, vi := f · vdxˆ + g · vdsxˆ = fividxˆ + gividsxˆ ˆ ˆ ˆ ˆ Ω Γ1 i=1 Ω i=1 Γ1 1 ˆ ˆ V :={v ∈ H (Ω) : v = 0 a.e. on Γ0}.

10 1.2. Linearized 3D Elasticity

In order to obtain well-posedness we strive for existence and uniqueness of a solution for our variational formulation (1.9). The following existence result applies to a large class of abstract problems of the form (1.9). Theorem 1.7 (Characterization lemma on existence, uniqueness and equivalence to minimizer). Let V be a real Hilbert space with norm k.k, let F : ∗ ∗ V → R be a continuous linear form (i.e. F ∈ V , where V denotes the dual space associated to any Hilbert space V ) and let a : V × V → R be a symmetric and continuous bilinear form that is V -elliptic, in the sense that there exist constants µ1, µ2 > 0 such that

a(u, v) ≤ µ2kukkvk for ∀u, v ∈ V (1.10) 2 a(v, v) ≥ µ1kvk for ∀v ∈ V . (1.11)

Then the problem

Find u ∈ V such that a(u, v) = hF, vi for ∀v ∈ V (1.12) has a unique solution, which also is the unique solution of the minimization problem

Find u ∈ V such that J(u) = inf J(v), (1.13) v∈V where 1 J : v 7→ a(v, v) − hF, vi. 2 The solution u depends continuously on the date F , i.e. 1 kukV ≤ kF kV ∗ . (1.14) µ1 Proof. By the V -ellipticity (1.11) and the continuity of the bilinear form (1.10) one gets

2 2 µ1kvk ≤ a(v, v) ≤ µ2kvk for ∀v ∈ V hence the symmetric bilinear form a(., .) can be seen as an inner product over 1/2 the space V , which therefore induces a norm k.ka : v 7→ (a(v, v)) that

11 1. Introduction is equivalent to the given norm k.k of our real Hilbert space V . Thus, the space V is also a Hilbert space if it is equipped with this inner product. By the Riesz representation theorem [43, 48, 50] there exists a unique element w ∈ V such that hF, vi = a(w, v) for ∀v ∈ V . We have therefore found a unique solution to our problem (1.12). On the other hand we have 1 J(u + v) = a(u + v, u + v) − hF, u + vi 2 1 = (a(u, u + v) + a(v, u + v)) − hF, u + vi 2 1 = (a(u, u) + a(u, v) + a(v, u) + a(v, v)) − hF, u − vi 2 1 1 = a(u, u) − hF, ui + a(u, v) − hF, vi + a(v, v) 2 2 1 = J(u) + a(u, v) − hF, vi + a(v, v) 2 Since u satisﬁes (1.12) one gets 1 µ J(u + v) − J(u) = a(v, v) ≥ 1 kvk2 for ∀v ∈ V , 2 2 which shows, that in this case u indeed is a minimizer of the functional J. Conversely, let u be a minimizer of J and let v ∈ V arbitrary ﬁxed. We then have ξ 0 ≤ J(u + ξv) − J(u) = ξ a(u, v) − hF, vi + a(v, v) for ∀ξ ∈ . 2 R

Assume a(u, v) 6= hF, vi. Since a(v, v) > 0 one can find ξ ∈ R with sufficiently small norm |ξ| such that J(u+ξv)−J(u) < 0, which contradicts the assertion above. Therefore, we draw the conclusion, that a(u, v) = hF, vi. [12, 17, 19, 50] Remark 1.8. The functional J is convex [12, Remark 2.6.] and it satisfies a coerciveness inequality [15, Theorem 4.7-7] 1 µ J(v) = a(v, v) − hF, vi ≥ 1 kvk2 − kF kkvk for ∀v ∈ V . 2 2

12 1.2. Linearized 3D Elasticity

Remark 1.9. Utilizing the celebrated Lax-Milgram lemma [12, 17, 50] a solution u ∈ V of (1.12) still exists if we drop the symmetry assumption on a(., .), however this class of problems is no longer associated to a minimization problem, which will be a central tool while discussing diﬀerent linearized plate models. After this general discussion of an abstract problem (1.12) we have to ver- ify that V is a suitable real Hilbert space enabling us to utilize Theorem 1.7. Concerning this matter we observe, that from repeated applications of the Cauchy-Schwarz inequality [25] the symmetric bilinear form a(., .) from Prob- lem 1.6 is continuous with respect to the norm k.k1,Ωˆ . We then observe, that since V is a closed subspace of the Hilbert space H1(Ω)ˆ [43], V indeed is a Hilbert space itself and since for physically sound materials λ, µ > 0 we have Z a(v, v) = (λ tr E(v) tr E(v) + 2µE(v): E(v)) dxˆ Ωˆ Z ≥ 2µ E(u): E(v)dxˆ, Ωˆ hence the V -ellipticity of a(., .) is given, as soon as one can show that on V the semi-norm 1 ˆ v ∈ H (Ω) 7→ kE(v)k0,Ωˆ is a norm, equivalent to k.k1,Ωˆ . This result can be derived from the following fundamental inequality [11, 12, 20].

Theorem 1.10 (Korn’s inequality). Let Ωˆ ⊂ R3 be a domain with a Lipschitz 1 ˆ 2 ˆ continuous boundary. For each v ∈ H (Ω) let Dv ∈ L (Ω)S. (a) Korn’s inequality without boundary conditions: Then there exists a constant c(Ω)ˆ > 0 such that 1/2 ˆ 2 2 1 ˆ kvk1,Ωˆ ≤ c(Ω) kvk0,Ωˆ + kE(v)k0,Ωˆ for ∀v ∈ H (Ω) (1.15)

and thus, on the space H1(Ω)ˆ , the mapping

1/2 2 2 v 7→ kvk0,Ωˆ + kE(v)k0,Ωˆ

is a norm, equivalent to the norm k.k1,Ωˆ .

13 1. Introduction

ˆ ˆ (b) Korn’s inequality with boundary conditions: Let Γ0 ⊂ Γ with ˆ ˆ ˆ area Γ0 > 0 be measurable. Then there exists a constant c(Ω, Γ0) > 0 such that

ˆ ˆ 1 ˆ ˆ kvk1,Ωˆ ≤ c(Ω, Γ0)kE(v)k0,Ωˆ for ∀v ∈ H (Ω) vanishing on Γ0. (1.16)

To prove Korn’s inequality [12, 16, 38] one has to gain a deeper insight into the connections between L2(Ω) and H−1(Ω), where Ω ⊂ Rn denotes some domain. It is clear that

2 −1 −1 v ∈ L (Ω) =⇒ v ∈ H (Ω) and ∂iv ∈ H (Ω) for ∀1 ≤ i ≤ n, (1.17)

∞ since one has for arbitrary 1 ≤ i ≤ n, ϕ ∈ Cc (Ω) Z Z

Theorem 1.11. Let Ω ⊂ Rn, n ∈ N be a domain, then the following holds

−1 −1 2 v ∈ H (Ω) and ∂iv ∈ H (Ω) for ∀1 ≤ i ≤ n =⇒ v ∈ L (Ω) With Korn’s inequality (1.16) established, we can now prove the desired V -ellipticity of our bilinear form a(., .). ˆ 3 ˆ ˆ ˆ Theorem 1.12. Let Ω ⊂ R be a domain, Γ0 be a subset of Γ = ∂Ω. Then 1 ˆ ˆ 1 ˆ V = {v ∈ H (Ω) : v = 0 a.e. on Γ0} is a closed subspace of H (Ω). If ˆ area Γ0 > 0 there exists a constant c > 0 such that

−1 c kvk1,Ωˆ ≤ kE(v)k0,Ωˆ ≤ ckvk1,Ωˆ for ∀v ∈ V . (1.18)

Therefore the semi-norm v 7→ kE(v)k0,Ωˆ indeed is a norm on V , which is equivalent to k.k1,Ωˆ . Proof. [15, Theorem 6.3-4. p. 292] On basis of Theorem 1.12 one usually motivates notations for the energy related to stresses and strains.

14 1.2. Linearized 3D Elasticity

Remark 1.13 (Energies of stresses and strains). For arbitrary v ∈ H1(Ω)ˆ one calls k vk = ( v, v)1/2 kvk := a(v, v)1/2 = ( v, v)1/2 D 0,Ωˆ D D 0,Ωˆ a CD D 0,Ωˆ the strain and stress energy norms, respectively. Assembling the previous results, we next establish the existence of a solution u ∈ H1(Ω)ˆ of the weak form of the linearized displacement-traction problem. Henceforth, this kind of solution will be referred to as a weak solution.

Theorem 1.14 (Existence of a weak solution). Let Ωˆ ⊂ R3 be a domain, let ˆ ˆ ˆ ˆ Γ0 ⊂ Γ = ∂Ω be a measurable set with area Γ0 > 0, let λ, µ > 0 and let the 2 ˆ 2 ˆ ˆ ˆ ˆ functions f ∈ L (Ω), g ∈ L (Γ1), where Γ1 = Γ \ Γ0, be given. Then there is a unique function u in the space 1 ˆ ˆ V := {v ∈ H (Ω) : v = 0 a.e. on Γ0} that satisﬁes a(u, v) = hF, vi for ∀v ∈ V , where Z a(u, v) := (Du):(CDv)dxˆ Ωˆ Z Z hF, vi := f · vdxˆ + g · vdsxˆ Ωˆ Γˆ1 In addition 1 J(u) = inf J(v) where J(v) = a(v, v) − hF, vi. v∈V 2 We call J(.) the potential energy. Proof. The continuity of the trace operator [24, Theorem 5.1] and [43, Theo- rem 7.34] imply that the linear form hF,.i is continuous on the space H1(Ω)ˆ 2 ˆ 2 ˆ ˆ if f ∈ L (Ω), g ∈ L (Γ1). Since λ, µ > 0 and meas Γ0 > 0, the symmetric and continuous bilinear form a(., .) is V -elliptic by Theorem 1.12. It is also bounded [50]. Hence the conclusion follows by Theorem 1.7. [15, Theorem 6.3-5 p.295]

15 1. Introduction

Remark 1.15. This theorem also holds for the weaker assumptions f ∈ 6/5 ˆ 4/3 ˆ L (Ω), g ∈ L (Γ1) on the data [15]. ˆ ˆ We ﬁnally show that, if Γ = Γ0, the weak solution found in Theorem 1.14 possesses additional regularity if the data (including the boundary of Ωˆ and f) also possesses additional regularity. Since in this case the boundary condition is u ≡ 0 on Γˆ, we are solving a linearized pure displacement problem, 1 ˆ and V = H0(Ω). Theorem 1.16 (Regularity of the weak solution for the linearized pure displacement problem). Let Ωˆ ⊂ R3 be a domain with a boundary Γˆ of class 2 p ˆ ˆ ˆ C , let f ∈ L (Ω), p ≥ 6/5 and let Γ = Γ0. Then the weak solution u ∈ 1 ˆ 2,p ˆ H0(Ω) of the linearized pure displacement problem is in the space W (Ω) and constitutes a strong solution, i.e.

− div (CDu) = f.

m,p Let m ∈ N. If the boundary Γˆ is of class Cm+2 and if f ∈ W (Ω)ˆ , the 1 ˆ m+2,p ˆ weak solution u ∈ H0(Ω) is in the space W (Ω). Proof. [15, Theorem 6.3-6 p. 296]

1.2.2. The Mixed Formulation According to the Hellinger-Reissner-Principle In most engineering applications one strives for highly reliable and detailed numerics on modelling stresses aside from the related displacements we have discussed in the previous section. This motivates the following mixed approach, which naturally incorporates an autonomous representation of resulting stresses instead of a-posteriori regaining them from a numerical approximation of a displacement ﬁeld [11, 12]. In Problem 1.6 we eliminated Σ, E by consecutively applying the constitutive and the kinematic equation to express these quantities in terms of the displacement. This has lead us to the so called principle of virtual work (1.9). For the following method, aside from the displacement we also keep the stress in our problem formulation, i.e. ﬁnd Σ : cl{Ωˆ} → M3, u : cl{Ωˆ} → R3, such that

−1 C Σ − E(u) =0 in Ωˆ (1.19)

16 1.2. Linearized 3D Elasticity

− div Σ =f in Ωˆ (1.20) ˆ u =0 on Γ0 (1.21) ˆ Σn =g on Γ1. (1.22)

Let T : cl{Ωˆ} → S3 be a test function. By multiplying (1.19) componentwise by T, integrating over Ωˆ and adding, one obtains Z Z −1 C Σ : Tdxˆ − T :(Du)dxˆ = 0. (1.23) Ωˆ Ωˆ

Let w : cl{Ωˆ} → R3 be a test function. By multiplying (1.20) componentwise by w, integrating over Ωˆ and adding, one obtains Z Z (div Σ) · wdxˆ = f · wdxˆ. (1.24) Ωˆ Ωˆ Following a single application of integration by parts on the left expression of (1.24) one arrives at Z Z Z (div Σ) · wdxˆ = − Σ :(Dw)dxˆ + Σn · wdsxˆ Ωˆ Ωˆ Γˆ (1.22) Z Z Z = − Σ :(Dw)dxˆ + g · wdsxˆ + Σn · wdsxˆ . Ωˆ Γˆ1 Γˆ0 ˆ This naturally leads to assuming w ≡ 0 a.e. on Γ0, therefore we end up with the following mixed problem.

Problem 1.17 (First mixed formulation of the Hellinger-Reissner-principle for linearized elasticity). Find a solution u ∈ W I, Σ ∈ V I of the saddle point problem

aI(Σ, T) + bI(T, u) = 0 for ∀T ∈ V I (1.25) bI(Σ, w) = hGI, wi for ∀w ∈ W I, (1.26) where we have

I −1 I a (Σ, T) :=(C Σ, T)0,Ωˆ for ∀T ∈ V I I b (T, v) := − (T, Dv)0,Ωˆ for ∀v ∈ W

17 1. Introduction

Z I I hG , wi := − (f, w)0,Ωˆ − g · wdsxˆ for ∀w ∈ W Γˆ1 and V I denotes a space of smooth enough matrix valued functions T : cl{Ωˆ} → I S3 and W denotes a space of smooth enough vector-valued functions w : ˆ 3 ˆ cl{Ω} → R , that vanish on Γ0. Remark 1.18 (Equivalence of (1.9) and (1.25), (1.26)). Our pure displacement variational formulation (1.9) reads a Z (Du, CDv)0,Ωˆ = (f, v)0,Ωˆ + g · v ∈ dsxˆ (1.27) Γˆ1

Let u be a solution of (1.27). Since u ∈ H1(Ω)ˆ we have

2 ˆ Σ := CDu ∈ L (Ω)S. (1.28)

Since by Remark 1.4 C is self-adjoint one already gets (1.26). On the other hand, one gains (1.25) from the weak formulation of (1.28). This framework is an application of the theory behind mixed variational problems [11, 12]. Hence, let (V , (., .)V ), (W , (., .)W ) be two Hilbert spaces and

a : V × V → R b : V × W → R c : W × W → R be continuous bilinear forms and let F ∈ V ∗,G ∈ W ∗. We then discuss the following mixed variational problem. Problem 1.19 (Prototype of a mixed variational problem). Find (u, w) ∈ V × W such that

a(u, v) + b(v, w) = hF, vi for ∀v ∈ V (1.29) b(u, χ) − c(w, χ) = hG, χi for ∀χ ∈ W (1.30)

To give an alternative formulation to (1.29), (1.30) we now introduce the associated operators: For any u, v ∈ V , w, χ ∈ W we set

A : V → V ∗ B : V → W ∗ B∗ : W → V ∗ C : W → W ∗

18 1.2. Linearized 3D Elasticity

hAu, vi := a(u, v) hBv, χi := b(v, χ) hB∗χ, vi := b(v, χ) hCw, χi = c(w, χ). By means of these operators we obtain the following representation of the mixed variational (1.29), (1.30) problem as an operator equation on the product space V × W . u AB∗ u Au + B∗w F A := = = =: F. w B −C w Bu − Cw G From this one naturally concludes, that Problem 1.19 can also be restated as a non-mixed variational problem on V × W . Problem 1.20 (Prototype of a mixed variational problem as standard variational problem). Find (u, w) ∈ V × W such that a(u, w; v, χ) =hF, v, χi for ∀(v, χ) ∈ V × W , where we have

a(u, w; v, χ) :=a(u, v) + b(v, w) + b(u, χ) − c(w, χ) hF, v, χi :=hF, vi + hG, χi. Note 1.21. Observe that a(., .) can never be V ×W -elliptic for non-negative bilinear forms c since a(0, χ; 0, χ) = −c(χ, χ) ≤ 0 for ∀χ ∈ W , hence neither the lemma of Lax-Milgram, nor Theorem 1.7 can be applied. In the light of Problem 1.20 one can also consider the following class of problems, which (under the conditions of Theorem 1.23) are equivalent to a mixed variational problem. Problem 1.22 (Prototype of a saddle point problem). Find (u, w) ∈ V ×W such that J(u, χ) ≤J(u, w) ≤ J(v, w) for ∀(v, χ) ∈ V × W , (1.31) where we have 1 J(v, χ) := a(u, w; v, χ) − hF, v, χi. 2 One calls (u, v) solving (1.31) saddle point of J(., .).

19 1. Introduction

Theorem 1.23 (Prototype of a mixed variational problem as saddle point problem). Let a, c be symmetric and non-negative bilinear forms, i.e. a(v, v) ≥ 0 for ∀v ∈ V and c(χ, χ) ≥ 0 for ∀χ ∈ W . Then (u, w) ∈ V × W constitutes a solution of (1.29), (1.30), if and only if (u, w) ∈ V × W solves (1.31). Proof. Assume (u, w) ∈ V × W is a saddle point of J(., .). This can be written as

J(u, w) ≤J(u + ξv, w) for ∀v ∈ V , ξ > 0 (1.32) J(u, w) ≥J(u, w + ξχ) for ∀χ ∈ W , ξ > 0 (1.33)

Similar as in the proof of the standard variational problem one has 1 J(u + ξv, w) = J(u, w) + ξ (a(u, v) + b(v, w) − hF, vi) + ξ2a(v, v) 2 Hence (1.32) can be equivalently written as 1 ξ (a(u, v) + b(v, w) − hF, vi) + ξ2a(v, v) ≥ 0 for ∀v ∈ V , ξ > 0, 2 and since a is non-negative, this is equivalent to

a(u, v) + b(v, w) − hF, vi =0 for ∀v ∈ V .

Analogously (1.33) is equivalent to

a(u, χ) + c(w, χ) − hG, χi for ∀χ ∈ W

The framework of Theorem 1.23 also motivates an interpretation of our mixed variational problem as an optimality system. Let us therefore assume that c(., .) is W -elliptic, therefore on W the induced norm k.kc : w 7→ 1/2 1/2 c(w, w) is equivalent to k.kW : w 7→ (w, w)W . Problem 1.24 (Prototype of an unconstrained optimization problem). Find u ∈ V such that

1 2 Jc(u) = inf Jc(v) where Jc(v) = J(v) + kBv − Gkc . (1.34) v∈V 2

20 1.2. Linearized 3D Elasticity

Theorem 1.25. Assuming that a, c are symmetric bilinear forms, a is non- negative and c is W -elliptic, then (u, w) ∈ V × W solves (1.29), (1.30) if and only if u ∈ V solves (1.34) and w ∈ W is the unique solution to

c(w, χ) = b(u, χ) − hG, χi for ∀χ ∈ W .

Proof. [11] The unconstrained optimization problem (1.34) might itself be interpreted as a penalty method for solving the following constrained optimization problem.

Problem 1.26 (Prototype of a constrained optimization problem). Find u ∈ V such that

J(u) = inf J(v). (1.35) v∈V b(v,χ)=hG,χi for ∀χ∈W

Recurring to Problem 1.17 we can further narrow the theory above by setting c(., .) ≡ 0 for the rest of this chapter. Apparently, the mixed variational problem (1.29), (1.30) then equals the optimality system of (1.35) [11, 26].

Problem 1.27 (Optimality system). Find (u, w) ∈ V × W such that

a(u, v) + b(v, w) = hF, vi for ∀v ∈ V (1.36) b(u, χ) = hG, χi for ∀χ ∈ W . (1.37)

As before, this mixed variational problem (1.36), (1.37) may also be given in terms of a non-mixed variational form on the product space V × W .

Problem 1.28 (Saddle point problem in non-mixed formulation). Find (u, w) ∈ V × W such that

a(u, w; v, χ) =hF, v, χi for ∀(v, χ) ∈ V × W , where we have

a(u, w; v, χ) :=a(u, v) + b(v, w) + b(u, χ) hF, v, χi :=hF, vi + hG, χi.

21 1. Introduction

Observe, that the Lagrangian functional associated to (1.35) is given by

J(v, χ) =J(v) + b(v, χ) − hG, χi 1 = a(v, v) − hF, vi + b(v, χ) − hG, χi 2 1 = a(v, v) + b(v, χ) − (hF, vi + hG, χi) . 2 | {z } | 1 {z } =hF,v,χi = 2 a(v,χ;v,χ)

Remark 1.29. In this context one often calls χ the Lagrange parameter or Lagrange multiplier associated to the constraint.

Theorem 1.30 (Solutions of mixed variational problems solve constrained optimization problems). Assuming that (u, w) ∈ V ×W is a saddle point of J(.), or equivalently, that (u, w) solves the mixed variational problem (1.36), (1.37). Then u is a solution of the constrained optimization problem (1.35).

Proof. [11]

Now that we have established a connection between (1.36), (1.37) and (1.35), one should ask, whether the reverse implication is also true, i.e. if our operator A constitutes an isomorphism. This equivalence, however, is not obvious [12, Chapter 4], [11].

Theorem 1.31 (Brezzi’s Splitting Theorem). Let V , W be real Hilbert ∗ ∗ spaces with inner products (., .)V , (., .)W , let F ∈ V ,G ∈ W and let a : V × V → R, b : V × W → R bilinear forms satisfying the following conditions:

(a) a is bounded on V × V , i.e. there exists a constant α2 > 0 such that

|a(u, v)| ≤ α2kukV kvkV for ∀(u, v) ∈ V × V .

(b) b is bounded V × W , i.e. there exists a constant β2 > 0 such that

|b(v, χ)| ≤ β2kvkV kχkW for ∀(v, χ) ∈ V × W .

22 1.2. Linearized 3D Elasticity

(c) a satisﬁes a double inf-sup condition on ker B = {v ∈ V : b(v, χ) = 0 for ∀χ ∈ W }, i.e. there exists a constant α1 > 0 such that a(u, v) inf sup ≥α1 u∈ker B\{0} v∈ker B\{0} kukV kvkV a(u, v) inf sup ≥α1. v∈ker B\{0} u∈ker B\{0} kukV kvkV

(d) b satisﬁes an inf-sup-condition, i.e. there exists a constant β1 > 0 such that b(v, χ) inf sup ≥ β1, (1.38) χ∈W \{0} v∈V \{0} kvkV kχkW

Then the mixed variational problem: Find (u, w) ∈ V × W such that

a(u, v) + b(v, w) = hF, vi for ∀v ∈ V b(u, χ) = hG, χi for ∀χ ∈ W has a unique solution satisfying 1 1 α2 kukV ≤ kF kV ∗ + 1 + kGkW ∗ α1 β1 α1 1 α2 α2 α2 ∗ ∗ kwkW ≤ 1 + kF kV + 2 1 + kGkW . β1 α1 β1 α1 Moreover the operator A : V × W → V ∗ × W ∗ given by

hA(u, w), (v, χ)i = a(u, v)+b(v, w)+b(u, χ) for ∀u, v ∈ V , w, χ ∈ W is well-deﬁned and constitutes an isomorphism, if and only if the conditions (a)-(d) are satisﬁed. Proof. [12, Satz 4.3.], [11]. Remark 1.32. Of course, (c) directly follows if a is elliptic on ker B = {v ∈ V : b(v, χ) = 0 for ∀χ ∈ W }, i.e. there exists α1 > 0 such that 2 a(v, v) ≥ α1kvkV for any v ∈ ker B.

23 1. Introduction

We are now able to discuss the ﬁrst mixed formulation of the Hellinger- Reissner-principle of Problem 1.17, which we complement by applying Green’s Formula on (1.23) instead of (1.24), Z Z Z − T :(Du)dxˆ = div T · udxˆ − Tn · udsxˆ (1.39) Ωˆ Ωˆ Γˆ (1.21) Z Z = (div Σ) · wdxˆ − Tn · udsxˆ . (1.40) Ωˆ Γˆ1 ˆ This naturally leads to assuming Tn ≡ 0 a.e. on Γ1, therefore we end up with the following mixed problem. Problem 1.33 (Second mixed formulation of the Hellinger-Reissner-princi- II II ple for linearized elasticity). Find a solution u ∈ W , Σ ∈ V g of the saddle point problem

II II II a (Σ, T) + b (T, u) = 0 for ∀T ∈ V 0 (1.41) bII(Σ, w) = hGII, wi for ∀w ∈ W II, (1.42) where we have

II −1 II a (Σ, T) :=(C Σ, T)0,Ωˆ for ∀T ∈ V II II b (T, v) :=(div T, v)0,Ωˆ for ∀v ∈ W II II hG , wi := − (f, w)0,Ωˆ for ∀w ∈ W

II II and V 0 , V g denote the linear space, the linear manifold of smooth enough ˆ 3 ˆ ˆ matrix valued functions T : cl{Ω} → S with Tn = 0 on Γ1, Tn = g on Γ1, respectively, and W II denotes a space of smooth enough vector-valued functions w : cl{Ωˆ} → R3. Observe, that the statements of Remark 1.18 also hold for (1.41), (1.42). We introduce the following spaces,

I 2 ˆ V :=L (Ω)S I 1 ˆ ˆ W :={w ∈ H (Ω) : w = 0 a.e. on Γ0} II ˆ V :=H(div, Ω)S II ˆ ˆ V 0 :={T ∈ H(div, Ω)S : Tn = 0 a.e. on Γ1}

24 1.2. Linearized 3D Elasticity

II ˆ ˆ V g :={T ∈ H(div, Ω)S : Tn = g a.e. on Γ1} W II :=L2(Ω)ˆ .

Remark 1.34. With a slight abuse of notation we use the boundary inte- R gral Γˆ Tn · udsxˆ instead of the duality product (Tn, u)Γˆ in (1.39), (1.40). ˆ II The imposition of Tn = 0 a.e. on Γ1 for T ∈ V has to be understood as 1 ˆ ˆ hTn, vi = 0 for ∀v ∈ H (Ω), v = 0 a.e. on Γ0. These traces are well de- ˆ 1/2 ˆ ﬁned on H(div, Ω)S as bounded linear functional on H (Γ). For a thorough discussion see [11].

Problem 1.17 already has the form of (1.36), (1.37), however, Problem 1.33 II has to be homogenized to be applicable to Theorem 1.31. Let Σg ∈ V g . II With the ansatz Σ = Σg + Σ0, Σ0 ∈ V 0 we obtain the following variational problem.

Problem 1.35 (Homogenized second mixed formulation of the Hellinger-Reiss- II II ner-principle for linearized elasticity). Find a solution v ∈ W , Σ0 ∈ V 0 of the saddle point problem

II II II II a (Σ0, T) + b (T, u) = −a (Σg, T) for ∀T ∈ V 0 (1.43) II II II II b (Σ0, w) = hG , wi − b (Σg, w) for ∀w ∈ W , (1.44) where aII(., .), hGII,.i are as in Problem 1.33.

Obviously the functionals F I,F II,GI,GII are linear and bounded, the functions aI(., .), aII(., .), bI(., .), bII(., .) are bilinear, aI(., .), aII(., .) are bounded on their respective spaces.

I −1 −1 |a (Σ, T)| ≤λmax(C )kΣk0,Ωˆ kTk0,Ωˆ = λmin(C) kΣk0,Ωˆ kTk0,Ωˆ II −1 −1 |a (Σ, T)| ≤λmax(C )kΣk0,Ωˆ kTk0,Ωˆ = λmin(C) kΣk0,Ωˆ kTk0,Ωˆ −1 ≤λmin(C) kΣkdiv,Ωˆ kTkdiv,Ωˆ .

We also have boundedness for bI(., .), bII(., .),

I |b (T, v)| ≤kTk0,Ωˆ kDvk0,Ωˆ ≤ kTk0,Ωˆ kvk1,Ωˆ , II |b (T, v)| ≤k div Tk0,Ωˆ kvk0,Ωˆ = kTkdiv,Ωˆ kvk0,Ωˆ .

25 1. Introduction

By Remark 1.4 C is positive definite, therefore the bilinear form aI(Σ, T) := −1 I (C Σ, T)0,Ωˆ is V -elliptic. In the light of Definition 1.13, one associates to 2 ˆ I 1/2 an arbitrary T ∈ L (Ω)S the stress energy norm T 7→ a (T, T) =: kTkaI . The inf-sup-condition for the first mixed problem directly follows by Korn’s inequality.

Lemma 1.36. Assume the conditions of Theorem 1.10 (b). Then there exists ˆ ˆ I a constant c = c(Ω, Γ0) > 0 such that for any χ ∈ W

I b (T, χ) ˆ ˆ sup ≥ c(Ω, Γ0)kχk1,Ωˆ . (1.45) T∈V I kTk0,Ωˆ

Proof. Let χ ∈ W I be arbitrary and ﬁxed. It is easy to see that one has 2 ˆ T := −Dχ ∈ L (Ω)S. With Korn’s inequality (b) we get ˆ ˆ kTk0,Ωˆ := kDχk0,Ωˆ ≥ c(Ω, Γ0)kχk1,Ωˆ . For χ ≡ 0 (1.45) trivially holds. Now let χ 6≡ 0 then we have

I (T, χ) k χk2 b (T, χ) D 0,Ωˆ D 0,Ωˆ ˆ ˆ sup ≥ ≥ ≥ c(Ω, Γ0)kχk1,Ωˆ . T∈V I kTk0,Ωˆ kTk0,Ωˆ kDχk0,Ωˆ

II For the second mixed problem we take T ∈ V 0 with div T ≡ 0. Hence we have the required ellipticity

II −1 2 −1 2 2 a (T, T) ≥ λmin(C )kTk0,Ωˆ = λmin(C ) kTk0,Ωˆ + k div Tk0,Ωˆ −1 2 = λmin(C )kTkdiv,Ωˆ . This time the inf-sup-condition requires more thinking.

II Lemma 1.37. For any χ ∈ W there exists a constant c = c(cK , cF ) > 0 only depending on the constants emerging from Friedrich’s and Korn’s inequality, such that

bII(T, χ) sup ≥ c(cK , cF )kχk0,Ωˆ . II kTk ˆ T∈V 0 div,Ω

26 1.2. Linearized 3D Elasticity

Proof. Let χ ∈ W II be arbitrary and ﬁxed. Through the density of smooth ∞ ˆ II functions with compact support Cc (Ω) in W [43, Thm. of Meyers-Serrin] we have 1 ∃w ∈ C∞(Ω)ˆ : kχ − wk ≤ kχk . c 0,Ωˆ 2 0,Ωˆ ˆ 3 Let ξ := inf{xˆ1 : xˆ ∈ Ω} and T := (τ i)i=1. We set

Z xˆ1 τ 1(xˆ) = w(t, xˆ2, xˆ3)dt τ i(xˆ) = 0, i = 2, 3. ξ

Then we have div T = ∂1τ 1 = w. It can be shown [12, p. 139] that kTk0,Ωˆ ≤ ckwk0,Ωˆ holds for some c > 0. From this follows

II b (T, χ) (w, χ)0,Ωˆ 1 sup ≥ ≥ kχk0,Ωˆ . II kTk ˆ (1 + c)kwk ˆ 2(1 + c) T∈V 0 div,Ω 0,Ω

Remark 1.38. Alternatively, one could show the inf-sup condition by veri- II 2 ˆ fying, that div : V 0 → L (Ω) is surjective, see [11]. Together with Theorem 1.31 we end up with two stable saddle point problems Problem 1.17 and Problem 1.33. From our discussion of the connection between saddle point problems and their aﬃliated minimization problems (Problem 1.26) we conclude, that our saddle point problems (1.41)-(1.42) and (1.25)-(1.26) characterize the following constrained minimization problems. Problem 1.39 (Principle of Castigliano/Principle of minimal complementary energy). Find Σ ∈ V II such that 1 { { { −1 J (Σ) = min J (T) where J (T) := ( T, T) ˆ II C 0,Ω T∈V g 2 (1.42)(− div Σ = f)

We call J {(.) the complementary energy. There exists a subtle connection between the potential and the complementary energy and their respective variational problems.

27 1. Introduction

I Theorem 1.40 (Two-Energies Principle). Let Σeq ∈ V satisfy (1.26) and let (u, Σ) ∈ V I × W I constitute a solution of (1.25), (1.26). Then the following identity holds for any v ∈ W I. 1 1 a(u − v, u − v) + aI(Σ − Σ , Σ − Σ ) = J(v) + J {(Σ ). (1.46) 2 2 eq eq eq Proof. Let v ∈ W I be arbitrary and ﬁxed. It follows from Theorem 1.14, that u ∈ W I(= V of the pure displacemnt formulation) satisﬁes

a(u, w) = hF, wi for ∀w ∈ W I as well as it follows from Problem 1.17 and Problem 1.6, that

aI(Σ, T) = 0 for ∀T ∈ V I, bI(T, w) = hGI, wi(= hF, wi) for ∀w ∈ W I.

I I Therefore, in particular, a(u, v) = hF, vi and a (Σ, Σeq) = a (Σ, Σ), and hence 1 1 a(u − v, u − v)+ aI(Σ − Σ , Σ − Σ ) 2 2 eq eq 1 1 = a(v, v) − a(u, v) + aI(Σ , Σ ) 2 2 eq eq 1 1 + a(u, u) + aI(Σ, Σ) − aI(Σ, Σ ) , 2 2 eq and since C−1Σ = Du we have a(u, u) = aI(Σ, Σ) and hence

1 1 = J(v) + J {(Σ ) + a(u, u) − aI(Σ, Σ) eq 2 2 | {z } =0 { =J(v) + J (Σeq)

[8, Lemma 3.2.] For the sake of notational brevity we introduce a new energy norm notation,

kΣ, uk2 := a(u, u) + aI(Σ, Σ) for ∀(Σ, u) ∈ V I × W I.

28 1.2. Linearized 3D Elasticity

Corollary 1.41. Let the assumptions of Theorem 1.40 hold, then

2 I I kΣ − Σeq, u − vk = kΣ, uk − 2hG , vi for ∀v ∈ W .

The internally stored energy of an elastic body can be determined through stresses or strains via the elasticity tensor C:

2 2 −1 k k := ( , ) kΣk −1 := (Σ, Σ) . E C E CE 0,Ωˆ C C 0,Ωˆ The following theorem is related to the second mixed formulation of the Hellinger-Reissner principle and oﬀers a more intuitive interpretation compared to Theorem 1.40, which was based on the ﬁrst mixed formulation of the Hellinger-Reissner principle. 2 ˆ −1/2 ˆ Theorem 1.42 (Two-Energies Principle). Let f ∈ L (Ω), g ∈ H (Γ1), v ∈ 1 ˆ ˆ ˆ {χ ∈ H (Ω) : χ = 0 a.e. on Γ0} and let Σeq ∈ H(div, Ω) satisfy (1.42), i.e. ˆ − div Σeq =f in Ω ˆ Σeqn =g on Γ1.

Let u ∈ V constitute a solution of (1.9), Σ = CDu. Then we have

2 2 2 k (v) − (u)k +kΣ − Σk −1 = kΣ − (v)k −1 . (1.47) E E C eq C eq CE C | {z 2 } =kv−uka Proof. By Green’s Formula Z Z Z T :(Dv)dxˆ = − div T · vdxˆ + v · (Tn)dsxˆ Ωˆ Ωˆ ∂Ωˆ we have

(E(v) − E(u), Σeq − Σ)0,Ωˆ Z = D(v − u) : Σeq − Σ dxˆ Ωˆ Z = − v − u · div Σeq − Σ dxˆ ˆ Ω | {z } =0 | {z } =0

29 1. Introduction

Z + v − u · Σeq − Σ n = 0 ˆ ∂Ω | {z } | {z } =0 on Γˆ0 =0 on Γˆ1 | {z } =0 With this property and the fact that

Σeq − CE(v) = (Σeq − CE(u)) + C(E(u) − E(v)) we ﬁnally get

2 −1 kΣ − (v)k −1 =(Σ − (v), (Σ − (v))) eq CE C eq CE C eq CE 0,Ωˆ = (Σeq − CE(u)) + C(E(u) − E(v)),

−1 C (Σeq − CE(u)) + (E(u) − E(v)) 0,Ωˆ 2 2 =kΣ − (u)k −1 + k (u) − (v)k eq CE C E E C + (Σeq − CE(u), E(u) − E(v))0,Ωˆ −1 + (C(E(u) − E(v)), C (Σeq − CE(u)))0,Ωˆ 2 2 =kΣ − (u)k −1 + k (u) − (v)k eq CE C E E C

+ 2(E(v) − E(u), Σeq − Σ)0,Ωˆ 2 2 =kΣ − (u)k −1 + k (u) − (v)k eq CE C E E C

Remark 1.43 (On diﬀerent interpretations of the Two-Energies Principle). Having a closer look at the structure of (1.47) reveals a subtle connection to J(.) and J {(.) by comparing the emerging terms to Theorem 1.40. The only deviation between Theorem 1.40 and Theorem 1.42 can be found at the right hand side of the respective assertions and the regularity of Σeq. However, the deﬁnition of Σeq gives us (Σeq, Dv)0,Ωˆ = (Dv, CDu)0,Ωˆ , therefore in this sense we indeed have two equivalent theorems.

Remark 1.44 (Regularity of the second mixed problem). For our second mixed problem we formally demand u ∈ L2(Ω)ˆ . However, as one would 1 ˆ ˆ expect we even implicitly satisfy u ∈ {v ∈ H (Ω) : v = 0 a.e. on Γ0} 2 [12]. First, from (1.42) we can conclude E(u) = Du = C−1Σ ∈ L (Ω)ˆ .

30 1.2. Linearized 3D Elasticity

Furthermore let i, j ∈ {1, 2, 3} be arbitrary and ﬁxed. We choose a test 3 function T = (τkl)k,l=1, whose only non-vanishing component is τij = τji and ∞ ˆ let τij ∈ Cc (Ω). Inserting this ansatz into (1.42) we get 1 Z Z ˆ −1 ˆ (ui∂jτij + uj∂iτij) dx = − C Σ ij τijdx. 2 Ωˆ Ωˆ

−1 Hence, the symmetric derivative (Du)ij exists weakly and equals (C Σ)ij ∈ L2(Ω)ˆ . Korn’s inequality Theorem 1.10 (a) then gives us u ∈ H1(Ω)ˆ . For the remaining implication on the boundary properties of u we apply Green’s ˆ Formula: Let T ∈ H(div, Ω)S be arbitrary and ﬁxed, Z Z Z u · (Tn)dsxˆ = (Du): Tdxˆ + u · div Tdxˆ ∂Ωˆ Ωˆ Ωˆ Z Z −1 = (C Σ): Tdxˆ + u · div Tdxˆ = 0. Ωˆ Ωˆ Since this is true for any test function in (1.41)-(1.42) by the virtue of the ˆ Fundamental Lemma of Calculus of Variations we have u = 0 a.e. on Γ0.

2. Models of Linearized Plate Theory

2.1. The Three-Dimensional Plate Model

For the remaining chapters of this thesis we will adopt our conventions and notations of Chapter 1 to ﬁt our framework on linearized elastic plate models. Let ωˆ ⊂ R2 be a domain with Lipschitz continuous boundary in the plane, which is spanned by the two vectors e1 and e2. Let e3 denote the transverse direction of our plate. Let γˆ0 ⊂ γˆ = ∂ωˆ be a measurable subset such that lengthγ ˆ0 > 0, γˆ1 :=γ ˆ \ γˆ0, let > 0 be an arbitrary parameter, and let

ˆ ˆ ˆ Ω :=ω ˆ × (−/2, /2) Γ+ :=ω ˆ × {/2} Γ− :=ω ˆ × {−/2} ˆ ˆ ˆ Γ :=γ ˆ × [−/2, /2] Γ0 :=γ ˆ0 × [−/2, /2] Γ1 :=γ ˆ1 × [−/2, /2] , see Figure 2.1. We deﬁne for each > 0, the set cl{Ωˆ } = cl{ωˆ} × [−/2, /2] to be the reference conﬁguration occupied by an elastic body in the absence of applied forces. Because the parameter is thought of as being “small” compared to the diameter of the set ωˆ, the body under consideration is called a plate, with thickness , and middle surface cl{ωˆ}, which is clamped on the ˆ ˆ ˆ portion Γ0 of its lateral face Γ . Furthermore, the boundary of the set Ω is ˆ ˆ ˆ partitioned into the lateral face Γ and the upper and lower faces Γ+ and Γ−, ˆ ˆ ˆ and the lateral face is itself partitioned into Γ = Γ0 ∪ Γ1. Note that we do ˆ ˆ not exclude the case Γ = Γ0, i.e. γˆ =γ ˆ0. The plate is subjected to applied body forces in its interior Ωˆ of density f : Ωˆ → R3 per unit volume. The plate is also exerted to applied surface forces acting on its upper and lower ˆ ˆ 3 ˆ faces of density g : Γ+ ∪ Γ− → R per unit area. To clamp our plate on Γ0 ˆ we impose u = 0 on Γ0. If lengthγ ˆ0 < lengthγ ˆ one calls a plate partially clamped, while the remaining portion γˆ1 remains free from external actions. Otherwise, the plate is said to be completely clamped i.e. if γˆ0 =γ ˆ. Under

33 2. Models of Linearized Plate Theory

ωˆ γˆ

Γˆ + 2 e 3 e 2 ωˆ × {0} 2

e1 ˆ Γ−

Figure 2.1.: Exemplary reference configuration of a general three- dimensional plate Ωˆ =ω ˆ × {0}. the influence of these forces, a point xˆ ∈ cl{Ωˆ } undergoes a displacement u(xˆ), i.e. it occupies the position xˆ + u(xˆ) =: ϕ(xˆ) =: x. In the following, we want to determine the displacement vector field u : cl{Ωˆ } → R3 satisfying a partial differential equation associated to linearized elasticity, which could then be approximated by numerical computations.

Remark 2.1. Discussing the theory of plates, one very often implicitly makes ˆ ˆ assumptions on the applied surface forces (tractions) g on Γ+, Γ− and on the volume forces f to mainly induce so-called plate bending. In many cases [3, 8, 13] this is translated into assuming forces acting only in the e3-direction, which obviously does not comply with our original three-dimensional plate problem. To handle more general forces one usually splits the general three-dimensional plate problem into a plate bending and a plate stretching problem, which can be done due to the linearity of our equations, see Section 2.2. The latter can then be handled by a comparably simple model (see Section A.1) whereas plate bending is more difficult to cope with, therefore a great variety of models and theories have been evolved of which we will discuss the Kirchhoff-Love and the Reissner-Mindlin plate bending model. Hence, since plate stretching does not pose difficulties from a computational point of view, the great ma- jority of published papers on linearized plate theory is entirely focused on the

34 2.1. The 3D Plate Model plate bending problem. Some treat both problems separated without mention- ing their combined potential to model the general problem. Only few papers start their discussion with the splitting into plate bending and stretching, i.e. [2, 16, 18, 47]. However, throughout this thesis our discussion of plates will not automatically incorporate any hidden assumptions on g, f . We finally assume that, for each > 0, the elastic material constituting the plate is homogeneous, isotropic, and that its reference configuration is in natural state [15, Sect. 3.6]. These assumptions imply that the behaviour of the material is, within the first order (with respect to the Green-St. Venant strain tensor), governed by only two of the constants λ, µ and ν,E called the Lamé-constants and Poisson-ratio, Young’s modulus of the material [15, Thm. 3.8-1] respectively. Experimental evidence shows that these constants of an actual elastic material satisfy λ, µ > 0 and ν ∈ [0, 1/2),E > 0. As γˆ is Lip- schitz continuous, Γˆ is Lipschitz continuous [47], and hence a unit outer normal vector n ∈ R3, |n| ≡ 1 and a counter-clockwise unit tangent vector τ ∈ R2, |τ | ≡ 1 can be defined a.e. on Γˆ and a.e. on γˆ, respectively. Problem 2.2 (Classical formulation of the linearized displacement-traction problem for plates). Find u : cl{Ωˆ } → R3 of the linear boundary value problem (according to Theorem 1.6) 1 E :=E(u) = u = (∇u + (∇u)>) D 2 Σ :=Σ(u ) = C E = C Du = (λ (tr E(u ))I + 2µ E(u )) − div Σ =f in Ωˆ (2.1) ˆ u =0 on Γ0 (2.2) ˆ Σ n =0 on Γ1 (2.3) ˆ ˆ Σ n =g on Γ+ ∪ Γ−, (2.4) By the same reasoning as in Chapter 1 one can deduce an associated variational problem. Problem 2.3. Find u ∈ V (Ωˆ ) satisfying the variational problem

a(u, v) = hF , vi for ∀v ∈ V (Ωˆ ), (2.5)

35 2. Models of Linearized Plate Theory where Z a (u , v ) := (Du ):(C Dv )dxˆ (2.6) Ωˆ Z Z hF , v i := f · v dxˆ + g · v dsxˆ (2.7) ˆ ˆ ˆ Ω Γ+∪Γ− ˆ 1 ˆ ˆ 1 ˆ V (Ω ) := {v ∈ H (Ω ): v = 0 a.e. on Γ0} ⊂ H (Ω ).

ˆ ˆ The case Γ = Γ0, i.e. the imposition of Dirichlet boundary conditions on the whole lateral boundary Γˆ will be referred to as clamped plate boundary ˆ ˆ conditions, whereas Γ = Γ1, i.e. the imposition of Neumann boundary conditions on the whole lateral boundary Γˆ will be called free plate boundary conditions.

Remark 2.4. As already noted in the introduction to this thesis, plate theory consists of the imposition of a speciﬁed class of boundary conditions, ˆ 1 ˆ 1 ˆ ˆ ˆ ˆ when V (Ω ) ⊂ H (Ω ) restricts H (Ω ) on Γ , not on Γ+ ∪ Γ−, and semi- discretization in the transverse direction.

By the same reasoning as in Section 1.2 one comes up with the analogon of Theorem 1.14 in our context of plates

Theorem 2.5 (Existence of a weak solution for plates). Let the reference ˆ ˆ ˆ ˆ ˆ ˆ 3 sets Ω , Γ , Γ0, Γ1, Γ+, Γ− ⊂ R and the constants λ , µ > 0 be deﬁned as 2 ˆ 2 ˆ ˆ above in this section. Let the functions f ∈ L (Ω ), g ∈ L (Γ+ ∪ Γ−). Then there is a unique function u in the space

ˆ 1 ˆ ˆ V (Ω ) := {v ∈ H (Ω ): v = 0 a.e. on Γ0} that satisﬁes (2.5), i.e.

a(u, v) = hF , vi for ∀v ∈ V (Ωˆ ), (2.8) where a(., .) and hF ,.i are as in (2.6) and (2.7), respectively. In addition, u also is the unique solution of the corresponding equivalent minimization problem 1 J (u) = inf J (v) where J (v) = a(v, v) − hF , vi. (2.9) v∈V (Ωˆ ) 2

36 2.2. Splitting into Plate Bending and Stretching

An analogue adaptation of the theorems related to the Hellinger-Reissner principle in Section 1.2.2 reveals no diﬃculties. Remark 2.6 (Geometry locking). The a-priori error estimate of Problem 2.3 implied by Cea’s lemma suﬀers from the unpleasant behaviour of Korn’s ˆ −1 inequality on Ω : cK = O( ) as → 0 [16, 18]. Hence, it may not be a good idea to imprudently discretize the plate problem by “normal” Galerkin approximation.

2.2. Splitting the Plate Problem into a Plate Bending- and a Plate Stretching Problem

Due to the linearity of our linearized elasticity problem (2.1)-(2.4) one comes up with the idea of splitting the general system, which does not have to satisfy any geometric assumptions on (f , g), into a problem complying with such assumptions and another problem, capturing the non-complying forces. This is done by taking odd and even parts of (f , g) with respect to the transversal coordinate xˆ3. [2, 16, 18, 35, 42]

Definition 2.7 (Odd and even parts of a function with respect to xˆ3). Let ψ(xˆ) be a tensor-valued, a vector-valued or a scalar function defined on either ˆ ˆ ˆ Ω or Γ+ ∪ Γ−. Then we define ψ(xˆ, xˆ ) + ψ(xˆ, −xˆ ) ψ(xˆ, xˆ ) − ψ(xˆ, −xˆ ) ψ (xˆ) := 3 3 ψ (xˆ) := 3 3 even 2 odd 2

Obviously one has ψ(xˆ) = ψeven(xˆ) + ψodd(xˆ).

3 3 3 3 Notation. Let a = (ai)i=1 ∈ R , A = (aij)i,j=1 ∈ M , then we set a := 2 2 2 2 (ai)i=1 ∈ R , A := (aij)i,j=1 ∈ M . Then     a1 a11 a12 a13 2 a A (ai3)i=1 a = a2 = A = a21 a22 a23 = 2 a3 (a3j)j=1 a33 a3 a31 a32 a33 One can now easily decompose the loads into a so called stretching and bending portion.

f =f s + f b g =gs + gb

37 2. Models of Linearized Plate Theory

f g even even f s := gs := f3,odd g3,odd f g odd odd f b := gb := f3,even g3,even

The problem (2.1)-(2.4) associated to the loads (f b, gb) will henceforth be called a plate bending problem, while the system (2.1)-(2.4) associated to the 1 loads (f s, gs) deﬁnes the plate stretching problem . In the next step, one also splits (u , Σ ) into (us, Σs) and (ub, Σb), where

u =us + ub Σ =Σs + Σb Σ (σ )2 u even even 3j j=1 odd us = Σs = 2 u3,odd ((σi3)i=1)odd σ3,even Σ (σ )2 u odd odd 3j j=1 even ub = Σb = 2 u3,even ((σi3)i=1)even σ3,odd

We denote the corresponding sets of admissible displacement ﬁelds by V s(Ωˆ ) b ˆ and V (Ω ) for the stretching and the bending portions, us, ub, respectively, i.e.

v V s(Ωˆ ) :={v ∈ V (Ωˆ ): v = even } (2.10) vodd v V b(Ωˆ ) :={v ∈ V (Ωˆ ): v = odd }. (2.11) veven

These subsets of V (Ωˆ ) are closed and orthogonal with respect to the inner product induced by the bilinear form a(., .) on V (Ωˆ ).

Lemma 2.8. Let V s(Ωˆ ), V b(Ωˆ ) be given by (2.10), (2.11), respectively. Then one has

V (Ωˆ ) =V s(Ωˆ ) ⊕ V b(Ωˆ ) a(u, v) =0 for ∀u ∈ V s(Ωˆ ), v ∈ V b(Ωˆ ). (2.12)

1This problem is also commonly addressed as the ﬂexural displacement or membrane problem.

38 2.2. Splitting into Plate Bending and Stretching

s ˆ b ˆ Proof. Let us ∈ V (Ω ), vb ∈ V (Ω ) be arbitrary and ﬁxed, then we conclude by Fubini’s theorem Z a (us, vb) = (Dus):(CDvb)dxˆ Ωˆ Z ( )2 Σ (σ )2 E even 3j j=1 odd odd 3j j=1 even = 2 : 2 dxˆ Ωˆ ((i3)i=1)odd 3,even ((σi3)i=1)even σ3,odd | {z } =:h Z Z /2 = hdxˆ3dxˆ. ωˆ −/2 Since the product of an odd and an even function is an odd function, h constitutes an odd function and hence (2.12) follows, [47, Subsect. 2.4.]. One similarly deduces a consequence for our linear form F ,

hF , v i =hFs , v i + hFb, v i Z Z = f s · v dxˆ + gs · v dsxˆ ˆ ˆ ˆ Ω Γ+∪Γ− Z Z + f b · v dxˆ + gb · v dsxˆ ˆ ˆ ˆ Ω Γ+∪Γ− =hFs , vsi + hFb, vsi + hFs , vbi + hFb, vbi.

Furthermore, by the same arguments as above one arrives at Z Z hF , v i = f · v dxˆ + f v ds s b even odd 3,odd 3,even xˆ ˆ ˆ ˆ Ω Γ+∪Γ− Z Z hF , vi = f · v dxˆ + f v ds . b s odd even 3,even 3,odd xˆ ˆ ˆ ˆ Ω Γ+∪Γ− This, as well as assertion (2.12) now provide us with a convenient simpliﬁ- cation of the general variational problem (2.5),

a (us + ub, vs + vb) =a (us, vs) + a (us, vb) + a (ub, vs) + a (ub, vb) (2.12) = a (us, vs) + a (ub, vb) =hF , vsi + hF , vbi

39 2. Models of Linearized Plate Theory

s ˆ b ˆ =hFs , vsi + hFb, vbi for ∀vs ∈ V Ω , vb ∈ V Ω . Hence, we have split the original system into two decoupled problems, therefore we can solve for the stretching- and the bending portions separately. See Figure 2.2 for an illustration of two exemplary plate problems and their respective separated stretching and bending problems. Problem 2.9 (Decoupled variational plate stretching and bending prob- s ˆ b ˆ lems). Find us ∈ V (Ω ), ub ∈ V (Ω ) satisfying the variational problems s ˆ a (us, vs) =hFs , vsi for ∀vs ∈ V (Ω ) b ˆ a (ub, vb) =hFb , vbi for ∀vb ∈ V (Ω ),

where a (., .) is given by (2.6) and hFs ,.i, hFb ,.i are given by Z Z hFs , v i = f s · v dxˆ + gs · v dsxˆ ˆ ˆ ˆ Ω Γ+∪Γ− Z Z hFb , v i = f b · v dxˆ + gb · v dsxˆ ˆ ˆ ˆ Ω Γ+∪Γ− Remark 2.10. The splitting of the plate problem into a plate bending and stretching problem is applicable to so-called monoclinic materials which admit a linear constitutive law of the form         Σ11 c11 c12 c13 c14 0 0 E11 E11 Σ22 c12 c22 c23 c24 0 0  E22 E22         Σ33 c13 c23 c33 c34 0 0  E33 E33   =     =: C˜   Σ12 c14 c24 c34 c44 0 0  E12 E12         Σ13  0 0 0 0 c55 c56 E13 E13 Σ23 0 0 0 0 c56 c66 E23 E23 for some positive deﬁnite C˜ ∈ S6. More general material laws have to be handled diﬀerently [16, 47].

2.3. Hierarchic Plate Models

The introductory chapter of [2] comprises a compact, yet felicitous ad-hoc statement on what plate theory is about.

40 2.3. Hierarchic Plate Models Example 1: Plate bending portion Example 2: Plate bending portion Example 1: Plate stretching portion Example 2: Plate stretching portion forces into plate bending and stretching portions. Example 1: Arbitrary acting forces Example 2: Transverse plate shearing Figure 2.2.: Here we give two intuitive examples to illustrate the implications of the splitting of the acting

41 2. Models of Linearized Plate Theory

A plate model seeks to approximate the solution of the elasticity problem (2.1)-(2.4) in terms of the solution of a system of partial diﬀerential equations on the two-dimensional domain ωˆ without requiring the solution of a three-dimensional problem. The passage from the three-dimensional problem to a plate model is known as dimensional reduction. In this regard, one usually creates a natural hierarchy of plate models by the semi-discretization of the plate problem (2.8) in e3-direction based on polynomial approximations along e3 and performing an energy projection of u on this semi-discrete subspace, see [47].

Remark 2.11 (Energy projection). Let V 0 ⊂ V be a closed subspace and u ∈ V a solution of Problem 1.6. The energy-projector P : V → V 0 is deﬁned via kP u − uka = infv∈V 0 kv − uka, which by virtue of Theorem 1.7 can be equivalently determined by solving

a(P u, v) = a(u, v) for ∀v ∈ V 0, i.e. the elasticity problem on the closed subspace V 0. More explicitly, by a hierarchic family of plate models we understand a sequence of two-dimensional models, whose related three-dimensional solutions converge to the exact three-dimensional solution of the plate problem (2.1)-(2.4), see Theorem 2.16, and any model of this sequence converges (after proper scaling) to the same limit, as → 0. (2.9) admits different model 3 orders m ∈ N0 for each single displacement component. Any plate model being a member of this hierarchic family is defined as the minimizer um of the following minimization problem. Problem 2.12 (The m-plate model). Find um ∈ V (m) such that 1 J (um) = min J (v), where J (v) := a(v, v) − hF , vi (2.13) v∈V (m) 2 V (m) ⊂ V (Ωˆ ) constitutes the set of all semi-discrete functions, whose dependence on the e3-coordinate is explicitly given by a linear combination of xˆ3-monomials, whose associated coefficient functions solely depend on the 2 e1-e2-coordinates xˆ ∈ ωˆ ⊂ R , i.e. mi m X m j > 3 ui (xˆ) := uij (xˆ)ˆx3 for i = 1, 2, 3, m = (m1, m2, m3) ∈ N0. (2.14) j=0

42 2.3. Hierarchic Plate Models

Hence, the resultant ansatz functions dispose of a general connection to the e3-coordinate, which will enable us to perform calculations related to the emergent xˆ3-polynomials explicitly. The solution according to (2.13), (2.14) is designated as the solution of the m-model. It is obvious, that S ˆ m 3 V (m) is dense in V (Ω ) and hence u → u as mi → ∞, i = 1, 2, 3 m∈N0 1/2 with the convergence in the energy norm k.ka : v 7→ a (v, v) . It can be shown, that V (m) ⊂ V (Ωˆ ) is a closed linear subspace [47, Prop. 3.1.], hence our statements on existence and uniqueness of (2.9), (2.8) carries over to (2.13). We will see, that hierarchic modelling leads to a singularly perturbed, elliptic system in the two-dimensional mid-surface ωˆ×{0} of the tree-dimensional structure Ωˆ [7].

Remark 2.13. An ansatz of the form (2.14) has already been proposed by Maurice Levy in 1877, but was not esteemed by his contemporaries, maybe because it was too much ahead of his time. The potential of Levy’s idea was subsequently incorporated into the analysis of [30] in 1949. For further historic aspects one refers to [42].

Remark 2.14. In some situations (e.g. other constitutive laws, such as orthotropic or laminated plates), semi-discrete approximation according to (2.14) does not deliver satisfactory results. Then one usually replaces the j monomials xˆ3 by other preferred basis functions, such as Legendre polynomials, or non-polynomial ansatz functions, see [47].

Remark 2.15. Hierarchic models permit the simultaneous use of diﬀerent model orders in various subregions of the plate, e.g. to handle boundary layers [3–5, 7, 8, 47].

We now complement our hierarchy of plate models by a separation of bending and stretching eﬀects as discussed in the previous section. The hierarchic separated plate models are obtained by restricting V s(Ωˆ ), V b(Ωˆ ) to V (m), which implies that Problem 2.12 can be equivalently split into a stretching m m and a bending part, us , ub , respectively. The respective contributions are therefore obtained independently of each other by solving

m s ˆ m s us ∈ V (m) ∩ V (Ω ) : a (us , v) = hFs , vi for ∀v ∈ V (m) (2.15) | {z } =:V s(m)

43 2. Models of Linearized Plate Theory

η = 1 η = 2 η = 3 η = 4 η = 5 η = 6 V s(m) (0, 0, 1) (2, 2, 1) (2, 2, 3) (4, 4, 3) (4, 4, 5) (6, 6, 5) V b(m) (1, 1, 0) (1, 1, 2) (3, 3, 2) (3, 3, 4) (5, 5, 4) (5, 5, 6)

Table 2.1.: Model orders m for common stretching and bending models in dependence on maximal polynomial degree η in transverse variable xˆ3.

m b ˆ m b ub ∈ V (m) ∩ V (Ω ) : a (ub , v) = hFb, vi for ∀v ∈ V (m). (2.16) | {z } =:V b(m) Again, V s(m), V b(m) constitute closed linear subspaces of H1(Ωˆ ) [47, 3 Prop. 3.1.], which by virtue of Theorem 2.5 implies, that for every m ∈ N0 m s m b there exist unique solutions us ∈ V (m) and ub ∈ V (m). Taking into account, that for the plate bending setting our loads, stresses and displacements satisfy certain symmetry assumptions, one reasons from our ansatz (2.14), that for the stretching and bending displacement contributions one has m Stretching: u1j ≡ 0 for ∀j ∈ 2N0 + 1, j ≤ m1 m u2j ≡ 0 for ∀j ∈ 2N0 + 1, j ≤ m2 m u3j ≡ 0 for ∀j ∈ 2N0, j ≤ m3 m Bending: u1j ≡ 0 for ∀j ∈ 2N0, j ≤ m1 m u2j ≡ 0 for ∀j ∈ 2N0, j ≤ m2 m u3j ≡ 0 for ∀j ∈ 2N0 + 1, j ≤ m3. Usually one classiﬁes the relations for the model orders in dependence of the maximal polynomial degree η with respect to xˆ3 according to Table 2.1, however, depending on the application of the plate model, one might consider other models and pairings of stretching and bending semi-discretizations. We conclude this section with some basic properties of the hierarchic models. Proposition 2.16 (Basic properties of the hierarchic models). (a) Optimality of the m-model ( Cea’s lemma) m m kus −us ka ≤ inf kus −vka kub −ub ka ≤ inf kub −vka . v∈V s(m) v∈V b(m)

44 2.3. Hierarchic Plate Models

(b) Let mi ≥ li, i = 1, 2, 3. Then we have

m l m l kus − us ka ≤ kus − uska kub − ub ka ≤ kub − ubka ,

i.e. an increase of the model order never increases the modelling error.

m kus − us ka →0 as mi → ∞, i = 1, 2, 3 m kub − ub ka →0 as mi → ∞, i = 1, 2, 3.

Proof. See [47, Prop. 3.2.].

(a) is a consequence of um being the energy projection of the three-dimensional solution u onto V (m). Therefore it follows by (2.8) and (2.15), (2.16) that Galerkin’s orthogonality holds, i.e.

a(u − um, v) = 0 for ∀v ∈ V (m).

The statement follows from this orthogonality and the Cauchy-Schwarz inequality,

m m m ku − u ka =a (u − u , u − u ) =a(u − um, u − v + v − um) =a(u − um, u − v) m ≤ku − u ka ku − vka for ∀v ∈ V (m).

(b) follows from (a) and the inclusion V (m) ⊃ V (l) for mi ≥ li, i = 1, 2, 3,

m m l ku − u ka ≤ inf ku − v ka ≤ inf ku − v ka vm∈V (m) vl∈V (l) l l ku − u ka ≤ inf ku − v ka , vl∈V (l)

m l which by subtracting these lines yields ku − u ka − ku − u ka ≤ 0. (c) follows from the density of the polynomials in H1(−1, 1). This implies that (V (m)) 3 is dense in V (Ωˆ ) which with (a) yields the assertion. m∈N0

45 2. Models of Linearized Plate Theory

While one can relatively easily see that the stretching problem can be well approximated by the plane stress model (as outlined in Section A.1.1), plate bending is more diﬃcult to cope with. As was already mentioned in the introductory statements to Chapter 1 the most common plate bending models are the Kirchhoﬀ-Love and the Reissner-Mindlin plate model [1, 2, 7, 12, 16, 18] which we are going to derive and justify in the subsequent section.

2.4. The Clamped Plate

In this section we intend to review the statements of [13] on the justification of the Kirchhoff-Love as well as the Reissner-Mindlin plate bending theory, based upon the two-energies principle of Prager and Synge, which probably was firstly introduced to plate theory in structural continuum mechanics by [36] almost six decades ago and was greatly extended to a more general framework by [2]. Our target is a proof, that the modelling error vanishes as the thickness of the structure Ωˆ tends to zero. Note, that in practice, however, > 0 is given and fixed. Hence one strives for a high asymptotic convergence order of our model error [47]. We approach this topic by consecutively discussing each part of the following procedure:

Step 1 Inspired by mechanical considerations on the bending behaviour of a “thin” elastic plate we heuristically introduce our ansatz functions for the diﬀerent plate models.

Step 2 We will derive statements on the asymptotic behaviour of the energy functional (2.9) as → 0.

Step 3 Finally, we determine a connection between the error of the full three- dimensional problem and our plate approximation with respect to some energy norm. For the norm of this error we end up with a convergence rate of O(1/2) as → 0.

46 2.4. The Clamped Plate

2.4.1. The Plate Hypotheses of the Kirchhoﬀ-Love and the Reissner-Mindlin Plate Model First and foremost in this section we will only cover plate bending, purely transversal loads and completely clamped plates, i.e.

γˆ0 =γ. ˆ

The second means that all acting forces are aligned with e3 and only depend > > ˆ ˆ on xˆ = (ˆx1, xˆ2) ∈ ωˆ as xˆ = (ˆx1, xˆ2, xˆ3) ∈ Ω i.e. for arbitrary xˆ ∈ Ω we have f (xˆ) = 2 (0, 0, p(xˆ))> g(xˆ, ±/2) = 3 0, 0, q±(xˆ)> , (2.17) where the loads have already been properly scaled to be able to elegantly reduce several terms later on. Forces and tractions with non-vanishing components in the other directions will be dealt with later on in Section 2.7 [2, 5, 7]. Hypothesis (Hypotheses of Mindlin and Reissner). [12, Def. 6.1. and the reference therein] [H1] Straight lines along the normal direction (with respect to ωˆ) are linearly deformed and their images again constitute straight lines.

[H2] The displacement along e3 does not depend on its e3-coordinate.

[H3] Points on ωˆ × {0} are only deformed in e3-direction.

3 [H4] For the stress Σ = (σij)i,j=1 in e3-direction we have σ33 ≡ 0. [H5] The material constants do not depend on the plate thickness , i.e. (λ, µ, ν,E) = (λ, µ, ν, E), C = C for ∀ > 0. From [H1]-[H3] we end up with the following ansatz. The Reissner-Mindlin (1,1,0)-Model For all xˆ ∈ Ωˆ we set

u1(xˆ) = − xˆ3θ1(xˆ) (2.18a)

u2(xˆ) = − xˆ3θ2(xˆ) (2.18b)

u3(xˆ) =w(xˆ). (2.18c) We will address w(xˆ) as the transversal displacement whereas θ := > (θ1, θ2) will be called torsion.

47 2. Models of Linearized Plate Theory

For the less general Kirchhoﬀ-Love plates we additionally make an assumption on the deformation of normals to the mid-surface ωˆ × {0}.

Hypothesis (Hypotheses of Kirchhoﬀ). [12, Def. 6.2. and the reference therein] Additionally to [H1]-[H5] we demand

[H6] After deformation normal directions (with respect to ωˆ) are again normal directions (with respect to ω), i.e. θ = ∇w.

Together with (2.18a)-(2.18c) one can conclude that [H6] gives us

The Kirchhoﬀ-Love (1,1,0)-Model For all xˆ ∈ Ωˆ we set

∂ u1(xˆ) = − xˆ3 w(xˆ) (2.19a) ∂xˆ1 ∂ u2(xˆ) = − xˆ3 w(xˆ) (2.19b) ∂xˆ2 u3(xˆ) =w(xˆ). (2.19c)

A thorough derivation can be found in [42, Chp. 3]. Furthermore, to illustrate [H6], see Figure 2.3.

Remark 2.17 (Historical considerations on the Kirchhoff-Love and the Reissner-Mindlin plate model). Even if the ansatz of Kirchhoff (1850) to approximate plate bending phenomena seems intuitive, its derivation has nonetheless posed considerable difficulties not only in the derivation of a differential equation (which we will most naturally achieve by considering the energy J(.)), but also in the imposition of boundary conditions, see [42, Chp. 3]. The incapacity of the Kirchhoff-Love model to adhere to shear phenomena and boundary layers has then consequently led to the introduction of the more general Reissner-Mindlin plate model (which Reissner himself refers to as Hencky’s procedure from 1947), see [42, Chp. 5].

2.4.2. Notations and Conventions for Diﬀerent Plate Models Now that we have established the hypothesis for the Kirchhoﬀ-Love and the Reissner-Mindlin plate models we prepend this section to accustom the

48 2.4. The Clamped Plate 1 θ 1 ˆ x ∂w ∂ 1 ˆ x 3 ˆ x 2 ˆ x relating the displacement of the mid-surface to the displace- [H6] 1 ˆ x ∂w ∂ ment of its normal in the case of a Kirchhoﬀ-Love (left) and a Reissner-Mindlin plate (right). Figure 2.3.: Illustration of hypothesis

49 2. Models of Linearized Plate Theory reader to the otherwise confusing derivation of energy functionals related to diﬀerent plate models. With regard to our notion of hierarchic plate models in (2.14) we refer to (2.18a)-(2.18c) and (2.19a)-(2.19c) as ansatz functions of (1,1,0) plate models. From now on we will therefore call (2.18a)-(2.18c) the ansatz functions of the (1,1,0)-Kirchhoﬀ-Love model whereas (2.19a)-(2.19c) will be designated as the ansatz functions of the (1,1,0)-Reissner-Mindlin model. As an example, the stress tensor associated to these models can now be most transparently written as

(1,1,0) (1,1,0) (1,1,0) (1,1,0) ΣK := Σ(uK ) respectively ΣR := Σ(uR ),

(1,1,0) (1,1,0) where uK and uR highlight the use of (2.19) and (2.18), respectively. For the justiﬁcation of our plate models it will be crucial to be well aware of the applied model. To discuss the question of the permissibility of hypothesis [H2] one introduces new models, which are more strongly connected to the original three-dimensional elasticity problem [2–4, 6, 12, 18]. This is achieved by increasing the polynomial dependence of the ansatz functions on xˆ3. The Kirchhoﬀ-Love (1,1,2)-Model For all xˆ ∈ Ωˆ we set ∂ u1(xˆ) = − xˆ3 w(xˆ) (2.20a) ∂xˆ1 ∂ u2(xˆ) = − xˆ3 w(xˆ) (2.20b) ∂xˆ2 2 u3(xˆ) =w(xˆ) +x ˆ3W (xˆ). (2.20c)

The Reissner-Mindlin (1,1,2)-Model For all xˆ ∈ Ωˆ we set

u1(xˆ) = − xˆ3θ1(xˆ) (2.21a)

u2(xˆ) = − xˆ3θ2(xˆ) (2.21b) 2 u3(xˆ) =w(xˆ) +x ˆ3W (xˆ). (2.21c)

Remark 2.18. According to Reissner, the very ﬁrst extension of the (1,1,0)- models to an (1,1,2)-ansatz has been introduced by [30] in 1949. In this regard, the author was very much surprised that the notion of hierarchic plate models of a general form (2.14) was already introduced by Maurice Levy in 1877 [42, Chp. 6].

50 2.4. The Clamped Plate

Remark 2.19 (Higher order models). By the virtue of an additional function W , the (1,1,2)-models (or other higher order models of Table 2.1) can give a substantially better representation of boundary layer phenomena resulting from the fully three-dimensional model at the lateral boundary Γˆ as → 0, see [3, 4, 16, 18]. For model analysis relying on sophisticated physical considerations similar to [H1]-[H6], this usually constitutes the point, when the advantages of mathematically motivated hierarchic plate models (asymptotic analysis) become obvious, for they easily oﬀer natural higher order models.

2.4.3. Derivation of Basic Plate Models In this subsection we will derive the energy functionals J (um) emerging from the explicit insertion of various ansatz functions (2.14) into the elastic energy (2.13) by utilising the given polynomial dependence on xˆ3 to analytically execute the respective integrations. In this regard it will turn out to be helpful to introduce the convenient notation E ν : 2 → 2, E 7→ E + (tr E)I . C S S 1 + ν 1 − 2ν

The (1,1,2)-Reissner-Mindlin Plate Model

(1,1,2) The insertion of uR given by (2.21a)-(2.21c) into the energy J (.) (2.13) leads us to

(1,1,2) 1 (1,1,2) (1,1,2) (1,1,2) J (u ) = (E , Σ ) − hF , u i, R 2 R R 0,Ωˆ R

(1,1,2) for which we need the symmetric gradient ER as well as the stress tensor (1,1,2) (1,1,2) ΣR associated to uR ,   −xˆ3∂1θ1 sym. sym. (1,1,2) 1 ER =  − 2 xˆ3(∂1θ2 + ∂2θ1) −xˆ3∂2θ2 sym.  1 1 2 1 1 2 2 (∂1w − θ1) + 2 xˆ3∂1W 2 (∂2w − θ2) + 2 xˆ3∂2W 2ˆx3W (1,1,2) tr ER =ˆx3(2W − div θ) (1.1) E ν Σ(1,1,2) = E(1,1,2) + (tr E(1,1,2))I . R 1 + ν R 1 − 2ν R | {z } 2µ

51 2. Models of Linearized Plate Theory

(1,1,2) We can now continue our computation of J (uR ) by the insertion of these quantities, 1 (1,1,2) (1,1,2) (1,1,2) 2 ν (1,1,2) 2 (E , Σ ) =µ kE k + k tr E k 2 R R 0,Ωˆ R 0,Ωˆ 1 − 2ν R 0,Ωˆ µ =µkxˆ θk2 + k∇w − θ +x ˆ2∇W k2 3D 0,Ωˆ 2 3 0,Ωˆ µν + 4µkxˆ W k2 + kxˆ (2W − div θ)k2 . 3 0,Ωˆ 1 − 2ν 3 0,Ωˆ One now performs the explicit computations of integrals with respect to the transverse coordinate by virtue of Fubini’s theorem, e.g.

Z Z Z /2 2 2 2 2 2 kxˆ3W k0,Ωˆ = xˆ3W dxˆ = xˆ3W dxˆ3dxˆ Ωˆ ωˆ −/2 Z 1 3 2 1 3 2 = W dxˆ = kW k0,ωˆ . 12 ωˆ 12 Other integrals with respect to the transverse coordinate are treated analogously. This leads to

1 (1,1,2) (1,1,2) µ 3 2 µ 2 (E , Σ ) = k θk + k∇w − θk 2 R R 0,Ωˆ 12 D 0,ωˆ 2 0,ωˆ µ µ + 3(∇w − θ, ∇W ) + 5k∇W k2 12 0,ωˆ 160 0,ωˆ µ µ ν + 3kW k2 + 3k2W − div θk2 . 3 0,ωˆ 12 1 − 2ν 0,ωˆ In our next step, we want to determine the structure of the load terms Z (1,1,2) (1,1,2) (1,1,2) hF , uR i :=(f , uR )0,Ωˆ + g · uR dsxˆ ˆ ˆ Γ+∪Γ− 2 2 = (p, w +x ˆ3W )0,Ωˆ Z 2 + 3 q+(w + W )dxˆ ωˆ 2 Z 2 + 3 q−(w + W )dxˆ ωˆ 2

52 2.4. The Clamped Plate

1 =3(p, w) + 5(p, W ) 0,ωˆ 12 0,ωˆ 3 + 3 − + (q , w)0,ωˆ + (q , w)0,ωˆ 1 1 + 5(q+,W ) + 5(q−,W ) 4 0,ωˆ 4 0,ωˆ 3 + − 5 = (p + q + q , w)0,ωˆ +O( ).

| 3 {z } =: (ptotal,w)0,ωˆ

Higher order load terms condensed in O(5) will henceforth be neglected, thus the resultant elastic energy of the (1,1,2)-Reissner-Mindlin model constitutes (1,1,2) an approximation of the original elastic energy, denoted by JR (w, θ,W ). We end up with the following model. Problem 2.20 (The (1,1,2)-Reissner-Mindlin plate bending model). Find 1 ˆ 1 ˆ 1 ˆ (1,1,2) (w, θ,W ) ∈ H0 (Ω ) × H0(Ω ) × H0 (Ω ) (uR ∈ V (1, 1, 2)) such that

(1,1,2) (1,1,2) JR (w,θ,W ) = min JR (v, φ,V ), 1 ˆ 1 ˆ 1 ˆ (v,φ,V )∈H0 (Ω )×H0(Ω )×H0 (Ω ) where µ µ µ J (1,1,2)(w,θ,W ) := 3k θk2 + k∇ w − θk2 + 5k∇ W k2 R 12 D 0,ωˆ 2 0,ωˆ 160 0,ωˆ µ µ + 3(∇ w − θ, ∇ W ) + 3kW k2 12 0,ωˆ 3 0,ωˆ ν µ + 3kdiv θ − 2W k2 − 3(p , w) . 1 − 2ν 12 0,ωˆ total 0,ωˆ

The (1,1,2)-Kirchhoﬀ-Love Plate Model

(1,1,2) The insertion of uK given by (2.20a)-(2.20c) leads us to the analogue (1,1,2) computation of J (uK ), which can readily be done by setting θ = ∇w in our calculations related to the (1,1,2)-Reissner-Mindlin model,

(1,1,2) 1 (1,1,2) (1,1,2) (1,1,2) J (u ) = (E , Σ ) − hF , u i K 2 K K 0,Ωˆ K 1 (1,1,2) (1,1,2) µ 3 2 2 µ 5 2 (E , Σ ) = k∇ wk + k∇W k 2 K K 0,Ωˆ 12 0,ωˆ 160 0,ωˆ µ µ ν + 3kW k2 + 3k2W − ∆wk2 . 3 0,ωˆ 12 1 − 2ν 0,ωˆ

53 2. Models of Linearized Plate Theory

One can further reformulate these terms by the application of a simple formula going back to [36],

2 2 2 2 ∇ w : ∇ w =(∂11w + ∂22w) + 2((∂12w) − (∂11w)(∂22w)) (2.22) ν ν 1 − ν ν 4W 2 + (z − 2W )2 = z2 + (2W − z)2 (2.23) 1 − 2ν 1 − ν 1 − 2ν 1 − ν with z := ∆w, which gives us the following structure of our energy Z (1,1,2) µ 3 2 2 J (uK ) = k∆ wk0,ωˆ + 2 (∂12w) − (∂11w)(∂22w) dxˆ 12 ωˆ ν 1 − ν ν + k∆ wk2 + k2W − ∆ wk2 1 − ν 0,ωˆ 1 − 2ν 1 − ν 0,ωˆ 3 + 2k∇ W k2 − hF , u(1,1,2)i. 40 0,ωˆ K Due to our given loads (2.17), our computations to determine the load terms (1,1,2) hF , uK i conclude with the same result,

(1,1,2) (1,1,2) 3 + − 5 hF , uK i =hF , uR i = (p + q + q , w)0,ωˆ +O( ).

| 3 {z } =: (ptotal,w)0,ωˆ As above, higher order load terms condensed in O(5) will henceforth be neglected, thus the resultant elastic energy of the (1,1,2)-Kirchhoﬀ-Love model constitutes an approximation of the original elastic energy, denoted (1,1,2) by JK (w, W ). We end up with the following model. Problem 2.21 (The (1,1,2)-Kirchhoﬀ-Love plate bending model). Find (w, W ) ∈ 2 ˆ 1 ˆ (1,1,2) H0 (Ω ) × H0 (Ω ) (uK ∈ V (1, 1, 2)) such that (1,1,2) (1,1,2) JK (w,W ) = min JK (v, V ), 2 ˆ 1 ˆ (v,V )∈H0 (Ω )×H0 (Ω ) where µ 1 1 − ν ν J (1,1,2)(w, W ) = 3 k∆ wk2 + k2W − ∆ wk2 K 12 1 − ν 0,ωˆ 1 − 2ν 1 − ν 0,ωˆ Z 3 2 2 2 + k∇ W k0,ωˆ + 2 (∂12w) − (∂11w)(∂22w) dxˆ 40 ωˆ 3 − (ptotal, w)0,ωˆ .

54 2.4. The Clamped Plate

The (1,1,0)-Reissner-Mindlin Plate Model

(1,1,0) The insertion of uR given by (2.18) leads us to an even shorter compu- (1,1,0) tation of J (uR ) by setting W ≡ 0 in our computations related to the (1,1,2)-Reissner-Mindlin model,

(1,1,0) 1 (1,1,0) (1,1,0) (1,1,0) J (u ) = (E , Σ ) − hF , u i R 2 R R 0,Ωˆ R 1 (1,1,0) (1,1,0) µ 3 2 µ 2 (E , Σ ) = k θk + k∇w − θk 2 R R 0,Ωˆ 12 D 0,ωˆ 2 0,ωˆ µν 1 + 3kdiv θk2 1 − 2ν 12 0,ωˆ

(1,1,0) The calculation of the load terms hF , uR i inherently disposes of the higher order terms previously incorporated into O(5) of the (1,1,2)-models above,

(1,1,0) 3 + − 3 hF , uR i = (p + q + q , w)0,ωˆ =: (ptotal, w)0,ωˆ .

This ends our derivation of the elastic energy of the (1,1,0)-Reissner-Mindlin (1,1,0) model, which we will henceforth refer to as JR (w, θ). We end up with the following model.

Problem 2.22 (The (1,1,0)-Reissner-Mindlin plate bending model). Find 1 ˆ 1 ˆ (1,1,0) (w, θ) ∈ H0 (Ω ) × H0(Ω ) (uR ∈ V (1, 1, 0)) such that

(1,1,0) (1,1,0) JR (w,θ) = min JR (v, φ), 1 ˆ 1 ˆ (v,φ)∈H0 (Ω )×H0(Ω ) where

µ µν 1 J (1,1,0)(w, θ) = 3k θk2 + 3kdiv θk2 R 12 D 0,ωˆ 1 − 2ν 12 0,ωˆ µ + k∇w − θk2 − 3(p , w) , 2 0,ωˆ total 0,ωˆ where the two terms in the ﬁrst line of the last expression denotes the bending energy whereas the second is called the shear energy.

55 2. Models of Linearized Plate Theory

The (1,1,0)-Kirchhoﬀ-Love Plate Model

(1,1,0) The insertion of uK given by (2.19) leads us to an even shorter compu- (1,1,0) tation of J (uK ) by setting W ≡ 0 in our computations related to the (1,1,2)-Kirchhoﬀ-Love model,

(1,1,0) 1 (1,1,0) (1,1,0) (1,1,0) J (u ) = (E , Σ ) − hF , u i K 2 K K 0,Ωˆ K 1 (1,1,0) (1,1,0) µ 3 1 − ν 2 (E , Σ ) = k∆wk 2 K K 0,Ωˆ 12 1 − 2ν 0,ωˆ Z µ 3 2 + 2 (∂12w) − (∂11w)(∂22w) dxˆ. 12 ωˆ

(1,1,0) Also for this iteration, the calculation of the load terms hF , uK i disposes of the higher order terms incorporated into O(5) of the (1,1,2)-models above,

(1,1,0) 3 hF , uK i := (ptotal, w)0,ωˆ . This ends our derivation of the elastic energy of the (1,1,0)-Kirchhoﬀ-Love (1,1,0) model, which we will henceforth refer to as JK (w). We end up with the following model. Problem 2.23 (The (1,1,0)-Kirchhoﬀ-Love plate bending model). Find w ∈ 2 ˆ (1,1,0) H0 (Ω ) (uK ∈ V (1, 1, 0)) such that

(1,1,0) (1,1,0) JK (w) = min JK (v), 2 ˆ v∈H0 (Ω ) where µ 1 − ν J (1,1,0)(w) := 3 k∆wk2 − 3(p , w) K 12 1 − 2ν 0,ωˆ total 0,ωˆ Z µ 3 2 + 2 (∂12w) − (∂11w)(∂22w) dxˆ. 12 ωˆ

2.4.4. Remarks on the (1,1,0)- and the (1,1,2)-Models Remark 2.24 (Boundary conditions for the Kirchhoﬀ-Love models). Our homogeneous Dirichlet conditions on Γˆ and (2.19a)-(2.19c) as well as (2.20a),

56 2.4. The Clamped Plate

(2.20b) give us ∇w = 0 a.e. on γˆ whereas (2.19c), (2.20c) imply W = w = 0 a.e. on γˆ. Imposing w = 0 a.e. on γˆ automatically gives us

∇w · t = 0 a.e. on γ.ˆ

It therefore suffices to have ∂ W = w = w = 0 a.e. on γ.ˆ ∂n This fact will be paramount for our discussion of different boundary conditions for plates in Section 2.5.1 and will also turn out to be too restrictive as one wants to discuss more general boundary conditions in Section 2.6.5. Furthermore from ∇w = 0 a.e. on γˆ it follows by consecutive application of integration by parts, that Z 2 (∂12w) − (∂11w)(∂22w) dxˆ = ωˆ Z Z = (∂2w)(∂12w)n1dsxˆ − (∂2w)(∂112w)dxˆ γˆ ωˆ Z − (∂11w)(∂22w)dxˆ ωˆ Z Z = (∂2w)(∂12w)n1dsxˆ − (∂11w)(∂2w)n2dsxˆ γˆ γˆ Z Z + (∂11w)(∂22w)dxˆ − (∂11w)(∂22w)dxˆ = 0, ωˆ ωˆ therefore the corresponding term can be eliminated in the representations of (1,1,0) (1,1,2) JK ,JK above. Remark 2.25 (The (1,1,0)-Kirchhoff-Love plate model). As we have a look −3 (1,1,0) at the scaled energy JK (w) one immediately notices a relevant shortcoming of this simplest Kirchhoff-Love model: For the incompressible limit 1−ν 2 ν → 1/2 the coefficient 1−2ν in front of the quadratic form k∆wk0,ωˆ becomes arbitrarily large, forcing this term to become very small. Since we discuss the case of homogeneous boundary conditions ∂ w = w = 0 a.e. on γˆ ∂n

57 2. Models of Linearized Plate Theory we end up with a very small solution w on ωˆ which by (2.19a)-(2.19c) means that we only have a very small displacement in e3-direction. This observation concurs with the plane strain state (as outlined in Section A.1.2) and stands in contrast to the fact, that we have imposed non-zero loads instead ˆ ˆ of homogeneous Dirichlet conditions on Γ+ ∪ Γ− [3, 4]. Remark 2.26 (The (1,1,0)-Reissner-Mindlin plate model). (a) Similar to the corresponding Kirchhoff-Love plate model we can observe ν 2 the coefficient 1−2ν exploding, forcing kdiv θk0,ωˆ to vanish as ν → 1/2. This time one cannot immediately infer obvious consequences for the solution, however, scrutinising the details of finding an optimal shear coefficient [6] as well as an extensive analysis of the boundary layer phe- nomenon for Reissner-Mindlin plates [3, 4] reveals far-reaching consequences for the approximation, when it comes to a discussion on the re- liability of statements deduced from the (1,1,0)-Reissner-Mindlin plate model, such as the analysis of boundary layers or the magnitude of the (1,1,0) displacement uR compared to u . (b) We give the usual strong formulation of the (1,1,0)-Reissner-Mindlin plate bending model [3–5, 7, 8, 34]. Problem 2.27 (Strong formulation of the (1,1,0)-Reissner-Mindlin plate model). Find θ ∈ C2(ˆω) ∩ C0(cl{ωˆ}), w ∈ C2(ˆω) ∩ C0(cl{ωˆ}) such that 2 div θ + θ − ∇w =0 in ωˆ (2.24) 12µ C D 2 −div θ + ∆w = p in ωˆ (2.25) µ total θ =0 on γˆ (2.26) w =0 on γˆ (2.27)

(1,1,0) (c) Observe, that JR (θ, w) can be written as 1 µ J (1,1,0)(θ, w) = 3( θ, θ) + k∇w−θk2 −3(p , w) , R 24 D C D 0,ωˆ 2 0,ωˆ total 0,ωˆ which leads to the following weak formulation of the (1,1,0)-Reissner- Mindlin plate bending model.

58 2.4. The Clamped Plate

Problem 2.28. Find (θ, w) ∈ V × W such that

(1,1,0) (1,1,0) (1,1,0) aR (θ, φ) + bR (φ, w) = hFR , φi for ∀φ ∈ V (1,1,0) (1,1,0) (1,1,0) bR (θ, v) + cR (w, v) = hGR , vi for ∀v ∈ W , where we have 2 a(1,1,0)(θ, φ) := ( θ, φ) + (θ, φ) R 12µ D C D 0,ωˆ 0,ωˆ (1,1,0) bR (φ, v) := − (φ, ∇v)0,ωˆ (1,1,0) cR (w, v) :=(∇w, ∇v)0,ωˆ (1,1,0) hFR , vi :=0 2 hG(1,1,0), wi := (p , w) . R µ total 0,ωˆ For this problem, we will discuss matching Hilbert spaces V , W in Section 2.6.5. Remark 2.29 (Alternative formulation of the energy functionals for the Kirchhoﬀ-Love models). Depending on one’s intentions or preferences, it (1,1,0) also makes clear sense to consider a diﬀerent representation of JK and (1,1,2) JK , 1 µ µ J (1,1,2)(w, W ) = 3( ∇w, ∇w) + 5k∇W k2 + 3kW k2 K 24 D C D 0,ωˆ 160 0,ωˆ 3 0,ωˆ 1 νµ + 3 kW k2 − (W, div ∇w) − hF , u(1,1,2)i 3 1 − 2ν 0,ωˆ 0,ωˆ K 1 J (1,1,0)(w) = 3( ∇w, ∇w) − hF , u(1,1,0)i. K 24 D C D 0,Ωˆ K The latter will be thoroughly utilized to derive a new mixed formulation as proposed by [33, 40, 49].

2.4.5. The Perturbed (1,1,2)-Kirchhoff-Love Plate Model For this section, we consider the following modification of the (1,1,2)-Kirchhoff- (1,1,2) Love model (Problem 2.21): Instead of minimizing JK (v, V ) with respect

59 2. Models of Linearized Plate Theory to v and V simultaneously, one could expect a decent approximation by performing the minimization of v and V sequentially. This is achieved by introducing a singularly perturbed problem, which ﬁnally will not give us (1,1,2) a minimum JK -energy solution for the displacement, but a displacement with (slightly) more energy. Let w be the solution of the problem: Find 2 w ∈ H0 (ˆω) such that

J˜(1,1,2)(w) = min J˜(1,1,2)(v), (2.28) K 2 K v∈H0 (ˆω) where

µ 1 −3J˜(1,1,2)(w) := k∆wk2 − (p , w) . (2.29) K 12 1 − ν 0,ωˆ total 0,ωˆ

1 We determine W ∈ H0 (ˆω) as the solution of the minimization problem 1 − ν ν 3 min k2V − ∆wk2 + 2k∇V k2 , 1 0,ωˆ 0,ωˆ V ∈H0 (ˆω) 1 − 2ν 1 − ν 40 which can equivalently be given as

80 1 − ν ν min kV − ∆wk2 + 2k∇V k2 . 1 0,ωˆ 0,ωˆ V ∈H0 (ˆω) 3 1 − 2ν 2(1 − ν)

˜(1,1,2) 1 Remark 2.30. Observe that for JK we end up with the coeﬃcient 1−ν in 2 front of the quadratic form k∆wk0,ωˆ , which remains bounded for admissible ν ∈ [0, 1/2), as well as for the incompressible limit case ν → 1/2. This observation concurs with the plain stress state (as outlined in Section A.1.1).

We will see that this error can be controlled (Lemma 2.32) and we will have convergence even for the following perturbed problem (Theorem 2.38 and Section 2.7)

Problem 2.31 (Perturbed (1,1,2)-Kirchhoﬀ-Love plate bending model). Let w be a solution of µ ∆2w =p in ωˆ (2.30) 6(1 − ν) total

60 2.4. The Clamped Plate

∂ w = w =0 on γ,ˆ (2.31) ∂n

−3 ˜(1,1,2) which is equivalent to the minimization problem associated to JK (.) in 1 (2.29). Then ﬁnd W ∈ H0 (ˆω) solving the variational problem

min αkV − φk2 + 2k∇V k2 (2.32) 1 0,ωˆ 0,ωˆ V ∈H0 (ˆω) ν 80 1 − ν φ := ∆w ∈ L2(ˆω) α := . (2.33) 2(1 − ν) 3 1 − 2ν

(1,1,2) 2 > We then set u˜ K := (−xˆ3∇w, w +x ˆ3W ) . Our goal is an asymptotic bound for the error between the solution of the (1,1,2) (1,1,2)-Kirchhoﬀ-Love plate bending model uK and the solution of the fully three-dimensional model u. This can be achieved by estimating the error of the perturbed problem above, i.e. the solution of (2.32) with respect to [2, Theorem 5.4]. Such an estimate strongly depends on the regularity of the solution of (2.30), (2.31). In the following we will assume ωˆ to be −1 convex or to be smoothly bounded and ptotal ∈ H (ˆω) [13, Remark 3.2] and consequently one has w ∈ H3(ˆω). In this framework we then have φ = c∆w ∈ H1(ˆω).The following lemma covers the situation for these assumptions.

Lemma 2.32. Let ωˆ be either convex or smoothly bounded and φ ∈ H1(ˆω). Let W be the unique solution of

min αkV − φk2 + 2k∇V k2 , ∈ (0, 1]. 1 0,ωˆ 0,ωˆ V ∈H0 (ˆω)

Then W satisﬁes the a-priori estimate

2 2 2 2 αkW − φk0,ωˆ + k∇W k0,ωˆ ≤ c(ˆω)kφk1,ωˆ , (2.34) where c(ˆω) > 0, c(ˆω) 6= c(ˆω, ).

Proof. [13, Lemma 3.1]

Remark 2.33. This theorem can be proven for much more general situations, e.g. reentrant corners [13, Lemma 3.1].

61 2. Models of Linearized Plate Theory

Remark 2.34 (The change from the (1,1,0)- to the (1,1,2)-Kirchhoﬀ-Love model). Let us consider (1,1,2) (1,1,2) 1 − ν ν ΣK = σK,33 =2µ xˆ3 2W − ∆w 33 1 − 2ν 1 − ν (1,1,0) (1,1,0) ν ΣK = σK,33 = − 2µ xˆ3∆w. 33 1 − 2ν 2 2 2 Since for any a1, a2 ∈ R one has (a1 + a2) ≤ 2(a1 + a2) we obtain from (2.34) for some generic c > 0 ν k2W − ∆wk + k∇W k ≤c1/2kφk . (2.35) 1 − ν 0,ωˆ 0,ωˆ 1,ωˆ

2 Due to φ = c1∆w and ∆φ = c1∆ w = c2ptotal for some c1, c2 > 0, one can apply (1.14) to kφk1,ωˆ , i.e. ν k2W − ∆wk + k∇W k ≤c1/2kp k . (2.36) 1 − ν 0,ωˆ 0,ωˆ total −1,ωˆ This gives us

(1,1,2) 1/2 (1,1,0) kσK,33 k0,Ωˆ = O( )kσK,33 k0,Ωˆ . For a thorough discussion on the comparison of stress tensors and energies of the (1,1,0)- to the (1,1,2)-Kirchhoﬀ-Love model by similar means as given in this remark, one refers to [13]. There, the following assertion is proven, 1 − 2ν J (1,1,2) ≥ J (1,1,0) 1 + O(1/2) . (2.37) K (1 − ν)2 K

2.4.6. Justiﬁcation of the (1,1,2)-Kirchhoﬀ-Love Plate Model by the Two-Energies Principle The internally stored energy of an elastic plate can be determined through stresses or strains via the elasticity tensor C: 2 2 −1 k k := ( , ) kΣk −1 := (Σ, Σ) . E C E CE 0,Ωˆ C C 0,Ωˆ The reader shall be reminded that the solutions of the fully three-dimensional problem will be denoted as

u , E = E(u ) = Du , Σ = Σ(u ) = CDu .

62 2.4. The Clamped Plate

For the reader’s convenience, we state Theorem 1.42 for the three-dimensional elastic plate, i.e. Ωˆ = Ωˆ .

Theorem 2.35 (Two-Energies Principle for Plates). Let f ∈ L2(Ωˆ ), g ∈ −1/2 ˆ −1/2 ˆ 1 ˆ ˆ H (Γ+) + H (Γ−), v ∈ {χ ∈ H (Ω ): χ = 0 a.e. on Γ } and let ˆ Σeq ∈ H(div, Ω ) satisfy

ˆ − div Σeq =f in Ω (2.38) ˆ ˆ Σeqn =g on Γ+ ∪ Γ−. (2.39)

Then we have

2 2 2 k (v) − (u )k +kΣ − Σ k −1 = kΣ − (v)k −1 . (2.40) E E C eq C eq CE C | {z } 2 =kv−u ka

As a direct consequence of the theorem above one obtains the following corollary.

Corollary 2.36. Let the assumptions from the theorem above prevail. Let additionally v, w ∈ {χ ∈ H1(Ωˆ ): χ = 0 a.e. on Γˆ},J (v) ≤ J (w). Then we have

2 2 2 k (v ) − (u )k + kΣ − Σ k −1 ≤ kΣ − (w )k −1 . E E C eq C eq CE C

In the following we will derive a statement on the error between the solution of the (1,1,2)-Kirchhoﬀ-Love plate model and the fully three-dimensional plate. This derivation is decomposed into two separate discussions on the “zero traction problem with volume forces” and the “pure traction problem without volume forces”, which – due to the linearity of our linearized elasticity problem – can then be combined to satisfy the general problem.

The Zero Traction Problem with Volume Forces Let us assume for this subsection that in (2.17) we have homogeneous Neu- mann data g ≡ 0 i.e. the case of body forces and zero traction. We now proceed as has already been proposed by [36] more than half a century ago,

63 2. Models of Linearized Plate Theory

who derived a suitable equilibrated strain tensor Σeq out of a solution w of (2.30), (2.31).

 2 3 2  12ˆx3M11 12ˆx3M12 −(6ˆx3 − 2 )d1 2 3 2 Σeq :=  12ˆx3M12 12ˆx3M22 −(6ˆx3 − 2 )d2  (2.41) 2 3 2 2 3 2 3 1 2 −(6ˆx3 − 2 )d1 −(6ˆx3 − 2 )d2 −(2ˆx3 − 2 xˆ3 )p with

1 µ ν M :=(M )2 := − ∇2w = − ∇2w + ∆wI ∈ 2 ij i,j=1 12C 6 1 − ν S µ ν = − (∂ w + ∆wδ )2 (2.42) 6 ij 1 − ν ij i,j=1 and µ 1 d :=(d )2 := div M = − ∇ ∆w i i=1 6 1 − ν µ 1 = − (∂ ∆w)2 . (2.43) 6 1 − ν i i=1 Let w constitute a solution of (2.30), (2.31), then we get µ 1 div d = − ∆2w = −p 6 1 − ν  2    12ˆx3(∂1M11 + ∂2M12) − 12ˆx3d1 0 2 div Σeq =  12ˆx3(∂1M12 + ∂2M22) − 12ˆx3d2  =  0  = −f 2 3 2 2 1 2 2 −(6ˆx3 − 2 )div d − (6ˆx3 − 2 )p − p  1 2 3 2  −(6 4 − 2 )d1 1 2 3 2 ˆ ˆ Σeqn =  −(6 4 − 2 )d2  = 0 on Γ+ ∪ Γ−. 1 3 1 1 2 −(2 8 − 2 2 )p Hence this choice of Σeq indeed satisﬁes (2.38), (2.39). The insertion of Σeq into (2.40) leads us to

(1,1,2) Σeq − CE(uK ) =  2 3 2 2  0 0 (6ˆx3 − 2 )∂1∆w − 6(1 − ν)ˆx3∂1W µ 1 2 3 2 2 =  0 0 (6ˆx3 − 2 )∂2∆w − 6(1 − ν)ˆx3∂2W  6 1 − ν 2 1 2 2 sym. sym. −xˆ3(2ˆx3 − 2 )∆ w

64 2.4. The Clamped Plate

ν 0 0  ν ν − 2µ xˆ (2W − ∆w) 0 ν 0 . (2.44) 1 − 2ν 3 1 − ν   0 0 1 − ν

Since the order of the lowest xˆ3-term in Σeq is 1 it immediately follows that kΣ k2 = O(3). By virtue of (2.36) and eq 0,Ωˆ

Z /2 2 3 2 2 5 (6ˆx3 − ) dxˆ3 = O( ) −/2 2 we obtain from (2.44)

(1,1,2) 1/2 kΣeq − CE(uK )k0,Ωˆ = O( )kΣeqk0,Ωˆ .

To derive a statement with respect to k.kC−1 one comes up with the following lemma.

2 ˆ Lemma 2.37. On L (Ω )S, k.kC−1 is equivalent to k.k0,Ωˆ i.e.

2 2 3ν 2 2 2µkSk −1 ≤ kSk ≤ 2µ(1 + )kSk −1 for ∀S ∈ L (Ωˆ ) . C 0,Ωˆ 1 − 2ν C S

Proof. Directly follows from the fact, that λmin(C) = 2µ and λmax(C) = 3ν 2µ + 3λ = 2µ(1 + 1−2ν ). Through this lemma we get

(1,1,2) 1/2 kΣeq − CE(uK )kC−1 = O( )kΣeqkC−1 . Finally, an application of Theorem 2.35 (two-energies principle) immediately gives us

(1,1,2) 2 2 2 k (u ) − (u )k + kΣ − Σ k −1 = ()kΣ k −1 . E K E C eq C O eq C Due to the presence of the second term within this expression we obtain

(1,1,2) 2 2 2 k (u ) − (u )k + kΣ − Σ k −1 = ()kΣ k −1 . E K E C eq C O C Hence the model error becomes small for thin linearly elastic plates. Let us sum up this result in the following theorem.

65 2. Models of Linearized Plate Theory

Theorem 2.38. Let ωˆ be either convex or smoothly bounded. Then the model error of the (1,1,2)-Kirchhoﬀ-Love plate model becomes small for thin linearly elastic plates, i.e.

(1,1,2) 1/2 kE(uK ) − E(u )kC + kΣeq − Σ kC−1 = O( )kΣ kC−1 .

Note that by the use of Lemma 2.32 we implicitly assumed a certain regularity of the computational domain ωˆ. One can derive an extended statement similar to Theorem 2.38 if one utilizes a more general version of Lemma 2.32 (e.g. for reentrant corners), as presented in [13].

The Pure Traction Problem without Volume Forces Let us assume for this subsection that from (2.17) we have zero volume forces f ≡ 0 and one sided traction q− ≡ 0, i.e. the case of no body forces and pure ˆ traction only on Γ+. The resulting calculation only require little adaptation: For the equilibrate strain tensor we keep the ansatz from the previous subsection and replace 3 1 (Σ ) := σ := (−2ˆx3 + xˆ 3 + 3)q+. eq 33 eq,33 3 2 3 2 Similar calculations then give us div d = −q+

 0  div Σeq =  0  = 0 2 3 2 2 3 2 + −(6ˆx3 − 2 )div d + (−6ˆx3 + 2 )q  0  Σeqn =  0  1 3 3 1 3 1 3 + (−2 8 + 2 2 + 2 )q  0    0  on Γˆ =   + 3q+   ˆ 0 on Γ−,

hence this choice of Σeq indeed satisﬁes (2.38), (2.39). We can therefore redo the remaining steps of the previous subsection.

66 2.4. The Clamped Plate

Remark 2.39. At this point, one might question the necessity of assuming (2.17). To achieve a convergence result we had to ﬁnd a equilibrated solution which had to fulﬁl conditions with respect to f and g (Theorem 2.35). This was done merely by using a solution of (2.30), (2.31). The author believes, that the presented ansatz of [13, 36] cannot be easily extended to more general boundary conditions than the case of (2.17).

2.4.7. Justiﬁcation of the (1,1,2)-Reissner-Mindlin Plate Model by the Two-Energies Principle The homogeneous Dirichlet boundary condition on Γˆ carry over to the (1,1,2)-Reissner-Mindlin plate model by setting

w ≡ 0 on γ,ˆ W ≡ 0 on γ,ˆ θ ≡ 0 on γ.ˆ

For the (1,1,2)-Reissner-Mindlin plate model we are no longer bound to [H6] (Kirchhoff Hypothesis), consequently we are more “flexible” while minimizing the energy, i.e. one has for the final solution

(1,1,2) (1,1,2) JR ≤ JK .

From Proposition 2.16 (b), Corollary 2.36 and Theorem 2.38 we analogously to the (1,1,2)-Kirchhoﬀ-Love model get

(1,1,2) 2 (1,1,2) 2 kuR − u ka =kE(uR ) − E(u )kC (1,1,2) 2 (1,1,2) 2 ≤kE(uK ) − E(u )kC = kuK − u ka , which directly implies the following corollary.

Corollary 2.40. Let ωˆ be either convex or smoothly bounded. Then the model error of the (1,1,2)-Reissner-Mindlin plate model becomes small for thin linearly elastic plates, i.e.

(1,1,2) 1/2 kE(uR ) − E(u )kC + kΣeq − Σ kC−1 = O( )kΣ kC−1 . (2.45)

This ends our discussion of plate bending models for completely clamped plates.

67 2. Models of Linearized Plate Theory

2.5. The Simply Supported Plate

As in Section 2.4, we will only cover plate bending and purely transversal loads (2.17), i.e. f (xˆ) = 2 (0, 0, p(xˆ))> g(xˆ, ±/2) = 3 (0, 0, q(xˆ))> . (2.46) In this section we substitute the completely clamped plate boundary condition for a new simple support boundary condition, which we will specify in the following subsection. The main feature of this boundary condition is the fact, that it can be phrased in a “hard” and a “soft” version, where the latter can be incorporated into the Reissner-Mindlin, but not the Kirchhoff- Love plate bending model. In this regard we will introduce an analogon of the Hellinger-Reissner mixed problem for the Kirchhoff-Love model, which can integrate this boundary condition [8]. From this mixed problem we will derive statements on the error between the Kirchhoff-Love and the Reissner- Mindlin model to the fully three-dimensional solution, which – in contrast to the -asymptotic estimates of the completely clamped plate in the previous section – are given with respect to the k.ka -energy norm.

2.5.1. The Imposition of Simple Support Boundary Conditions In our discussion on imposing correct boundary conditions for the Kirchhoff- Love plate model in Remark 2.24 we observed, that homogeneous Dirichlet boundary conditions u = 0 a.e. on Γˆ (clamped plate) can equivalently be written as ˆ u3 = u · n = u · τ = 0 a.e. on Γ , which then carry over to the Kirchhoff-Love plate model by setting ∂ ∂ w = w = w = 0 a.e. on γ.ˆ ∂n ∂τ ∂ In this regard, ∂τ w = 0 a.e. on γˆ could be dropped, since this condition is already implied by w = 0 a.e. on γˆ. Observe that this is not possible for the Reissner-Mindlin plate model. In the following, we are going to introduce the so called simple support boundary condition, whose discussion will reveal a severe shortcoming of the Kirchhoff-Love plate models.

68 2.5. The Simply Supported Plate

Deﬁnition 2.41 (Boundary conditions of simple support). Replacing (2.2), (2.3) by ( n · ( u n) = 0 a.e. on Γˆ Σn =0 a.e. on Γˆ C D (2.47) τ · (C D u n) = 0 a.e. on Γˆ ˆ u3 =0 a.e. on Γ (2.48) will be designated as soft simple support boundary conditions. The alternative imposition of n · (C D u n ) =0 a.e. on Γˆ (2.49) u · τ =0 a.e. on Γˆ (2.50) ˆ u3 =0 a.e. on Γ (2.51) will be designated as hard simple support boundary conditions.

u3 = 0 u3 = 0

Figure 2.4.: Schematic of the reference conﬁguration of a simply supported plate.

Remark 2.42 (Motivation). Our definition of simple support attempts to approximate the following setup of three-dimensional elasticity [3, 7, 8]: A simply supported plate has supports under its lateral face (i.e. along γˆ × {−/2}), which allow rotation, but no deflection, see Figure 2.4. Remark 2.43 (Plate modelling and different boundary conditions). It is common sense, that the global structure of the solution of the plate problem and in particular its local behaviour at its boundaries strongly correlate with the kind of imposed boundary conditions. Furthermore it can be shown, that any admissible boundary condition for the three-dimensional problem can be exactly represented by a hierarchic model of sufficiently high order. For more information refer to [47, Sect. 2.2.].

69 2. Models of Linearized Plate Theory

The Pure Displacement Formulation To guarantee existence and uniqueness for this kind of boundary conditions ˆ 1 ˆ ˆ we no longer have V (Ω ) = {v ∈ H (Ω ): v = 0 a.e. on Γ0}, therefore Theorem 1.10 (b) will no longer provide us with the desired V (Ωˆ )-ellipticity. Hence, to acquire existence and uniqueness of our minimization of the energy J (.) according to (2.9) over a suitable subset V (Ωˆ ) ⊂ H1(Ωˆ ) of admissible displacement ﬁelds, one has to comprehend the structure of the kernel of E(.).

Lemma 2.44. Let Ωˆ ⊂ R3 be a domain, then v ∈ RM := {w ∈ H1(Ω)ˆ : E(w) = 0mat} = {w ∈ H1(Ω)ˆ : Σ(w) = 0mat},

> if and only if v(xˆ) = Axˆ + b, A ∈ M3, A = −A , b ∈ R3. One calls RM the set of rigid body motions.

Proof. [11, 15]

Remark 2.45 (Space of three-dimensional rigid body motions). We have by Lemma 2.44

E(v) = 0mat ⇐⇒ v(xˆ) = a + b × xˆ for some a, b ∈ R3. One deﬁnes the set of homogeneous solutions (the kernel) of (2.1)-(2.4), replacing (2.2), (2.3) with (2.47), (2.48) or (2.49)-(2.51), respectively.

ˆ ˆ N SSS(Ω ) :={w ∈ RM : w3 ≡ 0 a.e. on Γ } ˆ ˆ N HSS(Ω ) :={w ∈ RM : w3 ≡ 0, w · τ ≡ 0 a.e. on Γ }.

It can be shown, that [8]

ˆ ˆ > 3 N SSS(Ω ) = N HSS(Ω ) = {xˆ 7→ (a1xˆ1 + a3xˆ2, a2xˆ2 − a3xˆ1, 0) : a ∈ R }. Remark 2.46. In the previous section we imposed pure Dirichlet conditions ˆ ˆ ˆ only, i.e. we considered (2.1)-(2.4) with Γ = Γ0 (or |Γ0| > 0). The set of homogeneous solutions is then given by

ˆ ˆ N 0(Ω ) :={w ∈ RM : w ≡ 0 a.e. on Γ }

70 2.5. The Simply Supported Plate

={0}.

On the other hand, for the pure Neumann problem, i.e. considering (2.1)- ˆ ˆ (2.4) with Γ = Γ1, one ends up with ˆ N 1(Ω ) :=RM . To enforce existence and uniqueness of our variational equation (2.8), one restricts the set of admissible solutions satisfying the essential boundary conditions to the orthogonal complement of the kernel of (2.8). Deﬁnition 2.47 (Spaces of geometrically admissible displacements for simply supported plates). The set of admissible displacement ﬁelds within which a unique minimizer of J (.) can be expected are ˆ 1 ˆ ˆ V SSS(Ω ) :={v ∈ H (Ω ): v3 = 0 a.e. on Γ , (v , w )0,Ωˆ = 0 ˆ for ∀w ∈ N SSS(Ω )} ˆ 1 ˆ ˆ V HSS(Ω ) :={v ∈ H (Ω ): v3 = 0 a.e. on Γ , ˆ v · τ = 0 a.e. on Γ , (v , w )0,Ωˆ = 0 ˆ for ∀w ∈ N HSS(Ω )}. We can now discuss the simply supported three-dimensional model by replacing our original space V (Ωˆ ) in Theorem 2.5 by these newly introduced ˆ ˆ spaces V SSS(Ω ), V HSS(Ω ). As in the proof of Theorem 1.14 it is clear by standard trace inequalities, that hF ,.i,F ∈ H−1(Ωˆ ) is bounded [11, 47] 2 ˆ 2 ˆ ˆ for f ∈ L (Ω ), g ∈ L (Γ+ ∪ Γ−). Also the proof for the boundedness of ˆ a (., .) can be left unchanged. On the other hand, V SSS(Ω )-ellipticity can again be achieved by the application of Korn’s inequality Theorem 1.10 (b). ˆ Lemma 2.48. The bilinear form a (., .) is V SSS(Ω )-elliptic, i.e. there exists a constant µ1 > 0 such that 2 ˆ a (v , v ) ≥ µ1kv k1,Ωˆ for ∀v ∈ V SSS(Ω ). Proof. [8] ˆ ˆ ˆ Since V SSS(Ω ) ⊃ V HSS(Ω ) the desired V HSS(Ω )-ellipticity follows from Lemma 2.48. By these considerations above, one ends up with the following theorem on the well-posedness of (2.9) for simple support.

71 2. Models of Linearized Plate Theory

Theorem 2.49 (Existence and uniqueness of a weak solution for simply supported plates).

ˆ ˆ 1 ˆ (a) V SSS(Ω ), V HSS(Ω ) are closed, linear subspaces of H (Ω ) and hence are Hilbert spaces.

2 ˆ ˆ 2 ˆ (b) Assume that g ∈ L (Γ+ ∪ Γ−), f ∈ L (Ω ) satisfy the compatibility condition

ˆ ˆ hF , vi = 0 for ∀v ∈ N SSS(Ω )( for ∀v ∈ N HSS(Ω )).

ˆ ˆ Then there exists a unique weak solution u ∈ V SSS(Ω ) (u ∈ V HSS(Ω )) of (2.8).

Proof. [47, Thm. 2.1.]

Discussing the diﬀerences between our two types of simple support for small > 0, the following theorem exists.

Theorem 2.50 (On the diﬀerence between HSS and SSS in the three-dimensional context). Let uSSS, uHSS be the solutions of the plate bending problems with the thickness > 0 for the cases of soft and hard simple support cases, respectively. Then

kuSSS − uHSSka → 0 as → 0, kuSSSka i.e. the diﬀerence between the two kinds of simple support measured in the relative stress energy norm converges to 0 as → 0.

Proof. Follows by arguments used in [8, 36] and Section 2.4. [7].

It is necessary to underline, that the statement of this theorem holds only with respect to the energy norm, as we will see later in Theorem 2.72. The two simple support boundary conditions mainly diﬀer in their behaviour of their associated solutions in the neighbourhood of Γˆ, whose size depends on the relation between the smoothness of γˆ and the plate’s thickness > 0 (compare to subsequent Section 2.6.5, Theorem 2.72, Theorem 2.53).

72 2.5. The Simply Supported Plate

The Hellinger-Reissner Principle By the same reasoning of Theorem 1.17 and Theorem 1.33 we can also determine solutions of our simply supported plate by considering a mixed problem.

Theorem 2.51 (First mixed formulation of the Hellinger-Reissner-principle for linearized elasticity and soft simple support). Find

ˆ 2 ˆ ,I ,I (u , Σ ) ∈ V SSS(Ω ) × L (Ω )S =: W SSS × V such that

a,I(Σ, T) + b,I(T, u) = 0 for ∀T ∈ V ,I (2.52) ,I ,I ,I b (Σ , w) = hG , wi for ∀w ∈ W SSS (2.53) ˆ u3 = 0 a.e. on Γ, where we have

,I −1 ,I a (Σ, T) :=(C Σ, T)0,Ωˆ for ∀T ∈ V ,I ,I b (T, v) := − (T, Dv)0,Ωˆ for ∀v ∈ W SSS Z ,I ,I hG , wi := − (f , w)0,Ωˆ − g · wdsxˆ for ∀w ∈ W SSS, ˆ ˆ Γ+∪Γ− has an unique solution.

Theorem 2.52 (Second mixed formulation of the Hellinger-Reissner-principle for linearized elasticity and soft simple support). Find

2 ˆ ˆ ˆ ˆ (u , Σ ) ∈L (Ω ) × {T ∈ H(div, Ω )S : Tn = 0 a.e. on Γ+ ∪ Γ− Tn = 0 a.e. on Γˆ} ,II ,II =: W × V SSS such that

a,II(Σ, T) + b,II(T, u) = 0 for ∀T ∈ V ,II (2.54) b,II(Σ, w) = hG,II, wi for ∀w ∈ W ,II (2.55) ˆ ˆ Σ n = g a.e. on Γ− ∪ Γ+

73 2. Models of Linearized Plate Theory

Σn = 0 a.e. on Γˆ, where we have

,II −1 ,II a (Σ, T) :=(C Σ, T)0,Ωˆ for ∀T ∈ V SSS ,II ,II b (T, v) :=(div T, v)0,Ωˆ for ∀v ∈ W ,II ,II hG , wi := − (f, w)0,Ωˆ for ∀w ∈ W , has an unique solution. To achieve existence and uniqueness we hand these mixed problems over to Theorem 1.7 for which the required boundedness and ellipticity of the respective product space formulations is satisﬁed [8, Thm. A.2].

Consistency Result for Higher Order Plate Bending Models On basis of any higher order plate bending model, one can show the following consistency result for the imposition of both kinds of boundary conditions. Theorem 2.53 (Consistency result for higher order plate bending models). > m Let m = (m1, m2, m3) , m1 ≥ 1, m2 ≥ 1, m3 ≥ 2. Denoting by u the plate model based on (2.13), (2.14), then for both types of simple support we have

m ku − u ka → 0 as → 0. (2.56) ku ka Proof. [7, 8]

2.6. The (1,1,0)-Plate Bending Models

Assuming m3 ≥ 2 in Theorem 2.53 is essential. Taking m3 = 0, ν = 0 the assertion (2.56) still holds, but taking ν > 0, (2.56) does not hold any more. Consequently in the case of m3 = 0 one cannot expect our m-model of Problem 2.12 to deliver reliable results (except for ν = 0), when dealing with simple support boundary conditions. In this regard, the reader shall be reminded of the introduction of the perturbed (1,1,2)-Kirchhoﬀ-Love model in Problem 2.31, which determines w by virtue of a minimization problem asso- ˜(1,1,2) ciated to JK (.). For this perturbed (1,1,2)-Kirchhoﬀ-Love model we have

74 2.6. Lower Order Plate Bending Models

(1,1,2) shown consistency (Theorem 2.38). One can show that u˜ K would still be consistent to the three-dimensional solution, if one removes the contribution 2 (1,1,2) of xˆ3W in u˜ K as → 0, i.e. we have found a consistent (1,1,0)-Kirchhoff- ˜(1,1,2) Love plate bending model associated to a modified energy JK (.). This observations could motivate the replacement of J (.) in (2.13) with some modified energy: Find a class of symmetric, positive definite tensors B, which satisfy [8]

(1,1,0) ku − u k B a → 0 as → 0, (2.57) ku ka where one defines um, m ∈ 3 as the solution of the following problem. B N0 Problem 2.54 (The m-plate model). Find um ∈ V (m) such that B 1 J (um) = min J (v), where J (v) := a (v, v) − hF , vi (2.58) B v∈V (m) B B 2 B ˆ and aB(u, v) := (Dv, BDv)0,Ωˆ ∀u, v ∈ V (Ω ). The existence and uniqueness of um is guaranteed by the symmetry and B positive definiteness of B. The problem of finding B has already been thoroughly studied [7]. Let us introduce the following class of symmetric, positive definite tensors B,  E νE  1−ν2 1−ν2 0 0 0 0 νE E  1−ν2 1−ν2 0 0 0 0   E   0 0 1−ν2 0 0 0  B =  E  , κ > 0. (2.59)  0 0 0 1+ν 0 0   κE   0 0 0 0 1+ν 0  κE 0 0 0 0 0 1+ν Theorem 2.55 (Consistency result for low order plate bending models). Let m = (1, 1, 0)> and let u(1,1,0) be the minimizer according to (2.58), (2.14) B with B given by (2.59). Then for both types of simple support, we have

(1,1,0) ku − u k B a → 0 as → 0. (2.60) ku ka Proof. [7, 8]

75 2. Models of Linearized Plate Theory

Note that for any κ > 0 in (2.59) assertion (2.60) is satisfied. The coefficient κ in (2.59) does not impair the validity of (2.60), but affects the magnitude of the error u(1,1,0) − u, hence an in this sense optimal choice of B κ > 0 is of great importance for any application. There are various recommendations for choosing κ, of which a fixed choice of κ = 5/6 probably is the most widely spread value going back to [41]. For a thorough theoretical analysis and an in-depth survey of the recommendations in literature, we refer to [6]. The minimization problem (2.58), (2.14) for this class of elasticity tensors in (2.59) associated to u(1,1,0) will again be designated as the Reissner- B B (1,1,0) Mindlin plate bending model uR . Subsequently, using B of (2.59) in (2.58) and specializing (2.14) to the ansatz of the (1,1,0)-Kirchhoff-Love model of Definition 2.19 with w ∈ H2(ˆω) (1,1,0) (1,1,0) we obtain a solution uK . Later in Section 2.7 we will show, that uK (1,1,0) (in contrast to uR ) does not depend on a particular choice of κ > 0. Theorem 2.56 (Consistency result for the lowest order Kirchhoff-Love plate (1,1,0) bending model). Let uK be defined as prior to this theorem. Then we have

(1,1,0) kuK − u ka → 0 as → 0. (2.61) ku ka

Proof. [7, 8]

Remark 2.57 (Domains and boundary layers). The m-model solution has a boundary layer, which has been analysed for the Reissner-Mindlin plate for smooth γˆ, see [3, 4], as well as for some other models, see [1, 34]. How- ever, for piecewise smooth γˆ, a rigorous analysis of the boundary layer is not available so far. The solution emerging from the Kirchhoﬀ-Love model has no boundary layer. From (2.60), (2.61) one concludes that the Reissner-Mindlin solution converges to the Kirchhoﬀ-Love solution, with respect to k.ka . Ob- serve, that this conclusion can only be drawn for the energy. In this regard it can be shown, that the k.k0,Ωˆ -convergence does not occur at the boundary [3].

Remark 2.58. We have seen (Theorem 2.53), that the (1,1,2)-Kirchhoﬀ- Love model is the lowest order model to use C [2, 7, 47]. For this and

76 2.6. Lower Order Plate Bending Models higher order models one can therefore assert statements such as Galerkin’s orthogonality, i.e.

a(u − um, v) = 0 for ∀v ∈ V (m).

Observe that for u(1,1,0) this statement as well as Proposition 2.16 do not B hold.

2.6.1. The (1,1,0)-Reissner-Mindlin Plate Model We have already derived a mixed problem associated to the (1,1,0)-Reissner- (1,1,0) Mindlin plate energy JR (.) in Remark 2.26, without knowing of the necessity of introducing B from (2.59) to Problem 2.12 to reach consistency with the three-dimensional model: We will now do the required calculations for the corrected (1,1,0)-model.

(1,1,0) (1,1,0) (1,1,0) ΣR :=BDuR = Σ(uR ) =  E νE    1−ν2 1−ν2 0 0 0 0 −xˆ3∂1θ1 νE E  1−ν2 1−ν2 0 0 0 0   −xˆ3∂2θ2   E     0 0 1−ν2 0 0 0   0  =  E   1   0 0 0 1+ν 0 0  − 2 xˆ3(∂1θ2 + ∂2θ1)  κE   1 (∂ w − θ )   0 0 0 0 1+ν 0   2 2 2  κE 1 (∂ w − θ ) 0 0 0 0 0 1+ν 2 1 1  E  − 1−ν2 xˆ3(∂1θ1 + ν∂2θ2) sym. sym. 1 E E =  − 2 1+ν xˆ3(∂1θ2 + ∂2θ1) − 1−ν2 xˆ3(ν∂1θ1 + ∂2θ2) sym. . 1 Eκ 1 Eκ 2 1+ν (∂1w − θ1) 2 1+ν (∂2w − θ2) 0

∗ Eν Eν Let λ := 1−ν2 6= (1+ν)(1−2ν) = λ and let us deﬁne a reduced shear corrected elasticity tensor

E ν T := 2µT + λ∗(tr T)I = T + (tr T)I . B 1 + ν 1 − ν

Then by Theorem 2.5 we have the corresponding energy 1 J (1,1,0)(θ, w) := ( u(1,1,0), u(1,1,0)) − hF , u(1,1,0)i R 2 D R BD R 0,Ωˆ R

77 2. Models of Linearized Plate Theory

1 (1,1,0) (1,1,0) (1,1,0) = (E , Σ ) − hF , u i 2 R R 0,Ωˆ R  

1  E (1,1,0) (1,1,0) Eν (1,1,0) (1,1,0)  =  (E , E ) ˆ + (E , (trE ) I) ˆ  2 1 + ν R R 0,Ω 1 − ν2 R R 0,Ω  | {z } | {z } | {z } 2µ λ∗ =−xˆ3(∂1θ1+∂2θ2)  

1 1 Eκ 2  (1,1,0) +  k∇w − θk ˆ  − hF , u i 2 2 1 + ν 0,Ω  R | {z } =µκ 1 E (1,1,0) (1,1,0) ν (1,1,0) (1,1,0) = (E , E ) + (E , (trE )I) 2 1 + ν R R 0,Ωˆ 1 − ν R R 0,Ωˆ | {z } =(ˆx3D θ,xˆ3B D θ)0,Ωˆ 1 + µκk∇w − θk2 − hF , u(1,1,0)i. 2 0,Ωˆ R Hence by (2.22) we end up with 1 µκ J (1,1,0)(θ, w) = 3( θ, θ) + k∇w − θk2 − hF , u(1,1,0)i, R 2 · 12 D B D 0,ωˆ 2 0,ωˆ R which shares many similarities with the original non-corrected energy of the (1,1,0)-Reissner-Mindlin model using C in Remark 2.26. Let (1,1,0) 1 V R,SSS :=H (ˆω) (1,1,0) 1 V R,HSS :={φ ∈ H (ˆω) : φ · τ = 0 a.e. on γˆ} (1,1,0) 1 W R :=H0 (ˆω) Problem 2.59 (The (1,1,0)-Reissner-Mindlin plate and soft simple support). (1,1,0) (1,1,0) Find (θ, w) ∈ V R,SSS × W R such that

(1,1,0) (1,1,0) (1,1,0) (1,1,0) aR (θ, φ) + bR (φ, w) = hFR , φi for ∀φ ∈ V R,SSS (2.62) (1,1,0) (1,1,0) (1,1,0) (1,1,0) bR (θ, v) + cR (w, v) = hGR , vi for ∀v ∈ W R , (2.63) where we have 2 a(1,1,0)(θ, φ) := ( θ, φ) + κ(θ, φ) R 12µ D BD 0,ωˆ 0,ωˆ

78 2.6. Lower Order Plate Bending Models

(1,1,0) bR (φ, v) := − κ(φ, ∇v)0,ωˆ (1,1,0) cR (w, v) :=κ(∇w, ∇v)0,ωˆ (1,1,0) hFR , vi :=0 2 hG(1,1,0), wi := (p , w) . R µ total 0,ωˆ

Remark 2.60. Note, that B = Cstress in Section A.1.1.

2.6.2. An Analogon of the Hellinger-Reissner Principle for the Reissner-Mindlin Plate Model One can now proceed similarly to Section 1.2.2 by introducing auxiliary vari- (1,1,0) ables for the second Piola Kirchhoﬀ stress tensor ΣR (constitutive law) and the plate bending quantity θ − ∇w. However, to achieve a convenient notational consistency between this thesis and other papers, these quantities will be designated by M and γ, respectively [8].

M =B D θ γ =θ − ∇w (1,1,0) 2 2 V R :=L (ˆω)S × L (ˆω) (1,1,0) (1,1,0) (1,1,0) (1,1,0) (1,1,0) (1,1,0) W R,SSS :=V R,SSS × W R W R,HSS :=V R,HSS × W R Theorem 2.61 (Mixed formulation of the (1,1,0)-Reissner-Mindlin plate (1,1,0) (1,1,0) and soft simple support). Find (M, γ) ∈ V R and (θ, w) ∈ W R,SSS such that

−1 2 (M, B K)0,ωˆ − (Dθ, K)0,ωˆ =0 for ∀K ∈ L (ˆω)S (2.64) 2 (γ, ζ)0,ωˆ − (θ − ∇w, ζ)0,ωˆ =0 for ∀ζ ∈ L (ˆω) (2.65) 2 − (M, φ) − (γ, φ) =0 for ∀φ ∈ V (1,1,0) (2.66) 12µ D 0,ωˆ 0,ωˆ R,SSS 2 (γ, ∇v) = (p , v) for ∀v ∈ W (1,1,0) (2.67) 0,ωˆ µ total 0,ωˆ R has an unique solution. The proof of existence and uniqueness is obtained via the related weak formulation on the product space, as discussed in Deﬁnition 1.20, for which

79 2. Models of Linearized Plate Theory one can show the conditions required for the application of Theorem 1.7, see [8, Thm. A.2]. Finally, we can give the Reissner-Mindlin analogue to the two-energies principle (Theorem 1.40). Theorem 2.62 (Two-Energies Principle for (1,1,0)-Reissner-Mindlin plates). (1,1,0) (1,1,0) (1,1,0) Let (M eq, γeq) ∈ V R satisfy (2.66)-(2.67) and let (M R , γR ) ∈ (1,1,0) (1,1,0) (1,1,0) (1,1,0) V R , (θR , wR ) ∈ W R,SSS constitute a solution of (2.64)-(2.67). (1,1,0) Then the following identity holds for any (θ, w) ∈ W R,SSS , that 1 a(1,1,0)(θ(1,1,0) − θ, w(1,1,0) − w; θ(1,1,0) − θ, w(1,1,0) − w) 2 R R R R R 1 + a(1,1,0)(M (1,1,0) − M , γ(1,1,0) − γ; M (1,1,0) − M , γ(1,1,0) − γ) 2 R R eq R R eq R (1,1,0) (1,1,0) = JR (w, θ) + JR (M eq, γeq). Proof. The proof works completely anologously to Theorem 1.40.

2.6.3. The Kirchhoﬀ-Love Plate Model (1,1,0) The computation of JK (.) with respect to B can readily be done by setting θ = ∇w in our calculations related to the corrected (1,1,0)-Reissner-Mindlin model, 1 J (1,1,0)(w) := ( u(1,1,0), u(1,1,0)) − hF , u(1,1,0)i K 2 D K BD K 0,Ωˆ K 1 = 3(∇2w, ∇2w) − 3(p , w) . 2 · 12 B 0,ωˆ total 0,ωˆ Note that κ no longer appears in this energy. As already mentioned, the implementation of simple support boundary conditions for the Kirchhoﬀ- Love plate model is achieved by naturally prescribing the Hilbert space

(1,1,0) 2 1 2 2 V K := {v ∈ H (ˆω): v = 0 a.e. on γˆ} = H0 (ˆω) ∩ H (ˆω) 6= H0 (ˆω).

Remark 2.63. Assuming that ωˆ ⊂ R2 is a polygonally bounded domain, the boundary condition of simple support for the Kirchhoﬀ-Love model in its classical formulation can be imposed by [7]

w = ∆w = 0 on γ.ˆ (2.68)

80 2.6. Lower Order Plate Bending Models

We can now give the canonical weak Kirchhoff-Love model formulation for simple support. Observe, that as expected, we do not differentiate between hard and soft simple support, hence it is not necessary to adopt our notation. Problem 2.64 (Weak formulation of the (1,1,0)-Kirchhoff-Love plate and (1,1,0) simple support). Find w ∈ V K such that

(1,1,0) (1,1,0) (1,1,0) aK (w, v) = hFK , φi for ∀v ∈ V K , (2.69) where we have 1 a(1,1,0)(w, v) := ( ∇w, ∇v) K 12µ D BD 0,ωˆ 2 hF (1,1,0), vi := (p , v) . K µ total 0,ωˆ Remark 2.65 (Numerical implications for soft simple support). According to [7], one can heuristically derive from the Kirchhoﬀ-Love model and hard simple support numerical conclusions for soft simple support situation of an elastic plate. Remark 2.66. • The reader shall be reminded that within the proof of (1,1,0) (1,1,0) Theorem 2.64, boundedness of hFK ,.i on V K is formally given ∗ (1,1,0) (1,1,0) for FK ∈ V K .

• If w constitutes a solution of the (1,1,0)-Kirchhoﬀ-Love model, then (1,1,0) θ := ∇w minimizes JR (.) on

(1,1,0) (1,1,0) {(v, φ) ∈ V K × V R,SSS : φ = ∇v}.

2.6.4. An Analogon of the Hellinger-Reissner Principle for the Kirchhoﬀ-Love Plate Model We now intend to explore similarities between the Reissner-Mindlin and the Kirchhoﬀ-Love plate model: Similar to the Reissner-Mindlin model in the previous subsection, let us introduce the following auxiliary variables [8].

M =BDθ θ =∇w

81 2. Models of Linearized Plate Theory

∗ (1,1,0) 2 (1,1,0) (1,1,0) (1,1,0) (1,1,0) V K,SSS :=L (ˆω)S × V R,SSS W K,SSS :=V R,SSS × V K ∗ (1,1,0) 2 (1,1,0) (1,1,0) (1,1,0) (1,1,0) V K,HSS :=L (ˆω)S × V R,HSS W K,HSS :=V R,HSS × V K

Theorem 2.67 (Mixed formulation of the (1,1,0)-Kirchhoﬀ-Love plate and (1,1,0) (1,1,0) soft simple support). Find (M, γ) ∈ V K,SSS and (θ, w) ∈ W K,SSS such that

−1 2 (M, B K)0,ωˆ − (Dθ, K)0,ωˆ =0 for ∀K ∈ L (ˆω)S (2.70) ∗ (1,1,0) hθ − ∇w, ζi0,ωˆ =0 for ∀ζ ∈ V R,SSS (2.71) 2 (M, φ) + hγ, φi =0 for ∀φ ∈ V (1,1,0) (2.72) 12µ D 0,ωˆ 0,ωˆ R,SSS 2 hγ, ∇vi = (p , v) for ∀v ∈ V (1,1,0) (2.73) 0,ωˆ µ total 0,ωˆ K has an unique solution. Theorem 2.68 (The equivalence of the (1,1,0)-Kirchhoff-Love plate to the mixed problem). The mixed problem as stated in Theorem 2.67 is well posed and the unique solution (M, γ, θ, w) satisfies θ = ∇w, M = BDθ and γ is defined by (2.72).

(1,1,0) Proof. Assuming that w = wK . Then M := BDw, θ := ∇w and γ (1,1,0) defined through (2.72) satisfy (2.70)-(2.73) per definition, since wK min- (1,1,0) (1,1,0) imizes JK (.) in V K . Well-posedness follows from Theorem 2.67. [8, Lemma 3.5] The proof of existence and uniqueness is obtained via the related weak formulation on the product space, as discussed in Definition 1.20, for which we have to show the conditions required for the application of Theorem 1.7, see [8, Thm. A.2]. One can show the following lemma on the connection between the mixed formulation of the Kirchhoff-Love to the Reissner-Mindlin plate model. Lemma 2.69. Let ωˆ be either a convex polygon or a smooth domain, and (1,1,0) (1,1,0) let (M, γ, θ, w) ∈ V K,HSS × W K,HSS be a solution of (2.70)-(2.73) for given −1 2 ptotal ∈ H (ˆω). Then one has γ = −∇(∆w) ∈ L (ˆω) and (M, γ, θ, w) is a solution of (2.64)-(2.67) with = 0 in (2.66).

82 2.6. Lower Order Plate Bending Models

three-dimensional HR formulation → plate HR formulation (1.25), (1.26) / (1.41), (1.42) (2.64)-(2.67) ↑ ↑ three-dimensional displacement → plate bending formulation formulation (1.9) (2.62), (2.63)

Table 2.2.: By choosing “matching” semi-discrete ansatz functions for the stress and the displacement within the Hellinger-Reissner principle one comes up with a model, equivalent to an analogue semi- discretization of the pure displacement ansatz.

Proof. [8, Lemma 3.7] Remark 2.70 (Alternative derivation of the Hellinger-Reissner-like formulations of the (1,1,0)-Reissner-Mindlin and Kirchhoﬀ-Love plates). Aside from introducing plate models as semi-discretization of Problem 2.3 according to Problem 2.12, one could have derived an even greater variety of plate models (including Theorem 2.61 and Theorem 2.67) by considering an autonomous semi-discrete ansatz of the plate’s stresses in the context of the Hellinger- I I Reissner principle, i.e. one replaces V and W with xˆ3-semi-discrete sub- I I I I spaces V (m1) and W (m2) satisfying DW (m2) ⊂ V (m1) to achieve I I existence and uniqueness. If one additionally has CDW (m2) ⊂ V (m1) I m1 m2 I m2 we end up with V (m1) 3 Σ = CDu ∈ CDW (m2), hence u is I determined as before as the minimizer in W (m2) of the potential energy of the pure displacement ansatz in Section 2. Thus, one calls such models minimum energy models or energy projection models. Similarly, one can derive minimum complementary energy models from the second mixed version of the Hellinger-Reissner principle. A complete analysis can be found in [1, 2]. We will soon see in Section 2.7, that for some theoretical results this more general ansatz is required for which one can deﬁne polynomial dependencies on xˆ3 of the entries of the plate model solution individually, which obviously is not possible for the method we propose in this thesis. See Table 2.2.

Remark 2.71 (Observations). Comparing the Kirchhoﬀ-Love to the Reissner- Mindlin ansatz spaces, one notices two things (see Table 2.3): (1,1,0) (1,1,0) 1. By replacing V K,SSS with V K,HSS, there actually is a diﬀerence in the imposition of hard and soft simple support boundary conditions even

83 2. Models of Linearized Plate Theory Smayo h nazsae nue yhr n otsml upr onayconditions boundary support simple soft and hard by induced spaces ansatz the of Summary 2.3.: Table esnrMnlnHSS: Reissner-Mindlin HSS: Kirchhoff-Love SSS: Reissner-Mindlin SSS: Kirchhoff-Love o h ie omlto fte(,,)Krho-oeadte(,,)Rise-idi plate (1,1,0)-Reissner-Mindlin the and (1,1,0)-Kirchhoff-Love model. the bending of formulation mixed the for L L L L 2 2 2 2 M (ˆ (ˆ (ˆ (ˆ ω ω ω ω ) ) ) ) S S S S { w ∈ H ( 1 H (ˆ L L ω 1 ) 2 2 γ (ˆ (ˆ (ˆ : ω ω ω w ) ) ) ) ∗ · τ 0 = } ∗ { { w w ∈ ∈ H H 1 1 H H (ˆ (ˆ ω ω 1 1 ) ) θ (ˆ (ˆ : : ω ω w w ) ) · · τ τ 0 = 0 = } } H H 2 2 (ˆ (ˆ ω ω H H ) ) 0 0 1 1 w ∩ ∩ (ˆ (ˆ ω ω H H ) ) 0 0 1 1 (ˆ (ˆ ω ω ) )

84 2.6. Lower Order Plate Bending Models

Figure 2.5.: Illustration of the disc ωˆ[0] and the ﬁrst four members of the [n] sequence (ˆω )n∈N of inscribed regular (n + 3)-polygons.

for the Kirchhoff-Love model in its weak mixed formulation. In this regard, w, θ and M are independent of a particular choice of a simple (1,1,0) support boundary condition. However, finding γ on both, V K,SSS and (1,1,0) V K,HSS does [8, Lemma 3.7., Rem. 3.2]. 2. From the point of ansatz spaces the Reissner-Mindlin and the Kirchhoff- Love model only differ in regularity of w and γ. This means that for the actual computation we would have a simpler algebraic structure for the Kirchhoff-Love model, but at the expense of an H2(ˆω)-function representation, which one usually strives to avoid.

2.6.5. Babuška’s Plate Paradox for Hard and Soft Simple Support We are now going to elaborate Babuška’s plate paradox as presented in [8, Section 4]. Hence, let

[0] 2 ωˆ := {xˆ ∈ R : |xˆ| < 1}. [n] Furthermore, let (ˆω )n∈N be the sequence of regular (n + 3)-polygons such that

[n] [n+1] [n+1] [0] cl{ωˆ } ⊂ ωˆ ⊂ cl{ωˆ } ⊂ ωˆ for ∀n ∈ N

85 2. Models of Linearized Plate Theory

[0] [n] ∀xˆ ∈ ωˆ ∃nxˆ ∈ N∀n ≥ nxˆ , n ∈ N : xˆ ∈ ωˆ , so – in a sense – we have “ωˆ[n] → ωˆ[0]” as n → ∞, see Figure 2.5. We then introduce a conforming notation for the deﬁnition of the related three- dimensional plate

ˆ [n] Ω[n] :=ω ˆ × (−/2, /2) for ∀n ∈ N0. To compare the different models, let us assume the imposition of the unit load, i.e. ptotal ≡ 1. Then, as already discussed in the previous subsection, for any fixed given plate thickness > 0 there exist unique solutions ,[n] (1,1,0),[n] (1,1,0),[n] (1,1,0),[n] u , (θR , wR ) and wK , n ∈ N0 corresponding, respectively, to the three-dimensional, the Reissner-Mindlin and the Kirchhoff-Love formulations of the plate bending problem with either hard or soft simple support boundary conditions imposed. One would expect, that the limit of solutions on the polygonal domains ωˆ[n], n ∈ N agrees with the solution on the disc ωˆ[0]. The following theorems show, that this assertion does not hold for the hard simply supported plate.

Theorem 2.72 (Babuška’s paradox for the three-dimensional plate model). There exists an 0 > 0 such that for any ∈ (0, 0]

,[n] ,[∞] ,[0] uSSS →uSSS = uSSSfor n → ∞ ,[n] ,[∞] ,[0] uHSS →uHSS 6= uHSSfor n → ∞,

,[n] ,[n] where uSSS, uHSS denote the three-dimensional solution for the soft and hard ˆ simple support on Ω[n], n ∈ N0, respectively. The convergence is to be under- ˆ stood in k.ka on Ω[n]. Proof. [7, 8]

Theorem 2.73 (Babuška’s paradox for the m-plate model). Let m = > (m1, m2, m3) , m1 ≥ 1, m2 ≥ 1, m3 ≥ 2 using C or m3 = 0 using B as in (2.59). Then there exists an 0 > 0 such that for any ∈ (0, 0]

m,[n] m,[∞] m,[0] uSSS →uSSS = uSSS for n → ∞ m,[n] m,[∞] m,[0] uHSS →uHSS 6= uHSS for n → ∞,

86 2.7. On Modiﬁed Lower Order Plate Bending Models

m,[n] m,[n] where uSSS , uHSS denote the m-model solution for the soft and hard simple ˆ support on Ω[n], n ∈ N0, respectively. The convergence is to be understood in ˆ k.ka on Ω[n]. Proof. [7, 8] We have already noted, that the Kirchhoﬀ-Love model only covers the hard simple support situation. In this regard, one comes up with the following statement.

(1,1,0) Theorem 2.74 (Babuška’s paradox for the Kirchhoﬀ-Love model). Let uK be deﬁned as in Theorem 2.56. Then there exists an 0 > 0 such that for any ∈ (0, 0]

(1,1,0),[n] (1,1,0),[∞] (1,1,0),[0] uK,SS → uK,SS 6= uK,SS for n → ∞,

(1,1,0),[n] where uK,SS denotes the Kirchhoff-Love model solution for simple support ˆ ˆ on Ω[n], n ∈ N0. The convergence is to be understood in k.k on Ω[n]. Proof. [7, 8] This consequence clearly shows, that the seemingly minor change of the hard to the soft simple support boundary condition can lead to a significant ˆ [n] difference between the limit solutions on Ω[n], ωˆ , when n ∈ N is large, ˆ [0] and the solution on Ω[0], ωˆ , respectively. Furthermore, the proofs of the theorems above employ the fact, that the Kirchhoff-Love model approximates the Reissner-Mindlin and the three-dimensional formulation very well for the case of hard simple support boundary conditions. This shows that the circular plate ωˆ[0] and the polygonal plate ωˆ[n] solutions are far apart in the entire region and not only in the immediate vicinity to the boundary γˆ, where boundary layer effects dominate [3, 4, 7, 8].

2.7. On Modiﬁed (1,1,0)-Plate Bending Models

In the following we will again consider the completely clamped plate, i.e. γˆ =γ ˆ0, but this time we are going to impose more general bending tractions

87 2. Models of Linearized Plate Theory

ˆ ˆ on Γ+, Γ− whose tangential components are non-vanishing and opposite to each other to induce a transverse shearing of the plate,

f (xˆ) =f (xˆ)ˆx =: p(xˆ)ˆx 0 (2.74) 1 3 3 2 f3(xˆ) =f0,3(xˆ) =: p3(xˆ) (2.75) g(xˆ, ) = − g(xˆ, − ) =: q(xˆ)2 (2.76) 2 2 g(xˆ, ) =g(xˆ, − ) =: q (xˆ)3. (2.77) 3 2 3 2 3 Observe that purely transversal loads, we considered in the previous sections, are contained in these bending loads. We are now going to derive the generalized bending load contributions for the Reissner-Mindlin and the Kirchhoﬀ-Love plate model by the application of integration by parts. Z (1,1,2) (1,1,2) (1,1,2) hF , uR i :=(f , uR )0,Ωˆ + g · uR dsxˆ ˆ ˆ Γ+∪Γ− 2 2 =(pxˆ3, −xˆ3θ)0,Ωˆ + ( p3, w +x ˆ3W )0,Ωˆ Z Z 2 + g · (−xˆ3θ)dsxˆ + g3(w +x ˆ3W )dsxˆ ˆ ˆ ˆ ˆ Γ+∪Γ− Γ+∪Γ− 1 1 = − 3(p, θ) + 3(p , w) + 5(p ,W ) 12 0,ωˆ 3 0,ωˆ 12 3 0,ωˆ + (2q, − θ) + (−2q, θ) + 2(3q , w) 2 0,ωˆ 2 0,ωˆ 3 0,ωˆ 1 + 2(3q ,W ) 2 3 0,ωˆ 1 1 = − 3(p, θ) + 3(p , w) + 5(p ,W ) 12 0,ωˆ 3 0,ωˆ 12 3 0,ωˆ 1 − 3(q, θ) + 23(q , w) + 5(q ,W ) 0,ωˆ 3 0,ωˆ 2 3 0,ωˆ 1 1 hF , u(1,1,2)i = − 3(p, ∇w) + 3(p , w) + 5(p ,W ) K 12 0,ωˆ 3 0,ωˆ 12 3 0,ωˆ 1 − 3(q, ∇w) + 23(q , w) + 5(q ,W ) 0,ωˆ 3 0,ωˆ 2 3 0,ωˆ IP 1 3 3 1 5 = (divp, w)0,ωˆ + (p3, w)0,ωˆ + (p3,W )0,ωˆ clamped 12 12 1 + 3(divq, w) + 23(q , w) + 5(q ,W ) 0,ωˆ 3 0,ωˆ 2 3 0,ωˆ

88 2.7. On Modiﬁed Lower Order Plate Bending Models

(1,1,2) Reduction of uK to a (1,1,0)-Plate Model In Section 2.4.5 we have introduced a simpliﬁed (1,1,2)-Kirchhoﬀ-Love plate bending model (Problem 2.31), which consists of two consecutive optimization problems in v and V . One could further simplify this model by arguing, 2 2 that the higher order term k∇V k0,ωˆ in (2.32) can be neglected. If one even eliminates the imposition of Dirichlet data on W , then one can obviously set W according to ν 2W − ∆w = 0 a.e. in ω.ˆ (2.78) 1 − ν

The resulting method is commonly known as the shear corrected (1,1,0)- Kirchhoﬀ-Love plate model of the clamped plate.

Problem 2.75 (Reduced (1,1,2)-Kirchhoﬀ-Love plate bending model). Let w be a solution of

µ 1 3 ∆2w = 3div p + 3p + 3div q + 23q in ωˆ (2.79) 6(1 − ν) 12 3 3 ∂ w = w =0 on γ.ˆ (2.80) ∂n

We then set 1 ν > u˜ (1,1,0)(xˆ) = − xˆ ∂ w(xˆ), −xˆ ∂ w(xˆ), w(xˆ) +x ˆ2 ∆w(ˆx) . K 3 1 3 2 3 2 1 − ν Remark 2.76. We have already seen in Section 2.4.5, that without the inclusion of ∆w into our three-dimensional representation of the Kirchhoﬀ-Love model we could not reach convergence to the three-dimensional solution in the energy norm k.ka . This fact was not only noted by [13, 36], but can also be detailed by an asymptotic expansion, see [3, 4, 7, 16, 18].

Remark 2.77. In Section 2.6 we have introduced the corrected (1,1,0)- Kirchhoﬀ-Love plate bending model. For the completely clamped plate, one can further simplify the corresponding energy functional, µ J (1,1,0)(w) = 3k∆wk − hF , u(1,1,0)i. K 2 · 6(1 − ν) 0,ωˆ K

89 2. Models of Linearized Plate Theory

This corrected energy leads us to the following biharmonic equation of the Kirchhoff-Love model for w ∈ H2(ˆω), µ 1 3∆2w = 3div p + 3p + 3div q + 23q in ω.ˆ (2.81) 6(1 − ν) 12 3 3 Observe, that we come up with the same biharmonic equation (2.81) as for the shear corrected clamped (1,1,0)-Kirchhoff-Love plate model (2.79), however, by the derivation of two different energy functionals.

(1,1,2) Reduction of uR to a (1,1,0)-Plate Model Our strategy above cannot be applied as easily to the Reissner-Mindlin model by a pure displacement ansatz, but to plate models derived from the Hellinger-Reissner principle, see Remark 2.70. The following problem is commonly addressed as the shear corrected (1,1,0)-Reissner-Mindlin plate bending model of the clamped plate. We refer to [2] for a complete derivation. Problem 2.78 (Reduced (1,1,2)-Reissner-Mindlin plate bending problem). Let (θ, w) be a solution of 3 5 E 5 3 − div θ + (θ − ∇w) = − 3q − p 12 C D 6 2(1 + ν) 6 12 ν5 − ∇ div q 60(1 − ν) ν5 − ∇q 5(1 − ν) 3 ν5 − ∇p˜ in ωˆ 12(1 − ν) 5 E 3 div(θ − ∇w) = div q + 23q + 2p in ωˆ 6 2(1 + ν) 6 3 3 θ =0 on γˆ w =0 on γ.ˆ

We then set Z /2 6ˆx 3 ˆ ˆ 3 p˜(x) := f3(x) 2 − dxˆ3 −/2 10

90 2.7. On Modiﬁed Lower Order Plate Bending Models

(1,1,0) u˜ R (xˆ) := − xˆ3θ1(xˆ), −xˆ3θ2(xˆ), w(xˆ) 2 45(1 − ν) − 84ν2 5(1 − ν) − 7ν2 + xˆ2 − 2q + 2div q 3 20 35(1 − ν)E 3 35(1 − ν)E 5(1 − ν) − 7ν2 ν + 2+3p˜(xˆ) + div θ . 7(1 − ν)E 2(1 + ν)(1 − 2ν)

Remark 2.79. In Section 2.6 we have introduced the corrected (1,1,0)- Reissner-Mindlin plate bending problem, which we can now state for the generalized loads and clamping boundary conditions,

3 3 − div θ + κµ (θ − ∇w) = − 3q − p in ωˆ 12 B D 12 3 3 κµdiv(θ − ∇w) =2 q3 + p3 in ωˆ θ =0 on γˆ w =0 on γ.ˆ

2.7.1. Refined Consistency Results We are now going to substantiate our statements introductory to this section, i.e. we want to quantify the relative model error for the displacement. According to [5], the Reissner-Mindlin plate model exhibits a substantially wider range of applicability compared to the Kirchhoff-Love plate model, meaning, that for a clearly specified class of bending load contributions, the relative displacement error of the Reissner-Mindlin plate converges to zero, while the same error for the Kirchhoff-Love model does not. For the rest of this section, let us assume p ≡ 0.

Theorem 2.80. Suppose that γˆ and the loads q(.) and p3(.) are suﬃciently smooth, satisfying

div q + 2q3 + p3 6= 0 on ω.ˆ (2.82)

Let k.kΩˆ denote one of the semi-norms in Table 2.4. Then there exists a constant c(ˆω, q, p3) > 0 (depending only on ωˆ and the loads) such that

(1,1,0) ku − u˜ K kΩˆ k ≤c(ˆω, q, p3) ku kΩˆ

91 2. Models of Linearized Plate Theory

(1,1,0) kvkΩˆ ku − u˜ K/R kΩˆ /ku kΩˆ

kvk0,Ωˆ O() kv3k0,Ωˆ O() 1/2 k∇ vk0,Ωˆ = |v|1,Ωˆ O( ) ∂v k k ˆ O() ∂xˆ3 0,Ω k∇v3k0,Ωˆ = |v3|1,Ωˆ O() ∂v3 1/2 k k ˆ O( ) ∂xˆ3 0,Ω 1/2 kvka O( )

Table 2.4.: Relative error convergence rates in various semi-norms for the reduced Kirchhoﬀ-Love and the reduced Reissner-Mindlin models assuming (2.82).

(1,1,0) ku − u˜ R kΩˆ k ≤c(ˆω, q, p3) , ku kΩˆ where k is given in Table 2.4.

Proof. [5, Thm. 1] and [4].

1 2 Theorem 2.81. Let q ∈ H (ˆω) and p3 ∈ L (ˆω) satisfying

div q + 2q3 + p3 = 0 on ω.ˆ (2.83)

Then there exists c(ˆω, q) > 0 (depending only on ωˆ and q(.)) such that

(1,1,0) ku − u˜ R kΩˆ 1/2 ≤ c(ˆω, q) . ku kΩˆ

Proof. [5, Thm. 2] The derivation of this assertion is based on the two- energies principle, Theorem 2.35.

It is clear, that no such theorem could be stated for the reduced Kirchhoﬀ- (1,1,0) Love model, since assuming (2.83) leads to u˜ K ≡ 0 for our clamped boundary conditions.

92 2.7. On Modiﬁed Lower Order Plate Bending Models

Remark 2.82 (Consequences for the reduced models). If q ∈ H1(ˆω) and 1 (1,1,0) (1,1,0) p3 ∈ H (ˆω), satisfying (2.82), both reduced models u˜ K , u˜ R are con- vergent (and the convergence rates are the same). However, as soon as (2.83) applies, the reduced Kirchhoff-Love model no longer converges, whereas (1,1,0) 1/2 u˜ R still gives convergence O( ) with respect to k.ka . Remark 2.83 (Practical significance of the two theorems above). As soon (1,1,0) as (2.83) is exactly satisfied, u˜ K does not give us any information about the behaviour of u, for any > 0. Even if (2.83) is not satisfied exactly, but (1,1,0) almost, one can not expect great accuracy for u˜ K . For closer details and actual examples satisfying (2.83) exactly and satisfying (2.83) almost, refer to [5, Sect. 4].

A. Additional Information in the Context of Linearized Elasticity and Dimensional Reduction

A.1. Elastic Discs1

A.1.1. The Plain Stress State Let us assume the framework of Section 2.1. We additionally assume that all acting forces only depend on xˆ and that their e3-components vanish. This situation is called plain stress state, whose hypothesis can be expressed through Deﬁnition A.1 (Plain Stress Hypothesis).   σ11(xˆ) σ12(xˆ) 0 ˆ Σ (xˆ) = Σ (xˆ) = σ21(xˆ) σ22(xˆ) 0 for ∀xˆ ∈ Ω (A.1) 0 0 0

By (1.2) we directly get i3(xˆ) = 3i(xˆ) = 0 for i = 1, 2 and through (1.3) we gain σ33 ≡ 0 by ν ν ≡ − ( + ) = − trE in Ωˆ 33 1 − ν 11 22 1 − ν Hence the constitutive equation (1.3) can be reduced to σ  1 ν 0    11 E 11 σ ν 1 0  22 = 2    22 1 − ν σ12 0 0 1 − ν 12 1This overview is based on [12, Chap. 6.5].

95 A. Additional Information on Dimensional Reduction which can be also written as E ν Σ = E + ( + )I 1 + ν 1 − ν 11 22 E ν = E + (trE)I (A.2) 1 + ν 1 − ν = : CstressE = B E 6= CE By (A.1),(A.2) we end up with > ˆ u (xˆ) = (u1(xˆ), u2(xˆ), xˆ333(xˆ)) for ∀xˆ ∈ Ω

A.1.2. The Plain Strain State Let us assume the framework of Section 2.1. We additionally assume that > u (xˆ) = (u1(xˆ), u2(xˆ), 0) . This situation is called plain strain state, whose hypothesis can be expressed through Deﬁnition A.2 (Plain Strain Hypothesis).   11(xˆ) 12(xˆ) 0 ˆ E (xˆ) = E (xˆ) = 21(xˆ) 22(xˆ) 0 for ∀xˆ ∈ Ω 0 0 0 Since E = Du we directly get > u (xˆ) = (u1(xˆ), u2(xˆ), 0) Since 33 ≡ 0 by (1.4) we get σ33 ≡ ν(σ11 + σ22). Now, (1.3) can be reduced to       σ11 1 − ν ν 0 11 E σ22 =  ν 1 − ν 0  22 (1 + ν)(1 − 2ν) σ12 0 0 1 − 2ν 12 which again can be written as E (1 + ν)(1 − 2ν) Σ = E + ( + )I 1 + ν Eν 11 22 E (1 + ν)(1 − 2ν) = E + (trE)I 1 + ν Eν = : CstrainE 6= CE

96 B. Notations in the Context of Linearized Elasticity and Plates

Scalar variables/functions Real valued variables and functions are designated by minuscules a(., ..), b(., ..), x, y, z, ν, , λ, µ, ξ ∈ R, whereas in- teger valued variables are solely designated by the latin minuscules i, j, k, n, m, p ∈ N. Exception: Young’s Modulus E, which also designates a real valued scalar variable.

Sets Subsets of R3 are designated by majuscules (i.e. Ω), whereas subsets of R2 are designated by minuscules (i.e. ω). Vector valued variables/functions Real valued vectors and vector functions are designated by boldface minuscules a, b, v, w, x, y, n ∈ Rn Matrix valued variables/functions Real valued matrices and matrix functions are designated by boldface majuscules A, B, C, Σ, T, E ∈ Rn×n Matrix Spaces

n M :={set of all real square matrices of order n} n n > S :={B ∈ M : B = B } n n S> :={B ∈ S : det B > 0} n n > > O :={P ∈ M : PP = P P = I}

Reference Conﬁguration ωˆ ⊂ R2, Ωˆ ⊂ R3, Ωˆ ⊂ R3. Domains with Lips- chitz continuous boundary representing an “undeformed” elastic body. Ωˆ represents a three-dimensional plate with thickness > 0.

97 B. Notations

Deformed Conﬁguration ω ⊂ R2, Ω ⊂ R3, Ω ⊂ R3. Domains with Lips- chitz continuous boundary representing an “deformed” elastic body. Ω represents a three-dimensional deformed plate with thickness > 0.

Operations on Sets Assuming Ω ⊂ Rd, d ∈ N constitutes a set. By writing ∂Ω, cl{Ω}, int{Ω}, diam Ω we deﬁne the boundary, closure, interiour and diameter of our set Ω respectively. By writing length ∂Ω, meas Ω we constitute the length of the boundary and the volume of Ω.

Operators Linear functionals and more general operators are designated by latin majuscules A, B, C, F, G. The kernel and the image of an operator A is given by ker A, im A. Writing A∗ gives the adjoint of operator A.

Hilbert Spaces and Duality Assuming (V , k.kV ) constitutes an arbitrary Hilbert space, we write V ∗ to give the dual of V to which we associate the natural dual norm

hF, vi ∗ kF kV ∗ := sup for ∀F ∈ V v∈V kvkV

Hölder Spaces [10, 25, 29, 48]

Lebesgue- and Sobolev Spaces [24, 25, 43] Let Ω ⊂ Rn, n ∈ N be a domain with boundary Γ := ∂Ω and outer unit normal vector n. For m m m m ∈ N,L (Ω),H (Ω) and H0 (Ω) denote the usual Lebesgue- and Sobolev spaces

Z 1/2 2 L (Ω) := {v :Ω → R such that kvk0,Ω := vdx < ∞} Ω 1 2 2 H (Ω) := {v ∈ L (Ω) : ∂iv ∈ L (Ω), 1 ≤ i ≤ n} 2 1 2 H (Ω) := {v ∈ H (Ω) : ∂ijv ∈ L (Ω), 1 ≤ i, j ≤ n}

where ∂i and ∂ij denote the partial derivatives of the ﬁrst and second order in the sense of distributions, and

1 1 H0 (Ω) := {v ∈ H (Ω) : v = 0 a.e. on Γ} 2 2 H0 (Ω) := {v ∈ H (Ω) : v = ∇v · n = 0 a.e. on Γ}

98 where the relations on Γ are to be understood in the sense of traces [24]. Boldface letters denote vector-valued or tensor-valued functions, also called vector ﬁelds or tensor ﬁelds, and their associated function spaces [15, Sect. 6.1.].

L2(Ω) := L2(Ω)3 H1(Ω) := H1(Ω)3 2 2 ={v ∈ L (Ω) : ∂iv ∈ L (Ω), 1 ≤ i ≤ 3} H2(Ω) := H2(Ω)3 1 2 ={v ∈ H (Ω) : ∂ijv ∈ L (Ω), 1 ≤ i, j ≤ 3} 1 1 3 H0(Ω) := H0 (Ω) ={v ∈ H1(Ω) : v = 0 a.e. on Γ} 2 2 3 H0(Ω) := H0 (Ω) ={v ∈ H2(Ω) : v = (∇v)n = 0 a.e. on Γ} the corresponding matrix spaces

2 2 3×3 L (Ω)M := L (Ω) 2 3 2 L (Ω)S :={Σ = (σij)i,j=1 : σij = σji ∈ L (Ω), 1 ≤ i, j ≤ 3} 1 1 3×3 H (Ω)M := H (Ω) 2 2 ={V ∈ L (Ω)M : ∂iV ∈ L (Ω)M, 1 ≤ i ≤ 3} 2 2 3×3 H (Ω)M := H (Ω) 1 2 ={V ∈ H (Ω)M : ∂ijV ∈ L (Ω)M, 1 ≤ i, j ≤ 3} 1 1 3×3 H0(Ω)M := H0 (Ω) 1 mat ={V ∈ H (Ω)M : V = 0 a.e. on Γ} 2 2 3×3 H0(Ω)M := H0 (Ω) 2 ={V ∈ H (Ω)M :(V )i = (∇(V )i)n = 0 a.e. on Γ} For handling mixed methods we will also work with the following additional Sobolev spaces [12, p. 139, 293]

H(div, Ω) ={v ∈ L2(Ω) : div v ∈ L2(Ω)}

99 B. Notations

H(rot, Ω) ={v ∈ L2(Ω) : rot v ∈ L2(Ω)} 2 2 H(div, Ω)M ={v ∈ L (Ω)M : div v ∈ L (Ω)} 2 2 H(rot, Ω)M ={v ∈ L (Ω)M : div v ∈ L (Ω)} 2 2 H(div, Ω)S ={v ∈ L (Ω)S : div v ∈ L (Ω)} 2 2 H(rot, Ω)S ={v ∈ L (Ω)S : div v ∈ L (Ω)}

Gradient/Jacobian (also understood in a weak sense1) ∇ : C1(Ω) → C0(Ω) 3 u 7→ ∇u = (∂iu)i=1 1 0 ∇ : C (Ω) → C (Ω)M 3 u 7→ ∇u = (∂jui)i,j=1

Hessian (also understood in a weak sense)

2 2 0 ∇ : C (ω) → C (ω)S 2 3 w 7→ ∇ w = ∇(∇w) = (∂ijw)i,j=1

Symmetric gradient (also understood in a weak sense)

1 0 D : C (Ω) → C (Ω)S 1 u 7→ E := E(u) := (∇u + ∇u>) 2 Divergence on vectors (also understood in a weak sense) div : C1(Ω) → C0(Ω) 3 X w 7→ div w = ∂iwi i=1

Divergence on matrices (also understood in a weak sense)

1 0 div : C (Ω)M → C (Ω) 3 !3 X T 7→ div T = ∂iτji i=1 j=1 1as discussed in [12, 15, 17, 31, 33, 40, 43, 49–51]

100 Laplace/Harmonic operator (also understood in a weak sense)

∆ : C2(Ω) → C0(Ω) 3 X w 7→ ∆ w = div ∇w = ∂iiw i=1

Bilaplace/Biharmonic operator (also understood in a weak sense)

∆2 : C4(Ω) → C0(Ω) w 7→ ∆2 w = ∆(∆ w)

Elasticity tensor

3 3 C : S → S E ν E 7→ E + (tr E)I 1 + ν 1 − 2ν

Reduced elasticity tensor

2 2 C : S → S E ν E 7→ E + (trE)I 1 + ν 1 − 2ν

Elasticity tensor of the plain stress state

2 2 Cstress : S → S E ν E 7→ E + (trE)I 1 + ν 1 − ν

Elasticity tensor of the plain strain state

2 2 Cstrain : S → S E (1 + ν)(1 − 2ν) E 7→ E + (trE)I 1 + ν Eν

101 B. Notations

Corrected reduced elasticity tensor

2 2 B : S → S E ν E 7→ E + (trE)I 1 + ν 1 − ν

Energy Norms Assuming a bilinear form c(., ..) is elliptic on the Hilbert 1/2 space (W , k.k). Then kχkc := c(χ, χ) constitutes a norm on W equivalent to k.k, whose associated dual norm is given by

hG, χi ∗ kGkc = sup for ∀G ∈ W χ∈W kχkc

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Eidesstattliche Erklärung

Ich, Ludwig Mitter, erkläre an Eides statt, dass ich die vorliegende Masterar- beit selbständig und ohne fremde Hilfe verfasst, andere als die angegebenen Quellen und Hilfsmittel nicht benutzt bzw. die wörtlich oder sinngemäß ent- nommenen Stellen als solche kenntlich gemacht habe. Die vorliegende Mas- terarbeit ist mit dem elektronisch übermittelten Textdokument identisch.

Linz, November 2017

———————————————— Ludwig Mitter