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2019 | Book

Introduction Read chapter Read first chapter

Structural Mechanics of Anti-Sandwiches

An Introduction

Author: Dr. Marcus Aßmus

Publisher: Springer International Publishing

Book Series : SpringerBriefs in Applied Sciences and Technology

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik"

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About this book

This book provides an extensive introduction to the mechanics of anti-sandwiches: non-classical composites with multiple homogeneous layers but widely differing parameters concerning their geometry and materials. Therefore, they require special attention in the context of structural mechanics.

The theoretical framework presented here is based on a five parametric, planar continuum, which is a pragmatic version of the COSSERAT shell. The direct approach used here is enlarged where constraints are introduced to couple layers and furnish a layer-wise theory. Restrictions are made in terms of linearity – geometrical and physical. After having defined appropriate variables for the kinematics and kinetics, linear elastic material behaviour is considered, where the constitutive tensors are introduced in the context of isotropy. The basics are presented in a clear and distinct manner using index-free tensor notation. This format is simple, concise, and practical.

Closed-form solutions of such boundary value problems are usually associated with serious limitations on the boundary conditions, which constitutes a serious disadvantage. To construct approximate solutions, a variational method is employed as the basis for computational procedures where the Finite Element Method is applied. Therefore, the introduction of the vector-matrix notation is convenient. Based on the plane considerations, a finite eight-node SERENDIPITY element with enlarged degrees of freedom is realised. To avoid artificial stiffening effects, various integration types are applied, and the solutions generated are subsequently verified with closed-form solutions for monolithic limiting cases.

Within this setting, it is possible to efficiently calculate the global structural behaviour of Anti-Sandwiches, at least up to a certain degree. The power of the proposed method in combination with the numerical solution approach is demonstrated for several case and parameter studies. In this regard, the optimal geometrical and material parameters to increase stiffness are analysed and the results for the kinematic and kinetic quantities are discussed.

Table of Contents

Frontmatter
Chapter 1. Introduction
Abstract
Laminates, Sandwiches and Anti-Sandwiches are classically classified as composite structures. Composite structures are multi-layered thin-walled structural elements which exhibit special geometrical features. For this purpose, plane dimensions \(L_\alpha \;\forall \,\alpha \in \{1,2\}\) and the overall thickness H are used.
Marcus Aßmus
Chapter 2. Theory of Planar Surface Continua
Abstract
Planar continua analogous to the direct approach will be introduced at this point, i.e. we will operate on a deformable surface instead of a voluminous body a priori. The description is following the Zhilinean path (Zhilin, Applied mechanics - foundations of shells theory (in russian). Publisher of the Polytechnic University, St. Petersburg, 2006, [11]) though using a more common and unique notation. However, the main hypotheses of the classical mechanics of continuous media hold true. The material surface \(\mathfrak S \) is a coherent and compact set of material space points \(\mathfrak M\). The boundary of this point set is indicated by the domain boundary \(\partial \mathfrak S\). First, let us start with a homogeneous body, i.e. all material points have the same characteristics. Another limitation is in the isotropy assumption, i.e. all directions are equal. Each material point has three translational and two rotational degrees of freedom as kinematic variables, with the rotations introduced as independent degrees of freedom. The continuum hypothesis applies, maintaining the continuity of material points during deformation. The relationship to the three dimensional body \(\mathfrak B\) in whose volume V the surface is embedded can be represented by the following expression, where the assumption \(h=\mathrm {const.}\) holds for the structural thickness.
Marcus Aßmus
Chapter 3. Multilayered Surface Continua
Abstract
The computation of composite structures, as introduced in Chap. 1, requires the extension of the presented concept of the surface continuum to multiple layers. This is especially true when the physical layer thicknesses differ widely and the mechanical properties of the layer materials are strongly divergent. This is the case with Anti-Sandwiches. In this sense, the present chapter introduces a so-called layer-wise theory. Each layer is considered individually, whereby the coupling is realized via kinematic constraints. However, apart from the consideration of the individual mid surfaces, this procedure also requires the consideration of interfaces of the physical structure.
Marcus Aßmus
Chapter 4. Vartiational Formulation
Abstract
In the previous chapters, the local behavior of an elastic, threelayered composite structure was derived, which resulted in the description of the initial boundary value problem. The principle of virtual work is a formulation equivalent to the balances of forces and moments, and represents a weak form of the balance (Bathe, Finite-elemente-methoden. Springer, Berlin, 2002, [1]). It is obtained by weighting the equations of motion with test functions equivalent to the vectors of degrees of freedom and subsequent partial integration over the area considered (Oñate, Structural analysis with the finite element method linear statics: vol 2. Beams, plates and shells. Springer, Dordrecht, 2013, [5]). The test function can be interpreted as infinitesimal deformation field (virtual displacements, virtual deflections, and virtual rotations). This field is arbitrary, but must satisfy the geometric boundary conditions and have to be continuously differentiable (Bathe, Finite-elemente-methoden. Springer, Berlin, 2002, [1]). There is no further assumption in this principle (Oden and Reddy, Variational methods in theoretical mechanics. Springer, Berlin, 1983, [4]). The following process serves as a basis for the numerical implementation.
Marcus Aßmus
Chapter 5. Finite Element Implementation
Abstract
The weak formulation of the boundary value problem serves as a starting point for the numerical solution of the problem. In this chapter, a spatial finite element discretization is used to generate a semidiscrete structural equation. Choosing a formulation considering the transverse shear distortions is also chosen due to the lower and easier-to-fulfill continuity requirements of the shear flexible finite elements (\(C^0\) continuity) compared to the requirement for shear-rigid elements (\(C^1\) continuity) (Oñate, Structural analysis with the finite element method linear statics: volume 2. Beams, plates and shells. Springer, Dordrecht, 2013, [6]). Thus only the zeroth derivative of the degrees of freedom has to be continuous. In contrast to the classical, shear-soft Bathe–Dvorkin element (Bathe and Dvorkin, Int J Numer Methods Eng 21(2):367–383, 1985, [2]), in-plane displacements are taken into account. The implementation is based on the global degrees of freedom introduced in the previous chapter.
Marcus Aßmus
Chapter 6. Convergence and Verification
Abstract
At this point, the convergence of the numerical solution is to be checked and the numerical solution to be verified with a closed-form solution based on two limiting cases. Ultimately, the suitability of the finite element can be proven. It is based on a simple load and support situation. As illustrated in Fig. 6.1 on the left-hand side, a homogeneous and orthogonal load is used as model problem for all considerations in this chapter. For the sake of simplicity, the structure to be examined is symmetric in the transverse direction. The cover layers thus have identical geometric dimensions and material properties. Regarding supports, only the deflections at the edges with normal vector \(\varvec{\nu }\) are prohibited. These edges are thus supported torque-free. The geometric dimensions and material characteristics are exemplary.
Marcus Aßmus
Chapter 7. Application
Abstract
At this point, the computational solution strategy introduced in this work is to be applied. The investigations are limited to common data for geometries and materials used. Thereby we focus on data of photovoltaic modules as they are a prominent application of an Anti-Sandwich (Aßmus, Global structural analysis at photovoltaic modules: theory, numerics, application (in German). Dissertation, Otto von Guericke University Magdeburg, 2018, [1]). A distinction is made between parameter and case studies. In the parameter studies the effects of the variation of geometrical and physical quantities are investigated, while in the case studies examinations are carried out at realistic loading scenarios from natural weathering.
Marcus Aßmus
Chapter 8. Summary and Outlook
Abstract
In present treatise an approach for structural analysis of Anti-Sandwiches is presented. In principle, theories for thin-walled structures are suitable for mechanical analysis at such configurations. Since mechanical properties and structural thicknesses of the different layers of an Anti-Sandwich differ widely, classical approaches for composite structures fail to predict correct results. Therefore, a layer-wise approach is chosen within the present discourse. Each layer is considered as a single continuum, while all equations are related to the middle surface of the structure.
Marcus Aßmus
Backmatter
Metadata
Title
Structural Mechanics of Anti-Sandwiches
Author
Dr. Marcus Aßmus
Copyright Year
2019
Electronic ISBN
978-3-030-04354-4
Print ISBN
978-3-030-04353-7
DOI
https://doi.org/10.1007/978-3-030-04354-4

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