Shell Structures: Theory and Applications (Vol. 2)

Proceedings of the 9th SSTA Conference, Jurata, Poland, 14-16 October 2009

Wojciech Pietraszkiewicz, Ireneusz Kreja, Wojciech Pietraszkiewicz, Ireneusz Kreja

• Publisher: CRC Press
• Year: 2009
• Print ISBN:
  9780415548830
• eBook ISBN:
  9781439859193
• Topic: Technology & Engineering
• Subtopic: Civil Engineering
• Index: Technology & Engineering

Shell Structures. Theory and Applications, Volume 2 contains 77 contributions from over 17 countries, reflecting a wide spectrum of scientific and engineering problems of shell structures. The papers are divided into six broad groups: 1. General lectures; 2. Theoretical modeling; 3. Stability; 4. Dynamics; 5. Numerical analysis; 6. Engineering

Theoretical modelling

On the shell and plate theories with surface stresses

H. Altenbach
Martin-Luther-Universität Halle-Wittenberg, Halle (Saale), Germany

VA. Eremeyev
South Scientific Center of RASci & South Federal University, Rostov-on-Don, Russian Federation

ABSTRACT: We discuss the methods of derivation of the governing plate and shell equations taking into account the surface stresses. The theory of elasticity with surface stresses is applied to the modeling of nanosized plates and shells. In particular, the surface stresses are responsible for the size effect. The surface stresses have an influence on the effective stiffness properties of the elastic plates and shells. Here we present the effective elastic stiffness properties of a shell taking into account the surface stresses.

1 INTRODUCTION

The development of nanotechnologies extends the field of application of the classical or non-classical theories of plates and shells towards the new thin-walled structures. In general, modern nanomaterials have physical properties which are different from the bulk material. The classical linear elasticity can be extended to the nanoscale by implementation of the theory of elasticity taking into account the surface stresses, cf. Duan et al. (2008) among others. In particular, the surface stresses are responsible for the size-effect, that means the material properties of a specimen depend on its size. For example, Young’s modulus of a cylindrical specimen increases significantly, when the cylinder diameter becomes very small. The surface stresses are the generalization of the scalar surface tension which is well-known phenomenon in the theory of capillarity. The investigations of the surface phenomena were initiated by Laplace, Young, Gibbs, see the reviews by Orowan (1970), Podstrigach & Povstenko (1985), Finn (1986), Rusanov (2005), and Duan et al. (2008). Let us mention here the works by Gurtin & Murdoch (1975), Podstrigach & Povstenko (1985), and Steigmann & Ogden (1999) among others where the mechanics with regard of surface stresses is developed.

The theory of elasticity with surface stresses was applied to the modifications of the two-dimensional theories of nanosized plates, see e.g. Lu et al. (2006), Huang (2008) and Eremeyev et al. (2009), where various theories of plates are formulated.

Let us note that there exist different approaches to formulate shell or plate theories. The discussion on the derivation of the plate and shell equations is very extensive, see Naghdi (1972), Reissner (1985), Libai & Simmonds (1998), Rubin (2000), Chróścielewski et al. (2004), Zhilin (2006), and Eremeyev & Zubov (2008) among others. The approaches can be classified, for example, by the starting point of the derivation. This can be the well-known three-dimensional continuum mechanics equations. In contrast, one can introduce à priory a two-dimensional deformable surface which is the basis for a more natural formulation of the two-dimensional governing equations. This so-called direct approach should be supplemented by the theoretical or experimental determination of the material parameters included in the constitutive equations.

Here we use the general theory of shell presented in Libai & Simmonds (1998), Chróścielewski et al. (2004) for the modification of the constitutive equations taking into account the surface stresses. We show that both the stress and the couple stress resultant tensors may be represented as a sum of two terms. The first term is the volume stress resultant while the second one is determined by the surface stresses and the shell geometry. In the linear case this modification reduces to the addition of new terms to the elastic stiffness parameters. The influence of these terms on the bending stiffness of a shell is discussed. We show that the surface elasticity makes a shell more stiffer in comparison with the shell without surface stresses.

2 BASIC EQUATIONS OF THE 6-PARAMETRIC ELASTIC SHELL THEORY

Following Libai & Simmonds (1998), and Chrósscielewski et al. (2004) we consider the static problem of the general nonlinear shell theory. This variant of the elastic shell theory is also named micropolar shell theory, see Eremeyev & Zubov (2008). A micropolar shell is kinematically equivalent to a two-dimensional Cosserat continuum in which the interaction between different parts of the shell is described by forces and moments only. The kinematical model of a shell bases on the introduction of a directed material surface ω(t), which is determined in the actual configuration by

Figure 1. Micropolar shell in an actual configuration.

where r(q¹, q², t) is the position vector defining the geometry of ω, q¹, q² Є ω, and d_k (q¹, q², t) is a set of orthormal vectors called directors, k = 1, 2, 3, see Figure 1. In the reference configuration the position vector and directors are introduced by

The considered shell theory was initiated by the Cosserat within framework of the direct approach, see the fundamental centenarian book by Cosserat (1990).

The strain energy function of an elastic micropolar shell is given b...

Table of contents

• Cover
• Title Page
• Copyright
• Table of Contents
• Preface
• Organization and Committees
• Acknowledgements
• General Lectures
• Theoretical Modelling
• Stability
• Dynamics
• Numerical Analysis
• Engineering Design
• Author Index