There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Analysis of thickness locking in classical, refined and mixed multilayered plate theories

Variational methods in elasticity and plasticity

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Piezoelectricity An Introduction to the Theory and Applications of Electromechanical Phenomena in Crystals

By means of Non-Uniform Rational B-Splines (NURBS) curves, it is possible to describe arbitrary shapes with holes and discontinuities. These peculiar shapes can be taken into account to describe the reference domain of several nanoplates, where a nanoplate refers to a flat structure reinforced with Carbon Nanotubes (CNTs). In the present paper, a micromechanical model based on the agglomeration of these nanoparticles is considered. Indeed, when this kind of reinforcing phase is inserted into a polymeric matrix, CNTs tend to increase their density in some regions. Nevertheless, some nanoparticles can be still scattered within the matrix. The proposed model allows to control the agglomeration by means of two parameters. In this way, several parametric studies are presented to show the influence of this agglomeration on the free vibrations. The considered structures are characterized also by a gradual variation of CNTs along the plate thickness. Thus, the term Functionally Graded Carbon Nanotubes (FG-CNTs) is introduced to specify these plates. Some additional parametric studies are also performed to analyze the effect of a mesh distortion, by considering several geometric and mechanical configurations. The validity of the current methodology is proven through a comparative assessment of our results with those available from the literature or obtained with different numerical approaches, such as the Finite Element Method (FEM). The strong form of the equations governing a plate is solved by means of the Generalized Differential Quadrature (GDQ) method.

Free vibration analysis of arbitrarily shaped Functionally Graded Carbon Nanotube-reinforced plates

http://apm.tstu.ru/KulikovRus/text/COST2012.pdf

Exact 3D stress analysis of laminated composite plates by sampling surfaces method

Advanced formulation for laminated composite shells: 3D stress analysis and rigid-body motions

http://apm.tstu.ru/KulikovRus/text/C&S2006.pdf

Geometrically exact assumed stress–strain multilayered solid-shell elements based on the 3D analytical integration

This paper presents a simple and effective method of solving the 3D elasticity problems for thick shells. It is based on the new concept of sampling surfaces (S-surfaces) inside the shell body. According to this concept, we introduce N not equally located S-surfaces parallel to the midsurface and choose displacements of these surfaces as basic shell unknowns. Such choice allows one to represent the governing equations of the proposed higher order shell model in a very compact form and to solve the 3D static problems for thick plates and shells efficiently.

On the use of sampling surfaces method for solution of 3D elasticity problems for thick shells

The exact representation of rigid-body motions in the displacement patterns of the first-order equivalent single-layer (ESL) and layerwise (LW) shell elements is considered. This consideration requires the development of the strain–displacement relationships of the ESL and LW shell theories with regard to their consistency with rigid-body motions. The fundamental unknowns consist of six displacements of the face surfaces of the shell in the ESL theory and 3(N + 1) displacements of the face surfaces of layers in the LW shell theory, where N is a number of layers. Such a choice of displacements makes it possible to deduce strain–displacement relationships, which are objective, that is, invariant under rigid-body motions. To overcome thickness locking, three types of the modified material stiffness matrix corresponding to the generalized plane-stress state are employed.

Svetlana V. Plotnikova

Papers

Exact 3D stress analysis of laminated composite plates by sampling surfaces method

Advanced formulation for laminated composite shells: 3D stress analysis and rigid-body motions

Geometrically exact assumed stress–strain multilayered solid-shell elements based on the 3D analytical integration

On the use of sampling surfaces method for solution of 3D elasticity problems for thick shells

Equivalent Single-Layer and Layerwise Shell Theories and Rigid-Body Motions—Part I: Foundations