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Cosserat Elasticity
Published in Lev Steinberg, Roman Kvasov, Cosserat Plate Theory, 2023
In this book, we use the usual Einstein notation that implies summation over a set of indexed terms in a formula. When an index variable appears twice in a single term, it implies summation of that term over all the values of the index. The index that is summed over we will call a “dummy index”; the index that is not summed over we will call a “free index”. We will assume that the expressions that contain Latin letters (i, j, k, etc.) as subindices are understood to be the components of the spatial vectors (i.e. vectors from ℝ3) and thus take values in the set {1,2,3}. We will also assume that the expressions that contain Greek letters (α, β, γ, etc.) as subindices are understood to be the components of the plane vectors (i.e. vectors from ℝ2) and thus take values in the set {1,2}.
Quasi-Phase Matching
Published in Peter E. Powers, Joseph W. Haus, Fundamentals of Nonlinear Optics, 2017
Peter E. Powers, Joseph W. Haus
Pursuant to the rules introduced in Section 2.2, εij transforms as ε′ij=RiαRjβεαβ, where the repeated indices indicate a summation according to the Einstein notation. εαβ represents the crystal before the transformation, and since εαβ starts out diagonal in the principal axis coordinate system, the only nonzero contributions to the sum are α = β so that ε′ij=RiαRjαεαα.
Optimum material distribution of porous functionally graded plates using Carrera unified formulation based on isogeometric analysis
Published in Mechanics of Advanced Materials and Structures, 2022
Farshad Rahmani, Reza Kamgar, Reza Rahgozar
Based on the Einstein notation, the repeated indices indicate summation. are functions which approximate the thickness direction z, and is the displacement components of in-plane coordinates x, y. The component M is the number of expansion terms in the z-direction. It can be concluded from Eq. (9) that the CUF can describe the shear deformation by any order, which is a significant feature of the CUF. Another advantage of CUF is that the governing equations are described by fundamental nuclei (FN), and they are independent of the order of expansion. In the present study, the Legendre polynomials are used to approximate the thickness, and NURBS basis functions are utilized to simulate the in-plane displacement field.
Carrera unified formulation for the micropolar plates
Published in Mechanics of Advanced Materials and Structures, 2021
In (3.1) according to the Einstein notation, the repeated subscript indicates summation. In general the choice of the number and functions and is arbitrary, that is, different base functions of any order can be taken into account to model the kinematic field of the plate along its thickness. The final equation becomes simple if functions and are polynomials, especially orthogonal polynomials. Coefficients of the expansions and as functions of the coordinates and coincided with the middle plane of the plate. The first subscript in the base functions and indicates the component of the displacements or rotations vectors, the second one indicates the series number of the function in the expansion.
Effect of Nonstructural Masses on Civil Structures by CUF-Based Finite Element Models
Published in International Journal for Computational Methods in Engineering Science and Mechanics, 2018
Alfonso Pagani, Matteo Filippi, Marco Petrolo, Giovanni Colonna, Erasmo Carrera
Classical beam models provide a reasonably good approximation of slender, solid section, homogeneous structures subject to bending phenomena but, in case of short, thin-walled, open cross-section beam analyses, the required degree of accuracy might not be reached. More sophisticated theories, which adopt richer kinematic fields to obtain more accurate 1D models, are needed. By means of the CUF, refined beam models having an arbitrary number of terms in the kinematic field can be developed. The kinematics of a CUF beam model can be summarised as follows:where indicates the functions of the cross section coordinates x and z, is the generalised displacement vector and M indicates the number of terms in the expansion. Since the Einstein notation has been adopted, the repeated subscript indicates summation. The basis functions adopted to model the displacement field across the section can be different and expanded to any order, since the choice of and M is arbitrary. The models known in the literature as Taylor expansion (TE) [15, 29, 30] are obtained considering the Taylor-like expansion polynomials as functions. It should be noted that (1) and (2) are particular cases of the linear (N = 1) TE model, which can be expressed aswhere the parameters on the right-hand side (, etc.) represent linear displacements and rotations of the beam axis. More details about TE models and the formulation of classical models as particular cases of TE can be found in [31, 32].