Preface Bibliography 1. Interpolation. Introduction The Taylor expansion The finite Taylor series with the remainder term Interpolation by polynomials The remainder of Lagrangian interpolation formula Equidistant interpolation Local and global interpolation Interpolation by central differences Interpolation around the midpoint of the range The Laguerre polynomials Binomial expansions The decisive integral transform Binomial expansions of the hypergeometric type Recurrence relations The Laplace transform The Stirling expansion Operations with the Stirling functions An integral transform of the Fourier type Recurrence relations associated with the Stirling series Interpolation of the Fourier transform The general integral transform associated with the Stirling series Interpolation of the Bessel functions 2. Harmonic Analysis. Introduction The Fourier series for differentiable functions The remainder of the finite Fourier expansion Functions of higher differentiability An alternative method of estimation The Gibbs oscillations of the finite Fourier series The method of the Green's function Non-differentiable functions Dirac's delta function Smoothing of the Gibbs oscillations by Fejer's method The remainder of the arithmetic mean method Differentiation of the Fourier series The method of the sigma factors Local smoothing by integration Smoothing of the Gibbs oscillations by the sigma method Expansion of the delta function The triangular pulse Extension of the class of expandable functions Asymptotic relations for the sigma factors The method of trigonometric interpolation Error bounds for the trigonometric interpolation method Relation between equidistant trigonometric and polynomial interpolations The Fourier series in the curve fitting 3. Matrix Calculus. Introduction Rectangular matrices The basic rules of matrix calculus Principal axis transformation of a symmetric matrix Decomposition of a symmetric matrix Self-adjoint systems Arbitrary n x m systems Solvability of the general n x m system The fundamental decomposition theorem The natural inverse of a matrix General analysis of linear systems Error analysis of linear systems Classification of linear systems Solution of incomplete systems Over-determined systems The method of orthogonalisation The use of over-determined systems The method of successive orthogonalisation The bilinear identity Minimum property of the smallest eigenvalue 4. The Function Space. Introduction The viewpoint of pure and applied mathematics The language of geometry Metrical spaces of infinitely many dimensions The function as a vector The differential operator as a matrix The length of a vector The scalar product of two vectors The closeness of the algebraic approximation The adjoint operator The bilinear identity The extended Green's identity The adjoint boundary conditions Incomplete systems Over-determined systems Compatibility under inhomogeneous boundary conditions Green's identity in the realm of partial differential operators The fundamental field operations of vector analysis Solution of incomplete systems 5. The Green's Function. Introduction The role of the adjoint equation The role of Green's identity The delta function -- The existence of the Green's function Inhomogeneous boundary conditions The Green's vector Self-adjoint systems The calculus of variations The canonical equations of Hamilton The Hamiltonisation of partial operators The reciprocity theorem Self-adjoint problems Symmetry of the Green's function Reciprocity of the Green's vector The superposition principle of linear operators The Green's function in the realm of ordinary differential operators The change of boundary conditions The remainder of the Taylor series The remainder of the Lagrangian interpolation formula

Linear Differential Operators. By C. Lanczos. Pp. 580. 80s. 1961. (Van Nostrand, London)

The aim of this paper is to find a computationally efficient and predictive model for the class of systems that we call ‘pantographic structures’. The interest in these materials was increased by the possibilities opened by the diffusion of technology of three-dimensional printing. They can be regarded, once choosing a suitable length scale, as families of beams (also called fibres) interconnected to each other by pivots and undergoing large displacements and large deformations. There are, however, relatively few ‘ready-to-use’ results in the literature of nonlinear beam theory. In this paper, we consider a discrete spring model for extensible beams and propose a heuristic homogenization technique of the kind first used by Piola to formulate a continuum fully nonlinear beam model. The homogenized energy which we obtain has some peculiar and interesting features which we start to describe by solving numerically some exemplary deformation problems. Furthermore, we consider pantographic structures, find the corresponding homogenized second gradient deformation energies and study some planar problems. Numerical solutions for these two-dimensional problems are obtained via minimization of energy and are compared with some experimental measurements, in which elongation phenomena cannot be neglected.

https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.2015.0790

Large deformations of planar extensible beams and pantographic lattices: Heuristic homogenization, experimental and numerical examples of equilibrium

This software is specialized in books discussing across different customers and nations, and ebook Introduction To Perturbation Techniques Download PDF could be also saved from here. With your large library of various books, your search demand Introduction To Perturbation Techniques can be saved in all digital types such as for instance ePub, PDF, and free with two easy steps. This really is one of the finest websites to get free e-books and declare to provide more than 37,000 below various types Introduction To Perturbation Techniques. There are numerous free, genuine internet sites where you could study publications on line or download books in a number of models, but just here you are able to acquire Introduction To Perturbation Techniques positively for free. This is an excellent site it has all the assortment of new and old e-books additionally it provides the publications which are free on one other sites like Kindle, it shows all of the free publications which on kindle it also offers a unique books Introduction To Perturbation Techniques Download PDF. This is a free search engine which allows you to search, critique and get an incredible number of PDF documents in to your devices with Introduction To Perturbation Techniques Download PDF available. Our website offers guidelines based in your interests and recent searches of Introduction To Perturbation Techniques Download PDF. The websites is definitely an start, editable selection collection to learn free Introduction To Perturbation Techniques Download PDF online. Your one-stop solution for examining Introduction To Perturbation Techniques on the web free. This book Introduction To Perturbation Techniques Download PDF will come in PDF structure and other for free download. Click on the keys below to get Introduction To Perturbation Techniques Download PDF. We allow you that proper as skillfully as simple pretentiousness to obtain those all. We provide Introduction To Perturbation Techniques and numerous guide collections from fictions to medical research in virtually any way. followed closely by them is that Introduction To Perturbation Techniques Download PDF which can be your partner. It will not waste your time. say you will me, the e-book may very expose you other problem to read. Only invest small mature to door this on-line detect Introduction To Perturbation Techniques Download PDF as competently as evaluation them wherever you are now. This is similarly among the facets by obtaining the soft documents with this Introduction To Perturbation Techniques online. You could maybe not need more epoch to spend to go to the commencement of the book as without trouble as seek out them. In some cases, you furthermore realize maybe not to discover the affirmation Introduction To Perturbation Techniques Download PDF that you are seeking for. It will enormously squander the time. It will not believe enough time once we inform before. You can reach it even if put-on another thing at home and even yet in your workplace. properly simple! Therefore, have you been question? Only workout exactly what we come up with the amount of money for under as well as evaluation Introduction To Perturbation Techniques that which you passed to read! It will not consent several age once we accustom before. You can total it even though appear in something else at your house and even yet in your workplace. superbly simple! So, are you

Introduction To Perturbation Techniques

We formulate a relaxed linear elastic micromorphic continuum model with symmetric Cauchy force stresses and curvature contribution depending only on the micro-dislocation tensor. Our relaxed model is still able to fully describe rotation of the microstructure and to predict nonpolar size effects. It is intended for the homogenized description of highly heterogeneous, but nonpolar materials with microstructure liable to slip and fracture. In contrast to classical linear micromorphic models, our free energy is not uniformly pointwise positive definite in the control of the independent constitutive variables. The new relaxed micromorphic model supports well-posedness results for the dynamic and static case. There, decisive use is made of new coercive inequalities recently proved by Neff, Pauly and Witsch and by Bauer, Neff, Pauly and Starke. The new relaxed micromorphic formulation can be related to dislocation dynamics, gradient plasticity and seismic processes of earthquakes. It unifies and simplifies the understanding of the linear micromorphic models.

/pdf/a-unifying-perspective-the-relaxed-linear-micromorphic-s3dvc7yf5l.pdf

A unifying perspective: the relaxed linear micromorphic continuum

A curved beam element based on the Timoshenko model and non-uniform rational B-splines (NURBS) interpolation both for geometry and displacements is presented. Such an element can be used to suitably analyse plane-curved beams and arches. Some numerical results will explore the effectiveness and accuracy of this novel method by comparing its performance with those of some accurate finite elements proposed in the technical literature, and also with analytical solutions: for the cases where such closed-form solutions were not available in the literature, they have been computed by exact integration of the governing differential equations. It is shown that the presented element is almost insensitive to both membrane- and shear-locking, and that such phenomena can be easily controlled by properly choosing the number of elements or the NURBS degree.

Isogeometric analysis of plane-curved beams:

It has been known since the pioneering works by Piola, Cosserat, Mindlin, Toupin, Eringen, Green, Rivlin and Germain that many micro-structural effects in mechanical systems can be still modeled by means of continuum theories. When needed, the displacement field must be complemented by additional kinematical descriptors, called sometimes microstructural fields. In this paper, a technologically important class of fibrous composite reinforcements is considered and their mechanical behavior is described at finite strains by means of a second-gradient, hyperelastic, orthotropic continuum theory which is obtained as the limit case of a micromorphic theory. Following Mindlin and Eringen, we consider a micromorphic continuum theory based on an enriched kinematics constituted by the displacement field u and a second-order tensor field ψ describing microscopic deformations. The governing equations in weak form are used to perform numerical simulations in which a bias extension test is reproduced. We show that second-gradient energy terms allow for an effective prediction of the onset of internal shear boundary layers which are transition zones between two different shear deformation modes. The existence of these boundary layers cannot be described by a simple first-gradient model, and its features are related to second-gradient material coefficients. The obtained numerical results, together with the available experimental evidences, allow us to estimate the order of magnitude of the introduced second-gradient coefficients by inverse approach. This justifies the need of a novel measurement campaign aimed to estimate the value of the introduced second-gradient parameters for a wide class of fibrous materials.

https://hal.archives-ouvertes.fr/hal-00838662/document

Modeling the onset of shear boundary layers in fibrous composite reinforcements by second-gradient theory

In this paper, we propose to use a second gradient, 3D orthotropic model for the characterization of the mechanical behavior of thick woven composite interlocks. Such second-gradient theory is seen to directly account for the out-of-plane bending rigidity of the yarns at the mesoscopic scale which is, in turn, related to the bending stiffness of the fibers composing the yarns themselves. The yarns’ bending rigidity evidently affects the macroscopic bending of the material and this fact is revealed by presenting a three-point bending test on \({0 ^{\circ}/90 ^{\circ} \,\,{\rm and}\,\, \pm45 ^{\circ}}\) specimens of composite interlocks. These specimens differ one from the other for the different relative direction of the yarns with respect to the edges of the sample itself. Both types of specimens are independently seen to take advantage of a second-gradient modeling for the correct description of their macroscopic bending modes. The results presented in this paper are essential for the setting up of a correct continuum framework suitable for the mechanical characterization of composite interlocks. The few second-gradient parameters introduced by the present model are all seen to be associated with peculiar deformation modes of the mesostructure (bending of the yarns) and are determined by inverse approach. Although the presented results undoubtedly represent an important step toward the complete characterization of the mechanical behavior of fibrous composite reinforcements, more complex hyperelastic second-gradient constitutive laws must be conceived in order to account for the description of all possible mesostructure-induced deformation patterns.

/pdf/thick-fibrous-composite-reinforcements-behave-as-special-4tytxmkwe7.pdf

Thick fibrous composite reinforcements behave as special second-gradient materials: three-point bending of 3D interlocks

In this paper, a new consistent dynamic model is proposed, aimed at studying linear vibrations induced in an elastic wire by a bilaterally constrained single mass moving with a constant velocity Starting from a variational formulation, through the Hadamard’s condition, a corrective term to the local linear stiffness is determined in the continuum model as a function of the moving mass velocity; in this way, the boundary conditions are properly found The representation of the solutions of the hyperbolic equations governing the motion of the wire presents some difficulties, which are solved by means of a suitable coordinate transformation in a time-invariant domain and a judicious choice of the set of shape functions, to be used in the discrete formulation of the problem This new description allows an easy estimation of high-order deformations that are neglected by a purely linear approach When the mass velocity is sufficiently high, displacements near the supports show high gradients: in these cases, it is necessary to use an unknown velocity or introduce an advanced mechanical model in order to correctly describe the motion of the mass Numerical examples confirm the stability of the proposed solution in all conditions examined

Dynamic modeling of taut strings carrying a traveling mass

A Timoshenko beam model embedded in a 3D space is introduced for buckling analysis of multi-store buildings, made by rigid floors connected by elastic columns. The beam model is developed via a direct approach, and the constitutive law, accounting for prestress forces, is deduced via a suitable homogenization procedure. The bifurcation analysis for the case of uniformly compressed buildings is then addressed, and numerical results concerning the Timoshenko model are compared with 3D finite element analyses. Finally, some conclusions and perspectives are drawn.

Flexural torsional buckling of uniformly compressed beam-like structures

The paper is devoted to discuss the effects of nonlinear damping on the post-critical behavior of the Ziegler’s column. The classical Ziegler’s double-pendulum is considered in regime of finite displacements, in which, moreover, nonlinear damping of Van der Pol-type is introduced at the hinges. A second-order Multiple Scale analysis is carried out on the equations of motion expanded up to the fifth-order terms. The nature of the Hopf bifurcation, namely, supercritical or subcritical, as well as the occurrence of the ‘hard loss of stability’ phenomenon, are investigated. The effects of the nonlinear damping on the amplitude of the limit-cycle are finally studied for different linearly damped columns.

Manuel Ferretti

Papers

Modeling the onset of shear boundary layers in fibrous composite reinforcements by second-gradient theory

Thick fibrous composite reinforcements behave as special second-gradient materials: three-point bending of 3D interlocks

Dynamic modeling of taut strings carrying a traveling mass

Flexural torsional buckling of uniformly compressed beam-like structures

Hard loss of stability of Ziegler’s column with nonlinear damping