Mechanics of Moving Materials

Abstract

PrefaceMechanics of moving materials is currently attracting considerable attention.Interest in research in the eld has grown in connection with the rapid development of process industry and precision machinery, especially in the eld of paper making.Specialized areas of investigation include dynamics and mechanical stability of travelling elastic materials, failure and reliability analysis, including modern fracture mechanics, and runnability optimization problems taking into account factors of uncertainty and incomplete data.Problems of mechanics of moving materials have not only practical but also theoretical importance. The investigation into new types of mathematical problems is interesting in itself. It is noteworthy that there are some nonconservative stability problems, and runnability optimization problems under fracture and stability constraints with uncertainties in positioning and sizes of initial defects (cracks), for which there are no systematic techniques of investigation. The most appropriate approach to tackle the complex problem varies depending on the case.This monograph is devoted to the exposition of new ways of formulating and solving problems of mechanics of moving materials. We present some research results concerning dynamics of travelling elastic strings, membranes, panels and plates. We study mechanical stability of axially moving elastic panels, accounting for the interaction between the structural element and its environment, such as axial potential ow. Most of the attention in this book is devoted to out-of-plane dynamics and stability analysis for isotropic and orthotropic travelling elastic and viscoelastic materials, with and without uid-structure interaction, using analytical and numerical approaches. Also such topics as fracture and fatigue are discussed in the context of moving materials. The last part of the book deals with some runnability optimization problems with physical constraints arising from the stability and fatigue analyses including uncertainties in the parameters. The approach taken in this monograph is to proceed analytically as far as is reasonable, and only then nish the investigation numerically.In this book, we oer a systematic and careful development of many aspects of mechanics of travelling materials, particularly for panels and plates.Some of the presented results are new, and some have appeared only in specialized journals or in conference proceedings. Some aspects of the theory presented here, such as the semi-analytical treatment of the uidstructure interaction problem of a travelling panel, studies of spectral problems with free boundary singularities, and optimization of problem parameters under crack growth and instability constraints have not been considered before to any extent.Important new results relate to optimization of runnability with a longevity constraint. Damage accumulation is modelled using the theory of fatigue crack growth, with the travelling material element subjected to cyclic load-3 ing. Uncertainties must be accounted for, beca...

Theoretical study on travelling web dynamics and instability under non-homogeneous tension

“…Some previous investigations show that for an axially moving elastic paper web under homogeneous tension and certain other conditions, the value of the divergence speed V with external media (see e.g. [16,17]) and for a two-dimensional model of the web considered as a moving plate under homogeneous tension [18].…”

“…To investigate the dynamic behaviour of the plate, the representation (19) is inserted into equation (18). One obtains…”

Estimates for Divergence Velocities of Axially Moving Orthotropic Thin Plates

“…This result is practical, since the critical velocity of the plate under constant tension can be found analytically (e.g. Banichuk et al, 2011). Second, the divergence velocity of a plate under a linear tension profile is also estimated from below by the divergence velocity of an ideal membrane with constant tension having a simple analytical formula.…”

“…In the recent studies concerning axially moving plates, material properties such as orthotropicity (Banichuk et al, 2011) or viscoelasticity (Marynowski, 2010) have been taken into consideration, and their effects on the plate behaviour have been investigated. In Banichuk et al (2011), divergence instability for travelling orthotropic rectangular plates, with two opposite edges simply supported and the other two edges free, was studied and an explicit expression for the limit velocity of stable axial motion was found.…”

“…In Banichuk et al (2011), divergence instability for travelling orthotropic rectangular plates, with two opposite edges simply supported and the other two edges free, was studied and an explicit expression for the limit velocity of stable axial motion was found. Hatami et al (2009) studied free vibration of the moving orthotropic rectangular plate in sub-and supercritical speeds, and flutter and divergence instabilities at supercritical speeds.…”