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This book discusses the stability of axially moving materials, which are encountered in process industry applications such as papermaking. A special emphasis is given to analytical and semianalytical approaches. As preliminaries, we consider a variety of problems across mechanics involving bifurcations, allowing to introduce the techniques in a simplified setting.

In the main part of the book, the fundamentals of the theory of axially moving materials are presented in a systematic manner, including both elastic and viscoelastic material models, and the connection between the beam and panel models. The issues that arise in formulating boundary conditions specifically for axially moving materials are discussed. Some problems involving axially moving isotropic and orthotropic elastic plates are analyzed. Analytical free-vibration solutions for axially moving strings with and without damping are derived. A simple model for fluid--structure interaction of an axially moving panel is presented in detail.

This book is addressed to researchers, industrial specialists and students in the fields of theoretical and applied mechanics, and of applied and computational mathematics.

### Chapter 1. Prototype Problems: Bifurcations of Different Kinds

Abstract
In this chapter, we present some prototype bifurcation problems that arise in the mechanics of rigid and deformable structural elements. These problems are typical for engineering applications and characterize the approaches that can be applied in the investigation of stability. Some methods of bifurcation theory will be presented in the context of stability studies of the considered one-dimensional mechanical problems.
Nikolay Banichuk, Alexander Barsuk, Juha Jeronen, Tero Tuovinen, Pekka Neittaanmäki

### Chapter 2. Bifurcation Analysis for Polynomial Equations

Abstract
This chapter is devoted to bifurcation problems based on some models described by polynomial equations with real coefficients. Bifurcation analysis, parametric representations of solutions and their asymptotic analysis and expressions are described within a framework of analytical approaches. The results presented in this chapter can be used to help locate the bifurcation points of the solution curves. The results also allow the development of very efficient procedures for sensitivity analysis of the dependences of solutions on the problem parameters.
Nikolay Banichuk, Alexander Barsuk, Juha Jeronen, Tero Tuovinen, Pekka Neittaanmäki

### Chapter 3. Nonconservative Systems with a Finite Number of Degrees of Freedom

Abstract
In this chapter we present some results on the stability and bifurcations of the systems with a finite number of degrees of freedom. We consider damping-induced destabilization in nonconservative systems. We start with a general theoretical treatment of the topic. As the model problem, we consider the double pendulum subject to both a follower force and gravitational loading. A special case of interest is treated with the theoretical framework. The chapter finishes with a thorough presentation and analysis of the model problem including the nonlinear dynamics, quasistatic equilibrium paths and their stability, and special cases of interest. In numerical examples, we show equilibrium paths and trajectory density visualizations of the time evolution of the nonlinear system. Sample-based uncertainty quantification is employed to capture both branches of a bifurcation in the same visualization.
Nikolay Banichuk, Alexander Barsuk, Juha Jeronen, Tero Tuovinen, Pekka Neittaanmäki

### Chapter 4. Some General Methods

Abstract
In this chapter, we take a brief general look into elastic stability in the setting of classical solid mechanics. We introduce the different types of stability loss, and then look at conditions under which merging of eigenvalues may occur. We consider a problem where applying symmetry arguments allows us to eliminate multiple (merged) eigenvalues, thus reducing the problem to determining a classical simple eigenvalue. We discuss a general technique to look for bifurcations in problems formulated as implicit functionals. Finally, we consider a variational approach to the stability analysis of an axially moving panel (a plate undergoing cylindrical deformation).
Nikolay Banichuk, Alexander Barsuk, Juha Jeronen, Tero Tuovinen, Pekka Neittaanmäki

### Chapter 5. Modeling and Stability Analysis of Axially Moving Materials

Abstract
This chapter considers the fundamentals of axially moving materials. We systematically develop and solve a simplified model for the small-amplitude free vibrations of a one-dimensional axially moving structure. The aim of the systematic presentation is to clearly expose the construction of the model, from physical principles through to the final linearized equations, which are then used to determine the stability of the physical system by linear stability analysis. We consider the construction of some commonly used linear material models, including the linear elastic solid, and two viscoelastic solids, the Kelvin-Voigt solid and the three-parameter solid. We highlight the connection between the beam and panel (plate undergoing cylindrical deformation) models. We derive the weak forms of the governing partial differential equations, and devote special attention to deriving appropriate boundary conditions for axially moving structures. With the help of a nondimensional parameterization, we identify scalings for which the physical system behaves identically. Finally, in the numerical examples, we present the results of a linear stability analysis for isotropic linear elastic and Kelvin-Voigt viscoelastic panels.
Nikolay Banichuk, Alexander Barsuk, Juha Jeronen, Tero Tuovinen, Pekka Neittaanmäki

### Chapter 6. Stability of Axially Moving Plates

Abstract
This chapter focuses on the stability analysis of axially moving materials, in the context of two-dimensional models. There are many similarities with the classical stability analysis of structures, such as the buckling analysis of plates. However, the presence of axial motion introduces inertial effects to the model. We consider the stability of an axially moving elastic isotropic plate travelling at a constant velocity between two supports and experiencing small transverse vibrations. We investigate the stability of the plate using an analytical approach. We also look at elastic orthotropic plates, and an elastic isotropic plate subjected to an axial tension distribution that varies in the width direction.
Nikolay Banichuk, Alexander Barsuk, Juha Jeronen, Tero Tuovinen, Pekka Neittaanmäki

### Chapter 7. Stability of Axially Moving Strings, Beams and Panels

Abstract
In this chapter, using analytical approaches, we consider the problems of dynamics and stability of moving elastic rods and strings, axially traveling between two supports at a constant velocity. Transverse, longitudinal and torsional vibrations of the moving structure are reduced to the same mathematical form, a hyperbolic second-order partial differential equation. The analysis is then extended to the axially traveling string with damping. An analytical free-vibration solution is obtained. It is seen external friction leads to stabilization, whereas internal friction in the traveling material will destabilize the system in a dynamic mode at the static critical point. Finally, we consider the effects of bending rigidity, which in the case of paper materials introduces a singular perturbation to the governing equation. We consider an implicit exact eigensolution for beams, the effect of elastic supports at the boundaries to the vibration behavior of a long traveling beam, and the stability of a beam traveling in a homogeneous gravitational field.
Nikolay Banichuk, Alexander Barsuk, Juha Jeronen, Tero Tuovinen, Pekka Neittaanmäki

### Chapter 8. Stability in Fluid—Structure Interaction of Axially Moving Materials

Abstract
This chapter considers fluid—structure interaction problems, where the vibrations of the structure are the main interest, but its motion is affected by the flow of the surrounding medium, such as air or water. We review some basic concepts of fluid mechanics, and then systematically derive a Green’s function based analytical solution of the flow component of a simple fluid—structure interaction problem in two space dimensions. As the structure component in the fluid—structure interaction problem we consider traveling ideal strings and panels. In the numerical results, we examine bifurcations in the natural frequencies of the coupled system, for strings, and for linear elastic and Kelvin-Voigt viscoelastic panels. Whereas the natural frequencies of the traveling ideal string have no bifurcation points, bifurcations appear in the model with fluid—structure interaction whenever there is an axial free-stream flow.
Nikolay Banichuk, Alexander Barsuk, Juha Jeronen, Tero Tuovinen, Pekka Neittaanmäki

### Chapter 9. Optimization of Elastic Bodies Subjected to Thermal Loads

Abstract
In this chapter, we consider three thermoelastic optimization problems. We look at the optimal thickness distribution for a beam of variable thickness, when the goal is to maximize its resistance to thermoelastic buckling, or in other words, to maximize the critical temperature at which buckling occurs. In the second problem, we allow the beam to be constructed inhomogeneously, looking for an optimal distribution of materials that maximizes the critical temperature. The third and final problem concerns heat conduction in locally orthotropic solid bodies. By locally orthotropic, we mean a particular type of inhomogeneity, where the principal directions (axes of orthotropy) may vary as a function of the space coordinates. We derive a guaranteed double-sided estimate for energy dissipation that occurs in heat conduction in a locally orthotropic body, without assuming anything about the material orientation field. This yields guaranteed lower and upper bounds for energy dissipation that always hold regardless of how the local material orientation is distributed in the solid body.
Nikolay Banichuk, Alexander Barsuk, Juha Jeronen, Tero Tuovinen, Pekka Neittaanmäki