The asymptotic limit of an eigenvalue problem related to the buckling of rolled elastic strips

We examine the structure of the marginal stability curves of an eigenvalue problem related to the buckling deformations observed during cold rolling of sheet metal. The instability in question is characterised by a centre “wave” pattern and arises as the interplay between the self-equilibrating residual stresses associated with the rolling process, on the one hand, and the traction force acting on the strip, on the other. When the latter effect dominates, we show that singular perturbation methods can be used to unravel a number of novel mathematical features of the linear bifurcation equation. We also provide simple quantitative formulae that facilitate an easy interpretation of the corresponding physical phenomena.     

Non-isothermal analysis of extrusion film casting using multi-mode Phan-Thien Tanner constitutive equation and comparison with experiments

Extrusion film casting (EFC) is an industrially important process which produces thousands of tons of polymer films, sheets, and coating used for various industrial as well as household applications. In this paper, we focus on an instability which occurs during certain polymer processing operations operating under predominantly elongational flow, such as extrusion film casting and fiber spinning. This instability, called the draw resonance, occurs in the form of sustained periodic fluctuations in the film dimensions. It appears when the process goes beyond the critical line speed of the EFC process. In this work, a conventional linear stability analysis is carried out for nonisothermal EFC process to determine the onset of the draw resonance. The polymer rheology is modeled by the Phan-Thien Tanner (PTT) multi-mode constitutive equation. For the implementation, a conventional shooting method approach is used. Extrusion film casting experiments were also carried out using a conventional linear low-density polyethylene (LLDPE) by varying process parameters such as draw ratio and aspect ratio, to observe the effect on the stability of the process. Linear stability analysis results under non-isothermal conditions are compared and validated with existing results from literature and with our own experimental data. This work displays the effect of multiple relaxation modes as well as the temperature influence on the stability of EFC process. Finally, results also indicate that the temperature highly affects the stability of the EFC process and cannot be ignored from modeling of EFC process.     

Draw resonance in film casting of viscoelastic fluids: A linear stability analysis

In order to understand the role of viscoelasticity on draw resonance in the isothermal film casting process, a steady state analysis and a linear stability analysis for three-dimensional flow disturbances have been conducted. The constitutive equation used is a modified convected Maxwell model, with shear-rate dependent viscosity and fluid characteristic time. The numerical results indicate that the flow is stable below a lower critical draw ratio and above an upper critical draw ratio. Shear thinning in viscosity reduces the lower critical draw ratio and somewhat increases the upper critical draw ratio—thereby enlarging the region of instability. Slower shear reduction in fluid characteristic time dramatically decreases the upper critical draw ratio but has no significant effect on the lower critical draw ratio; therefore, fluids with higher characteristic time are more stable.     

Spectral Properties of Dynamical Systems, Model Reduction and Decompositions

In this paper we discuss two issues related to model reduction of deterministic or stochastic processes. The first is the relationship of the spectral properties of the dynamics on the attractor of the original, high-dimensional dynamical system with the properties and possibilities for model reduction. We review some elements of the spectral theory of dynamical systems. We apply this theory to obtain a decomposition of the process that utilizes spectral properties of the linear Koopman operator associated with the asymptotic dynamics on the attractor. This allows us to extract the almost periodic part of the evolving process. The remainder of the process has continuous spectrum. The second topic we discuss is that of model validation, where the original, possibly high-dimensional dynamics and the dynamics of the reduced model – that can be deterministic or stochastic – are compared in some norm. Using the “statistical Takens theorem” proven in (Mezić, I. and Banaszuk, A. Physica D, 2004) we argue that comparison of average energy contained in the finite-dimensional projection is one in the hierarchy of functionals of the field that need to be checked in order to assess the accuracy of the projection.     

Stability of the combustion process in elastic combustion chambers

High-frequency instability phenomena in rigid combustion chambers have been studied theoretically in [1–3]. This phenomenon is attributed to the interaction between the combustion processes and combustion-product fluctuations in the chamber. One of the possible mechanisms of formation of high-frequency instability is examined in [3], where the combustion rate is represented in the form of a retarded pressure functional. In this case, the problem is reduced to studying the stability of a certain distributed self-oscillating time-lag system.If the oscillation frequencies of the combustion products are comparable to the natural vibrations of the shell which forms the combustion chamber, then it is natural to expect that the elasticity of the chamber walls will affect the combustion process. Coupled effects of acoustoelastic instability can arise, in whose development the vibrations of the chamber wall play a substantial role. These effects are particularly undesirable from the point of view of the vibrational stability of combustion chambers.In this paper, a theory of high-frequency instability of stationary combustion is developed with allowance for elastic deformations of the combustion chamber walls. The theory is based on the mechanism of vibrational combustion [1–3], according to which the combustion front is assumed to the concentrated, while the velocity jump at the front is expressed through a retarded pressure functional. It is assumed that the combustion product flow is one-dimensional and isentropic and that the chamber is cylindrical. The deformations of the chamber are described via the moment theory of shells. The existence is revealed of additional instability regions produced by the interaction between the elastic vibrations of the chamber walls and the acoustic oscillations of the combustion products. The influence of the relation between the elastic and acoustic frequencies and of the structural damping factor in the combustion chamber walls on the stability of the stationary combustion process is examined. The problem discussed is treated as a mathematical model for more complex asymmetric problems in which the elastic and acoustic frequencies can be of the same order.     

Characterization of material inhomogeneity by stationary values of strain energy density

The strain energy density theory is applied to analyze the fracture instability of a mechanical system whose behavior is governed by the interaction of geometry, load and material inhomogeneity. This is accomplished by obtaining the location of the global and local relative minima of the strain energy density function dW/dV denoted, respectively, by [(dW/dV)_min]_g and [(dW/dV)_min]_¢l. The former refers to a fixed global coordinate system for the entire solid while the latter corresponds to local coordinate systems referred to each material point. An unique length parameter “ℓ” representing the distance between [(dW/dV)_min^max]_g and [(dW/dV)_min^max]_ℓ can thus be found and serves as a measure of the degree of system instability tending toward failure by fracture.Numerical results are obtained and displayed graphically for the case of a solid containing an inclusion of dissimilar material. The changes that take place in material inhomogeneity, loading type and physical dimensions of the solid and inclusion are reflected through ℓ. The method suggests the compatibility of ℓ for each member of a multi-component structure in order to avoid premature failure of a single member.     

Mechanical behavior of non-crimp fabric PEEK/C thermoplastic composites

The thermoplastic resin Poly-Ether-Ether-Ketone (PEEK) was used to develop four new NCF composite materials. They refer to two different principal concepts, while each concept was investigated for two different material modifications. The tensile and compression behavior of the newly developed NCF materials was experimentally investigated. For comparison, same tests were also performed on APC-2/AS4 reference material. Prior to the mechanical tests, the quality of the produced laminates was evaluated by means of non destructive investigation (C-Scan tests) and optical microscopy analyses to obtain defects such as delaminations, porosities, micro-cracks etc. The results of the mechanical tests were exploited to obtain the “optimal” NCF fabrication process; the mechanical properties of the material solution considered to be “optimal” compare well to the respective properties of the reference material thus providing evidence for improved cost efficiency by the production of thermoplastic composite components.     

Stability of a two-phase process involving a planar phase boundary in a thermoelastic solid

This investigation is directed toward understanding the role of coupled mechanical and thermal effects in the linear stability of an isothermal antiplane shear motion which involves a steadily propagatingnormal planar phase boundary in anon-elliptic thermoelastic material. When the relevant process is static — so that the phase boundary does not move prior to the imposition of the disturbance —it is shown to be linearly stable. However, when the process involves a moving phase boundary it may be linearly unstable. Various conditions sufficient to guarantee the linear instability of the process are obtained. These depend on the monotonicity of thekinetic response function — a constitutively supplied entity which relates thedriving traction acting on a phase boundary to the local absolute temperature and the normal velocity of the phase boundary-and, in certain cases, on the spectrum of wave-numbers associated with the perturbation to which the process is subjected. Inertia is found to play an insignificant role in the qualitative features of the aforementioned sufficient conditions. It is shown, in particular, that instability can arise even when the normal velocity of the phase boundary is an increasing function of the driving traction if the temperature dependence in the kinetic response function is of a suitable nature. The instability which is present in this setting occurs only in thelong waves of the Fourier decomposition of the moving phase boundary, implying that the interface prefers to be highly wrinkled.     

Stability of deformation of viscoelastic-plastic bodies

There have been many studies of the stability of plastically deformable media. Specific problems are solved in [1, 2], etc. In these studies it has been assumed that the process of loss of stability can be investigated in the quasi-static formulation, i.e., an attempt is made to find the values of the external loads at which, together with the unperturbed equilibrium mode, the adjacent perturbed equilibrium state is possible, the transition from the unperturbed to the adjacent perturbed state being assumed to take place without unloading.The results thus obtained are in agreement with general experimental concepts.Below it is shown that the use of the model of a viscoelastic-plastic hardening body leads to a process of stability loss in which the material is plastically deformed, which justifies the use of the tangent-modulus formulation.It is established that if the external loads are conservative, then for viscoelastic-plastic bodies loss of stability will occur in the static instability mode.The stability of systems under creep conditions was previously examined in [3–8].     

Thermal instabilities in melt spinning of viscoelastic fibers

Nonisothermal melt spinning of viscoelastic fibers for which the viscosity varies in a step-like manner with respect to temperature is studied in this work. A set of one-dimensional equations based on the slender-jet approximation and the upper convected Maxwell model is used to describe the melt spinning process. The process is characterized by the force required to pull the fiber, the strength of external heating, and the draw ratio, the square of the ratio of the fiber diameter at the spinneret to that at the take-up roller. For low levels of elasticity and sufficiently strong external heating, there can be three pulling forces consistent with the same draw ratio, similar to the Newtonian case studied by Wylie et al. [31]. For higher levels of elasticity, the process exhibits a draw ratio plateau where the draw ratio hardly changes with the pulling force, reflecting a competition between thermal and elastic effects. As in the Newtonian case, external heating introduces a new instability – termed thermal instability – that is absent in isothermal systems. Linear stability analysis reveals that external heating improves stability for low levels of elasticity, but can worsen stability for higher levels of elasticity, which is again a consequence of the interplay between thermal and elastic effects. Nonlinear simulations indicate that the predictions of linear stability analysis carry over to the nonlinear regime, and show that unstable systems exhibit limit-cycle behavior. The results of the present work demonstrate a possible mechanism through which external heating can stabilize the melt spinning of viscoelastic fibers.     

Energy approach to crack growth in a fiber-reinforced brittle material modeled by an inclusion

Brittle materials randomly reinforced with a low volume fraction of strong, stiff and ductile fibers are considered, with specific reference to fiber-reinforced cements and concrete. Visible cracks in such materials are accompanied by a surrounding damage zone – together these constitute a very complex “crack system”. Enormous effort has been put into trying to understand the micromechanics of such systems. Almost all of these efforts do not deal with the “crack system” propagation behavior as a whole. The propagation process of such a “crack system” includes propagation of the visible crack and the growth of the damage zone. Propagation may take place by lengthening of the visible crack together with the concomitant lengthening of the surrounding damage zone, or simply by broadening of the damage zone while the visible crack length remains unchanged – or simultaneously by growth of both types. A phenomenological completely theoretical model (for an ideal material) is here proposed which can serve to examine the propagation process by means of energy principles, without recourse to the microscopic details of the process. An application of this theoretical approach is presented for the case of a damage zone evolving with a rectangular shape. This shape is chosen because it is expected that it will illustrate the nature of damage evolution and because the computational procedure necessary to follow the growth is the most straightforward.     

Stability and stabilization of motion of a uniaxial wheel transport platform

We consider a uniaxial wheel transport platform with a single-degree-of-freedom gyroscope moving without slipping either on a plane nonrotating horizontal surface or on the spherical rotating Earth surface. We obtain a general mathematical model which, in a special case, coincides with the model in the form of Chaplygin equations, which permits obtaining a physical interpretation of the Chaplygin equations. In the case of stationary motion where only the balance weight is controlled, we find the minimum value of the gyro angular momentum that ensures the system stability. An example with parameters of the breadboard model is used to consider the problem of the stationary motion stability and stabilization without gyro; the control matrix minimizing the quadratic performance functional is obtained. The characteristic curves of the transient process in the system are given.     

Steady shear banding in complex fluids

Shear banding (SB) is manifested by the abrupt “demixing” of the flow into regions of high and low shear rate. In this paper, we first relate analytically the rheological parameters of the fluid with the range of shear rates and stresses of SB occurrence. For this, we accept that the origin of shear banding is constitutive, and adopt a non-linear viscoelastic expression able to accommodate the double-valuedness of the stress with flow intensity, under certain conditions. We then implement the model for the case of pressure driven flow through a cylindrical pipe; we derive approximate expressions for the velocity profile in the two-banded regions (core and outer annular), the overall throughput in the presence or absence of “spurt”, and the radial location limits of the shear rate discontinuity.     

Orthotropic bone remodeling: case of plane stresses

Cancellous bone is constituted by a porous solid matrix filled with fluid. Matrix microstructure gives bone most of its mechanical strength properties. In our macroscopic approach, bone is seen as a continuous medium with a local (at our scale) time-dependent linearly elastic orthotropic behavior. Remodeling consists, by matrix material apposition or resorption, in microstructure modifications in order to optimize its mechanical characteristics. The proposed model is built on a time iterative procedure where the compliance tensor evolves such that, depending on the applied stresses, principal strains tend to fall within an admissible domain. The suggested remodeling laws in this work modify the elasticity “constants” as well as the orthotropy directions. The first results presented here correspond to the plane stresses case.     

Molecular dynamics simulation of deformation and failure of nanocrystals of bcc metals

Simulations of uniaxial and hydrostatic tension of Fe and Mo nanocrystal are made by molecular dynamics method. Stress versus strain are obtained while regularities of lattice rearrangement during nanocrystal plastic deformation are considered. Local instability of nanocrystal lattice, which is the cause for transition from elastic to plastic deformation of nanocrystal, is found. It is shown that local shear stresses is a driving force of nanocrystal lattice rearrangements under the conditions of both uniaxial and hydrostatic tension, so, local instability of nanocrystal of bcc metals should be considered as shear instability. Realization of “orthorhombic” path of deformation at 1 0 0 tension of Mo nanocrystal is specific case of above effect. It is demonstrated that unlike covalent nanocrystal, metallic nanocrystals display “heterogeneous” mechanism of crack nucleation, which essence is that cracks nucleate not in homogeneous elastically deformed lattice but in shear bands or near their boundaries, i.e., after non-homogeneous plastic deformation of nanocrystal.     

Bounds for the effective properties of heterogeneous plates

This paper presents new bounds for heterogeneous plates which are similar to the well-known Hashin–Shtrikman bounds, but take into account plate boundary conditions. The Hashin–Shtrikman variational principle is used with a self-adjoint Green-operator with traction-free boundary conditions proposed by the authors. This variational formulation enables to derive lower and upper bounds for the effective in-plane and out-of-plane elastic properties of the plate. Two applications of the general theory are considered: first, in-plane invariant polarization fields are used to recover the “first-order” bounds proposed by Kolpakov [Kolpakov, A.G., 1999. Variational principles for stiffnesses of a non-homogeneous plate. J. Meth. Phys. Solids 47, 2075–2092] for general heterogeneous plates; next, “second-order bounds” for n-phase plates whose constituents are statistically homogeneous in the in-plane directions are obtained. The results related to a two-phase material made of elastic isotropic materials are shown. The “second-order” bounds for the plate elastic properties are compared with the plate properties of homogeneous plates made of materials having an elasticity tensor computed from “second-order” Hashin–Shtrikman bounds in an infinite domain.     

Stability of a flow in a circular tube

Many studies, both theoretical and experimental, have been dedicated to the stability of flow in a circular tube (see, for example, review [1]). In every case mathematical investigation has not succeeded in obtaining an expression for hydrodynamic instability of such a flow for disturbances of sufficiently low amplitude. (An exception is [2].) Experiment also indicates the stability of such a flow [3], with a laminar mode being extended to Reynolds numbers of the order of tens of thousands. These facts are the basis for the assumption that the flow of a viscous incompressible liquid in a circular tube is stable for small perturbations. However, there is no analytical or even numerical proof of this hypothesis. Moreover, some studies, for example [2], indicate the instability of such a flow in relation to three-dimensional nonaxiosymmetric perturbations. The analysis of hydrodynamic stability with respect to three-dimensional disturbances of flow within a circular tube conducted in this study showed the stability of the flow over a wide range of wave numbers and Reynolds numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 20–24, January–February, 1973.     

A Nonlinear Stability Analysis for Thermoconvective Magnetized Ferrofluid Saturating a Porous Medium

The generalized energy method is developed to study the nonlinear stability analysis for a magnetized ferrofluid layer heated from below saturating a porous medium, in the stress-free boundary case. The mathematical emphasis is on how to control the nonlinear terms caused by magnetic body force. By introducing a suitable generalized energy functional, we perform a nonlinear energy stability (conditional) analysis. It is found that the nonlinear critical stability magnetic thermal Rayleigh number does not coincide with that of linear instability analysis, and thus indicates that the subcritical instabilities are possible. However, it is noted that, in case of non-ferrofluid, global nonlinear stability Rayleigh number is exactly the same as that for linear instability. For lower values of magnetic parameters, this coincidence is immediately lost. The effect of magnetic parameter, M ₃, and medium permeability, Da, on subcritical instability region has also been analyzed. It is shown that with the increase of magnetic parameter (M ₃) and Darcy number (Da), the subcritical instability region between the two theories decreases quickly. We also demonstrate coupling between the buoyancy and magnetic forces in nonlinear energy stability analysis as well as in linear instability analysis.     

Effect of inertia and gravity on the draw resonance in high-speed film casting of Newtonian fluids

The interplay between inertia and gravity is examined for Newtonian film casting in this study. Both linear and nonlinear stability analyses are carried out. Linear stability analysis indicates that while both inertia and gravity enhance the stability in film casting, inertia plays a more dominant role regarding the critical draw ratio. In contrast, the disturbance frequency is more sensitive to the effect of gravity. The nonlinear results show that at the critical draw ratio, the system oscillates harmonically, indicating the onset of a Hopf bifurcation. For a draw ratio above criticality, finite-amplitude disturbances are amplified, and sustained oscillation is achieved. It is found that the growth rate increases with draw ratio, but decreases with inertia and gravity, which suggests that initial transients tend to take longer to die out for a fluid with inertia and gravity. Transient post-critical calculations show that the nonlinearity can be effectively halted by inertia and gravity. The oscillation frequency (film-thickness amplitude) decreases (increases) with draw ratio. However, the film oscillates more frequently but less fiercely with stronger inertia and gravity effects. The rupture of the film is also examined, and is found to be delayed by inertia and gravity. Interestingly, although the oscillation amplitude is found to be weakest at the chill roll, it is at this location that the film tends to rupture first.     

Layers in the Presence of Conservation Laws

We study standing layers in systems where a reaction-diffusion equation couples to a scalar conservation law. Our results give spectral stability and instability results depending only on relative monotonicity of the two components of the system. We also prove the robustness of layers and their stability properties. Our results classify stability properties of layers in most such systems. Our method is based on tracking the point spectrum during a homotopy to a simple, decoupled system. Main difficulty is the possibility of eigenvalues disappearing in a branch point of the essential spectrum. This phenomenon is investigated using a Lyapunov?CSchmidt reduction method on exponentially weighted spaces combined with a matching procedure for the far-field.