Periodic Delay Effects on Cutting Dynamics

We consider a delay equation whose delay is perturbed by a small periodic fluctuation. In particular, it is assumed that the delay equation exhibits a Hopf bifurcation when its delay is unperturbed. The periodically perturbed system exhibits more delicate bifurcations than a Hopf bifurcation. We show that these bifurcations are well explained by the Bogdanov-Takens bifurcation when the ratio between the frequencies of the periodic solution of the unperturbed system (Hopf bifurcation) and the external periodic perturbation is 1:2. Our analysis is based on center manifold reduction theory.     

Vortex perturbations of the nonstationary motion of a liquid with a free boundary

The first studies on the stability of nonstationary motions of a liquid with a free boundary were published relatively recently [1–4]. Investigations were conducted concerning the stability of flow in a spherical cavity [1, 2], a spherical shell [3], a strip, and an annulus of an ideal liquid. In these studies both the fundamental motion and the perturbed motion were assumed to be potential flow. Changing to Lagrangian coordinates considerably simplified the solution of the problem. Ovsyannikov [5], using Lagrangian coordinates, obtained equations for small potential perturbations of an arbitrary potential flow. The resulting equations were used for solving typical examples which showed the degree of difficulty involved in the investigation of the stability of nonstationary motions [5–8]. In all of these studies the stability was characterized by the deviation of the free boundary from its unperturbed state, i.e., by the normal component of the perturbation vector. In the present study we obtain general equations for small perturbations of the nonstationary flow of a liquid with a free boundary in Lagrangian coordinates. We find a simple expression for the normal component of the perturbation vector. In the case of potential mass forces the resulting system reduces to a single equation for some scalar function with an evolutionary condition on the free boundary. We prove an existence and uniqueness theorem for the solution, and, in particular, we answer the question of whether the linear problem concerning small potential perturbations which was formulated in [5] is correct. We investigate two examples for stability: a) the stretching of a strip and b) the compression of a circular cylinder with the condition that the initial perturbation is not of potential type.     

Bifurcation conditions for ideal fibre-reinforced materials

A bifurcation of an equilibrium state for ideal fibre-reinforced material is discussed. It is assumed that the material is elastic, locally transversely isotropic, incompressible and inextensible in the direction of fibres. On a finite state of strain an arbitrary field of small displacements is superposed and a set of governing equations for the perturbed state is derived.As an example a stability problem of a rectangular block. Objected to a finite, homogeneous deformation is considered. A discussion of the results is focused on the influence on the stability of the pressure applied in the direction of fibres.Due to the assumption of inextensibility this pressure has no influence on the state of strain, but it is shown that it may cause a loss of stability.     

Equilibrium and stability of a z pinch in a multipole magnetic field

In the theoretical investigation of the dynamic stabilization of a current-carrying plasma filament by a high-frequency multipole magnetic field it is usually assumed that the cross section of the filament has a circular form in equilibrium [1, 2]. This considerably simplifies the calculations but it does not correspond to reality, since the surface of the plasma must be fluted in the multipole field. An attempt to estimate the influence of the deformation of the filament cross section on its stability against bending in the special case of quadrupole field was made in [3], in which the parameters were determined of the elliptical cross section corresponding to a plasma filament with current in a quadrupole field and an expression was found for the electrodynamic force acting on the filament in the case of long-wavelength kink perturbations. However, this force was calculated incorrectly in [3]. In the present paper a study is made of the equilibrium and stability of a current-carrying plasma filament against kink perturbations in the general case of a multipole stabilizing field. Under the assumption that the flute depth is small, the equilibrium form of the cross section of the current-carrying plasma filament in the multipole magnetic field is found and the components of the force exerted by the field on the perturbed filament are calculated. It is shown that the external field interacts with the current in the perturbed filament only in the case of a quadrupole field. The results are discussed in connection with the problem of multipole dynamical stabilization of a z pinch against kink perturbations.     

Buckling of a circular plate made of a shape memory alloy due to a reverse thermoelastic martensite transformation

We solve the axisymmetric buckling problem for a circular plate made of a shape memory alloy undergoing reverse martensite transformation under the action of a compressing load, which occurs after the direct martensite transformation under the action of a generally different (extending or compressing) load. The problem was solved without any simplifying assumptions concerning the transverse dimension of the supplementary phase transition region related to buckling. The mathematical problem was reduced to a nonlinear eigenvalue problem. An algorithm for solving this problem was proposed. It was shown that the critical buckling load under the reverse transition, which is obtained by taking into account the evolution of the phase strains, can be many times lower than the same quantity obtained under the assumption that the material behavior is elastic even for the least (martensite) values of the elastic moduli. The critical buckling force decreases with increasing modulus of the load applied at the preliminary stage of direct transition and weakly depends on whether this load was extending or compressing. In shape memory alloys (SMA), mutually related processes of strain and direct (from the austenitic into the martensite phase) or reverse thermoelastic phase transitions may occur. The direct transition occurs under cooling and (or) an increase in stresses and is accompanied by a significant decrease (nearly by a factor of three in titan nickelide) of the Young modulus. If the direct transition occurs under the action of stresses with nonzero deviator, then it is accompanied by accumulation of macroscopic phase strains, whose intensity may reach 8%. Under the reverse transition, which occurs under heating and (or) unloading, the moduli increase and the accumulated strain is removed. For plates compressed in their plane, in the case of uniform temperature distribution over the thickness, one can separate trivial processes under which the strained plate remains plane and the phase ratio has a uniform distribution over the thickness. For sufficiently high compressing loads, the trivial process of uniform compression may become unstable in the sense that, for small perturbations of the plate deflection, temperature, the phase ratio, or the load, the difference between the corresponding perturbed process and the unperturbed process may be significant. The results of several experiments concerning the buckling of SMA elements are given in [1, 2], and the statement and solution of the corresponding boundary value problems can be found in [3–11]. The experimental studies [2] and several analytic solutions obtained for the Shanley column [3, 4], rods [5–7], rectangular plates under direct [8] and reverse [9] transitions showed that the processes of thermoelastic phase transitions can significantly (by several times) decrease the critical buckling loads compared with their elastic values calculated for the less rigid martensite state of the material. Moreover, buckling does not occur in the one-phase martensite state in which the elastic moduli are minimal but in the two-phase state in which the values of the volume fractions of the austenitic and martensite phase are approximately equal to each other. This fact is most astonishing for buckling, studied in the present paper, under the reverse transition in which the Young modulus increases approximately half as much from the beginning of the phase transition to the moment of buckling. In [3–9] and in the present paper, the static buckling criterion is used. Following this criterion, the critical load is defined to be the load such that a nontrivial solution of the corresponding quasistatic problem is possible under the action of this load. If, in the problems of stability of rods and SMA plates, small perturbations of the external load are added to small perturbations of the deflection (the critical force is independent of the amplitude of the latter), then the critical forces vary depending on the value of perturbations of the external load [5, 8, 9]. Thus, in the case of small perturbations of the load, the problem of stability of SMA elements becomes indeterminate. The solution of the stability problem for SMA elements also depends on whether the small perturbations of the phase ratio and the phase strain tensor are taken into account. According to this, the problem of stability of SMA elements can be solved in the framework of several statements (concepts, hypotheses) which differ in the set of quantities whose perturbations are admissible (taken into account) in the process of solving the problem. The variety of these statements applied to the problem of buckling of SMA elements under direct martensite transformation is briefly described in [4, 5]. But, in the problem of buckling under the reverse transformation, some of these statements must be changed. The main question which we should answer when solving the problem of stability of SMA elements is whether small perturbations of the phase ratio (the volume fraction of the martensite phase q) are taken into account, because an appropriate choice significantly varies the results of solving the stability problem. If, under the transition to the adjacent form of equilibrium, the phase ratio of all points of the body is assumed to remain the same, then we deal with the “fixed phase atio” concept. The opposite approach can be classified as the “supplementary phase transition” concept (which occurs under the transition to the adjacent form of equilibrium). It should be noted that, since SMA have temperature hysteresis, the phase ratio in SMA can endure only one-sided small variations. But if we deal with buckling under the inverse transformation, then the variation in the volume fraction of the martensite phase cannot be positive. The phase ratio is not an independent variable, like loads or temperature, but, due to the constitutive relations, its variations occur together with the temperature variations and, in the framework of connected models for a majority of SMA, together with variations in the actual stresses. Therefore, the presence or absence of variations in q is determined by the presence or absence of variations in the temperature, deflection, and load, as well as by the system of constitutive relations used in this particular problem. In the framework of unconnected models which do not take the influence of actual stresses on the phase ratio into account, the “fixed phase ratio” concept corresponds to the case of absence of temperature variations. The variations in the phase ratio may also be absent in connected models in the case of specially chosen values of variations in the temperature and (or) in the external load, as well as in the case of SMA of CuMn type, for which the influence of the actual stresses on the phase compound is absent or negligible. In the framework of the “fixed phase ratio” hypothesis, the stability problem for SMA elements has a solution coinciding in form with the solution of the corresponding elastic problem, with the elastic moduli replaced by the corresponding functions of the phase ratio. In the framework of the supplementary phase transition” concept, the result of solving the stability problem essentially depends on whether the small perturbations of the external loads are taken into account in the process of solving the problem. The point is that, when solving the problem in the connected setting, the supplementary phase transition region occupies, in general, not the entire cross-section of the plate but only part of it, and the location of the boundary of this region depends on the existence and the value of these small perturbations. More precisely, the existence of arbitrarily small perturbations of the actual load can result in finite changes of the configuration of the supplementary phase transition region and hence in finite change of the critical values of the load. Here we must distinguish the “fixed load” hypothesis where no perturbations of the external loads are admitted and the “variable load” hypothesis in the opposite case. The conditions that there no variations in the external loads imply additional equations for determining the boundary of the supplementary phase transition region. If the “supplementary phase transition” concept and the “fixed load” concept are used together, then the solution of the stability problem of SMA is uniquely determined in the same sense as the solution of the elastic stability problem under the static approach. In the framework of the “variable load” concept, the result of solving the stability problem for SMA ceases to be unique. But one can find the upper and lower bounds for the critical forces which correspond to the cases of total absence of the supplementary phase transition: the upper bound corresponds to the critical load coinciding with that determined in the framework of the “fixed phase ratio” concept, and the lower bound corresponds to the case where the entire cross-section of the plate experiences the supplementary phase transition. The first version does not need any additional name, and the second version can be called as the "all-round supplementary phase transition" hypothesis. In the present paper, the above concepts are illustrated by examples of solving problems about axisymmetric buckling of a circular freely supported or rigidly fixed plate experiencing reverse martensite transformation under the action of an external force uniformly distributed over the contour. We find analytic solutions in the framework of all the above-listed statements except for the case of free support in the “fixed load” concept, for which we obtain a numerical solution.     

Interaction of cracks with bonds between their faces in an isotropic medium reinforced by a regular system of stringers

A problem of an elastic isotropic medium with a system of foreign (transverse with respect to crack alignment) rectilinear inclusions is considered. The medium is assumed to be attenuated by a periodic system of rectilinear cracks with zones where the crack faces interact with each other. These zones are assumed to be adjacent to the crack tips, and their sizes can be commensurable with the crack size. Interaction between the crack faces in the tip zone is modeled by introducing bonds (adhesion forces) between the cracks with a specified strain diagram. The boundary-value problem of the equilibrium of a periodic system of cracks with bonds between their faces under the action of external tensile loads and forces in the bonds is reduced to a nonlinear singular integrodifferential equation with a kernel of the Cauchy kernel type. The condition of critical equilibrium of the cracks with the tip zones is formulated with allowance for the criterion of critical tension of the bonds. A case of a stress state of the medium containing zones where the crack faces interact with each other is considered.     

When Heyman’s Safe Theorem of rigid block systems fails: Non-Heymanian collapse modes of masonry structures

Heyman’s Safe Theorem is the theoretical basis for several calculation methods in masonry analysis. According to the theorem, the existence of an internal force system which equilibrates the external loads guarantees that the masonry structure is in a stable equilibrium state, assuming that a few conditions on the material behaviour are satisfied: the stone blocks have infinite compressional resistance, and the contacts between them resist only compression and friction. This paper presents simple examples in which the Safe Theorem fails: collapse occurs in spite of the existence of an equilibrated force system. A theoretical analysis of the stability of assemblies of rigid blocks with frictional contacts is then introduced: the virtual work theorem is derived, and a refined formulation of the Safe Theorem is given.     

Bifurcation and post-buckling analysis of bimodal optimum columns

A mathematical formulation of column optimization problems allowing for bimodal optimum buckling loads is developed in this paper. The columns are continuous and linearly elastic, and assumed to have no geometrical imperfections. It is first shown that bimodal solutions exist for columns that rest on a linearly elastic (Winkler) foundation and have clamped-clamped and clamped-simply supported ends. The equilibrium equation for a non-extensible, geometrically nonlinear elastic column is then derived, and the initial post-buckling behaviour of a bimodal optimum column near the bifurcation point is studied using a perturbation method. It is shown that in the general case the post-buckling behaviour is governed by a fourth order polynomial equation, i.e., near the bifurcation point there may be up to four post-buckling equilibrium states emanating from the trivial equilibrium state. Each of these equilibrium states may be either supercritical or subcritical in the vicinity of the bifurcation point. The conditions for stability of these non-trivial post-buckling states are established based on verification of positive semi-definiteness of a two-by-two matrix whose coefficients are integrals of the buckling modes and their derivatives. In the end of the paper we present and discuss numerical results for the post-buckling behaviour of several columns with bimodal optimum buckling loads.     

Optimal Control of the Equilibrium State of a Prey-Predator Model

The control problem of the equilibrium state of prey-predator model has been studied. The equilibrium states of prey-predator model are found. The optimal control law is derived from the conditions that ensure the asymptotic stability of the equilibrium state of this model using the Lyapunov function. The general solution of the equations of the perturbed state as a function of time is obtained. Graphical and numerical simulation studies of the obtained results are presented.     

Vibrations of inclined cables under skew wind

A non-linear finite element model of inclined cables, i.e. cables with non-leveled supports, in the large displacement and deformation fields is proposed for computing the dynamic response to wind loads which blow in arbitrary direction. The initial equilibrium, assumed as the static configuration under self-weight and mean wind component, is defined by a continuous approach, following an iterative procedure which starts from the configuration under self-weight only. The proposed formulation, which accounts for longitudinal inertia forces, allows to spot the circumstances when the simplified small-sag approach, adopting longitudinal mode condensation, becomes too crude. Numerical simulations have been performed employing the Proper Orthogonal Decomposition to lower the computational effort.     

Application of perturbation idea to well-known Hencky problem: A perturbation solution without small-rotation-angle assumption

In existing studies, the well-known Hencky problem, i.e. the large deflection problem of axisymmetric deformation of a circular membrane subjected to uniformly distributed loads, has been analyzed generally on small-rotation-angle assumption and solved by using the common power series method. In fact, the problem studied and the method adopted may be effectively expanded to meet the needs of larger deformation. In this study, the classical Hencky problem was extended to the problem without small-rotation-angle assumption and resolved by using the perturbation idea combining with power series method. First, the governing differential equations used for the solution of stress and deflection in the perturbed system were established. Taking the load as a perturbation parameter, the stress and deflection were expanded with respect to the parameter. By substituting the expansions into the governing equations and corresponding boundary conditions, the perturbation solution of all levels were obtained, in which the zero-order perturbation solution exactly corresponds to the small-rotation-angle solution, i.e. the solution of the unperturbed system. The results indicate that if the perturbed and unperturbed systems as well as the corresponding differential equations may be distinguished, the perturbation method proposed in this study can be extended to solve other nonlinear differential equations, as long as the differential equation of unperturbed system may be obtained by letting a certain parameter be zero in the corresponding equation of perturbed system.     

General Melnikov Approach to Implicit ODE’s

Journal of Dynamics and Differential Equations - Existence of solutions connecting a singularity of a perturbed implicit differential equations is studied. It is assumed that the unperturbed...     

Stability of plane wave solutions of the two-space-dimensional nonlinear Schrödinger equation

The stability of plane, periodic solutions of the two-dimensional nonlinear Schrödinger equation to infinitesimal, two-dimensional perturbation has been calculated and verified numerically. For standing wave disturbances, instability is found for both odd and even modes; as the period of the unperturbed solution increases, the instability associated with the odd modes remains but that associated with the even mode disappears, which is consistent with the results of Zakharov and Rubenchik [8], Saffman and Yuen [4] and Ablowitz and Segur [1] on the stability of solitons. In addition, we have identified travelling wave instabilities for the even mode perturbations which are absent in the long-wave limit. Extrapolation to the case of an unperturbed solution with infinite period suggests that these instabilities]may also be present for the soliton. In other words, the soliton is unstable to odd, standing-wave perturbations, and very likely also to even, travelling-wave perturbations.     

On the adjacent equilibrium method in the stability theory of conservative elastic continua

The relationship of the adjacent equilibrium method, the regular perturbation method and the energy method for neutral equilibrium is studied. It is shown that unlike the adjacent equilibrium method, the regular perturbation method yields, for the problems under consideration, non-homogeneous perturbation equations and that adjacent states of equilibrium do not exist at the bifurcation point. These results are then compared with the result of the energy criterion for neutral equilibrium V₂[u] = 0. It is found that although the physical arguments are different in the three methods, the resulting stability equations are identical; thus showing why the adjacent equilibrium argument, even for cases when it is incorrect, yields correct critical loads. This is followed by a discussion of an incorrect derivation of a stability condition and a notion about a load type introduced in the stability literature, which are consequences of the assumption of the general existence of adjacent equilibrium states at bifurcation points.     

Stability of a rectangular plate under biaxial tension

The stability problem of a rectangular plate undergoing uniform biaxial in-plane tensile strain is solved using the three-dimensional equations of nonlinear elasticity. The surfaces of the plate are stress-free, and special boundary conditions that allow one to separate variables in the linearized equilibrium equations are specified on the lateral surfaces. For three particular models of incompressible materials, the critical curves are constructed and the instability region is determined in the plane of the loading parameters (the multiplicities of elongations of the plate material in the unperturbed equilibrium state). The numerical results show that for thin plates loaded by tensile stresses, the size and shape of the instability region depend only slightly on the relation among the length, width, and thickness of the plate. Based on the results obtained, a simple approximate stability criterion is proposed for an elastic plate under tensile loads. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 94–103, July–August, 2007.     

Secondary buckling of a flat plate under uniaxial compression: Part 2. Analysis of clamped plate by F.E.M. and comparison with experiments

In Part 1, a theoretical analysis was used to study the secondary buckling of a simply supported plate. In Part 2, a clamped plate is analysed by the finite element method. The stability criterion for a non-linear post-buckling equilibrium state is evaluated by the sign of the determinant of the stiffness matrix. It should be noted that the secondary buckling loads of clamped plates are unexpectedly smaller than those of simply supported plates and are only one and a half times the primary buckling loads. In previous analyses, only a quarter segment of the plate was considered by assuming a stable equilibrium state with symmetrical mode. However, instability can also be predicted by considering the unsymmetrical mode over the whole plate. Results of experimental analysis of secondary instability of clamped square plates under uniaxial compression agreed with the numerical results.     

Steady flow of a viscous fluid between coaxial rotating cylinders after loss of stability

Viscous fluid flow between rotating cylinders is the best known case in which a secondary steady (equilibrium) flow develops and reaches equilibrium after loss of stability. This flow, consisting of vortices which are periodic along the axis of rotation, the so-called Taylor vortices, is the result of essentially nonlinear interactions in the flow. It arises for sufficiently high rotational velocity of the inner cylinder. The first attempt at theoretical calculation of the flow was undertaken by Stuart [1], in which the form of solution was assumed from linear stability theory and the amplitude was found from the equation expressing the energy balance in integral form. The Stuart solution was improved by Davey [2], who took into account the appearance in the solution of the next harmonic and the distortion of the fundamental mode. Concrete calculations were carried out under the assumption that the vortex dimension equals the distance between the cylinders. The results agree in general with the experimental data. Individual calculations using the method of nets were made in [3], more detailed calculations weie made in [4], and the perturbation method was applied to this problem in [5].In the following, the method of [6, 7] is applied to the study of secondary flow of a viscous fluid between cylinders. The solution is found from a single system of nonlinear differential equations, which are derived, with a definite approximation, from the equations of motion (without account for the special relation for the amplitude).     

Stability of an infinite solid with a circular cylindrical crack under compression using the Treloar potential

The fracture stability of a circular cylindrical crack in an infinite incompressible solid subjected to an axial compression is considered. A state of subcritical initial strain is assumed. The failure criterion is based on the local stability loss. The investigation is carried out in a single form for the hyper-elastic bodies with an arbitrary type of an elastic potential. Critical loads are determined for axisymmetric forms of a stability loss in the region local to the crack. The linearized problem reduced to the eigenvalue problem is solved numerically. Numerical results are obtained for solids with Treloar potential.     

Stability of heterogeneous equilibrium in a system with liquid phases

Gibbs's method is used to study the equilibrium and stability in a heterogeneous system consisting of single-component liquid phases. The coefficient of surface tension is assumed to be a constant independent of the state of the phases. An expression is obtained for the second variation of the corresponding functional, and this expression can be used to analyze the stability of nucleating centers. It is shown that in the absence of external force fields and surface tension forces the equilibrium state of a closed thermodynamic system consisting of single-component liquid phases satisfying the classical Gibbs inequalities is always stable.Translated from Izvestlya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 88–94, March–April, 1982.