Criteria For Local Instability Research Articles

On the periodic motions of a one-degree-of-freedom oscillator

AbstractWe present a mechanical model for an oscillator with one degree of freedom under the influence of a flowing medium. Under fairly general conditions we show that the ensuing differential equation has at most two limit cycles and we give examples where exactly two limit cycles will occur. The implications of this result are that it is possible for a system of this kind to exhibit galloping even when the so-called Den Hartog criterion of local instability is not satisfied.

Strong-anisotropy-induced instability and discussion on kink band under complex loads

Compressive instability, such as the occurrence of kink bands, is common in composite materials and other highly anisotropic layered materials, limiting their use. Most previous research has focused on the microstructure and heterogeneity in studying the compressive instability of composites. In this paper, to highlight and quantify the influence of anisotropy, we investigate the deformation instability of effective homogeneous solids with strong anisotropy. Firstly, we re-write the local instability criterion from an energy perspective and verify its applicability. Even when the heterogeneity of the composites is eliminated, the derived criterion can reasonably predict the kink band instability and can be reduced to the well-known Rosen’s estimate. The strong stiffness anisotropy plays a dominant role in kink band instability. For unidirectional fiber-reinforced composites, increasing the modulus of the hard phase contributes little to an increase in the compressive strength because, as the anisotropy increases, the critical strain decreases and becomes approximately inversely proportional to the anisotropy degree. The kink band instability is also investigated under complex loads. We find that lateral tension increases the critical kink band load, while lateral compression or shear decreases the critical kink band load. To estimate the instability, we develop a series of instability phase diagrams that can be used as a guideline for predicting the compressive instability of composite materials. We have found that reducing the anisotropy of the material is the main way to increase the critical load of kink band instability, which is important for material design and engineering applications.

Material failure criterion in the finite element analysis of aluminium alloy plates under low-velocity impact

A material failure criterion, based on the combination of the revised Bressan-Williams-Hill (BWH) local instability criterion and damage evolution model, is presented to analyse numerically the fracture behaviour of aluminium alloy plates struck by an indenter with various nose shapes. The parameters used in the material failure criterion are easily calibrated for coarsely meshed shell structures. The failure modes of aluminium alloy plates under spherical, ellipsoidal, cylindrical and cuboidal indenters are discussed. The discrepancy between tensile and shear failures is considered in the failure criterion. The failure criterion is implemented into the finite element code LS-DYNA and verified against a series of experimental results. The revised material failure criterion can reasonably predict the onset of fracture of aluminium alloy plates with low mesh dependency.

Instability in charged Gauss–Bonnet–de Sitter black holes

We study the instability of the charged Gauss-Bonnet de Sitter black holes under gravito-electromagnetic perturbations. We adopt two criteria to search for an instability of the scalar type perturbations, including the local instability criterion based on the $AdS_2$ Breitenl\"{o}hner-Freedman (BF) bound at extremality and the dynamical instability via quasinormal modes by full numerical analysis. We uncover the gravitational instability in five spacetime dimensions and above, and construct the complete parameter space in terms of the ratio of event and cosmological horizons and the Gauss-Bonnet coupling. We show that the BF bound violation is a sufficient but not necessary condition for the presence of dynamical instability. While the physical origin of the instability without the Gauss-Bonnet term has been argued to be from the $AdS_2$ BF bound violation, our analysis suggests that the BF bound violation can not account for all physical origin of the instability for the charged Gauss-Bonnet black holes.

A damage-based failure model for coarsely meshed shell structures

This paper presents a fast and reliable method for failure prediction of coarsely meshed shell structures. The method is especially relevant when investigating the impact performance of offshore structures, typically stiffened panel structures where the size of the structure limits the possible detailing level during analysis.The method combines a local instability criterion with post-necking damage in order to numerically model the failure process in large shell elements. The failure model is based on power law plasticity with the stress-based Bressan–Williams–Hill (BWH) local instability criterion and a coupled damage model after incipient necking. The BWH criterion gives a robust estimation of incipient necking for coarsely meshed shell structures. After necking starts, a mesh-size dependent damage model is coupled to the element strength for material softening until failure, assuming that the strain localization occurs locally inside the element within a virtual neck. The model is incorporated in the explicit FE code LS-DYNA.The material model formulation is validated against experiments at several levels; from formability tests with varying strain states to medium and large scale impact experiments, giving a robust prediction of energy dissipation and material failure in the structure with low mesh dependency.The material model is calibrated from a single uniaxial test, and gives robust and consistent simulation results in which details of the localized necking phenomenon is included in the behavior of large shell elements. Thus, it is readily used for structural design of offshore structures in order to assess the technical safety level of the structure against collisions in all phases of the design process.

Dissipation effect on local and global stability of fluid-conveying pipes

In this article, the effect of dissipation on local and global stability of fluid conveying pipes is analyzed. The local approach refers to an infinite medium and uses wave propagation analyses without taking boundary conditions into account. The global approach refers to the same medium, but with finite length and associated with a given set of boundary conditions. The finite length system is generally studied by calculating its eigenmodes and eigenfrequencies. Criteria for local instability are derived in the first part of this paper, and dissipation is found to significantly affect local stability. Moreover, dissipation is found to have a stabilizing or destabilizing effect, depending on the other parameters. Next, numerical computations are presented for finite-length systems and results are analyzed and compared with local stability properties of the corresponding media. When the system is sufficiently long, it is found that critical velocity for global instability tends to a local criterion which can be that of local stability of the damped medium or a local transition criterion of the undamped medium, which is not necessarily the local instability criterion. Finally, a reasoning based on lengthscale ratios is developed. It allows to know which criterion is able to predict the global stability for long systems.

Local and global instabilities in nanosize rectangular prismatic gold specimens

Abstract We use molecular mechanics simulations with the tight-binding potential to study local and global instabilities in initially defect-free nanosize rectangular prismatic specimens of gold deformed in tension/compression and simple tension/compression. Whereas in simple tension/compression atoms on end faces are constrained to move axially but are free to move laterally and the cross-sectional dimensions of end faces can change, in tension/compression all three components of displacements of atoms on end faces are prescribed and the cross-section of an end face does not change. The three criteria used to delineate local instabilities in a specimen are: (i) a component of second-order spatial partial derivatives of the displacement field has large value relative to its average value in the body, (ii) the minimum eigenvalue of the Hessian of the potential energy of an atom is negative, (iii) a relatively high value of the common neighborhood parameter. A specimen becomes globally unstable when its potential energy decreases noticeably with a small increase in its deformations. It is found that the three criteria for local instability are met essentially simultaneously at the same atomic position. Deformations of interior points of a specimen are different when it is deformed in simple tension/compression from those in tension/compression. It is found that the initial unloaded configuration (or the reference configuration) of the minimum potential energy has significant in-plane stresses on the bounding surfaces and non-zero normal stresses at interior points. This initial stress distribution satisfies Cauchy’s equilibrium equations for a continuum. In deformations of a nanobar studied here, the yield stress defined as the average axial stress when the average axial stress vs. the average axial strain curve exhibits a sharp discontinuity depends upon the specimen size. It is shown possibly for the first time that deformations of the specimen are reversible if it is unloaded prior to yielding but have a permanent strain if unloaded after it has yielded. Because of residual stresses in the reference configuration, the average axial stress at yield in compression is nearly one-half of that in tension. The slope of the average axial stress vs. the average axial strain curve during unloading after it has yielded is the same as that during initial loading up to the yield point.

A local instability criterion for solid-state defects

The loss of mechanical stability is a general phenomenon that underlies a wide variety of features in the behavior of crystalline materials. In this paper, we present an instability criterion that is of general applicability, subject to the restriction of low temperatures. The criterion is applied to two situations, that of hydrostatically induced cavitation and of quasi-brittle crack growth due to mode I loading. As judged by the results of independent molecular statics simulations, the predictions of the criterion are quite accurate.

Short-range ferromagnetism and transport properties of amorphous (Gd, Y)xSi1−x alloys

We present a theoretical description and electrical conductivity measurements for amorphous (Gd, Y)xSi1−x alloys with 0.1 < x < 0.2. In our model, we take into account the strong topological disorder in the system, causing the appearance of regions with higher electron density (electron “ drops”) around nanoscale structural defects enriched with rare-earth ions (“clusters”). We calculate the local density of electron states in the drops and in the matrix and establish the criterion for local instability to ferromagnetism. In the framework of the “local phase transition” approach, we find that short-range ferromagnetic order is more favorable inside the drops than in the matrix and exists in a wide temperature range. We analyze recent measurements of the temperature and magnetic-field dependence of the electrical conductivity in these systems and show that the spin polarization of the electron states in the drops enhances the tendency towards the metal-insulator transition.

On the viability of local criteria for chaos

Recently, Zscze¸sny and Dobrowolski proposed a geometrical criterion for local instability based on the geodesic deviation equation. Although such a criterion can be useful in some cases, we show here that, in general, it is neither necessary nor sufficient for the occurrence of chaos. To this purpose, we introduce a class of chaotic two-dimensional systems with Gaussian curvature everywhere positive and, hence, locally stable. We show explicitly that chaotic behavior arises from some trajectories that reach certain non-convex parts of the boundary of the effective Riemannian manifold. Our result questions, once more, the viability of local, curvature-based criteria to predict chaotic behavior.

Geodesic Deviation Equation Approach to Chaos

A geodesic deviation equation is introduced. In an “adiabatic” approximation its exact solution is found. Perturbation theory in general case is formulated. A geometrical criterion of local instability which may lead to chaos is formulated.

The generalized local instability criterion and approximation method

We study the dynamics of Bianchi IX cosmology in the Jacobi formulation, and we demonstrate the dependence of local (or exponential) instability on the approximation. It is shown how the approximation method can preserve the property of sensitive dependence on the initial conditions—the key ingredient to deterministic chaos.

On problems of fracture of materials in compression along two internal parallel cracks

The fracture of materials under the action of compressive forces, directed along cracks which are parallel in plane cannot be described within the framework of the linear fracture mechanics. The criteria of fracture of the Griffith-Irvin or COC type, used in classical linear fracture mechanics, are not applicable in this problem, since these forces have no influence on stress intensity coefficients and on values of cracks opening. The problems of such a class may be described only by using new approaches. One of possible approaches is presented by the first author, which involves using linearized relations, derived from exact non-linear equations of deformable solid body mechanics. It should be remarked here that this approach has been widely used in problems of deformable bodies stability. As a criterion of the initiation of fracture the criterion of local instability near defects of the crack type is used. In these cases the process of loss of stability initiates the fracture process.

The local instability of simple relativistic systems

The invariant criterion of local instability of geodesic motion in terms of Riemannian (pseudo-Riemannian) geometry is proposed. The property of sensitive dependence on initial conditions is formulated as the local instability of nearby geodesics. In the study of sectional curvature, the bivector formalism is applied. We demonstrate the effectiveness of our method of investigating local instability in the behaviour of simple relativistic mechanical systems.

Simulation of shear band formation in plane strain tension and compression using FEM

A phenomenological constitutive relation using the conventional J 2-flow isotropic hardening yield function as well as a threshold shear stress based yield function that governs a ‘directionally preferred plastic component’ of shear strain, developed by the authors in an earlier work, is used for the simulation of the shear band formation. The constitutive relation which is expressed as an explicit function of elastic constants, deviatoric stress state, direction of the principal shear strain and material flow stresses, is shown to be fully capable of capturing the formation of the shear band even under material hardening conditions and does not demand any prior knowledge of the orientation of the shear band. This dual yield constitutive relation is incorporated in an FEM procedure capable of handling large strains and rotations and a numerical simulation of a tensile and compressive deformation of a plane strain specimen is performed. It is demonstrated that the formation of the shear band can be captured more easily as a natural outcome of the simulation, without resorting to any instability criterion or ‘enriched’ elements that are usually employed in the contemporary procedures. Nevertheless, the usefulness of a ‘local instability criterion’ (Ortiz et al. (1987), Comp. Meth. Appl. Mech. Eng. 61, 189) is demonstrated in this study. The model is verified using the experimental data of Anand and Spitzig (1980, J. Mech. Phys. Solids 28, 113) and the agreement between the results of the simulation and the experimental data is found to be reasonably good.

Instability criteria for the flow of an inviscid incompressible fluid.

We present a geometric estimate from below on the growth rate of a small perturbation of a three-dimensional steady flow of an ideal fluid and thus we obtain effective criteria for local instability for Euler's equations. We use these criteria to demonstrate the instability of several simple flows and to show that any flow with a hyperbolic stagnation point is unstable.

A criterion for local instability of a geodesic flow from the Jacobi equation in Fermi basis

We investigate ergodic properties of geodesic flows on compact Riemannian manifolds. A general criterion for local instability is formulated in terms of the averaged geodesic deviation equation. The negativity of the Ricci scalar provides a sufficient condition for local instability of a geodesic flow. This condition supplemented with compactness of the manifold leads to the so-called Anosov property of the geodesic flow. The effectiveness of our criterion is demonstrated in the case of multidimensional cosmological models. The chaotic behaviour of geodesics in spacetime is also discussed.

Spectra and growth rates of a generalized screw pinch

The theory of Weitzner for the bumpy θ pinch is extended to the case of a generalized screw pinch in which the equilibrium magnetic field is B = (εBr, εBθ, Bz). ε represents the slow periodic z variation of the axisymmetric equilibrium and the stability relative to their r variation. Perturbation expansions in ε and in the bumpiness of the flux surfaces are performed on the linearized equations of ideal magnetohydrodynamics. From the theory of Grad there should be four stable continua, two of which are found for modes with ω2 = O (ε2), while for the transverse modes a generalized Suydam criterion for local instability is derived. The (global) growth rates of the most unstable transverse modes are computed numerically for various Bθ and the results are compared to those of the bumpy θ pinch (where Bθ ≡ 0)) as well as to the ordinary screw pinch (where Br ≡ 0).