Energy Stability Criterion Research Articles

One-parameter spectral Galerkin methods for Timoshenko beam system with delay boundary feedback

This paper deals with Timoshenko beam (TB) system with delay boundary feedback (DBF). For external-force-free TB system with DBF, an energy stability criterion is established. For solving general TB system with DBF, a class of one-parameter spectral Galerkin (OPSG) methods with parameter θ∈[0,1] are suggested. It is proved under the appropriate conditions that OPSG methods can preserve the energy stability in the discrete sense. Based on a new projection operator and some analytical techniques, an L2-error estimate of the methods is derived. Finally, by performing several numerical experiments, the obtained theoretical results and the computational accuracy of the methods are further illustrated.

Whole‐body motion planning and control of a quadruped robot for challenging terrain

AbstractQuadruped robots working in jungles, mountains or factories should be able to move through challenging scenarios. In this paper, we present a control framework for quadruped robots walking over rough terrain. The planner plans the trajectory of the robot's center of gravity by using the normalized energy stability criterion, which ensures that the robot is in the most stable state. A contact detection algorithm based on the probabilistic contact model is presented, which implements event‐based state switching of the quadruped robot legs. And an on‐line detection of contact force based on generalized momentum is also showed, which improves the accuracy of proprioceptive force estimation. A controller combining whole body control and virtual model control is proposed to achieve precise trajectory tracking and active compliance with environment interaction. Without any knowledge of the environment, the experiments of the quadruped robot SDUQuad‐144 climbs over significant obstacles such as 38 cm high steps and 22.5 cm high stairs are designed to verify the feasibility of the proposed method.

Nonlinear stability analysis for double-diffusive convection when the viscosity depends on temperature

Models of double-diffusive convection in a fluid layer with the viscosity having linear dependence on the temperature are analyzed here. The density is assumed to have a quadratic dependence on temperature and a linear dependence on the species concentration. Conditional and unconditional nonlinear energy stability criteria for two different models are derived. A comparison of linear and nonlinear theories suggests that there are significant regions where the basic state is subject to a deep subcritical instability when the value of solute Rayleigh number is high.

Effect of Co-alloying Ti and V on microstructure, mechanical and tribological properties of (Wx,Tiy,V1-x-y)C–Co alloys: A combined theoretical and experimental study

In this study, the effects of co-alloying Ti and V on cemented carbide alloys were investigated. Theoretically, the effects of co-alloying Ti and V on structural stability were studied from the perspective of energy stability criteria and elastic stability criteria. The results show that the face-centered-cubic (FCC) structure (Wx,Tiy,V1-x-y)C carbides are stable in two domains, one is the WC-poor domain (the concentration of WC is less than 25 at.%), and the other is WC-rich domain (the concentration of WC ranges from 61.5 at.% to 75 at.%). The calculated hardness of FCC structure (Wx,Tiy,V1-x-y)C carbide has a maximum of about 27 GPa. As for the hexagonal-close-packed (HCP) structure (Wx,Tiy,V1-x-y)C carbides, they are stable in WC-rich domain (the concentration of WC is greater than 75 at.%). The calculated hardness of HCP structure (Wx,Tiy,V1-x-y)C carbide has a maximum of about 37.2 GPa. Experimentally, the HCP structure (Wx,Tiy,V1-x-y)C-8wt.%Co alloy was prepared by mechanical alloying and hot pressing sintering as the HCP structure (Wx,Tiy,V1-x-y) C carbide has higher hardness. The results show that, as the concentration of co-alloying Ti and V increases, the hardness of the (Wx,Tiy,V1-x-y)C-8 wt.%Co alloy increases, but the fracture toughness of the (Wx,Tiy,V1-x-y)C-8 wt.% Co alloy decreases. The effect of co-alloying Ti and V on the hardness can be attributed to the solid solution strengthening and grain refinement strengthening. The achieved hardness of the (Wx,Tiy,V1-x-y)C-8 wt.% Co alloy is higher than that of the conventional binary cemented carbide alloys. The tribological properties studies show that co-alloying Ti and V can effectively improve the wear resistance of (Wx,Tiy,V1-x-y)C-8 wt.% Co alloys. The main wear mechanism can be attributed to the adhesive wear.

The Elastic-Strain Energy Stability Criterion for Complex Concentrated Alloys

An empirical method is developed, based on the calculated theoretical energy, to predict the phase formation and its stability for complex concentrated alloys. The method prediction quality is compared with the traditional empirical rules based on atomic-size mismatch, enthalpy of mixing and valence electron for a database of 235 different alloys. Considering the different available empirical methods used to date, the elastic-strain energy vs. valence electron concentration criterion shows an improved ability to distinguish between single-phase solid solutions, mixtures of solid solutions and intermetallic phases. The criterion is especially strong for alloys that precipitate the µ phase. The theoretical elastic-strain-energy parameter can be combined with other known parameters, such as those noted above, to establish new criteria which can help predicting the design of novel high-entropy alloys with on-demand combination of mechanical properties.

Influence of carbon on stability, mechanical property, electronic structure, and lattice dynamics of silicon carbonitride

AbstractIn this study, first‐principles calculations were performed to study the stability, mechanical property, electronic structure and lattice dynamics of β‐Si3(Cx,N1−x)4 silicon carbonitride. The solubility of carbon in β‐Si3(Cx,N1−x)4 having a stable structure is shown to be about 15 at.%. Within the limit of solubility, an increase in carbon concentration in β‐Si3(Cx,N1−x)4 will lead to a decrease in the Young's modulus and density and an increase in the Poisson's ratio. The study of deformation behavior shows that the most likely slip system of is on prismatic plane rather than on basal plane. This feature of β‐Si3(Cx,N1−x)4 is similar to WC. In addition, the ductility and fracture toughness of β‐Si3(Cx,N1−x)4 can be optimized by controlling the carbon concentration. The improvement in ductility and fracture toughness can be attributed to the formation of metallic bonds by the incorporation of carbon atoms. The lattice dynamics study shows that the structural stability of β‐Si3(Cx,N1−x)4 is controlled by energy stability criteria under stress‐free condition. In the stressed state, the structural stability of β‐Si3(Cx,N1−x)4 is controlled by the elastic stability criteria. Subsequently, the β‐Si3(Cx,N1−x)4 solid solution was prepared by self‐propagating high‐temperature synthesis (SHS), and the carbon‐concentration‐dependent mechanical properties were consistent with the first‐principles calculations. The maximum fracture toughness of 10.4 MPa·m0.5 was obtained in β‐Si3(Cx,N1−x)4 at carbon concentration of 5 wt%, which means that the solid solution toughening can be used as a supplement to crack bridging toughening and phase transition toughening for ceramic toughening. The results obtained in this study reveal that the β‐Si3(Cx,N1−x)4 solid solution is a promising candidate for high‐speed ceramic bearings.

Surface instability of elastic half-spaces by using the energy method

Finding the complete set of stability conditions of an elastic half-space has been an open problem ever since Biot (Biot 1963 Appl. Sci. Res. 12 , 168–182 ( doi:10.1007/BF03184638 )) first studied the surface instability of half-spaces by seeking solutions of the incremental equilibrium equations. Towards solving this problem, a method based on the energy stability criterion is developed in the present work. A variational problem of minimizing the elastic energy associated with a half-space is formulated. The second variation condition is derived and is converted to an eigenvalue problem. For a half-space of neo-Hookean materials, the eigenvalue problem is solved, which leads to complete descriptions of stability and instability regions in the deformation space.

Stability of a strut with one end fixed and the other pinned only at the first bifurcation point

In the sense of the Lagrange–Dirichlet minimum energy stability criterion, the static stability of a strut with one end fixed and the other pinned only at the first bifurcation point is investigated analytically. The second variation of potential energy expressed by the deflection is semi-positive-definite only at the first bifurcation point and vanishes only in the ‘buckling mode’ in small deflection theory. The fourth variation of potential energy is positive in the ‘buckling mode’. The potential energy of the strut at the first bifurcation point is proved to hold a minimum. Based on the Lagrange–Dirichlet stability criterion, the strut at the first bifurcation point is stable.

On the qualitative theory of the rotating Boussinesq and quasi-geostrophic equations

The author studies the rotating Boussinesq equations describing the motion of a viscous incompressible stratified fluid in a rotating system which is relevant, e.g., a Lagrangian and Eulerian analysis of a geophysical fluid flow. This article focuses on the initial boundary value problem to the rotating Boussinesq equations. The equations are derived from conservation laws in continuum physics followed by the formulation of the problem as initial value problem on Hilbert spaces. By using Faedo-Galerkin approximation and semigroup technique, existence and uniqueness of solutions are proved. Additionally, the manuscript outlines how Lyapunov functions can be used to assess energy stability criteria. The author also addresses singular problems for which the equation has parabolic structure (rotating Boussinesq equations for the atmosphere and ocean) and the singular limit is hyperbolic (quasi-geostrophic equations for the atmosphere and ocean) in the asymptotic limit of small Rossby number. In particular, this approach gives as a corollary a constructive proof of the well-posedness of the problem of quasi-geostrophic potential vorticity equations governing modons or Rossby solitons.

An existence criterion for low-dimensional materials

The discovery of graphene and other two-dimensional (2-D) materials has stimulated a general interest in low-dimensional (low-D) materials. Whereas long time ago, Peierls (1935) and Landau's (1937) theoretical work demonstrated that any one- and two-dimensional materials could not exist in any finite temperature environment. Then, two basic issues became a central concern for many researchers: How can stable low-D materials exist? What kind of low-D materials are stable? Here, we establish an energy stability criterion for low-D materials, which seeks to provide a clear answer to these questions. For a certain kind of element, the stability of its specific low-D structure is determined by several derivatives of its interatomic potential. This atomistic-based approach is then applied to study any straight/planar, low-D, equal-bond-length elemental materials. We found that 1-D monatomic chains, 2-D honeycomb lattices, square lattices, and triangular lattices are the only four permissible structures, and the stability of these structures can only be understood by assuming multi-body interatomic potentials. Using this approach, the stable existence of graphene, silicene and germanene can be explained.

Energy Density Driven Galactic Formation and Evolution

© La Fortune This article is distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and redistribution in any medium, provided that the original author and source are credited. This letter is inspired by the findings of M. Rabinowitz in his recent publication, “Why Observable Space Is Solely Three Dimensional” (Rabinowitz 2014). His paper provides coherent reasoning why our physical universe can only have one time and three spatial dimensions. He also states that “flux,” i.e., electrostatics and by analogy, gravitation, must remain consistent with threedimensional space with forces obeying the inverse-square law.

Stability of pear-shaped configurations bifurcated from a pressurized spherical balloon

It is well-known that for most spherical rubber balloons the pressure versus volume curve associated with uniform inflation is N-shaped (the pressure increases rapidly to a maximum, falls to a minimum, and subsequently increases monotonically), and that somewhere along the descending branch of this curve the spherical shape may bifurcate into a pear shape through localized thinning near one of the poles. The bifurcation is associated with the (uniform) surface tension reaching a maximum. It is previously known that whenever a pear-shaped configuration becomes possible, it has lower energy than the co-existing spherical configuration, but the stability of the pear-shaped configuration itself is unknown. With the use of the energy stability criterion, it is shown in this paper that the pear-shaped configuration is unstable under pressure control, but stable under mass control. Our calculations are carried out using the Ogden material model as an example, but it is expected that the qualitative stability results should also be valid for other material models that predict a similar N-shaped behavior for uniform inflation.

Euler and mathematical methods in mechanics

This is an essay on the life of Leonhard Euler and his achievements in theoretical mechanics. We also discuss some topics related to the development of Euler’s ideas and methods, such as divergent series and asymptotics of solutions of nonlinear differential equations, ideal fluid dynamics and Hamiltonian systems, vortex theory of systems on Lie groups with left-invariant kinetic energy, energy stability criteria, and Euler’s problem of two centers of gravitation in curved spaces.

Stability of localized bulging in inflated membrane tubes under volume control

We study the stability of localized bulging solutions in an inflated hyperelastic membrane tube using an energy stability criterion. We first use this criterion to confirm a previously known result obtained using a dynamic stability criterion, namely that under pressure control all such bulging solutions are unstable. It is then shown that, under volume control, the solutions in the early stages of bulging are unstable whereas those in the later stages of bulging are stable. To be more precise, it is found that the unstable solutions correspond exactly to the snap-back section of the associated pressure–volume diagram.

Energy relaxation for transient convection in ferrofluids

The onset of transient instability driven by a coupling of thermal and magnetic effects in an initially quiescent ferrofluid layer is investigated using the energy method. Following the work of Kim [Phys. Lett. A 372, 4709 (2008)], an energy stability criterion is derived for the underlying dynamical system by taking into account the different boundary conditions and the Prandtl number effects. The critical onset time of the instability is determined as a function of the Rayleigh number, the Prandtl number, and the thermomagnetic parameter. For larger times, our analysis predicts that the energy stability theory and the linear theory yield essentially the same results irrespective of whether the fluid under consideration is a magnetically polarizable or a nonmagnetic fluid and subcritical instabilities are not possible. For the global nonlinear stability boundary in the impulsively heated ferrofluid layer, the minimum critical onset time is found to occur when the values of the Rayleigh number and the thermomagnetic parameter are same.

Structures and energetics of 98 atom Pd–Pt nanoalloys: potential stability of the Leary tetrahedron for bimetallic nanoparticles

The energetics of 98 atom bimetallic Pd-Pt clusters are studied using a combination of: a genetic algorithm technique (to explore vast areas of the configurational space); a basin-hopping atom-exchange routine (to search for lowest-energy homotops at fixed composition); and a shell optimisation approach (to search for high symmetry isomers). The interatomic interactions between Pd and Pt are modelled by the Gupta many-body empirical potential. For most compositions, the putative global minima are found to have structures based on defective Marks decahedra, but in the composition range from Pd46Pt52 to Pd63Pt35, the Leary tetrahedron (LT)--a structure previously identified for 98 atom Lennard-Jones clusters--is consistently found as the most stable structure. Based on the excess energy stability criterion, Pd56Pt42 represents the most stable cluster across the entire composition range. This structure, a Td-symmetry LT, exhibits multi-layer segregation with an innermost core of Pd atoms, an intermediate layer of Pt atoms and an outermost Pd surface shell (Pd-Pt-Pd). The stability of the Leary tetrahedron is compared against other low-energy competing structural motifs: the Marks decahedron (Dh-M), a "quasi" tetrahedron (a closed-packed structure) and two other closed-packed structures. The stability of LT structures is rationalized in terms of their spherical shape and the large number of nearest neighbours.

Stability and bifurcation of finite deformations of elastic cylindrical membranes—Part I. Stability analysis

The stability of an inflated and extended cylindrical elastic membrane is studied by using an energy stability criterion. Three types of loading devices are considered, that control the internal pressure, the mass of enclosed gas, and the volume of enclosed fluid, respectively. Four boundary conditions are considered for different end clamping conditions. The stability conditions are derived and compared for twelve combinations of these loading devices and boundary conditions.

On Predicting Burnout in the Jet-Disk Configuration

The prediction of boiling burnout in the jet-disk configuration by Lienhard and Eichhorn (based on the mechanical energy stability criterion) is used to recorrelate the data of Katto, Monde, and others. This new correlation includes the identification of the ranges of influence of viscosity (of a Reynolds number) and of gravity (of a Froude number.) The correlation brings the data together within a standard deviation of ±8.66 percent in the range in which viscosity and gravity exert no influence.

On Predicting Boiling Burnout with the Mechanical Energy Stability Criterion

The mechanical energy stability criterion has been used in the past to predict burnout in flow boiling and is extended here to describe pool boiling burnout. Its use replaces the need for a knowledge of the Helmholtz wavelengths, which is needed in the conventional hydrodynamic theory, with a need to know the surface area of departing bubbles. The hitherto unpublished data of Sun are used to predict the latter. The geometries investigated include the infinite horizontal flat plate, the cylinder, and the sphere.

Asymptotic Stability Analysis of Viscoelastic Systems

Based on the equivalent energy stability criteria formulated elsewhere by the writer and by others, three approximate techniques are presented herein for the asymptotic stability analysis of the equilibrium of viscoelastic or damped elastic systems subjected to nonconservative and gyroscopic forces. Two of them give sufficient stability loads while the third, which is analogous to the familiar Rayleigh-Ritz method, provides approximate critical loads. It is shown that the Ritz method yields reasonably good results if dissipative (damping) forces are finite. However, further effort must be made to improve the lower bound of the approximate critical loads, which here are shown to be, in general, smaller, or at the most equal to the minimum actual critical load.