On interfacial properties in gradient damaging continua

The bifurcation problem near an interface is considered for a heterogeneous body made of two different materials that damage following gradient constitutive relations. The roles of internal length scales on bifurcation are studied especially in the shortwavelength regime. It is shown that the interfacial complementing condition is always satisfied meaning that a minimum wavelength exists for the bifurcation mode. The regularization properties of gradient damage models are underlined. A simple plane strain problem is used to illustrate the results. The interface bifurcated modes are explicitly computed: their wavelengths turn out to be fixed by the gradient coefficient; the influence of the interface behaviour is also highlighted. To cite this article: A. Benallal, C. Comi, C. R. Mecanique 333 (2005).     

Bifurcations of relative equilibria of an oblate gyrostat with a discrete damper

We investigate relative equilibria of an oblate gyrostat with a discrete damper. Linear and nonlinear methods yield stability conditions for simple spins about the nominal principal axes. We use analytical and numerical methods to explore other equilibria, including bifurcations that occur for varying rotor momentum and damper parameters. These bifurcations are complex structures that are perturbations of the zero rotor momentum case. We use Lyapunov–Schmidt reduction to determine an analytic relationship between parameters to determine conditions for which a jump phenomenon occurs. This paper is declared a work of the U.S. Government and is not subject to copyright protection in the United States     

Transition bifurcation branches in non-linear water waves

We are concerned with the numerical computation of progressive free surface gravity waves on a horizontal bed. They are regarded as families of bifurcation branches (λ,A)_Q of constant discharge Q. Numerically we determine two transition values Q₁ and Q₂ with corresponding transition bifurcation branches that classify waves into three disjoint branch sets B₁, B₂ and B₃. Their members are families of waves (λ,A)_Q satisfying the conditions 0<Q² ?Q, Q <Q² ?Q and Q <Q² <B/27, respectively. The bifurcation patterns are analysed in some detail from the computed bifurcation diagram, which shows that in B₁ bifurcation is to the left and the amplitude A increases as the wavelength λ decreases; in B₂ bifurcation is to the right and turning points are observed nearly at breaking point. In B₃ bifurcation is to the right and A increases monotonically with λ.     

A feedback linearization design for the control of vehicle’s lateral dynamics

In this paper, the feedback linearization scheme is applied to the control of vehicle’s lateral dynamics. Based on the assumption of constant driving speed, a second-order nonlinear lateral dynamical model is adopted for controller design. It was observed in (Liaw, D.C., Chung, W.-C. in 2006 IEEE International Conference on Systems, Man, and Cybernetics, 2006) that the saddle-node bifurcation would appear in vehicle dynamics with respect to the variation of the front wheel steering angle, which might result in spin and/or system instability. The vehicle dynamics at the saddle node bifurcation point is derived and then decomposed as an affine nominal model plus the remaining term of the overall system dynamics. Feedback linearization scheme is employed to construct the stabilizing control laws for the nominal model. The stability of the overall vehicle dynamics at the saddle-node bifurcation is then guaranteed by applying Lyapunov stability criteria. Since the remaining term of the vehicle dynamics contains the steering control input, which might change system equilibrium except the designed one. Parametric analysis of system equilibrium for an example vehicle model is also obtained to classify the regime of control gains for potential behavior of vehicle’s dynamical behavior.     

Slightly asymmetric beams: Examples of a new class of structural optima

We demonstrate examples of beams with symmetric cross sections, loading and boundary conditions where, contrary to the engineer's intuition, slight perturbation of the symmetry (either in the cross section's geometry, or in the loading or boundary conditions) improves the overall structural response. We apply the classical Euler-Bernoulli beam model combined with linearly elastic or ideally plastic material law.     

Sufficient conditions for stability of Euler elasticas

Based on the conjugate point theory in calculus of variations, sufficient conditions on stability of all Euler elasticas for a column clamped at one end and guided at the other are proposed in this paper. For the Euler elasticas, an initial value problem is solved numerically by the Euler iteration. Sufficient conditions are satisfied that Euler elasticas with one half wave are stable and the others with more than one half wave are unstable in post-buckling. As load is smaller than the Euler critical force, it is sufficient theoretically that straight shape is stable. As load exceeds the Euler critical force, it is sufficient theoretically that straight shape is unstable.     

Bifurcation and stability analysis of an anaerobic digestion model

This paper presents the dynamic behaviour of the anaerobic digestion process, based on a simplified model. The hydraulic, biological and physicochemical processes such as those which underpin anaerobic digestion have more than one stable stationary solution and they compete with each other. Further, the attractive domains of the stable solutions vary with the key parameters. Thus, some initial transient process moving toward one stable solution could suddenly move towards another solution, at which a so-call catastrophe takes places (e.g. washout). The paper systematically analyses the stationary solutions with their associated stability, which provides insight and guidance for anaerobic digestion reactor design, operation and control.     

Hopf bifurcation from nonperiodic solutions of differential equations. I. Linear theory

In this and succeeding papers we consider some foundational elements of a theory of Hopf bifurcation from non-periodic solutions of ordinary differential equations.     

Transition to chaotic vibrations for harmonically forced perfect and imperfect circular plates

The transition from periodic to chaotic vibrations in free-edge, perfect and imperfect circular plates, is numerically studied. A pointwise harmonic forcing with constant frequency and increasing amplitude is applied to observe the bifurcation scenario. The von Kármán equations for thin plates, including geometric non-linearity, are used to model the large-amplitude vibrations. A Galerkin approach based on the eigenmodes of the perfect plate allows discretizing the model. The resulting ordinary-differential equations are numerically integrated. Bifurcation diagrams of Poincaré maps, Lyapunov exponents and Fourier spectra analysis reveal the transitions and the energy exchange between modes. The transition to chaotic vibration is studied in the frequency range of the first eigenfrequencies. The complete bifurcation diagram and the critical forces needed to attain the chaotic regime are especially addressed. For perfect plates, it is found that a direct transition from periodic to chaotic vibrations is at hand. For imperfect plates displaying specific internal resonance relationships, the energy is first exchanged between resonant modes before the chaotic regime. Finally, the nature of the chaotic regime, where a high-dimensional chaos is numerically found, is questioned within the framework of wave turbulence. These numerical findings confirm a number of experimental observations made on shells, where the generic route to chaos displays a quasiperiodic regime before the chaotic state, where the modes, sharing internal resonance relationship with the excitation frequency, appear in the response.     

Chaotic behavior of a parametrically excited nonlinear mechanical system

The chaotic dynamics of a single-degree-of-freedom nonlinear mechanical system under periodic parametric excitation is investigated. Besides the well known type-I and type-III intermittent transitions to chaos we give numerical evidence that the system can follow an alternative route to chaos via intermittency from an equilibrium state to a chaotic one, which was not found in the previous simulations of the dynamics of the system.     

Complete solution of the stability problem for elastica of Euler's column

The stability of postcritical equilibrium forms of a simply supported column loaded with an axial force is analyzed. Investigating the sign of the second variation of the column's total energy, we obtain the Sturm-Liouville boundary-value problem, which is solved numerically. The stability conditions are formulated in terms of eigenvalues of the problem. The complete solution to the column plane elastica is given. The ranges of the compressive force corresponding to stable equilibrium configurations of the column are established.     

Supercritical and subcritical buckling bifurcations for a compressible hyperelastic slab subjected to compression

We study the buckling bifurcation of a compressible hyperelastic slab under compression with sliding–sliding end conditions. The combined series-asymptotic expansions method is used to derive the simplified model equations. Linear bifurcation analysis yields the critical stress value of buckling, which gives a non-linear correction to the classical Euler buckling formula. The correction is due to the geometrical non-linearities coupled with the material non-linearities. Then through non-linear bifurcation analysis, the approximate analytical solutions for the post-buckling deformations are obtained. The amplitude of buckling is expressed explicitly in terms of the aspect ratio, the incremental dimensionless engineering stress, the mode of buckling and the material constants. Most importantly, we find that both supercritical and subcritical buckling could occur for compressible materials. The bifurcation type depends on the material constants, the geometry of the slab and the mode numbers.     

Homogeneous Equilibrium Configurations of a Hyperelastic Compressible Cube under Equitriaxial Dead-Load Tractions

Non-uniqueness, bifurcation and stability of homogeneous solutions to the equilibrium problem of a hyperelastic cube subject to equitriaxial dead-load tractions are investigated. Besides the basic and theoretical questions raised by the analysis, the study is motivated by the somewhat surprising feature of this nonlinear problem for which the symmetric load may give rise to asymmetric stable deformations. In reality, the equilibrium problem, formulated for general homogeneous compressible isotropic materials with polyconvex energy function, may exhibit primary and secondary bifurcations. A primary bifurcation occurs when there exist paths of equilibrium states that bifurcate from the primary path of three equal principal stretches. These bifurcation branches have two coinciding stretches and along them, through secondary bifurcations, other completely asymmetric bifurcation branches, which are characterized by all three stretches different, may risen. In this case, the cube transforms into an oblique parallelepiped. With increasing loads, they are also possible discontinuous paths of equilibria which evince prompt jumps in the deformation process. Of course, the set of asymmetric solutions admitted by the equilibrium problem depends on the specific form of the stored energy function adopted. In this paper, expressions governing the global development of asymmetric equilibrium branches are derived. In particular, conditions to have bifurcation points are individualized. For compressible neo-Hookean and Mooney-Rivlin materials a wide parametric analysis is carried out showing by means of graphs the most interesting branches. Finally, using the energy criterion, a detailed study is performed to assess the stability of the computed solutions.      

Singular analysis of bifurcation systems with two parameters

Bifurcation properties of dynamical systems with two parameters are investigated in this paper. The definition of transition set is proposed, and the approach developed is used to investigate the dynamic characteristic of the nonlin- ear forced Duffing system with nonlinear feedback controller. The whole parametric plane is divided into several persistent regions by the transition set, and then the bifurcation dia- grams in different persistent regions are obtained.     

Chaos in a coupled oscillators system with widely spaced frequencies and energy-preserving non-linearity

This paper is a sequel to Tuwankotta [Widely separated frequencies in coupled oscillators with energy-preserving nonlinearity, Physica D 182 (2003) 125-149.], where a system of coupled oscillators with widely separated frequencies and energy-preserving quadratic non-linearity is studied. We analyze the system for a different set of parameter values compared with those in Tuwankotta [Widely separated frequencies in coupled oscillators with energy-preserving nonlinearity, Physica D 182 (2003) 125-149.]. In this set of parameters, the manifold of equilibria are non-compact. This turns out to have an interesting consequence to the dynamics. Numerically, we found interesting bifurcations and dynamics such as torus (Neimark-Sacker) bifurcation, chaos and heteroclinic-like behavior. The heteroclinic-like behavior is of particular interest since it is related to the regime behavior of the atmospheric flow which motivates the analysis in Tuwankotta [Widely separated frequencies in coupled oscillators with energy-preserving nonlinearity, Physica D 182 (2003) 125-149.] and this paper.     

Buoyancy-driven instability in a vertical cylinder: Binary fluids with Soret effect. Part II: Weakly non-linear solutions

The buoyancy-driven instability of a monocomponent or binary fluid that is completely contained in a vertical circular cylinder is investigated, including the influence of the Soret effect for the binary mixture. The Boussinesq approximation is used, and weakly-non-linear solutions are generated via Galerkin's technique using an expansion in the eigensolutions of the associated linear stability problem. Various types of fluid mixtures and cylindrical domains are considered. Flow structure and associated heat transfer are computed and experimental observations are cited when possible.     

Bifurcation condition of crack pattern in the periodic rectangular array

Bifurcation condition of crack pattern in the periodic rectangular array plays an important role in determining the final failure pattern of rock material. An approximation for the critical crack size/spacing ratio is established for a uniformly growing periodic rectangular array yields to a non-uniform growing pattern of crack growth. Numerical results show that the critical crack size/spacing ratio λ_cr depends on the number of cracks, the crack spacing, the perpendicular distance between two adjacent rows, as well as the loading conditions. In general, λ_cr increases with the number of lines. It is observed that the critical crack size/spacing ratio λ_cr for the periodic rectangular array decreases with an increase in the perpendicular distance between two adjacent rows. It is clear that the critical crack size/spacing ratio λ_cr for the periodic rectangular array under shear stress increases with increasing the crack spacing.     

Bifurcation analysis of reduced rotor model based on nonlinear transient POD method

The singularity theory is applied to study the bifurcation behaviors of a reduced rotor model obtained by nonlinear transient POD method in this paper. A six degrees of freedom (DOFs) rotor model with cubically nonlinear stiffness supporting at both ends is established by the Newton's second law. The nonlinear transient POD method is used to reduce a six-DOFs model to a one-DOF one. The reduced model reserves the dynamical characteristics and occupies most POM energy of the original one. The singularity of the reduced system is analyzed, which replaces the original system. The bifurcation equation of the reduced model indicates that it is a high co-dimension bifurcation problem with co-dimension 6, and the universal unfolding (UN) is provided. The transient sets of six unfolding parameters, the bifurcation diagrams between the bifurcation parameter and the state variable are plotted. The results obtained in this paper present a new kind of method to study the UN theory of multi-DOFs rotor system.