Noise and stability in differential delay equations

We study the stability of linear stochastic differential delay equations in the presence of additive or multiplicative white and colored noise. Using a stochastic analog of the second Liapunov method, sufficient conditions for mean square and stochastic stability are derived.     

Nonlinear vibrations in beams and frames: The effect of the deformed equilibrium state

In the study of nonlinear vibrations of planar frames and beams with infinitesimal displacements and strains, the influence of the static displacements resulting from gravity effect and other conservative loads is usually disregarded. This paper discusses the effect of the deformed equilibrium configuration on the nonlinear vibrations through the analysis of two planar structures. Both structures present a two-to-one internal resonance and a primary response of the second mode. The equations of motion are reduced to two degrees of freedom and contain all geometrical and inertial nonlinear terms. These equations are derived by modal superposition with additional subsidiary conditions. In the two cases analyzed, the deformed equilibrium configuration virtually coincides with the undeformed configuration. Also, 2% is the maximum difference presented by the first two lower frequencies. The modes are practically coincident for the deformed and undeformed configurations. Nevertheless, the analysis of the frequency response curves clearly shows that the effect of the deformed equilibrium configuration produces a significant translation along the detuning factor axis. Such effect is even more important in the amplitude response curves. The phenomena represented by these curves may be distinct for the same excitation amplitude.     

A new two-point deformation tensor and its relation to the classical kinematical framework and the stress concept

Starting from the issue of what is the correct form for a Legendre transformation of the strain energy in terms of Eulerian and two-point tensor variables we introduce a new two-point deformation tensor, namely H=(F−F^−T)/2, as a possible deformation measure involving points in two distinct configurations. The Lie derivative of H is work conjugate to the first Piola–Kirchhoff stress tensor P. The deformation measure H leads to straightforward manipulations within a two-point setting such as the derivation of the virtual work equation and its linearization required for finite element implementation. The manipulations are analogous to those used for the Lagrangian and Eulerian frameworks. It is also shown that the Legendre transformation in terms of two-point tensors and spatial tensors require Lie derivatives. As an illustrative example we propose a simple Saint Venant–Kirchhoff type of a strain-energy function in terms of H. The constitutive model leads to physically meaningful results also for the large compressive strain domain, which is not the case for the classical Saint Venant–Kirchhoff material.     

Adaptive Control of Nonlinear Dynamical Systems Using a Model Reference Approach

In this paper we consider using a model reference adaptive control approach to control nonlinear systems. We consider the controller design and stability analysis associated with these type of adaptive systems. Then we discuss the use of model reference adaptive control algorithms to control systems which exhibit nonlinear dynamical behaviour using the example of a Duffing oscillator being controlled to follow a linear reference model. For this system we show that if the nonlinearity is small then standard linear model reference control can be applied. A second example, which is often found in synchronization applications, is when the nonlinearities in the plant and reference model are identical. Again we show that linear model reference adaptive control is sufficient to control the system. Finally we consider controlling more general nonlinear systems using adaptive feedback linearization to control scalar nonlinear systems. As an example we use the Lorenz and Chua systems with parameter values such that they both have chaotic dynamics. The Lorenz system is used as a reference model and a single coordinate from the Chua system is controlled to follow one of the Lorenz system coordinates.     

Analysis of viscoelastic contact with normal compliance,friction and wear diffusionAnalyse d'un problème viscoélastique avec compliance normale,frottement et diffusion d'usure

We consider a quasistatic problem of frictional contact between a viscoelastic body and a moving foundation. The contact is with wear and is modeled by normal compliance and a law of dry friction. The novelty in the model is that it allows for the diffusion of the wear debris over the potential contact surface. Such kind of phenomena arise in orthopaedic biomechanics and influence the properties of joint prosthesis. We derive a weak formulation of the problem and state that, under a smallness assumption on the problem data, there exists a unique weak solution for the model. To cite this article: M. Shillor et al., C. R. Mecanique 331 (2003).     

A “quasi-elastic” affine formulation for the homogenised behaviour of nonlinear viscoelastic polycrystals and composites

The derivation of the overall behaviour of nonlinear viscoelastic (or rate-dependent elastoplastic) heterogeneous materials requires a linearisation of the constitutive equations around uniform per phase stress (or strain) histories. The resulting Linear Comparison Material (LCM) has to be linear thermoviscoelastic to fully retain the viscoelastic nature of phase interactions. Instead of the exact treatment of this LCM (i.e., correspondence principle and inverse Laplace transforms) as proposed by the “classical” affine formulation, an approximate treatment is proposed here. First considering Maxwellian behaviour, comparisons for a single phase as well as for two-phase materials (with “parallel” and disordered morphologies) show that the “direct inversion method” of Laplace transforms, initially proposed by Schapery (1962), has to be adapted to fit correctly exact responses to creep loading while a more general method is proposed for other loading paths. When applied to nonlinear viscoelastic heterogeneous materials, this approximate inversion method gives rise to a new formulation which is consistent with the classical affine one for the steady-state regimes. In the transient regime, it leads to a significantly more efficient numerical resolution, the LCM associated to the step by step procedure being no more thermoviscoelastic but thermoelastic. Various comparisons for nonlinear viscoelastic polycrystals responses to creep as well as relaxation loadings show that this “quasi-elastic” formulation yields results very close to classical affine ones, even for high contrasts.     

任意外型薄板非线性后屈曲问题的边界元解

本文就薄板后屈曲问题建立一组新型的边界元计算公式，用这组公式求解能方便处理各种边界问题，另外文中将面内应力分解成基本部份和附加部份，并利用微分算子分解理论导得了挠度的一个不同形式的基本解，由于计算公式中，实现了面内位移和挠度的解耦，从而使迭代过程得到简化，文末还对圆板后屈曲路径进行了计算，得到了满意的结果。     

Rheology of poly(methyl methacrylate-<Emphasis Type="Italic">co</Emphasis>-styrene) particles suspended in water: effects of electrostatic surface layer

 Linear and nonlinear viscoelastic properties were examined for aqueous suspensions of monodisperse poly(methyl methacrylate-co-styrene) (MS) particles having the radius a ₀ =45 nm and the volume fractions φ=0.428−0.448. These particles had surface charges and the resulting electrostatic surface layer (electric double layer) had a thickness of t_s=5.7 nm. At low frequencies in the linear viscoelastic regime, the MS particles behaved approximately as the Brownian hard particles having an effective radius a _eff=a ₀ + t_s, and the dependence of their zero-shear viscosity η₀ on an effective volume fraction φ_eff (={a _eff/a ₀}³φ) agreed with the φ dependence of η₀ of ideal hard-core silica suspensions. In a range of φ_eff < 0.63, this φ_eff dependence was well described by the Brady theory. However, the φ_eff dependence of the high-frequency plateau modulus was weaker and the terminal relaxation mode distribution was narrower for the MS suspensions than for the hard-core suspensions. This result suggested that the electrostatic surface layer of the MS particles was soft and penetrable (at high frequencies). In fact, this “softness” was more clearly observed in the nonlinear regime: the nonlinear damping against step strain was weaker and the thinning under steady shear was less significant for the MS suspension than for the hard-core silica suspensions having the same φ_eff. These weaker nonlinearities of the concentrated MS particles with φ_eff∼ 0.63 (maximum volume fraction for random packing) suggested that the surface layers of those particles were mutually penetrating to provide the particles with a rather large mobility. Received: 10 July 2001 Accepted: 2 November 2001