On Hashin–Shtrikman-type bounds for nonlinear conductors

For linear composite conductors, it is known that the celebrated Hashin–Shtrikman bounds can be recovered by the translation method. We investigate whether the same conclusion extends to nonlinear composites in two dimensions. To that purpose, we consider two-phase composites with perfectly conducting inclusions. In that case, explicit expressions of the various bounds considered can be obtained. The bounds provided by the translation method are compared with the nonlinear Hashin–Shtrikman-type bounds delivered by the Talbot–Willis (1985) [2] and the Ponte Castañeda (1991) [3] procedures.     

A hyperbolic Lindstedt–Poincaré method for homoclinic motion of a kind of strongly nonlinear autonomous oscillators

A hyperbolic Lindstedt-Poincare method is presented to determine the homoclinic solutions of a kind of nonlinear oscillators, in which critical value of the homoclinic bifurcation parameter can be determined. The generalized Lienard oscillator is studied in detail, and the present method＇s predictions are compared with those of Runge-Kutta method to illustrate its accuracy.     

On Ericksen’s Theorem for Unconstrained Hyperelastic Materials

After reviewing the proof of Ericksens theorem for the set of unconstrained, homogeneous, isotropic, hyperelastic materials, Ericksens result is sharpened to cover the subset of materials that possess a natural configuration and satisfy the empirical inequalities. Mathematics Subject Classifications (2000) 74B20.     

Nonzero mean PDF solution of nonlinear oscillators under external Gaussian white noise

This paper is concerned with the nonzero mean stationary probability density function (PDF) solution for nonlinear oscillators under external Gaussian white noise. The PDF solution is governed by the well-known Fokker–Planck–Kolmogorov (FPK) equation and this equation is numerically solved by the exponential-polynomial closure (EPC) method. Different types of oscillators are further investigated in the case of nonzero mean response. Either weak or strong nonlinearity is considered to show the effectiveness of the EPC method. When the polynomial order equals 2, the results of the EPC method are identical with those given by equivalent linearization (EQL) method. These results obtained with the EQL method differ significantly from exact solution or simulated results. When the polynomial order is 4 or 6, the PDFs obtained with the EPC method present a good agreement with the exact solution or simulated results, especially in the tail regions. The numerical analysis also shows that the nonzero mean PDF of the response is nonsymmetrically distributed about its mean unlike the case of the zero mean PDF reported in the references.     

On Green’s function for a bimaterial elastic half-plane

The problem of a point force acting in a composite, two-dimensional, isotropic elastic half-plane is considered. An exact solution is obtained, using Mellin transforms and the Melan solution for a point force in a homogeneous half-plane.     

On Kato’s Method for Navier–Stokes Equations

We investigate Kato’s method for parabolic equations with a quadratic non-linearity in an abstract form. We extract several properties known from linear systems theory which turn out to be the essential ingredients for the method. We give necessary and sufficient conditions for these conditions and provide new and more general proofs, based on real interpolation. In application to the Navier–Stokes equations, our approach unifies several results known in the literature, partly with different proofs. Moreover, we establish new existence and uniqueness results for rough initial data on arbitrary domains in \mathbbR³{\mathbb{R}}^{3} and irregular domains in \mathbbRⁿ{\mathbb{R}}^{n}.     

PDF solution of nonlinear oscillators subject to multiplicative Poisson pulse excitation on displacement

A solution procedure for the stationary probability density function (PDF) of the responses of nonlinear oscillators subjected to Poisson white noises is formulated with exponential-polynomial closure (EPC) method. The effectiveness of the solution procedure is investigated with nonlinear oscillators subjected to both external and multiplicative Poisson white noises at different levels of system nonlinearity, excitation intensity, and impulse arrival rates. Numerical results show that the PDFs obtained with the EPC procedure are in good agreement with those from Monte Carlo simulation.     

‘Velocity’Finite Element Method for Dynamic Problem

After analysing the essential features of successive integration method taking displacementas variable by N.M.Newmark and E.L.Wilson et al,this paper presents a“Velocity”Element Method,taking velocity as variable for the solution of the initial value problem.A simplified scheme is offered for the non-damping system,and the stability is also discussed.Owing to the fact that this simplified scheme for non-damping and apparent static damping is explicitin form,it is unnecessary to solve the algebraic system of equations at every time interval,conse-quently the amount of computation is greatly reduced.For non-linear dynamic problems,this schememay be used to obtain fairly good initial values for iteration.An extended form of“elocity”Element is presented for the arbitrary damping system.For thenon-linear cases,the incremental Velocity iteration scheme is adopted and its convergence proved.Some discussions have been given on artificial damping and the effect of the parameter.Finally,the results of numeric     

‘Velocity’ Finite Element Method for Dynamic Problem

After analysing the essential features of successive integration method taking displacement as variable by N. M. Newmark and E. L. Wilson et al, this paper presents a Velocity Element Method, taking velocity as variable for the solution of the initial value problem.A simplified scheme is offered for the non-damping system, and the stability is also discussed. Owing to the fact that this simplified scheme for non-damping and apparent static damping is explicit in form, it is unnecessary to solve the algebraic system of equations at every time interval, consequently the amount of computation is greatly reduced. For non-linear dynamic problems, this scheme may be used to obtain fairly good initial values for iteration.An extended form of elocity Element is presented for the arbitrary damping system. For the non-linear cases, the incremental Velocity iteration scheme is adopted and its convergence proved. Some discussions have been given on artificial damping and the effect of the parameter.Finally, the results of numerical calculatio of some typical problem are given in the appendix.     

Design and implementation plan for indirect-drive highly nonlinear ablative Rayleigh–Taylor instability experiments on the National Ignition Facility

In the context of National Ignition Facility Basic Science program we propose to study on the NIF ablative Rayleigh–Taylor (RT) instability in transition from weakly nonlinear to highly nonlinear regimes. Based on the analogy between flame front and ablation front, highly nonlinear RT instability measurements at the ablation front can provide important insights into the initial deflagration stage of thermonuclear supernovae of type Ia. NIF provides a unique platform to study the rich physics of nonlinear and turbulent mixing flows in High Energy Density plasmas because it can accelerate targets over much larger distances and longer time periods than previously achieved on the NOVA and OMEGA lasers. In one shot, growth of RT modulations can be measured from the weakly nonlinear stage near nonlinear saturation levels to the highly nonlinear bubble-competition, bubble-merger regimes and perhaps into a turbulent-like regime. The role of ablation on highly-nonlinear RT instability evolution will be comprehensively studied by varying ablation velocity using indirect and direct-drive platforms. We present a detailed hydrocode design of the indirect-drive platform and discuss the implementation plan for these experiments which only use NIF diagnostics already qualified.     

On Aifantis’ strain gradient plasticity theory accounting for plastic spin

This paper examines the effect of codirectionality hypothesis on Aifantis’ distortion gradient plasticity theory. The system of microforces includes microstress, power-conjugate to the Burgers tensor rate. The proposed codirectionality hypothesis assumes, that the flow direction and the plastic microstress are in the same direction. It is obtained that the power expended by the microstress power-conjugate to the Burgers tensor rate, can be additively decomposed to power expended by scalar and vector microscopic stresses power-conjugate to the accumulated plastic distortion rate and gradient of plastic distortion rate respectively. Following the proposed codirectionality hypothesis, it is obtained that the microstress power-conjugate to the Burgers tensor rate is purely energetic. The obtained flow rule accounts for plastic spin and generalizes the Aifantis’ flow rule.     

‘Explicit’ and ‘implicit’ non-local continuous descriptions for a plate with circular inclusion in tension

Meccanica - Increasing application of composite structures in engineering field inherently speed up the studies focusing on the investigation of non-homogeneous bodies. Due to their capability on...     

Multifaceted nonlinear dynamics in $$\mathcal {PT}$$PT-symmetric coupled Liénard oscillators

Nonlinear Dynamics - We propose a generalized parity-time ($$\mathcal {PT}$$)-symmetric Liénard oscillator with two different orders of nonlinear position-dependent dissipation. We study the...     

Substructures’ coupling with nonlinear connecting elements

Nonlinear Dynamics - The analysis of complex structures is often very challenging since reliable data can only be obtained if the underlying model represents properly the real case. Thus,...     

Global solution for coupled nonlinear Klein-Gordon system

The globed solution for a coupled nonlinear Klein-Gordon system in two-dimensional space was studied. First, a sharp threshold of blowup and global existence for the system was obtained by constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow. Then the result of how small the initial data for which the solution exists globally was proved by using the scaling argument.     

sensor fault detection observer design for nonlinear systems in Takagi–Sugeno’s form

This paper addresses sensor fault detection observer design problems for discrete- and continuous-time nonlinear systems in Takagi–Sugeno’s (T–S) form. It is desired that the fault detection observer is as sensitive to fault and robust against disturbance as possible. To this end, sufficient conditions for stable T–S fuzzy model-based observer design with performance are derived in terms of linear matrix inequalities in both cases. An example on the backing-up problem of a truck-trailer is provided to illustrate the effectiveness of the proposed methodology.     

A robust ‘time-marching’ solver for one-dimensional nucleating steam flows

A mixed Lagrangian/Eulerian ‘time-marching’ solver capable of predicting one-dimensional nucleating steam flows is described. Simple nucleation and droplet growth models are employed which avoid the use of variable empirical factors and which have been validated using existing experimental data from nozzle experiments performed in the steam tunnel of the Central Electricity Research Laboratories. Theoretical predictions are compared against experimental results encompassing all flow regimes likely to be encountered in a one-dimensional analysis of flow in a low pressure steam turbine. These include supercritical heat addition cases which display both steady and unsteady shock wave formation.     

On the nonlinear dynamics of an ecoepidemic reaction–diffusion model

A reaction–diffusion ecoepidemic model of predator–prey type with a transmissible disease spreading among the predator species only is considered. The longtime behavior of solutions is analyzed and, in particular, absorbing sets in the phase space are determined. Conditions guaranteeing the non existence of non-constant equilibria have been found. Linear and non-linear stability conditions for biologically meaningful equilibria are determined.