Variation of the elasticoviscous characteristics of a heavily filled rubber after periodic shear deformation

The results of an investigation of the recovery of the elasticoviscous characteristics after thixotropic reduction by periodic deformation in the nonlinear zone are reported. The rate of recovery of the mechanical properties depends on the deformation frequency and the previous history of dynamic deformation is shown to affect the dependence of the elasticoviscous characteristics on the shear rate gradient. The effects of stationary and dynamic deformation regimes on the relaxation spectrum of the material are compared.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 528–533, May–June, 1969.     

High-frequency periodic deformation of polymers in the viscous-flow state

The propagation of high-frequency periodic vibrations in melts and solutions is considered in the case where the acoustic characteristics of the material are determined by the deformation amplitude. It is shown that in sufficiently large specimens the consequences of periodic deformation are concentrated near the vibration source, as a result of which a layer of material with strongly modified mechanical characteristics develops. It is suggested that when periodic high-frequency deformation is superimposed on steady-state flow, owing to the presence of a boundary layer there is effective slippages of the material at the vibration-emitting surfaces.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 4, pp. 722–728, July–August, 1973.     

Nonlinear periodic deformation of thixotropic media

The experimental procedure and the processing of results on the nonlinear periodic deformation of polymeric and dispersed system melts are examined. Experimental data are given on the effect of the amplitude of harmonic deformation on nonlinear stress distortions.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 4, pp. 689–696, July–August, 1972.     

Toward a solution to the high Weissenberg number problem

We present results of a numerical study of high Weissenberg number flows using the method of Trebotich, Colella and Miller, J. Comput. Phys. vol. 205 (2005). We consider benchmark viscoelastic flow of an Oldroyd-B fluid in four to one sudden contractions for We = 1, 10, Re = 1. We conclude that (i) underlying wave behavior in the solution for flows at critical and supercritical elastic Mach number prevents the solution from reaching a steady-state; and (ii) high resolution is needed to resolve structure in the solution downstream from the re-entrant corner in order to determine convergence. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Applications of F-expansion to periodic wave solutions for a new Hamiltonian amplitude equation

We present an extended F-expansion method for finding periodic wave solutions of nonlinear evolution equations in mathematical physics, which can be thought of as a concentration of extended Jacobi elliptic function expansion method proposed more recently. By using the F-expansion, without calculating Jacobi elliptic functions, we obtain simultaneously many periodic wave solutions expressed by various Jacobi elliptic functions for the new Hamiltonian amplitude equation introduced by Wadati et al. When the modulus m approaches to 1 and 0, then the hyperbolic function solutions (including the solitary wave solutions) and trigonometric function solutions are also given respectively. As the parameter ε goes to zero, the new Hamiltonian amplitude equation becomes the well-known nonlinear Schrödinger equation (NLS), and at least there are 37 kinds of solutions of NLS can be derived from the solutions of the new Hamiltonian amplitude equation.     

Studying the longitudinal flow of polymers with periodic deformation

Conclusions 1. At low rates of elongation the state of a polymer melt depends neither on the strain nor on the strain rate. The frequency dependence of the components of the complex longitudinal modulus, measured by superposition of a periodic deformation on a steady longitudinal flow, resemble, at low elongation rates, the frequency characteristics of the components of the complex shear modulus.2. Relaxation processes due to steady longitudinal flow affect the components of the complex longitudinal modulus measured by periodic deformation.3. A melt of a polydisperse polymer under conditions of longitudinal flow begins, at some definite strain rate, to respond to periodic deformation as a rubber-like body and continues to do so as the amount of elongation increases, which is manifested by the trend of the frequency dependence of the modulus component E', this trend becoming similar to that for a solid body.4. As the strain increases during elongation, the upper boundary of the flow range, which is determined by the periodic deformation, shifts toward low frequencies.5. The viscoelasticity characteristics of a material subject to elongation can be accurately enough described by nonlinear equations with the aid of the linear shear-relaxation spectrum.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 507–513, May–June, 1978.     

Reliable amplitude and frequency estimation for biased and noisy signals

Robust estimation of the amplitude, frequency and bias of unknown noisy sinusoidal signals is considered in this paper. It is only assumed that the measurements noise is bounded without any additional information such as stationarity, uncorrelation or type of distribution. In this context, the aim is to compute the set of all admissible values that are consistent with the measurements and with the error bound. The estimation problem is formulated as a Constraint Satisfaction Problem (CSP) where the amplitude, frequency and bias constitute the variables and a function relating them to the output is the constraint. Interval constraint propagation techniques are used to solve, in a guaranteed way, this problem. In order to illustrate the principle and the efficiency of the approach, numerical simulations are provided.     

The construction of a model of frequency independent damping using the amplitude characteristic of the absorption coefficient

Relations are obtained which connect the stresses and strains accompanying oscillations with the absorption coefficient and a procedure for modelling the frequency-independent scattering of energy is constructed based on them. Account is taken of the fact that damping lowers the resonance frequency and the accumulation of residual stresses and strains is not observed during oscillation. The models obtained using this procedure can be employed for any changes, including unsteady changes, in the load with respect to time. The case of a power dependence of the absorption coefficient on the amplitude, which is important in practice, is investigated in detail. The forced, transverse oscillations of a rod are considered as an example. Asymptotic and numerical solutions of the problem are obtained in the single-mode approximation.     

Harmonic distrotions under nonlinear periodic deformation of fluid polymers

Conclusions 1. It has been shown for a number of viscoelastic fluid systems that under nonlinear periodic deformation, the contribution of the third harmonic of the stress to the fundamental does not exceed 20% of the amplitude.2. In the case of clay soil and melt of filled polyethylene, the shape of the stress waves is essentially definable by the relative phase angle of the third harmonic of the stress and is practically independent of the deformation amplitude in a growing nonlinear range of deformation.3. In the case of the polyethylene melt, the amplitude dependence of the phase angles of the stress harmonics is in satisfactory agreement with the analysis of model I. With increasing deformation amplitude, the modulus vector of the first harmonic rotates counterclockwise and remains in the first trigonometric quadrant; the modulus vector of the third harmonic passes from the second to the third quadrant, and the modulus vector of the fifth harmonic passes from the second to the fourth quadrant via the third.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 893–898, September–October, 1978.     

Thixotropy of filled-polyethylene melts subjected to periodic shear deformation

As a result of the action of periodic finite-amplitude shear deformations filled high-pressure polyethylene melts exhibit thixotropic properties. The effect of the previous deformation history on the frequency dependence of the complex dynamic shear modulus is investigated.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR Riga. Translated from Mekhanika Polimerov, Vol. 4, No. 5, pp. 927–931, September–October, 1968.     

Variation of the deformation resistance of phenol-formaldehyde thermosets during cure

The dependence of the viscosity of phenol-formaldehyde thermosets on the previous thermal history has been investigated at a fixed shear rate.Moscow Institute of Chemical Machine Building. Institute of Problems of Mechanics, Academy of Sciences of the USSR, Moscow. Translated from Mekhanika Polimerov, No. 4, pp. 619–623, July–August, 1971.     

Bifurcation of periodic orbits near a frequency maximum in near-integrable driven oscillators with friction

We investigate analytically the effect of perturbations on an integrable oscillator in one degree of freedom whose frequency shows a maximum as a function of the energy, i.e. a system with nonmonotone twist. The perturbation depends on three parameters: one parameter describes friction such that the Jacobian is constant and less than one. A second and a third describe the variation of the frequency and of the strength of the driving force respectively. The main result is the appearance of two chains of saddle node pairs in the phase portrait. This contrasts with the bifurcation of one chain of periodic orbits in the case of perturbations of monotone twist systems. This result is obtained for a mapping, but it is demonstrated that the same formalism and results apply for time continuous systems as well. In particular we derive an explicit expression for the stroboscopic mapping of a particle in a potential well, driven by a periodic force and under influence of friction, thus giving a clear physical interpretation to the bifurcation parameters in the mapping.     

Example of a third-order periodic differential equation whose frequency spectrum contains a closed interval

We present an example of a linear homogeneous third-order differential periodic equation all of whose solutions are quasiperiodic functions and whose frequency spectra of zeros, signs, and roots contain the same closed interval of exact values.     

Tighter uncertainty principles for periodic signals in terms of frequency

In this study, we propose some new uncertainty principles for periodic signals with sharper lower bounds than those in the existing ones. The improved lower bounds, in particular, are related to the frequency of the signal. Three examples are employed to demonstrate sharpness of the new uncertainty principles. Copyright © 2014 John Wiley & Sons, Ltd.     

On the bending of plates with transverse shear deformation and mixed periodic boundary conditions

The method of matched asymptotic expansions is used to find a homogenized problem whose solution is an approximation to the solution of a mixed periodic boundary value problem in the theory of bending of thin elastic plates. A critical size for the fixed parts of the boundary is found such that the boundary condition of the homogenized problem is an intermediate case between that for the clamped edge plate and that for the free boundary plate.