Stress relaxation in a polymer disk

The stress relaxation in a polymer disk with a central hole, into which a rigid metal rod is press-fitted, is considered. The progressive redistribution of the stresses in the disk has been observed by a photoelastic method. In the theoretical solution of the problem the relaxation properties of the polymer are taken into account by means of a generalized nonlinear Maxwell equation with two terms of the relaxation spectrum. It is shown that the linear, as distinct from the nonlinear, theory is not capable of conveying the characteristics of the stress relaxation process in a rigid polymer disk.Institute of Chemical Physics, Academy of Sciences of the USSR, Moscow. Translated from Mekhanika Polimerov, No. 6, pp. 1064–1070, November–December, 1971.     

Stress relaxation in glassy polymers

Stress relaxation in amorphous linear polymers has been investigated below the glass transition point at various strain rates, temperatures, and strains. A model of a nonlinear viscoelastic body is proposed as a means of describing the effect. It is shown that stress relaxation is well described by the model considered and is correlated with the deformation characteristics of the polymers.A. F. Ioffe Physico-Technical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 6, pp. 987–994, November–December, 1971.     

Incubation and relaxation time for dislocation motion

Zusammenfassung Die Theorie quasistatischer Versetzungen ist in befriedigender Weise ausgebaut worden. Eine Anzahl von Unstimmigkeiten besteht jedoch in der Ansicht, wie Versetzungen auf dynamische Lasten reagieren. Insbesondere ergeben sich Schwierigkeiten bei der Berechnung von Versetzungsgeschwindigkeiten bei starker dynamischer Belastung. Die quasistatische Theorie sagt grosse Geschwindigkeiten voraus, während tatsächlich nur kleine Geschwindigkeiten gemessen werden. Beobachtungen über das Verhalten von Brüchen bei stossartiger Belastung lassen auf eine typische Inkubationszeit schliessen, bei der das Bruchverhalten aufhört, von der Belastungsgeschwindigkeit abhängig zu sein. Diese Inkubationszeit ist typisch von der Ordnung 10^–7 sec. Da plastische Brüche durch Bewegung und Anhäufung von Versetzungen verursacht werden, liegt der Schluss nahe, dass auch Versetzungen «immobil» werden, wenn die Stossdauer geringer wird als die typische Inkubationszeit. Mit dieser Auffassung lassen sich eine Reihe von Erscheinungen erklären, die nach der klassischen Theorie zu falschen Resultaten führen. Insbesondere wird gezeigt, dass das Viskositätsverhalten von Versetzungen nicht-newtonsch ist, was zu ganz anderen Ergebnissen führt, als wenn newtonsches Verhalten postuliert wird.     

Waveform relaxation as a dynamical system

In this paper the properties of waveform relaxation are studied when applied to the dynamical system generated by an autonomous ordinary differential equation. In particular, the effect of the waveform relaxation on the invariant sets of the flow is analysed. Windowed waveform relaxation is studied, whereby the iterative technique is applied on successive time intervals of length and a fixed, finite, number of iterations taken on each window. This process does not generate a dynamical system on since two different applications of the waveform algorithm over different time intervals do not, in general, commute. In order to generate a dynamical system it is necessary to consider the time map generated by the relaxation process. This is done, and -closeness of the resulting map to the time map of the underlying ordinary differential equation is established. Using this, various results from the theory of dynamical systems are applied, and the results discussed.

     

Investigation of the relaxation process in polycarbonate film

The temperature relaxation of polycarbonate film and the accompanying change in the birefringence path difference are considered. It is shown that at a temperature of about 170°C intense crystallization takes place in the film. The deformation modulus is a function of the density of the network and its degree of orientation. The variation of these parameters with the stretch ratio and temperature is determined experimentally. The creep of polycarbonate film is considered and correlated with the curves representing the fall in the number of chains of the molecular network in pure relaxation.Mendeleev All-Union Scientific-Research Institute of Metrology, Leningrad. Translated from Mekhanika Polimerov, No. 1, pp. 72–76, January–February, 1973.     

Rheology and molecular structure of a polyethylene melt. 3. Relaxation spectra and characteristic relaxation time

By comparing the relaxation spectra respectively calculated on the basis of the iterative and approximation methods, it is confirmed that the latter may be used to describe the viscoelastic properties of polyethylene melts. A method is proposed for identifying the form of the frequency relaxation spectrum by three characteristic numerical quantities. The effect of the molecular-mass, the molecular-mass distribution, and the long-chain branching characteristics on the form of the frequency relaxation spectrum is considered. The question of the geometrical similarity of the relaxation spectrum is discussed, and it is shown that, in general, these spectra are not linearly similar. The asymmetry of the spectrum depends solely on the index representing the rate of fall of the dynamic viscosity with increasing frequency of deformation. Methods of determining the characteristic relaxation time without constructing a frequency relaxation spectrum are proposed; ways of determining the characteristic relaxation time from experimental data are indicated.For communication 2, see [10].Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 322–330, March–April, 1976.     

Effect of temperature on the chemical relaxation process in rubbers

The effect of temperature on the chemical relaxation of rubbers has been investigated. With reference to nine industrial rubbers it has been shown that during storage in the stressed state (at =const) chemical relaxation is chiefly determined by two processes proceeding at different rates. Furthermore, the chemical relaxation of rubbers is determined not only by the rate constants of these two processes but also by the pre-exponential constants C_0i, the relationship between which varies with increase in temperature. It is possible to extrapolate the rate constants (k_i) and pre-exponential constants (C_0i) from elevated temperatures (110–50° C) to temperatures in the range 20–25° C. An equation is proposed that makes it possible to calculate the change in the elastic properties of rubbers in the stressed state at storage temperatures (20–25° C) by means of results obtained at elevated temperatures.Mekhanika Polimerov, Vol. 3, No. 3, pp. 448–454, 1967     

Strength of polyethylene and capron as a function of stretch ratio

The relation between the strengths and stretch ratios of a capronmonofilament and a high-pressure polyethylene film has been investigated for various stretching temperatures and stretching rates. The strength, measured at –196°C (_–196), is directly proportional to the stretch ratio (). For a given , _-196 does not depend on stretching temperature and stretching rate. The breaking load at –196° C (for a given initial cross section of the starting unoriented specimen) does not vary with . A possible explanation is offered.Mekhanika Polimerov, Vol. 3, No. 4, pp. 591–595, 1967     

Stress relaxation: Experiment,theory, and computer simulation

Experimental evidence on stress relaxation is analyzed first for a wide variety of classes of materials: metals and their alloys, synthetic and natural polymers, glasses and frozen non-polymeric organic liquids. Common features of curves (t) of relaxation of stress a as a function of time t are discussed, and the importance of the internal stress _i() noted. Theoretical approaches are then reviewed, with particular attention to the cooperative model and its modifications; that model corresponds well to the experimental results. Some simulation results obtained with the method of molecular dynamics are reported for ideal metal lattices, metal lattices with defects, and for polymeric systems. In agreement with both experiments and the cooperative theory, the simulated (log t) curves exhibit three regions: initial, nearly horizontal, starting at ₀; central, descending approximately linearly; and final, corresponding to _i. In agreement with the theory, the slope of the simulated central part is proportional to the initial effective stress ₀*= ₀– _i. The time range taken by the central part is strongly dependent on the defect concentration: the lower the defect concentration, the shorter the range. Imposition in the beginning of a high strain destroys largely the resistance of a material to deformation, resulting in low values of the internal stress _i. On the joint basis of experimental, theoretical, and numerical results, we explain the mechanism of stress relaxation in terms of deformations occuring in the immediate environment of the defects. Simulations show several common features in the behavior of metals and polymers. Apart from the defect concentration, the amount of free volumev _f is also important.Published in Mekhanika Kompozitnykh Materialov, Vol. 31, No. 5, pp. 591–606, September–Ocotober, 1995.     

Stress relaxation in physically nonlinear polymer materials under simultaneous tension and torsion

Conclusions 1. Application of our method of determining the parameters of physically nonlinear polymer materials from tests in uniaxial tension or in torsion ensures a satisfactory agreement between theoretical and experimental stress-relaxation curves corresponding to other modes of loading.2. Theoretical stress-relaxation curves for shear (tension) calculated on the basis of parameters which have been determined from approximations of stress-relaxation curves for tension (torsion) fall closely within the confidence interval for an individual measurement (with =95%).3. The agreement between theoretical and experimental stress-relaxation curves for a plane state of stress is 2–3 times worse, in terms of the mean-square relative difference, than the agreement between the respective curves for uniaxial tension.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 5, pp. 797–803, September–October, 1977.     

A novel Lagrangian relaxation approach for a hybrid flowshop scheduling problem in the steelmaking-continuous casting process

One of the largest bottlenecks in iron and steel production is the steelmaking-continuous casting (SCC) process, which consists of steel-making, refining and continuous casting. The SCC scheduling is a complex hybrid flowshop (HFS) scheduling problem with the following features: job grouping and precedence constraints, no idle time within the same group of jobs and setup time constraints on the casters. This paper first models the scheduling problem as a mixed-integer programming (MIP) problem with the objective of minimizing the total weighted earliness/tardiness penalties and job waiting. Next, a Lagrangian relaxation (LR) approach relaxing the machine capacity constraints is presented to solve the MIP problem, which decomposes the relaxed problem into two tractable subproblems by separating the continuous variables from the integer ones. Additionally, two methods, i.e., the boundedness detection method and time horizon method, are explored to handle the unboundedness of the decomposed subproblems in iterations. Furthermore, an improved subgradient level algorithm with global convergence is developed to solve the Lagrangian dual (LD) problem. The computational results and comparisons demonstrate that the proposed LR approach outperforms the conventional LR approaches in terms of solution quality, with a significantly shorter running time being observed.     

Quantum dynamics as a stochastic process in phase space

It is shown that quantum dynamics is equivalent to a stochastic process in phase space. The process is described by normalized but not necessarily positive probability distributions (“pseudoprobabilities”). Evolution of the simultaneous probability distribution of momentum and coordinate exactly conincides with the dynamics of the Wigner function of a quantum system. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 3, pp. 454–458, March, 1997.     

Approximating k-cuts using network strength as a Lagrangean relaxation

Given an undirected, edge-weighted connected graph, the k-cut problem is to partition the vertex set into k non-empty connected components so as to minimize the total weight of edges whose end points are in different components.     

Constrained linear filtering as a multistage process

Linear filtering in the presence of contraints is viewed as a multistage process. This appeoach gives some new results and leads to a simple building block method for the constrained filter.     

Cultural evolution as a nonstationary stochastic process

We present an individual based model of cultural evolution, where interacting agents are coded by binary strings standing for strategies for action, blueprints for products or attitudes and beliefs. The model is patterned on an established model of biological evolution, the Tangled Nature Model (TNM), where a “tangle” of interactions between agents determines their reproductive success. In addition, our agents also have the ability to copy part of each other's strategy, a feature inspired by the Axelrod model of cultural diversity. Unlike the latter, but similarly to the TNM, the model dynamics goes through a series of metastable stages of increasing length, each characterized by mutually enforcing cultural patterns. These patterns are abruptly replaced by other patterns characteristic of the next metastable period. We analyze the time dependence of the population and diversity in the system, show how different cultures are formed and merge, and how their survival probability lacks, in the model, a finite average life‐time. Finally, we use historical data on the number of car manufacturers after the introduction of the automobile to the market, to argue that our model can qualitatively reproduce the flurry of cultural activity which follows a disruptive innovation. © 2015 Wiley Periodicals, Inc. Complexity 21: 214–223, 2016     

Perfect plasticity as a zero relaxation limit of plasticity with isotropic hardening

This work presents an approximation result for elasto‐perfect plasticity in the dynamical setting of the problem. We prove that the model of perfect plasticity with isotropic hardening converges to the Prandlt–Reuss model. Copyright © 2001 John Wiley & Sons, Ltd.