Thermomechanical response of a viscoelastic rod driven by a sinusoidal displacement

The forced vibrations of a rod of thermoviscoelastic material are studied. The rod is considered to be laterally insulated but not constrained, such that a one-dimensional analysis may be employed. Temperature dependence of the material properties and the resulting thermomechanical coupling effects are included. The vibrations are forced by the imposition of a sinusiodal displacement of known amplitude and frequency at one end of the rod. This problem corresponds to a dissipative material bonded to the surface of a relatively rigid, vibrating structure.Initial transient behavior is not considered. A steady-state response is found by means of a finite difference formulation. Material properties of a Lockheed solid propellant are used.The presence of critical frequencies, characterized by high stresses and temperatures, is found for small amplitudes of vibration. Nonlinearities and instabilities lead to a lack of one-to-one correspondence between stress and displacement boundary conditions. No relationship is found between the critical frequencies of the driven rod and the natural frequencies of a rod with an equivalent temperature profile.     

Thermomechanical behavior of a rectangular viscoelastic prism exposed to repeated stretching and contraction

Vibrational heat production is an important problem in studying the efficiency of viscoelastic elements of structures experiencing cyclic loads. The design of heat regimes constitutes one of the fundamental problems in the construction of such types of vibrational-proof systems as laminated rods, plates, and shells [1] and fiberglass and rubber-metal products, in particular, shock absorbers [2, 3]. Calculation of the critical parameters beyond which a rapid growth in temperature occurs (the phenomenon of thermal explosion), which leads to partial or complete loss of the supporting power of the product as a result of softening of the material, is of particular interest. A variational method has been used [4] to calculate heat production in a twodimensional shock absorber. The boundary conditions are satisfied on the basis of the St. Venant principle. In the current work, the stress-strain state, self-heating temperature field, and thermal instability of a long rectangular prism being periodically loaded (plane deformation) are investigated.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 1 pp, 149–156, January–February, 1976.     

Vibrational heating of a nonlinearly viscoelastic rod interacting with a viscoelastic element

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 27, No. 5, pp. 95–102, May, 1991.     

Thermomechanical behavior of a layer under the influence of a harmonic electromagnetic field

Institute of Applied Problems of Mechanics and Mathematics, Academy of Sciences of the Ukrainian SSR, Lvov. Translated from Prikladnaya Mekhaika, Vol. 25, No. 12, pp. 93–101, December, 1989.     

Vibrations of a viscoelastic rod with small non-linearities

Summary A partial differential equation is considered whose dominant linear term involves third order time derivatives and which contains a nonlinear term multiplied by a small parameter. The solutions of initial-boundary-value problems for the equation are investigated using an extension of the Krylov-Bogoliubov-Mitropolskii asymptotic method. The vibrations of a viscoelastic rod lead to an equation of the type considered when the material constitutive relation is that of a standard linear solid together with additional small nonlinearities of behaviour, and this particular physical problem is investigated in detail. It is shown that under appropriate initial conditions only a single mode of vibration is excited to lowest order, and some numerical results are given which illustrate the effects of the nonlinearity on these single mode solutions.

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Sommario Viene considerata una equazione differenziale alle derivate parziali il cui termine lineare dominante implica derivate del terzo ordine del tempo e che contiene un termine non lineare moltiplicato per un parametro di piccola entità. Vengono studiate le soluzioni dell'equazione con le condizioni iniziali e al contorno usando una estensione del metodo asintotico di Krylov-Bogoliubov-Mitropolskii. Le vibrazioni in una sbarra viscoelastica portano ad una equazione del tipo considerato quando la soluzione costitutiva è quella di un normale solido lineare con l'aggiunta di piccole non-linearità di comportamento. Viene studiato nel dettaglio questo particolare problema fisico. Viene mostrato che, con appropriate condizioni iniziali, viene eccitato solo il primo modo di vibrare. Vengono inoltre forniti alcuni risultati numerici che illustrano gli effetti della non linearità per questo particolare modo.

     

Stability of a viscoelastic rod on dynamic loading

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 4, pp. 153–157, July–August, 1992.     

Propagation of a stress pulse in a viscoelastic rod

Experimental work is reported on the propagation of a stress pulse in a viscoelastic waveguide. The data obtained are compared with results of analysis using one-dimensional wave-propagation theory. The waveguide used in this work is a low-density polyethylene rod 1/2 in. in diameter and 30-in. long. Stress input to the waveguide and the resulting particle velocity at three stations are measured using a crystal stress transducer, two Faraday-principle velocity transducers and a capacitor transducer. The experiment is described mathematically as a boundary-value problem formulated in terms of the one-dimensional equation of motion, the strain-displacement relationship, a hereditary constitutive equation and the stress-boundary condition. Fourier transform and inversion yield an integral expression for velocity which is evaluated numerically at three stations using measured values for the stress-boundary condition, material attenuation and phase velocity. The analytical results compare favorably with the experimental data. The one-dimensional theory appears adequate to describe pulse propagation of this type. The attenuation and phase velocity used here are found to be a linear function and a logarithmic increasing function of frequency respectively.     

Thermomechanical instability of a heat releasing granular layer

A study is made of a homogeneous granular layer of thickness h and porosity in which heat is released uniformly with bulk power q. The heat is taken up by a uniform gas flow through the layer in the transverse direction of the y axis at velocity v ° produced by a given pressure difference across the layer. This scheme corresponds to the simplest model of a gas-cooled nuclear reactor, in which the heat release does not depend on the thermogasdynamic state. The case of a chemical reactor with heat release of Arrhenius type is considered in [1, 2].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 45–53, March–April, 1979.     

Longitudinal and torsional oscillations of a rod in a viscoelastic fluid

The flow of a viscoelastic fluid due to the torsional and longitudinal oscillations of an infinite circular rod is examined. The idealized equation of state to characterise this liquid is of the implicit Oldroyd-B model, for which momentum equations are solved analytically. The effect of the Weissenberg number and the viscosity ratio on the flow field are discussed. Also, the axial shear force and torque on the rod are computed. Received: 14 March 1997 Accepted: 14 May 1998