Inversion and quasi-static problems in linear viscoelasticity

Within the theory of linear viscoelasticity, we seek solutions to the inversion problem of the constitutive equation respectively inL ² and in the spaceS of the tempered distributions. Successively we study the quasi-static problem inS. Both problems admit one and only one solution if the relaxation function satisfies Graffi's inequality. Finally we show that the inversion problem and the quasi-static one are deeply connected and that every counterexample about the existence or uniqueness of the solutions for the first problem also provides a counterexample for the latter.

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Sommario Nell'ambito della viscoelasticità lineare, si studia il problema dell'invertibilità della equazione costitutiva rispettivamente inL ² e nello spazioS delle distribuzioni temperate. Successivamente si studia il problema quasi statico inS. Entrambi i problemi ammettono una ed una sola soluzione se la funzione di rilassamento soddisfa la disuguaglianza di Graffi. Infine si mostra che i due problemi sono strettamente correlati in quanto ogni problema quasi statico è riconducibile ad un problema di invertibilità.

     

Molecular architecture and linear viscoelasticity of homogeneous ethylene/styrene copolymers

We have studied correlations between molecular features and the linear rheology of two series of carefully synthesized ethylene/styrene copolymers. The materials were polymerized using two single-site catalyst systems; a commercial constrained geometry catalyst and an ansa-metallocene single-site catalyst designed on purpose. The copolymers obtained are characterized by a narrow molecular weight distribution and a broad range of different styrene content. The thermorheological properties and the shape of the viscoelastic fingerprint of the samples obtained from the ansa-metallocene catalyst are those featured by linear polymers. The flow activation energy in these materials depends on the amount of comonomer, but the values are much higher than those corresponding to aliphatic copolymers with the same amount of comonomer, likely due to the hindrance effect of the styrene units in flow behavior. The results obtained for the samples polymerized by the constrained geometry catalyst indicate a complex molecular architecture. In fact, the samples show characteristic features like thermorheological complexity, increased values of the flow activation energy, and a bimodal linear viscoelastic fingerprint in the terminal zone. From these results it can be argued that these materials are actually blends of linear and branched species, these later with the highest relaxation times and flow activation energy values.     

Ray expansions for the solution of dynamic problems of linear viscoelasticity

Voronezh' Civil-Engineering Institute. Translated from Prikladnaya Mekhanika, Vol. 27, No. 1, pp. 89–95, January, 1991.     

A numerical analytic method for solving boundary-value problems of linear anisotropic viscoelasticity

The paper proposes an approach to solving boundary-value problems for linear viscoelastic orthotropic bodies based on the method of operator continued fractions and the boundary-element method. A problem-solving algorithm and a procedure to estimate the possible error are outlined. The solution for a viscoelastic orthotropic plate with a rigid circular inclusion under uniaxial tension is obtained as an example and compared with available ones     

Uniqueness in linear elastostatics for problems involving unbounded bodies

The problem of the uniqueness of elastostatic solutions to various boundary value problems involving unbounded two- and three-dimensional bodies is considered. The general boundary value problem is first considered for anisotropic bodies on which the elasticity tensor is uniformly positive definite. The displacement problem is then considered for the cases where the body is homogeneous and isotropic and the elasticity tensor is strongly-elliptic. Finally, the traction problem on homogeneous, isotropic bodies is considered with fairly little restriction placed on the Lamé moduli. In the first two cases no restriction is placed on the geometry of the body other than the normal assumptions allowing for the use of the divergence theorem. For the traction problem, however, it is assumed that one component of the outward normal to the boudary of the body almost never vanishes. In each case it is shown that the desired uniqueness results hold with fairly mild restrictions placed on the displacement or stress fields (depending on the problem) in a neighborhood of infinity. Where possible, the results are shown to hold even though the solutions may possess the sort of discontinuities and singularities commonly found in problems involving cracks, corners, and bonded interfaces.Related results involving the insolvability of certain non-trivial boundary value problems and the behavior of certain elastic states are, where appropriate, developed.     

Viscoelastic problem for piecewise homogeneous piezoelectric plates

The electromagnetoviscoelastic problem is solved for piecewise-homogeneous plates. The problem is reduced to solving a sequence of electromagnetoelastic problems with complex potentials. General representations of approximation functions for multiply connected domains and boundary conditions for their determination are given. An analytical solution of the problem for a plate with one inclusion and an approximate solution for a plate with a finite number of inclusions are obtained. The change in the electromagnetoelastic state is investigated numerically as a function of time, the properties of the plate and inclusion materials, and the distance between the inclusions.     

Analogies between linearized and linear problems for compressible elastic bodies

International Applied Mechanics -     

One method of solving boundary-value problems of the linear theory of viscoelasticity

A form of integral resolvent viscoelastic operators with arbitrary analytic difference-type kernel is found using an improved method of calculating irrational functions. The approach is illustrated by determining the viscoelastic characteristics of the plane stressed state of an orthotropic plate. Translated from Prikladnaya Mekhanika, Vol. 34, No. 12, pp. 77–85, December, 1998.     

New estimates for work in linear viscoelasticity

Archive for Rational Mechanics and Analysis -     

Normal modes for piecewise linear vibratory systems

A method to construct the normal modes for a class of piecewise linear vibratory systems is developed in this study. The approach utilizes the concepts of Poincaré maps and invariant manifolds from the theory of dynamical systems. In contrast to conventional methods for smooth systems, which expand normal modes in a series form around an equilibrium point of interest, the present method expands the normal modes in a series form of polar coordinates in a neighborhood of an invariant disk of the system. It is found that the normal modes, modal dynamics and frequency-amplitude dependence relationship are all of piecewise type. A two degree of freedom example is used to demonstrate the method.     

A unified approach to quasi-static shakedown problems for elastic-plastic solids with piecewise linear yield surface

Summary The paper concerns shakedown analysis of elastic-plastic bodies subjected to quasi-statically varying loads within a given domain. Using a perturbation method, a general inequality is given, from which, by simply specializing the perturbing terms, the generalized Melan theorem as well as bounds on various deformation parameters (such as displacements or plastic strain intensities) are derived. The solution of the «perturbed» shakedown problem in finite or holonimic terms permits the bound to be the most stringent and expressible in «local» terms instead of integral terms. A simple application concludes the paper.

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Sommario La memoria considera problemi di analisi a shakedown (o adattamento) relativamente a solidi elastoplastici sottoposti a carichi quasi-statici i quali variano restando all'interno di un dato dominio. Mediante l'uso di un metodo di perturbazione si fornisce una disuguaglianza generale dalla quale — particolarizzando opportunamente i termini della perturbazione — si deducono il teorema di adattamento di Melan nonché delimitazioni a priori su alcuni parametri della deformazione (come spostamenti o intensità delle deformazioni plastiche). La soluzione del problema perturbato, espressa in termini olonomi, consente di rendere tali delimitazioni stringenti al massimo possibile ed esprimibili in termini locali anziché in termini integrali. Una semplice applicazione conclude il lavoro.

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The results presented in this paper were obtained in the course of a research project sponsored by the National (Italian) Research Council, C.N.R., PAdIS Committee.