Hemivariational inequalities for linear and non linear elastic materials

Summary In the present paper we study hemivariational inequalities arising in linear and nonlinear elastic bodies which are subjected to boundary conditions of nonmonotone multivalued type. After the formulation of the boundary conditions and the derivation of the variational principles we study a hemivariational inequality for a plane elastic body concerning the approximation of its solution. The proposed method is illustrated by an example dealing with the adhesive contact problem of two cylinders and another one concerning an adhesively supported beam.

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Sommario Nel presente lavoro si studiano diseguaglianze semivariazionali che nascono in relazione a corpi elastici lineari e non lineari, sottoposti a condizioni al contorno di tipo a più valori non monotono. Dopo la formulazione delle condizioni al contorno e la derivazione dei principi variazionali si studia una diseguaglianza semivariazionale per un corpo elastico piano, con particolare riguardo per l'approssimazione della sua soluzione. Il metodo proposto viene illustrato con un esempio che tratta il problema del contatto con adesione di due cilindri.

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On leave from School of Technology. Aristotle University Thessaloniki, Greece. Fulbright Visiting Scholar.     

Microstructural effects in non linear stratified composites

In this paper, we analyze the microstructural effects on non linear elastic and periodic composites within the framework of asymptotic homogenization. We assume that the constitutive laws of the individual constituents derive from strain potentials. The microstructural effects are incorporated by considering the higher order terms, which come from the asymptotic series expansion. The complete solution at any order requires the resolution of a chain of cell problems in which the source terms depend on the solution at the lower order. The influence of these terms on the macroscopic response of the non linear composite is evaluated in the particular case of a stratified microstructure. The analytic solutions of the cell problems at the first and second order are provided for arbitrary local strain–stress laws which derive from potentials. As classically, the non-linear dependence on the applied macroscopic strain is retrieved for the solution at the first order. It is proved that the second order term in the expansion series also exhibits a non linear dependence with the macroscopic strain but linearly depends on the gradient of macroscopic strain. As a consequence, the macroscopic potential obtained by homogenization is a quadratic function of the macroscopic strain gradient when the expansion series is truncated at the second order. This model generalizes the well known first strain gradient elasticity theory to the case of non linear elastic material. The influence of the non local correctors on the macroscopic potential is investigated in the case of power law elasticity under macroscopic plane strain or antiplane conditions.     

On periodic flow of non linear viscoelastic liquids

Summary After brief reference to energetical principles in order to find suitable constitutive equations for non-linear viscoelastic fluids, the behaviour of stress components is examined, for a simple shear motion with sinusoidal time time varying displacements of a liquid with an oscillating fading memory.

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Sommario Dopo aver richiamato brevemente un possibile procedimento per la determinazione della equazione costitutiva per liquidi viscoelastici nel caso di grandi deformazioni, si esamina il comportamento delle componenti del tensore degli sforzi nel caso di un moto periodico a filetti rettilinei e paralleli, supposto che il liquido abbia una memoria debole e oscillante nel tempo.

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Research supported by C.N.R., Gruppo nazionale per la fisica matematica.     

Waves features of a non linear theory of porous media

Summary In this paper a one-phase porous medium is considered within the non linear theory proposed by G. Szefer [4].Among the other things by requiring, for the fundamental system of equations, the existence of exceptional waves, as usual in several non linear theories, we are able to characterize possible classes of constitutive laws for the elastic response functions and the porosity of the medium.

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Sommario Nella presente nota si considera un mezzo poroso mono-fase seguendo la teoria non lineare dei mezzi porosi proposta da G. Szefer. Richiedendo, com'è solito nelle teorie non lineari, l'esistenza di onde eccezionali, si caratterizzano classi di legami costitutivi per le funzioni di risposta elastica e per la porosità del mezzo.

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This work is supported by C.N.R. through the G.N.F.M.     

Some examples of nonuniqueness of solutions to the equation of linear elastodynamics

Sunto Si mostra, tramite esempi, come l'equazione che governa il moto di un corpo linearmente elastico unidimensionale possa ammettere, nell'ipotesi che il rapporto tra i campi di elasticità e densità diverga in un punto del corpo, infinite soluzioni corrispondenti ad una stessa assegnazione dei dati iniziali ed al contorno e delle forze.     

The centrifugal pendulum as a non linear torsional vibration absorber

Summary In this paper we study the centrifugal pendulum as a torsional vibrations absorber of an engine. A non-autonomous two degrees of freedom mechanical system with viscous-Coulomb damping is considered. The motion equations are written and we analyse for small oscillations of the pendulum, the effect of the non linear damping on the response curves in amplitude. Several numerical examples are ratio and the damping coefficients on the resonance peaks.

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Sommario In questo lavoro si studia il pendolo centrifugo come ammortizzatore di vibrazioni torsionali di un motore. Si considera un sistema meccanico non autonomo a due gradi di libertà con dissipazione coulombiana-viscosa. Si scrivono le equazioni del moto e si analizza, per piccole oscillazioni del pendolo, l'effetto dell'attrito non lineare sulle curve di risposta in ampiezza. Vengono presentati numerosi esempi numerici per illustrare l'influenza del rapporto di massa e dei coefficienti di attrito sui picchi di risonanza.

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This work has been realized within the activities of the Italian Council for the Research, C.N.R., G.N.F.M.     

A quadratic programming approach for certain classes of non linear structural problems

Summary The search for the deviation from the linear elastic, of the response of a nonlinear discrete structure to given loads, is formulated for various classes of cases. All the problems considered are shown to be reducible to a single mathematical model: the quadratic program with sign constraints only. The Kubn-Tucker theorem for quadratic programming directly supplies the appropriate extremum principles, for which some physical interpretations are proposed. Other results of quadratic programming theory prove to be useful in the discussion of the structural problems considered.

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Sommario Si considerano certe categorie di strutture costituite da un numero discreto di parti a comportamento non lineare e si formula il problema della ricerca dello scostamento del loro stato di sforzo e deformazione per dati carichi, da quello che si avrebbe in regime elastico lineare. Nei vari casi considerati il problema si riduce ad un programma quadratico con variabili soggette solo a restrizioni di segno. Il teorema di Kuhn e Tucker per programmazione quadratica fornisce gli opportuni principi di estremo, per i quali si propongono interpretazioni fisiche. Altri teoremi di programmazione quadratica si rivelano utili nel discutere la soluzione dei problemi formulati.

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This research was conducted during the author's stay at the Engineering Department of Cambridge University, England, with the financial support of a N.A.T.O. fellowship 1965–66. The author wishes to thank Professor Sir John Baker, Head of the Department, for the kind hospitality, and the C.N.R. (National Research Council) for the award of the fellowship.     

Proposed nomenclature for linear viscoelastic behavior

Summary There is proposed a brief systematic nomenclature for linear viscoelastic behavior; this nomenclature is based upon earlier reports of the American Society of Rheology and the Netherlands Society of Rheology.

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Zusammenfassung Es wird ein kurzes systematisches Nomenklatur-System für lineares viskoelastisches Verhalten vorgeschlagen; dieses System stellt eine gedrungene Zusammenfassung von zwei diesbezüglichen Rapporten der Amerikanischen Rheologischen Gesellschaft und der Niederländischen Rheologischen Gesellschaft dar.

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This report is a brief summary of two proposals on rheological nomenclature and definitions.     

Structure of the attracting set of a piecewise linear Hénon mapping

Detailed structure of the attracting set of the piecewise linear Hénon mapping (x,y)→(1−a|x|+by,x) with a=8/5 and b=9/25 is described in this paper using the method of dual line mapping. Let A and B denote the fixed saddles in the first quadrant, and in the third quadrant, respectively. It is claimed that (1) the attracting set is the closure of the unstable manifold of saddle B, which includes the unstable manifold of A as its subset, and (2) the basin of attraction is the closure of the stable manifold of A, bounded by the stable manifold of B, which is in the limiting set of the stable manifold of A. Relations of the manifolds of the periodic saddles with the manifolds of the fixed point are given. Symbolic dynamics notations are adopted which renders possible the study of the dynamical behavior of every piece of the manifolds and of every homoclinic or heteroclinic point.     

Mechanical behavior of sandwich panels with tetrahedral and Kagome truss cores fabricated from wires

Wires are great candidates as the raw material for truss periodic cellular metals because they can display high strength as in piano wires, are easy to fabricate, and can be controlled to be defect free. New approaches based on tri-axial weaving of wires to create ideal trusses, i.e., tetrahedral and Kagome truss have been presented. The mechanical properties of the sandwich panels with the truss cores fabricated by using the new approaches under compression and bending loadings are analyzed by elementary beam theory and experiments. The relative density, stiffness, and strength of the sandwich panels are estimated by the derived equations and compared with the measured results. The failure mechanisms of the sandwich panels are analyzed, and also benefits and shortcomings of each approach with respect to mechanical performance and production are discussed.     

Existence and uniqueness theorems for boundaryvalue problems involving incrementally non linear models

Sufficient conditions for existence and uniqueness of rate boundary value probleminvolving an incrementally non-linear constitutive equation useful for geomaterials are proved.The models considered are incrementally non-linear models not descending from a rate potentialand thus are not included into the Hill theory framework. Results are given in the framework ofsmall strain assumption. The main assumption is the positiveness of the second-order work. Theexistence theorem is a generalisation of the Lax-Milgram theorem, whereas uniqueness theoremis based on the well-known Hill exclusion functional.     

Creep recovery behavior of metallocene linear low-density polyethylenes

The rheological behavior of two metallocene linear low-density polyethylenes (mLLDPE) is investigated in shear creep recovery measurements using a magnetic bearing torsional creep apparatus of high accuracy. The two mLLDPE used are homogeneous with respect to the comonomer distribution. The most interesting feature of the two mLLDPE is that their molecular mass distributions are alike. Therefore, as one of the mLLDPE contains long-chain branches, the influence of long-chain branching on the elastic properties of polyethylene melts could be investigated. It was found that long-chain branches increase the elasticity of the melt characterized by the steady-state recoverable compliance. The long-chain branched mLLDPE has a flow activation energy of 45 kJ/mol which is distinctly higher than that of the other mLLDPE. The shear thinning behavior is much more pronounced for the long-chain branched mLLDPE. A discrepancy between the weight average molecular mass M _w calculated from size exclusion chromatography measurements by the universal calibration method and the zero shear viscosities of the two mLLDPE was observed. These observations are discussed with reference to the molecular architecture of the long-chain branched mLLDPE. The rheological properties of the long-chain branched mLLDPE are compared with those of a classical long-chain branched LDPE. It is surprisingly found that the rheological behavior is very much the same for these two products although their molecular mass distributions and presumedly the branching structures differ largely. Received: 15 February 1999 Accepted: 10 June 1999     

Micromechanical modeling of linear viscoelastic behavior of heterogeneous materials

Study of effective behavior of heterogeneous materials, starting from the properties of the microstructure, represents a critical step in the design and modeling of new materials. Within this framework, the aim of this work is to introduce a general internal variables approach for scale transition problem in linear viscoelastic case. A new integral formulation is established, based on the complete taking into account of field equations and differential constitutive laws of the heterogeneous problem, in which the effects of elasticity and viscosity interact in a representative volume element. Thanks to Green’s techniques applied to space convolution’s term, a new concentration relation is obtained. The step of homogenization is then carried out according to the self-consistent approximation. The results of the present model are illustrated and compared with those provided by Hashin’s and Rougier’s ones, considered as references, and by internal variables models such as those of Weng and translated fields.     

Linear and non linear rheology of a wormlike micellar system in presence of sodium tosylate

In this experimental work, we investigate the influence of an organic counterion, sodium tosylate, on the rheological properties of an aqueous solution of CTAB at the concentration of 0.05M. With this system we can clearly see shear thickening for small salt concentrations C _s and only shear thinning behavior at higher C _s characterized by a linear evolution of η=f(γ) in a log-log representation. In these evolutions it is only in a very small domain of concentrations of the salt (near C _s =0.035M) that we can observe a nearly constant plateau of the shear stress against shear rate. The values of σ₀ (characterizing the stress plateau), G ₀ (the plateau modulus) and τ_R (the relaxation time) obtained by dynamical rheological measurements, allow to compare experimental results obtained to predicted values of the theory of Cates corresponding to the occurrence of shear induced banding structures. Received: 22 July 1997 Accepted: 3 February 1998