Micro-mechanical study of interactions between polymers and filler aggregates

Filled rubber-like materials exhibit a complex, history-dependent hysteretic behavior which is mostly due to damage of micro-structures inside the rubber matrix. In this paper, we study the contribution of filler aggregates inside the elastomer to this damage behavior. To this end, a recently proposed multi-scale model of single aggregates [1] is applied. The network decomposition concept adopted there is further extended here to an additional network [1] which takes into account elasticity of filler aggregates and polymer chains in their vicinity. This network is described by means of a 3D statistical volume element (SVE) obtained by homogenization over the aggregate size distribution within the rubber matrix. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Non-linear near-resonance oscillations of an elastic incompressible layer

Plane one-dimensional waves of small amplitude, propagating transverse to an incompressible elastic layer and reflected successively from its boundaries, are considered. The oscillations are caused by small periodic (or close to periodic) external action on one of the layer boundaries, when the period of the external action is close to the period of natural oscillations of the layer. One of the boundaries of the elastic layer is fixed, while the other performs small specified two-dimensional motion in its plane. In such a near-resonance situation, non-linear effects occur which may build up over time. A system of equations is obtained which describes the slow change in the functions characterizing the oscillations of the medium in each period of the external action. It is assumed that all the quantities depend both on real time, any change of which in the approach considered is limited to one period, and on “slow” time, for which one period of real time serves as a small quantity. It is assumed that the evolution of the solution occurs when the slow time changes, while the role of real time is similar to the role of a spatial variable. This system of equations is obtained by the method of averaging over a period of the quantities representing nonlinear terms and the effect of the boundary conditions in the equations. It contains derivatives with respect to the real and slow times and also values of the functions characterizing the solution averaged over a period of the real time. If the averaged values are known, the equations have a hyperbolic form and their solutions can be both continuous and contain weak and strong discontinuities.     

Optimal designing of orthotropic cylindrical shells fastened to elastic filler

Conclusions A method has been proposed for optimally designing an orthotropic cylindrical shell rigidly fastened to an elastic and isotropic filler of finite dimensions. The design takes into account simultaneous action of pressure, body forces, and heat on the structure. The optimum design has been calculated for the case of temperature-dependent elastic properties and strength characteristics of the tape. The method allows also for limitation on the strength of the filler. The convergence of the iteration process schematically shown in Fig. 2 is quite fast. Indeed, for the given design variant, the condition of manufacturability (1) is satisfied with a sixfold margin in the third approximation (n=3) already.Translated from Mekhanika Kompozitnykh Materialov, No. 1, pp. 91–94, January–February, 1984.     

Non-linear reflection of a plane wave in an elastic medium

Exact formulae are derived for the reflected and refracted waves which occur for the inclined incidence of a plane horizontally polarized transverse wave of arbitrary profile on a horizontal interface between two elastic half-spaces experiencing non-linear friction when they move with respect to one another. A smooth function of general form is adopted as the friction function, which depends on the difference between the horizontal velocities of the elements of the boundaries of the half-spaces considered. It is shown that if the friction function depends non-monotonically on the relative velocity of displacement of the sides of a slit, then even when the profile of the incident wave is smooth, the reflected and refracted waves may contain strong discontinuities.     

Structural damping in an equi-strength cylindrical shell with an elastic filler

We present a method for rational application of the deformation properties of a shell system with an elastic filler: design of a shell with variable thickness while preserving the load-bearing ability of the system as a whole. For the equi-strength shell thereby obtained we state and solve the mixed contact problem taking account of dry friction with nonmonotone loading, making it possible to estimate the structural hysteresis in the system. Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 37, 1994, pp. 86–91.     

Non-linear hypotheses of deformation of flat cross sections of elastic sandwich beam.

The main goal of the paper is a comparison of three non-linear hypotheses of deformation of flat cross sections of an elastic sandwich beam. Three-layered simply supported beam is under uniformly distributed load (pressure). Studied beam has two faces and a metal foam core. Mechanical properties of the isotropic metal foam core of the beam vary across the thickness of the beam. Fields of displacements (geometric models of non-linear hypotheses) are described. The results of solution of the deflection problem are presented at Figures. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Optimized synthesis of cylindrical shells of reinforced plastics with elastic filler for stable performance

A cylindrical shell of reinforced plastic with an elastic filler is optimized. Limited local stability and certain geometrical constraints are taken into account. The method of projectable gradients, according to J. B. Rosen, is used for optimizing the design. Optimum reinforcement structures are obtained for a shell, depending on the stiffness of the elastic filler.     

Effect of elastic filler on the fatigue failure of thermoplastic polyurethane film at low temperature

Conclusion It was established by mathematical modeling of the curves of spectral transmissivity and by comparing them with experiments that in the mixture of polyurethane with caoutchouc an increase of the volume fraction of filler entails changes of the characteristic dimensions of its particles. With small volume fractions of filler (less than 10%), in consequence of the predominantly small size of the impurities, the mechanism of quasibrittle failure is realized without development of bulk damage to the mixture. When the mixture contains 20–30% filler, satisfactory static elastic and strength properties are retained, and in case of fatigue a considerable amount of damage accumulates and the mechanism of inhibiting macrocracks on the boundaries of impurities begins to act. When the proportion of filler increases further, the elastic and strength properties of the mixture are rapidly impaired, and as a consequence the material becomes practically unusable in operation.Translated from Mekhanika Kompozitnykh Materialov, No. 6, pp. 1040–1044, November–December, 1988.     

An eigenvalue approach to the limiting behavior of time series aggregates

Many time series variables such as rainfall, industrial production, and sales exist only in some aggregated forms. To see the implication of time series aggregation it is important to know the limiting behavior of the time series aggregates. From the relationship of autocovariances between the underlying time series variable and its aggregates, we show that the limiting behavior of time series aggregates is closely related to the eigenvalues and the eigenvectors of the aggregation operator. Specifically, the vector of admissible autocorrelations of the limiting model for the time series aggregates is the eigenvector associated with the largest eigenvalue of the aggregation transformation. This provides an interesting and simple method for deriving the limiting model for time series aggregates. Systematic sampling of time series can be treated similarly. The method is illustrated with an empirical example.     

Effect of aggregation of a dispersed rigid filler on the elastic characteristics of a polymer composite

Conclusions We proposed a method for describing the effective elastic characteristics of a polymer composite with a rigid aggregating filler. An important feature of such a medium is the variable coupling of the inclusion phase in relation to its volume content. A change in the degree of coupling of the filler is accounted for by introducing an additional parameter. We examined a method of determining the coupling parameter from the results of statistical modeling of the geometry of the medium. Using the example of a calcite-HDPE composite, we showed that aggregation has a significant effect on the dependence of the elastic modulus on the volume content of filler; satisfactory agreement was obtained between the theoretical and experimental data.Translated from Mekhanika Kompozitnykh Materialov, No. 1, pp. 14–22, January–February, 1986.     

Initial supercritical behavior of buckled transversely isotropic elastic medium

The paper is concerned with a transversely isotropic homogeneous elastic medium subjected to uniform compression in the isotropy plane. The medium becomes unstable in the sense of Hadamard [1] at a definite level of initial strain. The critical strain is established to be uniquely determinate from the system of equations of bifurcation of equilibrium; however, there are many modes of buckling corresponding to this strain. A solution of the system of equations of bifurcation is built in the form of doubly periodic functions sinr ₁ x ₁sinr ₂ x ₂. The uncertainty of the mode of buckling consists in the fact that the wave numbers r ₁ and r ₂ remain arbitrary. In order to determine the relationship between the wave numbers we examine the initial supercritical behavior of the material. It turns out that the only possible modes are the chess-board mode (with r ₁ = r ₂) and the corrugation-type mode (when one of the wave numbers r ₁ or r ₂ vanishes). The initial supercritical equilibrium is shown as being stable.     

Parametric oscillations of an orthotropic cylindrical shell with an elastic filler I. Formulation of the problem

A linear model of a Timoshenko-type shell with an elastic filler is proposed for solving the problem of the parametric oscillations of the system. A class of operators is derived, which act in a Hilbert space, for which the problem of the determination of a spectrum of eigen-values permits an iterative scheme of solution. Thus, the method for solving problems of parametric oscillations of structures proposed by Bolotin is confirmed.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 3, pp. 470–478, May–June, 1974.     

Some notes on the relation between the elastic moduli of anisotropic cristallites and quasi-isotropic polycrystalline aggregates

Zusammenfassung Drei Methoden zur Berechnung des mittleren Elastizitätsmoduls vielkristalliner kubischer Metalle werden formuliert als eine Summierung über die Beiträge der Kristallite zur elastischen Energie. Dadurch kommt man zu einer zwanglosen Erklärung für die Abweichungen der berechneten Mittelwerte vom beobachteten Modul des Vielkristalls.     

On the influence of the prebuckling behavior of elastic structures on their initial postbuckling behavior

Depending on whether or not bending and/or torsion are part of the prebuckling deformation state of elastic structures, the initial postbuckling behavior is either completely general or characterized by specific restrictions [1]. In this context, the term ‘completely general’ means (a) no restriction to monotonic improvement or deterioration of the initial postbuckling behavior in consequence of a monotonic change of a design parameter and (b) the impossibility of ‘zero-stiffness postbuckling’ characterized by a postbuckling path at a constant load level. The mathematical reason for restrictions of the postbuckling behavior is a constraint condition for the eigenvector ν ₁ appearing in the classical static stability condition (1) where denotes the tangent stiffness matrix [2]. This condition contains the first and the second derivative of with respect to the dimensionless load parameter λ. The theoretical findings will be verified numerically. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The behavior of elastic heat conductors with second-order response functions

Second-order forms for the material response functions of an elastic heat conductor are derived by approximating the response functions by polynomials in the appropriate invariants. Solutions based upon these forms of the response functions are exact for special materials and approximate for general materials. The second-order dependence on temperature of isothermal elasticity solutions is found, and the results are shown to agree well with experimental data taken on rubber. Within the second-order theory for incompressible and isotropic bodies, the problems of biaxial stretching of a plate with transverse heat flow and the simultaneous extension and shear of a cylindrical annulus with radial heat flow are solved.

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Résumé Les termes du deuxième ordre des fonctions de réponse d'un conducteur de chaleur élastique sont dérivés dans une approximation basée sur le développement en polynomes d'invariants appropriés de ces fonctions. Les solutions qui en résultent sont exactes pour certains matériaux et approchées dans le cas général. Une dépendance du second ordre en température de l'élasticité isotherme est établie et il est montré que, dans le cas du caoutchouc, les résultats sont en bon accord avec les données expérimentales. Les problèmes de l'état de tension bidimensionnel d'une plaque soumise à un flux de chaleur transversal et de l'allongement et du cisaillement simultanés d'un anneau cylindrique parcouru par un flux radial sont résolus dans le cadre de la théorie du second ordre pour les corps incompressibles et isotropes.