One-dimensional propagation of discontinuities in non-homogeneous linear viscoelastic semi-infinite media

One-dimensional wave-propagation is analysed in a non-homogeneous, isotropic, linearly viscoelastic semi-infinite medium, by the theory of singular surfaces. The characteristics of the medium at any point are assumed to be dependent upon the position of the point. The solution for the stress field, which is valid even after the wave-front has passed, is obtained in the form of Taylor’s series by prescribing the time dependent stress boundary condition in the form of a Maclaurin’s series. Furthermore, it is shown that the higher order discontinuities satisfy the same propagation conditions as the stress waves and all the discontinuities decay as they traverse the material. An application of inhomogeneity varying exponentially with position has been analysed. Diagrams for the above example with arbitrarily chosen parameters are presented.     

A non linear integral constitutive equation for viscoelastic fluids

An intégral constitutive equation is written using a particular reference frame, built with unit vectors along the principal axes of the rate-of-deformation tensor, and using the associated intrinsic rate-of-rotation. This equation is easier to handle in calculations than corotational or codeformational models. The material functions for a rheological model including the first six terms of the constitutive equation have been studied in steady and unsteady shear flows, as well as in elongational flows. Material functions are readily written from six memory functions and no inconsistency comes out.     

Propagation of small disturbances on large deformations in a viscoelastic fluid

Summary The propagation characteristics (i.e., wave velocity & amplitude strength) of small disturbances existing in a large base motion of a BKZ viscoelastic liquid are considered. A small motion superposed on a large deformation is taken to be a wave of small amplitude propagating across a base flow. The analysis treats a wave as a singular surface across which there is a jump discontinuity of a kinematical variable. Specific attention is directed to second order waves where the perturbed displacement and its first derivatives are continuous but higher order derivatives are not. The growth and decay of amplitude of small plane second order shear waves propagating across steady simple shearing flow is studied in detail. Using the Zapas form for the BKZ material function, it is found that the amplitude of these shear waves will always decay. The rate of decay is dependent on the rate of shear of the base flow.

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Die Ausbreitung von endlichen Verformungen überlagerten kleinen Störungen in einer viskoelastischen Flüssigkeit

Zusammenfassung Die Ausbreitungseigenschaften (d.h. Wellengeschwindigkeit und Amplitude) von kleinen, einer endlichen Grundbewegung überlagerten Störungen in einer viskoelastischen Flüssigkeit werden betrachtet. Die kleine Störung wird als eine der Grundströmung überlagerte Welle von kleiner Amplitude angenommen. In der Untersuchung wird die Welle als eine singuläre Fläche mit einer Unstetigkeit in einer kinematischen Größe behandelt. Besondere Beachtung gilt Wellen 2. Ordnung, mit stetiger Verschiebung und deren 1. Ableitung, aber nicht stetigen höheren Ableitungen. Wachstum und Abklingen der Amplitude von einfachen stationären Scherströmungen überlagerten ebenen Scherwellen 2. Ordnung werden im Detail untersucht. Bei Verwendung der von Zapas angegebenen Form der BKZ-Werkstoffunktion zeigt sich, daß die Amplitude dieser Scherwellen stets abklingt.Die Abklinggeschwindigkeit hängt von der Schergeschwindigkeit der Grundströmung ab.

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With 7 Figures     

Crack propagation in linear viscoelastic solids: some new results

An extension of a theory of linear viscoelastic fracture is derived which removes a former restriction that Poisson's ratio is a constant. The new formulation thus includes both material property characteristics, i.e. the bulk (dilatation) and shear (distortion) linear viscoelastic compliances. Characteristic crack growth laws, assuming a Dugdale region ahead of the crack, are developed for creep laws which depend linearly upon time but are thought to be representative of more general materials. Examples include a single crack, a linear array of cracks, and a specialized cyclic load application.

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Résumé Une extension de la théorie de la rupture viscoélastique est déduite; cette extension se dégage de la restriction antérieure suivant laquelle le rapport de Poisson est une constante. La nouvelle formulation comporte donc deux caractéristiques des propriétés des matériaux, à savoir les compliances viscoélastiques linéaires d'ensemble (dilatation) et de cisaillement (distorsion). Les lois de croissance caractéristiques de la fissure supposant une région de Dugdale en amont de celle-ci sont développées pour des lois de fluage dépendant linéairement du temps; on pense cependant qu'elles sont représentatives de matériaux plus généraux. Des exemples comportant une fissure simple, une série simple de fissures alignées et une application spécifique de sollicitations cycliques sont fournies.

     

A note on plane wave propagation in a linear viscoelastic solid

A new formulation of the governing equations for one spatial dimension wave propagation in a linear viscoelastic solid with more than one discrete relaxation time is proposed. The resulting system of three equations is treated as a strictly hyperbolic system of first order hyperbolic partial differential equations and the method of characteristics is adapted to obtain numerical solutions. Results are presented for a solid with two discrete relaxation times and are compared with those obtained from a predictor-corrector scheme.     

Wave propagation in linear viscoelastic composites modelled as interpenetrating solid continua

This work concerns the behavior of shock waves and higher order discontinuities which propagate in a viscoelastic composite which is modelled as a mixture ot two interpenetrating solid continua. The response of the composite is assumed to be linear. The propagation conditions for shock and higher order waves are derived and it is shown that, in general, six waves of any order may propagate in any direction. It is shown that the amplitudes of all waves decay as they propagate. Detailed results are presented for isotropic composites.     

Fatigue crack propagation from small holes in linear arrays

The propagation of fatigue cracks emanating from linear arrays of circular holes has been analytically investigated. An approximate fracture mechanics analysis is developed for plates containing cracked holes in rows which are either normal to or parallel to a uniaxial tensile load and the crack growth law % MathType!MTEF!2!1!+- % feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf % ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr % 0-vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaiaadsgaieGaca % WFHbacbaGaa43laiaa+rgacaWFUbGaa8xpaiaa-feacaGFOaGaeuiL % dqKaa83saiaa+LcadaahaaWcbeqaaiaa-jeaaaaaaa!41B0! \[ da/dn = A(\Delta K)^B \] is integrated over the K field of each array from a crack nucleus size to the critical crack length. Propagation lifetimes thus obtained are found to decrease with decreasing hole spacing for collinear arrays normal to the loading axis but to increase with decreasing spacing for holes aligned parallel to the stress axis. In addition, the absolute hole size, applied stress range, and the materials crack growth law are shown to affect the lifetimes of the various arrays.

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Zusammenfassung Es wird eine analytische Studie vorgelegt über die Fortpflanzung von Alterungsrissen, welche aus einer Anordnung von Rundlöchern herstammen. Für Platten mit gerissenen Löchern, welche in zu einer uniaxialen Spannung senkrecht oder parallel liegenden Reihen angeordnet sind, wurde eine auf die Bruchmechanik aufbauende angenäherte Analyse entwickelt. Das Gesetz der Rißfortpflanzung % MathType!MTEF!2!1!+- % feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf % ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr % 0-vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaiaadsgaieGaca % WFHbacbaGaa43laiaa+rgacaWFUbGaa8xpaiaa-feacaGFOaGaeuiL % dqKaa83saiaa+LcadaahaaWcbeqaaiaa-jeaaaaaaa!41B0! \[ da/dn = A(\Delta K)^B \]wurde \:uber die Kennzahl K der verschiedenen Arten von Anordnungen integriert und dies im Bereich von den Anfangsabmessungen des Risses bis zur kritischen L\:ange.Die sich aus dieser Berechnung ergebende Lebensdauern bei Rißfortpflanzung zeigen eine Verminderung für senkrecht zur Beanspruchungsaxe liegende kolineare Anordnung wenn man den Lochabstand verkleinert.Außerdem wird gezeigt, daß die für verschiedene Anordnungen ermittelte Lebensdauer von der absoluten Abmessung der Löcher, der Amplitude der angelegten Spannungen und dem dem Werkstoff eigenen Rißfortpflanzungsgesetz beeinflußt wird.

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Résumé On a étudié de manière analytique la propagation des fissures de fatigue provenant d'arrangements de trous circulaires. Pour des plaques comportant des trous fissurés en rangées orientées normalement ou parallèlement à une tension uniaxiale, on a développé une analyse approximative basée sur la mécanique de la rupture; la loi de propagation des fissures % MathType!MTEF!2!1!+- % feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf % ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr % 0-vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaaiaadsgaieGaca % WFHbacbaGaa43laiaa+rgacaWFUbGaa8xpaiaa-feacaGFOaGaeuiL % dqKaa83saiaa+LcadaahaaWcbeqaaiaa-jeaaaaaaa!41B0! \[ da/dn = A(\Delta K)^B \] a \'et\'e int\'egr\'ee sur la caract\'eristique K de chaque type d'arrangements, entre la dimension de la fissure initiale et longueur critique.Les durées de vie en propagation déduite de cette opération font état d'une réduction lorsque diminue la distance entre trous pour des arrangements colinéaires normaux par rapport à l'axe de sollicitation.En outre, on montre que la durée de vie correspondant à divers arrangements de trous est affectée par la dimension absolue des trous, l'amplitude des contraintes appliquées, et la loi de propagation des fissures propre au matériau.

     

On the theory of diffusion in linear viscoelastic media

Summary A recently developed viscoelastodiffusive theory is further examined in terms of its mathematical foundations and obvious generalizations. A derivation of the basic equations under quite general assumptions is provided and an equivalence between a hereditary version and a differential operator version of the theory is established. Both relaxation and creep types of viscoelastodiffusive behavior is considered. The concepts of chemical potential, fading memory, and a correspondence principle are discussed. The general development is concluded with an illustrative example.With 2 Figures     

Thermal convection waves in viscoelastic fluids

We demonstrate that undamped and weakly damped thermal convection waves, in particular, thermal and transverse waves, can be propagated in viscoelastic heat-conducting fluids. We have found the frequency spectrum and the wavelengths for such waves. We indicate the possibility of developing a mechanical generator of thermal oscillations.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 16, No. 5, pp. 780–785, May, 1969.     

Crack propagation,resulting from a monotonic increasing applied stress,in a linear viscoelastic material

A theory of crack propagation, resulting from the application of a monotonic increasing applied stress, in a linear viscoelastic material, is derived based upon an energy balance fracture criterion. It is shown that for a Maxwell solid the crack growth law can be derived either from a global energy balance taking full account of the energy dissipation resulting from viscoelastic flow, or from a local energy balance taking account of the dissipation in the failure zones. The local energy balance method allows the derivation of the crack growth law for more general linear viscoelastic solids. The theory predicts the well known Griffith condition for fracture when the material is simply linear elastic. For a crack having failure zones in a linear viscoelastic solid the growth law for a constant applied stress is where c(t) is the time dependent half-crack length, 641-1 is the yield or crazing stress in the failure zone, K(t) is the time dependent stress intensity factor, is the fracture energy, is Poisson's ratio and J(t) is the uniaxial creep function of the viscoelastic material. This growth law is valid if either J(t)0 for all times t>0 (i.e. a Maxwell solid) or if 641-1 641-2 641-3

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Résumé Une théorie de propagation d'une fissure résultant de l'application d'une contrainte monotonique croissante dans un matériau linéaire viscoélastique est dérivée sur la base d'un critère d'équilibre d'énergie de rupture. On montre que, pour un solide de Maxwell, la loi de croissance d'une fissure peut être déduite soit d'un équilibre global de l'énergie tenant complètement compte de la dissipation de l'énergie qui résulte de l'écoulement viscoélastique ou d'un équilibre local de l'énergie tenant compte de la dissipation de l'énergie dans les zônes de rupture. La méthode de l'équilibre d'énergie locale permet de dériver une loi de croissance de la fissure pour des solides viscoélastiques linéaires de caractère plus général. La théorie prédit la condition bien connue de Griffith pour la fracture dans le cas d'un matériau de linéarité élastique simple. Dans le cas d'une fissure qui présente des zônes de rupture dans un solide viscoélastique, la loi de croissance pour une contrainte appliquée est fournie. Dans cette loi, c(t) est la longueur de la demi-fissure dépendant du temps, 654-1 est la limite élastique dans la zône fissurée, K(t) est le facteur d'intensité de contrainte dépendant du temps, est l'énergie de rupture, est le module de Poisson et J(t) est la fonction uniaxiale de fluage dans le matériau viscoélastique. Cette loi de croissance est valable pour des conditions particulières de J(t).

     

On the mechanics of crack closing and bonding in linear viscoelastic media

The mechanics of quasi-static crack closing and bonding of surfaces of the same or different linear viscoelastic materials is described. Included is a study of time-dependent joining of initially curved surfaces under the action of surface forces of attraction and external loading. Emphasis is on the use of continuum mechanics to develop equations for predicting crack length or contact size as a function of time for relatively general geometries; atomic and molecular processes associated with the healing or bonding process are taken into account using a crack tip idealization which is similar to that used in the Barenblatt method for fracture. Starting with a previously develiped correspondence principle, an expression is derived for the rate of movement of the edge of the bonded area. The effects of material time-dependence and the stress intensity factor are quite different from those for crack growth. A comparison of intrinsic and apparent energies of fracture and bonding is made, and criteria are given for determining whether or not bonding can occur. Examples are given to illustrate use of the basic theory for predicting healing of cracks and growth of contact area of initially curved surfaces. Finally, the effect of bonding time on joint strength is estimated from the examples on contact area growth.

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Résumé On décrit le mécanisme de fermeture d'une fissure quasi-statique et de collage des surfaces de matériaux visco-élastiques linéaires indentiques ou différents. On couvre également l'étude de la liaison dans le temps de surfaces à courbure initiale sous l'effet des forces de surface résultant de l'attraction et de charges extérieures. L'accent est placé sur l'utilisation de la mécanique des milieux continus pour l'établissement d'équations permettant de prédire la longueur de fissuration et la dimension des zones et contact en fonction du temps, dans le cas de géométries relativement générales. On prend en compte les processus atmostiques et moléculaires associés aux phénomènes de cicatrisation ou de collage en recourant à une idéalisation de l'extrémité de la fissure comparable à celle utilisée dans la méthode d'analyse des ruptures due à Barenblatt. En partant d'un principe de correspondances établi précédemment, on déduit une expansion décrivant la vitesse de déplacement du bord de la zone collée. On constate que les effets de la dépendance des propriétés du matériau en fonction du temps et du facteur d'intensité de contraintes sont différents de ceux exercés sur la croissance d'une fissure. On procède à une comparaison des énergies intrinsèques et apparentes de rupture et de collage, et on donne des critères pour déterminer si un collage peut ou non avoir lieu. Des exemples illustrent l'emploi de la théorie de base pour décrire la cicatrisation des fissures et la croissance de la surface de contact de surfaces initialement courbes. Enfin, on estime l'effet du temps de collage sur la résistance du joint, à partir des exemples de croissance de la surface de contact.

     

Rayleigh wave propagation in a viscoelastic half-space

Summary Rayleigh waves excited by an impulsive force imbedded in a linear viscoelastic half-space are synthesized by applying an approximate inversion of the Fourier transform which yields reliable results. The method is general enough and can be applied to general models of viscoelasticity described by the Boltzmann superposition principle, with a relaxation or creep function given analytically or numerically in the time or frequency domain. Illustrations are given in cases of simple and complicated models of viscoelasticity.     

On the use of a spectrum-based model for linear viscoelastic materials

A new spectrum-based model for describing the behavior of time-dependent materials is presented. In this paper, unlike most prior modeling techniques, the time-dependent response of viscoelastic materials is not expressed through the use of series. Instead, certain criteria have been imposed to select a spectrum function that has the potential of describing a wide range of material behavior. Another consequence of choosing the spectrum function of the type used in this paper is to have a few closed form analytic solutions in the theory of linear viscoelasticity. The Laplace transform technique is used to obtain the necessary formulae for viscoelastic Lame' functions, relaxation and bulk moduli, creep bulk and shear compliance, as well as Poisson's ratio. By using the Elastic–Viscoelastic Correspondence Principle (EVCP), material constants appearing in the proposed model are obtained by comparing the experimental data with the solution of the integral equation for a simple tensile test. The resulting viscoelastic functions describe the material properties which can then be used to express the behavior of a material in other loading configurations. The model's potential is demonstrated and its limitations are discussed.     

Wave propagation in fluid filled nonlinear viscoelastic tubes

Summary The present work considers one dimensional wave propagation in an infinitely long, straight and homogeneous nonlinear viscoelastic tube filled with an incompressible, inviscid fluid. In order to include the geometric dispersion in the analysis, the tube wall inertia effects are added to the pressure-area relation. Using the reductive perturbation technique, the propagation of weakly nonlinear waves in the long-wave approximation is examined. In the long-wave approximation, a general equation is obtained, and it is shown that by a proper scaling this equation reduces to the well-known nonlinear evolution equations. Intensifying the effect of nonlinearity in the perturbation process, the modified forms of these evolution equations are also obtained. In the absence of nonlinear viscoelastic effects all the equations reduce to those of the linear viscoelastic tube.     

Plane defect-field wave propagation in a viscoelastic medium

The wave solutions of the system of equations describing the defect field in a viscoelastic medium are studied within the framework of the defect field theory. The refraction and absorption coefficients and the velocities of propagation of the elastic continuum and the defect continuum waves are determined. Features of the correlation between various waves are analyzed.     

Experiments on crack propagation in a viscoelastic material

The problem of plane stress crack growth in a rectangular linearly viscoelastic plate under a pair of equal and opposite forces applied to the crack surface is studied experimentally. A fatigue crack is generated at the end of a slot cut on the specimen. After application of two opposite forces to the specimen with a fatigue crack, the crack grows gradually. It is found that under sufficiently small load, the speed of crack growth decreases as time increases, and crack arrest occurs finally. However when the applied load is sufficiently large, the crack grows with increasing speed until a catastrophic failure of the specimen takes place. Between these two extreme cases, the crack grows with an oscillatory speed. A theory based on the models of an infinite plate and double cantilevers is proposed for the explanation of the behaviour of crack growth under various loads. Our experimental results check fairly well with theoretical predictions.     

Transient wave propagation in a viscoelastic sandwich plate

Summary This paper examines the propagation of coupled P and S waves in a three layer sandwich plate consisting of a viscoelastic core bounded by identical elastic outer layers. The waves are initiated by an impulsive normal line load applied to the upper surface of the plate. The response of the top surface is computed for various values of the creep and relaxation time constants of the core. The results indicate that, for some values of the time constants, the viscoelastic core provides effective damping of the transients, while for others the damping is virtually non-existent. In each case the dominant contribution to the far-field solution comes from the Rayleigh wave.     

Flow of viscoelastic fluids under high shearing stresses

Some characteristics of viscoelastic fluid flow under high shearing stresses are analyzed here.Notation _w tangential shearing stress at a channel wall - h half-height of a channel - L channel length - =w/¯w flow velocity, dimensionless - ¯w mean flow velocity at a given section - fluidity of viscoelastic fluid - P_xx-P_yy first difference of normal stresses, N/m² - =y/h distance from channel wall, dimensionless Translated from Inzhenerno-Fizicheskii Zhurnal, vol. 20, No. 5, pp. 918–920, May, 1971.     

Axisymmetric wave propagation in a solid viscoelastic sphere

Two problems of linear wave propagation in a viscoelastic solid sphere are solved. The waves are generated by two types of impact on the surface of the sphere. The deformation has symmetry with respect to an axis through the center of the sphere. The solution is based on a superposition principle which reduces the general solution to a static elastic solution, an elastic solution of an eigenvalue problem and an integral equation of the Volterra type involving time only. The solutions are given in double infinite series involving spherical Bessel functions, Legendre polynomials and Legendre functions of the first kind and order one.