Theoretical estimation of fracture toughness of fibrous composites

A method of estimating the fracture surface energy of fibre-reinforced materials is discussed. The surface work is shown to increase with increasing fibre content, strength and diameter, and decrease with increasing fibre modulus and matrix flow stress (or hardness).Relatively short fibres should be used if high toughness is required, and the maximum toughness that can be achieved is limited by the amount of crack opening that can be permitted. Under certain conditions, incorporation of fibres into a material can lead to embrittlement.Symbols used c half length of crack - d fibre diameter - D average separation of nearest neighbour fibres - E Young's modulus - G shear modulus of matrix - K fracture toughness - L length of fibre on either side of crack - n number of fibres per unit area - P force on fibre - R pressure exerted by matrix on fibres - u displacement - U work - x distance Greek symbols 8 G/d ² E _fr log(2/p 3) - surface energy or work - tensile strain - _fm E _mr/E_fr R - tensile stress - shear stress - coefficient of friction - Poisson's ratio Suffixes c composite - e elastic - f fibre - m matrix     

On the calculation of plastic energy dissipation rate during stable crack growth

An approximate analysis is presented for the calculation of the plastic energy dissipation rate during stable growth of a centrally located through crack. in a sheet subjected to gradually increasing uniaxial tension normal to the crack plane.It is shown that the plastic energy dissipation rate is a function of the slow growth parameter (_p/)·(d/da)+(_p/a), where _p is the plastic enclave width in the plane of the crack and d/da is the rate of increase of the gross stress with respect to stable growth. At the point of instability this parameter becomes equal to _p/a. By assuming that this parameter is zero at the point of instability, a simple expression is obtained for the plastic energy dissipation rate.The analysis excludes the energy dissipation rate resulting from energy changes in an inner fracture zone in the immediate neighborhood of the crack tip in which it is presumed that fracture processes such as vacancy formation, crack initiation by dislocation pile ups etc., are active. The analysis is not applicable in this inner zone as deformation is not homogeneous.

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Zusammenfassung Es wurde eine mehr oder weniger genaue Analyse für die Kalkulation eines plastischen Energiezerstreuungsverhältnisses während stabilem Wachstums eines zentral lokalisierten Durchrißes in einer einfachen Anspannungsplatte präsentiert.Man zeigte, daß das plastische Energiezerstreuungsverhältnis eine Funktion des langsam wachsenden Parameters ist, wobei _p die eingeschlossene, plastische Weite in der Rißebene und d/da das Wachstumsverhältnis des Rohdruckes mit Rücksicht auf das stabile Wachstum, darstellt.Diese Analyse schließt das Energiezerstreuungsverhältnis vom Energiewechsel in einer inneren Bruchzone in der unmittelbaren Nähe der Rißspitze aus. Man nahm nicht an, daß these Veränderung homogen, sondern auf Vakanzen, Verschiebungen, etc. begründet sei.

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Résumé On présente une analyse approchée pour le calcul de la vitesse de dissipation de l'érgie de déformation plastique au cours de l'extension stable d'une fissure située au centre d'une tôle et traversant celle-ci de part en part, lorsque cette tôle est sujette à une contrainte uniaxiale, normale au plan de la fissure, et graduellement croissante.On montre que cette vitesse de dissipation est fonction du paramètre d'extension lente: où _p est la largeur de l'enclave de déformation plastique dans le plan de la fissure, et d/da le taux d'accroissement de la tension nominale par rapport à l'extension de la fissure dans des conditions stables.Au point d'instabilité, ce paramètre devient égal à _p/a, en supposant que sa valeur soit nulle, on aboutit à une expression simple de la vitesse de dissipation de 1'énergie de déformation plastique.L'analyse ne considère pas la vitesse de dissipation qui résulte de modifications de l'énergie au sein d'une zone de rupture interne, et située dans le voisinage immédiat de la pointe de la fissure. Dans cette zone, l'on présume que les processus de fissuration tels que la formation de lacunes, ou l'amorçage d'une fissure par empilements de dislocations, etc., sont particulièrement actifs. L'analyse n'est pas applicable dans cette zone interne, car les déformations n'y sont pas homogènes.

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Notations 2a Crack length - t Thickness of sheet - W Width of sheet - Gross stress applied at infinity normal to the crack plane - A, ₀, n Parameters in a Ramburg-Osgood representation of the octahedral shear stresss-hear strain curve - _oct Octahedral shear stress at any point near the crack tip - _oct Octahedral shear strain at any point near the crack tip. - _infoct ^supel Octahedral shear stress at any point near the crack tip given by elastic analysis - _oli Octahedral shear stress at yield - _oli/₀ - _infoct ^supel /_oli - \5m _oct/_oli - E Young's modulus - E _s Secant modulus - G Shear modulus - G _s Secant modulus of the octahedral stress-strain curve - u _p Plastic energy density at any point inside the plastic enclave at the crack tip - U _p Plastic energy dissipated in the plastic enclave per unit thickness - v Poisson's ratio - r, Polar co-ordinates with the crack tip as origin - K Stress intensity factor - _p Plastic enclave width given by Irwin's formula - f ₁, f ₂, f ₃ Functions of , defining the stress field near the crack tip - f _e f _inf1 ^sup2 –f ₁ f ₂+f _inf2 ^sup2 +3f _inf3 ^sup2 - _oct Limiting octahedral shear stress beyond which homogeneous plastic deformation is not possible since fracture processes such as vacancy formation etc., become active. - \5m_{0 infoct} ^supu /_oli - C (_p/)·(d/da)+(_p/da); slow growth parameter - B (G·A/_o1i)· ⁿ      

Decreasing of the initial shear modulus with increasing axial strain explained by means of a plastic-hypoelastic model

Summary The purpose of this work is to examine in detail the possibility to explain the decreasing of the initial shear modulus with increasing axial strain, observed first by Feigen, by means of the plastic-hypoelastic stress-strain relation suggested by Lehmann and by the author of the present paper.Notations _ij components of the infinitesimal strain tensor dilatation - strain deviator - _ij components of the stress tensor - spherical part of the stress tensor - stress deviator - ²= _{ij ij} second invariant of the stress deviator - = ₃₃ axial strain - e= ₁₃ shear component of the strain tensor - =2 ₁₃ shear strain - = ₃₃ axial stress - s= ₁₃ shear stress - T _ij components of Cauchy's stress tensor - F _ij components of the deformation gradient - L _ij components of the velocity gradient (Eulerian coordinates) - components of the rate of deformation tensor - components of the spin tensor - components of the rate of deformations deviator - components of Cauchy's stress deviator - T=T ₃₃ axial Cauchy's stress With 7 Figures     

The preparation of alumina fibre by sol-gel processing

Attempts have been made to prepare alumina fibre from the colloidal sol and polymerized alkoxides. The aluminium chloride or aluminium nitrate systems were found to be potential methods for producing continuous alumina fibre: the aluminium nitrate system had a better sintering behaviour than the aluminium chloride system. The aluminium isopropoxide system, however, was unsuitable for preparing alumina fibre but was suitable for the preparation of monoliths, membranes, powders, and multicomponent ceramics. The thermal changes of these precursors were studied by transmission electron microscopy, Fourier-transform infrared spectroscopy and X-ray diffraction. The results demonstrated the different routes of phase transformation as the temperature increases. The aluminium chloride system exhibits two routes for phase transformation: (a) boehmite -Al₂O₃, and (b) gibbsite -Al₂O₃.     

An asymptotic solution for large Prandtl number free convection

Summary The set of ordinary differential equations governing free convection boundary layer flow past an isothermal semi-infinite vertical flat plate is solved for large Prandtl numbers by means of the method of matched asymptotic expansions. The analysis leads to an expression for heat transfer which contains the Prandtl number explicitly and which is very accurate for sufficiently large values of the Prandtl number. On the other hand the analysis also has qualitative assets. Before choosing the mathematical method of solution, the physical aspects of the large Prandtl number free convection boundary layer are investigated. The mathematical solution serves to enlarge our understanding of the physical implications of a free convection boundary layer in a large Prandtl number fluid.Nomenclature a_ij coefficient defined by - b_ij coefficient defined by F_j()=b_0j+b_1j +b_2j ²+.... - c coefficient defined by equation (3) - c_p specific heat - f non-dimensional stream function of inner expansion (7) - f_n n-th perturbation of f - F non-dimensional stream function of outer expansion (15) - g non-dimensional stream function (1) - ¯g acceleration due to gravity - Gr_x local Grashof number:g(T_w–T)x³/v² - h non-dimensional temperature (2) - k coefficient of heat conduction - Nu_x local Nusselt number: - T temperature - T_w wall-temperature - T ambient temperature - u longitudinal velocity - x co-ordinate measuring distance from the leading edge - y co-ordinate measuring distance normal to the plate Greek symbols coefficient of thermal expansion - _i expansion parameter (21) - expansion parameter (22) - _i expansion parameter (33) - expansion parameter (34) - expansion parameter: ^–1/2 - inner similarity co-ordinate (9) - non-dimensional temperature of inner expansion (8) - _n n-th perturbation of - non-dimensional temperature of outer expansion (16) - _n n-th perturbation of - similarity co-ordinate (3) - kinematic viscosity - outer similarity co-ordinate (17) - density - Prandtl number:c_p/k - stream function     

On the re-inforcement of thermoplastics by imperfectly aligned discontinuous fibres

Studies of deformation behaviour of short fibre reinforced thermoplastics are complicated by the facts that usually a wide range of fibre lengths are present in moulded test pieces and that the fibres are not systematically oriented with respect to any test direction. An equation has been derived for the stress/strain curve of such a material. This has been used to determine fibre/matrix bond strengths in two glass/nylon 6.6 and two glass/polypropylene composites from measured stress/strain curves and fibre length distributions.It is concluded that major improvements in the properties of these materials will only be achieved by modifying processing to retain longer fibres.List of symbols E _c Modulus of composite - E _f Modulus of fibre - E _m Modulus of matrix - V _f Volume fraction of fibre - _c Stress in the composite - _f Peak stress in a fibre - Average stress in a fibre - _uf Ultimate strength of fibres - _m Stress in matrix at fibre failure strain - _uc Ultimate strength of the composite - _c Strain in composite - _uc Ultimate strain of the composite - Shear strength of the fibre matrix bond - L Fibre length - L Critical fibre length at a composite strain e - L _c Critical fibre length for fibre failure - r Fibre radius     

Die Entwicklung von Längswirbeln in zeitlich anwachsenden Grenzschichten an konkaven Wänden

Zusammenfassung Messungen des Anwachsens von Längswirbeln in zeitlich anwachsenden Grenzschichten an konkav gekrümmten Wänden (Görtler-Taylor-Wirbel) ergaben drei deutlich getrennte Bereiche: Es traten zunächst Wirbel mit der Wellenläge 0,9 auf (=Grenzschichtdicke, =Höhe einer Zelle, die zwei gegensinnig drehende Wirbel enthält). Je nach Größe der mit der Verdrängungsdicke ₁ der Grenzschicht gebildeten Reynolds-Zahl erschienen dann kurze Zeit später Wirbel mit 2,5, wenn war. Im Bereiche dagegen traten stattdessen bei den hier durchgeführten Versuchen immer Wirbel mit der Wellenlänge 6,5 auf. Bei werden die ersten Tollmien-Schlichting-Wellen mit der Wellenlänge _TS 6· angefacht. In ihren wandnahen Bereichen der Wellentäler könnten sich dann die oben genannten Längswirbel der Wellenlänge 6,5· ausbilden, die die zwei-in eine dreidimensionale Störung allseits gleicher Größenordnung verwandeln können.

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The development of longitudinal vortices in boundary layers growing with time along concave walls

Summary Measurements of the growth of longitudinal vortices in boundary layers growing with time along concave walls (Görtler-Taylor vortices) rendered three distinctly separated regions. First, vortices with a wave-length 0.9 appeared (-boundary layer thicness, =height of a cell containing two counterrotating vortices). Then, depending on the Reynolds number R _{a 1}/v ₁=displacement thickness), vortices with 2.5 appeared shortly afterwards, provided . In the region , however, the wave-length was 6.5. For the first Tollmien-Schlichting waves with _TS 6 were excited. In the wave-throughs close to the wall the abovementioned longitudinal vortices with wave length 6.5 may then be formed. This might transform the two-dimensional into a three-dimensional flow of equal order of magnitude in all directions.

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Zeichenerklärungen R _a Innenradius - Re _a Reynolds-Zahl gebildet mit dem InnenradiusR _a - Reynolds-Zahl gebildet mit der Verdrängungsdicke ₁ - kritische Taylor-Zahl - h Standhöhe der Flüssigkeit im Zylinder - t Zeit - z Anzahl - Steigungswinkel der Geraden - Grenzschichtdicke - ₁ Verdrängungsdicke - Wellenlänge (enthält ein gegensinnig rotierendes Längswirbelpaar) - v kinematische Zähigkeit - Winkelgeschwindigkeit Indizes K Knickpunkt der Geradensteigung - L unterhalb des Knickpunktes der Geradensteigung - TS Tollmien-Schlichting - e Einsatz der Wirbelentstehung     

A simple quadrature method for computing laminar boundary layers

Summary An approximate method is developed for calculating two-dimensional steady laminar boundary layers. Simple quadrature formulae have been obtained to determine energy thickness and a shape factor which subsequently determines the position of the point of separation; the only requirement being knowledge of the free-stream velocity. The relations between the shape factor and the other important characteristics of the boundary layer are also given.The method is extended to compressible flow assuming that the surface is adiabatic, that the viscosity is proportional to temperature, and that the Prandtl number is unity. Calculations and comparison of the results with other known solutions show that the quadrature method is not only simple to use, but also that it supplies closer approximation than do most other approximate methods.

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Eine einfache Quadraturmethode zur Berechnung laminarer Grenzschichten

Zusammenfassung Es wird ein Verfahren zur Berechnung zweidimensionaler stationärer laminarer Grenzschichten entwickelt. Einfache Quadraturformeln, die nur die Kenntnis der Geschwindigkeitsverteilung der Außenströmung erfordern, dienen der Berechnung der Energieverlustdicke und eines Geschwindigkeitsformparameters. Letzterer bestimmt die Lage des Ablösungspunktes. Zwischen dem Formparameter und anderen wichtigen Größen der Grenzschicht bestehen einfache Zusammenhänge.Bei Anwendung der Methode auf kompressible Strömungen wird angenommen, daß sich die Wand wärmeisoliert (adiabat) verhält, die Zähigkeit proportional der Temperatur ist und die Prandtl-Zahl den Wert Eins besitzt. Durchgeführte Rechnungen zeigen nicht nur die einfache Handhabung des Quadratur-Verfahrens, sondern durch Vergleich mit bekannten Ergebnissen auch eine bessere Übereinstimmung als die meisten anderen Näherungsmethoden.

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List of Symbols a speed of sound - c constant - c _p specific heat at constant pressure - G integral of the free-stream velocity - H modified shape factor - known shape factors - k, m, n, p exponents - l reference length - Mach number - p pressure - Prandtl number - R gas constant - Reynolds number with reference to energy thickness - Reynolds number for a reference state - T temperature - u velocity component in thex-direction - x, y coordinate axes parallel and perpendicular to the surface - x _S point of separation - function appearing in shearing stress - function appearing in energy dissipation ( value for the flat plate) - ₁, ₂, ₃ displacement, momentum and energy thickness, respectively - function appearing in dissipation integral - viscosity - ratio of specific heats, =1.4 (air) - thermal conductivity - kinematic viscosity - density - _w shearing stress at the wall With 3 FiguresDedicated to Professor Kurt Magnus in honor of his sixtieth birthday.     

A theory for the free shrinkage of steel fibre reinforced cement matrices

The paper presents a theoretical model to predict the free shrinkage of cement matrices reinforced with randomly oriented discrete steel fibres. The model is based on the consideration that the equivalent aligned length of a random fibre is responsible for restraining the shrinkage of a thick matrix cylinder of diameter equal to the fibre spacing, through the fibre-matrix interfacial bond strength. The validity of the model is established by means of extensive experimental data for different types of steel fibres in cement, mortar or concrete matrices. The theoretical model is also used to determine the values of coefficient of friction,, and the average bond strength,, of the fibre-matrix interface. It is shown that is a basic property of the matrix and fibre interface, which is affected by the surface roughness and mechanical deformation of the fibres., however, is greatly influenced by the shrinkage of the matrix and volume fraction of fibres. Finally, an empirical expression is derived to determine the shrinkage of steel fibre reinforced cement matrices based on the shrinkage of unreinforced matrices and fibre properties.     

Higher approximations for supersonic flow past slowly oscillating bodies of revolution

Summary Supersonic flow past slowly oscillating pointed bodies of revolution is studied. Starting from the complete nonlinear potential equation an elementary linearized solution is discussed and it is shown how this solution together with the method of matched asymptotic expansions can be used to derive an elementary second-order slender body theory. This approach is further demonstrated for the oscillating cone and its range of validity is evaluated by comparison with other theoretical methods.

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Zusammenfassung Es wird die Überschallströmung um langsam schwingende spitze Rotationskörper untersucht. Ausgehend von der vollständigen nichtlinearen Potentialgleichung wird zuerst eine elementare linearisierte Lösung besprochen und gezeigt, wie diese Lösung im Verein mit der Method of matched asymptotic expansions zur Herleitung einer elementaren Schlankkörpertheorie zweiter Ordnung verwendet werden kann. Die Theorie wird am Beispiel des schwingenden Kegels näher erläutert und mit anderen Methoden verglichen.

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Symbols a Velocity of sound - c _N Normal force coefficient - Damping coefficient - F _(x) Dipole distribution - k Reduced frequency - M Mach number - R (x) Meridian profile - t Time - x, r, Cylindrical coordinates - - Ratio of specific heats - Amplitude of oscillation - Thickness ratio - Perturbation potential - Zero angle of attack potential - æ - Velocity potential - Out-of-phase potential - - In-phase potential - - Source coordinate With 4 Figures     

Theory of the hall effect in strongly correlated Fermi systems

The anomalous properties of the Hall constant in the normal state of high-T_c superconductors are investigated within the Hubbard model. In Mori theory, the frequency dependent Hall constant is given as the sum of its infinite frequency limit and a memory function contribution. The first term (R _H ) was already considered by Shastry et al.¹ In perturbation theory and in the limit of infinite spatial dimensions, the memory function contribution causes the Hall constant to change sign as a function of doping () and to decrease as a function of temperature, if we allow U to be 2W (U: Coulomb repulsion; W: bandwidth). In the limit U , the memory function is calculated via its moments. For frequencies W U, this leads to a decrease of R _H by a factor of (1 + )/2.     

The influence of brittle boundary layers on the strength of fibrous metallic composites

The brittle boundary layers often caused during the production of composites or by their treatment at higher temperatures, may change the mechanical properties. On the steel wire/aluminium system the growth of the intermetallic boundary phase and its influence on the strength of the composite were investigated. Hence followed a maximum strength at small layer thicknesses. By means of fracture investigations new models were developed which allow the calculation of the dependence of strength behaviour on layer thickness.List of symbols E _f Young's modulus of fibre - E _b Young's modulus of boundary layer - _c external load - _f tensile stress in the fibre - _m tensile stress in the matrix - _b tensile stress in the boundary layer - _uc,f,b ultimate strength of the composite, the fibre or the boundary layer, respectively - averaged stress in the fibre - _bf shear stress in the boundary layer-fibre interface - ₀ shear strength of the boundary layer-fibre-interface - _uf ultimate strain of the fibre - fraction of the layer which has grown into the matrix - Weibull parameter - ^–1 characteristic length of stress transfer between fibre and boundary layer - d diameter of the boundary layer - 2l length of the boundary layer segments - r _f fibre radius - u(x) displacement field - v _{f, b,m} volume fraction of fibres, boundary layer or matrix respectively.     

Steady state creep of a composite material with short fibres

The shear within a matrix volume is assumed to be an important process during the creep of composite material reinforced with short rigid fibres. The rate of elongation of such a composite with certain fibre distributions can be estimated. The agreement with a few experimental data is reasonably good.List of main symbols V _f volume fraction of fibres in composite - aspect ratio of a fibre - L length of a fibre - h transverse size of a fibre - h interfibre spacing - m, _m , _m constants for creep for a matrix material - n, _f, _f constants of creep for a fibre material _{f m} - v rate of relative motion of two fibres - _* ultimate strength of a fibre - ^* the first critical value of aspect ratio - ^** the second critical value of aspect ratio This work was carried out when the author was a guest worker at the National Physical Laboratory, Teddington, Middlesex, UK.     

Die Herleitung von Schalengleichungen über ein erweitertes Variationsprinzip bei Berücksichtigung der Querschubverzerrungen

Ohne ZusammenfassungBezeichnungen L Bezugsgrößen für dimensionslose Koordinaten - L charakteristische Schalenabmessung - t Schalendicke - Schalenparameter - körperfeste, krummlinige, dimensionslose Koordinaten der Schalenmittelfläche - Dimensionslose Koordinate in Richtung der Schalennormalen - i, j,...=1,2,3 Indizierung des dreidimensionalen Euklidischen Raumes - ,,...=1,2 Indizierung des zweidimensionalen Riemannschen Raumes - (...), Partielle Differentiation nach der Koordinate - (...), Kovariante Differentiation für Tensorkomponenten des zweidimensionalen Raumes nach der Koordinate - (...)| Kovariante Differentiation für Tensorkomponenten des dreidimensionalen Raumes nach der Koordinate - Variationssymbol - a ,a ₃ Basisvektoren der Schalenmittelfläche - V Verschiebungsvektor - U ,U ₃ Verschiebungskomponenten des Schalenraumes - v ,w,w ,W Verschiebungskomponenten der Schalenmittelfläche - Verhältnis der Metriktensoren des Schalenraumes und der Schalenmittelfläche - _ik Verzerrungstensor des Raumes - _{(, )}, Symmetrische Verzerrungstensoren der Schalenmittelfläche - _{[, ]} Antimetrischer Term des Verzerrungsmaßes - ^, Spannungstensor - n ,m ,q Tensorkomponenten der Schnittgrößenvektoren - p ,p,c Tensorielle Lastkomponenten     

Damped response of an axially excited rotating liquid bridge in zero-gravity

Summary The response of a solidly rotating finite liquid bridge due to axial excitation exhibits for frictionless liquid at the resonances singularities. For the experimenter in a spacelabmission the actual resonance amplitude is of quite some importance. For this reason damping, that has to be measured in ground tests, has been introduced into the results of the response.Notation a radius of the liquid bridge - h length of the liquid bridge - I ₀,I ₁ modified Besselfunctions - J ₀,J ₁ Besselfunctions - r, ,z polar coordinates - t time - excitation amplitude - elliptic case - hyperbolic case - abbreviation - damping factor of liquid - (z, t) free surface displacement - =² – ² surface tension - surface tension - liquid density - ₀ rotational speed of liquid bridge - forcing frequency of axial excitation - natural frequency of liquid bridge With 2 Figures     

Free vibrations of a circular plate with attached concentrated mass,spring and dashpot

Summary The natural vibrations of a circular plate with attached concentrated mass, spring and dashpot have been obtained by means ofYoung's analysis [1]. The results are presented in terms of eigen-functions of the plate alone. The case of a plate carrying two masses and resting on elastic foundation has also been studied. Some particular cases have been deduced.

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Zusammenfassung Die Eigenschwingungen einer Kreisplatte mit lokal befestigter Einzelmasse, Feder und Dämpfer werden nach derYoungschen Methode [1] ermittelt. Die Ergebnisse werden als Entwicklung nach den Eigenfunktionen der reinen Plattenschwingung dargestellt. Der Fall der elastisch gebetteten Platte mit zwei Einzelmassen wird ebenfalls studiert. Einige Sonderfälle werden hergeleitet.

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Nomenclature a radius of circular plate - h plate thickness - k ₁ spring constant - k _c generalized spring constant - modulus of elastic foundation - decay constant - c dashpot strength - D , flexural rigidity of plate - E Young's modulus - v Poisson's ratio - p natural frequency of plate alone - natural frequency of composite system - w deflection mode of plate - r, cylindrical coordinates - mass density - r - (l/D)^1/4 - - - F _m ,L _m ,G _m ,M _m unknown constants With 5 Figures     

Thermodynamics of the quasiequilibrial growth of crazes

By comparing the morphology and physical properties (averaged over the scale of 1 to 10m) of a crazed and uncrazed polymer, it can be concluded that crazing is a new phase development in the initially homogeneous material. The present study is based on recent work on the general thermodynamic explanation of the development of a damaged layer of material. The treatment generalizes the model of a crack-cut in mechanics. The complete system of equations for the quasiequilibrial craze growth follows from the conditions of local and global phase equilibrium, mechanical equilibrium and a kinematic condition. Constitutive equations of craze growth-equations are proposed that are between the geometric characteristics of a craze and generalized forces. It is shown that these forces, conjugated with the geometric characteristics of a craze, can be expressed through the known path independent integrals (J, L, M,). The criterion of craze growth is developed from the condition of global phase equilibrium. F Helmholtz's free energy - G Gibb's free energy (thermodynamic potential) - f density ofF - g density ofG - T absolute temperature - S density of entropy - strain tensor - components of - stress tensor - components of - _y stress along the boundary of an active zone (yield stress) - _b stress along the boundary of an inert zone - applied stress - value of at the moment of craze initiation - K stress intensity factor - C tensor of elastic moduli - C ^–1 tensor of compliance - internal tensorial product - V volume occupied by sample - V ₁ volume occupied by original material - V ₂ volume occupied by crazed material - V boundary ofV - (V) vector-function localized on V - (x) characteristic function of an area - (x) variation of(x) - (x) a finite function - tensor of alternation - components of the boundary displacement vector - l components of the vector of translation - n components of the normal to a boundary - _k components of the vector of rotation - e symmetric tensor of deviatoric deformation of an active zone - expansion of an active zone - J ⁽ⁱ⁾ ,L _k ⁽ⁱ⁾ ,M ⁽ⁱ⁾,N ⁽ⁱ⁾ partial derivatives ofG ⁽ⁱ⁾ with respect tol , _k, ande , respectively - [ ] jump of the parameter inside the brackets - thickness of a craze - 2l length of a craze - 2b length of an active zone - l _c distance between the geometrical centres of the active zone and the craze - ^* craze thickness on the boundary of an active and the inert zone - l ^* craze parameter (length dimension) - A craze parameter (dimensionless) - ^* extension of craze material     

Axi-symmetric natural frequencies and response of a spinning liquid column under strong surface tension

Summary The response of a solidly rotating anchored finite liquid column consisting of frictionless liquid is subjected to axial harmonic excitation. The response of the free liquid surface elevation and velocity distribution has been determined analytically in the elliptic (>2 ₀) and hyperbolic frequency range (>2 ₀). For the liquid surface displacement the response has been evaluated numerically as a function of the forcing frequency/2 ₀. In addition the first natural stuck-edge frequency has been determined and compared with the slipping case.List of symbols a radius of liquid bridge - h length of liquid bridge - I ₀,I ₁ modified Besselfunctions - J ₀,J ₁ Besselfunctions - p liquid pressure - r, ,z cylindrical polar coordinates - t time - u, v, w velocity distribution in rotating liquid - Weber number - z₀ axial excitation amplitude - elliptic case (>2 ₀) - hyperbolic case (>2 ₀) - liquid density - surface tension - liquid surface displacement - acceleration potential - ₀ rotational speed - axial forcing frequency - natural frequency of rotating system - _0n natural frequency of harmonic axial response     

On the torsional waves in an elastic bar with long-range cohesive forces

Summary With reference to some of the results obtained in [1], the equation of motion and the equations of nonlocal stress components in the form of Kroener-Eringen are transformed into the Fourier space. A lengthy calculation using a separable form of the nonlocal elastic moduli leads to the governing equation of the problem with two types of solutions. In each case the nonlocal phase velocity and, respectively, the nonlocal group velocity of the very short waves turn out to be by about 36% less than the velocities of their classical counterparts.Notation a atomic spacing - c ₂ classical speed of shear waves - c _2nonloc nonlocal speed of shear waves - c _a2nonloc,c _b2nonloc nonlocal speeds in rods - k wave number - r radius vector - R radius of the bar - v hoop displacement - Fourier transform ofv - V volume of the rod - z axial coordinate - delta sequence - , Lamé's constants - , nonlocal moduli - transform of - A wave length - * mass density - frequency     

Response of a spinning liquid column to pitch excitation

Summary The response of a solidly rotating liquid bridge consisting of inviscid liquid is determined for pitch excitation about its undisturbed center of mass. Free liquid surface displacement and velocity distribution has been determined in the elliptic (>2₀) and hyperbolic (<2₀) excitation frequency range.List of symbols a radius of liquid column - h length of column - I ₁ modified Besselfunction of first kind and first order - J ₁ Besselfunction of first kind and first order - r, ,z cylindrical coordinates - t time - u, v, w velocity distribution in radial-, circumferential-and axial direction resp. - mass density of liquid - free surface displacement - velocity potential - ₀ rotational excitation angle - ₀ velocity of spin - forcing frequency - _1n natural frequency - surface tension - acceleration potential - for elliptic range >2₀ - for hyperbolic range >2₀