EPMA mapping of small particles of α-AlFeSi and β-AlFeSi in AA6063 alloy billets

It is well known that the spatial distribution and the spatial density of the particles o-AlFeSi and -AlFeSi in the billets of Al-Mg-Si alloys, such as AA6063 alloys affect the quality of anodizing performance of their extrusions. For this reason it is very important to control the spatial distribution and the spatial density of both AlFeSi particles at extrusion plants. The X-ray diffraction method (XRD) has been used for discrimination between -AlFeSi and -ALFeSi particles. However it is not an appropriate method for determining the spatial distributions of particles in the alloys. As an alternative method an electron probe microanalyzer (EPMA) has been used for determining the spatial distributions of each element in the microstructures. However, unfortunately it is difficult to discriminate between the particles composed of the same elements like -AlFeSi and -AlFeSi particles. Thus, we tried to develop a convenient method to discriminate between -AlFeSi and -AlFeSi particles in the microstructure of AA6063 alloys and developed the EPMA mapping of -AlFeSi and -AlFeSi particles. First, in order to discriminate between the two particles, we tried to use the relative X-ray intensity ratio, the I _Fe/I _Si ratio instead of the Fe/Si mass ratio. Then, we calculated the value of the I _Fe/I _Si ratio from -AlFeSi and -AlFeSi by using Monte Carlo calculations and obtained the critical value of the I _Fe/I _Si ratio, to distinguish between -AlFeSi and -AlFeSi. After that, using the discrimination value, we developed the EPMA mapping program (EPMA method) to observe the distributions of -AlFeSi and -AlFeSi, and to calculate the areas (%) of -AlFeSi and -AlFeSi. Finally, we checked the correlation between the EPMA and the XRD methods. Consequently, the two methods were in good agreement. Today, this EPMA method instead of the XRD method is successfully used in the quality control of 6063 aluminum alloy billets after heat treatment at our aluminum extrusion works.     

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.     

The tensile strength and ductility of continuous fibre composites

The plastic instability approach has been applied to the tensile behaviour of a continuous fibre composite. It is shown that the combination of two components with different strengths and degrees of work-hardening produces a new material with a new degree of work-hardening, which may be determined by the present analysis. Expressions for the elongation at rupture and the strength of a composite have been obtained and the results of the calculation are compared with some experimental data.List of symbols V _f volume fraction of fibres in composite - , , true strain of fibre, matrix and composite - s true stress - , , nominal stress on fibre, matrix and composite - _*, _*, _* critical stress of fibre, matrix and composite (ultimate tensile strength) - _*, _* critical strain of separate fibre and matrix - _* critical strain of composite - Q external load - A cross-sectional area - A ₀ initial value of area     

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     

The structure of the γ′ extended solid solution in a splat-cooled Ag-50 at.% Cu alloy

The structure of the extended solid solution in electron transparent areas of a splatcooled Ag-50 at.% Cu alloys was examined by transmission electron microscopy. This phase was usually found to be spinodally decomposed at large grain sizes (–1m in diameter), in contrast to X-ray diffraction data indicating that the solid solution was undecomposed. A solidification model for rapidly quenched eutectic alloys is proposed to account for the observed structure of the splat-cooled alloy. A transformation curve for the spinodal decomposition of is also calculated and related to predictions derived from the solidification model.     

Negatively isotone optimal policies for random walk type Markov decision processes

Summary This paper considers a random walk type Markov decision process in which the state spaceI is an integer subset of IR ^m , and the action spaceK is independent ofi I. The natural order, overI, and a quasi order,, overK, is assumed, together with aconditional convexity assumption on the returns {r _i ^k }, and certain other assumptions about these rewards and the transition probabilities in relationship to the orders and.A negatively isotone policy is one for whichi i(i))(i) (i.e.(i) (i) or(i) i)). It is shown that, under specified conditions, a negatively isotone optimal policy exists. Some consideration is given to computational implications in particular relationship to Howard's policy space method.

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Zusammenfassung Wir betrachten einen Markovschen Entscheidungsprozeß vom random walk Typ. Der ZustandsraumI sei eine Teilmenge des IR^m, wobeii I ganzzahlige Komponenten habe. Die MengeK der zulässigen Aktionen ini I sei unabhängig voni I. Sei die natürliche Ordnung aufI und sei eine Quasiordnung aufK. Die Erträge {r _i ^k }seienbedingt konvex, darüberhinaus seien weitere Voraussetzungen über diese Erträge und die Übergangswahrscheinlichkeiten in Bezug auf die Ordnungen und erfüllt. Eine Politik heißt negativ isoton, falls ausi i folgti(i) (d. h.(i) (i) oder(i)(i)). Wir zeigen, daß unter gewissen Voraussetzungen einenegativ isotone optimale Politik existiert: Auch diskutieren wir einige Folgerungen für die Numerik, insbesondere hinsichtlich Howards Politikiteration.

     

Effect of the weight of the admixture on the turbulence structure of a two-phase jet

The effect of gravity on the turbulence structure of an inclined two-phase jet is evaluated according to the Prandtl theory of mixing length.Notation C_x drag coefficient for a particle - D_p particle diameter - g_i components of the acceleration g due to gravity acting on a particle in the direction of jet flow (g_i=g sin ) and in the direction normal to it (g_i=g cos ) - V_poi ^±, V_goi ^± fluctuation components of the velocities of the particles and gas, respectively, at the end of a mole formation - V_fi free-fall velocity of a particle - l _u mixing length - m_p particle mass - t _p length of time of particle-mole interaction - V_pi ^±, V_gi ^± positive and negative fluctuation velocities of particles and of the gas respectively, with the components u_p ^±, u_g ^±, v_p ^±, v_g ^±, k=V_goi/V_fi - V_i ^± relative velocity of the gas - jet inclination angle relative to the earth's surface - empirical constant - _u, jet boundaries in terms of velocity and concentration - _u=y/ _u dimensionless velocity ordinate - =y/ dimensionless concentration ordinate - admixture concentration - u_m, _m velocity and the concentration of the admixture at the jet axis - _g dynamic viscosity of the gas - _s, _g densities of the particle material and of the gas - _g, _p shearing stresses in the gas and in the gas of particles - _m, ₀ shearing stresses in the mixture and in pure gas, respectively Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 40, No. 3, pp. 422–426, March, 1981.     

The age hardening effect in Ti-6 Al-4 V due to ω and α precipitation in the β grains

The structure at room temperature of a quenched TA6V titanium alloy has been investigated. This structure is + or + according to the treatment temperature; it is always metastable. During ageing the grains decomposed by the reaction + + +; this decomposition was accompanied by a large increase of the 0.2% yield stress. No structural modification was observed in. The and phase of TA6V were separately investigated in the form of single-phase alloys. The hardness of was insensitive to ageing, while was considerably hardened by and; we deduced that the strengthening of the minor phase during ageing is mainly responsible for the hardening of TA6V.     

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.     

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      

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.     

Analytical Theory of DC SQUIDS Operating in the Presence of Thermal Fluctuations

A comprehensive analytical theory of symmetric DC SQUIDs is presented taking into account the effects of thermal fluctuations. The SQUID has a reduced inductance < 1/ where = 2LI_c/₀, L is the loop inductance, ₀ is the flux quantum, and I_c is the critical current of the identical Josephson junctions which are assumed to be overdamped. The analysis, based on the two dimensional Fokker–Planck equation, has been successfully performed in first order approximation with considered a small parameter. All important SQUID characteristics (circulating current, current-voltage curves, transfer function, and energy sensitivity) are obtained. In the limit 1( = 2k_BT/I_c0 is the noise parameter, k_B is the Boltzmann constant, and T is the absolute temperature) the theory reproduces the results of numerical simulations performed for the case of small thermal fluctuations. It was found that for < 1 the SQUID energy sensitivity is optimum when is higher than 1/, i.e., outside the range for which the present analysis is valid. However, for 1 the energy sensitivity has a minimum at L = L_F , where L_F = ( ₀ /2) ²/k_B , and therefore, in this case, the optimal reduced DC SQUID inductance is _opt = 1/, i.e., within the range for which the present analysis is valid. In contrast to the case of an RF SQUID, for a DC SQUID the transfer function decreases not only with increasing L/L_F but also with increasing (as 1/). As a consequence, the energy sensitivity of a DC SQUID with < 1/ degrades more rapidly (as ⁴ ) with the increase of than that of an RF SQUID does (as ² ).     

Coupling of order parameter collective modes to spin density in3He-B

We derive a general expression for the dynamic spin susceptibility of³He-B which is valid for all magnetic fields. The coupling of real and imaginary modes by particle-hole asymmetry is taken into account. Then we calculate the contribution of the mode at frequency =2 – 1/4 ( is the effective Larmor frequency) to the transverse susceptibility. The spectral weight of this mode in magnetic resonance absorption is proportional to (/)^1/2 (–)², where and are particle-hole asymmetry parameters. From the experimental coupling strength of the real squashing mode to sound we estimate (–)²10^–4. The dynamic susceptibility satisfies the sum rules of Leggett. Finally we point out the difficulties in calculating the transverse NMR frequency of³He-B. These difficulties arise from theS _z =0 Cooper pairs and from the coupling ofJ _z =±1 modes forJ=1 andJ=2.     

Effect of fibre misalignment on fracture behaviour of fibre-reinforced composites

When a matrix crack encounters a fibre that is inclined relative to the direction of crack opening, geometry requires that the fibre flex is bridging between the crack faces. Conversely, the degree of flexing is a function of the crack face separation, as well as of (1) the compliance of the supporting matrix, (2) the crossing angle, (3) the bundle size, and (4) the shear coupling of the fibre to the matrix. At some crack face separation the stress level in the fibre bundle will cause it to fail. Other bundles, differing in size and orientation, will fail at other values of the crack separation. Such bridging contributes significantly to the resistance of the composite to crack propagation and to ultimate failure. The stress on the composite needed to produce a given crack face separation is inferred by analysing the forces and displacements involved. The resulting model computes stress versus crack-opening behaviour, ultimate strengths, and works of failure. Although the crack is assumed to be planar and to extend indefinitely, the model should also be applicable to finite cracks.Glossary of Symbols a radius of fibre bundle - C 2 _f /aE _f - ^* critical failure strain of fibre bundle - _b bending strain in outer fibre of a bundle - _c background strain in composite - _f axial strain in fibre - _s strain in fibre bundle due to fibre stretching = _f - () strain in composite far from crack - E Young's modulus of fibre bundle - E _c Young's modulus of composite - E _f Young's modulus of fibre - E _m Young's modulus of matrix - f() number density per unit area of fibres crossing crack plane in interval to + d - F total force exerted by fibre bundle normal to crack plane - F _s component of fibre stretching force normal to crack plane - F _b component of bending force normal to crack plane - G _m shear modulus of matrix - h crack face opening relative to crack mid-point - h _m matrix contraction contribution to h - h _f fibre deformation contribution to h - h _max crack opening at which bridging stress is a maximum - I moment of inertia of fibre bundle - k fibre stress decay constant in non-slip region - k ₀ force constant characterizing an elastic foundation (see Equation 7) - L exposed length of bridging fibre bundle (see Equation 1a) - L _f half-length of a discontinuous fibre - m, n parameters characterizing degree of misalignment - N number of bundles intersecting a unit area of crack plane - P _b bending force normal to bundle axis at crack midpoint - P _s stretching force parallel to bundle axis in crack opening - Q() distribution function describing the degree of misalignment - s _f fibre axial tensile stress - s _f ^* fibre tensile failure stress - S stress supported by totality of bridging fibre bundles - S _max maximum value of bridging stress - v fibre displacement relative to matrix - v elongation of fibre in crack bridging region - u _coh non-slip contribution to fibre elongation - U fibre elongation due to crack bridging - v overall volume fraction of fibres - v _f volume fraction of bundles - v _m volume fraction matrix between bundles - w transverse deflection of bundle at the crack mid-point - x distance along fibre axis, origin defined by context - X distance between the end of discontinuous fibre and the crack face - X ^* threshold (minimum) value of X that results in fibre failure instead of complete fibre pullout - y displacement of fibre normal to its undeflected axis - Z() area fraction angular weighting function - tensile strain in fibre relative to applied background strain - ^* critical value of to cause fibre/matrix debonding - angle at which a fibre bundle crosses the crack plane - (k ₀/4EI)^1/4, a parameter in cantilever beam analysis - v_m Poisson's ratio of matrix - L (see Equation 9) - shear stress - _* interlaminar shear strength of bundle - _d fibre/matrix interfacial shear strength - _f frictional shear slippage stress at bundle/matrix interface - angular deviation of fibre bundle from mean orientation of all bundles - angle between symmetry axis and crack plane     

Numerical solution to the solidification of aluminium in the presence of various fibres

In an attempt to understand the experimentally observed solidification microstructures in metal matrix composites, the influence of SiC, graphite and alumina fibres on the solidification of aluminium has been studied numerically. Irregular geometries of the composite material were mapped into simple rectangles through numerical conformal mapping techniques to analyse the influence of a single fibre or a row of fibres on a unidirectionally advancing planar solid-liquid interface. The fibres were assumed to be circular in cross-section and the direction of the interface movement was perpendicular to the length of the fibres. The study showed that for fibres with lower thermal conductivity than aluminium, the interface first goes through acceleration as it approaches and ascends the fibre and then deceleration as it descends the fibre. The acceleration and deceleration phenomena of the interface increases as the thermal conductivity ratio of fibre to liquid aluminium decreases. With low thermal conductivity ratios (K _f/K _L1), the interface is orthogonal to the fibre surface. When the conductivity of the fibre is lower than that of the melt, the interface becomes convex facing the fibre; this mode would lead to pushing of the fibre ahead if it was free to move, as has been experimentally observed in cast microstructures of metal matrix composites. The temperature versus solidification time plots of two points, one in the fibre and the other in aluminium, show that the fibre with a conductivity lower than the matrix is at a temperature higher than the melt; the temperature difference between the two points increases with increasing solidification rate for all the positions of the interface before it touches the fibre. The three-fibre study shows that as the number of fibres increases, the curvature of the interface increases upon approaching the subsequent fibres. The relationship between these numerical computations and experimental observations has been discussed.Nomenclature a reference length = diameter of the fibre - h - K thermal conductivity; in Equation 4 it is defined as K = (K + K _f)/2 for the common boundary between fibre and the freezing medium. For all the rest of the points K = K in Equation 4 - L latent heat of fusion - r non-dimensional variable in radial direction - S non-dimensional distance travelled by the interface - Ste Stefan number = - T non-dimensional temperature - t non-dimensional time - x a non-dimensional spatial coordinate of physical plane - y a non-dimensional spatial coordinate of physical plane - thermal diffusivity - non-dimensional axial coordinate of the mapped plane - non-dimensional vertical coordinate of the mapped plane - a polar coordinate - l liquid - m melting - s solid - O constant wall temperature - i initial - f fibre - * dimensional variables     

A note on the solution of a differential equation arising in boundary-layer theory

Summary The differential equation f + ff + f² = 0 (where dashes denote differentiation with respect to the independent variable ) subject to the boundary conditions f(0)=0, f()=0 and either f(0)=1 or f(0)=–1 is considered. It is shown that by using pf as dependent variable and =C–f (where C=f()) as independent variable and then expanding in powers of , a very good approximation to the solution can be obtained using only a few terms in the expansion.     

Thermal stresses in a thin circular disc of orthotropic material due to an instantaneous point heat source

Summary In this paper, we have analysed the transient plane thermal stress problem of a circular disc of orthotropic material with instantaneous point heat source. The variation of with time along different radius vectors is exhibited graphically and compared with that of the isotropic case.

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Wärmespannungen in einer dünnen Kreisscheibe aus orthotropem Material zufolge einer punktförmigen instantanen Wärmequelle

Zusammenfassung In dieser Arbeit wurde das instationäre ebene Wärmespannungsproblem einer Kreisscheibe aus orthotropem Material zufolge einer punktförmigen instantanen Wärmequelle untersucht. Die Veränderung von über die Zeit für verschiedene Radien ist graphisch dargestellt und wird mit dem isotropen Fall verglichen.

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Nomenclature r, polar coordinates - T temperature rise - ² ratio of conductivities - ² thermal diffusivity in -direction - J _n Bessel function ofn-th order - t time - p introduced in equation (2) - h heat transfer coefficient - a radius of circular disc - f(r, ) temperature distribution at initial state - r ₀, ₀ a point on the disc - T ₀ strength of point heat source - Dirac delta function - F stress function in two dimensions - ₁, ₂ coefficient of thermal expansion - a ₁₁,a ₁₂,a ₂₂,a ₆₆ elastic constants With 9 Figures     

Deformation of a carbon-epoxy composite under hydrostatic pressure

This paper describes the behaviour of a carbon-fibre reinforced epoxy composite when deformed in compression under high hydrostatic confining pressures. The composite consisted of 36% by volume of continuous fibres of Modmur Type II embedded in Epikote 828 epoxy resin. When deformed under pressures of less than 100 MPa the composite failed by longitudinal splitting, but splitting was suppressed at higher pressures (up to 500 MPa) and failure was by kinking. The failure strength of the composite increased rapidly with increasing confining pressure, though the elastic modulus remained constant. This suggests that the pressure effects were introduced by fracture processes. Microscopical examination of the kinked structures showed that the carbon fibres in the kink bands were broken into many fairly uniform short lengths. A model for kinking in the composite is suggested which involves the buckling and fracture of the carbon fibres.List of symbols d diameter of fibre - E _f elastic modulus of fibre - E _m elastic modulus of epoxy - G _m shear modulus of epoxy - k radius of gyration of fibre section - l length of buckle in fibre - P confining pressure (= ₂ = ₃) - R radius of bent fibre - V _f volume fraction of fibres in composite - _t, _c bending strains in fibres - angle between the plane of fracture and ₁ - ₁ principal stress - ₃ confining pressure - _c strength of composite - _f strength of fibre in buckling mode - _n normal stress on a fracture plane - _m strength of epoxy matrix - shear stress - tangent slope of Mohr envelope - slope of pressure versus strength curves in Figs. 3 and 4.     

Interfacial debonding and fibre pull-out stresses

Based on a theoretical model developed previously by the authors in Part II of this series for a single fibre pull-out test, a methodology for the evaluation of interfacial properties of fibre-matrix composites is presented to determine the interfacial fracture toughness G _c, the friction coefficient , the radial residual clamping stress q _o and the critical bonded fibre length z _max. An important parameter, the stress drop , which is defined as the difference between the maximum debond stress _d ^* and the initial frictional pull-out stress _fr, is introduced to characterize the interfacial debonding and fibre pull-out behaviour. The maximum logarithmic stress drop, In(), is obtained when the embedded fibre length L is equal to the critical bonded fibre length z _max. The slope of the In()-L curve for L bigger than z _max is found to be a constant that is related to the interfacial friction coefficient . The effect of fibre anisotropy on fibre debonding and fibre pull-out is also included in this analysis. Published experimental data for several fibre-matrix composites are chosen to evaluate their interfacial properties by using the present methodology.On leave at the Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong.     

Use of adjoint functions in investigations of heat conduction and transfer processes

An examination is made of the use of adjoint functions in heat conduction and convection theory. Formulas of perturbation theory are obtained for steady and unsteady cases, an interpretation of the physical meaning of adjoint temperature is given, and some applications of the theory are discussed.Notation (r,) thermal conductivity - t(r,) temperature - t *(r,) adjoint temperature - q_V(r,) density of heat release sources - p(r,) a parameter of adjoint equation - r generalized coordinate - time - (r_s, ) heat transfer coefficient - I linear functional of temperature - (r,;r₀,₀) and *(r,; r₀,₀) Green's function for t(r, ) and t *(r, ) - C(r,) volume specific heat - W(r, ) vector distribution of flow velocities - V, S volume and surface areas of body - R radius of HRE - r, radial and angular coordinates - F_in, F_out inlet and outlet flow areas of channel