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     

Flow visualization glass-ceramic: Preliminary experimental and modelling results

A glass-ceramic material was developed to act as a flow visualization material. Preliminary experiments indicate that aperiodic, thermally induced, convective flows can be sustained at normal processing conditions. These flows and the stress and temperature gradients induced are most likely responsible for the anomalous behaviour seen in these materials and the difficulties encountered in their development and in their production on industrial and experimental scales. A simple model describing the dynamics of variable-viscosity fluids was developed and was shown to be in qualitative agreement with more sophisticated models as well as with experimental results. The model was shown to simulate the dependence of the critical Rayleigh number for the onset of convection on the viscous properties of the fluid at low T, and also to simulate quenching behaviour when the temperature differences were high.Nomenclature C _p Heat capacity - D, E, F Expansion coefficients - H Height of the roll cell - Pr Prandtl number - R _a Rayleigh number - R _c Critical Rayleigh number for the onset of convection in a constant-viscosity fluid - S Dimensionless stream function - T Temperature - T _m Mean temperature - T ₀ Bottom surface temperature - T _r Reference temperature - a Aspect ratio of cell - g Acceleration due to gravity - k Thermal conductivity - k ₁ Function related to ²v/T ² - k ₂ Function related to ⁴v/T ⁴ - r Rayleigh number ratioR _a/R _c - t Time - w Dimensionless vertical coordinate - w _m Mean cell height - x Horizontal coordinate - y Dimensionless horizontal coordinate - z Vertical coordinate - , Constants - _t Thermal expansion coefficient - Constant in viscosity function - T Temperature difference between top and bottom surfaces - _i Viscosity coefficients - Kinematic viscosity - _m Mean kinematic viscosity - Dimensionless kinematic viscosity - Thermal diffusivity - Non-linear temperature function - Dimensionless non-linear temperature function - _o - Stream function - Dimensionless time - Eigenvalues     

Theory of RF SQUIDs Operating in the Presence of Large Thermal Fluctuations

A comprehensive analytical theory is presented for non-hysteretic RF SQUIDs operating in the adiabatic mode in the presence of large thermal fluctuations. When 1 ( = 2LI_c/₀ is the hysteresis parameter, L is the SQUID inductance, I_c is the critical current of the Josephson junction, and ₀ is the flux quantum) the theory is applicable also for RF SQUIDs operating in the non-adiabatic mode. In contrast to previous theories in which the noise is treated perturbatively and which therefore are applicable only if the product 1 ( = 2k_BT/ ₀ I_c is the noise parameter, k_B is the Boltzmann constant, and T is the absolute temperature)—the case of small thermal fluctuations—the present theory is valid for around unity or higher. In the limit 0 the theory reproduces the results of small thermal fluctuations theories. It has been found that in the presence of large thermal fluctuations the screening current in the SQUID inductance is suppressed by a factor that increases with increasing . Taking into account this new basic fact, all SQUID characteristics (output signal, transfer function, noise spectral density and energy sensitivity) have been recalculated and a good agreement with experimental data has been obtained. It has been also found that RF SQUIDs can be operated with substantially higher values of the inductance and of the noise parameter than DC SQUIDs. These two aspects, which are of particular importance at liquid nitrogen temperature, make high T_c RF SQUIDs very attractive.     

Stress distributions along a short fibre in fibre reinforced plastics

This paper develops an analysis for predicting the normal stress and interfacial shearing stress distribution along a single reinforcing fibre of a randomly oriented chopped-fibre composite, such as sheet moulding compound (SMC), from a knowledge of the constituent properties and the length-to-diameter ratio of the fibres. The analysis is useful in analysing the tensile strength of SMC, and as a guide to increasing the tensile strength by altering the elastic characteristics. The model is based on a generalized shear-lag analysis. Numerical values of the normal stress and interfacial shearing stress are presented as functions of various parameters. It is observed that the maximum normal stress occurs at the middle of the fibre and the maximum shear stress occurs at the end. The analysis is restricted to loading which does not result in buckling of the fibre; i.e., axial loads on the fibre can be at most only slightly compressive.List of symbols a _f Ratio of the fibre length to diameter (aspect ratio, l _f/d _f) - E _a Young's modulus of the composite (defined in Equation 21) - E _f Young's modulus of the fibre material - E _m Young's modulus of the matrix material - G _f Shear modulus of the fibre material - G _m Shear modulus of the matrix material - l Half the length of the matrix sheath which surrounds the fibre - l _f Half of the length of the fibre - Q Defined in Equation 14. - R Ratio of the length of the fibre to the matrix in a representative volume element; a parameter 0R[(1/V _f–1) ] - r _a Radius of the composite body (we assume r _ar _m, r _f) - r _f Radius of the fibre - r _m Radius of the matrix sheath which surrounds the fibre - u _a Displacement of the composite along the fibre direction - u _f Displacement of the fibre along the fibre direction - V _f Fibre volume fraction - (XYZ) Co-ordinate system with Z-axis parallel to the direction of the applied load (Fig. 1a) - (xyz) Co-ordinate system which is rotated by about the X-axis (Fig. 1a) - (¯x¯y¯z) Co-ordinate system which is rotated by about the z-axis (Fig. 1b) - Fibre orientation angle measured from the Z-axis - _m Engineering shear strain in the matrix - Defined in Equation 8 - Polar angle measured from the x–z plane - Defined in Equation 9 - Applied normal stress - _a Normal stress in the composite along the fibre axis - _f Normal stress in the fibre along the fibre axis - _m Normal stress in the matrix along the fibre axis - Shear stress on the fibre—matrix interface     

Non fermi liquid ground states in strongly correlated f-electron materials

Experimental efforts to characterize and develop an understanding of non Fermi liquid (NFL) behavior at low temperature in f-electron materials are reviewed for three f-electron systems: M_1–xU_xPd₃ (M = Sc, Y), U_1–xTh_xPd₂Al₃, and UCu_5–xPd_x. The emerging systematics of NFL behavior in f-electron systems, based on the present sample of nearly ten f-electron systems, is updated. Many of the f-electron systems exhibit the following temperature dependences of the electrical resistivity p, specific heat C, and magnetic susceptibility for T T₀, where T_o is a characteristic temperature: P(T) 1 –aT/T ₀, where a < 0 or > 0, C(T)/T (-1/T _o) In (T/bT ₀), and (T) 1 –c(T/T_o)^1/2. In several of the f-electron systems, the characteristic temperature T_o can be identified with the Kondo temperature T_k.     

Stress relaxation in zirconium carbide. Report 1. Test method and stress relaxation curves

A method of stress relaxation tests with use of four-point bending is described. Experimental data is given on pure stress relaxation in ZrC _1.00 with different grain sizes (d= 6–55m in the area of the ductile-to-brittle temperature (1600–2200C)Translated from Problemy Prochnosti, No. 1, pp. 76–83, January, 1994.     

Investigation of the evaporation process and sonic limits in sodium heat pipes

The maximum heat fluxes in the heating zone and the sonic limits of power transfer in sodium heat pipes and vapor chambers with composite (channel) wicks are investigated experimentally.Notation P vapor pressure - T temperature - density - molecular weight - L latent heat of vaporization - q specific heat flux in heating zone - Q specific heat transfer along heat pipe - G mass flow of vapor in pipe cross section - W average vapor flow velocity in pipe - surface porosity of wick - R universal gas constant - f coefficient of evaporation - momentum-flux coefficient - coefficient of friction - d_v diameter of heat pipe vapor channel Indices 0 liquid surface - v vapor at liquid surface - w wall surface - in entrance to heating zone - so critical cross section Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 33, No. 5, pp. 832–837, November, 1977.     

Interlaminar Fracture Toughness of CF/PEI and GF/PEI Composites at Elevated Temperatures

An experimental study has been conducted to assess temperature effects on mode-I and mode-II interlaminar fracture toughness of carbon fibre/polyetherimide (CF/PEI) and glass fibre/polyetherimide (GF/PEI) thermoplastic composites. Mode-I double cantilever beam (DCB) and mode-II end notched flexure (ENF) tests were carried out in a temperature range from 25 to 130°C. For both composite systems, the initiation toughness, G _IC,ini and G _IIC,ini, of mode-I and mode-II interlaminar fracture decreased with an increase in temperature, while the propagation toughness, G _IC,prop and G _IIC,prop, displayed a reverse trend. Three main mechanisms were identified to contribute to the interlaminar fracture toughness, namely matrix deformation, fibre/matrix interfacial failure and fibre bridging during the delamination process. At delamination initiation, the weakened fibre/matrix interface at elevated temperatures plays an overriding role with the delamination growth initiating at the fibre/matrix interface, rather than from a blunt crack tip introduced by the insert film, leading to low values of G _IC,ini and G _IIC,ini. On the other hand, during delamination propagation, enhanced matrix deformation at elevated temperatures and fibre bridging promoted by weakened fibre/matrix interface result in greater G _IC,prop values. Meanwhile enhanced matrix toughness and ductility at elevated temperatures also increase the stability of mode-II crack growth.     

Thermal gradient effects on the vibration of prestressed rectangular plates

Summary The effect of a thermal gradient on the transverse vibration of a prestressed rectangular plate is investigated by the method of matched asymptotic expansions. This class of heated plate is characterised by changing its Young's modulus with temperature. Analytical results for the eigenvalues are presented for fully-clamped and fully-hinged rectangular plates when the bending rigidity is small compared to the in-plane loading. To leading order in (where ² denotes the normalized bending rigidity), the eigenvalues of an ideal membrane are obtained, independent of thermal effects.Thermal gradient effects occur in the first order correction of eigenvalues for a clamped plate while the eigenvalues of a hinged plate are affected by thermal gradient only to second order. In particular, Schneider's results are recovered when thermal gradient effects are absent.Nomenclature W' bending deflection - D(x) flexural rigidity - D ₀ reference flexural rigidity - x',y' rectangular co-ordinate - E modulus of elasticity - E ₁ reference modulus of elasticity - t' time - h' height of plate - a' length of plate - b' width of plate - T temperature - T ₀ reference temperature - slope of variation ofE withT - parameter - L characteristic length - N ₀ characteristic in-plane force - m mass per unit area - characteristic frequency - outer solution - inner solution - small parameter     

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.     

Simultaneous determination of the thermal conductivity and diffusivity of metals using an electronic heater

A nonstationary method allowing simultaneous determination of the thermal conductivity and diffusivity of metals using an electronic heater is described.Notation q_e heat flux density - t temperature - x distance to thermocouple - heating time - t₀ temperature at time zero - thermal conductivity of material examined - a thermal diffusivity of material - c coefficient describing the energy losses - X₁, x₂, x₃ distance to first, second, and third thermocouples, respectively - ₁, ₂, ₃ times during which specimen is heated to temperature t at points X₁, x₂, X₃, respectively     

On the analysis of tunneling data for A15 superconductors

We examine the applicability of the standard McMillan inversion of Eliashberg's equations for superconductors with a nonconstant electronic density of statesN(). We do this usign simple models forN() and a realistically shapedN() taken from recent band structure work for Nb₃Sn. It turns out that peak structure inN() near _F may lead to gross errors in the derived Eliashberg function ²F() when the energy dependence ofN() is omitted in the inversion procedure. For Nb₃Sn, this leads to a 40% overestimate of when ²F() is evaluated via the standard McMillan program.     

Heat capacity and magnetic properties in the (PdCO y )H x -system at low temperatures

Results of heat capacity measurements on (PdCo _y )H _x (y 2.5 at %; 0 x 0.8) in the temperature range from 2 to 12 K are reported. The dependence of the coefficient of the electronic heat capacity on the Co content and on the x value is considered. For the binary alloy PdCo _y a decreasing spin wave contribution to the heat capacity (T ^3/2) is found, which agrees well with the known ferromagnetic properties of the alloy. In contrast, for the phase of (PdCo _y )H _x a contribution to the heat capacity can be separated, probably caused by the thermal excitation of magnetic clusters. The temperature dependence of this cluster contribution can be well explained by means of an Einstein function. Conclusions are drawn with regard to the intercluster interaction and to the number of atoms per cluster.     

Electronic and two-magnon light scattering in oxide superconducting crystals

From investigations of two-magnon Raman scattering (RS) under high pressures up to 430 kbar in Eu₂CuO₄ and YBa₂Cu₃O_6.2 crystals, it was shown that the dependence of the superexchange integralJ on the distance between Cu and O atoms in CuO₂ planesa is anomalously weak (Ja^–n, n=3±0.5). The large value ofJ indicates strong initial overlapping of Cu and O wave functions in high-T _c , materials. It was found that an increase in free carrier concentration results in a rapid increase of magnon damping and the disappearance of the two-magnon peak from RS spectra. A detailed study of electron Raman scattering has been carried out in superconducting and insulating YBa₂Cu₃O_6–x , single crystals. The spectral redistribution at frequencies<600 cm^–1 in different polarizations indicate that the superconducting gap is strongly anisotropic. In the normal (metallic) phase the behavior of the imaginary part of the response functionR() in the polarization (xx) corresponds to the model of a marginal Fermi liquid, and in the polarization (xx), this behavior is independent of the temperature. In insulating crystals,R() is independent of temperature toT200 K in both polarizations.     

Synthesis and Properties of Neptunium(V) Propionates

From propionate-containing solutions of Np(V), the crystalline compounds NpO₂OOCC₂H₅·nH₂O with n = 1 or 2 and M(NpO₂)₂(OOCC₂H₅)₃·H₂O with M = NH₄ or Cs were separated. Their X-ray powder patterns were recorded and behavior at heating was studied. The IR and electronic absorption spectra suggest that, in these compounds, the NpO₂ ⁺ ions are coordinated with each other to form cation-cation bonds involving all yl oxygen atoms.     

Compressive strength of coated rigid-rod polymer fibres

One limitation to the use of high-strength/high-modulus rigid-rod polymer fibres like poly-(p-phenylene benzobisthiazole) (PBZT) and poly-(p-phenylene benzobisoxazole) (PBZO) in composite structures is their low compressive strength. Various theories have been developed to predict compressive strength of rigid-rod fibres. In this study the critical buckling stress for rigid-rod fibres with stiff external coatings has been theoretically modelled assuming that the failure mode in compression is the microbuckling of the fibrils in shear. Our model predicts that significant improvement in fibre compressive strength will occur only when relatively thick coatings, with thickness to diameter (t/D) ratios in excess of > 0.05, are used. Experimentally measured compressive strength of aluminium coated PBZT fibres shows values in good agreement to the theory at t/D ratios of 0.006 and below. Factors related to the selection of suitable coating materials and problems associated with establishing coating performance are identified.Nomenclature P axial compressive load - P _f axial compressive load on the fibre - P _c axial compressive load on the coating - P _cr ⁱ critical buckling load in the ith case - _cr critical buckling stress - _co compressive strength of the uncoated fibre - _c compressive strength of the coated fibre - v(x) lateral deflection of a buckled fibril or coating - V _m amplitude of the lateral deflection in the mth mode - m number of half-sine waves in the deflection mode - x coordinate distance along axial direction - y coordinate distance along radial direction - coordinate distance along circumferential direction - l length of the buckling unit - N number of fibrils in the fibre - D fibre diameter - d fibril diameter - t coating thickness - I _f moment of inertia of the fibril - A _f cross-sectional area of the fibril - E _f tensile modulus of the fibre - E _c tensile modulus of the coating material - E tensile modulus of the coated fibre - G torsional shear modulus of the fibre - v_c Poisson's ratio of the coating material - _f density of the fibre - _c density of the coating material - density of the coated fibre - U _f strain-energy change in the fibre - U _c strain-energy change in the coating - T _f external work done on the fibre - T _c external work done on the coating - d/D - t/D     

Ultrasonic attenuation of longitudinal and shear waves in superconducting lead

Longitudinal and shear wave ultrasonic attenuations have been measured in high-purity Pb on two single crystals obtained from the same ingot. The measurements were done at low temperatures, at different frequencies, and in transverse magnetic fields, up to a field of 7.3 kG. The propagation directions in the two crystals were along [100] and [110]. For some propagation and polarization directions the _s / _n ratio is found to be frequency-independent, while for others, large divergences in the _s / _n ratios at different frequencies are observed. A sharp decrease of _s / _n nearT _c is observed for a particular longitudinal wave propagation, but not in any shear wave propagation. In some cases _s / _n is found to be abnormally high and this feature is associated with a peak in attenuation _n and a relatively high _n at 7.2 K. None of the _s / _n curves fits closely to any BCS energy gap. For longitudinal waves the high magnetic field (H) dependence of the normal state attenuation was found to agree qualitatively with the free electron theory for propagation along [100], but not for propagation along [110]. For shear waves the high-field attenuations do not extrapolate to zero asH tends to infinity. For all propagation and polarization directions the high-field attenuations show 1/H ² field dependence.     

Effect of swelling in the extrusion of an elastic liquid from a capillary

The article demonstrates that the profile of a jet extruded from a capillary can be constructed from the data of uniform stretching.Notation V₁ mean velocity of the liquid in the capillary - r₁ radius of the capillary - q flow rate - P pressure in the preinlet zone of the capillary - F₁ and F₀ extrusion force - ₁ and ₂ elastic deformations in parallel Maxwellian elements upon stretching - p, z radial and longitudinal coordinates, respectively - v_z(p) velocity of steady-state flow in the capillary - _i modulus of elasticity in the i-th Maxwellian element - swelling coefficient - t time - l length of the specimen at instant t - elastic deformation - x rate of deformation - ₀, ₁ deformation - L length from which onward the radius of the jet extruded from the capillary practically does not change - r radius of the jet - t₁ time from which onward the radius of the specimen in retardation practically does not change - d₀,l ₀ diameter and length, respectively, of the undeformed specimen - l _r length to which the extended specimen tends after removal of the load - b length of the capillary - ₁ tensile stress (for uniform stretching at t = 0, for extrusion from the capillary at z = 0) - l _H length of the stretched specimen at the instant of beginning reading of t Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 2, pp. 343–350, August, 1980.     

Microstructure and fracture behaviour of short and long fibre-reinforced polypropylene composites

Microstructure and fracture mechanical behaviour of injection-moulded, longer glass fibrereinforced polypropylene (Verton* aspect ratio 320) were studied as a function of fibre volume fraction and compared to that of shorter fibre-filled polypropylene (aspect ratio 70). Toughness was measured using instrumented notched lzod and falling weight impact tests, as well as compact tension specimens. It was found that the addition of longer fibres generally increased the toughness of the material, although more significant increases were seen in the impact tests than were seen in the compact tension test. For the latter results, a correlation between toughness improvement and microstructural details was performed on the basis of the microstructural efficiency concept, a semi-empirical approach of the formK _c,C = (a* +nR)K _c,M, where,K _c,C andK _c,M are the fracture toughnesses of the composite and the matrix, respectively,_a* is a matrix stress correction factor,n is a scaling parameter andR is a fibre reinforcement effectiveness factor. The latter corrects for differences in the composite microstructures, and incorporates effective fibre orientation factors, layering of injection moulded parts, and fibre volumes in the different layers.Nomenclature a crack length - a ^* matrix toughness correction factor - A cross-sectional area - B thickness of the sample plaques - C thickness of the composite core regions - E _peak energy adsorbed up to the maximum force in the impact load-displacement curve - E _t tensile modulus - F _max maximum force in impact force-displacement curves - f _p fibre orientation factor - f _pe effective orientation factor - f _pe,C effective orientation parameter, core region - f _{pe, s} effective orientation parameter, surface region - F critical load in the tensile test load-displacement curves - K _c critical stress intensity factor/fracture toughness - K _L fracture toughness of the composite materials - K _d dynamic fracture toughness - K _L fracture toughness of the matrix - L test with crack parallel to the mould filling direction - M microstructural efficiency factor - n scaling parameter for reinforcement effectiveness factor (energy absorbtion ratio) - R reinforcement effectiveness factor - S thickness of the composite surface regions - T test with crack perpendicular to the mould filling direction - V _f fibre volume fraction - V _m matrix volume fraction (= 1 —V _f) - W specimen width - W _f fibre weight fraction - W _m matrix weight fraction (= 1 —W _f) - X _n number average fibre length - X _v volume average fibre length - Y(a/ W) polynomial correction for compact tension specimens - variable in effective orientation factor formula - variable in effective orientation factor formula - _B strain to break - _c density of the composite - _f fibre density - _m matrix density - _F fracture strength - fibre angle with respect to a reference direction     

Geometrical factors for parts of an infinite cylindrical surface

A method is described for calculating geometrical factors, allowing for attenuation of radiation by the medium, and computing radiative heat transfer between coaxial strips forming part of a cylinder.Notation mean generalized geometrical factor - local generalized geometrical factor for a closed infinite cylindrical surface - F surface area - l length of ray between elements dF_m - D and H diameter and length of cylinder - k ray attenuation coefficient - equivalent solid angle [1] determined from the condition _mm=1 at =0 for a closed surface - f () indicatrix of the specific radiation intensity (a quantity analogous to the specific luminous intensity [1]) - angle between the normal to an element of surface and the ray linking the surface elements - and angles in a section of the surface - angle between the radius and the chord subtending the projections of dF_mon the section - angle between the radius and the chord subtending the projection of element dF_mon the section and the edge of the surface - acute angle between rayl and the generator of the cylinder - ₀ half subtended angle for the arc of a section of the surface - S₂ and M generalized local geometrical factors in the notation of [2] (s₂ and is shown in Table 2 - M local factor for an infinite cylinder with a semicircular cross section for a point in the middle of the plane part of the surface; a table of values of M is given in [2]) - n number of strips