Short-rod elastic-plastic fracture toughness test using miniature specimens

The standard ASTM-E399 plane-strain fracture toughness (K _IC) test requires (1) the test specimen dimensions to be greater than a minimum size and, (2) fatigue precracking of the specimen. These criteria render many materials impractical to test. The short-rod elastic-plastic plane-strain fracture toughness test proposed by Barker offers a method of testing not requiring fatigue precracking and furthermore, it appears that test specimens smaller than that stipulated by ASTM can be used to obtain validK _IC values. In this study, the use of a modified miniature short-rod fracture toughness test specimen was investigated. Our miniature short-rod specimen is approximately 7 mm long and 4 mm diameter. These mini specimens are well suited for the purpose of testing biomaterials. The value of the minimum stress intensity factor coefficient (Y _m ^* ) for the mini short-rod specimens was determined experimentally using specimens machined from extruded acrylic rod stock. An elastic-plastic fracture toughness analysis using the mini specimens gave values ofK _IC for extruded acrylic (nominally PMMA) equal to 0.67 ± 0.06 MPa m^1/2. The problem of testing non-flat crack growth resistance curve materials (such as PMMA) using the short-rod fracture toughness test method is discussed. A modification to the test procedure involving the use of aY ^* value corresponding to a short crack length is suggested as a method of overcoming this difficulty.Nomenclature a crack length - a ₀ initial crack length - a ₁ length of the chevron notch on the mini short-rod specimen - a _m critical crack length — crack length atY _m ^* - C specimen compliance - C dimensionless specimen compliance = CED - D mini short-rod specimen diameter - E Young's modulus - K ₁ stress intensity factor - K _1C plane-strain fracture toughness - K _max fracture toughness calculated usingP _max - P load applied to the test specimen during a short-rod fracture toughness test - P _c load applied to the test specimen atY _m ^* - P _max maximum load applied to the specimen during a short-rod fracture toughness test - p plasticity factor - W mini short-rod specimen width - Y ^* stress intensity factor coefficient - Y _m ^* minimum of the stress intensity factor coefficient - dimensionless crack length =a/W - ₀ dimensionless initial crack length = ₀/W - ₁ dimensionless chevron notch length =a ₁/W - _m dimensionless critical crack length =a _m/W     

Optimum notch configurations for the chevron-notched four-point bend specimens

The failure sequence following crack formation in a chevron-notched four-point bend specimen is examined in a parametric study using the Bluhm slice synthesis model. Premature failure resulting from crack formation forces which exceed those required to propagate a crack beyond is examined together with the critical crack length and critical crack front length. An energy based approach is used to establish factors which forecast the tendency of such premature failure due to crack formation for any selected chevron-notched geometry. A comparative study reveals that, for constant values of ₁ and ₀, the dimensionless beam compliance and stress intensity factor are essentially independent of specimen width and thickness. The chevron tip position ₀ has its primary effect on the force required to initiate a sharp crack. Small values for ₀ maximize the stable region length, however, the premature failure tendency is also high for smaller ₀ values. Improvements in premature failure resistance can be realized for larger values of ₀ with only a minor reduction in the stable region length. The stable region length is also maximized for larger chevron base positions, ₁, but the chance for premature failure is also raised. Smaller base positions improve the premature failure resistance with only minor decreases in the stable region length. Chevron geometries having a good balance of premature failure resistance, stable region length, and crack front length are 0.20 ₀0.03 and 0.70₁0.80NASA Resident Research Associate at Lewis Research Center.     

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     

An evaluation of the short rod technique to measure the fracture toughness of polymers

The investigation of the fundamental variables which influence the fracture toughness of structural plastics is greatly hampered by a large amount of scatter and uncertainty associated with the fracture toughness measurement. A major part of the problem is due to a lack of adherence to ASTM Standard E399, mainly with regard to the requirement for a fatigue crack. A razor-blade arrested crack, which is often blunted, is common practice in the plastics field. It is also common to ignore size (plane strain) and precise machining requirements. The short rod (SR) method was evaluated as a potentially more precise and simpler fracture toughness measurement. This toughness measurement is made on a slowly moving and presumably sharp crack, and the geometry of the sample enforces plane strain conditions. Toughness measurements on compact tension (CT) specimens via ASTM E399 were performed on one-inch (25 mm) samples of poly(methyl methacrylate) (PMMA), polystyrene (PS), polycarbonate (PC) and polysulphone (PSO). Also, a constant compliance method using a contoured double cantilever beam (CDCB) was used to evaluate the toughness of PS, PC, and PSO, but in general we did not achieve stable crack growth. The used samples were then fabricated into SR specimens and their toughness measured. The CT and CDCB methods agreed with each other for PSO and PC, but for PS the CDCB method gave high values. It is argued that the SR method should be compared to the other methods without using a plasticity correction. Then the SR method agrees well with the CT method for PSO and PS and is 15% higher for PC. The PMMA SR results were invalid. Differences between the methods are explained in terms of crack blunting, rate effects, non-homogeneity, residual stresses and the global nature of the crack front. The SR method has promise for polymer evaluation but more experience and evaluation is needed. The method is unique in the ability to study the effects of thermal history and of the environment on fracture toughness.Nomenclature a Distance from load line in the crack plane to the crack tip - a ₀ Distance from load line in the crack plane to the chevron tip. Equal to 10.11 + 0.287 mm - A Calibration constant:A = 22 atr = 0.555,A = 50.9r ² - 56.25r + 37.58 forr > 0.555,A = 77.87r ² -86.05r + 45.77 forr < 0.555. These relationships for A, supplied to us by Terra Tek, were found to be valid forr = 0.4 to 0.75 - B SR specimen diameter, equal to 19.1 ± 0.23 mm in this study or to specimen thickness for CT and CDCB specimens - C Correction for SR specimen sizes which do not exactly meet specifications of compliance calibration - C _a,C _b,C _w,C C corrections fora, B, W and , respectively - F SR load atr = 0.555 - K Stress intensity factor (MPam^1/2). - K _c Critical value of K at point of instability - K _1c K _c, in opening mode - K _1A Arrest toughness - K _sr (=K _Isr) Short-rod determined toughness. When used in conjunction withK _srp, refers to toughness value without thep correction - K _srp (=K _isrp) K _sr withp correction. A prime (as used inK_srp) is used to differentiate between toughness atr _c and at the downloadings - m Constant-compliance constant for CDCB specimen derived from specimen geometry and beam theory - m Correction tom obtained from experimental compliance calibration. Corrects for side grooves - p Ratio of difference in strain for two SR downloading curves at load = 0 to that at load =F. Must be less than 0.2. See Fig. 3 - p p correction due to plasticity alone - r SR initial compliance divided by     

A three-dimensional graphical aid to analyze fatigue crack nucleation and propagation phases under fatigue limit conditions

A three-dimensional diagram is presented in which the fatigue limit of notched components is plotted as a function of the notch stress concentration factor K _t and the ² a/a ₀ ratio, , a and a ₀ being a shape factor, the notch depth and the El Haddad–Smith–Topper length parameter, respectively. Intersections with the planes normal to the axes allow the display of the different influence of crack nucleation and propagation on the fatigue limit of notched components.     

About endurance limit of ductile inhomogeneous materials

The theory describing the fatigue mechanism in elasto-plastic material containing pores or inclusions has been developed. An attempt at quantitative determination of the effect of endurance limit reduction by analysis of sizes of plastic zones formed near the inclusions, and their cracking has been done. The geometrical configuration, consisting of a round inclusion from which a nucleating crack emerged, was considered, and the stress intensity factor of such configuration was analysed. Based on a threshold value of K below which crack propagation ceases, the critical value of loading stress was determined. Theoretical results were compared with results from experiments, showing quite good agreement.Glossary of Symbols a rack length (mm) - A plastic zone range (mm) - B width of specimen (mm) - D pore diameter (mm) - H materials hardening coefficient - K stress intensity coefficient (N mm^–3/2) - K _I,K _II stress intensity coefficient for first and second mode of fracture - KIZ equivalentK _I coefficient for zigzag crack. (N mm^–3/2) - KTH threshold value ofK (N mm^–3/2) - KTHM KTH in microscale (N mm^–3/2) - L length of flat crack (mm) - Na real length of zigzag (mm) - N _f fatigue life in cycles - P loading force variation (N) - RA reduction of area of sample - S loading stress (MPa) - W height of specimen (mm) - Y yield stress of matrix material (MPa) - , , coefficients inA/D=f(S/Y) formula - K stress concentration coefficient - , , coefficients inK=f(S, A/D) formula - _f, ₁ coefficients - _p plastic strain components - , parallel and perpendicular to crack front surface development correction coefficients - surface development coefficient     

Generalization of the temperature-strain characteristics for polyvinyl chloride under tension

A universal temperature-time characteristic of the stress-strain relation has been derived for polyvinyl chloride under uniaxial tension.Notation P true stress - strain - l ₀ initial length of specimen - l length of specimen at any instant of time - total elongation factor - _p=max elongation factor at maximum stress on P () curve - v elongation rate - strain rate - t time - T absolute temperature - E_t relaxation modulus - H relaxation spectrum - relaxation time - T_S reference temperature, °K - a _T reference temperature coefficient Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 22, No. 6, pp. 1031–1035, June, 1972.     

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.     

Effects of rate,temperature and absorption of organic solvents on the fracture of plain and glass-filled polystyrene

The rate/temperature dependent fracture behaviour of plain and glass-filled polystyrene has been investigated over the crack speed (a) range of 10^–6 to 10^–2 m sec^–1 and in the temperature (T) range of 296 to 363 K. TheK _c (a, T) relationships obtained, whereK _c is the stress intensity factor at fracture, are shown to follow those given by the Williams/Marshall relaxation crack growth model and the toughness-biased rate theory. Crack propagation in both materials is shown to be controlled by a-relaxation molecular process associated with crazing. Crack instabilities observed in plain polystyrene are analysed successfully in terms of isothermal-adiabatic transitions at the crack tip. Fracture initiation experiments are also conducted in which the effects of organic liquids on the fracture resistances of both plain/glass-filled polystyrene have been determined. Good correlations betweenK _i ² (K _i being the crack initiation stress intensity factor) and _s, solvent solubility parameter, of various liquid environments have been obtained, which give a minimumK _i ² value at _{s p}, where _p is the solubility parameter of the polymer. For a given temperature, liquid environment and crack speed, the glass-filled polystyrene is shown to possess greater resistances to crack propagation than plain polystyrene.     

Thermal expansion of strontium tungstate

Accurate lattice parameters of strontium tungstate, an isotype of scheelite, have been determined as a function of temperature by the X-ray powder method in the temperature range 28 to 355° C. Both the lattice parameters are found to increase with temperature. Using these data, the two coefficients of thermal expansion, _a along the a-axis and _a along the c-axis, have been calculated. The temperature dependence of the coefficients could be expressed by the following equations: _a =5.88×10^–6–25.63×10^–10 T + 59.49×10^–12 T ² _c =13.20×10^–6–18.18×10^–10T+71.45×10^–12 T ². Here T is the temperature in °C.     

The effect of a circumferential stiffener on the stress in a pressurized cylindrical shell with a longitudinal crack

From a numerical analysis of the Marguerre complex stress function in an integral representation, the effect of a continuous uniform circumferential stiffener upon the stress singularities at the crack tips of a nearby longitudinal crack and the maximum load concentration in the stiffener are calculated and presented in graphical parametric form.

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Zusammenfassung Eine geschlossene zylindrische Hülle unter gleichmässigem inneren Druck hat einen Riß entlang einer Portion einer von ihren Erzeugern. Die Hülle hat auch einen kontinuierlich angefesteten Ringversteifer ringsum ihren Umfang und dicht am Riß. Eine Integraldarstellung ist gefunden worden für die komplexe Spannungsfunktion der Flach-Hülle-Theorie von Marguerre. Die Integralgleichungen sind abgeleitet und gelöst worden durch den IBM 7094 Komputer. Eine Darstellung ist gegeben der Wirkung des Versteifers auf die Spannungseigenheiten an den Riß-Spitzen, und auch der Höchstbelastungskonzentration im Versteifer.

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Résumé Un corps creux cylindrique fermé scus pression intérieure uniforme est supposé d'avoir une fissure dans le sens d'une part d'un de ses générateurs. Le corps creux soit aussi pourvu d'un contrefort annulaire continuellement attaché au corps sur sa circonférence près de la fissure. Une représentation intégrale a été trouvée pour la fonction de tension complexe de la théorie de Marguerre du corps creux sans profondeur. Les équations intégrales sont dérivées et résolues par l'ordinateur IBM 7094. Une démonstration est donnée de l'effet du contrefort sur les particularités de la tension aux pointes de la fissure, aussi bien de la. concentration minimum de charge dans le contrefort.

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Notation A() S+(3+)(1–)[1/3(1–²)]^–1/2B: composite singularity strength defined by Copley [6]. - constant cross-sectional area of the stiffener - a / - B bending singularity strength for =0. (See footnote to (30)) - B , B values of B at x=, x= for 0 - b (Rh)^1/2[3/4(1–²)]^1/4: length scale in shell, with respect to which distances are made dimensionless - c half crack length - CA (),CA() . See footnote to (30) - CS ,CS, CB, CB correction factors S /S, S /S, B /B, B /B - d cut-off distance for integrations on -axis - E modulus of elasticity of shell - E _s modulus of elasticity of stiffener - G Green's function, (22) - h thickness of shell - M _x, M _y dimensionless bending moment resultants - M _x, M _y (p ₀b²)(M _x, M _y): bending moment resultants - M _xy dimensionless twisting moment resultant - M _xy p ₀b²M_xy: twisting moment resultant - M _inf ^sup* see (23a) - N _x, N _y, N _xy dimensionless stress resultants - N _x, N _y, N _xy (p0R)·(N _x, N _y, N _xy): membrane stress resultants - N ₀ tension in stiffener at infinity - N tension in stiffener for perturbation problem - N _inf ^sup* see (23b) - p ₀ pressure inside shell - Q _x, Q _y dimensionless transverse shear resultants - Q _y, Q _x (p ₀b)·(Q _x, Q _y): transverse shear resultants - R radius of shell - S stretching singularity strength for =0 (see footnote to (30)) - S , S values of S at x=, x= for 0 - V Kirchhoff transverse shear (see Fig. 2) - V _x, V _y see Appendix - u, dimensionless displacements in x, y-directions - , v ·(u, ): displacements in x, y-direction - _inf ^sup* see (23c) - dimensionless displacement in z-direction - w {it{p₀R²}/(Eh)}: displacement in z-direction - x dimensionless coordinate along generator - x bx: coordinate along generator - y dimensionless coordinate perpendicular to x-direction in plane of shallow shell - z bx: coordinate perpendicular to x- and y-directions - z dimensionless coordinate normal to shell - z (b ²/R)z: coordinate normal to shell - dimensionless distance of near crack tip from stiffener - b: distance of near crack tip from stiffener - dimensionless distance of far crack tip from stiffener - b: distance of far crack tip from stiffener - strain in stiffener - _x, _y dimensionless strains in x, y-directions in shell - poR/(Eh)( _x , _y ):strains in x, y-directions in shell - (/bh)(E _s/E):defines rigidity of stiffener - · - c/b: dimensionless half crack length - dimensionless stress function - p ₀ Rb ²: stress function - +i: complex stress function This work was supported in part by the National Aeronautics and Space Administration under Grant NsG-559, and by the Division of Engineering and Applied Physics, Harvard University.Now Mary E. Fama.     

Direct and inverse heat-conduction problems for a semiinfinite rod for a partial outflow of heat through the surface

Analytical solutions of the direct and inverse problems of nonstationary heat conduction in a thin semiinfinite rod are given for the case of radiative heat fluxes at the lateral surfaces and a partial outflow of heat by convection and radiation through the end of the rod.Notation thermal diffusivity - x₁ coordinate along the length of the rod - t₁ time - t=t₁/d² dimensionless time (Fourier number) - x=X₁/d relative coordinate - T_o initial temperature - Boltzmann constant - Sk=aT_c ³d/ Stark number - Bi=d/ reduced Biot number - emissivity Translated from Inzhenero-Fizicheskii Zhurnal, Vol. 47, No. 1, pp. 148–153, July, 1984.     

Application of a new iterative method for elastic-plastic stress analysis

A new iterative method for elastic-plastic stress analysis based on a new approximation of the constitutive equations is proposed and compared with standard methods on the accuracy and the computational time in a test problem. The proposed method appears to be better than the conventional methods on the accuracy and comparable with others on the computational time. Also the present method is applied to a crack problem and the results are compared with experimental ones. The agreement of both results are satisfactory.List of symbols u = (u ₁, u ₂) displacements u ^(H) = u ⁽ⁿ⁺¹⁾ - u ⁽ⁿ⁾ u _k ⁽ⁿ⁾ = u _(k ^{(n + 1)} - u ⁽ⁿ⁾ (n, k = 0, 1, 2, ...) - = ₁₁, ₂₂, ₁₂) stresses - = (₁₁, ₂₂, ₁₂) strains - = (₁₁, ₂₂, ₁₂) center of yield surface - D elastic coeffficient matrix, C = D ^–1 - von Mises yield function. The initial yielding is given by f() = _Y - f {f/} - ^* transposed f - H hardening parameter (assumed to be a positive constant for kinematic hardening problems) - time derivative of - [K] total elastic stiffness matrix - T traction vector - = [B] relation between nodal displacements and strains     

Thermal expansion of epoxy-fiberglass composite specimens

The thermal expansion behavior of three epoxy-fiberglass composite specimens was measured from 20 to 120°C (70 to 250°F) using a fused quartz push-rod dilatometer. Billets produced by vacuum-impregnating layers of two types of fiberglass cloth with an epoxy were core-drilled to produce cylindrical specimens. These were used to study expansion perpendicular and parallel to the fiberglass layers. This type of composite is used to separate the copper conductors that form a helical field coil in the Advanced Toroidal Facility, a plasma physics experiment operated by the Fusion Energy Division at Oak Ridge National Laboratory. The coil is operated in a pulsed mode and expansion data were needed to assess cracking and joint stresses due to expansion of the copper-composite system. The dilatometer is held at a preselected temperature until steady state is indicated by stable length and temperature data. Before testing the composite specimens, a reliability check of the dilatometer was performed using a copper secondary standard. This indicated thermal expansion coefficient () values within ±2% of expected values from 20 to 200°C. The percentage expansion of the composite specimen perpendicular to the fiberglass layers exceeded 0.8% at 120°C, whereas that parallel to the fiberglass layers was about 0.16%. The expansion in the perpendicular direction was linear to about 70°C, with an value of over 55×10^–6 °C^–1. Anomalous expansion behavior was noted above 70°C. The expansion in the direction parallel to the fiberglass layers corresponds to an value of about 15×10^–6 °C^–1. The lower values in the parallel direction are consistent with the restraining action of the fiberglass layers. The values decreased with the specimen density and this is consistent with literature data on composite contraction from 20 to –195°C.Nomenclature Thermal expansion coefficient, °C^–1 - L L(T ₂)–L(T ₁), cm - T T ₂–T ₁, °C - L ₀ Length at room temperature, cm - L(T _i ) Length at temperature T _i , cm - T _i Temperature, °C - T ₀ Room temperature, °C Paper presented at the Ninth International Thermal Expansion Symposium, December 8–10, 1986, Pittsburgh, Pennsylvania, U.S.A.     

The failure of fibre composites and adhesively bonded fibre composites under high rates of test

The dynamic effects which are commonly encountered during high-rate DCB tests with fibre composite and adhesively bonded fibre composite arms have been studied in detail. This paper, Part II of the series, follows Part I, which described the experimental aspects of the high-rate testing. Part III will report the results from mode II and mixed-mode I/II tests on the fibre-composite materials.Nomenclature a crack length - a ₀ initial crack length - a crack speed - ä crack acceleration - c longitudinal wave speed - h thickness of single arm of test specimen - p crack length perturbation (i.e. the measured value of the crack length minus the value predic ted by steady-state theory) - p crack velocity perturbation - crack acceleration perturbation - t time - t ₀ time taken for crack to initiate during the mode I test - u ₀ load-line vertical displacement of single arm of test specimen (/2 in Part I) - u(x) vertical displacement of specimen at distance x from the load-line - u(x) vertical displacement rate of specimen at distance x from the load-line - x distance along the test specimen from the load-line - A constant relating the steady state crack length to root time - B width of specimen - C compliance of the specimen (u ₀/P) - E ₁₁ axial modulus of the fibre-composite beam - G mode I energy release rate - G _Ic mode I critical energy release rate or fracture toughness - G ₁ half the value of G _Ic during steady-state propagation (i.e. calculated for half the beam as shown in Fig. 1) - G ₂ half the value of G _Ic at crack initiation - P end load applied to specimen - U _ext external work done - U _s strain energy - U _k kinetic energy - V velocity of a single arm of test specimen (i.e. half the measured test velocity) - dynamic term, governed by the ratio of the energy to initiate versus that to propagate a crack - _I mode I crack shear deflection and root rotation correction term - crack length correction term, evaluated by the negative intercept on the a versus t ^1/2 plot - dynamic term controlling the form of the computed perturbations - Poisson's ratio for the fibre-composite beams density of the fibre-composite beams - time, normalized by the initiation time, t ₀ and thus equivalent to (t/t ₀) - values of at which crack arrest occurs. n = 1,2,3... - ratio of distance along beam to crack length (x/a)     

Influence lines for bending under a ring load of a free shell,a shell embedded in a soft medium and a shell containing a soft core

Summary Theoretical expressions for stresses and displacements have been derived for bending under a ring load of a free shell, a shell embedded in a soft medium, and a shell containing a soft core. Numerical work has been done for typical cases with anElliot 803 Digital Computer and influence lines are drawn therefrom.

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Einflußlinien für die Biegung einer freien Schale, einer Schale in einer weichen Bettung und einer Schale mit weichem Kern

Zusammenfassung Für die Biegung einer freien Schale, einer weich gebetteten Schale und einer Schale mit weichem Kern unter einer Ringlast werden Ausdrücke für die Spannungen und Verschiebungen hergeleitet. Die Ergebnisse wurden für einige typische Fälle mit einem DigitalrechnerElliot 803 numerisch ausgewertet. Die sich ergebenden Einflußlinien wurden graphisch dargestellt.

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Nomenclature A (),B () Functions of - a, t Mean radius and thickness of the shell - E _S , _S Young's modulus andPoisson's ratio of the shell - G _c , _c Shear modulus andPoisson's ratio of the casting or core - I ₀ (r),I ₁ (r) ModifiedBessel functions of the first kind and order zero and one respectively - K ₀ (r),K ₁ (r) ModifiedBessel functions of the second kind and order zero and one respectively - p Ring load, lb/in - U, W Displacement components in the casing or core in thez andr direction - u, w Displacement components of a middle surface point in the shell - _r , _rz Radial and shearing stress components - Independent variable of infinite integrals - k [3(1–S ²)a ²/t ²]^1/4 With 13 Figures     

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     

Analysis of heat transfer in the furnace of the P-67 boiler P-67 furnace and improvement of its design

The results of experimental study of heat transfer in the furnace of the P-67 boiler (under the Russian trademark) burning Kansk-Achinsk coal are presented. Means of improving the design of the furnace device are proposed.Notation N energy unit power, MW - _fur furnace air excess coefficient - r gas recirculation degree - q_in incident radiation flux density, kW/m² - a spacing between the combustion chamber walls, m - b_b burner width, m - n number of the burner row, starting from below - and inclination angles of the burners located in front of and behind the combustion chamber axis - d diameter of the conventional circumference, tangential to which the burners are directed, m - q_ch heat-stress of the radiant heat absorbing surface of the active combustion zone, MW/m² - furnace-mean thermal efficiency of deflecting walls Krasnoyarsk Institute of Non-Ferrous Metals, Siberian Branch of the All-Union Heat Engineering Institute, Krasnoyarsk. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 64, No. 3, pp. 275–278, March, 1993.     

Similarity solutions in axisymmetric mixed-convection boundary-layer flow

The similarity equations for mixed-convection axisymmetric boundary-layer flow are considered. The equations involve a buoyancy parameter and a curvature parameter . The equations are solved numerically and it is found that for large , and of O(1), an asymptotic solution is approached, the nature of which is discussed. When is also large, of O(^1/4), the problem, at leading order, becomes independent of the mainstream and the free-convection limit is obtained. This problem is also discussed, including the behaviour for large values of ₀, the free-convection curvature parameter. For < 0 we find that the solution can be continued past the point where the wall heat transfer becomes zero (where previous mixed-convection similarity solutions in plane geometry were terminated) with the solution ending as 0^–. The nature of this limit is also discussed. For < 0 it is also found that there are solutions only in _b = < 0 with two branches of solution bifurcating out of = _b , and values of _b are computed for a range of . The behaviour of the solution for large values of the curvature parameter , and of O(1), is discussed where it is shown that the solution proceeds in inverse powers of log .     

The athermal α → ω transformation in Zr-2 at% Nb alloy

The athermal transformation in Zr-2 at.% Nb alloy has been investigated by transmission electron microscopy. Analysis of the selected-area diffraction pattern has shown that the orientation relationships between the omega and the parent-phase in quenched Zr-2 at.% Nb alloy are the same as have been previously observed for the reaction in pure zirconium. Thus it was deduced that the direct transition has taken place in the alloy during cooling. The-originated -particles were visualized using the dark-field technique. The formation of the athermal omega in the-region of-stabilized Zr-Nb alloy is discussed in terms of the relative positions of the free energy equilibrium curvesT ₀ ,T ₀ ,T ₀ and the correspondingM _s ,M _s andT _s start curves. It is concluded that the omega phase can occur over a much wider range of alloy compositions than is usually recognized on the basis of transformation data.