Effective conductivity of a composite of poly-dispered spherical particles in a linear continuum

Rayleigh's method is used to find the electric potentials of a composite of poly-dispered spherical particles in a linear continuum in an external electric field. Based on the solutions of potentials, analytical formula for the effective electric conductivity is derived. Based on the formula, several factors, such as the number of spherical inclusions, the spatial distribution of the spheres, the contrast ratio _i / _h (where, _i and _h are the conductivities of the spherical inclusion and the host medium, respectively) and volume fraction of the inclusions, are discussed. Our results show that at high volume fraction, the effective conductivity is also affected by the spatial distribution of the inclusions.     

Crack-tip damaged zones in rubber-toughened amorphous polymers: a micromechanical model

The various stages of crack propagation in rubber-toughened amorphous polymers (onset and arrest, stable and unstable growth) are governed by the rate of energy dissipation in the cracktip damaged zone; hence the relationship between the applied stress intensity factorK ₁ and the damaged zone size is of utmost importance. The size of the crack-tip damaged zone has been related toK ₁ via a parameter which is characteristic of the material in given conditions: this factor is proportional to the threshold stress for damage initiation in a triaxial stress field, and has been denoted by ^*. Theoretical values of ^* have been calculated by means of a micromechanical model involving the derivation of the stresses near the particles and the application of damage initiation criteria. The morphology, average size and volume fraction of the rubbery particles have been taken into account together with the nature of the matrix. The calculated values of ^* have been successfully compared with the experimental ones, for a wide set of high-impact polystyrenes (HIPS) and rubber-toughened poly(methyl methacrylate) (RTPMMA).Nomenclature PS; HIPS polystyrene; high-impact polystyrene - PMMA; RTPMMA poly(methyl methacrylate); rubber-toughened PMMA - MI; CS/H; CS/R particle morphologies (multiple inclusion; hard core - rubber shell; rubber core - rigid shell) - K _r;K _g bulk moduli of rubber and glassy materials - G _r;G _g shear moduli of the same materials - v _p particle volume fraction - L mean centre-to-centre distance between neighbouring particles - B; H; W standard names for the dimensions of the compact tension specimen - R _y size of the crack-tip plastic zone in a homogeneous material - h half thickness of the crack-tip damaged zone - r; polar coordinates around the crack tip (Fig. 1) - r;r _p distance from particle centre; particle radius - p normalized distance from the particle (Equation 5) - K ₁;K _1c;K _1p stress intensity factor; critical values ofK ₁ at the onset of and during crack growth - G _1c plane strain energy release rate - _y yield stress in uniaxial tension - _th macroscopic threshold stress for the onset of local damage initiation in a composite material - ^* characteristic parameter (Equation 3) - ⁰; ₁ ⁰ ; ₂ ⁰ ; ₃ ⁰ applied stress tensor and its three principal stresses - ⁰ uniaxial applied stress - ; ₁; ₂; ₃ local stress tensor and its three principal stresses - A tensor which elements are the ratios of those of over those of ⁰ (Equation 4) - v Poisson's coefficient of the matrix - g triaxiality factor of the crack-tip stress field - _e; _p Mises equivalent stress; dilatational stress (negative pressure) - I ₁;I ₂ invariants of the stress tensor - U ₁;U ₂ material parameters for argon and Hannoosh's craze initiation criterion (Equation 12)     

Fracture stress difference of notched polycarbonate between atmospheric pressure and a hydrostatic pressure

Experimental data on fracture stress of polycarbonate (PC) with and without various artificial notches have been obtained at atmospheric pressure and a high hydrostatic pressure (400 MPa). The difference in fracture stress, _F, between both pressures was directly proportional to the intensity of pressure,P, and was inversely proportional to the stress concentration factor of the notch,K _n such that _F following the form of the Kaieda-Oguchi formula, _F. By using the combined stress concentration factor,K _nc, of superposed notch and craze, and by considering the change in elastic modulus due to pressure, the experimental data agreed with the modified Kaieda-Oguchi formula. The stress concentration factor of the craze was calculated by using the Dugdale model.     

Notch tip stress distribution in strain hardening materials

Using the results of elastic-plastic stress analyses for notched bars, it is shown that a modified form of slip-line field solution can satisfactorily explain the variation of longitudinal stress ahead of notch tips in strain hardening materials.

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Résumé En utilisant les résultats d'analyses de contrainte élastoplastique dans le cas de barres entaillées, on montre qu'il est possible d'utiliser une forme simplifiée de solution du champ des lignes de glissement pour expliquer de façon satisfaisante la variation des contraintes longitudinales en avant d'extrémités d'entaille dans des matériaux susceptibles d'un écrouissage.

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Nomenclature _yy longitudinal tensile stress in the notch tip plastic zone - _xx transverse stress in the x-direction - _zz transverse stress in the z-direction - k yield stress in shear - ₀ yield stress in tension - ₀ ^* strain hardened yield stress (flow stress) - _{0/* c} flow stress at notch tip - _total total strain _pl plastic strain _l principal strain - _{1 c} maximum principal strain at notch tip - _1pl plastic strain in they-direction - _{1 cp1} E_{1 pl} at notch tip - _eff effective plastic strain - _{c eff eff} at notch tip - ₀ yield strainC Stress decay constant in the notch tip region - /e_pl linear strain hardening rate - n strain hardening exponent in power hardening law - 2 flank angle of notch - distance from notch tip - p notch tip radius - k _I applied stress intensity for Mode I loading - E Young's modulus - V _c crack tip opening displacement     

Charged defects-controlled conductivity in Ge-ln-Se glasses

The variation of the d.c. electrical conductivity, , with composition and temperature was investigated for glasses of the Ge-In-Se system. The results indicate a decrease in the activation energy for electrical conductivity, E, and an increase in on introduction of indium into Ge-Se glasses. The changes in E and with composition (selenium content in the glasses) are identical for the Ge_x In₅ Se_95–x and Ge_x In₈Se_92–x families. The results have been traced to the conduction controlled by charged defects in these chalcogenide glasses. The changes in E and have been explained by a shift in the Fermi level, being brought by the introduction of indium.     

On the measurement and physical meaning of the cleavage fracture stress in steel

Elastic-plastic two-dimensional (2D) and three-dimensional (3D) finite element models (FEM) are used to analyze the stress distributions ahead of notches of four-point bending (4PB) and three-point bending (3PB) specimens with various sizes of a C-Mn steel. By accurately measuring the location of the cleavage initiation sites, the local cleavage fracture stress _f and the macroscopic cleavage fracture stress _F is accurately measured. The _f and _F measured by 2D FEM are higher than that by 3D FEM. _f values are lower than the _F, and the _f values could be predicted by _f=(0.8––1.0)_F. With increasing specimen sizes (W,B and a) and specimen widths (B) and changing loading methods (4PB and 3PB), the fracture load P _f changes considerably, but the _F and _f remain nearly constant. The stable lower boundary _F and _f values could be obtained by using notched specimens with sizes larger than the Griffiths–Owen specimen. The local cleavage fracture stress _f could be accurately used in the analysis of fracture micromechanism, and to characterize intrinsic toughness of steel. The macroscopic cleavage fracture stress _F is suggested to be a potential engineering parameter which can be used to assess fracture toughness of steel and to design engineering structure.     

Optimal design of a thick-walled pipeline cross-section against creep rupture

Summary Cylinder under combined loadings (pressure, bending, axial force) is subject to non-linear creep described by Norton-Odqvist creep law. In view of bending a circularly-symmetric cross-section is no longer optimal in this case. Hence we optimize the shape of the cross-section; minimal area being the design objective under the constraint of creep rupture. Kachanov-Sdobyrev hypothesis of brittle creep rupture is applied. The solution is based on the perturbation method (expansions into double series of small parameters), adjusted to optimization problems.Notation A cross-sectional area - C, , creep rupture constants - K, n, _C , _C creep constants - F dimensionless creep modulus - M bending moment - N axial force - a(),b() internal and external radii of the cross-section - j creep modulus - p internal pressure - r, ,z cylindrical coordinates - s _r ,s ,s _z ,t _r dimensionless stresses - t _R time to rupture - stress function - , () dimensionless internal and external radii - _e effective strain rate - _kl strain rates - rate of curvature - rate of elongation of the central axis - dimensionless radius - _e effective stress - _I maximal principal stress - _S Sdobyrev's reduced stress - _r , , _z , _r components of the stress tensor - measure of material continuity - measure of deterioration With 7 Figures     

Hydrodynamics of rising bubbles and drops

Conclusion An analysis of the problem concerning the interaction of moving single formations (bubbles and drops) with the surrounding carrying medium shows the possibility of describing real physical processes within the frameworks of the models of ideal and viscous liquids. Generalization of experimental and theoretical data is made by constructing charts of flow modes. A successful selection of the dimensionless numbersR andR as coordinates made it possible to arrange in an ordered fashion in the diagram all of the media by means of the parameter M, with the data for each specific medium being located on a certain straight line whose slope is determined by M. Vast computational data both of the present author and of other investigators for the region of small and intermediate values of dimensionless parameters, converted into theR ,R functions, made it possible to fully display the region inaccessible for investigation when the effect of all the parameters is significant and their number cannot be reduced. The use of these very parameters for correlating the data of the problem of steady rise of a drop made it possible to simply demonstrate the influence both of the presence of the medium itself and its motion on the drop rise. Comparison of the data with the problem of bubble rise can be made by simple superposition of diagrams. The logical clarity of the parametersR , which is equal to the ratio of the equivalent radiusa of a bubble (drop) to the capillary constant of the surrounding medium , allowed the determination of a certain critical sizea after which waves appear on the surface at the back side of the bubble (drop) when the value ofa is higher than 2-3. Another form of instability of steady rise is associated with the formation of a stagnant zone and with the separation of flow from the surface of a bubble or drop characteristic for surrounding media with small values of M. The use of the parametersR andR will definitely significantly simplify the analysis of more complex problems, for example, the problems of the motion of bubbles and drops in tubes with liquid and also the change in the external conditions (reduced or increased gravitation).Institute of Thermophysics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 66, No. 1, pp. 93–123, January, 1994.     

The influence of grain size on the creep of uranium dioxide

The creep of uranium dioxide has been investigated as a function of grain size. At high stresses, when creep is controlled by dislocation movement, grain boundaries exert a strengthening effect and this strengthening is correlated with the Hall-Petch equation. The degree of strengthening diminishes with increases in temperature. At lower stresses, when creep is controlled by mass transport, grain boundaries exert a weakening effect owing to the reduction in diffusion path length as grain size is reduced. In this range behaviour is correlated with the Nabarro-Herring equation with stress replaced by an effective stress _E=–₀ where ₀ is a threshold stress for diffusional creep associated with the limitation of the ability of boundaries to emit and absorb vacancies. ₀ appears to decrease as grain size is increased.     

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     

Dynamic emission of dislocations from a crack tip — a computer simulation

The behavior of dynamic emission of dislocations from the tip of a stationary crack under mode II or mode III loading is examined. A critical stress intensity factor, K _D is assumed for dislocation emission. After emission, the dislocation moves with a velocity which varies with the effective shear stress to the third power. The effective shear stress is due to the applied stress , modified by the presence of the crack and all other dislocations minus the lattice friction stress, ^F. The effects of K _D, , and ^F on the rate of emission, the plastic zone strain rate, the plastic zone size, the dislocation distribution, and the dislocation-free zone are reported.

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Résumé On a examiné le comportement d'une émission dynamique de dislocations depuis l'extrémité d'une fissure stationnaire. On suppose que l'émission de dislocations est caractérisée par un facteur critique d'intensité de constrainte K _D. Après son émission, la dislocation se meut avec une vitesse qui varie avec le cube de la constrainte effective de cisaillement. Celle-ci résulte de la constrainte appliquée , modifiée par la présence de la fissure et de toutes les autres dislocations, et sous déduction de la contrainte de friction du réseau, ^F.On étudie les effects de K _D, et ^F sur la vitesse d'émission des dislocations, la vitesse de déformation plastique, la taille de la zone plastique, la distribution des dislocations, et sur la zone qui en est dépourvue.

     

The influence of antimony-additions on the creep behaviour of a 25wt% Cr-20wt% Ni austenitic stainless steel

The effect of antimony on the creep behaviour (dislocation creep) of a 25 wt% Cr-20 wt% Ni stainless steel with ~ 0.005 wt% C was studied with a view to assessing the segregation effect. The antimony content of the steel was varied up to 4000 ppm. The test temperature range was 1153 to 1193 K, the stress range, 9.8 to 49.0 MPa, and the grain-size range, 40 to 600m. The steady state creep rate, , decreases with increasing antimony content, especially in the range of intermediate grain sizes (100 to 300m). Stress drop tests were performed in the secondary creep stages and the results indicate that antimony causes dislocations in the substructure to be immobile, probably by segregating to them, reducing the driving stress for creep.Nomenclature _a Creep stress in a constant load creep test without stress-drop - _A Initial applied stress in stress-drop tests - Stress decrement - ( _A-) Applied stress after a stress decrement, - t _i Incubation time after stress drop (by the positive creep) - _C Strain-arrest stress - _i Internal stress - _{s s}-component (= _i- _c) - Steady state creep rate (average value) in a constant load creep test - Strain rate at time,t, in a constant load creep test - New steady state creep rate (average value) after stress drop from _A to ( _A-) - Strain rate at time,t, after stress drop.     

Absence of Phase-Fluctuation Contribution to the Low-Frequency Optical Conductivity in d-Wave Superconductors at Zero Temperature

The behavior of the low-frequency optical conductivity _reg() in superconducting cuprates is, at the present, an open and interesting issue. In particular, since the zero-temperature and zero-frequency limit of _reg() attains a value much larger than the universal value expected within a self-consistent T-matrix calculation, an intriguing possibility is that the collective mode can also contribute to _reg(). By taking into account the effect of dissipation on the collective mode in a d-wave superconductor, we evaluate the phase-fluctuation contribution to _reg(0) within the formalism of the phase-only action. We show that even though the collective mode contributes to _reg() at finite frequencies, approaching the zero-temperature and zero-frequency regime the corrections at _reg() due to phase fluctuations vanish.     

A study of the Bailey-Orowan equation of creep

A new method was developed to study the Bailey-Orowan equation of creep, _c=r/h, where _c is the creep rate,r is the recovery rate andh is the work-hardening coefficient. The method was to vary the strain rate,, around the creep rate, _c, and to measure the corresponding stress rate,. In a plot of stress rate against strain rate, a straight line was obtained. The slope of the straight line was equal toh, and the intersection of the straight line with the stress axis was equal to –r, as in the equation=–r+h. The creep test under a constant stress is a special case of this equation when the stress rate,, is zero. The above measurement was carried out within a very small stress variation, less than 1% of the total stress, so that the values ofr andh were not disturbed. The creep test was performed on Type 316 stainless steel. The creep rate was shown to be equal to the ratior/h, but the value ofh was approximately equal to Young's modulus at the testing temperature, rather than, as is commonly believed, to the work-hardening coefficient.     

On internal stress and activation volume in polymers

The two-site model is developed for the analysis of stress relaxation data. It is shown that the product of d In (– )/d and (- _i) is constant where is the applied stress, _i is the (deformation-induced) internal stress and = d/dt. The quantity d In ( )/d is often presented in the literature as the (experimental) activation volume, and there are many examples in which the above relationship with (- _i) holds true. This is in apparent contradiction to the arguments that lead to the association of the quantity d In (– )/d with the activation volume, since these normally start with the premise that the activation volume is independent of stress. In the modified theory presented here the source of this anomaly is apparent. Similar anomalies arise in the estimation of activation volume from creep or constant strain rate tests and these are also examined from the standpoint of the site model theory. In the derivation presented here full account is taken of the site population distribution and this is the major difference compared to most other analyses. The predicted behaviour is identical to that obtained with the standard linear solid. Consideration is also given to the orientation-dependence of stress-aided activation.     

Distribution characteristics of mechanical properties and correlation between the respective properties on S35C carbon steel

Mechanical properties of tensile strength, , upper yield stress, _SU, lower yield stress, _SL elongation, , area reduction, , Vickers hardness, H _v, and impact absorbed energy, E, were examined using 50 specimens of S35C carbon steel, which were machined from two bars supplied from the same charged and heat-treated material. Distribution characteristics of these properties are discussed, and the correlation between each pair of them is investigated from a statistical viewpoint. The main conclusions obtained are summarized as follows; distribution characteristics of _B, _SL, , , H _v and E are well approximated by a normal distribution, but those of asu are not approximated as well by this type of distribution. In the latter case, a Weibull distribution is preferable to represent the distribution pattern. No significant correlation was observed between each pair of the above mechanical properties. Consequently, individual properties have the inherent distribution characteristics independent of the other properties.     

Fatigue of metallic glasses

The fatigue behaviour of Ni₄₉Fe₂₉P₁₄B₆Si₂, Ni₄₈Fe₂₉P₁₄B₆Al₃ and Pd_77.5Cu₆Si_16.5 metallic glasses is examined. In the finite lifetime regime the relationship between stress amplitude ( _a), fracture stress ( _f), mean stress ( _m) and cycles to failure (N _f) is _a=A( _f–_m) (2N _f) ^b , whereA andb are 16.9 and –0.40 respectively for reduced gauge section Ni₄₉ strips (for _m 140 kg mm^–2) and 27.0 and –0.44 for Pd base wires. These results are unusual in thatA 1. Consequently, a sharp discontinuity exists near _a( _f– _m) –1. In a simple tensile test failure occurs at _f(=_y) and 2N_f=1; for peak stresses only a percent or so less than _f the sample will withstand hundreds of cycles of stress. For uniform cross-section glassy metal filaments, a fatigue limit is observed at stress ratios ( _a/ _f) in the vicinity of 0.07 to 0.15. The fatigue limit for reduced section specimens is a factor of 2 higher. Fatigue failure of the Ni-Fe strips may occur under partially or fully plane stress or plane strain conditions, depending on sample thickness and stress. Final failure of the Pd_77.5Cu₆Si_16.5 wires always occurs by general yielding of the remaining section.     

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.     

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.