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)     

Effect of magnesium content on the stress exponent and effective stress in the steady state creep of Al-Mg alloys

The stress exponent of steady state creep,n, and the internal ( _i) and effective stresses ( _e) have been determined using the strain transient dip test for a series of polycrystalline Al-Mg alloys creep tested at 300° C and compared with previously published data. The internal or dislocation back stress, _i, varied with applied stress,, but was insensitive to magnesium content of the alloy, being represented by the empirical equation _i=1.084 ^1.802. Such an applied stress dependence of _i can be explained by using an equation for _i of the form _i (dislocation density)^1/2 and published values for the stress dependence of dislocation density. Values of the friction stress, _f, derived using the equation _e/=(1–c) (1– _f/), indicate that _f is not dependent on the magnesium content. A constant value of _f can best be rationalized by postulating that the creep dislocation structure is relatively insensitive to the magnesium content of the alloy.On leave from Engineering Materials Department, University of Windsor, Windsor, Ontario N9B 3P4, Canada.     

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.     

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.     

Fatigue properties of pure metals

The report considers the stress fatigue limit _D, the fatigue to tensile strength ratio _D/R_m, and the deformation fatigue limit _D of pure metals. For FCC and some HCP metals there exists a linear correlation between fatigue limit and tensile strength. The fatigue limit and the fatigue ratio _D/R_m depend upon the homologous temperature. The fatigue ratios of BCC metals are higher than the fatigue ratios of other metals at room temperature as is also true for metals with nearly the same homologous temperature. The deformation fatigue limit decreases rapidly when T_h - 0.5 and T_h 1. When T_h = 1 then _D = 0, _D = 0 and _D/R_m = 0. The deformation fatigue limits of metals belonging to the same subgroup of the periodic element system and having, similar homologous temperatures at room temperature are very similar.

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Zusammenfassung Der Artikel behandelt die Dauerfestigkeit _D des Ermuedungskoeffizienten _D/R_m und die Deformation _D, die der Dauerfestigkeit entspricht. Fuer kubisch flachzentrierte Metalle und einige hexagonal Metalle existiert eine lineare Korrelation zwischen Dauerfestigkeit _D and Zugfestigkeit R_m. Die Dauerfestigkeit _D und der Ermuedungskoeffizient _D/R_m koennen als Funktion der homologischen Temperatur betrachtet werden. Der Ermuedungskoeffizient der kubisch raumzentrierten Metalle ist groesser als der der anderen in Raumtemperatur. Dieses stimmt auch fuer Metalle mit aehnlichen hornologischen Temperaturen. Die Deformation _D, die der Dauerfestigkeit entspricht, faellt schnell ab, wenn T_h > 0.5 und T_h 1, Wenn T_h = 1, darn ist _D = 0, _D = 0 und _D/R_m = 0. Die Metalle, die zu derselben Gruppe des periodischen Systems der Elemente gehoeren and aehnliche homologische Temperaturen (in Raumtemperatur) besitzen, haben sehr aehnliche Werte der Deformation _D.

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Résumé L'article parle de la contrainte de la limite de fatigue _D le facteur de fatigue _D/R_m et la déformation _D correspondante à la limite de fatigue des métaux pures. Pour les métaux du système cubique aux faces centrées (c. f. c.) et quelques uns du système hexagonale (hex. c.) il existe une correlation linéare entre la limite de fatigue _D et de la résistance a la fraction R_m. La limite de fatigue _D et le facteur de fatigue _D/R_m dependent de la température homologue. Les facteurs de la fatigue des métaux du système cubique centre (c.c.) sort plus grands que ceux des autres métaux dans la température de chambre, aussi por des métaux qui ont presque la même température homologue. La déformation correspondante à la limite de fatigue tombe rapidement quand T_h > 0.5 et T_h 1. Quand T_h = 1, alors _D = 0, _D = 0 et _D/R_m = 0. Cettes déformations _D pour les métaux du même groupe du système périodique des éléments sont proches si les températures homologues dans la température de chambre sont analogues.

     

Ionic and electronic conductivity in some simple lithium salts

The electrical conductivity () and thermoelectric power (S) of Li₃VO₄, Li₃PO₄ and Li₃BO₃ solidified melts are presented in the temperature range from 415 K to the melting point of each solid. The ionic ( _i,) and electronic ( _e) contributions to have been separated over the entire temperature range with the help of a time-dependence study of the d.c. electrical conductivity. Superionic phases in all three solids have been observed below their melting points in which the conductivity is almost purely ionic. The value of the phase transition temperature below which the solid transforms from superionic to normal phase has been obtained. It has been shown that in the normal phase, these solids are mixed conductors. Data for the temperature variations of both _i, and _e are also presented and discussed.     

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.     

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.     

Thermal conductivity of argon and argon-neon mixtures at high temperature

The thermal conductivity of neon and argon-neon mixtures is studied in the temperature range 400–1500°K. This is the first recording of such data above 793°K.Notation T_g true temperature drop in gas layer °K - ¯T temperature, °K, Q, effective thermal flux, W - Q_t, Qr thermal flux transmitted by thermal conductivity and radiation, respectively, W - T_sh correction for temperature shift, % - thermal conductivity of gas mixture, W/m·°K - x_i a molar concentration of neon - _i, _i, _ij, _ij potential function parameters for inter molecular interaction of homogeneous and inhomogeneous molecules - slope of exponential repulsion term Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 33, No. 5, pp. 848–856, November, 1977.     

Frequency and temperature dependence of the millimeter wave conductivity of epitaxial YBa2Cu3O7 films

A millimeter wave spectrometer for frequencies between 100 and 350 GHz consisting of continuously tunable backward wave oscillators as sources and a quasioptical interferometer in the Mach-Zehnder configuration was used to measure the transmittivity in phase and amplitude of YBa₂Cu₃O₇ thin films on NdGaO₃ substrates. From the measured spectra we derived the real and imaginary part of the dynamic conductivity= ₁+i ₂ in the superconducting state as a function of temperature. The ₁(T) and ₂(T) values at 300 GHz were compared to corresponding values at 19 GHz determined by surface impedance measurements of the same films using a shielded dielectric resonator. Our observed frequency dependence of both ₁(T) and ₂(T) is consistent with a strong reduction of the quasiparticle scattering rate ^–1(T) with decreasing temperature belowT _c .     

A note on the resistance to brittle fracture in various ceramic materials

The square of the ratio of the abraded bending strength, _d, to the unabraded bending strength, , is proposed as a measure of the resistance to crack propagation in ceramic materials. Data for various porcelains, glass-ceramics, and glasses showed that _d is essentially constant and that (_d/)² decreased rapidly with increase of the unabraded strength.     

Photoplastic effects in Cd0.2Hg0.8Te

We irradiated Cd_0.2Hg_0.8Te samples at room temperature in the plastic range, with a CO₂ laser beam the wavelength of which (=10 500 nm) is 20% longer than the absorption threshold. We observed a positive photoplastic effect (PPE) of the order _PPE/4 to 5%.     

Electrical conductivity of α-H x V2O5 (X=0.00-0.27): Dependence of hydrogen concentration and orientation

The dependence of the electrical conductivity, , on the hydrogen concentration and crystallographic orientation has been investigated using single crystals of H_xV₂O₅, which were grown by the Bridgeman method and doped with hydrogen within the solid solubility in the phase by the spillover technique. The temperature dependence of showed the feature of diffusive hopping of thermally activated electrons above 180 K and variable range hopping below 180 K. The dependence of on the crystallographic orientation was little different from that of V₂O₅. The change in with the hydrogen concentration was not monotonic; increases withx up tox }-0.06, but decreases abovex }-0.06. This behaviour can be explained based on the competition between the increase in the carrier density and the depression of the mobility of carriers with increasingx.     

Stress relaxation in hot-drawn low density polyethylene

The tensile stress relaxation behaviour of hot-drawn low density polyethylene, (LDPE), has been investigated at room temperature at various draw ratios. The drawing was performed at 85° C. The main result was an increase in relaxation rate in the draw direction, especially at low draw ratios when compared to the relaxation behaviour of the isotropic material. This is attributed to a lowering of the internal stress. The position of the relaxation curves along the log time axis was also changed as a result of the drawing, corresponding to a shift to shorter times. The activation volume, , varied with the initial effective stress ₀ ^* according to ₀ ^* 10kT, where ₀ ^* =₀ — _i, is the difference between the applied initial stress, ₀, and the internal stress _i. This result supports earlier findings relating to similarities in the stress relaxation behaviour of different solids.     

Study of dielectric and electrical properties of mortar in the early hydration period at microwave frequencies

The dielectric constant, , and electrical conductivity, , of mortars with various sand-cement ratios,s/c, were measured for the first 30 h hydration using microwave techniques in the frequency range 8.2–12.4 GHz. The and of the mortars were found to increase linearly with increasing water-solid ratiow/(s + c), but decrease with increasings/c. It was found that as long as thes/c values were the same, the rate of changes in and of the mortars were the same. It appears that thes/c is the key factor controlling the rates of changes in dielectric and electrical parameters of cement hydration in mortar. The relationship between compressive strength and dielectric and electrical properties of mortars was also discussed.     

Shock response of stainless steel at high temperature

Measurements of the dynamic tensile strength of HR-2 (Cr-Ni-Mn-N) stainless steel have been carried out over the initial temperature range of 300 K–1000 K at shock stress of 8 GPa, the corresponding spall strength _f and Hugoniot elastic limit _HEL are determined from the wave profiles. In the temperature range of 300 K–806 K, _f and _HEL decrease linearly with increasing temperature T, i.e., _f = 5.63-4.32 × 10^–3T, _HEL = 2.08-1.54 × 10^–3T, but when heated to 980 K, _HEL increases from 0.84 GPa at 806 K to 0.93 GPa at 980 K and _f keeps at an almost fixed value of 2.15 GPa. The TEM analysis on recovery samples identified the existence of intermatallic compound Ni₃Al and the carbide Cr₂₃C₆ in the sample of 806 K, another intermatallic compound Ni₃Ti was found in the sample of 980 K. All these products emerge along crystal boundary. While no such products were found in the samples of 300 K and 650 K.     

Transport property and battery discharge characteristic studies on 1−x(0.75Agl∶0.25AgCl)∶ xAl2O3 composite electrolyte system

Various experimental studies on a new fast Ag⁺ ion-conducting composite electrolyte system: (1–x) (0.75Agl0.25AgCl)xAl₂O₃ are reported. Undried Al₂O₃ particles of size <10 m were used. The conventional matrix material Agl has been replaced by a new mixed 0.75Agl0.25AgCl quenched and/or annealed host compound. Conductivity enhancements 10 from the annealed host and 3 times from the quenched host obtained for the composition 0.7(0.75Agl0.25AgCl)0.3Al₂O₃, can be explained on the basis of the space charge interface mechanism. Direct measurements of ionic mobility as function of temperature together with the conductivity were carried out for the best composition. Subsequently, the mobile ion concentration n values were calculated from and a data. The value of heat of ion transport q* obtained from the plot of thermoelectric power versus 1/T supports Rice and Roth's free ion theory for superionic conductors. Using the best composition as an electrolyte various solid state batteries were fabricated and studied at room temperature with different cathode preparations and load conditions.     

Effect of the temperature dependence of thermal properties on the thermal shock tests of ceramics

Thermal stress generated during a thermal shock is closely related to the fracture of ceramics. An attempt has been made to obtain thermal stress in a specimen by numerical calculation. The temperature dependence of thermal conductivity and diffusivity were introduced to realize the practical thermal conditions. The maximum thermal stress, _max ^* , was recognized at the Fourier number, but differed from the temperature dependence. Correlative equations of _max ^* and _max ^* with the Biot number, _i, under cooling or heating tests, have been proposed. These equations resulted in the exact _max ^* and _max ^* compared with the previous equations, in which temperature dependence was ignored. The thermal shock resistance parameter was expressed by the correlative equations of _max ^* in order to suggest adequate experimental conditions and specimen size. A comparison of the measured and calculated time to failure of the specimen led to confirmation of the fracture criterion. The measured time disagreed with the calculated one, if the fracture by thermal shocking was not predominant. The correlative equations were also useful to select the kind of ceramics subjected to thermal shocking.     

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     

Stress-strain behaviour of nitrogen bearing austenitic stainless steels in the temperature range 298–473 K

The stress-strain behaviour of three nitrogen-bearing low-nickel austenitic stainless steels has been investigated via a series of tensile tests in the temperature range 298–473 K at an initial strain rate of 1.6×10^–5s^–1. Experimental stress-strain data were analysed employing Rosenbrock's minimization technique in terms of constitutive equations proposed by Hollomon, Ludwik, Voce and Ludwigson. Ludwigson's equation has been found to describe the flow behaviour accurately, followed by Voce's equation. The resultant strain-hardening parameters were analysed in terms of variations in temperature. A linear relationship between ultimate tensile stress and the Ludwigson parameters has been established. The influence of nitrogen on the Ludwigson modelling parameters has also been explained.Nomenclature True stress - _t True strain - _f True fracture strain - Strain rate - T Temperature - K _H, n _H Hollomon parameters - K _L, n _L Ludwik parameters - K _1L, k _2L, n _1L, n _2L Ludwigson parameters - _s, K _V, n _V Voce parameters - _{u relation} Uniform strain computed from a particular relation - _L Transient strain - ₀ Flow stress at zero plastic strain (Ludwik) - _L Transient stress - _y Yield stress - _u Ultimate tensile stress