Crack layer analysis of fatigue crack propagation in rubber compounds

A methodology to characterize the resistance of rubber compounds to crack propagation (fracture toughness) is presented. A constitutive model based on the crack layer theory is utilized to extract the specific energy of damage ^*, a material parameter characteristic of the material's resistance to crack propagation and the dissipative characteristic, . The model expresses the rate of crack propagation as% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaamaalaaabaGaam% izaGqaciaa-fgaaeaacaWGKbGaa8Ntaaaaaaa!3AFA!\[\frac{{da}}{{dN}}\]= % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaeq% OSdiMaamOsamaaDaaaleaacaaIXaaabaGaaGOmaaaaaOqaaiaadMha% caGGQaGaamOuamaaBaaaleaacaaIXaaabeaakiabgkHiTiaadQeada% WgaaWcbaGaaGymaaqabaaaaaaa!41A5!\[\frac{{\beta J_1^2 }}{{y*R_1 - J_1 }}\]where da/dN is the cyclic rate of fatigue crack propagation (FCP), J ₁ is the energy release rate (tearing energy) and R ₁ is the resistance moment which accounts for the amount of damage associated with the crack advance. Microscopic examination revealed that crack tip microcracking is the dominant damage mechanism. Hence, R ₁ was evaluated as the area (m²) of microcracking surfaces per unit crack advance.Fatigue crack propagation data for a particular rubber compound have been analyzed using the present model. The proposed equation describes the entire FCP history in the compound. According to this model, ^* and for the compound investigated, are found to be 9.3 kJ m^-2 and 9.7×10^-9 m⁴/J-cycle, respectively.

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Résumé On présente une méthodologie pour caractériser la résistance de composés de caoutchouc à la propagation des fissures du point de vue de la ténacité à la rupture. Un modèle constitutif basé sur la théorie de la couche de fissuration est utilisé pour obtenir l'énergie spécifique d'endommagement ^*, un paramètre du matériau représentatif de sa résistance à la propagation d'une fissure, et une caractéristique de dissipation . Le modèle exprime la vitesse de propagation d'une fissure de fatigue par cycle da/dN en fonction de ces deux paramètres, de la vitesse de relaxation de l'énergie de cisaillement J ₁, et du moment résistif R ₁ qui tient compte de état de l'endommagement associé à la progression de la fissure. Un examen microscopique révèle que la microfissuration à l'extrémité de la fissure est le mécanisme déterminant de l'endommagement. Dès lors, on évalue R ₁ en fonction de l'aire de microfissuration (en m²) par unité de progression de la fissure.Des données de propagation de fissure de fatigue sont analysées à l'aide du présent modèle pour un composé de caoutchouc particulier. L'équation proposée décrit l'entièreté de la propagation de la fissure dans le composé. Des valeurs numériques pour ^* et pour de respectivement 9,3 kJ m^-2 et 9,7×10^-9 m^-4/J-cycle sont trouvées.

     

Oxidation of an Al2O3-γAlON ceramic composite

The oxidation of dispersed aluminium oxynitride particles in an alumina matrix has been studied. The kinetics law of this reaction is linear and the activation energy is 420±40 kJ mol^–1 (100±10 kcal mo^–1). A-alumina layer is formed and leads to-alumina above 1200° C. The-alumina formation produces surface compressive stresses, and thus the mechanical properties ( _f, HV) are improved. We have proved that the formation of-alumina in the Al₂O₃-AION composite can lead to the best properties for this ceramic. A-alumina layer has a very interesting effect on the wear resistance of this material.     

Temperature coefficients of capacitance of solids

An earlier treatment of temperature coefficient of capacitance, _c [1] has been extended to include most solids. Materials are divided into those with given ranges of permittivity, , and temperature coefficient of polarisability. It appears that, for low dielectric loss, high permittivity glasses, like simple ionic compounds, always have a positive _c, whereas paraelectrics and polymers have negative _c. Ferroelectrics can have any value of _c. Limitations in _c and for given classes of solid are discussed.     

Nonlinear dynamic response of isotropic thin rectangular plates on elastic foundations

Summary An approximate analytical solution of the large deflection dynamic response of isotropic thin rectangular plates resting on Winkler, Pasternak and nonlinear Winkler foundations is presented. Von Kármán type governing equations in terms of the transverse deflection and stress function are employed. The deflection is approximated by a one term shape function satisfying the boundary conditions. The Galerkin's method is used to get the differential equation for the deflection at the centre. Closed form solutions are presented for the nonlinear free vibration response and for the responses under uniformly distributed static and step function loads. Response under sinusoidal pulse load is also obtained. Clamped and simply supported plates with movable and immovable inplane conditions at the edges are considered.Notations A Amplitude - A _i Coefficients in differential equation for ø - a, b, h Sides of plate alongx andy directions and its thickness - c,c Damping factor, dimensionless damping:c( ^* ha ⁴/D)^1/2 - D Flexural rigidity:Eh ³/[12(1-v ²)] - E, v Young's modulus, Poisson's ratio - F Stress function - g, k, k ₁ Foundation parameters - G, K, K ₁ Dimensionless foundation parameters:ga ²/D, ka ⁴/D, k ₁ a ⁴ h ²/D - H Depth of single layer foundation - I, M Immovable and movable inplane conditions at the edges - N _x ,N _y ,N _xy Inplane stress resultants - P _x ,P _y Resultant forces on an edge inx andy directions - q, Q, Q _o Uniformly distributed load,Q=qa ⁴/Eh ⁴, step load - S, C Simply supported and clamped edges - t, Time, dimensionless time: = [D/ ^* ha ⁴]^1/2 t - T, T _o Period for amplitudeA, linear period - u, v; w Inplane displacements; transverse displacement - x, y Rectangular Cartesian co-ordinates - ₁, ₃ Linear and cubic parameters of static response - , ₀; ^* Mass densities of plate and foundation; effective mass density - Nonlinearity parameter of nonlinear vibrations - Aspect ratio of the plate;a/b - × Damping ratio - , _m Central deflection, maximum central deflection - ₀ ^*, ₀ Linear frequency, dimensionless linear frequency: ₀ ^*[ ^* ha ⁴/D]^1/2 - (·) ()/() With 6 Figures     

Nonlinear axisymmetric response of orthotropic thin truncated conical and spherical caps

Summary An approximate analytical solution of the large deflection axisymmetric response of polar orthotropic thin truncated conical and spherical shallow caps is presented. Donnell type equations are employed. The deflection is approximated by a one term mode shape satisfying the boundary conditions. The Galerkin's method is used to get the governing equation for the deflection at the hole. Nonlinear free vibration response and the response under uniformly distributed static and step function loads are obtained. The effect of various parameters is investigated.Notations A, A ^* Inward and outward amplitudes - a, b, h Base radius, inner radius and thickness of the cap - D M h ³/[12(–v ² )] - E ,E Young's moduli - H ^*,H Apex height, dimensionless apex heght:H ^*/h - N _, Stress resultants - p ^1/2 - q Uniformly distributed load - Q,Q₀ Dimensionless load: , dimensionless step load - Q, Q ₀ Dimensionless load: , step load - t, Time, dimensionless time: t - T _A Ratio of nonlinear periodT for inward amplitudeA and the linear periodT _L - w ^* Normal displacement at middle surface - w Dimensionless displacement:w ^*/h - ₁ Linear parameter of static response - Orthotropic Parameter:E /E - Mass density - ₂,₃ Quadratic and cubic nonlinearity parameters - b/a - v ,v Poisson's ratios - Dimensionless radius:r/a - ^*, Stress function, dimensionless stress function: - ₀ ^* ,₀ Linnear frequency, dimensionless frequency: With 7 Figures     

Correlation of high-pressure thermal conductivity,viscosity, and diffusion coefficients for n-alkanes

Thermal conductivity, viscosity, and self-diffusion coefficient data for liquid n-alkanes are satisfactorily correlated simultaneously by a method based on the hard-sphere theory of transport properties. Universal curves are developed for the reduced transport properties ^*, ^*, and D ^* as a function of the reduced volume. A consistent set of equations is derived for the characteristic volume and for the parameters R , R , and R _D, introduced to account for the nonsphericity and roughness of the molecules. The temperature range of the above scheme extends from 110 to 370 K, and the pressure range up to 650 MPa.     

Improvements in the transverse properties of composites

Fracture surface energies of initiation ( _I ^c) for a transverse fracture process in glass-reinforced epoxy composites have been measured and the results calculated by three different treatments and are compared with the average fracture surface energies ( _F ^c) for the complete fracture process.Changes in these two fracture properties are studied as a function of the volume fraction of the fibres, and the relation between the surface energies is established as a factor which determines the nature of the fracture process. When _I{sic}– _F ^c>0 a catastrophic failure is expected, whereas a controlled fracture is observed for _I ^c– _F ^c<0.     

Generic Guiding Law Between Irreversibility Field and Anisotropy—Chemical Control of Critical Current of High Temperature Superconductors

Systematic studies on the irreversibility field H _irr, and anisotropy factor ² of high temperature superconductors (HTSC) were performed using single crystals with high quality. The generic scaling law have been found to hold for all the HTSC systems examined, i.e., H _irr[Oe]=4×10^7–2 (1–T/T _c)1.5 at T0.7T _c. In addition, of each HTSC material is roughly expressed as 2=2 exp(0.78d[Å]) at the carrier optimally-doped state. Based on the generic scaling law, the behaviors of the variously doped superconductors, such as Bi(Pb)2212 and Hg(Re)1223, are discussed in terms of the critical current.     

Magnetic field dependence of the density of states of YBa2Cu3O6.95: Implications for the vortex structure

The total specific heat of YBa₂Cu₃O_6.95 single crystals includes contributions from phonons and spin-1/2 particles, as well as electronic contributions. The electronic specific heat is described by a quadratic term T² in zero field and a linear term [(0)+(H)]T which is increased when a magnetic field H is applied perpendicular to the CuO₂ planes. In agreement with d-wave superconductivity, we find that _n/T_c and (H)_n(H/H_c2)^1/2, where _n is the coefficient of the normal-state linear term. The H^1/2 dependence of the density of states at the Fermi level was predicted by G. Volovik for lines of nodes in the gap: the quasiparticles which contribute to this density of states are close to the nodes in momentum space and are located outside the vortex core.     

γ ? α2 Phase transformation in fractured high temperature stress rupture Ti-48Al-2Nb(at.%)

The ₂ phase transformation in fractured high temperature stress rupture Ti-48Al-2Nb(at.%) alloy has been studied by analytical electron microscopy. ₂ and phases were found at grain boundaries. ₂ layers that suspended in layers and interfacial ledge higher than 2d ₍₁₁₁₎ at /₂ interfaces were observed in the lamellar grains. These facts indicated that ₂ phase transformation and dynamic recrystallization have occurred during high temperature stress rupture deformation. It can be concluded that deformation induced ₂ phase transformation and dynamic recrystallization resulted in the presence of particles at grain boundaries. A structural and compositional transition area between deformation-induced ₂(or ) and its adjacently original (or ₂) phases was found by HREM and EDS and is suggested as a way to transform between and ₂ phase during high temperature stress rupture deformation. The transition area was formed by slide of partial dislocations on close-packed planes and diffusion of atoms.     

Low-temperature specific heat anomalies in the group V transition metals

Anomalies previously reported in the specific heat curve of normal-state niobium at 3 and 9.5 K prompted new measurements on single crystals of niobium, tantalum, and vanadium from their superconductingT _c 's up to 20 K. The upper anomaly was confirmed in Nb and found to occur at 10.3 K. At this temperature theC/T vs. T ² curve changed abruptly from a line with constants ₂ =7.67 mJ/K ² mole and ₂ =241 K to one with ₃ =9.16 mJ/K ² mole and ₃ =250 K. The NbT _c was 9.275 K. Anomalies similar to that occurring at 3 K in the niobium curve were discovered to exist in tantalum and vanadium as well, but at the higher temperatures of 7.19 and 7.47 K, respectively. The tantalum data yielded line constants of ₁ =5.42 mJ/K mole, ₁ =238 K, ₂ =4.36 mJ/K ² mole, and ₂ =228 K and aT _c of 4.475 K. For vanadium ₁ =397 K is higher than previous specific heat values of ₁ =382 K, and in agreement with that obtained from elastic constant measurements (399 K). The discontinuities in the slopes of the specific heat curves are analyzed in terms of anomalies in the electron and phonon spectra of the materials investigated.     

A process model for active brazing of ceramics

In the present investigation process modelling techniques have been applied to describe reaction layer growth during active brazing of ceramics. As a starting point, the classical solution for parabolic growth of transformation products is considered. Specific computational features are then explicitly built into the model to allow for transient effects during heating and cooling as well as changes in the growth kinetics due to depletion of the active element during brazing. This approach gives considerable scope for optimization of both process and joint properties through adjustment of the filler metal composition and the temperature-time programme under which brazing takes place. The aptness of the process model is illustrated in an accompanying paper (Part II).Nomenclature A, C B reaction products in diffusion couple - C A ceramic component - C reactive element in braze alloy - C _B concentration of element B at a given position within the reaction layer, C B (mole m^–3) - C _B ^b bulk concentration of element B in ceramic, A B (mole m^–3) - C _B ⁱ (1), C _B ⁱ (2) concentration of element B in reaction layer at C B /A B and braze metal/C B interface, respectively (mole m^–3) - C _C concentration of element C at a given position within the reaction layer, C B (mole m^–3) - C _C ⁰ initial concentration of element C in braze metal (mole m^–3) - C _C ^b bulk concentration of element C in braze metal (mole m^–3) - C _C ⁱ (1), C _C ⁱ (2) concentration of element C in reaction layer at braze metal/C B and C B /A interface, respectively (mole m^–3) - D ₀, D ₀ ^* constants in expression for diffusion coefficient (m² s^–1) - D _B intrinsic diffusivity of B in C B (m² s^–1) - D _C intrinsic diffusivity of C in C B (m² s^–1) - J _B molar flux of element B (mole m^–2 s^–1) - J _C molar flux of element C (mole m^–2 s^–1) - k ₀ constant in expression for k_p (m² s^–1) - k ₀ ^* rate constant referring to infinite diffusion couple analogue (m² s^–1) - k _p parabolic growth rate constant (m² s^–1) - L half width of braze metal zone (m) - m proportionality constant (equal to the ratio between C _C ⁱ (1) and C _C ^b ) - Q _app apparent activation energy for diffusion of C in C B (J mole^–1) - Q _app ^* apparent activation energy for diffusion of B in C B (J mole^–1) - R universal gas constant (8.314 J mole^–1 K^–1) - t time (s) - t ₀ incubation time (s) - t ₁, t ₂ limits of integration (s) - t _i isothermal hold time (s) - t _i time increment used in the numerical integration procedure (s) - T absolute temperature (K or °C) - T _c chosen reference temperature (K or °C) - T _i isothermal hold temperature (K or °C) - X thickness of reaction layer (m) - X _c contribution of the cooling leg of the brazing cycle to the total reaction layer thickness (m) - X _h contribution of the heating leg of the brazing cycle to the total reaction layer thickness (m) - X _i contribution of the isothermal hold period to the total reaction layer thickness (m) - X _lim limiting thickness of reaction layer, C B (m) - X _i increase in reaction layer thickness due to a small time increment t _i (m) - y ₁, y ₂, y ₃ molar partitioning factors - , , , , , ) molar stoichiometric factors - molar volume of reaction product, C B (m³ mole^–1)     

A molecular theory of the fracture toughness of low molecular weight polymers

Recent experiments by Robertson show that the fracture toughness G _IC of glassy polystyrene PS does not decrease to the ideal brittle value 2 (where is the surface energy for PS) at molecular weights M _w below M _c the entanglement molecular weight. Instead G _IC is more than an order of magnitude above 2 at M _c and decreases slowly below M _c. It is postulated that a small craze exists at the crack tip in such low molecular weight glassy polymers. However, since entanglements do not occur single molecules must span this craze; if they do not the craze becomes unstable and the crack advances. Under these conditions a critical craze surface displacement exists and G _C can be computed to be G _IC=S _c(–1) R ^21/2, where and S _c are the craze fibril extension ratio and craze surface drawing stress observed in high molecular weight crazes (both quantities should be only weak functions of M _w) and R ^21/2 is the root mean square end-to-end distance of the PS molecule in the glass from neutron scattering measurements. The fracture toughness is predicted to decrease as M _w ^1/2 ; this prediction and the absolute magnitude of G _IC are in excellent agreement with experiment.     

Polymorphism in Sr3Fe2O7−x

The compound Sr₃Fe₂O_7–x , with variable iron valence, was investigated by X-ray powder techniques, both at room and at high temperatures. If the material is examined in massive form, a single phase called -Sr₃Fe₂O_7–x appears as previously reported in the literature. This -phase is tetragonal and exhibits the lattice parameters: a=3.874 and c=40.314 Å. Two other phases, called and -Sr₃Fe₂O_7–x , respectively, can be obtained on heating the finely powdered material when laid on a flat platinum support. The form is stable up to 1275° C, while the form is revealed only above 1275° C and changes always into -Sr₃Fe₂O_7–x when quenched. Both and phases are tetragonal, with a=4.001 and c= 58.251 for the form and a=4.013, c=57.092 Å for the form. The transition involves a true phase equilibrium, while the transformation is possible only by means of a suitable mechanical treatment of the material.     

Abrasive wear and craze breakdown in polystyrene

The role of craze breakdown during the fracture process of abrasive wear in glassy polystyrene was investigated. At first, the wear resistance, _w, was compared with the craze breakdown strain as a function of molecular weight and diluent concentration. It was found that _w increases with molecular weight and decreases with the diluent concentration. Although craze breakdown strain also increases with molecular weight and decreases with the diluent concentration, the wear data do not converge into a single curve in a plot against the craze breakdown strain. Selected specimens were then studied by micro-indentation and micro-scratching experiments. An analysis of the scratch patterns and contact load at the polymer surface indicated that a critical stress criterion, rather than a critical strain criterion, may be suitable for the onset of the failure process in brittle polymer wear. With this criterion, the critical load for crack opening, _c, can be related to the craze breakdown strain and Young's modulus, and the observed deviation between the craze breakdown strain and _w can be explained.     

Ultrasonic determination of the elastic and nonlinear acoustic properties of transition-metal carbide ceramics: TiC and TaC

Pulse-echo overlap measurements of ultrasonic wave velocity have been used to determine the elastic stiffness moduli and related elastic properties of ceramic transition-metal carbides TiC and TaC as functions of temperature in the range 135–295 K and hydrostatic pressure up to 0.2 GPa at room temperature. The carbon concentration of each ceramic has been determined using an oxidation method: the carbon-to-metal atomic ratios are both 0.98. In general, the values determined for the adiabatic bulk modulus (B ^S), shear stiffness (), Young's modulus (E), Poisson's ratio () and acoustic Debye temperature (_D) for the TiC and TaC ceramics agree well with the experimental values determined previously. The temperature dependences of the longitudinal stiffness (C _L) and shear stiffness measured for both ceramics show normal behaviour and can be approximated by a conventional model for vibrational anharmonicity. Both the bulk and Young's moduli of the ceramics increase with decreasing temperature and do not show any unusual effects. The results of measurements of the effects of hydrostatic pressure on the ultrasonic wave velocity have been used to determine the hydrostatic pressure derivatives of elastic stiffnesses and the acoustic-mode Grüneisen parameters. The values determined at 295 K for the hydrostatic pressure derivatives (C _L/P)_{P = 0}, (/P)_{P = 0} and (B ^S/P)_{P = 0} for TiC and TaC ceramics are positive and typical for a stiff solid. The adiabatic bulk modulus B ^S and its hydrostatic pressure derivative (B ^S/P)_{P = 0} of TiC are in good agreement with the results of recent high pressure X-ray diffraction measurements and theoretical calculations. The longitudinal (_L), shear (_S) and mean (^el) acoustic-mode Grüneisen parameters of TiC and TaC ceramics are positive: the zone-centre acoustic phonons stiffen under pressure. The shear _S is much smaller than the longitudinal _L. The relatively larger values estimated for the thermal Grüneisen parameter ^th in comparison to ^el for the TiC and TaC ceramics indicate that the optical phonons have larger Grüneisen parameters. Hence knowledge of the elastic and nonlinear acoustic properties sheds light on the thermal properties of ceramic TiC and TaC.     

Optimal choice of descent steps in gradient methods of solution of inverse heat-conduction problems

Modifications are proposed for the methods of steepest descent and conjugate gradients for the solution of multiparameter inverse problems in heat conduction.Notation A, B, L linear operators - u element of the solution space U - f exact initial data - f error in the initial data - value of the error in the initial data - A^–1 inverse operator - u^(k)() k-th derivative of the function u - _i() polynomials of degree i–1 - A^*, B^*, L^* operators conjugate to the operators A, B, L - J(g) discrepancy functional - J'g gradient of the discrepancy functional - _n ⁱ depth of descent with respect to the i-th component of the antigradient of the discrepancy in the n-th iteration - _m length of the observation interval Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 2, pp. 264–269, August, 1980.     

On the strong coupling effects in superfluid 3He. II

Introducing a generalized model for the momentum dependence of the normal-state effective quasiparticle vertex function , we have calculated the strong coupling effects in superfluid ³He as well as the transport coefficients in the normal liquid ³He. Using those values of that account for the observed values of the normal-state transport coefficients, we have predicted the possible ranges for the values of the jump in specific heat at T _c in the BW, ABM, and A₁ states of superfluid ³He. We have also found that, compared to the normal-state transport properties, the strong coupling effects in superfluid ³He are much more sensitive to the forms of .rk supported in part by grant No. A4630 of the National Research Council of Canada.     

Fracture initiation and crack propagation of acrylonitrile-butadiene-styrene (ABS) in organic solvents

The effects of organic liquid environments on the fracture behaviour of acrylonitrile-butadiene-styrene (ABS) have been investigated. Fracture initiation experiments showed thatK _i ² , (K _i being the stress intensity factor at crack/craze initiation), could be meaningfully correlated with the solvent solubility parameter ( _s) of the different liquid environments and had a minimum value at _s= _p, where _p was the solubility parameter of ABS. For the range of organic liquids used, hydrogen bonding did not have any significant effects on the correlations. It was demonstrated that theK _i ² – _s correlations could also be usefully extended to other materials such as plain and glass-filled polystyrenes. At a common crack speed (å), the fracture toughness (R) values in crazing liquids (i.e. alcohols) were greater than those in cracking solvents (i.e. acetone, benzene, toluene, etc.) which usually caused a dissolution effect on the plastic. From crack propagation experiments, and using fracture mechanics analyses, definiteR(å) andK _c(å) relationships for ABS immersed in toluene, carbon tetrachloride and methanol were determined. These experimental results showed that crack propagation was relaxation controlled and agreed well with a recent theoretical analysis due to Williams and Marshall for environmental crack and craze growth in polymers. Finally, SEM pictures were presented to show the remarkable differences in the fracture morphologies of ABS in both crazing and cracking liquid environments.     

A generalized equation for surface tension from the triple point to the critical point

A three-parameter generalized equation is proposed for surface tension from the triple point to the critical point. This equation not only fits the data well but also is good for interpolation between the normal boiling point and the critical point. This equation is also good for extrapolation to the triple point. This equation has been tested using the surface tension of water from the triple point to the critical point. The constants of this equation obtained using orthobaric surface tensions are given for a number of compounds. The isobaric surface tensions determined at a pressure of 1 atm do not differ significantly from the orthobaric surface tensions. Such data also have been used in obtaining equations from the triple to the critical point.Nomenclature T _c Critical temperature, K - T _t Triple point, K - T _m Melting point, K - T _r Reduced temperature, K - X (T _c-T)/T _c - Surface tension, dyne · cm^–1;10^–3N · m^–1 - _m Surface tension at the melting point - _f Surface tension at T _r=0.9 - _t Surface tension at the triple point - Relative deviation 100[ _obsd– _calcd]/ _obsd - Standard deviation [( _obsd– _calcd)²/(No. points—No. parameters)]^0.5