Effect of finite extensibility on the viscoelastic properties of a styrene–butadiene rubber vulcanizate in simple tensile deformations up to rupture

Stress–strain and rupture data were determined on an unfilled styrene–butadiene vulcanizate at temperatures from ?45 to 35°C and at extension rates from 0.0096 to 9.6 min^?1. The data were represented by four functions: (1) the well-known temperature function (shift factor) a_T; (2) the constant strain rate modulus, F(t,T), reduced to temperature T₀ and time t/a_T, i.e., T₀F(t/a_T)/T; (3) the time-dependent maximum extensibility, λ_m(t/a_T); and (4) a function Ω(χ) where χ = (λ ? 1)λ_m⁰/λ_m, in which λ is the extension ratio and λ_m⁰ is the maximum extensibility under equilibrium conditions. The constant strain rate modulus characterizes the stress–time response to a constant extension rate at small strains, within the range of linear response; λ_m is a material parameter needed to represent the response at large λ; and Ω(χ) represents the stress–strain curve of the material in a reference state of unit modulus and λ_m = λ_m. The shift factor a_T was found to be sensibly independent of extension. At all values of t/a_T for which the maximum extensibility is time-independent, the relaxation rate was also found to be independent of λ. These observations indicate that the monomeric friction coefficient is strain-independent over the ranges of T and λ covered in the present study. It was found that λ_m⁰ = 8.6 and that the largest extension ratio at break, (λ_b)_max, is 7.3. Thus, rupture always occurs before the network is fully extended.     

Comment on “on the possibility of a three-temporal Lorentz transformation”

In a recent paper it has been suggested that, strictly speaking, light speed invariance is not fully consistent with ordinary space geometry unless our concept of time is drastically modified. This note argues against that paper.     

Some solutions for Yang's equations for self-dual SU(2) gauge fields

It is shown that a large class of solutions of Yang's equations for self-dual SU(2) gauge fields can be obtained from the solutions of two-dimensional and four-dimensional Laplace equations.     

Perfectly conducting ferrofluid in general relativity

The growth of magnetic energy density is considered in a collapsing ferrofluid with infinite electrical conductivity whose magnetic permeability is not a constant. It is shown that the variation of the magnetic permeability affects the growth of magnetic energy. The case of isotropic collapse and a specific case of anisotropic collapse are examined.     

A note on ratio and product type estimators

Summary The use of ratio and product estimators for the mean of a finite population is well known. This paper proposes transformed estimators obtained through parametric linear combination of the ratio, or product, and the usual unbiased estimator of the mean for any sample design. To the first degree of approximation, the proposed estimators have smaller mean square error than that of the ratio, product and the usual unbiased estimator, for suitable choice of the parameter. The superiority of the proposed estimators over others for small samples has been studied empirically.     

Heterocyclic synthesis using ethyl carboethoxyformimidate

Ethyl carboethoxyformimidate is shown to be a versatile reagent for the synthesis of a variety of mono- and bicyclic heterocyclic systems.     

Thermal entrance region heat transfer for MHD laminar flow in parallel-plate channels with unequal wall temperatures

The problem of thermal entry heat transfer for Hartmann flow in parallel-plate channels with uniform but unequal wall temperatures considering viscous dissipation, Joule heating and axial conduction effects is approached by the eigenfunction expansion method. The series expansion coefficients for the nonorthogonal eigenfunctions are obtained by using a method for nonorthogonal series described by Kantorovich and Krylov [21]. Numerical results are obtained for the case with entrance condition parameter _o=1 and open circuit condition K=1. The parametric values of Ha=0, 2, 6, 10 and Br=0, –1 are considered for Hartmann and Brinkman numbers, respectively.

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Zusammenfassung Das Problem der Wärmeübertragung im thermischen Einlauf einer Hartmannströmung im ebenen Spalt mit einheitlichen, aber ungleichen Wandtemperaturen wurde unter Berücksichtigung viskoser Dissipation, Joulescher Heizung und axialer Wärmeleitung mit Hilfe einer Entwicklung nach Eigenfunktionen behandelt. Die Koeffizienten der Entwicklung für nichtorthogonale Eigenfunctionen wurde nach einer Methode für nichtorthogonale Reihen nach Kantorovicz und Krylow [21] berechnet. Numerische Ergebnisse werden für den Eintrittsparameter _o=1 und die Bedingung für den offenen Stromkreis K=1 erhalten. Die Parameterwerte Ha=0, 2, 6, 10 und Br=0, –1 werden für die jeweiligen Werte der Hartmann- und der Brinckman-Zahl betrachtet.

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Nomenclature a one-half of channel height - ¯B,B₀ magnetic field Induction vector and magnitude of applied magnetic field - Br Brinkman number, _f U_m ²/(k_c) - C_n,D_n coefficients in the series expansion of _e, see eq. 16 - c_p specific heat at constant pressure - ,E₀ electric field intensity vector and component - E_n,O_n even and odd eigenfunctions - Ha Hartmann number, (/_f)^1/2 B_o a - h₁,h₂ local heat transfer coefficients at lower and upper plates - ¯J,J_y electric current density vector and component - K external loading parameter, E_o/(B_o U_m) - k thermal conductivity - Nu₁, Nu₂ local Nusselt numbers, h₁,a/k and h₂a/k, respectively - P fluid pressure - Pe Peclet number, PrRe - Pr Prandtl number, C_{p f}/k - q₁,q₂ rates of heat transfer per unit area,^–k(T/Z)Z=–a'^–k(T/Z) Z=a respectively - Re Reynolds number, U_ma/u_f - T,T₀,T₁,T₂ fluid temperature, uniform entrance temperature, uniform but different lower and upper plate temperatures, respectively - T_b,T_m bulk temperature and (T₁+T₂)/2 - U,U_m,u axial, mean and dimensionless velocities, respectively - ¯V velocity vector - X,Z axial and transverse coordinates - x,z dimensionless coordinates - _n,_n even and odd eigenvalues - ,₀,_b dimensionless fluid, entrance and bulk temperatures, respectively - _c,_e,_f characteristic temperature difference (T₂-T_m), and dimensionless fluid temperatures, defined by eq. (10) - _e,_f magnetic permeability and viscosity of fluid - fluid density - electric conductivity - viscous dissipation function - (1-)/2