Experimental investigation of the effects of stress rate and stress level on fracture in polystyrene

The effects of stress rate and stress level on fatigue crack propagation in compression-moulded single-edge notched specimens (0.25 mm in thickness) of polystyrene are reported. Values of the stress rate are obtained from the formula = 2v(_max – _max),, wherev is the frequency and _max, _min are the maximum and minimum stresses of the fatigue cycle. Different levels of are achieved by changing the frequency while keeping _max, _min at fixed values. The effect of the stress level is investigated by keeping and _min constant and varying _max andv. The results show that when the kinetic data are plotted as l/t against the energy release rateG ₁, a relatively small effect of the stress rate is observed. If the same data are treated as l/N againstG ₁, a decrease in l/N with test frequency is seen. The increase in the level of _max results in a higher crack speed. The critical crack length is found to be practically the same for all stress-rate experiments. A decrease in the critical crack length is observed with the increase in stress level. Analysis of craze distribution around the crack path shows that the extent of crazing decreases with the increase in stress rate and increases with the increase in stress level. For all experimental conditions, the ratio of the second moment to the square root of the fourth moment of the histograms of craze density along directions normal to the crack path is found to be constant throughout the slow phase of crack propagation. This result supports a self-similarity hypothesis of damage evolution proposed in the crack layer model.     

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.     

Theory of superconductivity. II. Excited Cooper pairs. Why does sodium remain normal down to 0 K?

Based on a generalized BCS Hamiltonian in which the interaction strengths (V ₁₁,V ₂₂,V ₁₂) among and between electron (1) and hole (2) Cooper pairs are differentiated, the thermodynamic properties of a type-I superconductor below the critical temperatureT _c are investigated. An expression for the ground-state energy,W-W ₀, relative to the unperturbed Bloch system is obtained:W-W ₀=–1/4[N ₁(0) ₁ ² +N ₂(0) ₂ ² ], whereN _j(0) represent the electron and hole densities of states at the Fermi energy _F, and _j are the solutions of the simultaneous equations, with _D denoting the Debye frequency. The usual BCS formulas are obtained in the limits: (all)V _jl=V₀,N ₁(0) =N ₂(0). Any excitations generated through the BCS interaction Hamiltonian containingV _jl must involve Cooper pairs of antiparallel spins and nearly opposite momenta. The nonzero momentum orexcited Cooper pairs belowT _c are shown to have an excitation energy band minimum lower than the quasi-electrons, which were regarded as the elementary excitations in the original BCS theory. The energy gap _g(T) defined relative to excited and zero-momentum Cooper pairs (whenV _jl>0) decreases from _g(0) to 0 as the temperatureT is raised from 0 toT _c. If electrons only are available as in a monovalent metal like sodium (V ₁₂=0), the energy constant ₁ is finite but the energy gap vanishes identically for allT. In agreement with the BCS theory, the present theory predicts that a pure nonmagnetic metal in any dimensions should have a Cooper-pair ground state whose energy is lower than that of the Bloch ground state. Additionally it predicts that a monovalent metal should remain normal down to 0K, and that there should be no strictly one-dimensional superconductor.     

Effect of cold rolling on the superconducting and electronic properties of two amorphous alloys; Nb50Zr35Si15 and Nb70Zr15Si15

The effect of cold rolling on the superconducting properties was examined for amorphous Nb₅₀Zr₃₅Si₁₅ and Nb₇₀Zr₁₅Si₁₅ superconductors. Cold rolling to 10 to 20% reduction in thickness results in a rise of superconducting transition temperature (T _c) and a decrease in transition width (T _c), upper critical field gradient near , critical current density [J _c(H)] and normal electrical resistivity (_n). Changes of about 7% forT _c 33% for T _c, 12% for and 70% forJ _c(H) are found. The rise ofT _c upon cold rolling was considered to originate from the increase in the electron-phonon coupling constant () due to an increase in the electronic density of states at the Fermi level [N(E _f)] and a decrease in the phonon frequency (), while the decreases in T_c,J _c(H) and _n were attributed to the decrease in fluxoid pinning force due to an increase in homogeneity in the amorphous structure. From the results described above, the following two conclusions were derived: (a) cold rolling causes changes in electronic and phonon-states in the quenched amorphous phase, and (b) deformation upon cold rolling occurs not only in the coarse deformation bands observable by optical microscopy, but also on a much finer scale comparable to the coherence length (7.7 nm).     

Heat flow model of rapid solidification with nonhomogeneous thermal contact

A heat flow model is presented of the solidification process of a thin melt layer on a heat conducting substrate. The model is based on the two-dimensional heat conduction equation, which was solved numerically. The effect of coexisting regions of good and bad thermal contact between foil and substrate is considered. The numerical results for thermal parameters of the Al-Cu eutectic alloy show considerable deviations from one-dimensional solidification models. Except for drastic differences in the magnitude of the solidification rate near the foil-substrate interface, the solidification direction deviates from being perpendicular to the substrate and large lateral temperature gradients occur. Interruption of the thermal contact may lead to back-melting effects. A new quantity, the effective diffusion length, is introduced which allows some conclusions to be drawn concerning the behaviour of the frozen microstructure during subsequent cooling.Nomenclature _i ,a _i Thermal diffusivity _i = _i /c _{i i} ,a _i = _i / ₁ - c _i Specific heat capacity - d Foil thickness - D Solid state diffusion coefficient - e_x, e_z Unit vectors - H Latent heat of fusion - h ,h Foil-substrate heat transfer coefficients - i Index: 1, melt; 2, solidified foil; 3, substrate - _i ,k _i Thermal conductivityk _i = _i / ₁ - n Normal unit vector - Nu ,Nu Nusselt numbers for regions of badNu(x,) and good thermal contact, respectivelyNu =h Nu _d / ₁,,Nu(x, )=h(x,)d/ ₁ - R Universal gas constant - , s Position of the liquid-solid interface ¯s/d=s=s _xe_x+s _ze_z - Local solidification rate /d = s =s _xe_x +s _ze_z - t Real time - T _i Temperature field - T ₀ Ambient temperature - T _f Melting temperature - u _i Dimensionless temperature fieldu _i (x, z,)=T _i (x,z,)/T _f - u ₀ Dimensionless ambient temperatureu ₀=T ₀/T _f - _i Local cooling rate within the foil _i = du _i /d - W Stefan numberW=H/c ₁ T _f - ,x Cartesian coordinate parallel to the foil-substrate interfacex= /d - ₀,x ₀ Lateral extension of foil sectionx ₀= ₀/d - ₁,x ₁ Lateral contact lengthx ₁= ₁/d - ,z Cartesian coordinate perpendicular to the foil-substrate interfacez= /d - ₀,z ₀ Substrate thicknessz ₀= ₀/d - E Activation energy of diffusion - T Initial superheat of the melt - u Dimensionless initial superheat u=T/T _f - (x) Step function - _eff Dimensionless effective diffusion length - _i Mass density - Dimensionless time=t ₁/d ² - _f, _f(x, z) Total and local dimensionless freezing time, respectively     

Anisotropic model of an emitting electric arc

An analytical procedure is proposed for obtaining exponential dimensionless equations to calculate the characteristics of an electric arc in a plasmatron channel that is based on using an exponential approximation of the temperature dependence of electric conductivity with different exponents for longitudinal and transverse coordinates.Notation density - C _p specific heat at constant pressure - electrical conductivity - thermal conductivity - V velocity - T temperature - E electric-field strength - r, z radial and longitudinal coordinates - thermal conductivity function - G gas flow rate - Q bulk radiation density - R, D radius and diameter of channel - I strength of current - r/R - z/R - m R/r _*=1/r _* - S S–S _* - b _Q Q ₀/S ₀ - a, k,k ,n ,C ₀₀, constants - J ₀,J ₁,Y ₀ Bessel functions - ₁, _n roots of characteristic equations - j current density - q heat flux density - Po, Pe Pomerantsev and Pecklet numbers, respectively - Nu Nusselt number - _K , _U , _q ,K _Q ,K _S similarity numbers - U voltage - heat-transfer coefficient Indexes z longitudinal component - * boundary of conducting band - 0 base value - 1 value on the wall - 00 axial value - I, II conducting and nonconducting bands, respectively Academic Scientific Complex A. L. Luikov Institute of Heat and Mass Transfer of the Academy of Sciences of Belarus, Minsk. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 68, No. 5, pp. 820–826, September–October, 1995.     

The critical curve in an antiferromagnetic superconductor with nonmagnetic impurities

The critical curve of a transition of the second kind in an antiferromagnetic superconductor (AFS) with nonmagnetic impurities has been studied. The AFS is described by using the mean-field model given by Nass, Levin, and Grest and assuming a one-dimensional electron band. We find that the points on the critical curve satisfy the thermodynamic stability condition for 0₁/₀5.04 and 0.49H_Q/₀1.64.Here ₁ is the inverse lifetime of a conduction electron for nonmagnetic impurity scattering,H _Q is the antiferromagnetic molecular field, ₀ is the zero-temperature order parameter of a superconductor in the absence ofH _Q and impurities. Further, ₁ and H_Q denote the values of these quantities for points on the critical curve. For ₁/₀>5.04 and H_Q/₀>1.64, the phase transition from the superconducting to the normal state is always of the second kind. Some thermal properties of the system near the critical curve have also been investigated and we find that these depends dramatically on the impurity concentration.     

Effect of metal electrode on thermoelectric power in bismuth telluride compounds

The measurements of apparent effective Seebeck coefficients S _a, thermoelectric powers E and I-V characteristics were made on a copper-semiconductor-metal contact junction, where the semiconductor is consisted of the p- and n-type bismuth telluride compounds. The S _a measured by heating either of copper and metal alternatively to produce the temperature differences of T = ±6 K changed slightly with the kind of metal electrodes, but it changed very little when the direction of the temperature gradient was reversed. The averaged S _a values over all kinds of metal electrodes agreed closely with the Seebeck coefficients S measured by the conventional technique using two alumel-chromel thermocouples as an electrode. The thermoelectric power E generated by imposing the temperature differences of T = ±6 K on a thermoelement tended to increase with increase of S _a and reached large values in noble metal electrodes of Ag and Au. The E was found to achieve enhancements of up to 10% or even more, when one end of a thermoelement contacts with Au electrode and the external electrical resistance is zero. Thus, the selection of the optimal metal electrode is necessary to make the thermoelectric conversion efficiency as high as possible.     

Higher approximations for supersonic flow past slowly oscillating bodies of revolution

Summary Supersonic flow past slowly oscillating pointed bodies of revolution is studied. Starting from the complete nonlinear potential equation an elementary linearized solution is discussed and it is shown how this solution together with the method of matched asymptotic expansions can be used to derive an elementary second-order slender body theory. This approach is further demonstrated for the oscillating cone and its range of validity is evaluated by comparison with other theoretical methods.

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Zusammenfassung Es wird die Überschallströmung um langsam schwingende spitze Rotationskörper untersucht. Ausgehend von der vollständigen nichtlinearen Potentialgleichung wird zuerst eine elementare linearisierte Lösung besprochen und gezeigt, wie diese Lösung im Verein mit der Method of matched asymptotic expansions zur Herleitung einer elementaren Schlankkörpertheorie zweiter Ordnung verwendet werden kann. Die Theorie wird am Beispiel des schwingenden Kegels näher erläutert und mit anderen Methoden verglichen.

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Symbols a Velocity of sound - c _N Normal force coefficient - Damping coefficient - F _(x) Dipole distribution - k Reduced frequency - M Mach number - R (x) Meridian profile - t Time - x, r, Cylindrical coordinates - - Ratio of specific heats - Amplitude of oscillation - Thickness ratio - Perturbation potential - Zero angle of attack potential - æ - Velocity potential - Out-of-phase potential - - In-phase potential - - Source coordinate With 4 Figures     

On the Method of Determination of Strain Energy Release Rate During Fatigue Delamination in Composite Materials

In this paper, the problem of calculation of the energy release rate for a fatigue test on composite material has been investigated. The application of the Linear Elastic Failure Mechanics (LEFM) leads to the use of varation of the energy release rate ( G). As the energy release rate is a function of the load squared, the variation of G becomes either a function of variation of the load squared ( G = f((P²))) or a function of the square of the load variation ( G = f(( P)²)).In this paper, we determine, by different fatigue tests, which of the two theoretical results is the best to describe the experiments. These fatigue tests have been made on DCB test-specimen in mode I with different R ratios (R = P_max / P_min) and different maximum loads. The material was a unidirectionnal glass-epoxy.The results show that considering G as a function of ( P)² seems more appropriated to describe a cracking test in fatigue.     

Bounding-surface plasticity: Stress space versus strain space

Summary A bounding-surface plasticity model is formulated in stress space in a general enough manner to accommodate a considerable range of hardening mechanisms. Conditions are then established under which this formulation can be made equivalent to its strain-space analogue. Special cases of the hardening law are discussed next, followed by a new criterion to ensure nesting. Finally, correlations with experimental data are investigated.Notation (a) centre of the stress-space (strain-space) loading surface; i.e., backstress (backstrain) - ^* (a ^*) centre of the stress-space (strain-space) bounding surface - (a ) target toward which the centre of the stress-space (strain-space) loading surface moves under purely image-point hardening - (b) parameter to describe how close the loading surface is to nesting with the bounding surface in stress (strain) space; see (H10) - (c) elastic compliance (stiffness) tensor - (d) parameter to describe how close the stress (strain) lies to its image point on the bounding surface; see (H10) - (D) generalised plastic modulus (plastic compliance); see (1) - function expressing the dependence of the generalised plastic modulus on (plastic complianceD ond) - ^* (D ^*) analogue to (D) for the bounding surface - function expressing the dependence of ^* on (D ^* ond) - () strain (stress) - ' (') deviatoric strain (stress) - ^P ( ^R ) plastic strain (stress relaxation); see Fig. 1 - () image point on the bounding surface corresponding to the current strain (stress) - _iso (f _iso) at the point of invoking consistency, the fraction of local loading-surface motion arising from a change of radius; i.e., fraction of isotropic hardening in the stress-space theory - _kin (f _kin) at the point of invoking consistency, the fraction of local loading-surface motion arising from a change in the backstress (backstrain); i.e., fraction of kinematic hardening in the stress-space theory - _nor (f _nor) at the point of invoking consistency, the fraction of backstress (backstrain) motion directed toward the image stress (strain); i.e., the image-point fraction of the kinematic hardening in the stress-space theory - _ima (f _ima) at the point of invoking consistency, the fraction of backstress (backstrain) motion directed toward the image stress (strain); i.e., the image-point fraction of the kinematic hardening in the stress-space theory - function relating _iso to , , and (f _iso tob,d, andl) - function relating _kin to , , and (f _kin onb,d, andl) - function relating _nor to , , and (f _nor onb,d, andl) - function relating _ima to , , and (f _ima onb,d, andl) - the fraction of outwardly normal bounding-surface motion at the Mróz image point which arises from a change of radius - the fraction of outwardly normal bounding-surface motion at the Mróz image point which arises from a change in the centre - function relating _iso ^* to (f _iso ^* tod) - function relating _kin ^* to (f _kin ^* tod) - (l) parameter to describe the full extent of plastic loading up to the present, giving the arc length of plastic strain (stress relaxation) trajectory; see (H10) - function relating the direction for image-point translation of the loading surface to various other tensorial directions associated with the current state; see (H5). With 6 Figures     

Localized single-particle excitations and the density of states for superconducting composites

The problem of localized single-particle excitations and the density of states (DOS) for an inhomogeneous system consisting of a spherical superconductor (with radius a and order parameter ₁) embedded in another superconductor (order parameter ₂) of infinite size is considered. With the assumption of constant values of ₁ and ₂, the Bogoliubov equations are solved for general values of l (the orbital angular momentum quantum number). For a fixed value of ₁/₂ and different values of ₂/E _F, the dependence of the excitation energy (l=0)/₂ on the particle sizek _F a is shown (k _F is the Fermi wave vector andE _F is the Fermi energy). Fork _F a=300, 450, and 800 and a fixed value of ₂/E _F, the variations in the DOS by changing ₁/₂ are also shown.     

Interplay between thermal fluctuations and uniformly distributed Tc inhomogeneities: Application to inhomogeneous Bi2Sr2Ca1Cu2O8 crystals

The effective medium approach proposed some years ago by Maza and Vidal is used here to take into account the influence of small T_c-inhomogeneities, at long length scales and uniformly distributed, on the measured (or effective) in-plane paraconductivity, _e ^ab , and fluctuation-induced magnetoconductivity, .     

Influence of Microstructure on the Strength and Cyclic Crack Resistance of Cast Irons

We study the influence of microstructure of high-strength cast irons of ferritic, ferritic-pearlitic, and pearlitic classes on the characteristics of strength and cyclic crack resistance and establish the relationship between the characteristics of strength and cyclic crack resistance and the chemical composition and microstructural parameters of the matrix and graphite inclusions of high-strength cast irons. Indeed, the ultimate strength _u = 750–850 MPa, fatigue threshold K _th = 8–10 MPa , and cyclic fracture toughness K _fc = 50–60 MPa are guaranteed by the pearlitic matrix and the following parameters of the graphite phase: the content of graphite f _p 10%, the degree of its spheroidization of about 95%, the size of globules d _gl 20–50 m, and the distance between them 40–70 m, obtained in high-strength cast irons containing (wt.%): 3.2–3.5 C, 1.9–2.5 Si, 0.35–0.4 Mn, and 0.05 Mg.     

Asymptotic expansions for constant-composition dew-bubble curves near the critical locus

Explicit functional representations are developed for constant-composition dew and bubble curves near critical according to the modified Leung-Griffiths theory. The pressure and temperature incrementsP=P–P _c andT= T–T _c, where c denotes critical, are linearly transformed to new variablesP andT. In the transformed space, the coexistence curves are no longer double-valued and can be expressed as a nonanalytic expansion, where the coefficients are functions of the critical properties and their derivatives. A similar asymptotic expansion is developed forT in terms of the density increment=– _c. In the approximation that the critical exponents=0 and=1/3, the critical point in temperature-density space is shown to be a point of maximum concave upward curvature, rather than an inflection point as previously conjectured.Paper presented at the Tenth Symposium on Thermophysical Properties, June 20–23, 1988, Gaithersburg, Maryland, U.S.A.Formerly National Bureau of Standards     

Response of a spinning liquid column to pitch excitation

Summary The response of a solidly rotating liquid bridge consisting of inviscid liquid is determined for pitch excitation about its undisturbed center of mass. Free liquid surface displacement and velocity distribution has been determined in the elliptic (>2₀) and hyperbolic (<2₀) excitation frequency range.List of symbols a radius of liquid column - h length of column - I ₁ modified Besselfunction of first kind and first order - J ₁ Besselfunction of first kind and first order - r, ,z cylindrical coordinates - t time - u, v, w velocity distribution in radial-, circumferential-and axial direction resp. - mass density of liquid - free surface displacement - velocity potential - ₀ rotational excitation angle - ₀ velocity of spin - forcing frequency - _1n natural frequency - surface tension - acceleration potential - for elliptic range >2₀ - for hyperbolic range >2₀     

A dilatometric method to measure the thermal diffusivity of nonmetallic liquids

A new method to measure the thermal diffusivity of liquids is presented. It requires determination of the time dependence of the thermal expansion of the liquid when it is subjected to a heat source at the top of the cell containing the liquid. The high accuracy of the method (about 3%) is due to an essential reduction of convective currents and also to the absence of temperature detectors, which generally introduce unwanted perturbations on the thermal Field.Nomenclature Thermal conductivity - c Specific heat - Density - c = specific heat x density - h Newton coefficient - Thermal diffusivity - T, 0 Temperature - tV Electric signal - Calibration coefficient - _exp, _th Volume change of the liquid     

Heat Transport in a Pure Fluid Near the Critical Point: Steady State and Relaxation Dynamics

The steady-state conditions and relaxation dynamics of a fluid layer during heat transport experiments near the liquid-vapor critical point are studied theoretically. The application is to ³ He along the critical isochore and under normal gravity—where experimental data are available—as well as zero gravity. First, the steady-state density stratification from gravity and from heat flow is calculated. The resulting observable thermal conductivity is obtained as a function of the temperature difference T (or the heat current Q) across the fluid layer and becomes a strong non-linear function of T as T_c is approached. The calculated is compared with that from experiments both above and below T_c. Second, the spatial profiles of temperature and density and their temporal evolutions are calculated above T_c as T is established after Q is switched on, or conversely as T decays to zero after Q has been switched off. From the calculations, done both from a closed-form expression and from simulations, the observable and asymptotic relaxation times for reaching the steady state are calculated as a function of Q, and compared with the experiments above T_c.     

Torsional boundary layer effects in shells of revolution undergoing large axisymmetric deformation

Numerical and asymptotic solutions are developed to the equations governing large torsional, axisymmetric deformation of rubberlike shells of revolution. The shell equations include large-strain geometric and material nonlinearities, transverse shear deformation, transverse normal stress and strain, and torsion. Both analyses allow ready incorporation of different strain-energy density functions. In the asymptotic analysis, the interior solution corresponds to that of nonlinear membrane theory and contains a primary boundary layer. The edge-zone solution gives a secondary boundary layer that, for large strain, divides into a bending-twisting moment component and a torsional-membrane component. The boundary layer behavior is illustrated for a clamped neo-Hookean cylinder subjected to internal pressure and axial torque.List of symbols Latin symbols a General dependent variable - a ^(mn) Terms of the asymptotic expansion of a(x) - b Characteristic length - c Scalar curvature components in the normal direction - c , c , , c Cosine of , respectively - C Material constant with units of a Young's modulus - e _i Deformed local orthonormal basis associated with (, s, n)(x ₁, x ₂, x ₃) coordinates - Undeformed cylindrical coordinate basis - Intermediate coordinate basis - g Shear correction factor - H Horizontal stress resultants - l ₁ Strain invariant - k Scalar curvature components - L Undeformed cylinder length - M Moment resultants - M _r, M , M _z Moment resultant components in the basis - N Membrane stress resultants - p Internal pressure - p _H, p _v Horizontal and vertical surface loads, respectively - p _i Thickness-averaged surface tractions - Q Transverse shear stress resultants - , r Radial coordinate prior to, after deformation - R Undeformed cylinder radius - , s Meridional coordinate prior to, after deformation - s , s _x, , s Sine of , respectively - , S Reference surface prior to, after deformation - S ₁, S ₂ Shear stress resultants parallel to the reference surface - S ₃ Average transverse normal stress resultant - t Undformed shell thickness - T Axial torque - V Vertical stress resultants - w Two-dimensional strain-energy density function - w _n Terms in expansion for w - W Three-dimensional strain-energy density function - x Undeformed axial coordinate in cylinder - , z Axial coordinate prior to, after deformation     

Pseudo-Gap Phenomenon in the Sn-Doped Mercury-Based Cuprate Superconductors

The Sn-doped mercury-based copper oxide high temperature superconductors, with nominal composition (Hg_{1 – x} Sn _x )Ba₂Ca₂Cu₃O_{8 +} , have been successfully fabricated. These samples were used to study the pseudo-gap phenomenon in their normal phase. By means of ln[1/(T) – 1/ _N (T)] versus 1/T plots we found three characteristic temperatures T ^*, T _S, and T _F, which describe the role of pseudo-gap phenomenon. From the stated plots, we are able to estimate the magnitude of the pseudo-gap for our samples. The pseudo-gap _PG changes into the spin pairing dipolar superconducting gap _SPDS, when the Fermi kinetic energy shrinks to zero at the critical temperature T _c. It is proved that the spin pairing dipolar superconducting gap , where, _T being the total superconducting gap of the cuprate superconductors, which has the value determined by . It is also proved that the spin pairing dipolar superconducting gap is smaller than the dipolar pseudo-gap, _PG.