Mixed Mode Stress Intensity Factor Determination of Multiple Cracks in Hollow Circular Pipe

Failure of pressure vessels and piping due to high temperature applications occurs due to the formation of fatigue cracks caused by cyclic load. It is well known that, the consequences of collapses of pipes causing enormous disruption of daily life. Thus there is a need to design and manufacture the pipes with precision and care. The major cause of crack nucleation in pipes is due to corrosion and internal fluid pressure. The crack-tip stresses are determined using stress intensity factor (SIF). In the present work an attempt has been made to determine the SIF for multiple cracks in a circular pipe subjected to internal fluid pressure. Two surface cracks of same size were introduced at the inner wall of the tube. The crack depth ratio (a/t) ranging between 0.1 and 0.5 and crack aspect ratio (a/c) of 0.6 and 1.0 was considered. Internal fluid pressure of 100 MPa was applied at the inner surface of the pipe and the corresponding SIF was measured. SIF values were calculated with consideration of mode-II and mode-III fracture in order to predict the exact SIF. As available SIF solutions of cracked pipes are limited to mode-I fracture, present work presents the influence of additional influence of mode-II and mode-III fracture. It is observed that, as crack depth ratio increases, SIF also increases considerably for semi-circular cracks. Higher SIF values were observed at the crack surface region [S/S ₀ = ±1] compared to crack middle [S/S ₀ = 0] region. A crossover in SIF was noted at a crack depth ratio of 0.3. At higher crack depths, SIF values decrease at the crack surface region due to additional influence of mode-II and mode-III fracture. In contrast to semi-circular cracks, SIF values are higher at the crack surface region for semi-elliptic cracks irrespective of the crack depths.     

Mixed-Mode Crack Propagation in Cruciform Joint using Franc2D

The focus of this research was on determining the cracking behavior when parameter such as the biaxiality ratio was varied. The crack propagation under mixed-mode loading was simulated by means of finite element method. The stress intensity factors have been calculated by the linear elastic fracture mechanics approach using fracture analysis code-2D (Franc2D). The crack growth under opening mode-I was considered because the crack growth occurs mainly along the direction where the mode-I stress component becomes the maximum. The numerical integration of Paris’ equation was carried out. The effect of normal and transverse applied load (σ _x, and σ _y, respectively) on crack propagation was presented. It was found that the fatigue crack growth was faster at a smaller biaxial stress ratio (λ), i.e., higher σ _y on the horizontal crack plan. Moreover, fatigue strength values decrease as λ decreases. The results confirm the use of fracture mechanics approach in biaxial fracture.     

Surface Effect on Spatial Length-Scales in a Two-Gap Superconductor

The healing of two-band superconductivity near its interface is studied. It is demonstrated that the restoration of superconductivity gaps in the immediate vicinity of the interface is governed by two length scales: the first one diverges at critical temperature T _c , while the second one diverges at T _c+<T _c . By moving away from the boundary, the temperature dependencies of characteristic lengths change so that singularity at T _c+ becomes removed in the bulk by arbitrary weak interband coupling. The asymptotes for the spatial behavior of gaps have been found analytically near the surface and approaching the bulk state.     

The H-T and P-T Phase Diagram of the Superconducting Phase in Pd:Bi<Subscript>2</Subscript>Te<Subscript>3</Subscript>

We study the magnetic field vs. temperature (H – T) and pressure vs. temperature (P – T) phase diagrams of the T _c ≈ 5.5 K superconducting phase in Pd _x Bi₂Te₃ (x ≈ 1) using electrical resistivity versus temperature measurements at various applied magnetic fields (H) and magnetic susceptibility versus temperature measurements at various applied magnetic fields (H) and pressure (P). The H – T phase diagram has an initial upward curvature as observed in some unconventional superconductors. The critical field extrapolated to T = 0 K is H _c (0) ≈ 6–10 kOe. The T _c is suppressed approximately linearly with pressure at a rate d T _c /d P ≈ ?0.28 K/GPa.     

The influence of the diffusion cooling on the noise band of the superconductor NbN hot-electron bolometer operating in the terahertz range

Results of an experimental study of the noise temperature (T _n ) and noise bandwidth (NBW) of the superconductor NbN hot-electron bolometer (HEB) mixer as a function of its temperature (T _b ) are presented. It was determined that the NBW of the mixer is significantly wider at temperatures close to the critical ones (T _c ) than are values measured at 4.2 K. The NBW of the mixer measured at the heterodyne frequency of 2.5 THz at temperature T _b close to T _c was ~13 GHz, as compared with 6 GHz at Tb = 4.2 K. This experiment clearly demonstrates the limitation of the thermal flow from the NbN bridge at T _b ? T _c for mixers manufactured by the in situ technique. This limitation is close in its nature to the Andreev reflection on the superconductor/ metal boundary. In this case, the noise temperature of the studied mixer increased from 1100 to 3800 K.     

Pressure Dependence of T<Emphasis Type="Italic">c</Emphasis> in High Temperature Superconductors: Role of Interlayer Interactions

We have studied the role of interlayer interactions (W) in the pressure dependence of T _c of layered superconductors. The expressions for dT _c /dP are obtained by including the effects of layered structure within the framework of two different proposed models, namely the negative-U Hubbard model and the Hirsch model. We observe that the inclusion of interlayer interaction provides better explanation of pressure dependence of T _c . Our numerical results show that the systems having one CuO₂ layer per unit cell may be well described by small values of W while the larger values of W accounts for the systems having two or more superconducting layers in a unit cell. The calculated values of dT _c /dP vs. W are found to be in good agreement with those of experimental results obtained for various high T _c superconductors of cuprate family.     

Specific Heat Jump at T<Emphasis Type="Italic">c</Emphasis> of High T<Emphasis Type="Italic">c</Emphasis> Superconductors: Effect of Van Hove Singularity

In the BCS framework, exact expressions for the ratio between the jump in the specific heat at T _c and the normal phase specific heat are derived within the Van Hove singularity scenario. Analytical results are obtained for an isotropic s-wave and anisotropic d-wave pairing symmetries. Graphical solutions of the ratio as functions of ω _D /T _c and E _F /T _c , where ω _D is the cutoff energy and E _F is the Fermi energy, show significant deviations from the BCS value of 1.43.     

Low-temperature acoustic properties of single-crystal hafnium

Five independent adiabatic elastic tensor components c _ij of single-crystal Hf (0.3% Zr) have been precisely measured in the range 78–300 (c ₁₁ and c ₃₃) or 78–160 K (c ₁₂, c ₁₃, and c ₄₄) at 50 MHz using an ultrasonic pulse technique. The corresponding linear absorption coefficients α _ij (T) have been measured in the same temperature ranges. The c _ij data have been used to determine adiabatic compressibilities β_∥ and β_⊥, and Young’s moduli E _∥ and E _⊥ along and across the c axis, respectively. Near 140 K, we have revealed an anomaly in c ₃₃(T) due to changes in the phonon spectrum of single-crystal α-Hf. The observed anomalies in temperature-dependent α₁₁ and α₃₃ are tentatively attributed to grown-in dislocations.     

Extended BCS-Bose Crossover

Applying the generalized Bose-Einstein condensation (GBEC) formalism, we extend the BCS-Bose crossover theory by explicitly including hole Cooper pairs (2hCPs). From this, follows a phase diagram with two pure phases, one with 2hCPs and the other with electron Cooper pairs (2eCPs), plus a mixed phase with arbitrary proportions of 2eCPs and 2hCPs. The special-case phase when there is perfect symmetry, i.e., with ideal 50-50 proportions between 2eCPs and 2hCPs, corresponds to the usual BCS-Bose crossover. Explicitly including 2hCPs yields an extended BCS-Bose crossover which predicts improved T _c /T _F values for some conventional superconductors (i.e., with electron-phonon dynamics) when compared with experiment. To do this, we employ the BCS dimensionless coupling constant λ _{B C S} via the BCS gap equation and compare with the Bogoliubov et al. upper limit λ _{B C S} ≤ 1/2. Another phase diagram presented exhibits experimental T _c /T _F values for some conventional superconductors for arbitrary proportions between 2eCPs and 2hCPs as function of Δn = n/n _f ? 1, where n is the electron concentration and n _f that of unbound electrons at T = 0. The extended crossover is compared with experimental T _c /T _F values as well as to the gap-to- T _c ratio.     

Published Tunneling Results of Binnig et al Interpreted as Related to Surface Superconductivity in SrTiO<Subscript>3</Subscript>

In 1980, Binnig et al. reported tunneling measurements on Nb-doped SrTiO₃, and interpreted their results as indicating two-band superconductivity in the bulk of SrTiO₃. However, (1) effective masses determined from tunneling results in the normal state by Sroubek in 1969 and 1970 are much smaller than those determined by most other methods. The much smaller masses were attributed to properties of a surface layer by the present author in 1971. (2) The only other reports of two-band superconductivity in bulk SrTiO₃ can be used to infer much smaller values for the band separations than found by Binnig et al. In this paper, we give an alternative explanation of the results of Binnig et al. in terms of superconductivity in a surface layer. We obtain fair fits to the band gaps versus Fermi energy for the two bands in the three samples where two surface subbands are occupied and to the temperature dependence of the gaps in one crystal, using a model with three adjustable interaction parameters, an adjustable energy for the phonons which dominate the pairing, and an adjustable ratio of the mean-field T _c to the actual T _c . We show results for a combined fit to the low-temperature band gaps and to the T-dependence in one crystal. The phonon energy which gives the best fit is 21 meV. This is probably an appropriate average over the three longitudinal polar modes and acoustic modes in the material. A large value of about two is found for the ratio T _{c m f} /T _c , and we conjecture that this arises because a band with a small Fermi energy, not seen in the tunneling results, plays a part in increasing T _{c m f} /T _c .     

Peculiarities of phase transitions in polytypes of monoclinic TlInS<Subscript>2</Subscript>

The existence of two polytypes at room temperatures, C-TlInS₂ and 2C-TlInS₂, with different monoclinic cell parameters, c and 2c, has been revealed. Significant differences in crystal lattice dynamics of these polytypes have been found. In particular, two phase transitions (PTs) have been detected for the polytype C-TlInS₂ as the temperature varies: a second-order PT from paraphase to incommensurate phase at T _i = 215 K and a first-order ferroelectric PT accompanied by a quadrupling of the parameter c at T _c = 197 K. No PT accompanied by an increase in unit cell parameter c has been found in the polytype 2C-TlInS₂, but a global temperature hysteresis characteristic of crystals with an incommensurately modulated structure has been detected at T = 180–230 K.     

Superconductivity in Ca<Subscript>0.5</Subscript>La<Subscript>0.5</Subscript>FBiSe<Subscript>2</Subscript>

Through the measurement of resistivity, magnetic susceptibility, and Hall effect, we discovered a novel BiSe₂-based superconductor Ca_0.5La_0.5FBiSe₂ with T _c of 3.9 K. A strong diamagnetic signal below T _c in susceptibility χ(T) is observed indicating the bulk superconductivity. The negative Hall coefficient throughout the whole temperature regime implies the dominant electron-type carriers in the sample. Different to most of BiS₂-based compounds where superconductivity develops from a semiconducting-like normal state, its resistivity in the present compound exhibits a metallic behavior down to T _c . Together with the enhanced T _c , the metallic character of the normal state implies that the electronic structure of Ca_0.5La_0.5FBiSe₂ may be different to those in the other BiS₂-based compounds.     

On the Role of Fermi Energy in Determining Properties of Superconductors: a Detailed Comparative Study of Two Elemental Superconductors (Sn and Pb), a Non-cuprate (MgB<Subscript><Emphasis Type="Bold">2</Emphasis></Subscript>) and Three Cuprates (YBCO,Bi-2212 and Tl-2212)

Fermi energies (E _Fs) of high- T _c superconductors (SCs) have of late been evincing considerable interest because they are believed to be the cause of their high T _cs and gap structures. Since Bardeen-Cooper-Schrieffer (BCS) equations for elemental and generalized-BCS equations for non-elemental SCs are derived under the blanket of the assumption E _F/ k θ > > 1 (k = Boltzmann constant, θ = Debye temperature), they cannot shed light on the E _Fs of these SCs. This fact leads us to address the gaps (Δ₀s) and T _cs of both types of SCs via recently derived equations which incorporate E _F as a variable. For the specification of the E _F of any SC, we now need another of its properties. Choosing j ₀, the critical current density of the SC at T = 0, and following an idea due to Pines, we present for both types of SCs new equations for j ₀ that depend solely on the following properties of the SC: E _F, θ, gram-atomic volume, electronic specific heat constant and a dimensionless construct \(y=k\theta \sqrt {2m\ast } \text {/}P_{\text {0}} \sqrt {E_{\mathrm {F}} } \text {,}\) where m* is the effective mass of superconducting electrons and P ₀ their critical momentum. Appeal to the experimental values of Δ₀, T _c and j ₀ of any SC then not only leads to values of E _F, m* and P ₀ but also provides plausible clues about how its j ₀—and therefore T _c—may be increased.     

Certification of YBa<Subscript>2</Subscript>Cu<Subscript>3</Subscript>O<Subscript>7−<Emphasis Type="Italic">x</Emphasis></Subscript> Single Crystals

A method for evaluation of the critical temperature T _c and the width of the superconducting transition ΔT _c in HTSC single crystals has been developed. By this method, the first derivative of the temperature dependence of the resistivity, \(\frac{\partial \rho (T)}{\partial T}\), is constructed. A technique for synthesis of YBa₂Cu₃O_7?x single crystals with highly reproducible physical-mechanical properties has been described. A standard sample with T _c=94 K and ΔT _c=0.25 K has been synthesized and certified.     

Symmetry of Two-Electron States in Unconventional Superconductors

A group-theoretical approach is developed to constructing two-electron states in the basis of one-electron states of crystals. Such states correspond to Cooper pairs in superconductors and pseudogap states (at T* > T _c ). The theory is applied to unconventional superconductors: heavy-fermion and high-T _c materials. Comparison of theoretical results with experimental data is used to gain insight into the structure of two-electron states in these materials.     

Characteristics of Superconducting State in Vanadium: the Eliashberg Equations and Semi-analytical Formulas

The superconducting state in vanadium characterizes with the critical temperature (T _c ) equal to 5.3 K. The Coulomb pseudopotential, calculated with the help of the Eliashberg equations, possesses anomalously high value μ ^?(3Ω_max) = 0.259 or μ ^?(10Ω_max) = 0.368 (Ω_max denotes the maximum phonon frequency). Despite the relatively large electron-phonon coupling constant (λ = 0.91), the quantities such as the order parameter (Δ), the specific heat (C), and the thermodynamic critical field (H _c ) determine the values of the dimensionless ratios not deviating much from the predictions of the BCS theory: R _Δ = 2Δ(0)/k _B T _c = 3.68, R _C = ΔC(T _c ) /C ^N (T _c ) = 1.69, and \(R_{H}=T_{c}C^{N}\left (T_{c}\right )\slash {H^{2}_{c}}\left (0\right )=0.171\). This result is associated with the reduction of the strong-coupling and the retardation effects by the high value of the Coulomb pseudopotential. It has been shown that the results of the Eliashberg formalism can be relatively precisely reproduced with the help of the semi-analytical formulas, if the value of μ ^? is determined on the basis of the T _c -Allen-Dynes expression (μ A D? = 0.198). The attention should be paid to the fact that in the numerical and in the semi-analytical approach the comparable values of the thermodynamic parameters for the same μ ^? have been obtained only in the vicinity of the point μ ^? = 0.1.     

Comparative Experimental and Density Functional Theory (<Emphasis Type="Italic">DFT</Emphasis>) Study of the Physical Properties of MgB<Subscript>2</Subscript> and AlB<Subscript>2</Subscript>

In the present study, we report an intercomparison of various physical and electronic properties of MgB₂ and AlB₂. In particular, the results of phase formation, resistivity ρ(T), thermoelectric power S(T), magnetization M(T), heat capacity (C _P ), and electronic band structure are reported. The original stretched hexagonal lattice with a=3.083 Å, and c=3.524 Å of MgB₂ shrinks in c-direction for AlB₂ with a=3.006 Å, and c=3.254 Å. The resistivity ρ(T), thermoelectric power S(T) and magnetization M(T) measurements exhibited superconductivity at 39 K for MgB₂. Superconductivity is not observed for AlB₂. Interestingly, the sign of S(T) is +ve for MgB₂ the same is ?ve for AlB₂. This is consistent with our band structure plots. We fitted the experimental specific heat of MgB₂ to Debye–Einstein model and estimated the value of Debye temperature (Θ _D) and Sommerfeld constant (γ) for electronic specific heat. Further, from γ, the electronic density of states (DOS) at Fermi level N(E _F) is calculated. From the ratio of experimental N(E _F) and the one being calculated from DFT, we obtained value of λ to be 1.84, thus placing MgB₂ in the strong coupling BCS category. The electronic specific heat of MgB₂ is also fitted below T _c using α-model and found that it is a two gap superconductor. The calculated values of two gaps are in good agreement with earlier reports. Our results clearly demonstrate that the superconductivity of MgB₂ is due to very large phonon contribution from its stretched lattice. The same two effects are obviously missing in AlB₂, and hence it is not superconducting. DFT calculations demonstrated that for MgB₂, the majority of states come from σ and π 2p states of boron on the other hand σ band at Fermi level for AlB₂ is absent. This leads to a weak electron phonon coupling and also to hole deficiency as π bands are known to be of electron type, and hence obviously the AlB₂ is not superconducting. The DFT calculations are consistent with the measured physical properties of the studied borides, i.e., MgB₂ and AlB₂.     

Inducement of Itinerant Electron Transport in Charge-Ordered Pr<Subscript>0.6</Subscript>Ca<Subscript>0.4</Subscript>MnO<Subscript>3</Subscript> by Ba Doping

The effects of Ba ²⁺ doping on the electrical and magnetic properties of charge-ordered Pr_0.6Ca_0.4MnO₃ were investigated through electrical resistivity and AC susceptibility measurements. X-ray diffraction data analysis showed an increase in unit cell volume with increasing Ba ²⁺ content indicating the possibility of substituting Ba ²⁺ for the Ca-site. Electrical resistivity measurements showed insulating behavior and a resistivity anomaly at around 220 K. This anomaly is attributed to the existence of charge ordering transition temperature, \(T^{\mathrm {R}}_{\text {CO}}\) for the x = 0 sample. The Ba-substituted samples exhibited metallic to insulator transition (MI) behavior, with transition temperature, T _MI, increasing from ～98 K (x = 0.1) to ～122 K (x = 0.3). AC susceptibility measurements showed ferromagnetic to paramagnetic (FM-PM) transition for Ba-substituted samples with FM-PM transition temperature, T _c, increasing from ～121 K (x = 0.1) to ～170 K (x = 0.3), while for x = 0, an antiferromagnetic to paramagnetic transition behavior with transition temperature, T _N, ～170 K was observed. In addition, inverse susceptibility versus T plot showed a deviation from the Curie–Weiss behavior above T _c, indicating the existence of the Griffiths phase with deviation temperature, T _G, increasing from 160 K (x = 0.1) to 206 K (x = 0.3). Magnetoresistance, MR, behavior indicates intrinsic MR mechanism for x = 0.1 which changed to extrinsic MR for x > 0.2 as a result of Ba substitution. The weakening of charge ordering and inducement of ferromagnetic metallic (FMM) state as well as increase in both T _c and T _MI are suggested to be related to the increase of tolerance factor, τ, and increase of e _g ?electron bandwidth as average ionic radius at A-site, <r _A> increased with Ba substitution. The substitution may have reduced MnO₆ octahedral distortion and changed the Mn–O–Mn angle which, in turn, promotes itinerancy of charge carrier and enhanced double exchange mechanism. On the other hand, increase in A-site disorder, which is indicated by the increase in σ ² is suggested to be responsible for the widening of the difference between T _c and T _MI.     

Effect of CdTe Addition on the Electrical Properties and AC Susceptibility of YBa<Subscript>2</Subscript>Cu<Subscript>3</Subscript>O<Subscript>7−<Emphasis Type="Italic">δ</Emphasis></Subscript> Superconductor

The effect of CdTe addition on YBa₂Cu₃O_7?δ (CdTe) _x (x = 0–0.10) has been studied. XRD patterns showed the presence of impurities including CdTe for x ≥ 0.05 wt%. The resistance versus temperature curves showed metallic behavior for all samples. The onset temperature T _{c onset} for all samples is between 90 and 92 K. The superconducting transition width, ΔT _c = 4 K for all samples except for x = 0.1 wt% where ΔT _c = 6 K. The superconducting transition determined by AC susceptibility measurement also showed little change in \(T_{\mathrm {c}\chi ^{\prime }}\) (between 89 and 92 K). However, the peak temperature T _p of the imaginary part of the susceptibility χ″, showed a drastic decrease for samples with x > 0.05 wt%. This indicated that the superconducting grains were strongly decoupled for x > 0.05 wt% due to the impurities as observed in the XRD pattern. The intergrain critical current density, J _c at T _p for the x = 0 sample is J _c (82 K) = 17 A cm^?2. These results can be useful in the fabrication of semiconductor/YBCO superconductor hybrid systems.