Transient and steady state creep behaviour of polycrystalline MgO

Stress change experiments during compressive creep tests at high stresses on polycrystalline MgO at 1596 K have shown that the creep rate at any instant during transient and steady state creep is predicted by the ratio,r/h, wherer is the rate of recovery (=??σ/t6t) andh is the coefficient of strain hardening (=?σ/?ε). Over most of transient and steady state creep, whenh is constant and the decrease in creep rate, \(\dot \in\) , is a direct result of a decrease inr, the variation of the creep strain,ε, with time,t, is accurately described as $$ \in = \in _0 + \in _T (1 - e^{ - mt} ) + \dot \in _s t$$ whereε ₀ is the instantaneous strain on loading,ε _T the transient creep strain,m relates to the rate of exhaustion of transient creep and \(\dot \in _s\) is the steady creep rate. Deviations from this equation occur during the initial 10 to 15% of the transient creep life, whenh increases rapidly. The variations in \(\dot \in\) ,r andh during transient and steady state creep are explained in terms of a model for creep in which the rate-determining process is the diffusion controlled growth of the three-dimensional dislocation network within subgrains to form dislocation sources allowing slip to occur.     

Work hardening and recovery during compressive creep of polycrystalline MgO

High precision equipment has been used to study the effects of small stress changes during steady state creep of magnesia at 1596 K. When the stress,σ ₁, is reduced by a small amount,Δσ, the creep rate decreased to zero for a period,Δt, before accelerating to a new steady value. Calculating the rate of recovery,r(=?δσ/δt) asΔσ/Δt and the coefficient of strain hardeningh(=δσ/δε) asΔσ/Δε (whereΔε is the instantaneous strain recorded on increasing the stress byΔσ) gave the ratio,r/h, which predicts accurately the observed steady creep rate, \(\dot \in _s \) . It is proposed that when \(\dot \in _s \) ∝ σ ³, creep is recovery controlled. The results are explained in terms of a model for creep in which the rate controlling process is the growth of the 3-D dislocation network within subgrains, to form dislocation sources allowing slip to occur.     

Creep strength of discontinuous fibre composites

A unidirectional, discontinuous fibre composite is considered under conditions of steady state creep in the direction of reinforcement. The composite consists of noncreeping, discontinuous, perfectly aligned, uniformly distributed fibres which are perfectly bonded to a matrix obeying a power relation between stress and strain rate. Expressions for the interface stress, the creep velocity profile adjacent to the fibres and the creep strength of the composite are derived. Previous results for the creep strength,σ _c obtained for composites of the same type are briefly reviewed and compared with the present result. It is shown that all results reduce to the same general expression $$\sigma _c = \alpha V_{f^{\sigma _0 } } \left( {\frac{{\dot \in }}{{\dot \in _{0 } }}} \right)1/n_{\rho ^{1 + 1/n} }$$ in whichρ is the fibre aspect ratio, \(\dot \in\) is the composite creep rate,V _f is the fibre volume fraction,σ ₀,ε ₀ andn are the constants in the matrix creep law. The creep strength coefficient α is found to be very weakly dependent onV _f and practically independent ofn whenn is greater than about 6.     

Generalized fracture mechanics of ductile steels

The generalized fracture mechanics approach is applied to two ductile steels, namely mild steel and 18/8 stainless steel in plane stress. The theory defines a fracture parameter \(\mathcal{T}\) , which is a truly plastic analogue of theJ contour integral and, for an edge crack specimen, is given by $$\mathcal{T} = k_1 ( \in _0 )cW_{0_c } $$ wherek ₁ is an explicit function,c is the crack length andε ₀, W_0c are respectively the strain and input energy density at fracture, remote from the crack. The functionk ₁(ε _o) is derived experimentally and the constancy of \(\mathcal{T}\) with respect to crack length and applied load is demonstrated. The variation of \(\mathcal{T}\) with crack extension during slow growth is investigated, as is the rate dependence of \(\mathcal{T}\) in mild steel.     

A dislocation network model of recovery-controlled creep

A new model of recovery-controlled creep deformation, based on the jerky glide motion of dislocations between obstacles, is proposed. A three-dimensional distribution of dislocation links is visualized such that only links which attain a certain threshold size,λ _a, through recovery can glide rapidly until they are again arrested at the next obstacle. The rate of mobilization of arrested dislocations is shown to be directly proportional to the annihilation rate, ?_a. The strain rate, γ, during transient creep is related to the annihilation rate, the obstacle spacingL and the Burgers vectorb of the dislocations according to the expression $$\gamma = \alpha _1 \psi (t)\dot \varrho bL$$ where α₁ is a geometrical constant and ψ(t) is a time-dependent parameter whose value is determined by the instantaneous (free) dislocation density as well as some salient features of the dislocation distribution. At steady state, ψ(t) translates into a constant which is stress and temperature independent. The average effective dislocation velocity is also shown to be directly proportional to the annihilation rate. The model is used to rationalize the familiar creep transients which are usually observed when the stress is altered abruptly during recovery creep.     

1-generator quasi-cyclic codes over finite chain rings

Let R be an arbitrary commutative finite chain ring with $1\ne 0$ . 1-generator quasi-cyclic (QC) codes over R are considered in this paper. Let $\gamma $ be a fixed generator of the maximal ideal of R, $F=R/\langle \gamma \rangle $ and $|F|=q$ . For any positive integers m, n satisfying $\mathrm{gcd}(q,n)=1$ , let $\mathcal{R}_n=R[x]/\langle x^n-1\rangle $ . Then 1-generator QC codes over R of length mn and index m can be regarded as 1-generator $\mathcal{R}_n$ -submodules of the module $\mathcal{R}_n^m$ . First, we consider the parity check polynomial of a 1-generator QC code and the properties of the code determined by the parity check polynomial. Then we give the enumeration of 1-generator QC codes with a fixed parity check polynomial in standard form over R. Finally, under the condition that $\mathrm{gcd}(|q|_n,m)=1$ , where $|q|_n$ denotes the order of q modulo n, we describe an algorithm to list all distinct 1-generator quasi-cyclic codes with a fixed parity check polynomial in standard form over R of length mn and index m.     

Phase stability of \upbeta-MoSi2−x prepared by the Na flux method against thermal, oxidative, and mechanical treatments

The single $\upbeta$ -MoSi₂ phase was prepared by the Na flux method and its stability against thermal, oxidative, and mechanical treatments was investigated. The X-ray diffraction results show that the single $\upbeta$ phase is formed at 600 °C within 1 h using pre-mixed Mo and Si powders with a Si/Mo molar ratio of 2.00–2.25. By energy-dispersive X-ray spectroscopy, the produced powder is found to be Si-deficient with a Si/Mo molar ratio of 1.87–1.96. The differential thermal analysis shows that the $\upbeta$ phase transforms into the $\upalpha$ -MoSi₂ phase at 815 °C at 10 K/min with the segregation of a small amount of Mo₅Si₃. The transformation heat is ?5.5 kJ/mol and the activation energy calculated by the Kissinger method is 290 kJ/mol. Thermogravimetry reveals that the $\upbeta$ -MoSi_2?x powder oxidizes significantly at 400–600 °C via the pest oxidation mechanism while it is resistant to oxidation at 700 °C for 5 h similarly to the $\upalpha$ -MoSi₂ phase. At last, mechanical milling on the $\upbeta$ -MoSi_2?x powder with a planetary ball mill up to 216 h demonstrates that this powder is stable under a severe mechanical treatment.     

Evaluation of microstructural instability during the differential strain rate test of a superplastic alloy

Stress (σ)-strain rate ( \(\dot \varepsilon \) ) data of banded and elongated grain microstructures of the Pb-Sn eutectic alloy were analysed over 298 to 443 K to evaluate microstructural instability during differential strain rate tests in the superplastic region. With reference to a stable equiaxed microstructure exhibiting uniqueσ- \(\dot \varepsilon \) relation, banded structure is more susceptible to strain hardening while the elongated grain microstructure exhibits either strain softening or strain hardening depending on the test temperature. This flow behaviour is considered in terms of a change in grain size, represented by the cube root of the grain volume. Activation energy for grain growth calculated from the differential strain rate test data indicates that the activation energy depends on strain rate and type of microstructure.     

An approximate approach for the calculation ofM s in iron-base alloys

Having estimated the critical driving force associated with martensitic transformation,ΔG ^α→M, as $$\Delta G^{\alpha \to M} = 2.1 \sigma + 900$$ whereσ is the yield strength of austenite atM _s, in MN m^?2, we can directly deduce theM _s by the following equation: $$\Delta G^{\gamma \to {\rm M}} |_{M_S } = \Delta G^{\gamma \to \alpha } + \Delta G^{\alpha \to M} = 0.$$ The calculatedM _s are in good agreement with the experimental results in Fe-C, Fe-Ni-C and Fe-Cr-C, and are consistent with part of the data in Fe-Ni, Fe-Cr and Fe-Mn alloys. Some higher “M _s” determined in previous works may be identified asM _a,M _s of surface martensite or bainitic temperature. TheM _s of pure iron is about 800 K. TheM _s in Fe-C can be approximately expressed as $$M_S (^\circ {\text{C}}) = 520 {\text{--- }}\left[ {{\text{\% C}}} \right]{\text{ }}x 320.$$ In Fe-X, the effect of the alloying element onM _s depends on its effect onT ₀ and on the strengthening of austenite. An approach for calculation of ΔG ^γ→α in Fe-X-C is suggested. Thus dM _s/dx _c in Fe-X-C is found to increase with the decrease of the activity coefficient of carbon in austenite.     

An extension of the (strong) primitive normal basis theorem

An extension of the primitive normal basis theorem and its strong version is proved. Namely, we show that for nearly all \(A = {\small \left( \begin{array}{cc} a&{}b \\ c&{}d \end{array} \right) } \in \mathrm{GL}_2(\mathbb {F}_{q})\) , there exists some \(x\in \mathbb {F}_{q^m}\) such that both \(x\) and \((-dx+b)/(cx-a)\) are simultaneously primitive elements of \(\mathbb {F}_{q^m}\) and produce a normal basis of \(\mathbb {F}_{q^m}\) over \(\mathbb {F}_q\) , granted that \(q\) and \(m\) are large enough.     

Stick-slip behaviour of model granular materials in drained triaxial compression

Drained triaxial axisymmetric compression tests are performed on water-saturated short cylindrical samples of nearly monodisperse glass beads, initially assembled in a loose state by a moist tamping technique. Both deviator stress $q$ and volumetric strain $\epsilon _v$ , measured as functions of axial strain $\epsilon _a$ , for different strain rates, are affected by stick-slip events of very large amplitude, while the classical behavior of loose, contractant granular assemblies, approaching the critical state for large $\epsilon _a$ , corresponds to the upper envelop of the stress-strain behaviour. Those events consist in $(i)$ a very fast (slip) part in which a drop of $q$ coincides with a jump of $\epsilon _v$ (contraction), while loss of control of $\epsilon _a$ and generation of pore pressure signal a dynamic collapse of the material structure triggered by an instability; and then $(ii)$ a quasi-static (stick) part in which the sample regains its strength and, over a short strain interval, behaves similarly to a denser system that dilates before reaching its critical state. A unique stress-dilatancy relation applies to all stick-slip events. Apparent internal friction angles and effects of strain rate and confining pressure are discussed, and it is argued that stick-slip instabilities originate in physico-chemical aging phenomena coupled to contact mechanics.     

Strain-rate sensitivity index of thermoplastics

Strain-rate sensitivity index, m, values of several thermoplastics (HDPE, PP, PMMA, PS, PVC, PC, and PA) were determined at ambient temperature by both variable strain-rate and stressrelaxation methods. Specimens were loaded in tension in the elastic portion of the stress-strain curve at various strain rates and the load was recorded as a function of elongation. Index values were determined from the relation $m = [\partial \ln (\sigma )]/[\partial \ln (\dot e)]_{e, T} $ . Specimens were also loaded in tension at constant strain rate to the proportional limit, loading was halted, and the load was recorded as a function of time at constant strain. A numerical algorithm was implemented to minimize the root-mean-square difference between an empirical equation and the experimental data, i.e. $$\Phi (n, \tau ) = \left( {1/N\sum\limits_i {\{ P_o \exp [ - (t_i /\tau )^n ] - P(t_i )\} } ^2 } \right)^{1/2} $$ The characteristic time parameter, (τ), and the rate-of-decay parameter, n, were found when Φ(n,τ) was minimized. Index values were determined from the relation $m = {\text{[}}\partial {\text{ln}} {\text{(}}P{\text{)]/[}}\partial {\text{ln(}} {\text{ - }} \dot P{\text{)]}}_{e,{\text{ }}T} $ . A marked difference in index values derived from both experimental methods indicates that different processes are operative in each case. Index values are qualitatively evaluated in terms of cohesive energy density, side-chain group molar volume, and main-chain group flexibility.     

Primitive elements with prescribed trace

Let \(q\) be a power of a prime number \(p\) . Let \(n\) be a positive integer. Let \(\mathbb {F}_{q^n}\) denote a finite field with \(q^n\) elements. In this paper, we consider the existence of the some specific elements in the finite field \(\mathbb {F}_{q^n}\) . We get that when \(n\ge 29\) , there are elements \(\xi \in \mathbb {F}_{q^n}\) such that \(\xi +\xi ^{-1}\) is a primitive element of \(\mathbb {F}_{q^n}\) , and \(\mathrm{Tr}(\xi ) = a, \mathrm{Tr}(\xi ^{-1}) = b\) for any pair of prescribed \(a, b \in \mathbb {F}_q^*\) .     

Is There a Unified Description of Conductivity of Layered Cuprates ?

We present a novel approach to the analysis of the normal state in-plane $\sigma _{ab} $ and out-of-plane σ_c conductivities of anisotropic layered crystals such as oxygen deficient YBa ₂ Cu ₃ O _x . It can be shown that the resistive anisotropy is determined by the ratio of the phase coherence lengths in the respective directions; i.e., $\sigma _{ab} /\sigma _c = \ell _{ab}^2 /\ell _c^2 $ . From the idea that at all doping levels and temperatures T the out-of-plane transport in these crystals is incoherent, follows that $\ell _c $ is T-independent, equal to the spacing $\ell _0 $ between the neighboring bilayers. Thus, the T-dependence of $\ell _{ab} $ is given by the measured anisotropy, and $\sigma _{ab} (\ell _{ab} )$ dependence is obtained by plotting $\sigma _{ab} {\text{ }}vs{\text{ }}\ell = {\text{ (}}\sigma _{ab} /\sigma _c )^{1/2} \ell _0 $ .The analysis of several single crystals of YBa ₂ Cu ₃ O _x (6.35 < x < 6.93) shows that for all of them $\sigma _{ab} (\ell ) $ is described by a universal dependence $\sigma _{ab} /\overline \sigma = f(\ell /\overline \ell ) $ with doping dependent parameters $\overline \sigma {\text{ }}and{\text{ }}\overline \ell $ .     

Surface and grain-boundary energies in yttria-stabilized zirconia (YSZ-8 mol%)

The multiphase equilibration technique has been used to measure the equilibrium angles that develop at the interphase boundaries of a solid-liquid-vapour system after annealing and also the surface (γ_sv)and the grain-boundary, (γ_ss) energies of polycrystalline yttria-stabilized zirconia (8 mol% Y₂O₃). The data was recorded in the temperature range 1573–1873 K. Linear temperature functions were obtained for the surface energy $$\gamma _{SV} (Jm^{ - 2} ) = 1.927 - 0.428x10^{ - 3} T$$ and for the grain-boundary energy $$\gamma _{SS} (Jm^{ - 2} ) = 1.215 - 0.358x10^{ - 3} T$$      

Surface,grain-boundary and interfacial energies in Al2O3 and Al2O3-Sn,Al2O3-Co systems

Using the multiphase equilibrium method for the measurement of contact angles, the surface and grain-boundary energies of polycrystalline Al₂O₃ in the temperature range of 1473 to 1923 K were determined. Linear temperature functions were obtained by extrapolation for both quantities between absolute zero and the melting point of Al₂O₃. The temperature dependence of the surface and grain boundary energies can be expressed as $$\gamma _{{\rm A}l_2 O_3 } = 2.559 - 0.784 \times 10^{ - 3} T(J m^{ - 2} )$$ and $$\gamma _{{\rm A}l_2 O_3 - Al_2 O_3 = } 1.913 - 0.611 \times 10^{ - 3} T(J m^{ - 2} )$$ respectively. The interfacial energies of Al₂O₃ in contact with the molten metals tin and cobalt revealed a linear dependence on temperature.     

Features in the formation of Ge/Si multilayer nanostructures under ion-beam-assisted crystallization

A method of combined ion-beam crystallization of the Ge/Si multilayer nanostructures is proposed. Using atomic-force microscopy and electron microscopy, we observed the formation of an array of germanium quantum dots with lateral dimensions 〈a〉 = 12–15 nm at the following conditions: silicon-substrate temperature T = 330–350°C, ion-beam energies E _Ge + = 30–40 eV, $E_{Ar^ + }^0 = 230 - 240$ eV (primary pulsed defect formation mode), $E_{Ar^ + } = 130 - 140$ eV (permanent diffusion stimulation mode), and ion-beam fluences $f_{Ge^ + } = 1.5 \times 10^{14} cm^{ - 2} s^{ - 1} $ , $f_{Ar^ + } = 5 \times 10^{12} cm^{ - 2} s^{ - 1} $ . The Raman spectroscopy data indicate the experimental possibility of low-temperature ion-stimulated growth of the spacer layers of silicon (T = 420–450°C, $E_{Ar^ + } = 80 - 90$ eV, $E_{Si^ + } = 30 - 40$ eV, $f_{Si^ + } = 3.5 \times 10^{14} cm^{ - 2} s^{ - 1} $ ) and the formation of multilayer structures with Ge _x Si_{1 ? x} quantum dots (x > 0.85).     

Dislocation structures in a { \bar{1} 104}/��11 \bar{2} 0�� low-angle tilt grain boundary of alumina (��-Al2O3)

An alumina (??-Al₂O₃) bicrystal with a ( $ \bar{1} $ 104)/[11 $ \bar{2} $ 0] 2^o low-angle tilt grain boundary was fabricated by diffusion bonding at 1500 °C in air, and the grain boundary was observed by transmission electron microscopy (TEM). High-resolution TEM observations revealed that the grain boundary consists of at least two kinds of dislocations. One is a perfect dislocation which has a Burgers vector of 1/3[ $ \bar{1} $ 2 $ \bar{1} $ 0]. The other is dissociated into two partial dislocations with a stacking fault on the (0001) plane, and each partial dislocation has a 1/6[ $ \bar{1} $ 101] edge component. It is suggested from structural considerations that the dissociated-dislocation pair originates from a b = 1/3[02 $ \bar{2} $ 1] perfect dislocation (i.e., 1/3[02 $ \bar{2} $ 1] ?? 1/6[02 $ \bar{2} $ 1] + 1/6[02 $ \bar{2} $ 1]). This dissociation produces a stacking fault in the anion sublattice. The stacking fault energy is estimated to be roughly 1.3 Jm^?2 based on the elastic theory. The authors discuss the dislocation structures and the stacking fault formed on the (0001) plane in detail.