Buckling of a circular plate made of a shape memory alloy due to a reverse thermoelastic martensite transformation

We solve the axisymmetric buckling problem for a circular plate made of a shape memory alloy undergoing reverse martensite transformation under the action of a compressing load, which occurs after the direct martensite transformation under the action of a generally different (extending or compressing) load. The problem was solved without any simplifying assumptions concerning the transverse dimension of the supplementary phase transition region related to buckling. The mathematical problem was reduced to a nonlinear eigenvalue problem. An algorithm for solving this problem was proposed. It was shown that the critical buckling load under the reverse transition, which is obtained by taking into account the evolution of the phase strains, can be many times lower than the same quantity obtained under the assumption that the material behavior is elastic even for the least (martensite) values of the elastic moduli. The critical buckling force decreases with increasing modulus of the load applied at the preliminary stage of direct transition and weakly depends on whether this load was extending or compressing. In shape memory alloys (SMA), mutually related processes of strain and direct (from the austenitic into the martensite phase) or reverse thermoelastic phase transitions may occur. The direct transition occurs under cooling and (or) an increase in stresses and is accompanied by a significant decrease (nearly by a factor of three in titan nickelide) of the Young modulus. If the direct transition occurs under the action of stresses with nonzero deviator, then it is accompanied by accumulation of macroscopic phase strains, whose intensity may reach 8%. Under the reverse transition, which occurs under heating and (or) unloading, the moduli increase and the accumulated strain is removed. For plates compressed in their plane, in the case of uniform temperature distribution over the thickness, one can separate trivial processes under which the strained plate remains plane and the phase ratio has a uniform distribution over the thickness. For sufficiently high compressing loads, the trivial process of uniform compression may become unstable in the sense that, for small perturbations of the plate deflection, temperature, the phase ratio, or the load, the difference between the corresponding perturbed process and the unperturbed process may be significant. The results of several experiments concerning the buckling of SMA elements are given in [1, 2], and the statement and solution of the corresponding boundary value problems can be found in [3–11]. The experimental studies [2] and several analytic solutions obtained for the Shanley column [3, 4], rods [5–7], rectangular plates under direct [8] and reverse [9] transitions showed that the processes of thermoelastic phase transitions can significantly (by several times) decrease the critical buckling loads compared with their elastic values calculated for the less rigid martensite state of the material. Moreover, buckling does not occur in the one-phase martensite state in which the elastic moduli are minimal but in the two-phase state in which the values of the volume fractions of the austenitic and martensite phase are approximately equal to each other. This fact is most astonishing for buckling, studied in the present paper, under the reverse transition in which the Young modulus increases approximately half as much from the beginning of the phase transition to the moment of buckling. In [3–9] and in the present paper, the static buckling criterion is used. Following this criterion, the critical load is defined to be the load such that a nontrivial solution of the corresponding quasistatic problem is possible under the action of this load. If, in the problems of stability of rods and SMA plates, small perturbations of the external load are added to small perturbations of the deflection (the critical force is independent of the amplitude of the latter), then the critical forces vary depending on the value of perturbations of the external load [5, 8, 9]. Thus, in the case of small perturbations of the load, the problem of stability of SMA elements becomes indeterminate. The solution of the stability problem for SMA elements also depends on whether the small perturbations of the phase ratio and the phase strain tensor are taken into account. According to this, the problem of stability of SMA elements can be solved in the framework of several statements (concepts, hypotheses) which differ in the set of quantities whose perturbations are admissible (taken into account) in the process of solving the problem. The variety of these statements applied to the problem of buckling of SMA elements under direct martensite transformation is briefly described in [4, 5]. But, in the problem of buckling under the reverse transformation, some of these statements must be changed. The main question which we should answer when solving the problem of stability of SMA elements is whether small perturbations of the phase ratio (the volume fraction of the martensite phase q) are taken into account, because an appropriate choice significantly varies the results of solving the stability problem. If, under the transition to the adjacent form of equilibrium, the phase ratio of all points of the body is assumed to remain the same, then we deal with the “fixed phase atio” concept. The opposite approach can be classified as the “supplementary phase transition” concept (which occurs under the transition to the adjacent form of equilibrium). It should be noted that, since SMA have temperature hysteresis, the phase ratio in SMA can endure only one-sided small variations. But if we deal with buckling under the inverse transformation, then the variation in the volume fraction of the martensite phase cannot be positive. The phase ratio is not an independent variable, like loads or temperature, but, due to the constitutive relations, its variations occur together with the temperature variations and, in the framework of connected models for a majority of SMA, together with variations in the actual stresses. Therefore, the presence or absence of variations in q is determined by the presence or absence of variations in the temperature, deflection, and load, as well as by the system of constitutive relations used in this particular problem. In the framework of unconnected models which do not take the influence of actual stresses on the phase ratio into account, the “fixed phase ratio” concept corresponds to the case of absence of temperature variations. The variations in the phase ratio may also be absent in connected models in the case of specially chosen values of variations in the temperature and (or) in the external load, as well as in the case of SMA of CuMn type, for which the influence of the actual stresses on the phase compound is absent or negligible. In the framework of the “fixed phase ratio” hypothesis, the stability problem for SMA elements has a solution coinciding in form with the solution of the corresponding elastic problem, with the elastic moduli replaced by the corresponding functions of the phase ratio. In the framework of the supplementary phase transition” concept, the result of solving the stability problem essentially depends on whether the small perturbations of the external loads are taken into account in the process of solving the problem. The point is that, when solving the problem in the connected setting, the supplementary phase transition region occupies, in general, not the entire cross-section of the plate but only part of it, and the location of the boundary of this region depends on the existence and the value of these small perturbations. More precisely, the existence of arbitrarily small perturbations of the actual load can result in finite changes of the configuration of the supplementary phase transition region and hence in finite change of the critical values of the load. Here we must distinguish the “fixed load” hypothesis where no perturbations of the external loads are admitted and the “variable load” hypothesis in the opposite case. The conditions that there no variations in the external loads imply additional equations for determining the boundary of the supplementary phase transition region. If the “supplementary phase transition” concept and the “fixed load” concept are used together, then the solution of the stability problem of SMA is uniquely determined in the same sense as the solution of the elastic stability problem under the static approach. In the framework of the “variable load” concept, the result of solving the stability problem for SMA ceases to be unique. But one can find the upper and lower bounds for the critical forces which correspond to the cases of total absence of the supplementary phase transition: the upper bound corresponds to the critical load coinciding with that determined in the framework of the “fixed phase ratio” concept, and the lower bound corresponds to the case where the entire cross-section of the plate experiences the supplementary phase transition. The first version does not need any additional name, and the second version can be called as the "all-round supplementary phase transition" hypothesis. In the present paper, the above concepts are illustrated by examples of solving problems about axisymmetric buckling of a circular freely supported or rigidly fixed plate experiencing reverse martensite transformation under the action of an external force uniformly distributed over the contour. We find analytic solutions in the framework of all the above-listed statements except for the case of free support in the “fixed load” concept, for which we obtain a numerical solution.     

Oscillating gap 2π pulse in resonantly absorbing lattice

The interaction of a laser pulse with resonant Bragg lattice is studied theoretically for arbitrary initial conditions on the field, inverse population, and polarization of a medium. It is shown that the oscillating 2π self-induced transparency Bragg pulse can form if the Bragg conditions are exactly met. Various regimes are described for the oscillation dynamics of the gap 2π pulse.     

Electrical conduction of emulsion explosives

Results of experiments to obtain a spatial distribution of electrical conduction of emulsion explosives for different content of a sensitizer are presented.     

Synthesis and Characterization of α,ω‐bi[2,4‐dinitrophenyl (DNP)] poly(2‐methoxystyrene) Functional Polymers. Initial Evaluation of the Interaction of the Functional Polymers with RBL Mast Cells

Two series of functional polymers, α,ω‐bi[2,4‐dinitrophenyl][poly(ethylene oxide)‐b‐poly(2‐methoxystyrene)‐b‐poly(ethylene oxide)] (DNP‐PEO‐P2MS‐PEO‐DNP) and α,ω‐bi[2,4‐dinitrophenyl caproic][poly(ethylene oxide)‐b‐poly(2‐methoxystyrene)‐b‐poly(ethylene oxide)] (CDNP‐PEO‐P2MS‐PEO‐CDNP), were synthesized by anionic living polymerization. The polymers were characterized by FT‐IR, ¹H‐NMR and Gel Permeation Chromatography (GPC). The molecular weight distributions for the lower molecular weight functional polymers were slightly broad (1.3–1.5). However, the molecular weight distributions for higher molecular weight polymers were narrower (1.1–1.2). Differential scanning calorimetry (DSC) studies showed thermal transitions indicative of the presence of microphases in the polymer solid state. The polymers were white powders and soluble in tetrahydrofuran. The binding affinity of DNP‐PEO‐P2MS‐PEO‐DNP ligands towards anti DNP IgE was determined by titrations with fluorescently labeled FITC‐IgE. A water soluble CDNP‐PEO‐P2MS‐PEO‐CDNP/DMEG (dimethoxyethylene glycol) complex binds and achieves steady state binding with solution IgE within a few seconds. This strongly suggests that CDNP functional polymers with improved water solubility have potential in therapeutics. Higher molecular weight (water insoluble) CDNP‐PEO‐P2MS‐PEO‐CDNP polymers were electrosprayed as fibers (500 nm) on silicon surface. Fluorescence spectroscopy clearly showed that RBL mast cells were interacting with the fibers suggesting that the cell‐surface receptors were clustered along the fiber surface. These observations suggest that the functional polymers hold promise for developing an antibody detection device.     

Studies on the Complexation of Transition Metal Ions with Macrocyclic Compounds in Mixed Solvents by Competitive Potentiometry and Polarography

Complexation of Cu(II), Zn(II), Ni(II), Co(II), Pb(II), Cd(II), Cr(III) and Fe(III) ions with 15-crown-5, benzo-15-crown-5, 18-crown-6, dibenzo-18-crown-6, dicyclohexano-18-crown-6, dibenzo-24-crown-8, 5,6,14,15-dibenzo-1,4-dioxa-8,12-diazacyclopentadeca-5,14-diene, 1,4,10-trioxa-7,13-diazacyclopentadecane and cyclam has been investigated in 75% (v/v) DMF+water using the silver(I) ion as an auxiliary cation at 0.05?M ionic strength adjusted with tetrabutylammonium perchlorate (TBAP) by competitive potentiometry. Stability constant values obtained for the metal ion–aza macrocyclic complexes are higher than those for the oxa crowns. Shifts in the peak potential and reduction in the peak current in the differential pulse polarography (DPP) method were also used to determine the stability constants of some of the metal ions with aza and tosylated-aza macrocycles using TBAP as supporting electrolyte in 75% (v/v) DMF+water and 90% (v/v) DMSO+water media. Stability constant values determined both by shifts in the potential and by reduction in the peak current were found to be in good agreement with each other.     

Archetypical Conductive Polymer Structure for Specific Interaction with Proteins

A series of 2,4-dinitrophenyl (DNP) functionalized polypyrrole terpolymers, capable of specific binding to IgE antibodies (proteins), have been synthesized using oxidizing initiator, ammonium persulfate and were characterized by ¹H-NMR, IR, DSC, Light Scattering etc. The terpolymers were composed of monomers and macromonomers: monomer (pyrrole), macromonomer A (pyrrole with pendant ethylene glycol) and macromonomer B (pyrrole with pendant DNP) with specific functionality of conductance, processiblity and binding, respectively. The terpolymers are found to be semiconductive, (5 × 10^?6 S cm^?1) by itself without the addition of doping agents. Molecular dynamics simulation of terpolymer shows that the DNP-(2,4-dinitrophenyl) functional group extends out from the polymer backbone and thus, is available for binding. The DNP functional groups on the terpolymer achieve steady state binding with anti-DNP IgE proteins at nanomolar concentrations in solution. The terpolymer was blended with sulfonated polystyrene and processed in fibers which exhibited effective specific binding to fluorescently tagged IgE proteins and therefore, possessed the potential to be an active component in biosensor.     

Comparative Study of the Effects of Light Controlled Germination Conditions on Saponarin Content in Barley Sprouts and Lipid Accumulation Suppression in HepG2 Hepatocyte and 3T3-L1 Adipocyte Cells Using Barley Sprout Extracts

Barley sprouts (BS) contain physiologically active substances and promote various positive physiological functions in the human body. The levels of the physiologically active substances in plants depend on their growth conditions. In this study, BS were germinated using differently colored LED lights and different nutrient supplements. Overall, there were 238 varied BS samples analyzed for their total polyphenol and flavonoid contents. Principal component analysis (PCA) was performed to determine the relationship between the germinated samples and their total polyphenol and flavonoid contents, and those with high levels were further analyzed for their saponarin content. Based on the PCA plot, the optimal conditions for metabolite production were blue light with 0.1% boric acid supplementation. In vitro experiments using the ethanol extract from the BS cultured in blue light showed that the extract significantly inhibited the total lipid accumulation in 3T3-L1 adipocytes and the lipid droplets in HepG2 hepatocytes. These findings suggest that specific and controlled light source and nutrient conditions for BS growth could increase the production of secondary metabolites associated with inhibited fat accumulation in adipocytes and hepatocytes.     

A separable approximation to the two-body t-matrix for short-range central local potentials

The S-wave potential in momentum space, V₀(p, p′), corresponding to a local two-body central short-range potential may be expanded in separable form with the help of suitable quadrature formulae. With a two-term separable expansion for a variety of nuclear potentials, the resulting t-matrix is found to be in close agreement with the corresponding result with a six-term expansion for small values of p and p′ which are important in the calculation of the trinucleon binding energy.     

Two new cycloartane saponins from the roots of Astragalus membranaceus

Two new cycloartane-type triterpenoid saponins were isolated from the roots of Astragalus membranaceus (FISCH.) BGE. (Leguminosae) cultivated in Kangwon province, Korea. These saponins were named astramembranosides A and B and were established to be cycloastragenol 6,25-di-O-beta-D-glucopyranoside (astramembranoside A) and cyclocanthogenin 3-O-beta-D-glucopyranosyl(1-->2)-beta-D-xylopyranoside (astramembranoside B) on the basis of chemical and spectral evidence. In addition, 12 known saponins were also isolated from the same materials. Although cycloastragenol 3-O-xyloside and agroastragalosides I and II have already been isolated from A. membranaceus adventitious roots, these three saponins together with brachyoside B and azukisaponin V methyl ester were isolated for the first time from this plant.