An internal-variable rod model for stress-induced phase transitions in a slender SMA layer. I: Asymptotic equations and a two-phase solution

In part I of this series of two papers, an internal-variable rod model is proposed to study the stress-induced phase transitions in a slender shape memory alloy (SMA) layer. To study the mechanical responses of SMAs, two independent energy functions are adopted: the Helmholtz free energy and the rate of mechanical dissipation. A phase state variable is introduced to describe the phase transition process. Starting from the 2-D governing system and by using the coupled series-asymptotic expansion method, one single equilibrium equation is derived, which involves the leading order term of the axial strain and the phase state functions. Further by using the phase transition criteria, the evolution laws of the phase state functions corresponding to the outer loop of the stress–strain response are derived. As a result, the governing ODEs for the purely loading and purely unloading processes are obtained, which are called the asymptotic rod equations. The two-phase solution of the asymptotic rod equations in an infinitely long layer is then constructed. An explicit solution for the phase volume fraction of the corresponding inhomogeneous deformation is deduced, which appears to be the first analytical expression for this important quantity in stress-induced phase transitions. The key parameters in terms of original material constants for the phase transitions and deformations are also identified.     

On the thermodynamics of rate-type fluids and phase transitions. II. Phase transitions

The general thermodynamic restrictions on a shearless rate-type fluid obtained in part I of this paper are used together with a so called extended Holtzmann law to get specific forms of the constitutive functions. Next a van der Waals type equilibrium function is adopted. Under these conditions the internal energy is pressure independent and together with the entropy verify the whole set of conditions required by the two laws of thermodynamics. The explicit forms of the internal energy and of the entropy are also presented. It is shown that the system of partial differential equations governing the motion of the rate type fluid remains hyperbolic in a larger domain of the state space than the corresponding elastic system does and the rate-type system is still capable to describe phase transitions. An availability function (of density, temperature and pressure) relative to a reference equilibrium state is constructed. It is shown that the availability remains positive at any state if the reference density is taken outside the Maxwells interval at that reference temperature. Based on this property and on an energy identity/inequality, energy estimates are given and the way the equilibrium is attained, when appropriate, is also discussed.     

Approximate solutions of the heat conduction equation in the one-dimensional case with phase transitions on the outer surface of a body

Approximate solutions are presented for the heat conduction equation for a semi-infinite rod or half-space with sublimation at its outside boundary. The formulas presented are in a form convenient for practical computations.     

An analytical model for the recording of transitions in thin magnetic recording media with linearized hysteresis loops

Analytical expressions are derived for the magnetization distributions recorded in thin-film recording media possessing "linear' hysteresis loops. The results are presented in a form somewhat familiar to workers in this field.     

A theoretical model for the bending of a laminated beam with SMA fiber embedded layer

A theoretical model for the bending of a laminated beam bonded with shape memory alloy (SMA) fiber reinforced layer is presented. The constitutive relations of the SMA layer are obtained by using the method of micromechanics. The bending of the laminated beam is then discussed, and the relationship between bending moment, curvature and temperature are provided. The governing relationships obtained in this paper can be used for theoretical predications of thermomechanical properties of beam-like SMA actuators.     

Nonequilibrium phase transitions and S-shaped current-voltage characteristics in a model semiconductor-metal system

A model of nonequilibrium phase transition in a semiconductor-metal system was considered using a finite-element method. It is shown that, beginning with a certain fraction of the metal component in the heterophase system, the S-shaped current-voltage characteristics exhibit a change: the system features a hysteresis but possesses no singular points.     

Quantum coherence and phase transitions in granular superconductors with dissipation. II. Effect of disorder

We study the effect of disorder on a superconducting phase transition in granular arrays with dissipation. It is shown that, for a quite weak Gaussian disorder, this effect does not lead to a qualitative change of the phase diagram. Local quantum fluctuations suppress Cooper pair tunneling for a part of Josephson junctions and thus effectively increase disorder. For strongly disordered systems, the critical resistance which corresponds to a dissipative phase transition is found to be universal for a wide range of parameters.     

On the tracking of resonance and antiresonance of a piezoelectric resonator. II. Accurate models of the phase locked loop

For pt.I see ibid., vol.38, no.3, p.231 (May 1991). The previous model of the phase-locked loop for the tracking of resonance and antiresonance has been extended so that the observed transient waveforms of the voltage-controlled oscillator (VCO) may be reproduced for a wide range of off-resonant initial frequencies. The use of an accurate VCO response function and accurate admittance phase angles has brought significant improvements. The remaining discrepancies have been removed nearly completely by making an integrator within the loop filter slightly imperfect.     

Study of a spherical head model. Analytical and numerical solutions in fluid–structure interaction approach

This paper deals with the analytical and the numerical computation of elastoacoustic vibration modes. We determine in the present study the eigenfrequencies and the modal shapes of the system of brain, cerebro-spinal fluid (CSF) and skull. Two models are presented in this work: an elastic-acoustic model assuming a rigid skull and an elastic-acoustic-elastic model assuming a deformable skull. The analytical results are compared with the numerical solution obtained using the software Comsol Multiphysics. It is shown that eigenfrequencies and more significantly the modal shapes are strongly influenced by the interaction between solid phases (brain and skull) and the cerebro-spinal fluid. Finally the influence of the CSF compressibility and thickness on the natural frequencies was investigated.     

An integrated model for statistical and vision monitoring in manufacturing transitions

Manufacturing transitions have been increasing due to higher pressures for product variety. One dimension of this variety is color. A major quality control challenge is to regulate the color by capturing data on color in real‐time during the operation and to use it to assess the opportunities for good parts. Control charting, when applied to a stable state process, is an effective monitoring tool to continuously check for process shifts or upsets. However, the presence of transition events can impede the normal performance of a traditional control chart. In this paper, we present an integrated model for statistical and vision monitoring using a tracking signal to determine the start of the transition and a confirmation signal to ensure that any process oscillation has concluded. We also developed an automated color analysis and forecasting system (ACAFS) that we can adjust and calibrate to implement this methodology in different production processes. We use a color transition process in plastic extrusion to illustrate a transition event and demonstrate our proposed methodology. Copyright © 2003 John Wiley & Sons, Ltd.     

Approximate analytical temperature calculation for a fuel rod with defects in the contact layer

A solution has been obtained in infinite-series form for the two-dimensional thermal conduction equation for a rod with boundary conditions of the third kind and a heat-transfer coefficient varying around the perimeter.     

Analytical solutions for time-resolved fluorescence lifetime imaging in a turbid medium such as tissue.

An analytical solution is developed to quantify a site-specific fluorophore lifetime perturbation that occurs, for example, when the local metabolic status is different from that of surrounding tissue. This solution may be used when fluorophores are distributed throughout a highly turbid media and the site of interest is embedded many mean scattering distances from the source and the detector. The perturbation in lifetime is differentiated from photon transit delays by random walk theory. This analytical solution requires a priori knowledge of the tissue-scattering and absorption properties at the excitation and emission wavelengths that may be obtained from concurrent time-resolved reflection measurements. Additionally, the solution has been compared with the exact, numerically solved solution. Thus the presented solution forms the basis for practical lifetime imaging in turbid media such as tissue.     

An alternative model for elastic thermal stresses in two materials joined by a graded layer

The elastic stress distribution resulting from thermal mismatch in a film/graded layer/substrate system is a critical issue. Recently by a three-variable method Hsueh and Lee formulated a closed-form analytical solution [Hsueh CH, Lee S. Modeling of elastic thermal stresses in two materials joined by a graded layer. Composites: Part B 2003;34:747–52] in order to overcome the complexity of the traditional analytical models. The paper is devoted to developing an alternative analytical model for graded beams by Zhang’s two-variable method for multilayered beams. To illustrate applications of the present solution, specific results are calculated for the GaAs/graded GaAs–Si/Si systems. The transition phenomenon between tensile stress and compressive stress in the graded beam is investigated in detail for different graded parameters or different thickness. The present results agree well with existing analytical results. The differences between the present two-variable model and Hsueh’s three-variable model are also discussed.     

A closed loop ML algorithm for phase aberration correction in phased array imaging systems. II. Performance analysis [Ultrasound medical imaging

For pt.I see ibid., vol.44, pp.259-70 (1997). The performance of the maximum likelihood closed loop circuits, proposed in part I for phase aberration correction in phased array imaging systems, is analysed. This analysis is helpful in designing and also in determining the tracking mode phase error variance performance of both closed loop circuits (1-D and 2-D). It is an approximate analysis, suitable under small errors conditions, and considered accurate for high values of the signal to noise ratio.     

Long current loops as regular solutions of the equation for coupling currents in a flat two-layer superconducting cable

A wave-like equation describing coupling currents in a flat multi-strand two-layer superconducting cable has been solved to find slowly decaying non-uniform current loops. The maximum time constant of the loop decay is evaluated analytically. Relation between the uniform and non-uniform time constants and its impact on the loss measurement results are discussed.     

Analytical optical solution of the extension of the depth of field using cubic-phase wavefront coding. Part II. Design and optimization of the cubic phase

In this paper we use our derived approximate representation of the modulation transfer function to analytically solve the problem of the extension of the depth of field for two cases of interest: uniform quality imaging and task-based imaging. We derive the optimal result for each case as a function of the problem specifications. We also compare the two different imaging cases and discuss the advantages of using our optimization approach for each case. We also show how the analytical solutions given in this paper can be used as a convenient design tool as opposed to previous lengthy numerical optimizations.