A two-phase obstacle-type problem for the p-Laplacian

2008 ◽  
Vol 35 (4) ◽  
pp. 421-433 ◽  
Author(s):  
Anders Edquist ◽  
Erik Lindgren
Keyword(s):  
Type Problem ◽  
Two Phase

scholarly journals A two phase boundary obstacle-type problem for the bi-Laplacian

2022 ◽  
Vol 214 ◽  
pp. 112583
Author(s):  
Donatella Danielli ◽  
Alaa Haj Ali
Keyword(s):  
Phase Boundary ◽  
Type Problem ◽  
Two Phase

scholarly journals On a two-phase Serrin-type problem and its numerical computation

2020 ◽  
Vol 26 ◽  
pp. 65
Author(s):  
Lorenzo Cavallina ◽  
Toshiaki Yachimura
Keyword(s):  
Unique Solvability ◽  
Implicit Function Theorem ◽  
Divergence Form ◽  
Sufficient Condition ◽  
Implicit Function ◽  
Type Problem ◽  
Two Phase ◽  
Piecewise Constant ◽  
Constant Coefficients ◽  
Steepest Descent Algorithm

We consider an overdetermined problem of Serrin-type with respect to an operator in divergence form with piecewise constant coefficients. We give sufficient condition for unique solvability near radially symmetric configurations by means of a perturbation argument relying on shape derivatives and the implicit function theorem. This problem is also treated numerically, by means of a steepest descent algorithm based on a Kohn–Vogelius functional.

A remark on the two-phase obstacle-type problem for the p-Laplacian

2018 ◽  
Vol 11 (3) ◽  
pp. 325-334 ◽  
Author(s):  
Jun Zheng ◽  
Binhua Feng ◽  
Peihao Zhao
Keyword(s):  
Free Boundary ◽  
Hausdorff Measure ◽  
Type Problem ◽  
Two Phase ◽  
Locally Finite

AbstractIn this paper, we give a remark on the two-phase obstacle-type problem for the p-Laplacian when {1<p<2} and p is close to 2. We prove that the free boundary where the gradient vanishes has locally finite {(n-1)}-Hausdorff measure.

scholarly journals Gelfand-type problem for two-phase porous media

2014 ◽  
Vol 470 (2163) ◽  
pp. 20130573 ◽  
Author(s):  
Peter V. Gordon ◽  
Vitaly Moroz
Keyword(s):  
Porous Media ◽  
Heat Exchange ◽  
Thermal Explosion ◽  
Natural Extension ◽  
Type Problem ◽  
Two Phase ◽  
Single Temperature ◽  
Different Temperatures ◽  
Gelfand Problem ◽  
Interphase Heat Exchange

We consider a generalization of the Gelfand problem arising in Frank-Kamenetskii theory of thermal explosion. This generalization is a natural extension of the Gelfand problem to two-phase materials, where, in contrast to the classical Gelfand problem which uses a single temperature approach, the state of the system is described by two different temperatures. We show that similar to the classical Gelfand problem the thermal explosion occurs exclusively owing to the absence of stationary temperature distribution. We also show that the presence of interphase heat exchange delays a thermal explosion. Moreover, we prove that in the limit of infinite heat exchange between phases the problem of thermal explosion in two-phase porous media reduces to the classical Gelfand problem with renormalized constants.

Effect of Shock Loading on the Microstructure of Uniloy 326

1974 ◽  
Vol 32 ◽  
pp. 504-505
Author(s):  
K. P. Staudhammer ◽  
L. E. Murr
Keyword(s):  
Grain Size ◽  
Shock Loading ◽  
Current Investigation ◽  
Two Phase ◽  
05 C ◽  
Shock Deformation ◽  
Heat Treated

The effect of shock loading on a variety of steels has been reviewed recently by Leslie. It is generally observed that significant changes in microstructure and microhardness are produced by explosive shock deformation. While the effect of shock loading on austenitic, ferritic, martensitic, and pearlitic structures has been investigated, there have been no systematic studies of the shock-loading of microduplex structures.In the current investigation, the shock-loading response of millrolled and heat-treated Uniloy 326 (thickness 60 mil) having a residual grain size of 1 to 2μ before shock loading was studied. Uniloy 326 is a two phase (microduplex) alloy consisting of 30% austenite (γ) in a ferrite (α) matrix; with the composition.3% Ti, 1% Mn, .6% Si,.05% C, 6% Ni, 26% Cr, balance Fe.

Exsolution and inversion in pigeonite from the Whin Sill, Northern England

1978 ◽  
Vol 36 (1) ◽  
pp. 474-475
Author(s):  
P.P.K. Smith
Keyword(s):  
High Temperature ◽  
Low Temperature ◽  
Homogeneous Nucleation ◽  
Silicate Mineral ◽  
Antiphase Boundaries ◽  
Two Phase ◽  
Primitive Form ◽  
The Core ◽  
Nucleation Mechanisms ◽  
And Inversion

Grains of pigeonite, a calcium-poor silicate mineral of the pyroxene group, from the Whin Sill dolerite have been ion-thinned and examined by TEM. The pigeonite is strongly zoned chemically from the composition Wo8En64FS28 in the core to Wo13En34FS53 at the rim. Two phase transformations have occurred during the cooling of this pigeonite:- exsolution of augite, a more calcic pyroxene, and inversion of the pigeonite from the high- temperature C face-centred form to the low-temperature primitive form, with the formation of antiphase boundaries (APB's). Different sequences of these exsolution and inversion reactions, together with different nucleation mechanisms of the augite, have created three distinct microstructures depending on the position in the grain.In the core of the grains small platelets of augite about 0.02μm thick have farmed parallel to the (001) plane (Fig. 1). These are thought to have exsolved by homogeneous nucleation. Subsequently the inversion of the pigeonite has led to the creation of APB's.

Shock consolidation of Ni-Ti alloy powder

1986 ◽  
Vol 44 ◽  
pp. 430-431
Author(s):  
Naresh N. Thadhani ◽  
Thad Vreeland ◽  
Thomas J. Ahrens
Keyword(s):  
Alloy Powder ◽  
Amorphous Material ◽  
Ti Alloy ◽  
Two Phase ◽  
Near Surface ◽  
Optical Micrograph ◽  
Shock Compaction ◽  
Powder Particles ◽  
Ti Powder ◽  
Rapidly Solidified

A spherically-shaped, microcrystalline Ni-Ti alloy powder having fairly nonhomogeneous particle size distribution and chemical composition was consolidated with shock input energy of 316 kJ/kg. In the process of consolidation, shock energy is preferentially input at particle surfaces, resulting in melting of near-surface material and interparticle welding. The Ni-Ti powder particles were 2-60 μm in diameter (Fig. 1). About 30-40% of the powder particles were Ni-65wt% and balance were Ni-45wt%Ti (estimated by EMPA).Upon shock compaction, the two phase Ni-Ti powder particles were bonded together by the interparticle melt which rapidly solidified, usually to amorphous material. Fig. 2 is an optical micrograph (in plane of shock) of the consolidated Ni-Ti alloy powder, showing the particles with different etching contrast.

A Comparison of Tem and Apfim to the Interpretation of Modulated Microstructures

1985 ◽  
Vol 43 ◽  
pp. 70-71
Author(s):  
M.G. Burke ◽  
M.K. Miller
Keyword(s):  
Low Temperature ◽  
Microstructural Characterization ◽  
Superlattice Reflection ◽  
High Volume ◽  
Atom Probe ◽  
Dark Field ◽  
Miscibility Gap ◽  
Second Phase ◽  
Two Phase ◽  
Probe Field

Interpretation of fine-scale microstructures containing high volume fractions of second phase is complex. In particular, microstructures developed through decomposition within low temperature miscibility gaps may be extremely fine. This paper compares the morphological interpretations of such complex microstructures by the high-resolution techniques of TEM and atom probe field-ion microscopy (APFIM).The Fe-25 at% Be alloy selected for this study was aged within the low temperature miscibility gap to form a <100> aligned two-phase microstructure. This triaxially modulated microstructure is composed of an Fe-rich ferrite phase and a B2-ordered Be-enriched phase. The microstructural characterization through conventional bright-field TEM is inadequate because of the many contributions to image contrast. The ordering reaction which accompanies spinodal decomposition in this alloy permits simplification of the image by the use of the centered dark field technique to image just one phase. A CDF image formed with a B2 superlattice reflection is shown in fig. 1. In this CDF micrograph, the the B2-ordered Be-enriched phase appears as bright regions in the darkly-imaging ferrite. By examining the specimen in a [001] orientation, the <100> nature of the modulations is evident.

A TEM study of the influence of microstructure on the superplastic behavior of a U-5.8 wt.% Nb alloy

1986 ◽  
Vol 44 ◽  
pp. 568-569
Author(s):  
G. Mackiewicz Ludtka
Keyword(s):  
Grain Size ◽  
Heating Rate ◽  
Phase Field ◽  
Single Phase ◽  
Superplastic Behavior ◽  
Two Phase ◽  
Single Phase Field

Historically, metals exhibit superplasticity only while forming in a two-phase field because a two-phase microstructure helps ensure a fine, stable grain size. In the U-5.8 Nb alloy, superplastici ty exists for up to 2 h in the single phase field (γ1) at 670°C. This is above the equilibrium monotectoid temperature of 647°C. Utilizing dilatometry, the superplastic (SP) U-5.8 Nb alloy requires superheating to 658°C to initiate the α+γ2 → γ1 transformation at a heating rate of 1.5°C/s. Hence, the U-5.8 Nb alloy exhibits an anomolous superplastic behavior.

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