Nanomaterials have attracted considerable interest with numerous technological developments in the last decade. Nanomaterials exhibit different physicochemical properties compared with their bulk counterparts because the diameters of the nanoparticles are less than the Bohr exciton radius. Cerium oxide based materials have been extensively studied for various technological applications. In the present study, the application of nanocrystalline cerium oxide for improvement of high–temperature–oxidation resistance of stainless steel has been studied. The role of coating of nanocrystalline cerium oxide towards improvement of high–temperature–oxidation resistance has been investigated and compared with that of the micrometre–sized cerium oxide particles. It was observed that nanocrystalline ceria improved high–temperature–oxidation resistance of AISI 304 stainless steel to a large extent compared with the micrometre–sized ceria coating. Nanocrystalline ceria coating decreased the isothermal parabolic rate constant of oxidation by more than two orders of magnitude compared with that of the bare alloy. The resistance to oxide scale spallation was also found to improve with the coating of cerium oxide nanoparticles. Secondary ion mass spectroscopy (SIMS) study of nanocrystalline ceria–coated and oxidized specimens revealed the presence of nanoceria at the outermost oxide surface, indicating a change in the oxide scale growth mechanism from outward cation diffusion to inward oxygen diffusion in the bare alloy at high temperature in dry air. The oxide scale morphology was studied using a scanning electron microscope (SEM) while a focused ion beam (FIB) technique was used to study the oxide–alloy interface. X–ray photoelectron spectroscopy (XPS) study of nanocrystalline ceria showed the presence of Ce3+ and Ce4+ oxidation states. It is proposed that the presence of the Ce3+ oxidation state in nanocrystalline ceria improves the oxidation resistance of stainless steel, and the related mechanisms are discussed.
{"title":"Nanocrystalline ceria imparts better high–temperature protection","authors":"S. Patil, S. Kuiry, S. Seal","doi":"10.1098/rspa.2004.1352","DOIUrl":"https://doi.org/10.1098/rspa.2004.1352","url":null,"abstract":"Nanomaterials have attracted considerable interest with numerous technological developments in the last decade. Nanomaterials exhibit different physicochemical properties compared with their bulk counterparts because the diameters of the nanoparticles are less than the Bohr exciton radius. Cerium oxide based materials have been extensively studied for various technological applications. In the present study, the application of nanocrystalline cerium oxide for improvement of high–temperature–oxidation resistance of stainless steel has been studied. The role of coating of nanocrystalline cerium oxide towards improvement of high–temperature–oxidation resistance has been investigated and compared with that of the micrometre–sized cerium oxide particles. It was observed that nanocrystalline ceria improved high–temperature–oxidation resistance of AISI 304 stainless steel to a large extent compared with the micrometre–sized ceria coating. Nanocrystalline ceria coating decreased the isothermal parabolic rate constant of oxidation by more than two orders of magnitude compared with that of the bare alloy. The resistance to oxide scale spallation was also found to improve with the coating of cerium oxide nanoparticles. Secondary ion mass spectroscopy (SIMS) study of nanocrystalline ceria–coated and oxidized specimens revealed the presence of nanoceria at the outermost oxide surface, indicating a change in the oxide scale growth mechanism from outward cation diffusion to inward oxygen diffusion in the bare alloy at high temperature in dry air. The oxide scale morphology was studied using a scanning electron microscope (SEM) while a focused ion beam (FIB) technique was used to study the oxide–alloy interface. X–ray photoelectron spectroscopy (XPS) study of nanocrystalline ceria showed the presence of Ce3+ and Ce4+ oxidation states. It is proposed that the presence of the Ce3+ oxidation state in nanocrystalline ceria improves the oxidation resistance of stainless steel, and the related mechanisms are discussed.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":"59 1","pages":"3569 - 3587"},"PeriodicalIF":0.0,"publicationDate":"2004-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74278041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Finance is one of the fastest growing areas in mathematics. In some senses it is not a discipline in its own right, but rather an application area in which mathematicians with backgrounds in probability theory, statistics, optimal control, convex and functional analysis and partial differential equations can bring to bear experiences and results from their own fields to problems of real world interest. In this survey we begin with the simplest possible financial model, and then give an account of the Black–Scholes option pricing formula, in which the key ideas are the replication of option pay–offs and pricing under the risk–neutral measure. Then we move on to discuss other important problems in finance, including the general theory for semi–martingale price processes, pricing in incomplete markets, interest–rate models and credit risk. The emphasis is on techniques and methodologies from stochastic processes.
{"title":"Review Paper. A survey of mathematical finance","authors":"D. Hobson","doi":"10.1098/rspa.2004.1386","DOIUrl":"https://doi.org/10.1098/rspa.2004.1386","url":null,"abstract":"Finance is one of the fastest growing areas in mathematics. In some senses it is not a discipline in its own right, but rather an application area in which mathematicians with backgrounds in probability theory, statistics, optimal control, convex and functional analysis and partial differential equations can bring to bear experiences and results from their own fields to problems of real world interest. In this survey we begin with the simplest possible financial model, and then give an account of the Black–Scholes option pricing formula, in which the key ideas are the replication of option pay–offs and pricing under the risk–neutral measure. Then we move on to discuss other important problems in finance, including the general theory for semi–martingale price processes, pricing in incomplete markets, interest–rate models and credit risk. The emphasis is on techniques and methodologies from stochastic processes.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":"110 1","pages":"3369 - 3401"},"PeriodicalIF":0.0,"publicationDate":"2004-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91261764","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
I. Eltayeb, E. A. Hamza, J. Jervase, E. Krishnan, D. Loper
We investigate the stability of a vertical interface separating two semi–infinite fluids with differing composition of light material and permeated by a magnetic field. Both fluids possess finite kinematic viscosity, ν, thermal diffusivity, κ, magnetic diffusivity, η, and negligible material diffusion. The stability depends on six dimensionless parameters: the Prandtl number, σ (where σ = ν/ κ), the magnetic Prandtl number, σm = ν/η, the Chandrasekhar number, Qc, the Reynolds number, Re, and the ratios, Bv, Γ of the vertical and normal components to the lateral component of field. A comprehensive study of the dependence of the stability on the parameters is made when Re is small. The presence of a horizontal magnetic field tends to reduce the growth rate of the non–magnetic modes and can also give rise to new modes of instability. The addition of a vertical component of field can completely counteract the stabilizing influence of the horizontal component. For field strengths in excess of some value dependent on σ, σm and Bv, the non–magnetic unstable mode is replaced by one of two magnetic modes, one of which is a roll aligned with the field and the other inclined to it. The helicity and α–effect of the small–scale unstable motions are also discussed.
{"title":"Compositional convection in the presence of a magnetic field. I. A single interface","authors":"I. Eltayeb, E. A. Hamza, J. Jervase, E. Krishnan, D. Loper","doi":"10.1098/rspa.2004.1375","DOIUrl":"https://doi.org/10.1098/rspa.2004.1375","url":null,"abstract":"We investigate the stability of a vertical interface separating two semi–infinite fluids with differing composition of light material and permeated by a magnetic field. Both fluids possess finite kinematic viscosity, ν, thermal diffusivity, κ, magnetic diffusivity, η, and negligible material diffusion. The stability depends on six dimensionless parameters: the Prandtl number, σ (where σ = ν/ κ), the magnetic Prandtl number, σm = ν/η, the Chandrasekhar number, Qc, the Reynolds number, Re, and the ratios, Bv, Γ of the vertical and normal components to the lateral component of field. A comprehensive study of the dependence of the stability on the parameters is made when Re is small. The presence of a horizontal magnetic field tends to reduce the growth rate of the non–magnetic modes and can also give rise to new modes of instability. The addition of a vertical component of field can completely counteract the stabilizing influence of the horizontal component. For field strengths in excess of some value dependent on σ, σm and Bv, the non–magnetic unstable mode is replaced by one of two magnetic modes, one of which is a roll aligned with the field and the other inclined to it. The helicity and α–effect of the small–scale unstable motions are also discussed.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":"23 1","pages":"3505 - 3528"},"PeriodicalIF":0.0,"publicationDate":"2004-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83708099","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ocean waves propagating beneath a sea–ice sheet encounter a variety of inhomogeneities, which normally arise because of the dynamic nature of the ice veneer over large physical scales. Zones of thinner, thicker, rougher or ridged ice, changes of material property, and abrupt transitions into and from open water, for example, each have their own distinctive scattering kernel that modifies the incoming wave energy spectrum as it progresses further into the ice cover. Here we present a theoretical analysis of wave propagation beneath sea–ice, where the ice is allowed to vary spatially. Isolated irregularities such as pressure ridges, rafted regions, ice islands that have become trapped in the sea–ice, open and refrozen leads, etc., are considered, as well as groups of such features, with their peripheries either welded to the surrounding ice sheet or separated from it by a free crack. Reflection and transmission coefficients, plotted as functions of wave period or wavelength, reveal considerable fine structure, including in some cases a comb of wave frequencies at which perfect transmission occurs. The work generalizes and extends work by Squire & Dixon, Williams & Squire and Evans & Porter, which all deal with abrupt transitions, to properly allow for inhomogeneity in the ice cover. For the multiple, randomly shaped, oriented and spaced irregularities observed in a real ice sheet, good agreement is found between the full solution, a wide–spacing approximation that neglects the evanescent parts of the wave field in subsequent interactions, and a simple serial approach where interactions between features are neglected and the effect of each irregularity is computed in sequence.
在冰盖下传播的海浪遇到了各种各样的不均匀性，这通常是由于大物理尺度上冰盖的动态性质造成的。例如，更薄、更厚、更粗糙或脊状冰的区域，物质性质的变化，以及进入和离开开阔水域的突然转变，每一个都有自己独特的散射核，当入射波进一步进入冰盖时，散射核会改变入射波的能谱。在这里，我们提出了一个理论分析，在海冰下，波的传播是允许空间变化的。孤立的不规则现象，如压力脊、漂流区、被困在海冰中的冰岛、开放和重新冻结的铅等，以及这些特征的群体，它们的外围要么与周围的冰盖焊接在一起，要么被自由裂缝与冰盖分离。以波周期或波长的函数表示的反射和透射系数揭示了相当精细的结构，包括在某些情况下出现完美透射的梳状波频率。这项工作概括和扩展了Squire & Dixon, Williams & Squire和Evans & Porter的工作，这些工作都处理突变，以适当地考虑冰盖的不均匀性。对于在真实冰盖中观测到的多种、随机形状、定向和间隔的不规则性，在完整解、忽略波场在随后相互作用中消失部分的宽间距近似和忽略特征之间相互作用并按顺序计算每个不规则性影响的简单串行方法之间发现了很好的一致性。
{"title":"Oblique scattering of plane flexural–gravity waves by heterogeneities in sea–ice","authors":"T. Williams, V. Squire","doi":"10.1098/rspa.2004.1363","DOIUrl":"https://doi.org/10.1098/rspa.2004.1363","url":null,"abstract":"Ocean waves propagating beneath a sea–ice sheet encounter a variety of inhomogeneities, which normally arise because of the dynamic nature of the ice veneer over large physical scales. Zones of thinner, thicker, rougher or ridged ice, changes of material property, and abrupt transitions into and from open water, for example, each have their own distinctive scattering kernel that modifies the incoming wave energy spectrum as it progresses further into the ice cover. Here we present a theoretical analysis of wave propagation beneath sea–ice, where the ice is allowed to vary spatially. Isolated irregularities such as pressure ridges, rafted regions, ice islands that have become trapped in the sea–ice, open and refrozen leads, etc., are considered, as well as groups of such features, with their peripheries either welded to the surrounding ice sheet or separated from it by a free crack. Reflection and transmission coefficients, plotted as functions of wave period or wavelength, reveal considerable fine structure, including in some cases a comb of wave frequencies at which perfect transmission occurs. The work generalizes and extends work by Squire & Dixon, Williams & Squire and Evans & Porter, which all deal with abrupt transitions, to properly allow for inhomogeneity in the ice cover. For the multiple, randomly shaped, oriented and spaced irregularities observed in a real ice sheet, good agreement is found between the full solution, a wide–spacing approximation that neglects the evanescent parts of the wave field in subsequent interactions, and a simple serial approach where interactions between features are neglected and the effect of each irregularity is computed in sequence.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":"31 1","pages":"3469 - 3497"},"PeriodicalIF":0.0,"publicationDate":"2004-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76514042","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The thermal cook–off response of energetic materials (ignition resulting from direct, bulk thermal heating) is important from a safety point of view, but also challenges our understanding of these materials. Explosives are not designed to be cooked off, and, especially in the case of slow cook–off, by the time the material ignites it is substantially different, both chemically and physically, from its original state. In attempting to model such a process numerically, it has generally been assumed that combustion proceeds, from an ignition point, in a more or less planar manner, as has been observed many times in pristine energetic materials at room temperature. To investigate directly the spread of reaction following cook–off in one such energetic material (PBX 9501), small discs of PBX 9501 were heated with a confining glass or sapphire window through which the early stages of the combustion process could be observed directly by high–speed photography. The resulting combustion was found to vary with temperature of ignition, but in all cases was quite different to the laminar burn model. The results of these tests are presented, together with some possible explanations of the behaviour and discussion of the implications to modelling this response.
{"title":"Thermal cook–off response of confined PBX 9501","authors":"P. Dickson, B. Asay, B. Henson, L. Smilowitz","doi":"10.1098/rspa.2004.1348","DOIUrl":"https://doi.org/10.1098/rspa.2004.1348","url":null,"abstract":"The thermal cook–off response of energetic materials (ignition resulting from direct, bulk thermal heating) is important from a safety point of view, but also challenges our understanding of these materials. Explosives are not designed to be cooked off, and, especially in the case of slow cook–off, by the time the material ignites it is substantially different, both chemically and physically, from its original state. In attempting to model such a process numerically, it has generally been assumed that combustion proceeds, from an ignition point, in a more or less planar manner, as has been observed many times in pristine energetic materials at room temperature. To investigate directly the spread of reaction following cook–off in one such energetic material (PBX 9501), small discs of PBX 9501 were heated with a confining glass or sapphire window through which the early stages of the combustion process could be observed directly by high–speed photography. The resulting combustion was found to vary with temperature of ignition, but in all cases was quite different to the laminar burn model. The results of these tests are presented, together with some possible explanations of the behaviour and discussion of the implications to modelling this response.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":"20 1","pages":"3447 - 3455"},"PeriodicalIF":0.0,"publicationDate":"2004-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83690808","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The existence and uniqueness of solutions to multivalued stochastic differential equations of the second order on Riemannian manifolds are proved. The class of problem is motivated by rigid body and multibody dynamics with friction and an application to the spherical pendulum with friction is presented.
{"title":"Second–order multivalued stochastic differential equations on Riemannian manifolds","authors":"F. Bernardin, M. Schatzman, C. Lamarque","doi":"10.1098/rspa.2004.1312","DOIUrl":"https://doi.org/10.1098/rspa.2004.1312","url":null,"abstract":"The existence and uniqueness of solutions to multivalued stochastic differential equations of the second order on Riemannian manifolds are proved. The class of problem is motivated by rigid body and multibody dynamics with friction and an application to the spherical pendulum with friction is presented.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":"61 1","pages":"3095 - 3121"},"PeriodicalIF":0.0,"publicationDate":"2004-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85548061","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We reconsider the problem of the boundary–layer flow of a non-Newtonian fluid whose constitutive law is given by the classical Ostwald–de Waele power–law model. The boundary–layer equations are solved in similarity form. The resulting similarity solutions for shear–thickening fluids are shown to have a finite–width crisis resulting in the prediction of a finite–width boundary layer. A secondary viscous adjustment layer is required in order to smooth out the solution and to ensure correct matching with the far–field boundary conditions. In the case of shear–thinning fluids, the similarity forms have solutions whose decay into the far field is strongly algebraic. Smooth matching between these inner algebraically decaying solutions and an outer uniform flow is achieved via the introduction of a viscous diffusion layer.
{"title":"On the boundary–layer equations for power–law fluids","authors":"J. Denier, P. Dabrowski","doi":"10.1098/rspa.2004.1349","DOIUrl":"https://doi.org/10.1098/rspa.2004.1349","url":null,"abstract":"We reconsider the problem of the boundary–layer flow of a non-Newtonian fluid whose constitutive law is given by the classical Ostwald–de Waele power–law model. The boundary–layer equations are solved in similarity form. The resulting similarity solutions for shear–thickening fluids are shown to have a finite–width crisis resulting in the prediction of a finite–width boundary layer. A secondary viscous adjustment layer is required in order to smooth out the solution and to ensure correct matching with the far–field boundary conditions. In the case of shear–thinning fluids, the similarity forms have solutions whose decay into the far field is strongly algebraic. Smooth matching between these inner algebraically decaying solutions and an outer uniform flow is achieved via the introduction of a viscous diffusion layer.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":"20 1","pages":"3143 - 3158"},"PeriodicalIF":0.0,"publicationDate":"2004-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82361754","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Singularity formation phenomena for 2mth–order quasilinear parabolic equations are studied by using energy estimates related to Saint–Venant's principle. Sharp estimates of propagation of singularities generated by boundary global and regional blow-up regimes are established.
{"title":"Structure of boundary blow-up for higher-order quasilinear parabolic equations","authors":"V. Galaktionov, A. Shishkov","doi":"10.1098/rspa.2004.1297","DOIUrl":"https://doi.org/10.1098/rspa.2004.1297","url":null,"abstract":"Singularity formation phenomena for 2mth–order quasilinear parabolic equations are studied by using energy estimates related to Saint–Venant's principle. Sharp estimates of propagation of singularities generated by boundary global and regional blow-up regimes are established.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":"16 1","pages":"3299 - 3325"},"PeriodicalIF":0.0,"publicationDate":"2004-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73431077","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Many solid materials exhibit stress–induced phase transformations. Such phenomena can be modelled with the aid of the nonlinear elasticity theory with appropriate choices of the strain–energy function. It is known that if a two–phase deformation (with gradient F) in a finite elastic body is a local energy minimizer, then given any point p of the surface of discontinuity, the piecewise–homogeneous deformation corresponding to the two values F±(p) of F(p) is a global energy minimizer. Thus, instability of the latter state would imply instability of the former state. In this paper we investigate the stability properties of such piecewise–homogeneous deformations. More precisely, we are concerned with two joined half–spaces that correspond to two different phases of the same material. We first show how such a two–phase deformation can be constructed. Then the stability of the piecewise–homogeneous deformation is investigated with the aid of two test criteria. One is a kinetic stability criterion based on a quasi–static approach and on the growth/decay behaviour of the interface in the undeformed configuration when it is perturbed; the other, referred to as the energy criterion, is used to determine whether the deformation is a minimizer of the total energy with respect to perturbations of the interface in both the current and undeformed configurations. We clarify the differences between the two criteria, and provide a compact formula which can be used to establish the stability/instability of any two–phase piecewise–homogeneous deformations.
{"title":"Review Paper. Characterization and stability of two–phase piecewise–homogeneous deformations","authors":"Yibin Fu, A. Freidin","doi":"10.1098/rspa.2004.1361","DOIUrl":"https://doi.org/10.1098/rspa.2004.1361","url":null,"abstract":"Many solid materials exhibit stress–induced phase transformations. Such phenomena can be modelled with the aid of the nonlinear elasticity theory with appropriate choices of the strain–energy function. It is known that if a two–phase deformation (with gradient F) in a finite elastic body is a local energy minimizer, then given any point p of the surface of discontinuity, the piecewise–homogeneous deformation corresponding to the two values F±(p) of F(p) is a global energy minimizer. Thus, instability of the latter state would imply instability of the former state. In this paper we investigate the stability properties of such piecewise–homogeneous deformations. More precisely, we are concerned with two joined half–spaces that correspond to two different phases of the same material. We first show how such a two–phase deformation can be constructed. Then the stability of the piecewise–homogeneous deformation is investigated with the aid of two test criteria. One is a kinetic stability criterion based on a quasi–static approach and on the growth/decay behaviour of the interface in the undeformed configuration when it is perturbed; the other, referred to as the energy criterion, is used to determine whether the deformation is a minimizer of the total energy with respect to perturbations of the interface in both the current and undeformed configurations. We clarify the differences between the two criteria, and provide a compact formula which can be used to establish the stability/instability of any two–phase piecewise–homogeneous deformations.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":"23 1","pages":"3065 - 3094"},"PeriodicalIF":0.0,"publicationDate":"2004-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81767676","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper, which is a continuation of C. A. Stuart & G. Vuillaume (2003 Proc. R. Soc. Lond. A459, 1863–1889), is concerned with the study of the buckling of a tapered rod. This physical phenomenon leads to the nonlinear eigenvalue problem: {A(s)u'(s)}'+μsinu(s)=0s∈(0,1), u(1)= lim s→0 A(s)u'(s)=0 ∫ 0 1 A(s)u' (s) 2 ds <∞, where A(s) ε C([0,1]) is such that A(s) > 0 for all s > 0 and lims→0A(s)/sp = L for some constants p ⩾ 0 and L ε (0,∞). We deal with the critical case p= 2 and study the set of all the solutions of the problem. In Stuart & Vuillaume (2003) and under additional assumptions on A, we found a set of points {μi , i ε I ⫅ N* ={1,2,3,...}} ⊂ R+ such that a global branch of non–trivial solutions emanates from each μi , iε I. In this paper, we provide more detailed information about these global branches of solutions.
本文是C. a . Stuart & G. Vuillaume (2003 Proc. R. Soc)的延续。Lond。A459, 1863-1889)，是关于锥形棒的屈曲的研究。这种物理现象导致非线性特征值问题:{A(s)u'(s)}'+μsinu(s)=0s∈(0,1)，u(1)= lims→0A(s) u'(s)=0∫1 A(s)u'(s) 2 ds 0对于所有s > 0和lims→0A(s)/sp = L对于某些常数p大于或等于0和L ε(0，∞)。我们处理了临界情况p= 2，并研究了问题所有解的集合。在Stuart & Vuillaume(2003)中，在A的附加假设下，我们发现了点{μi, i ε i⫅N* ={1,2,3，…}}∧R+使得非平凡解的全局分支从每个μi， ε i中发散出来，在本文中，我们提供了关于这些解的全局分支的更详细的信息。
{"title":"Buckling of a critically tapered rod: properties of some global branches of solutions","authors":"C. Stuart, G. Vuillaume","doi":"10.1098/rspa.2004.1355","DOIUrl":"https://doi.org/10.1098/rspa.2004.1355","url":null,"abstract":"This paper, which is a continuation of C. A. Stuart & G. Vuillaume (2003 Proc. R. Soc. Lond. A459, 1863–1889), is concerned with the study of the buckling of a tapered rod. This physical phenomenon leads to the nonlinear eigenvalue problem: {A(s)u'(s)}'+μsinu(s)=0s∈(0,1), u(1)= lim s→0 A(s)u'(s)=0 ∫ 0 1 A(s)u' (s) 2 ds <∞, where A(s) ε C([0,1]) is such that A(s) > 0 for all s > 0 and lims→0A(s)/sp = L for some constants p ⩾ 0 and L ε (0,∞). We deal with the critical case p= 2 and study the set of all the solutions of the problem. In Stuart & Vuillaume (2003) and under additional assumptions on A, we found a set of points {μi , i ε I ⫅ N* ={1,2,3,...}} ⊂ R+ such that a global branch of non–trivial solutions emanates from each μi , iε I. In this paper, we provide more detailed information about these global branches of solutions.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":"13 1","pages":"3261 - 3282"},"PeriodicalIF":0.0,"publicationDate":"2004-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75779126","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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