Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences最新文献

{"title":"Does the pressure‐induced a‐e phase transition occur for all low–alloy steels?","authors":"R. Hammond, W. Proud","doi":"10.1098/rspa.2004.1314","DOIUrl":"https://doi.org/10.1098/rspa.2004.1314","url":null,"abstract":"When designing many products, especially vehicles, it is important to take into consideration impact properties. This paper presents fundamental dynamic data for two examples of bainitic steels. Hugoniots and shear–strength variation with longitudinal stress is reported. These were similar for both materials and in agreement with values for mild steel. The ferrite in the lower–temperature bainite was found to undergo a shock–induced phase transition at 13 GPa, whereas no phase transition was observed in the upper bainite. Further research is being conducted to account for this difference by studying the microstructure and the effect of impact on it. The lower–temperature bainite was found to behave in a brittle fashion, fragmenting extremely, whereas upper bainitic samples were usually recovered intact indicating ductile behaviour. The dynamic tensile or spall strength was also measured in these materials over a range of impact conditions. For the upper bainite, the spall strength dropped slightly with increasing longitudinal stress. However, for the lower–temperature bainite, there was a significant drop in spall strength above the longitudinal stress at which the phase transition occurs. Microstructural studies were also undertaken.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82029121","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}

{"title":"Theory of deformation in small volumes of material","authors":"D. Dunstan, A. Bushby","doi":"10.1098/rspa.2004.1306","DOIUrl":"https://doi.org/10.1098/rspa.2004.1306","url":null,"abstract":"It has long been controversial as to whether small volumes of material are, or should be, more resistant to plastic deformation than is implied by the yield strength of bulk material. We generalize the established theory of critical thickness for strained layers to show that there is an increase in the initial yield stress for geometrical reasons wherever there is a strain gradient. The theory is in quantitative agreement, without free fitting parameters, with the classic experimental data on the torsion of thin wires and on the bending of thin beams.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72695597","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}

{"title":"Resonant porous penetrative convection","authors":"B. Straughan","doi":"10.1098/rspa.2004.1292","DOIUrl":"https://doi.org/10.1098/rspa.2004.1292","url":null,"abstract":"An interesting convection situation is studied in a porous medium. We show that there is a range in which the convection may switch from the lower part of the layer to being predominantly in the upper part of the layer. Both linear instability and nonlinear energy stability thresholds are derived.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74312869","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}

{"title":"Periodic conduction problems: the fast multipole method and convergence of integral equations and lattice sums","authors":"G. Rodin, J. Overfelt","doi":"10.1098/rspa.2004.1318","DOIUrl":"https://doi.org/10.1098/rspa.2004.1318","url":null,"abstract":"This paper presents a version of the fast multipole method (FMM) for integral equations describing conduction through three–dimensional periodic heterogeneous media. The proposed method is based on the standard rather than periodic fundamental solution, and therefore it is very close to the original FMM. In deriving the method, particular attention is paid to convergence of arising integral equations and lattice sums. It is shown that convergence can be achieved without introducing artificial compensatory sources or boundary conditions.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85415030","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}

{"title":"Shock-like free-surface perturbations in low-surface-tension, viscous, thin-film flow exterior to a rotating cylinder","authors":"E. Hinch, M. Kelmanson, P. Metcalfe","doi":"10.1098/rspa.2004.1327","DOIUrl":"https://doi.org/10.1098/rspa.2004.1327","url":null,"abstract":"Continuing from the work of Hinch & Kelmanson (2003 Proc. R. Soc. Lond. A459, 1193–1213), the lubrication approximation is used to investigate the drift and decay of free–surface perturbations in the viscous flow exterior to a circular cylinder rotating about its horizontal axis in a vertical gravitational field. Non–dimensional parameters corresponding to gravity, γ = ρgbar{h}2/3ωμa, and surface tension, α = γh3/3ωμa4, are used to characterize the flow, where ω and a are respectively the angular velocity and radius of the cylinder, μ, ρ, σ and h are respectively the kinematic viscosity, density, surface tension and mean film thickness of the fluid, and g is the acceleration due to gravity. Within the parameter hierarchy γ2 << α << γ << 1, Hinch & Kelmanson (2003) discovered a complex interaction between rotation, gravity and surface tension, leading to a four–time–scale cascade over which drift and decay of free–surface perturbations occur. However, when α = o(γ2), the low–harmonic asymptotics of Hinch & Kelmanson (2003) cannot represent the shock–like solutions manifest in numerical simulations. Accordingly, the case of vanishingly small surface tension is investigated herein, and the resulting shock–like solutions are analysed. When the surface tension is identically zero, the resulting Hamiltonian problem may be solved explicitly via the method of characteristics, action–angle variables and strained–coordinate asymptotic expansions, which reveal a shock–formation time–scale of ω2μ3a3/3 g3h6. The strained (fast) time–scale which can be deduced a priori via action–angle variables, is consistent with that obtained via the independent asymptotic approach of Hinch & Kelmanson (2003), and the (slow) shock time–scale T = 30γ3t is derived and confirmed via spectral numerical integrations of the full lubrication approximation with vanishingly small, non–zero surface tension. With β = α/30γ3 << 1, a shock thickness of order O(β1/3) is discovered, and the leading–order transient in the surface elevation is found to satisfy a Kuramoto–Sivashinsky evolution equation, which is solved via multiple scales for the extreme cases β << 1 and β >> 1, and numerically otherwise. A universal scaling of the transient results is discovered which gives good agreement with the quasi–steady shock solution, even when the transient shock thickens in response to its decreasing amplitude. Depending upon critical values of α/γ2, β and γ, the transient solution is discovered to decay in one of only four possible sequences comprising one or more of T−1, T−½ and exp(−81αγ2t). Physical data indicate that all four decay sequences are observable in practice.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88462583","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}

{"title":"Mechanisms of deep penetration of soft solids, with application to the injection and wounding of skin","authors":"Oliver A. Shergold, N. Fleck","doi":"10.1098/rspa.2004.1315","DOIUrl":"https://doi.org/10.1098/rspa.2004.1315","url":null,"abstract":"Micromechanical models are developed for the deep penetration of a soft solid by a flat–bottomed and by a sharp–tipped cylindrical punch. The soft solid is taken to represent mammalian skin and silicone rubbers, and is treated as an incompressible, hyperelastic, isotropic solid described by a one–term Ogden strain energy function. Penetration of the soft solid by a flat–bottomed punch is by the formation of a mode–II ring crack that propagates ahead of the penetrator tip. The sharp–tipped punch penetrates by the formation of a planar mode–I crack at the punch tip, followed by wedging open of the crack by the advancing punch. For both modes of punch advance the steady–state penetration load is calculated by equating the work done in advancing the punch to the sum of the fracture work and the strain energy stored in the solid. For the case of a sharp penetrator, this calculation is performed by considering the opening of a plane–strain crack by a wedge using a finite–element approach. Analytical methods suffice for the flat–bottomed punch. In both models the crack dimensions are such that the load on the punch is minimized. For both geometries of punch tip, the predicted penetration pressure increases with diminishing punch radius, and with increasing toughness and strain–hardening capacity of solid. The penetration pressure for a flat–bottomed punch is two to three times greater than that for a sharp–tipped punch (assuming that the mode–I and mode–II toughnesses are equal), in agreement with experimental observations reported in a companion paper.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88994228","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}

{"title":"Isolated microstructures on linear elastic strains","authors":"Kewei Zhang","doi":"10.1098/rspa.2004.1334","DOIUrl":"https://doi.org/10.1098/rspa.2004.1334","url":null,"abstract":"We show in the geometrically linear multi–well models for material microstructure that ‘scaling’ or distance between the locations of the wells can be a factor to separate microstructures. We give conditions that prevent large–scale microstructures. At the same time, scale preserves local microstructure for wells located close to one another. We also give some quantitative estimates of the scalings needed to separate microstructures.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86089709","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}

{"title":"On the Camassa-Holm equation and a direct method of solution I. Bilinear form and solitary waves","authors":"A. Parker","doi":"10.1098/rspa.2004.1301","DOIUrl":"https://doi.org/10.1098/rspa.2004.1301","url":null,"abstract":"Over the last decade, a variety of techniques has been used to find exact solutions (both analytic and other) of the Camassa–Holm equation. The different approaches have met with varying measures of success in eliciting the important class of soliton solutions. In this, the first of two papers, we show how Hirota's bilinear transform method can be used to obtain analytic solutions of the Camassa–Holm equation. A bilinear form of the Camassa–Holm equation is presented and used to derive the solitary–wave solution, which is examined in various parameter regimes. A limiting procedure is then used to recover the well–known non–analytic ‘peakon’ solution from the solitary wave. The results reported here provide a basis for constructing explicitly the erstwhile elusive N–soliton solutions of the Camassa–Holm equation in a sequel paper.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84100348","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}

{"title":"Simulation of nonlinear interactions between waves and floating bodies through a finite-element-based numerical tank","authors":"G. Wu, Z. Z. Hu","doi":"10.1098/rspa.2004.1302","DOIUrl":"https://doi.org/10.1098/rspa.2004.1302","url":null,"abstract":"Fully nonlinear water–wave interactions with a floating structure are investigated through a numerical towing tank. A wave maker is installed on one end of the tank while a numerical beach based on a combination of damping zone and Sommerfeld condition is adopted on the other side of the tank. A floating body is placed at a location in the tank, where it will be set into motion by the waves generated by the wave maker. The simulation is based on the velocity potential theory together with the finite–element method. The mesh used follows the deformation of the free surface and the body motion. Its structure is adjusted and the distribution of elements is completely rearranged when the motion is big to avoid an over–distorted grid. Auxiliary functions are introduced to decouple the nonlinear mutual dependence between the hydrodynamic force and the body motion. Extensive numerical results are provided for vertical circular cylinders and a simplified floating production, storage and offloading, for which meshes are obtained through an efficient scheme based on a two–dimensional tri–tree method.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84569478","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}

{"title":"Near-critical free-surface rotating flow over topography","authors":"G. G. Vilenski, E. Johnson","doi":"10.1098/rspa.2004.1317","DOIUrl":"https://doi.org/10.1098/rspa.2004.1317","url":null,"abstract":"Free–surface rotating flow over localized topography is studied in the weakly three–dimensional nonlinear long–wave dispersive limit. The analysis is based on the forced rotating Kadomtsev–Petviashvili (frKP) equation. For small forcing, steady supercritical flow is described analytically. Finite–amplitude topographic effects are described numerically for both supercritical and subcritical flows. The pressure drag on the flow is described as a function of obstacle height, Rossby number and the detuning parameter measuring the difference between the flow speed and that of linear long gravity waves.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82420173","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}