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The quarterly journal of mechanics and applied mathematics最新文献

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‘Killing Mie Softly’: Analytic Integrals for Complex Resonant States “柔和地杀死Mie”：复杂谐振态的解析积分
The quarterly journal of mechanics and applied mathematics Pub Date : 2020-02-01 DOI: 10.1093/qjmam/hbaa004
R C Mcphedran;B Stout
{"title":"‘Killing Mie Softly’: Analytic Integrals for Complex Resonant States","authors":"R C Mcphedran;B Stout","doi":"10.1093/qjmam/hbaa004","DOIUrl":"10.1093/qjmam/hbaa004","url":null,"abstract":"SUMMARY We consider integrals of products of Bessel functions and of spherical Bessel functions, combined with a Gaussian factor guaranteeing convergence at infinity. These integrals arise in wave and quantum mechanical scattering problems of open systems containing cylindrical or spherical scatterers, particularly when those problems are considered in the framework of complex resonant modes. Explicit representations are obtained for the integrals, building on those in the 1992 paper by McPhedran, Dawes and Scott. Attention is paid to those sums with a distributive part arising as the Gaussian tends towards the unit function. In this limit, orthogonality and normalisability of complex modes are investigated.","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"73 1","pages":"119-139"},"PeriodicalIF":0.0,"publicationDate":"2020-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qjmam/hbaa004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49145393","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}
引用次数: 3
Asymptotic Field Near the Tip of a Debonded Anticrack in an Anisotropic Elastic Material 各向异性弹性材料中反裂纹尖端附近的渐近场
The quarterly journal of mechanics and applied mathematics Pub Date : 2020-02-01 DOI: 10.1093/qjmam/hbaa001
Xu Wang;Peter Schiavone
{"title":"Asymptotic Field Near the Tip of a Debonded Anticrack in an Anisotropic Elastic Material","authors":"Xu Wang;Peter Schiavone","doi":"10.1093/qjmam/hbaa001","DOIUrl":"10.1093/qjmam/hbaa001","url":null,"abstract":"We use the sextic Stroh formalism to study the asymptotic elastic field near the tip of a debonded anticrack in a generally anisotropic elastic material under generalised plane strain deformations. The stresses near the tip of the debonded anticrack exhibit the oscillatory singularities \u0000<tex>$r^{-3/4pm ivarepsilon }$</tex>\u0000 and \u0000<tex>$r^{-1/4pm ivarepsilon }$</tex>\u0000 (where \u0000<tex>$varepsilon $</tex>\u0000 is the oscillatory index) as well as the real power-type singularities \u0000<tex>$r^{-3/4}$</tex>\u0000 and \u0000<tex>$r^{-1/4}$</tex>\u0000. Two complex-valued stress intensity factors and two real-valued stress intensity factors are introduced to respectively scale the two oscillatory and two real power-type singularities. The corresponding three-dimensional analytic vector function is derived explicitly, and the material force on the debonded anticrack is obtained. Our solution is illustrated using an example involving orthotropic materials.","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"73 1","pages":"76-83"},"PeriodicalIF":0.0,"publicationDate":"2020-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qjmam/hbaa001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41816537","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}
引用次数: 1
Time to Approach Similarity 接近相似性的时间
The quarterly journal of mechanics and applied mathematics Pub Date : 2020-02-01 DOI: 10.1093/qjmam/hbz019
Joseph J Webber;Herbert E Huppert
{"title":"Time to Approach Similarity","authors":"Joseph J Webber;Herbert E Huppert","doi":"10.1093/qjmam/hbz019","DOIUrl":"https://doi.org/10.1093/qjmam/hbz019","url":null,"abstract":"In a recent article, Ball and Huppert (J. Fluid Mech., 874, 2019) introduced a novel method for ascertaining the characteristic timescale over which the similarity solution to a given time-dependent nonlinear differential equation converges to the actual solution, obtained by numerical integration, starting from given initial conditions. In this article, we apply this method to a range of different partial differential equations describing propagating gravity currents of fixed volume as well as modifying the techniques to apply to situations for which convergence to the numerical solution is oscillatory, as appropriate for gravity currents propagating at large Reynolds numbers. We investigate properties of convergence in all of these cases, including how different initial geometries affect the rate at which the two solutions agree. It is noted that geometries where the flow is no longer unidirectional take longer to converge. A method of time-shifting the similarity solution is introduced to improve the accuracy of the approximation given by the similarity solution, and also provide an upper bound on the percentage disagreement over all time.","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"73 1","pages":"1-23"},"PeriodicalIF":0.0,"publicationDate":"2020-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qjmam/hbz019","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49963671","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}
引用次数: 3
Hypersingular Integral Equations Arising in the Boundary Value Problems of the Elasticity Theory 弹性力学边值问题中的超奇异积分方程
The quarterly journal of mechanics and applied mathematics Pub Date : 2020-02-01 DOI: 10.1093/qjmam/hbz022
S M Mkhitaryan;M S Mkrtchyan;E G Kanetsyan
{"title":"Hypersingular Integral Equations Arising in the Boundary Value Problems of the Elasticity Theory","authors":"S M Mkhitaryan;M S Mkrtchyan;E G Kanetsyan","doi":"10.1093/qjmam/hbz022","DOIUrl":"10.1093/qjmam/hbz022","url":null,"abstract":"The exact solutions of a class of hypersingular integral equations with kernels \u0000<tex>$left( {s-x} right)^{-2}$</tex>\u0000, \u0000<tex>$left( {sin frac{s-x}{2}} right)^{-2}$</tex>\u0000, \u0000<tex>$left( {sinh frac{s-x}{2}} right)^{-2},cos frac{s-x}{2}left( {sin frac{s-x}{2}} right)^{-2}$</tex>\u0000, \u0000<tex>$cosh frac{s-x}{2}left( {sinh frac{s-x}{2}} right)^{-2}$</tex>\u0000 are obtained where the integrals must be interpreted as Hadamard finite-part integrals. Problems of cracks in elastic bodies of various canonical forms under antiplane and plane deformations, where the crack edges are loaded symmetrically, lead to such equations. These problems, in turn, lead to mixed boundary value problems of the mathematical theory of elasticity for a half-plane, a circle, a strip and a wedge.","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"73 1","pages":"51-75"},"PeriodicalIF":0.0,"publicationDate":"2020-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qjmam/hbz022","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46065714","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}
引用次数: 2
A Constructive Proof of Helmholtz’s Theorem 亥姆霍兹定理的构造性证明
The quarterly journal of mechanics and applied mathematics Pub Date : 2019-11-01 DOI: 10.1093/qjmam/hbz016
B. D. L. C. Ysern, J. S. D. Lis
{"title":"A Constructive Proof of Helmholtz’s Theorem","authors":"B. D. L. C. Ysern, J. S. D. Lis","doi":"10.1093/qjmam/hbz016","DOIUrl":"https://doi.org/10.1093/qjmam/hbz016","url":null,"abstract":"\u0000 It is a known result that any vector field ${boldsymbol{u}}$ that is locally Hölder continuous on an arbitrary open set $Omegasubset mathbb{R}^3$ can be written on $Omega$ as the sum of a gradient and a curl. Should $Omega$ be unbounded, no conditions are required on the behaviour of ${boldsymbol{u}}$ at infinity. We present a direct, self-contained proof of this theorem that only uses elementary techniques and has a constructive character. It consists in patching together local solutions given by the Newtonian potential that are then modified by harmonic approximations—based on solid spherical harmonics—to assure convergence near infinity for the resulting series.","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82053971","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}
引用次数: 2
Stress Amplification/Shielding Phenomena of Spherically Anisotropic and Radially Inhomogeneous Linear Elastic Hollow Spheres 球面各向异性和径向非均匀线弹性空心球的应力放大/屏蔽现象
The quarterly journal of mechanics and applied mathematics Pub Date : 2019-11-01 DOI: 10.1093/qjmam/hbz017
M. Chung
{"title":"Stress Amplification/Shielding Phenomena of Spherically Anisotropic and Radially Inhomogeneous Linear Elastic Hollow Spheres","authors":"M. Chung","doi":"10.1093/qjmam/hbz017","DOIUrl":"https://doi.org/10.1093/qjmam/hbz017","url":null,"abstract":"\u0000 Exact solutions of radially symmetric deformation of a spherically anisotropic and radially inhomogeneous linear elastic hollow sphere subjected to uniform radial tractions on the surfaces are derived. The power-law function is assumed to represent the radially inhomogeneity. Stress amplification/shielding phenomena are fully investigated and the benefits of using functionally graded materials are indicated. For a solid sphere under external uniform loadings, the conditions in which infinite stresses occur at the centre of the sphere regardless of applied traction magnitudes are specified. Also, circumferential stresses might have opposite sign of the applied loadings. Cavitation and blackhole phenomena at the centre of the sphere are also discussed.","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79687124","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}
引用次数: 5
Asymptotic Analysis of Frictional Contact Problem for Piezoelectric Shallow Shell 压电浅壳摩擦接触问题的渐近分析
The quarterly journal of mechanics and applied mathematics Pub Date : 2019-11-01 DOI: 10.1093/QJMAM/HBZ014
M. E. Mezabia, A. Ghezal, D. A. Chacha
{"title":"Asymptotic Analysis of Frictional Contact Problem for Piezoelectric Shallow Shell","authors":"M. E. Mezabia, A. Ghezal, D. A. Chacha","doi":"10.1093/QJMAM/HBZ014","DOIUrl":"https://doi.org/10.1093/QJMAM/HBZ014","url":null,"abstract":"\u0000 The objective of this work is to study the asymptotic justification of a new two-dimensional model for the equilibrium state of a piezoelectric linear shallow shell in frictional contact with a rigid foundation. More precisely, we consider the Signorini problem with Tresca friction of a piezoelectric linear shallow shell in contact with a rigid foundation. Then, we establish the convergence of the mechanical displacement and the electric potential as the thickness of the shallow shell goes to zero.","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85316425","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}
引用次数: 2
Mechanical Performance of a Thermoelectric Composite in the Vicinity of an Elliptic Inhomogeneity 椭圆不均匀区附近热电复合材料的力学性能
The quarterly journal of mechanics and applied mathematics Pub Date : 2019-11-01 DOI: 10.1093/QJMAM/HBZ012
K. Song, H. P. Song, P. Schiavone, C. Gao
{"title":"Mechanical Performance of a Thermoelectric Composite in the Vicinity of an Elliptic Inhomogeneity","authors":"K. Song, H. P. Song, P. Schiavone, C. Gao","doi":"10.1093/QJMAM/HBZ012","DOIUrl":"https://doi.org/10.1093/QJMAM/HBZ012","url":null,"abstract":"\u0000 Thermal stress induced by an uneven temperature field and mismatched thermal expansion is known to be a dominating factor in the debonding mechanism that threatens reliability and ultimately leads to failure in thermoelectric (TE) composites. Accordingly, we analyse the stress distributions in a TE composite induced by the presence of an elliptic inhomogeneity embedded in the surrounding matrix material. Using complex variable methods, we obtain closed-form representations of the thermal–electric and thermal–elastic fields and find that the temperature field around the inhomogeneity is reduced dramatically by the application of a remote electric current density without affecting the temperature difference across the inhomogeneity–matrix interface. This ensures the conversion efficiency of the TE composite while improving its reliability. Numerical results illustrate that a suitable choice of electric current density can prevent interfacial debonding via the suppression of the maximum positive normal stress on the interface.","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"61 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84681562","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}
引用次数: 13
A Geometrical Regularity Criterion in Terms of Velocity Profiles for the Three-Dimensional Navier–Stokes Equations 三维Navier-Stokes方程速度分布的几何规则准则
The quarterly journal of mechanics and applied mathematics Pub Date : 2019-11-01 DOI: 10.1093/qjmam/hbz018
C. Tran, Xinwei Yu
{"title":"A Geometrical Regularity Criterion in Terms of Velocity Profiles for the Three-Dimensional Navier–Stokes Equations","authors":"C. Tran, Xinwei Yu","doi":"10.1093/qjmam/hbz018","DOIUrl":"https://doi.org/10.1093/qjmam/hbz018","url":null,"abstract":"\u0000 In this article, we present a new kind of regularity criteria for the global well-posedness problem of the three-dimensional Navier–Stokes equations in the whole space. The novelty of the new results is that they involve only the profiles of the magnitude of the velocity. One particular consequence of our theorem is as follows. If for every fixed $tin (0,T)$, the ‘large velocity’ region $Omega:={(x,t)mid |u(x,t)|>C(q)left|mkern-2muleft|{u}right|mkern-2muright|_{L^{3q-6}}}$, for some $C(q)$ appropriately defined, shrinks fast enough as $qnearrow infty$, then the solution remains regular beyond $T$. We examine and discuss velocity profiles satisfying our criterion. It remains to be seen whether these profiles are typical of general Navier–Stokes flows.","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79973368","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}
引用次数: 2
Channel Flow Past A Near-Wall Body 通道流过近壁体
The quarterly journal of mechanics and applied mathematics Pub Date : 2019-08-01 DOI: 10.1093/QJMAM/HBZ009
F. Smith, P. Servini
{"title":"Channel Flow Past A Near-Wall Body","authors":"F. Smith, P. Servini","doi":"10.1093/QJMAM/HBZ009","DOIUrl":"https://doi.org/10.1093/QJMAM/HBZ009","url":null,"abstract":"\u0000 Near-wall behaviour arising when a finite sized body moves in a channel flow is investigated for high flow rates. This is over the interactive-flow length scale that admits considerable upstream influence. The focus is first on quasi-steady two-dimensional flow past a thin body in the outer reaches of one of the viscous wall layers. The jump conditions near the front of the body play an important part in the whole solution which involves an unusual multi-structured flow due to the presence of the body: flows above, below, ahead of and behind the body interact fully. Analytical solutions are presented and the repercussions for shorter and longer bodies are then examined. Second, implications are followed through for the movement of a free body in a dynamic fluid–body interaction. Particular key findings are that instability persists for all body lengths, the growth rate decreases like the $1/4$ power of distance as the body approaches the wall, and lift production on the body is dominated by high pressures from an unexpected flow region emerging on the front of the body.","PeriodicalId":92460,"journal":{"name":"The quarterly journal of mechanics and applied mathematics","volume":"35 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90752828","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}
引用次数: 7
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