Summary The article presents a new complex variables-based approach for analytical evaluation of threedimensional integrals involved in boundary element method (BEM) formulations. The boundary element is assumed to be planar and its boundary may contain an arbitrary number of straight lines and/or circular arcs. The idea is to use BEM integral representations written in a local coordinate system of an element, separate in-plane components of the fields involved, arrange them in certain complex combinations, and apply integral representations for complex functions. These integral representations, such as Cauchy–Pompeiu formula (a particular case of Bochner– Martinelli formula) are the corollaries of complex forms of Gauss’s theorem and Green’s identity. They reduce the integrals over the area of the domain to those over its boundary. The latter integrals can be evaluated analytically for various density functions. Analytical expressions are presented for basic integrals involved in the single- and double-layer potentials for potential (harmonic) and elasticity problems.
{"title":"The use of complex integral representations for analytical evaluation of three-dimensional BEM integrals - Potential and elasticity problems","authors":"S. Mogilevskaya, D. Nikolskiy","doi":"10.1093/QJMAM/HBU015","DOIUrl":"https://doi.org/10.1093/QJMAM/HBU015","url":null,"abstract":"Summary The article presents a new complex variables-based approach for analytical evaluation of threedimensional integrals involved in boundary element method (BEM) formulations. The boundary element is assumed to be planar and its boundary may contain an arbitrary number of straight lines and/or circular arcs. The idea is to use BEM integral representations written in a local coordinate system of an element, separate in-plane components of the fields involved, arrange them in certain complex combinations, and apply integral representations for complex functions. These integral representations, such as Cauchy–Pompeiu formula (a particular case of Bochner– Martinelli formula) are the corollaries of complex forms of Gauss’s theorem and Green’s identity. They reduce the integrals over the area of the domain to those over its boundary. The latter integrals can be evaluated analytically for various density functions. Analytical expressions are presented for basic integrals involved in the single- and double-layer potentials for potential (harmonic) and elasticity problems.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"67 1","pages":"505-523"},"PeriodicalIF":0.9,"publicationDate":"2014-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/QJMAM/HBU015","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"61239380","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Scattering by a semi-infinite lattice and the excitation of Bloch waves","authors":"N. Tymis, Ian Thompson","doi":"10.1093/QJMAM/HBU014","DOIUrl":"https://doi.org/10.1093/QJMAM/HBU014","url":null,"abstract":"","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"67 1","pages":"469-503"},"PeriodicalIF":0.9,"publicationDate":"2014-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/QJMAM/HBU014","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"61239826","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper considers the wide-spacing approximation when applied to twodimensional linear problems involving the scattering and radiation of water waves by structures in a fluid of infinite depth. The approximation is set into the context of a formal expansion procedure and an extended approximation is obtained. The extended approximation significantly improves the accuracy of calculations for submerged structures when compared with the standard approximation procedure based on plane-wave interactions.
{"title":"An extended wide-spacing approximation for two-dimensional water-wave problems in infinite depth","authors":"P. Mciver","doi":"10.1093/QJMAM/HBU013","DOIUrl":"https://doi.org/10.1093/QJMAM/HBU013","url":null,"abstract":"This paper considers the wide-spacing approximation when applied to twodimensional linear problems involving the scattering and radiation of water waves by structures in a fluid of infinite depth. The approximation is set into the context of a formal expansion procedure and an extended approximation is obtained. The extended approximation significantly improves the accuracy of calculations for submerged structures when compared with the standard approximation procedure based on plane-wave interactions.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"67 1","pages":"445-468"},"PeriodicalIF":0.9,"publicationDate":"2014-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/QJMAM/HBU013","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"61239816","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The effects of an outer flow on the unsteady free convection boundary layer near a stagnation point in a heat-generating porous medium","authors":"J. Merkin","doi":"10.1093/QJMAM/HBU012","DOIUrl":"https://doi.org/10.1093/QJMAM/HBU012","url":null,"abstract":"","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"67 1","pages":"419-444"},"PeriodicalIF":0.9,"publicationDate":"2014-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/QJMAM/HBU012","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"61239751","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
AbstractA complete methodology, based on a two-scale asymptotic approach, that enables the ho-mogenisation of elastic lattices at non-zero frequencies is developed. Elastic lattices are distin-guished from scalar lattices in that two or more types of coupled waves exist, even at low frequen-cies. Such a theory enables the determination of e ective material properties at both low and highfrequencies. The theoretical framework is developed for the propagation of waves through latticesof arbitrary geometry and dimension. The asymptotic approach provides a method through whichthe dispersive properties of lattices at frequencies near standing waves can be described; the theoryaccurately describes both the dispersion curves and the response of the lattice near the edges ofthe Brillouin zone. The leading order solution is expressed as a product between the standingwave solution and long-scale envelope functions that are eigensolutions of the homogenised partialdi erential equation. The general theory is supplemented by a pair of illustrative examples fortwo archetypal classes of two-dimensional elastic lattices. The eciency of the asymptotic ap-proach in accurately describing several interesting phenomena is demonstrated, including dynamicanisotropy and Dirac cones.
{"title":"High frequency homogenisation for elastic lattices","authors":"D. Colquitt, R. Craster, M. Makwana","doi":"10.1093/QJMAM/HBV005","DOIUrl":"https://doi.org/10.1093/QJMAM/HBV005","url":null,"abstract":"AbstractA complete methodology, based on a two-scale asymptotic approach, that enables the ho-mogenisation of elastic lattices at non-zero frequencies is developed. Elastic lattices are distin-guished from scalar lattices in that two or more types of coupled waves exist, even at low frequen-cies. Such a theory enables the determination of e ective material properties at both low and highfrequencies. The theoretical framework is developed for the propagation of waves through latticesof arbitrary geometry and dimension. The asymptotic approach provides a method through whichthe dispersive properties of lattices at frequencies near standing waves can be described; the theoryaccurately describes both the dispersion curves and the response of the lattice near the edges ofthe Brillouin zone. The leading order solution is expressed as a product between the standingwave solution and long-scale envelope functions that are eigensolutions of the homogenised partialdi erential equation. The general theory is supplemented by a pair of illustrative examples fortwo archetypal classes of two-dimensional elastic lattices. The e\u000eciency of the asymptotic ap-proach in accurately describing several interesting phenomena is demonstrated, including dynamicanisotropy and Dirac cones.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"68 1","pages":"203-230"},"PeriodicalIF":0.9,"publicationDate":"2014-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/QJMAM/HBV005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"61239773","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Summary A theory of elastically rigid finite deformation plasticity emphasizing the role of material symmetry is developed. The fields describing lattice rotation, dislocation density and plastic spin, irrelevant in the case of isotropy, are found to be central to the present framework. A plane strain characteristic theory for anisotropic plasticity is formulated wherein the solutions, as well as the nature of their discontinuities, show remarkable deviation from the classical isotropic slipline theory.
{"title":"Plane strain problem in elastically rigid finite plasticity","authors":"Anurag Gupta, D. Steigmann","doi":"10.1093/QJMAM/HBU007","DOIUrl":"https://doi.org/10.1093/QJMAM/HBU007","url":null,"abstract":"Summary A theory of elastically rigid finite deformation plasticity emphasizing the role of material symmetry is developed. The fields describing lattice rotation, dislocation density and plastic spin, irrelevant in the case of isotropy, are found to be central to the present framework. A plane strain characteristic theory for anisotropic plasticity is formulated wherein the solutions, as well as the nature of their discontinuities, show remarkable deviation from the classical isotropic slipline theory.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"67 1","pages":"287-310"},"PeriodicalIF":0.9,"publicationDate":"2014-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/QJMAM/HBU007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"61239639","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Summary Using the method of asymptotic homogenization in the low-frequency wave propagation regime we derive the effective antiplane elastic properties of a cracked solid which in the absence of cracks would be orthotropic. The solid is a periodic medium defined by a periodic cell containing N cracks of infinite extent in thez direction but with arbitrary shape and orientation in thexy plane. Effective properties are defined in terms of the solution to a so-called cell problem. We propose two convenient schemes by which the cell problem can be solved, one based on a nearly periodic Green’s function, the other based on doubly periodic multipole expansions. Using these methods we compare results with existing approximate expressions for periodic arrays of straight cracks and discuss their regimes of validity by assessing their departure from the exact results obtained by the present method. We go on to consider more complex distributions such as elliptical cavities, non-straight cracks and effects of an orthotropic host phase. In particular we show that specific types of regular arrays of cracks in an orthotropic host medium can induce a macroscopically isotropic response to antiplane shear waves propagating in the xy plane.
{"title":"EFFECTIVE ANTIPLANE ELASTIC PROPERTIES OF AN ORTHOTROPIC SOLID WEAKENED BY A PERIODIC DISTRIBUTION OF CRACKS","authors":"T. Williams, W. Parnell","doi":"10.1093/QJMAM/HBU008","DOIUrl":"https://doi.org/10.1093/QJMAM/HBU008","url":null,"abstract":"Summary Using the method of asymptotic homogenization in the low-frequency wave propagation regime we derive the effective antiplane elastic properties of a cracked solid which in the absence of cracks would be orthotropic. The solid is a periodic medium defined by a periodic cell containing N cracks of infinite extent in thez direction but with arbitrary shape and orientation in thexy plane. Effective properties are defined in terms of the solution to a so-called cell problem. We propose two convenient schemes by which the cell problem can be solved, one based on a nearly periodic Green’s function, the other based on doubly periodic multipole expansions. Using these methods we compare results with existing approximate expressions for periodic arrays of straight cracks and discuss their regimes of validity by assessing their departure from the exact results obtained by the present method. We go on to consider more complex distributions such as elliptical cavities, non-straight cracks and effects of an orthotropic host phase. In particular we show that specific types of regular arrays of cracks in an orthotropic host medium can induce a macroscopically isotropic response to antiplane shear waves propagating in the xy plane.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"67 1","pages":"311-342"},"PeriodicalIF":0.9,"publicationDate":"2014-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/QJMAM/HBU008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"61239682","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Summary In this article, we solve a canonical problem of the consolidation of a laterally loaded radiallydrained poroelastic cylinder by the series expansion method, or Fourier method. This method is simpler and more intuitive than the widely used Laplace transform technique because it does not require inversion of the Laplace transformed solution. Time evolution of the fluid pressure and volumetric strain is examined and, in addition, an upper bound on the pressure overshoot, occurring during the consolidation, is obtained for the loaded cylinder and sphere.
{"title":"Problem of the consolidation of a laterally loaded radially-drained poroelastic cylinder","authors":"A. Suvorov","doi":"10.1093/QJMAM/HBU001","DOIUrl":"https://doi.org/10.1093/QJMAM/HBU001","url":null,"abstract":"Summary In this article, we solve a canonical problem of the consolidation of a laterally loaded radiallydrained poroelastic cylinder by the series expansion method, or Fourier method. This method is simpler and more intuitive than the widely used Laplace transform technique because it does not require inversion of the Laplace transformed solution. Time evolution of the fluid pressure and volumetric strain is examined and, in addition, an upper bound on the pressure overshoot, occurring during the consolidation, is obtained for the loaded cylinder and sphere.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"67 1","pages":"159-173"},"PeriodicalIF":0.9,"publicationDate":"2014-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/QJMAM/HBU001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"61239538","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Circumferential stress waves in a non-linear cylindrical annulus in a new class of elastic materials","authors":"M. Kambapalli, K. Kannan, K. Rajagopal","doi":"10.1093/QJMAM/HBU003","DOIUrl":"https://doi.org/10.1093/QJMAM/HBU003","url":null,"abstract":"","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"35 1","pages":"193-203"},"PeriodicalIF":0.9,"publicationDate":"2014-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/QJMAM/HBU003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"61239584","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This is a pre-copyedited, author-produced PDF of an article accepted for publication in The Quarterly Journal of Mechanics & Applied Mathematics following peer review. The version of record R. Porter and D. V. Evans Connections between potentials describing wave scattering by complementary arrangements of vertical barriers. Q J Mechanics Appl Math 2014 67 (2): 175-192. is available online at: http://qjmam.oxfordjournals.org/content/67/2.toc.
{"title":"Connections between potentials describing wave scattering by complementary arrangements of vertical barriers","authors":"R. Porter, D. Evans","doi":"10.1093/QJMAM/HBU002","DOIUrl":"https://doi.org/10.1093/QJMAM/HBU002","url":null,"abstract":"This is a pre-copyedited, author-produced PDF of an article accepted for publication in The Quarterly Journal of Mechanics & Applied Mathematics following peer review. The version of record R. Porter and D. V. Evans Connections between potentials describing wave scattering by complementary arrangements of vertical barriers. Q J Mechanics Appl Math 2014 67 (2): 175-192. is available online at: http://qjmam.oxfordjournals.org/content/67/2.toc.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"67 1","pages":"175-192"},"PeriodicalIF":0.9,"publicationDate":"2014-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/QJMAM/HBU002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"61239574","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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