We study the equilibrium of a mechanical system composed by two rods that bend under the action of a pressure difference; they have one fixed endpoint and are partially in contact. This system can be viewed as a bi-valve made by two smooth leaflets that lean on each other. We obtain the balance equations of the mechanical system exploiting the principle of virtual work and the contact point is identified by a jump condition. The problem can be simplified exploiting a first integral. In the case of quadratic energy, another first integral exists: its peculiarity is discussed and a further reduction of the equations is carried out. Numerical integration of the differential system shows how the shape of the beams and the position of the contact point depend on the applied pressure. For small pressure, an asymptotic expansion in a small parameter allows us to find an approximate solutions of polynomial form which is in surprisingly good agreement with the solution of the original system of equations, even beyond the expected range of validity. Finally, the asymptotics predicts a value of the pressure that separates the contact from the no-contact regime of the beams that compares very well with the one numerically evaluated.
{"title":"Equilibrium of Two Rods in Contact Under Pressure","authors":"S. Turzi, M. Zoppello, Davide Carlo Ambrosi","doi":"10.1093/QJMAM/HBAA016","DOIUrl":"https://doi.org/10.1093/QJMAM/HBAA016","url":null,"abstract":"\u0000 We study the equilibrium of a mechanical system composed by two rods that bend under the action of a pressure difference; they have one fixed endpoint and are partially in contact. This system can be viewed as a bi-valve made by two smooth leaflets that lean on each other. We obtain the balance equations of the mechanical system exploiting the principle of virtual work and the contact point is identified by a jump condition. The problem can be simplified exploiting a first integral. In the case of quadratic energy, another first integral exists: its peculiarity is discussed and a further reduction of the equations is carried out. Numerical integration of the differential system shows how the shape of the beams and the position of the contact point depend on the applied pressure. For small pressure, an asymptotic expansion in a small parameter allows us to find an approximate solutions of polynomial form which is in surprisingly good agreement with the solution of the original system of equations, even beyond the expected range of validity. Finally, the asymptotics predicts a value of the pressure that separates the contact from the no-contact regime of the beams that compares very well with the one numerically evaluated.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/QJMAM/HBAA016","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"61233605","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}
An important element of the asymptotic description of flows having a moving liquid/gas interface which intersects a solid boundary is a function denoted $Q_i left( alpha right)$ by Hocking and Rivers (The spreading of a drop by capillary action, J. Fluid Mech. 121 (1982) 425–442), where $0 < alpha < pi$ is the contact angle of the interface with the wall. $Q_i left( alpha right)$ arises from matching of the inner and intermediate asymptotic regions introduced by those authors and is required in applications of the asymptotic theory. This article describes a new numerical method for the calculation of $Q_i left( alpha right)$, which, because it explicitly allows for the logarithmic singularity in the kernel of the governing integral equation and uses quadratic interpolation of the non-singular factor in the integrand, is more accurate than that employed by Hocking and Rivers. Nonetheless, our results show good agreement with theirs, with, however, noticeable departures near $alpha = pi $. We also discuss the limiting cases $alpha to 0$ and $alpha to pi $. The leading-order terms of $Q_i left( alpha right)$ in both limits are in accord with the analysis of Hocking (A moving fluid interface. Part 2. The removal of the force singularity by a slip flow, J. Fluid Mech. 79 (1977) 209–229). The next-order terms are also considered. Hocking did not go beyond leading order for $alpha to 0$, and we believe his results for the next order as $alpha to pi $ to be incorrect. Numerically, we find that the next-order terms are $Oleft( {alpha ^2} right)$ for $alpha to 0$ and $Oleft( 1 right)$ as $alpha to pi $. The latter result agrees with Hocking, but the value of the $Oleft( 1 right)$ constant does not. It is hoped that giving details of the numerical method and more precise information, both numerical and in terms of its limiting behaviour, concerning $Q_i left( alpha right)$ will help those wanting to use the asymptotic theory of contact-line dynamics in future theoretical and numerical work.
具有与固体边界相交的移动液/气界面的流动的渐近描述的一个重要元素是由hockking和Rivers表示为$Q_i left( alpha right)$的函数(由毛细作用引起的液滴扩散，J.流体力学。121(1982) 425-442)，其中$0 < alpha < pi$为接触面与壁面的接触角。$Q_i left( alpha right)$是由这些作者介绍的内渐近区域和中间渐近区域的匹配产生的，是渐近理论应用中所必需的。本文描述了一种新的计算$Q_i left( alpha right)$的数值方法，由于它明确地允许控制积分方程核的对数奇异性，并在被积函数中使用非奇异因子的二次插值，因此比霍金和里弗斯采用的方法更精确。尽管如此，我们的结果与他们的结果非常一致，然而，在$alpha = pi $附近有明显的偏差。我们还讨论了极限情况$alpha to 0$和$alpha to pi $。$Q_i left( alpha right)$在两个极限处的首阶项与霍金(A)运动流体界面的分析一致。第2部分。滑移流对力奇点的去除[j] .流体力学。79(1977) 209-229)。下一阶项也被考虑在内。霍金没有超越$alpha to 0$的领先顺序，我们认为他的下一个顺序$alpha to pi $的结果是不正确的。数值上，我们发现$alpha to 0$的下一阶项为$Oleft( {alpha ^2} right)$, $Oleft( 1 right)$为$alpha to pi $。后一种结果与霍金的结论一致，但$Oleft( 1 right)$常数的值却不一致。希望给出关于$Q_i left( alpha right)$的数值方法的细节和更精确的信息，包括数值和其极限行为，将有助于那些想要在未来的理论和数值工作中使用接触线动力学的渐近理论的人。
{"title":"Calculation of a key function in the asymptotic description of moving contact lines","authors":"J. Scott","doi":"10.1093/qjmam/hbaa012","DOIUrl":"https://doi.org/10.1093/qjmam/hbaa012","url":null,"abstract":"\u0000 An important element of the asymptotic description of flows having a moving liquid/gas interface which intersects a solid boundary is a function denoted $Q_i left( alpha right)$ by Hocking and Rivers (The spreading of a drop by capillary action, J. Fluid Mech. 121 (1982) 425–442), where $0 < alpha < pi$ is the contact angle of the interface with the wall. $Q_i left( alpha right)$ arises from matching of the inner and intermediate asymptotic regions introduced by those authors and is required in applications of the asymptotic theory. This article describes a new numerical method for the calculation of $Q_i left( alpha right)$, which, because it explicitly allows for the logarithmic singularity in the kernel of the governing integral equation and uses quadratic interpolation of the non-singular factor in the integrand, is more accurate than that employed by Hocking and Rivers. Nonetheless, our results show good agreement with theirs, with, however, noticeable departures near $alpha = pi $. We also discuss the limiting cases $alpha to 0$ and $alpha to pi $. The leading-order terms of $Q_i left( alpha right)$ in both limits are in accord with the analysis of Hocking (A moving fluid interface. Part 2. The removal of the force singularity by a slip flow, J. Fluid Mech. 79 (1977) 209–229). The next-order terms are also considered. Hocking did not go beyond leading order for $alpha to 0$, and we believe his results for the next order as $alpha to pi $ to be incorrect. Numerically, we find that the next-order terms are $Oleft( {alpha ^2} right)$ for $alpha to 0$ and $Oleft( 1 right)$ as $alpha to pi $. The latter result agrees with Hocking, but the value of the $Oleft( 1 right)$ constant does not. It is hoped that giving details of the numerical method and more precise information, both numerical and in terms of its limiting behaviour, concerning $Q_i left( alpha right)$ will help those wanting to use the asymptotic theory of contact-line dynamics in future theoretical and numerical work.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qjmam/hbaa012","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44172402","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}
In this paper, we consider capillary-gravity waves propagating on the interface of two dielectric fluids under the influence of normal electric fields. The density of the upper fluid is assumed to be much smaller than the lower one. Linear and weakly nonlinear theories are studied. The connection to the results in other limit configurations is discussed. Fully nonlinear computations for travelling wave solutions are achieved via a boundary integral equation method. Periodic waves, solitary waves and generalised solitary waves are presented. The bifurcation of generalised solitary waves is discussed in detail.
{"title":"Capillary-gravity waves on the interface of two dielectric fluid layers under normal electric fields","authors":"A. Doak, T. Gao, J. Vanden-Broeck, Josh Kandola","doi":"10.1093/qjmam/hbaa009","DOIUrl":"https://doi.org/10.1093/qjmam/hbaa009","url":null,"abstract":"In this paper, we consider capillary-gravity waves propagating on the interface of two dielectric fluids under the influence of normal electric fields. The density of the upper fluid is assumed to be much smaller than the lower one. Linear and weakly nonlinear theories are studied. The connection to the results in other limit configurations is discussed. Fully nonlinear computations for travelling wave solutions are achieved via a boundary integral equation method. Periodic waves, solitary waves and generalised solitary waves are presented. The bifurcation of generalised solitary waves is discussed in detail.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"73 1","pages":"231-250"},"PeriodicalIF":0.9,"publicationDate":"2020-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qjmam/hbaa009","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42396502","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}
A history-dependent cohesive zone model is considered in the linear elasticity framework with the cohesive stresses governed by the fracture condition formulated in terms of a nonlinear Abel-type integral operator. A possibility for the cohesive stresses to possess a weak singularity has been examined by utilizing asymptotic modeling approach. It has been shown that the balance of the leading-term asymptotic representations in the model equations is possible for nonsingular cohesive stresses only.
{"title":"A Singular Nonlinear History-Dependent Cohesive Zone Model: Is it Possible?","authors":"I. Argatov","doi":"10.1093/qjmam/hbaa007","DOIUrl":"https://doi.org/10.1093/qjmam/hbaa007","url":null,"abstract":"\u0000 A history-dependent cohesive zone model is considered in the linear elasticity framework with the cohesive stresses governed by the fracture condition formulated in terms of a nonlinear Abel-type integral operator. A possibility for the cohesive stresses to possess a weak singularity has been examined by utilizing asymptotic modeling approach. It has been shown that the balance of the leading-term asymptotic representations in the model equations is possible for nonsingular cohesive stresses only.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"73 1","pages":"201-215"},"PeriodicalIF":0.9,"publicationDate":"2020-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qjmam/hbaa007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48119069","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}
Four distinct solutions exist for the potential distribution around two equal circular parallel conducting cylinders, charged to the same potential. Their equivalence is demonstrated, and the resulting analytical identities are discussed. The identities relate the Jacobi elliptic function $sn$, the Jacobi theta functions $theta _1 ,~theta _2 $ and infinite series over trigonometric and hyperbolic functions.
{"title":"Four solutions of a two-cylinder electrostatic problem, and identities resulting from their equivalence","authors":"J. Lekner","doi":"10.1093/qjmam/hbaa010","DOIUrl":"https://doi.org/10.1093/qjmam/hbaa010","url":null,"abstract":"\u0000 Four distinct solutions exist for the potential distribution around two equal circular parallel conducting cylinders, charged to the same potential. Their equivalence is demonstrated, and the resulting analytical identities are discussed. The identities relate the Jacobi elliptic function $sn$, the Jacobi theta functions $theta _1 ,~theta _2 $ and infinite series over trigonometric and hyperbolic functions.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"73 1","pages":"251-260"},"PeriodicalIF":0.9,"publicationDate":"2020-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/qjmam/hbaa010","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49133000","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 Legendre–Hadamard condition in Cosserat elasticity theory","authors":"M. Shirani, D. Steigmann, P. Neff","doi":"10.1093/QJMAM/HBAA013","DOIUrl":"https://doi.org/10.1093/QJMAM/HBAA013","url":null,"abstract":"The Legendre-Hadamard necessary condition for energy minimizers is derived in the framework of Cosserat elasticity theory.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2020-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41899081","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}
Pub Date : 2018-12-10eCollection Date: 2018-01-01DOI: 10.18332/tid/99539
Sharifa Ezat Wan Puteh, Roslina Abdul Manap, Tidi Maharani Hassan, Izzah Syazwani Ahmad, Idayu Badilla Idris, Fariza Md Sham, Andrea Ban Yu Lin, Chun Ian Soo, Rashidi Mohamed Pakri Mohamed, Ahmad Irdha Mokhtar, Hazli Zakaria, Jing Lee, Amer Siddiq Amer Nordin, Suthahar Ariaratnam, Mohd Zaliman Mohd Yusoff
Introduction: E-cigarette use is an emerging phenomenon with increasing recognition and acceptance globally. This study aims to create a profile of e-cigarette users among university students in Malaysia.
Methods: The study was conducted using a cross-sectional research involving six universities in Malaysia. A semi-structured questionnaire was distributed to 1302 randomly selected students, who either smoked cigarettes and/or e-cigarettes. The 2011 version of Global Adult Tobacco Surveys (GATS) tool was used to record the respondents' sociodemographic data.
Results: The study revealed that 74.9% of the respondents smoked e-cigarettes; 40.3% used both cigarettes and e-cigarettes (dual users), and 34.5% were exclusive e-cigarette users. The exclusive use of e-cigarettes was related to gender (OR=0.18, 95% CI: 0.09-0.39). Also, male respondents were the majority users (95%). Of the respondents, 75.2 % were Malays, 98.0% single and most believed they have no health problems (92.1%). Further findings revealed the occurrence of adverse effects, dizziness 14.4%, cough 14.1%, and headaches 12.4%. Overall, 57.8% of the respondents used e-cigarettes as a smoking cessation tool, while others consider e-cigarettes a self-image enhancing tool or as part of social activities.
Conclusions: Further research on the use of e-cigarettes should be conducted on a large number of respondents in other settings to augment the findings of this study, and also guide policy making on and prevention practice of e-cigarette use, among the general student population in Malaysia.
{"title":"The use of e-cigarettes among university students in Malaysia.","authors":"Sharifa Ezat Wan Puteh, Roslina Abdul Manap, Tidi Maharani Hassan, Izzah Syazwani Ahmad, Idayu Badilla Idris, Fariza Md Sham, Andrea Ban Yu Lin, Chun Ian Soo, Rashidi Mohamed Pakri Mohamed, Ahmad Irdha Mokhtar, Hazli Zakaria, Jing Lee, Amer Siddiq Amer Nordin, Suthahar Ariaratnam, Mohd Zaliman Mohd Yusoff","doi":"10.18332/tid/99539","DOIUrl":"10.18332/tid/99539","url":null,"abstract":"<p><strong>Introduction: </strong>E-cigarette use is an emerging phenomenon with increasing recognition and acceptance globally. This study aims to create a profile of e-cigarette users among university students in Malaysia.</p><p><strong>Methods: </strong>The study was conducted using a cross-sectional research involving six universities in Malaysia. A semi-structured questionnaire was distributed to 1302 randomly selected students, who either smoked cigarettes and/or e-cigarettes. The 2011 version of Global Adult Tobacco Surveys (GATS) tool was used to record the respondents' sociodemographic data.</p><p><strong>Results: </strong>The study revealed that 74.9% of the respondents smoked e-cigarettes; 40.3% used both cigarettes and e-cigarettes (dual users), and 34.5% were exclusive e-cigarette users. The exclusive use of e-cigarettes was related to gender (OR=0.18, 95% CI: 0.09-0.39). Also, male respondents were the majority users (95%). Of the respondents, 75.2 % were Malays, 98.0% single and most believed they have no health problems (92.1%). Further findings revealed the occurrence of adverse effects, dizziness 14.4%, cough 14.1%, and headaches 12.4%. Overall, 57.8% of the respondents used e-cigarettes as a smoking cessation tool, while others consider e-cigarettes a self-image enhancing tool or as part of social activities.</p><p><strong>Conclusions: </strong>Further research on the use of e-cigarettes should be conducted on a large number of respondents in other settings to augment the findings of this study, and also guide policy making on and prevention practice of e-cigarette use, among the general student population in Malaysia.</p>","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"11 1","pages":"57"},"PeriodicalIF":2.2,"publicationDate":"2018-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6659562/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83244506","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article addresses efficient implementation of the method of images for acoustic multiple scattering models (MSM) with perfectly reflecting flat boundaries. Time-harmonic problems are first solved in the polar coordinate system for circular scatterers; then the models are extended to the cylindrical coordinate system with (semi-)infinitely long circular cylinders. The MSM in this article is based on the method of separation of variables and Graf’s addition theorem. Derivations are provided for ‘image conditions’ which relate the unknown coefficients of outgoing waves from image scatterers with those of real counterparts. The method of images is applied to wedge-shaped domains with apex angles of π/n rad for a positive integer n. Image conditions make the MSM numerically more efficient: the system of linear equations for unknown coefficients is formulated 2n times faster; its memory requirements are reduced by 4n2 times for direct solvers. The proposed model is applied to a benchmark wedge in ocean environment with n=64. Good agreement is observed between the MSM with image conditions and the boundary element method. Furthermore, half- and quarter-space measurements in an anechoic chamber are in accordance with the correct use of image conditions. Incorrect image conditions reported elsewhere for polar coordinates are also discussed.
{"title":"Image conditions for polar and cylindrical coordinate separation-of-variables acoustic multiple scattering models with perfectly reflecting flat boundaries","authors":"Ho-Chul Shin","doi":"10.1093/QJMAM/HBY005","DOIUrl":"https://doi.org/10.1093/QJMAM/HBY005","url":null,"abstract":"This article addresses efficient implementation of the method of images for acoustic multiple scattering models (MSM) with perfectly reflecting flat boundaries. Time-harmonic problems are first solved in the polar coordinate system for circular scatterers; then the models are extended to the cylindrical coordinate system with (semi-)infinitely long circular cylinders. The MSM in this article is based on the method of separation of variables and Graf’s addition theorem. Derivations are provided for ‘image conditions’ which relate the unknown coefficients of outgoing waves from image scatterers with those of real counterparts. The method of images is applied to wedge-shaped domains with apex angles of \u0000π/n rad for a positive integer n. Image conditions make the MSM numerically more efficient: the system of linear equations for unknown coefficients is formulated 2n times faster; its memory requirements are reduced by 4n2 times for direct solvers. The proposed model is applied to a benchmark wedge in ocean environment with n=64. Good agreement is observed between the MSM with image conditions and the boundary element method. Furthermore, half- and quarter-space measurements in an anechoic chamber are in accordance with the correct use of image conditions. Incorrect image conditions reported elsewhere for polar coordinates are also discussed.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"71 1","pages":"273-296"},"PeriodicalIF":0.9,"publicationDate":"2018-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/QJMAM/HBY005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44707800","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}
The 2D scattering of in-plane elastic waves by a circle is considered when the surrounding medium is isotropic and the medium inside the circle is anisotropic (orthotropic). The equations inside the circle are transformed to polar coordinates and then depend explicitly on the azimuthal angle through trigonometric functions. Making expansions in trigonometric series in the azimuthal coordinate give a coupled system of ordinary differential equations in the radial coordinate that is solved by power series expansions. With the solution inside the circle complete the scattering problem is solved essentially as in the classical case. The elements of the transition (T) matrix of the circle are given explicitly for low frequencies (long wavelengths). For low frequencies some numerical examples are given showing the strong influence of anisotropy.
{"title":"Scattering of In-plane Elastic Waves by an Anisotropic Circle","authors":"A. Boström","doi":"10.1093/QJMAM/HBX029","DOIUrl":"https://doi.org/10.1093/QJMAM/HBX029","url":null,"abstract":"The 2D scattering of in-plane elastic waves by a circle is considered when the surrounding medium is isotropic and the medium inside the circle is anisotropic (orthotropic). The equations inside the circle are transformed to polar coordinates and then depend explicitly on the azimuthal angle through trigonometric functions. Making expansions in trigonometric series in the azimuthal coordinate give a coupled system of ordinary differential equations in the radial coordinate that is solved by power series expansions. With the solution inside the circle complete the scattering problem is solved essentially as in the classical case. The elements of the transition (T) matrix of the circle are given explicitly for low frequencies (long wavelengths). For low frequencies some numerical examples are given showing the strong influence of anisotropy.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"71 1","pages":"139-155"},"PeriodicalIF":0.9,"publicationDate":"2018-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/QJMAM/HBX029","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45172669","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}
Two model problems of plane elasticity on subsonic steady-state motion of a thin rigid body in an elastic medium are analyzed. Both models concern a finite body symmetric with respect to the plane of motion and assume that the body contacts with the surrounding medium according to the Coulomb friction law. The body, while moves, leaves a trailing semi-infinite cracklike cavity moving at the body speed. The first model also assumes that ahead of the body a finite crack-like cavity is formed, and it is moving at the same speed. The second model does not admit the existence of this finite cavity. Both problems reduce to two sequently solved Riemann–Hilbert problems with piece-wise constant coefficients. Analysis of the solution to these problems obtained by quadratures reveals that the normal and tangential traction components and the normal velocity are continuous for any point of separation of the medium from the body. A criterion for the separation point based on the analysis of the sign of the normal traction component is proposed. Numerical results for the length of the fore crack (the first model), the normal traction and the resistance force for some ogive-nose penetrators are reported.
{"title":"Subsonic Frictional Cavitating Penetration of a Thin Rigid Body Into an Elastic Medium","authors":"Y. Antipov","doi":"10.1093/QJMAM/HBY003","DOIUrl":"https://doi.org/10.1093/QJMAM/HBY003","url":null,"abstract":"Two model problems of plane elasticity on subsonic steady-state motion of a thin rigid body in an elastic medium are analyzed. Both models concern a finite body symmetric with respect to the plane of motion and assume that the body contacts with the surrounding medium according to the Coulomb friction law. The body, while moves, leaves a trailing semi-infinite cracklike cavity moving at the body speed. The first model also assumes that ahead of the body a finite crack-like cavity is formed, and it is moving at the same speed. The second model does not admit the existence of this finite cavity. Both problems reduce to two sequently solved Riemann–Hilbert problems with piece-wise constant coefficients. Analysis of the solution to these problems obtained by quadratures reveals that the normal and tangential traction components and the normal velocity are continuous for any point of separation of the medium from the body. A criterion for the separation point based on the analysis of the sign of the normal traction component is proposed. Numerical results for the length of the fore crack (the first model), the normal traction and the resistance force for some ogive-nose penetrators are reported.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"71 1","pages":"221-243"},"PeriodicalIF":0.9,"publicationDate":"2018-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1093/QJMAM/HBY003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47065754","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：