Summary This article analyses static buckling of the so-called Gao beam nonlinear model. It considers pure buckling problems in which the vertical loads are omitted. The analysis, using minimisation of energy and the concept of a modified Rayleigh quotient, leads to new results regarding the critical load necessary for buckling, and the existence and number of post-buckling solutions. Computational results are provided for cases with fixed axial loading. Furthermore, the authors explore the impact of the system parameters on the solutions, which are summarised in a table. The new findings in this research are unique and help to better understand the behaviour of the static and dynamic Gao beam.
{"title":"Post-Buckling Solutions for the Gao Beam","authors":"H Netuka, J Machalová","doi":"10.1093/qjmam/hbad007","DOIUrl":"https://doi.org/10.1093/qjmam/hbad007","url":null,"abstract":"Summary This article analyses static buckling of the so-called Gao beam nonlinear model. It considers pure buckling problems in which the vertical loads are omitted. The analysis, using minimisation of energy and the concept of a modified Rayleigh quotient, leads to new results regarding the critical load necessary for buckling, and the existence and number of post-buckling solutions. Computational results are provided for cases with fixed axial loading. Furthermore, the authors explore the impact of the system parameters on the solutions, which are summarised in a table. The new findings in this research are unique and help to better understand the behaviour of the static and dynamic Gao beam.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135924485","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 We study the elastic field in a three-phase composite composed of an internal spherical homogeneous elastic inhomogeneity, an intermediate functionally graded interphase layer and an outer unbounded homogeneous elastic matrix subjected to an arbitrary uniform remote loading. The shear modulus of the interphase layer obeys a power law distribution along the radial direction. We accomplish the design of harmonic and neutral spherical elastic inhomogeneities. Specifically, the shear modulus of the matrix can be judiciously chosen in such a way that the insertion of the harmonic spherical inhomogeneity does not disturb the original constant mean stress in the surrounding matrix. The shear modulus of the matrix and relative thickness of the interphase can also be suitably chosen such that the insertion of the neutral spherical inhomogeneity does not disturb the original uniform deviatoric stresses in the surrounding matrix.
{"title":"Harmonic And Neutral Spherical Elastic Inhomogeneities with A Functionally Graded Interphase Layer","authors":"Xu Wang, Peter Schiavone","doi":"10.1093/qjmam/hbad006","DOIUrl":"https://doi.org/10.1093/qjmam/hbad006","url":null,"abstract":"Summary We study the elastic field in a three-phase composite composed of an internal spherical homogeneous elastic inhomogeneity, an intermediate functionally graded interphase layer and an outer unbounded homogeneous elastic matrix subjected to an arbitrary uniform remote loading. The shear modulus of the interphase layer obeys a power law distribution along the radial direction. We accomplish the design of harmonic and neutral spherical elastic inhomogeneities. Specifically, the shear modulus of the matrix can be judiciously chosen in such a way that the insertion of the harmonic spherical inhomogeneity does not disturb the original constant mean stress in the surrounding matrix. The shear modulus of the matrix and relative thickness of the interphase can also be suitably chosen such that the insertion of the neutral spherical inhomogeneity does not disturb the original uniform deviatoric stresses in the surrounding matrix.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135306039","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}
Nikolaos L Tsitsas, Hamad M Alkhoori, Akhlesh Lakhtakia
Summary A boundary-value problem was formulated for perturbation of an electrostatic field by a coated dielectric sphere made of two distinct linear anisotropic dielectric (LAD) materials. Specific affine transformations were employed to represent the electric potential inside the core and the coating in terms of the solutions of the Laplace equation. A transition matrix was found to relate the source potential and the perturbation potential in the exterior region. The formulation can be straightforwardly extended to concentrically multilayered spheres made of several homogeneous LAD materials.
{"title":"Theory of Perturbation of Electrostatic Field By A Coated Anisotropic Dielectric Sphere","authors":"Nikolaos L Tsitsas, Hamad M Alkhoori, Akhlesh Lakhtakia","doi":"10.1093/qjmam/hbad005","DOIUrl":"https://doi.org/10.1093/qjmam/hbad005","url":null,"abstract":"Summary A boundary-value problem was formulated for perturbation of an electrostatic field by a coated dielectric sphere made of two distinct linear anisotropic dielectric (LAD) materials. Specific affine transformations were employed to represent the electric potential inside the core and the coating in terms of the solutions of the Laplace equation. A transition matrix was found to relate the source potential and the perturbation potential in the exterior region. The formulation can be straightforwardly extended to concentrically multilayered spheres made of several homogeneous LAD materials.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136072128","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 work, we propose a new two-scale finite-strain thin plate theory for highly heterogeneous plates described by a repetitive periodic microstructure. For this type of theory, two scales exist, the macroscopic one is linked to the entire plate and the microscopic one is linked to the size of the heterogeneity. We consider the case when the plate thickness is comparable to in-plane heterogeneities. We assume that the nonlinear macroscopic part of the model is of Kirchhoff–Love type. We obtain the nonlinear homogenised model by performing simultaneously both the homogenisation and the reduction of the initial three-dimensional plate problem to a two-dimensional one. Since nonlinear equations are difficult to solve, we linearise this homogenised Kirchhoff–Love plate theory. Finally, we discuss the treatment of edge effects in the vicinity of the lateral boundary of the plate.
{"title":"Homogenisation of Nonlinear Heterogeneous Thin Plate When the Plate Thickness and In-Plane Heterogeneities are of the Same Order of Magnitude","authors":"E. Pruchnicki","doi":"10.1093/qjmam/hbad004","DOIUrl":"https://doi.org/10.1093/qjmam/hbad004","url":null,"abstract":"\u0000 In this work, we propose a new two-scale finite-strain thin plate theory for highly heterogeneous plates described by a repetitive periodic microstructure. For this type of theory, two scales exist, the macroscopic one is linked to the entire plate and the microscopic one is linked to the size of the heterogeneity. We consider the case when the plate thickness is comparable to in-plane heterogeneities. We assume that the nonlinear macroscopic part of the model is of Kirchhoff–Love type. We obtain the nonlinear homogenised model by performing simultaneously both the homogenisation and the reduction of the initial three-dimensional plate problem to a two-dimensional one. Since nonlinear equations are difficult to solve, we linearise this homogenised Kirchhoff–Love plate theory. Finally, we discuss the treatment of edge effects in the vicinity of the lateral boundary of the plate.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48827087","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 model problem of scattering of a sound wave by an infinite plane structure formed by a semi-infinite acoustically hard screen and a semi-infinite sandwich panel perforated from one side and covered by a membrane from the other is exactly solved. The model is governed by two Helmholtz equations for the velocity potentials in the upper and lower half-planes coupled by the Leppington effective boundary condition and the equation of vibration of a membrane in a fluid. Two methods of solution are proposed and discussed. Both methods reduce the problem to an order-2 vector Riemann–Hilbert problem. The matrix coefficients have different entries, have the Chebotarev–Khrapkov structure and share the same order-4 characteristic polynomial. Exact Wiener–Hopf matrix factorization requires solving a scalar Riemann–Hilbert on an elliptic surface and the associated genus-1 Jacobi inversion problem solved in terms of the associated Riemann θ-function. Numerical results for the absolute value of the total velocity potentials are reported and discussed.
{"title":"Scattering by a Perforated Sandwich Panel: Method of Riemann Surfaces","authors":"Y. Antipov","doi":"10.1093/qjmam/hbad003","DOIUrl":"https://doi.org/10.1093/qjmam/hbad003","url":null,"abstract":"\u0000 The model problem of scattering of a sound wave by an infinite plane structure formed by a semi-infinite acoustically hard screen and a semi-infinite sandwich panel perforated from one side and covered by a membrane from the other is exactly solved. The model is governed by two Helmholtz equations for the velocity potentials in the upper and lower half-planes coupled by the Leppington effective boundary condition and the equation of vibration of a membrane in a fluid. Two methods of solution are proposed and discussed. Both methods reduce the problem to an order-2 vector Riemann–Hilbert problem. The matrix coefficients have different entries, have the Chebotarev–Khrapkov structure and share the same order-4 characteristic polynomial. Exact Wiener–Hopf matrix factorization requires solving a scalar Riemann–Hilbert on an elliptic surface and the associated genus-1 Jacobi inversion problem solved in terms of the associated Riemann θ-function. Numerical results for the absolute value of the total velocity potentials are reported and discussed.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42584509","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 While several articles have been written on water waves on flows with constant vorticity, little is known about the extent to which a non-constant vorticity affects the flow structure, such as the appearance of stagnation points. In order to shed light on this topic, we investigate in detail the flow beneath solitary waves propagating on an exponentially decaying sheared current. Our focus is to analyse numerically the emergence of stagnation points. For this purpose, we approximate the velocity field within the fluid bulk through the classical Korteweg-de Vries asymptotic expansion and use the Matlab language to evaluate the resulting stream function. Our findings suggest that the flow beneath the waves can have 0, 1 or 2 stagnation points in the fluid body. We also study the bifurcation between these flows. Our simulations indicate that the stagnation points emerge from a streamline with a sharp corner.
{"title":"Solitary waves on flows with an exponentially sheared current and stagnation points","authors":"Marcelo V Flamarion, Roberto-J R Ribeiro","doi":"10.1093/qjmam/hbac021","DOIUrl":"https://doi.org/10.1093/qjmam/hbac021","url":null,"abstract":"Summary While several articles have been written on water waves on flows with constant vorticity, little is known about the extent to which a non-constant vorticity affects the flow structure, such as the appearance of stagnation points. In order to shed light on this topic, we investigate in detail the flow beneath solitary waves propagating on an exponentially decaying sheared current. Our focus is to analyse numerically the emergence of stagnation points. For this purpose, we approximate the velocity field within the fluid bulk through the classical Korteweg-de Vries asymptotic expansion and use the Matlab language to evaluate the resulting stream function. Our findings suggest that the flow beneath the waves can have 0, 1 or 2 stagnation points in the fluid body. We also study the bifurcation between these flows. Our simulations indicate that the stagnation points emerge from a streamline with a sharp corner.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136377346","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 We investigate the small-amplitude deformations of a long thin-walled elastic tube having an initially axially uniform elliptical cross-section. The tube is deformed by a (possibly non-uniform) transmural pressure. At leading order, its deformations are shown to be governed by a single partial differential equation (PDE) for the azimuthal displacement as a function of the axial and azimuthal co-ordinates and time. Previous authors have obtained solutions to this PDE by making ad hoc approximations based on truncating an approximate Fourier representation. In this article, we instead write the azimuthal displacement as a sum over the azimuthal eigenfunctions of a generalised eigenvalue problem and show that we are able to derive an uncoupled system of linear PDEs with constant coefficients for the amplitude of the azimuthal modes as a function of the axial co-ordinate and time. This results in a formal solution of the whole system being found as a sum over the azimuthal modes. We show that the $n$th mode’s contribution to the tube’s relative area change is governed by a simplified second-order PDE and examine the case in which the tube’s deformations are driven by a uniform transmural pressure. The relative errors induced by truncating the series solution after the first and second terms are then evaluated as a function of both the ellipticity and pre-stress of the tube. After comparing our results with Whittaker et al. (A rational derivation of a tube law from shell theory, Q. J. Mech. Appl. Math. 63 (2010) 465–496), we find that this new method leads to a significant simplification when calculating contributions from the higher-order azimuthal modes, which in turn makes a more accurate solution easier to obtain.
我们研究了具有初始轴向均匀椭圆截面的长薄壁弹性管的小振幅变形。管被(可能不均匀的)跨壁压力变形。在导阶，它的变形被证明是由一个单一的偏微分方程(PDE)控制的方位角位移作为轴向和方位角坐标和时间的函数。以前的作者已经通过基于截断近似傅立叶表示的临时近似获得了该PDE的解。在本文中，我们将方位角位移写成广义本征值问题的方位角本征函数的和，并表明我们能够推导出一个解耦合的线性偏微分方程系统，其方位角模态的振幅是轴向坐标和时间的函数。这导致整个系统的形式解被发现为方位角模态的和。我们表明，第n阶模态对管的相对面积变化的贡献是由简化的二阶偏微分方程控制的，并研究了管的变形是由均匀的跨壁压力驱动的情况。在第一项和第二项之后截断级数解所引起的相对误差，然后作为管的椭圆度和预应力的函数进行了评估。在将我们的结果与惠特克等人的结果进行比较后(从壳理论中合理推导管定律，Q. J. Mech。达成。数学。63(2010)465-496)，我们发现这种新方法在计算高阶方位角模式的贡献时显著简化，这反过来又使更准确的解更容易获得。
{"title":"A new solution for the deformations of an initially elliptical elastic-walled tube","authors":"D J Netherwood, R J Whittaker","doi":"10.1093/qjmam/hbac018","DOIUrl":"https://doi.org/10.1093/qjmam/hbac018","url":null,"abstract":"Summary We investigate the small-amplitude deformations of a long thin-walled elastic tube having an initially axially uniform elliptical cross-section. The tube is deformed by a (possibly non-uniform) transmural pressure. At leading order, its deformations are shown to be governed by a single partial differential equation (PDE) for the azimuthal displacement as a function of the axial and azimuthal co-ordinates and time. Previous authors have obtained solutions to this PDE by making ad hoc approximations based on truncating an approximate Fourier representation. In this article, we instead write the azimuthal displacement as a sum over the azimuthal eigenfunctions of a generalised eigenvalue problem and show that we are able to derive an uncoupled system of linear PDEs with constant coefficients for the amplitude of the azimuthal modes as a function of the axial co-ordinate and time. This results in a formal solution of the whole system being found as a sum over the azimuthal modes. We show that the $n$th mode’s contribution to the tube’s relative area change is governed by a simplified second-order PDE and examine the case in which the tube’s deformations are driven by a uniform transmural pressure. The relative errors induced by truncating the series solution after the first and second terms are then evaluated as a function of both the ellipticity and pre-stress of the tube. After comparing our results with Whittaker et al. (A rational derivation of a tube law from shell theory, Q. J. Mech. Appl. Math. 63 (2010) 465–496), we find that this new method leads to a significant simplification when calculating contributions from the higher-order azimuthal modes, which in turn makes a more accurate solution easier to obtain.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"140 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135013259","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}
We study the problem of diffraction by a right-angled no-contrast penetrable wedge by means of a two-complex-variable Wiener–Hopf approach. Specifically, the analyticity properties of the unknown (spectral) functions of the two-complex-variable Wiener–Hopf equation are studied. We show that these spectral functions can be analytically continued onto a two-complex dimensional manifold, and unveil their singularities in C2. To do so, integral representation formulae for the spectral functions are given and thoroughly used. It is shown that the novel concept of additive crossing holds for the penetrable wedge diffraction problem, and that we can reformulate the physical diffraction problem as a functional problem using this concept.
{"title":"Diffraction by a Right-Angled No-Contrast Penetrable Wedge: Analytical Continuation of Spectral Functions","authors":"Valentin D. Kunz, R. Assier","doi":"10.1093/qjmam/hbad002","DOIUrl":"https://doi.org/10.1093/qjmam/hbad002","url":null,"abstract":"\u0000 We study the problem of diffraction by a right-angled no-contrast penetrable wedge by means of a two-complex-variable Wiener–Hopf approach. Specifically, the analyticity properties of the unknown (spectral) functions of the two-complex-variable Wiener–Hopf equation are studied. We show that these spectral functions can be analytically continued onto a two-complex dimensional manifold, and unveil their singularities in C2. To do so, integral representation formulae for the spectral functions are given and thoroughly used. It is shown that the novel concept of additive crossing holds for the penetrable wedge diffraction problem, and that we can reformulate the physical diffraction problem as a functional problem using this concept.","PeriodicalId":56087,"journal":{"name":"Quarterly Journal of Mechanics and Applied Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2022-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45248831","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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