Kheng Lim Goh, Richard M. Aspden, K. Mathias, D. Hukins
The finite–element method was used to calculate the axial stress in an elastic fibre embedded in an elastic matrix to model a fibre–composite material. Axisymmetric models were created for cylindrical, ellipsoidal, paraboloidal and conical fibres embedded in a matrix and characterized by a fibre axial ratio, q. The effects of varying q, from 200 to 1000, and the ratio of the Young moduli of the fibre and the matrix, Ef/Em, from 50 to 104, were investigated. For a cylindrical fibre, the axial stress distribution along the fibre axis was similar for all values of q and Ef/Em; it was greatest at the centre, decreased steadily over most of the fibre length and fell rapidly to zero near the fibre end. For fixed q, the magnitude of this stress increased with increasing Ef/Em, whereas for fixed Ef/Em the variation with q was small. There was good qualitative agreement between these data and previous analytical models. The axial stress in the conical fibre was a minimum at the fibre centre and rose gradually to a maximum close to the fibre end. This was most pronounced for small values of q and at large values of Ef/Em. Stress distributions for the paraboloid and ellipsoid lay between those for the cylinder and the cone. For small values of Ef/Em, both the magnitudes and the axial distributions of axial stress were almost indistinguishable for all shapes of fibre and all values of q studied.
{"title":"Finite–element analysis of the effect of material properties and fibre shape on stresses in an elastic fibre embedded in an elastic matrix in a fibre–composite material","authors":"Kheng Lim Goh, Richard M. Aspden, K. Mathias, D. Hukins","doi":"10.1098/rspa.2003.1264","DOIUrl":"https://doi.org/10.1098/rspa.2003.1264","url":null,"abstract":"The finite–element method was used to calculate the axial stress in an elastic fibre embedded in an elastic matrix to model a fibre–composite material. Axisymmetric models were created for cylindrical, ellipsoidal, paraboloidal and conical fibres embedded in a matrix and characterized by a fibre axial ratio, q. The effects of varying q, from 200 to 1000, and the ratio of the Young moduli of the fibre and the matrix, Ef/Em, from 50 to 104, were investigated. For a cylindrical fibre, the axial stress distribution along the fibre axis was similar for all values of q and Ef/Em; it was greatest at the centre, decreased steadily over most of the fibre length and fell rapidly to zero near the fibre end. For fixed q, the magnitude of this stress increased with increasing Ef/Em, whereas for fixed Ef/Em the variation with q was small. There was good qualitative agreement between these data and previous analytical models. The axial stress in the conical fibre was a minimum at the fibre centre and rose gradually to a maximum close to the fibre end. This was most pronounced for small values of q and at large values of Ef/Em. Stress distributions for the paraboloid and ellipsoid lay between those for the cylinder and the cone. For small values of Ef/Em, both the magnitudes and the axial distributions of axial stress were almost indistinguishable for all shapes of fibre and all values of q studied.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86318663","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}
In this paper we investigate the linear stability and properties of the planar travelling non–adiabatic combustion front for the cases of zero and non–zero ambient temperature. The speed of the front is estimated numerically using the shooting and relaxation methods. It is shown that for given parameter values the solution either does not exist or there are two solutions with different values of the front speed, which are referred to as ‘fast’ and ‘slow’. The Evans function approach extended by the compound–matrix method is employed to numerically solve the linear–stability problem for the travelling–wave solution. We demonstrate that the ‘slow’ branch of the solutions is unstable, whereas the ‘fast’ branch can be stable or exhibits Hopf or Bogdanov–Takens instability, depending on the parameter values.
{"title":"Evans function stability of non-adiabatic combustion waves","authors":"V. Gubernov, G. Mercer, H. Sidhu, R. Weber","doi":"10.1098/rspa.2004.1285","DOIUrl":"https://doi.org/10.1098/rspa.2004.1285","url":null,"abstract":"In this paper we investigate the linear stability and properties of the planar travelling non–adiabatic combustion front for the cases of zero and non–zero ambient temperature. The speed of the front is estimated numerically using the shooting and relaxation methods. It is shown that for given parameter values the solution either does not exist or there are two solutions with different values of the front speed, which are referred to as ‘fast’ and ‘slow’. The Evans function approach extended by the compound–matrix method is employed to numerically solve the linear–stability problem for the travelling–wave solution. We demonstrate that the ‘slow’ branch of the solutions is unstable, whereas the ‘fast’ branch can be stable or exhibits Hopf or Bogdanov–Takens instability, depending on the parameter values.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86939848","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}
We consider the simulation of deformation of polycrystalline materials by grain–boundary diffusion creep. For a given network of grain boundaries intersecting at nodes, with appropriate boundary conditions, we can calculate the rate at which material will be dissolved or deposited along each grain boundary and hence predict the rate at which each grain will move to accommodate this dissolution/deposition. We discuss two numerical methods for simulating the network changes over a finite time–interval, based on using the movement of adjacent grain boundaries over a small time–interval to estimate the velocities of the nodes. (The second of these methods has enabled us to speed up solution by 100 times in typical experiments compared with a naive forward–Euler approach.) We show that the accuracy with which the node velocities can be estimated is dependent only on the precision of the machine with which they are computed and deduce that, for all practical purposes, the lack of precise node velocity values does not detract from the quality of our solution. Finally, we consider the underlying stability of the problem under various different boundary conditions and conclude that our methods have the potential for providing useful insight into the effect of grain size and shape on deformation in polycrystalline materials.
{"title":"Simulation of grain-boundary diffusion creep: analysis of some new numerical techniques","authors":"J. Ford, N. Ford, J. Wheeler","doi":"10.1098/rspa.2004.1287","DOIUrl":"https://doi.org/10.1098/rspa.2004.1287","url":null,"abstract":"We consider the simulation of deformation of polycrystalline materials by grain–boundary diffusion creep. For a given network of grain boundaries intersecting at nodes, with appropriate boundary conditions, we can calculate the rate at which material will be dissolved or deposited along each grain boundary and hence predict the rate at which each grain will move to accommodate this dissolution/deposition. We discuss two numerical methods for simulating the network changes over a finite time–interval, based on using the movement of adjacent grain boundaries over a small time–interval to estimate the velocities of the nodes. (The second of these methods has enabled us to speed up solution by 100 times in typical experiments compared with a naive forward–Euler approach.) We show that the accuracy with which the node velocities can be estimated is dependent only on the precision of the machine with which they are computed and deduce that, for all practical purposes, the lack of precise node velocity values does not detract from the quality of our solution. Finally, we consider the underlying stability of the problem under various different boundary conditions and conclude that our methods have the potential for providing useful insight into the effect of grain size and shape on deformation in polycrystalline materials.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90065515","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}
This paper reports results of numerical simulations of stratified tidal flow over the Knight Inlet sill. A non–hydrostatic, two–dimensional model is used, which incorporates a no–slip bottom boundary condition through the use of a vertical eddy viscosity/diffusivity parametrization that is non–zero only near the bottom. In inviscid model runs, a large lee wave is rapidly formed, which quickly breaks, leading to the formation of a high–drag state and a strong downslope jet in the early stages of the ebb tide. The use of a no–slip bottom boundary condition results in boundary–layer separation from near the top of the sill. This significantly reduces the amplitude of the lee wave during the initial stages of the flow development. For most model runs, a large lee wave is ultimately formed and the separation point moves down the lee of the sill to a position immediately downstream of the lee wave. The transition to this high–drag state is significantly delayed compared with inviscid model runs. Weakened stratification immediately above the sill, inclusion of an eddy viscosity/diffusivity above the bottom and a pool of dense water on the downstream (seaward) side of the sill can all contribute to a delay in the transition to a high–drag state, and can eliminate it entirely. For one model run using a vertical eddy viscosity parametrization above the bottom, a reduction of the vertical diffusivity eliminated the formation of a high–drag state. This suggests that at least in some cases entrainment into the lee wave can cause its growth and result in the formation of a high–drag state.
{"title":"On boundary–layer separation and internal wave generation at the Knight Inlet sill","authors":"K. Lamb","doi":"10.1098/rspa.2003.1276","DOIUrl":"https://doi.org/10.1098/rspa.2003.1276","url":null,"abstract":"This paper reports results of numerical simulations of stratified tidal flow over the Knight Inlet sill. A non–hydrostatic, two–dimensional model is used, which incorporates a no–slip bottom boundary condition through the use of a vertical eddy viscosity/diffusivity parametrization that is non–zero only near the bottom. In inviscid model runs, a large lee wave is rapidly formed, which quickly breaks, leading to the formation of a high–drag state and a strong downslope jet in the early stages of the ebb tide. The use of a no–slip bottom boundary condition results in boundary–layer separation from near the top of the sill. This significantly reduces the amplitude of the lee wave during the initial stages of the flow development. For most model runs, a large lee wave is ultimately formed and the separation point moves down the lee of the sill to a position immediately downstream of the lee wave. The transition to this high–drag state is significantly delayed compared with inviscid model runs. Weakened stratification immediately above the sill, inclusion of an eddy viscosity/diffusivity above the bottom and a pool of dense water on the downstream (seaward) side of the sill can all contribute to a delay in the transition to a high–drag state, and can eliminate it entirely. For one model run using a vertical eddy viscosity parametrization above the bottom, a reduction of the vertical diffusivity eliminated the formation of a high–drag state. This suggests that at least in some cases entrainment into the lee wave can cause its growth and result in the formation of a high–drag state.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90197260","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}
Analytic approximations for percolation points in two–dimensional and three–dimensional particulate arrays have been reported for only a very few, simple particle geometries. Here, an analytical approach is presented to determine the percolative properties (i.e. statistical cluster properties) of permeable ellipsoids of revolution. We generalize a series expansion technique, previously used by other authors to study arrays of spheres and cubes. Our analytic solutions are compared with Monte Carlo simulation results, and show good agreement at low particle aspect ratio. At higher aspect ratios, the analytic approximation becomes even more computationally intensive than direct simulation of a number of realizations. Additional simulation results, and simplified, closed–form bounding expressions for percolation thresholds are also presented. Limitations and applications of the asymptotic expressions are discussed in the context of materials design and design of sensor arrays.
{"title":"Analytical approximation of the percolation threshold for overlapping ellipsoids of revolution","authors":"Y. Yi, A. Sastry","doi":"10.1098/rspa.2004.1279","DOIUrl":"https://doi.org/10.1098/rspa.2004.1279","url":null,"abstract":"Analytic approximations for percolation points in two–dimensional and three–dimensional particulate arrays have been reported for only a very few, simple particle geometries. Here, an analytical approach is presented to determine the percolative properties (i.e. statistical cluster properties) of permeable ellipsoids of revolution. We generalize a series expansion technique, previously used by other authors to study arrays of spheres and cubes. Our analytic solutions are compared with Monte Carlo simulation results, and show good agreement at low particle aspect ratio. At higher aspect ratios, the analytic approximation becomes even more computationally intensive than direct simulation of a number of realizations. Additional simulation results, and simplified, closed–form bounding expressions for percolation thresholds are also presented. Limitations and applications of the asymptotic expressions are discussed in the context of materials design and design of sensor arrays.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90960816","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}
A model is presented for concentrated sediment transport that is driven by strong, fully developed turbulent shear flows over a mobile bed. Balance equations for the average mass, momentum and energy for the two phases are phrased in terms of concentration–weighted (Favre averaged) velocities. Closures for the correlations between fluctuations in concentration and particle velocities are based on those for collisional grain flow. This is appropriate for particles that are so massive that their fall velocity exceeds the friction velocity of the turbulent fluid flow. Particular attention is given to the slow flow in the region of high concentration above the stationary bed. A failure criterion is introduced to determine the location of the stationary bed. The proposed model is solved numerically with a finite–difference algorithm in both steady and unsteady conditions. The predictions of sediment concentration and velocity are tested against experimental measurements that involve massive particles. The model is further employed to study several global features of sheet flow such as the total sediment transport rate in steady and unsteady conditions.
{"title":"On two-phase sediment transport: sheet flow of massive particles","authors":"T. Hsu, J. Jenkins, P. Liu","doi":"10.1098/rspa.2003.1273","DOIUrl":"https://doi.org/10.1098/rspa.2003.1273","url":null,"abstract":"A model is presented for concentrated sediment transport that is driven by strong, fully developed turbulent shear flows over a mobile bed. Balance equations for the average mass, momentum and energy for the two phases are phrased in terms of concentration–weighted (Favre averaged) velocities. Closures for the correlations between fluctuations in concentration and particle velocities are based on those for collisional grain flow. This is appropriate for particles that are so massive that their fall velocity exceeds the friction velocity of the turbulent fluid flow. Particular attention is given to the slow flow in the region of high concentration above the stationary bed. A failure criterion is introduced to determine the location of the stationary bed. The proposed model is solved numerically with a finite–difference algorithm in both steady and unsteady conditions. The predictions of sediment concentration and velocity are tested against experimental measurements that involve massive particles. The model is further employed to study several global features of sheet flow such as the total sediment transport rate in steady and unsteady conditions.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83925396","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}
We derive absolute–stability results of Popov and circle–criterion type for infinite–dimensional systems in an input–output setting. Our results apply to feedback systems in which the linear part is the series interconnection of an input–output stable linear system and an integrator, and the nonlinearity satisfies a sector condition which, in particular, allows for saturation and deadzone effects. We use the input–output theory developed to derive state–space results on absolute stability applying to feedback systems in which the linear part is the series interconnection of an exponentially stable, well–posed infinite–dimensional system and an integrator.
{"title":"Absolute-stability results in infinite dimensions","authors":"R. Curtain, H. Logemann, O. Staffans","doi":"10.1098/rspa.2003.1261","DOIUrl":"https://doi.org/10.1098/rspa.2003.1261","url":null,"abstract":"We derive absolute–stability results of Popov and circle–criterion type for infinite–dimensional systems in an input–output setting. Our results apply to feedback systems in which the linear part is the series interconnection of an input–output stable linear system and an integrator, and the nonlinearity satisfies a sector condition which, in particular, allows for saturation and deadzone effects. We use the input–output theory developed to derive state–space results on absolute stability applying to feedback systems in which the linear part is the series interconnection of an exponentially stable, well–posed infinite–dimensional system and an integrator.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79686181","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}
A partial–differential–equation model of neurite growth is developed. This model is the first of its kind and uses a continuum mechanical approach to model the effects of active transport, diffusion and species degradation of the oligomer tubulin, which is used in the elongation of a single neurite. The model problem is mathematically difficult since it must be solved on a dynamically growing domain. The development and implementation of a spatial transformation to a neurite length coordinate simplifies the problem. Existence and uniqueness of solutions to the steady–state problem are found and shown to be equivalent to solving a nonlinear equation for the steady–state length. This expression is not directly solvable except in certain degenerate cases. However, one system parameter is naturally small and permits solutions in terms of asymptotic series. We identify three growth regimes analytically and verify them numerically. It is then evident that a neuron may easily regulate the extent of its own neuritic growth by increasing or decreasing its tubulin production relative to the active transport/degradation fraction.
{"title":"Mathematical formulation and analysis of a continuum model for tubulin-driven neurite elongation","authors":"D. McLean, B. Graham","doi":"10.1098/rspa.2004.1288","DOIUrl":"https://doi.org/10.1098/rspa.2004.1288","url":null,"abstract":"A partial–differential–equation model of neurite growth is developed. This model is the first of its kind and uses a continuum mechanical approach to model the effects of active transport, diffusion and species degradation of the oligomer tubulin, which is used in the elongation of a single neurite. The model problem is mathematically difficult since it must be solved on a dynamically growing domain. The development and implementation of a spatial transformation to a neurite length coordinate simplifies the problem. Existence and uniqueness of solutions to the steady–state problem are found and shown to be equivalent to solving a nonlinear equation for the steady–state length. This expression is not directly solvable except in certain degenerate cases. However, one system parameter is naturally small and permits solutions in terms of asymptotic series. We identify three growth regimes analytically and verify them numerically. It is then evident that a neuron may easily regulate the extent of its own neuritic growth by increasing or decreasing its tubulin production relative to the active transport/degradation fraction.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81652630","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}
During the course of a Stokes phenomenon, an asymptotic expansion can change its form as a further series, prefactored by an exponentially small term and a Stokes multiplier, appears in the representation. The initially exponentially small contribution may nevertheless grow to dominate the behaviour for other values of the asymptotic or associated parameters. In this paper we introduce the concept of a‘higher–order Stokes phenomeno’, at which a Stokes multiplier itself can change value. We show that the higher–order Stokes phenomenon can be used to explain the apparent sudden birth of Stokes lines at regular points and how it is indispensable to the proper derivation of expansions that involve three or more possible asymptotic contributions. We provide an example of how the higher–order Stokes phenomenon can have important effects on the large–time behaviour of partial differential equations.
{"title":"On the higher–order Stokes phenomenon","authors":"C. Howls, P. J. Langman, A. Olde Daalhuis","doi":"10.1098/rspa.2004.1299","DOIUrl":"https://doi.org/10.1098/rspa.2004.1299","url":null,"abstract":"During the course of a Stokes phenomenon, an asymptotic expansion can change its form as a further series, prefactored by an exponentially small term and a Stokes multiplier, appears in the representation. The initially exponentially small contribution may nevertheless grow to dominate the behaviour for other values of the asymptotic or associated parameters. In this paper we introduce the concept of a‘higher–order Stokes phenomeno’, at which a Stokes multiplier itself can change value. We show that the higher–order Stokes phenomenon can be used to explain the apparent sudden birth of Stokes lines at regular points and how it is indispensable to the proper derivation of expansions that involve three or more possible asymptotic contributions. We provide an example of how the higher–order Stokes phenomenon can have important effects on the large–time behaviour of partial differential equations.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83573386","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}
In this paper we investigate in–plane harmonic vibrations of a semi–infinite elastic strip with prescribed edge stresses. Low–frequency decay conditions are established demonstrating the deviation from the classical Saint–Venant principle in quadratic terms in frequency. In the case of the symmetric motion (strip extension), the proposed correction is expressed explicitly in terms of given end data, whereas for the antisymmetric motion (strip bending) this also involves unknown edge displacements. Further applications are defined including those related to dynamic analysis of plates and shells excited by statically self–equilibrated edge loads. The derivation is based on a perturbation approach using the Laplace transform technique. We also address methodological aspects dealing with a continuous eigenspectrum and the two–parametric nature of the problem.
{"title":"Low–frequency decay conditions for a semi–infinite elastic strip","authors":"E. Babenkova, J. Kaplunov","doi":"10.1098/rspa.2003.1275","DOIUrl":"https://doi.org/10.1098/rspa.2003.1275","url":null,"abstract":"In this paper we investigate in–plane harmonic vibrations of a semi–infinite elastic strip with prescribed edge stresses. Low–frequency decay conditions are established demonstrating the deviation from the classical Saint–Venant principle in quadratic terms in frequency. In the case of the symmetric motion (strip extension), the proposed correction is expressed explicitly in terms of given end data, whereas for the antisymmetric motion (strip bending) this also involves unknown edge displacements. Further applications are defined including those related to dynamic analysis of plates and shells excited by statically self–equilibrated edge loads. The derivation is based on a perturbation approach using the Laplace transform technique. We also address methodological aspects dealing with a continuous eigenspectrum and the two–parametric nature of the problem.","PeriodicalId":20722,"journal":{"name":"Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2004-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80181428","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}
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