As was customary at that time, Srinivasa Ramanujan was born in the home of his maternal grandparents on 22 December 1887 in the south Indian town of Erode. After a few months, his mother brought him home to Kumbakonam (figure 1), approximately 120 miles east of Erode and 160 miles south-southwest of Madras (now Chennai). A brother, sister and brother were born in 1889, 1891 and 1894, respectively, but each died within a few months of birth. The two surviving younger brothers (1898–1946; 1905–1978) wrote an interesting but somewhat disconnected account of Ramanujan’s life that contains personal information that we would not have known otherwise [1]. At the time of Ramanujan’s birth, Kumbakonam had a population of about 53 000. The family was quite poor; Ramanujan’s father worked for 20 rupees a month as a clerk for a cloth merchant in Kumbakonam, and his mother took in student boarders from the local high school and government college. Ramanujan’s family home was small and humble, much like the other houses on the dirt street in front of their home. It had essentially one room flanked by a very small kitchen at the back of the home and a small storage room at the front. When the author visited the home in 1984, the only visible sign that this was once the home of the most famous mathematician in Indian history was a picture of Ramanujan cut from a newspaper and taped above the home’s entrance behind a small porch in front of the home. Although the author did not ask how many lived in the home, it appeared to him that a set of grandparents, two parents and seven children lived there. Facing Ramanujan’s home and turning to the left, one sees the famous Sarangapani Temple only about two blocks away. Kumbakonam is famous for its many temples. Ramanujan’s home has now been converted into a museum dedicated to the memory of Ramanujan.
{"title":"Living with Ramanujan for 40 years","authors":"B. Berndt","doi":"10.1098/rsta.2018.0437","DOIUrl":"https://doi.org/10.1098/rsta.2018.0437","url":null,"abstract":"As was customary at that time, Srinivasa Ramanujan was born in the home of his maternal grandparents on 22 December 1887 in the south Indian town of Erode. After a few months, his mother brought him home to Kumbakonam (figure 1), approximately 120 miles east of Erode and 160 miles south-southwest of Madras (now Chennai). A brother, sister and brother were born in 1889, 1891 and 1894, respectively, but each died within a few months of birth. The two surviving younger brothers (1898–1946; 1905–1978) wrote an interesting but somewhat disconnected account of Ramanujan’s life that contains personal information that we would not have known otherwise [1]. At the time of Ramanujan’s birth, Kumbakonam had a population of about 53 000. The family was quite poor; Ramanujan’s father worked for 20 rupees a month as a clerk for a cloth merchant in Kumbakonam, and his mother took in student boarders from the local high school and government college. Ramanujan’s family home was small and humble, much like the other houses on the dirt street in front of their home. It had essentially one room flanked by a very small kitchen at the back of the home and a small storage room at the front. When the author visited the home in 1984, the only visible sign that this was once the home of the most famous mathematician in Indian history was a picture of Ramanujan cut from a newspaper and taped above the home’s entrance behind a small porch in front of the home. Although the author did not ask how many lived in the home, it appeared to him that a set of grandparents, two parents and seven children lived there. Facing Ramanujan’s home and turning to the left, one sees the famous Sarangapani Temple only about two blocks away. Kumbakonam is famous for its many temples. Ramanujan’s home has now been converted into a museum dedicated to the memory of Ramanujan.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83702880","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}
As a methodology complementary to acoustic analogy, the asymptotic approach to aeroacoustics seeks to predict aerodynamical noise on the basis of first principles by probing into the physical processes of acoustic radiation. The present paper highlights the principal ideas and recent developments of this approach, which have shed light on some of the fundamental issues in sound generation in shear flows. The theoretical work on sound wave emission by nonlinearly modulated wavepackets of supersonic and subsonic instability modes in free shear flows identifies the respective physical sources or emitters. A wavepacket of supersonic modes is itself an efficient emitter, radiating directly intensive sound in the form of a Mach wave beam, the frequencies of which are in the same band as those of the modes in the packet. By contrast, a wavepacket of subsonic modes radiates very weak sound directly. However, the nonlinear self-interaction of such a wavepacket generates a slowly modulated mean-flow distortion, which then emits sound waves with low frequencies and long wavelengths on the scale of the wavepacket envelope. In both cases, the acoustic waves emitted to the far field are explicitly expressed in terms of the amplitude function of the wavepacket. The asymptotic approach has also been applied to analyse generation of sound waves in wall-bounded shear flows on the triple-deck scale. Several subtleties have been found. The near-field approximation has to be worked out to a sufficiently higher order in order just to calculate the far-field sound at leading order. The back action of the radiated sound on the flow in the viscous sublayer and the main shear layer is accounted for by an impedance coefficient. This effect is of higher order in the subsonic regime, but becomes a leading order in the transonic and supersonic regimes. This article is part of the theme issue ‘Frontiers of aeroacoustics research: theory, computation and experiment’.
{"title":"First-principle description of acoustic radiation of shear flows","authors":"Xuesong Wu, Zhongyu Zhang","doi":"10.1098/rsta.2019.0077","DOIUrl":"https://doi.org/10.1098/rsta.2019.0077","url":null,"abstract":"As a methodology complementary to acoustic analogy, the asymptotic approach to aeroacoustics seeks to predict aerodynamical noise on the basis of first principles by probing into the physical processes of acoustic radiation. The present paper highlights the principal ideas and recent developments of this approach, which have shed light on some of the fundamental issues in sound generation in shear flows. The theoretical work on sound wave emission by nonlinearly modulated wavepackets of supersonic and subsonic instability modes in free shear flows identifies the respective physical sources or emitters. A wavepacket of supersonic modes is itself an efficient emitter, radiating directly intensive sound in the form of a Mach wave beam, the frequencies of which are in the same band as those of the modes in the packet. By contrast, a wavepacket of subsonic modes radiates very weak sound directly. However, the nonlinear self-interaction of such a wavepacket generates a slowly modulated mean-flow distortion, which then emits sound waves with low frequencies and long wavelengths on the scale of the wavepacket envelope. In both cases, the acoustic waves emitted to the far field are explicitly expressed in terms of the amplitude function of the wavepacket. The asymptotic approach has also been applied to analyse generation of sound waves in wall-bounded shear flows on the triple-deck scale. Several subtleties have been found. The near-field approximation has to be worked out to a sufficiently higher order in order just to calculate the far-field sound at leading order. The back action of the radiated sound on the flow in the viscous sublayer and the main shear layer is accounted for by an impedance coefficient. This effect is of higher order in the subsonic regime, but becomes a leading order in the transonic and supersonic regimes. This article is part of the theme issue ‘Frontiers of aeroacoustics research: theory, computation and experiment’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91027543","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}
Within the new model of surface elasticity, the propagation of anti-plane surface waves is discussed. For the proposed model, the surface strain energy depends on surface stretching and on changing of curvature along a preferred direction. From the continuum mechanics point of view, the model describes finite deformations of an elastic solid with an elastic membrane attached on its boundary reinforced by a family of aligned elastic long flexible beams. Physically, the model was motivated by deformations of surface coatings consisting of aligned bar-like elements as in the case of hyperbolic metasurfaces. Using the least action variational principle, we derive the dynamic boundary conditions. The linearized boundary-value problem is also presented. In order to demonstrate the peculiarities of the problem, the dispersion relations for surface anti-plane waves are analysed. We have shown that the bending stiffness changes essentially the dispersion relation and conditions of anti-plane surface wave propagation. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 2)’.
{"title":"Strongly anisotropic surface elasticity and antiplane surface waves","authors":"V. Eremeyev","doi":"10.1098/rsta.2019.0100","DOIUrl":"https://doi.org/10.1098/rsta.2019.0100","url":null,"abstract":"Within the new model of surface elasticity, the propagation of anti-plane surface waves is discussed. For the proposed model, the surface strain energy depends on surface stretching and on changing of curvature along a preferred direction. From the continuum mechanics point of view, the model describes finite deformations of an elastic solid with an elastic membrane attached on its boundary reinforced by a family of aligned elastic long flexible beams. Physically, the model was motivated by deformations of surface coatings consisting of aligned bar-like elements as in the case of hyperbolic metasurfaces. Using the least action variational principle, we derive the dynamic boundary conditions. The linearized boundary-value problem is also presented. In order to demonstrate the peculiarities of the problem, the dispersion relations for surface anti-plane waves are analysed. We have shown that the bending stiffness changes essentially the dispersion relation and conditions of anti-plane surface wave propagation. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 2)’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"121 3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80190335","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}
The paper describes a fault-tolerant design of a special two-dimensional beam lattice. The morphology of such lattices was suggested in the theoretical papers (Cherkaev and Ryvkin 2019 Arch. Appl. Mech. 89, 485–501; Cherkaev and Ryvkin 2019 Arch. Appl. Mech. 89, 503–519), where its superior properties were found numerically. The proposed design consists of beam elements with two different thicknesses; the lattice is macro-isotropic and stretch dominated. Here, we experimentally verify the fault-tolerant properties of these lattices. The specimens were three-dimensional-printed from the VeroWhite elastoplastic material. The lattice is subjected to uniaxial tensile loading. Due to its morphology, the failed beams are evenly distributed in the lattice at the initial stage of damage; at this stage, the material remains intact, preserves its bearing ability, and supports relatively high strains before the final failure. At the initial phase of damage, the thinner beams buckle; then another group of separated thin beams plastically yield and rupture. The fatal macro-crack propagates after the distributed damage reaches a critical level. This initial distributed damage stage allows for a better energy absorption rate before the catastrophic failure of the structure. The experimental results are supported by simulations which confirm that the proposed fault-tolerant material possesses excellent energy absorption properties thanks to the distributed damage stage phenomenon. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 2)’.
本文描述了一种特殊二维梁格的容错设计。理论论文(Cherkaev and Ryvkin 2019 Arch)提出了这种晶格的形态。达成。机械89、485-501;Cherkaev和Ryvkin 2019拱门。达成。机械，89,503-519)，在那里它的优越性能被发现的数值。提出的设计由两种不同厚度的梁单元组成;晶格是宏观各向同性和拉伸主导的。在这里，我们通过实验验证了这些格的容错特性。这些样本是用VeroWhite弹塑性材料三维打印出来的。晶格受到单轴拉伸载荷。由于其形态的原因，在损伤初始阶段，失效梁均匀分布在晶格中;在此阶段，材料保持完整，保持其承载能力，并在最终破坏之前承受相对较高的应变。在损伤初期，较薄的梁发生屈曲;然后另一组分离的薄梁塑性屈服并断裂。当分布损伤达到临界水平后，致命宏裂纹开始扩展。这种初始分布损伤阶段允许在结构发生灾难性破坏之前有更好的能量吸收率。实验结果与仿真结果相吻合，表明该容错材料由于存在分布式损伤阶段现象，具有良好的能量吸收性能。本文是主题“结构化媒体中动态现象的建模和定位(第二部分)”的一部分。
{"title":"Fault-tolerant elastic–plastic lattice material","authors":"M. Ryvkin, V. Slesarenko, A. Cherkaev, S. Rudykh","doi":"10.1098/rsta.2019.0107","DOIUrl":"https://doi.org/10.1098/rsta.2019.0107","url":null,"abstract":"The paper describes a fault-tolerant design of a special two-dimensional beam lattice. The morphology of such lattices was suggested in the theoretical papers (Cherkaev and Ryvkin 2019 Arch. Appl. Mech. 89, 485–501; Cherkaev and Ryvkin 2019 Arch. Appl. Mech. 89, 503–519), where its superior properties were found numerically. The proposed design consists of beam elements with two different thicknesses; the lattice is macro-isotropic and stretch dominated. Here, we experimentally verify the fault-tolerant properties of these lattices. The specimens were three-dimensional-printed from the VeroWhite elastoplastic material. The lattice is subjected to uniaxial tensile loading. Due to its morphology, the failed beams are evenly distributed in the lattice at the initial stage of damage; at this stage, the material remains intact, preserves its bearing ability, and supports relatively high strains before the final failure. At the initial phase of damage, the thinner beams buckle; then another group of separated thin beams plastically yield and rupture. The fatal macro-crack propagates after the distributed damage reaches a critical level. This initial distributed damage stage allows for a better energy absorption rate before the catastrophic failure of the structure. The experimental results are supported by simulations which confirm that the proposed fault-tolerant material possesses excellent energy absorption properties thanks to the distributed damage stage phenomenon. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 2)’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"48 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81149415","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 heterogeneous composite materials, the behaviour of the medium on larger scales is determined by the microgeometry and properties of the constituents on finer scales. To model the influence of the microlevel processes in composite materials, they are described as materials with memory in which the constitutive relations between stress and strain are given as time-domain convolutions with some relaxation kernel. The paper reveals the relationship between the viscoelastic relaxation kernel and the spectral measure in the Stieltjes integral representation of the effective properties of composites. This spectral measure contains all information about the microgeometry of the material, thus providing a link between the relaxation kernel and the microstructure of the composite. We show that the internal resonances of the microstructure determine the characteristic relaxation times of the fading memory kernel and can be used to introduce a set of internal variables that captures dissipation at the microscale. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 2)’.
{"title":"Internal resonances and relaxation memory kernels in composites","authors":"E. Cherkaev","doi":"10.1098/rsta.2019.0106","DOIUrl":"https://doi.org/10.1098/rsta.2019.0106","url":null,"abstract":"In heterogeneous composite materials, the behaviour of the medium on larger scales is determined by the microgeometry and properties of the constituents on finer scales. To model the influence of the microlevel processes in composite materials, they are described as materials with memory in which the constitutive relations between stress and strain are given as time-domain convolutions with some relaxation kernel. The paper reveals the relationship between the viscoelastic relaxation kernel and the spectral measure in the Stieltjes integral representation of the effective properties of composites. This spectral measure contains all information about the microgeometry of the material, thus providing a link between the relaxation kernel and the microstructure of the composite. We show that the internal resonances of the microstructure determine the characteristic relaxation times of the fading memory kernel and can be used to introduce a set of internal variables that captures dissipation at the microscale. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 2)’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"81 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85544777","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 presents a generalization of a recent iterative approach to solving a class of 2 × 2 matrix Wiener–Hopf equations involving exponential factors. We extend the method to square matrices of arbitrary dimension n, as arise in mixed boundary value problems with n junctions. To demonstrate the method, we consider the classical problem of scattering a plane wave by a set of collinear plates. The results are compared to other known methods. We describe an effective implementation using a spectral method to compute the required Cauchy transforms. The approach is ideally suited to obtaining far-field directivity patterns of utility to applications. Convergence in iteration is fastest for large wavenumbers, but remains practical at modest wavenumbers to achieve a high degree of accuracy. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 2)’.
{"title":"Applying an iterative method numerically to solve n × n matrix Wiener–Hopf equations with exponential factors","authors":"Matthew J. Priddin, A. Kisil, Lorna J. Ayton","doi":"10.1098/rsta.2019.0241","DOIUrl":"https://doi.org/10.1098/rsta.2019.0241","url":null,"abstract":"This paper presents a generalization of a recent iterative approach to solving a class of 2 × 2 matrix Wiener–Hopf equations involving exponential factors. We extend the method to square matrices of arbitrary dimension n, as arise in mixed boundary value problems with n junctions. To demonstrate the method, we consider the classical problem of scattering a plane wave by a set of collinear plates. The results are compared to other known methods. We describe an effective implementation using a spectral method to compute the required Cauchy transforms. The approach is ideally suited to obtaining far-field directivity patterns of utility to applications. Convergence in iteration is fastest for large wavenumbers, but remains practical at modest wavenumbers to achieve a high degree of accuracy. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 2)’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90577279","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 study dynamical phenomena in a harmonic graphene (honeycomb) lattice, consisting of equal particles connected by linear and angular springs. Equations of in-plane motion for the lattice are derived. Initial conditions typical for molecular dynamic modelling are considered. Particles have random initial velocities and zero displacements. In this case, the lattice is far from thermal equilibrium. In particular, initial kinetic and potential energies are not equal. Moreover, initial kinetic energies (and temperatures), corresponding to degrees of freedom of the unit cell, are generally different. The motion of particles leads to equilibration of kinetic and potential energies and redistribution of kinetic energy among degrees of freedom. During equilibration, the kinetic energy performs decaying high-frequency oscillations. We show that these oscillations are accurately described by an integral depending on dispersion relation and polarization matrix of the lattice. At large times, kinetic and potential energies tend to equal values. Kinetic energy is partially redistributed among degrees of freedom of the unit cell. Equilibrium distribution of the kinetic energies is accurately predicted by the non-equipartition theorem. Presented results may serve for better understanding of the approach to thermal equilibrium in graphene. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 2)’.
{"title":"Equilibration of energies in a two-dimensional harmonic graphene lattice","authors":"I. Berinskii, V. Kuzkin","doi":"10.1098/rsta.2019.0114","DOIUrl":"https://doi.org/10.1098/rsta.2019.0114","url":null,"abstract":"We study dynamical phenomena in a harmonic graphene (honeycomb) lattice, consisting of equal particles connected by linear and angular springs. Equations of in-plane motion for the lattice are derived. Initial conditions typical for molecular dynamic modelling are considered. Particles have random initial velocities and zero displacements. In this case, the lattice is far from thermal equilibrium. In particular, initial kinetic and potential energies are not equal. Moreover, initial kinetic energies (and temperatures), corresponding to degrees of freedom of the unit cell, are generally different. The motion of particles leads to equilibration of kinetic and potential energies and redistribution of kinetic energy among degrees of freedom. During equilibration, the kinetic energy performs decaying high-frequency oscillations. We show that these oscillations are accurately described by an integral depending on dispersion relation and polarization matrix of the lattice. At large times, kinetic and potential energies tend to equal values. Kinetic energy is partially redistributed among degrees of freedom of the unit cell. Equilibrium distribution of the kinetic energies is accurately predicted by the non-equipartition theorem. Presented results may serve for better understanding of the approach to thermal equilibrium in graphene. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 2)’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88193680","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}
R. Zhou, K. Pang, A. Bisht, A. Roy, S. Suwas, V. Silberschmidt
A phenomenological approach, based on a combination of a damage mechanism and a crystal plasticity model, is proposed to model a process of strain localization in Ti–6AI–4V at a high strain rate of 103 s−1. The proposed model is first calibrated employing a three-dimensional representative volume element model. The calibrated parameters are then employed to investigate the process of onset of strain localization in the studied material. A suitable mesh size is chosen for the proposed model by implementing a mesh-sensitivity study. The influence of boundary conditions on the initiation of the strain localization is also studied. A variation of crystallographic orientation in the studied material after the deformation process is characterized, based on results for different boundary conditions. The study reveals that the boundary conditions significantly influence the formation of shear bands as well as the variation of crystallographic orientation in the studied material. Results also indicate that the onset of strain localization can affect considerably the material's behaviour. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 2)’.
{"title":"Modelling strain localization in Ti–6Al–4V at high loading rate: a phenomenological approach","authors":"R. Zhou, K. Pang, A. Bisht, A. Roy, S. Suwas, V. Silberschmidt","doi":"10.1098/rsta.2019.0105","DOIUrl":"https://doi.org/10.1098/rsta.2019.0105","url":null,"abstract":"A phenomenological approach, based on a combination of a damage mechanism and a crystal plasticity model, is proposed to model a process of strain localization in Ti–6AI–4V at a high strain rate of 103 s−1. The proposed model is first calibrated employing a three-dimensional representative volume element model. The calibrated parameters are then employed to investigate the process of onset of strain localization in the studied material. A suitable mesh size is chosen for the proposed model by implementing a mesh-sensitivity study. The influence of boundary conditions on the initiation of the strain localization is also studied. A variation of crystallographic orientation in the studied material after the deformation process is characterized, based on results for different boundary conditions. The study reveals that the boundary conditions significantly influence the formation of shear bands as well as the variation of crystallographic orientation in the studied material. Results also indicate that the onset of strain localization can affect considerably the material's behaviour. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 2)’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"63 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74496394","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}
G. Carta, D. Colquitt, A. Movchan, N. Movchan, I. Jones
In this paper, we demonstrate a new approach to control flexural elastic waves in a structured chiral plate. The main focus is on creating one-way interfacial wave propagation at a given frequency by employing double resonators in a doubly periodic flexural system. The resonators consist of two beams attached to gyroscopic spinners, which act to couple flexural and rotational deformations, hence inducing chirality in the system. We show that this elastic structure supports one-way flexural waves, localized at an interface separating two sub-domains with gyroscopes spinning in opposite directions, but with otherwise identical properties. We demonstrate that a special feature of double resonators is in the directional control of wave propagation by varying the value of the gyricity, while keeping the frequency of the external time-harmonic excitation fixed. Conversely, for the same value of gyricity, the direction of wave propagation can be reversed by tuning the frequency of the external excitation. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 2)’.
{"title":"One-way interfacial waves in a flexural plate with chiral double resonators","authors":"G. Carta, D. Colquitt, A. Movchan, N. Movchan, I. Jones","doi":"10.1098/rsta.2019.0350","DOIUrl":"https://doi.org/10.1098/rsta.2019.0350","url":null,"abstract":"In this paper, we demonstrate a new approach to control flexural elastic waves in a structured chiral plate. The main focus is on creating one-way interfacial wave propagation at a given frequency by employing double resonators in a doubly periodic flexural system. The resonators consist of two beams attached to gyroscopic spinners, which act to couple flexural and rotational deformations, hence inducing chirality in the system. We show that this elastic structure supports one-way flexural waves, localized at an interface separating two sub-domains with gyroscopes spinning in opposite directions, but with otherwise identical properties. We demonstrate that a special feature of double resonators is in the directional control of wave propagation by varying the value of the gyricity, while keeping the frequency of the external time-harmonic excitation fixed. Conversely, for the same value of gyricity, the direction of wave propagation can be reversed by tuning the frequency of the external excitation. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 2)’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85440891","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}
The equations of motion are derived for a system of point masses on the (hyper)surface Sn of a sphere embedded in Rn+1 for any dimension n > 1. Owing to the symmetry of the surface, the equations take a particularly simple form when using the Cartesian coordinates of Rn+1. The constraint that the distance of the jth mass ∥rj∥ from the origin remains constant (i.e. each mass remains on the surface) is automatically satisfied by the equations of motion. Moreover, the equations are a Hamiltonian system with a conserved energy as well as a host of conserved angular momenta. Several examples are illustrated in dimensions n = 2 (the sphere) and n = 3 (the glome). This article is part of the theme issue ‘Topological and geometrical aspects of mass and vortex dynamics’.
{"title":"Point mass dynamics on spherical hypersurfaces","authors":"D. Dritschel","doi":"10.1098/rsta.2018.0349","DOIUrl":"https://doi.org/10.1098/rsta.2018.0349","url":null,"abstract":"The equations of motion are derived for a system of point masses on the (hyper)surface Sn of a sphere embedded in Rn+1 for any dimension n > 1. Owing to the symmetry of the surface, the equations take a particularly simple form when using the Cartesian coordinates of Rn+1. The constraint that the distance of the jth mass ∥rj∥ from the origin remains constant (i.e. each mass remains on the surface) is automatically satisfied by the equations of motion. Moreover, the equations are a Hamiltonian system with a conserved energy as well as a host of conserved angular momenta. Several examples are illustrated in dimensions n = 2 (the sphere) and n = 3 (the glome). This article is part of the theme issue ‘Topological and geometrical aspects of mass and vortex dynamics’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":"127 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-11-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84412318","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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