Mock modular forms have found applications in numerous branches of mathematical sciences since they were first introduced by Ramanujan nearly a century ago. In this proceeding, we highlight a new area where mock modular forms start to play an important role, namely the study of three-manifold invariants. For a certain class of Seifert three-manifolds, we describe a conjecture on the mock modular properties of a recently proposed quantum invariant. As an illustration, we include concrete computations for a specific three-manifold, the Brieskorn sphere Σ(2, 3, 7). This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.
{"title":"Three-manifold quantum invariants and mock theta functions","authors":"Miranda C. N. Cheng, Francesca Ferrari, G. Sgroi","doi":"10.1098/rsta.2018.0439","DOIUrl":"https://doi.org/10.1098/rsta.2018.0439","url":null,"abstract":"Mock modular forms have found applications in numerous branches of mathematical sciences since they were first introduced by Ramanujan nearly a century ago. In this proceeding, we highlight a new area where mock modular forms start to play an important role, namely the study of three-manifold invariants. For a certain class of Seifert three-manifolds, we describe a conjecture on the mock modular properties of a recently proposed quantum invariant. As an illustration, we include concrete computations for a specific three-manifold, the Brieskorn sphere Σ(2, 3, 7). This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87750232","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}
Mock theta functions appeared out of the blue in Ramanujan's last letter to Hardy. What would lead Ramanujan to consider the possibility of such functions in the first place? This paper seeks to provide a plausible answer to this question. This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.
{"title":"How Ramanujan may have discovered the mock theta functions","authors":"G. Andrews","doi":"10.1098/rsta.2018.0436","DOIUrl":"https://doi.org/10.1098/rsta.2018.0436","url":null,"abstract":"Mock theta functions appeared out of the blue in Ramanujan's last letter to Hardy. What would lead Ramanujan to consider the possibility of such functions in the first place? This paper seeks to provide a plausible answer to this question. This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82185976","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 review the Ramanujan conjecture in classical and modern settings and explain its various applications in computer science, including the explicit constructions of the spectrally extremal combinatorial objects, called Ramanujan graphs and Ramanujan complexes, points uniformly distributed on spheres, and Golden-Gate Sets in quantum computing. The connection between Ramanujan graphs/complexes and their zeta functions satisfying the Riemann hypothesis is also discussed. This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.
{"title":"The Ramanujan conjecture and its applications","authors":"Wen-Ching Winnie Li","doi":"10.1098/rsta.2018.0441","DOIUrl":"https://doi.org/10.1098/rsta.2018.0441","url":null,"abstract":"In this paper, we review the Ramanujan conjecture in classical and modern settings and explain its various applications in computer science, including the explicit constructions of the spectrally extremal combinatorial objects, called Ramanujan graphs and Ramanujan complexes, points uniformly distributed on spheres, and Golden-Gate Sets in quantum computing. The connection between Ramanujan graphs/complexes and their zeta functions satisfying the Riemann hypothesis is also discussed. This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80234423","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}
Srinivasa Ramanujan, the so-called Man Who Knew Infinity, was one of the most influential, as well as most enigmatic, mathematicians in the recent history of mathematics. With a letter written to G. H. Hardy in 1913, the impoverished Hindu college dropout, self-taught in mathematics, reaching for worlds beyond the shores of India, introduced himself to the history of science. He had spent his youth sitting on cool stone floors in the neighbourhood temple, surrounded by Hindu deities, his mind wandering the world of mathematics. After absorbing the mysterious equations in the letter, Hardy invited Ramanujan to study in England, an extraordinary offer for an Indian under colonial rule. Together they innovated vast tracts of mathematics, before Ramanujan returned to India in fragile health. Tragically, he died at 32 from a misdiagnosed illness, leaving behind three enigmatic notebooks. Ramanujan’s notebooks and research papers have continued to inspire developments in modern mathematics and physics. His formulae and observations now play central roles in fields extending well beyond the realm of pure mathematics. For these reasons, we felt the need to honour the legacy of this great man. To celebrate the centenary of Srinivasa Ramanujan’s election as a Fellow of the Royal Society,1 we organized a public discussion meeting at which leading scientists spoke about Ramanujan’s legacy to mathematics and science. This meeting was held on 15–16 October 2018 at Carlton House. Fifteen distinguished scientists spoke about Ramanujan’s mathematics and his extraordinary legacy across Computer Science, Electrical Engineering, Mathematics and Physics. They were:
斯里尼瓦萨·拉马努金，被称为“知道无限的人”，是近代数学史上最具影响力，也是最神秘的数学家之一。1913年，在给g·h·哈代(G. H. Hardy)的一封信中，这位贫穷的印度大学辍学生自学数学，向印度海岸以外的世界探索，向科学史介绍了自己。他的青年时代是坐在附近寺庙凉爽的石头地板上度过的，周围都是印度教的神像，他的思想徘徊在数学的世界里。在理解了信中神秘的方程式后，哈代邀请拉马努金去英国学习，这对殖民统治下的印度人来说是一个非同寻常的提议。在身体虚弱的拉马努金回到印度之前，他们共同创造了大量的数学领域。不幸的是，他32岁时死于一种误诊的疾病，留下了三本神秘的笔记本。拉马努金的笔记和研究论文继续激励着现代数学和物理学的发展。他的公式和观察现在在远远超出纯数学领域的领域中发挥着核心作用。由于这些原因，我们感到有必要尊重这位伟人的遗产。为了庆祝斯里尼瓦萨·拉马努金当选英国皇家学会会员一百周年，我们组织了一次公开讨论会议，会上主要科学家谈到了拉马努金对数学和科学的贡献。本次会议于2018年10月15日至16日在卡尔顿大厦举行。15位杰出的科学家谈到了拉马努金的数学以及他在计算机科学、电子工程、数学和物理领域的非凡遗产。他们是:
{"title":"Srinivasa Ramanujan: in celebration of the centenary of his election as FRS","authors":"K. Ono","doi":"10.1098/rsta.2019.0386","DOIUrl":"https://doi.org/10.1098/rsta.2019.0386","url":null,"abstract":"Srinivasa Ramanujan, the so-called Man Who Knew Infinity, was one of the most influential, as well as most enigmatic, mathematicians in the recent history of mathematics. With a letter written to G. H. Hardy in 1913, the impoverished Hindu college dropout, self-taught in mathematics, reaching for worlds beyond the shores of India, introduced himself to the history of science. He had spent his youth sitting on cool stone floors in the neighbourhood temple, surrounded by Hindu deities, his mind wandering the world of mathematics. After absorbing the mysterious equations in the letter, Hardy invited Ramanujan to study in England, an extraordinary offer for an Indian under colonial rule. Together they innovated vast tracts of mathematics, before Ramanujan returned to India in fragile health. Tragically, he died at 32 from a misdiagnosed illness, leaving behind three enigmatic notebooks. Ramanujan’s notebooks and research papers have continued to inspire developments in modern mathematics and physics. His formulae and observations now play central roles in fields extending well beyond the realm of pure mathematics. For these reasons, we felt the need to honour the legacy of this great man. To celebrate the centenary of Srinivasa Ramanujan’s election as a Fellow of the Royal Society,1 we organized a public discussion meeting at which leading scientists spoke about Ramanujan’s legacy to mathematics and science. This meeting was held on 15–16 October 2018 at Carlton House. Fifteen distinguished scientists spoke about Ramanujan’s mathematics and his extraordinary legacy across Computer Science, Electrical Engineering, Mathematics and Physics. They were:","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87107054","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 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":null,"pages":null},"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}
Time-resolved simulations present a new opportunity for studying the disturbances responsible for the broadband interaction noise created by the fan stage. In this paper, two vane configurations from the source diagnostic test at the approach rotor speed were computed with PowerFLOW's very large-eddy simulation (VLES) method using two solution strategies: a coarser mesh near the rotor and a trip to trigger turbulent transition on the rotor; and a much finer mesh near the rotor with no trip. The simulated data allow for an investigation of the potential effect from the vane configuration and an in-depth study of the mean and turbulent flow in the interstage gap. A challenge related to post-processing of high-resolution simulations is discussed. Comparison of the flow quantities with previously obtained Reynolds Averaged Navier-Stokes simulation results indicates that little advantage is gained by running a lattice Boltmann method (LBM)/VLES to simply recover the gap flow parameters for use with a lower-order fan broadband interaction noise calculation method. The true benefit of the LBM/VLES is that the noise calculation can be directly and simultaneously completed with the flow simulation. This article is part of the theme issue 'Frontiers of aeroacoustics research: theory, computation and experiment'.
{"title":"Analysis of fan-stage gap-flow data to inform simulation of fan broadband noise.","authors":"Sheryl Grace, Ignacio Gonzalez-Martino, Damiano Casalino","doi":"10.1098/rsta.2019.0080","DOIUrl":"10.1098/rsta.2019.0080","url":null,"abstract":"<p><p>Time-resolved simulations present a new opportunity for studying the disturbances responsible for the broadband interaction noise created by the fan stage. In this paper, two vane configurations from the source diagnostic test at the approach rotor speed were computed with PowerFLOW's very large-eddy simulation (VLES) method using two solution strategies: a coarser mesh near the rotor and a trip to trigger turbulent transition on the rotor; and a much finer mesh near the rotor with no trip. The simulated data allow for an investigation of the potential effect from the vane configuration and an in-depth study of the mean and turbulent flow in the interstage gap. A challenge related to post-processing of high-resolution simulations is discussed. Comparison of the flow quantities with previously obtained Reynolds Averaged Navier-Stokes simulation results indicates that little advantage is gained by running a lattice Boltmann method (LBM)/VLES to simply recover the gap flow parameters for use with a lower-order fan broadband interaction noise calculation method. The true benefit of the LBM/VLES is that the noise calculation can be directly and simultaneously completed with the flow simulation. This article is part of the theme issue 'Frontiers of aeroacoustics research: theory, computation and experiment'.</p>","PeriodicalId":20020,"journal":{"name":"Philosophical Transactions of the Royal Society A","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2019-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6801395/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78782220","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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":null,"pages":null},"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}
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