Can a filament transmit the curvatures across the constituting modules and control them at one of its end? Inspired by the observation of protofilament - constituent biopolymer of microtubule - this question is addressed by a constructive approach. In our model a simple allosteric element in each module couples with the neighboring modules at its interfaces, which gives rise to a single degree of freedom to control the global shape of the filament. The model can be analyzed in analogy with discrete-time dynamical systems having a bifurcation of trans-critical type.
{"title":"Allosteric propagation of curvature along filament","authors":"Ken Sekimoto","doi":"arxiv-2407.10826","DOIUrl":"https://doi.org/arxiv-2407.10826","url":null,"abstract":"Can a filament transmit the curvatures across the constituting modules and\u0000control them at one of its end? Inspired by the observation of protofilament -\u0000constituent biopolymer of microtubule - this question is addressed by a\u0000constructive approach. In our model a simple allosteric element in each module\u0000couples with the neighboring modules at its interfaces, which gives rise to a\u0000single degree of freedom to control the global shape of the filament. The model\u0000can be analyzed in analogy with discrete-time dynamical systems having a\u0000bifurcation of trans-critical type.","PeriodicalId":501482,"journal":{"name":"arXiv - PHYS - Classical Physics","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720063","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}
Euler-Poisson equations belong to the class of first-order differential equations for determining the integral lines of a given vector field. The general solution to these equations can be written as a power series of the evolution parameter. We calculated the sum of these series for the case of a free symmetric body, obtaining its rotation matrix through the elementary functions.
{"title":"General solution to Euler-Poisson equations of a free symmetric body by direct summation of power series","authors":"Guilherme Corrêa Silva","doi":"arxiv-2407.10326","DOIUrl":"https://doi.org/arxiv-2407.10326","url":null,"abstract":"Euler-Poisson equations belong to the class of first-order differential\u0000equations for determining the integral lines of a given vector field. The\u0000general solution to these equations can be written as a power series of the\u0000evolution parameter. We calculated the sum of these series for the case of a\u0000free symmetric body, obtaining its rotation matrix through the elementary\u0000functions.","PeriodicalId":501482,"journal":{"name":"arXiv - PHYS - Classical Physics","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720062","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 study establishes a non-Bloch band theory for time-modulated discrete mechanical systems. We consider simple mass-spring chains whose stiffness is periodically modulated in time. Using the temporal Floquet theory, the system is characterized by linear algebraic equations in terms of Fourier coefficients. This allows us to employ a standard linear eigenvalue analysis. Unlike non-modulated linear systems, the time modulation makes the coefficient matrix non-Hermitian, which gives rise to, for example, parametric resonance, non-reciprocal wave transmission, and non-Hermitian skin effects. In particular, we study finite-length chains consisting of spatially periodic mass-spring units and show that the standard Bloch band theory is not valid for estimating their eigenvalue distribution. To remedy this, we propose a non-Bloch band theory based on a generalized Brillouin zone. The proposed theory is verified by some numerical experiments.
{"title":"Non-Bloch band theory for time-modulated discrete mechanical systems","authors":"Kei Matsushima, Takayuki Yamada","doi":"arxiv-2407.09871","DOIUrl":"https://doi.org/arxiv-2407.09871","url":null,"abstract":"This study establishes a non-Bloch band theory for time-modulated discrete\u0000mechanical systems. We consider simple mass-spring chains whose stiffness is\u0000periodically modulated in time. Using the temporal Floquet theory, the system\u0000is characterized by linear algebraic equations in terms of Fourier\u0000coefficients. This allows us to employ a standard linear eigenvalue analysis.\u0000Unlike non-modulated linear systems, the time modulation makes the coefficient\u0000matrix non-Hermitian, which gives rise to, for example, parametric resonance,\u0000non-reciprocal wave transmission, and non-Hermitian skin effects. In\u0000particular, we study finite-length chains consisting of spatially periodic\u0000mass-spring units and show that the standard Bloch band theory is not valid for\u0000estimating their eigenvalue distribution. To remedy this, we propose a\u0000non-Bloch band theory based on a generalized Brillouin zone. The proposed\u0000theory is verified by some numerical experiments.","PeriodicalId":501482,"journal":{"name":"arXiv - PHYS - Classical Physics","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720064","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}
Shiyuan LuoIRDL, Yongxin JiangIRDL, Sandrine ThuillierIRDL, Philippe CastanyISCR, Liangcai Zeng
A simplified 3D FE model based on McCormick's model is developed to numerically predict the spatiotemporal behaviors of the PLC effect in Ti-12Mo alloy tensile tests at 350 degrees C with strain rates from the order of $10^{-4}$ s$^{-1}$ to $10^{-2}$ s$^{-1}$. The material parameter identification procedure is firstly presented in details, and the simulated results are highly consistent with experimental ones, especially in terms of stress drop magnitudes and PLC band widths. The distribution of simulated stress drop magnitudes at a constant tensile velocity (0.01 mm/s) follows a normal distribution and its peak value is in the range of 26-28 MPa. Furthermore, the simulated band width slightly fluctuates with the increase of true strain and its average value is about 1.5 mm. Besides, the staircase behavior of strain-time curves and the hopping propagation of the PLC band are observed in Ti-12Mo alloy tensile process, which are related to the strain localization and stress drop magnitudes.
{"title":"Numerical Analysis on the Spatiotemporal Characteristics of the Portevin-Le Chatelier Effect in Ti-12Mo Alloy","authors":"Shiyuan LuoIRDL, Yongxin JiangIRDL, Sandrine ThuillierIRDL, Philippe CastanyISCR, Liangcai Zeng","doi":"arxiv-2407.09054","DOIUrl":"https://doi.org/arxiv-2407.09054","url":null,"abstract":"A simplified 3D FE model based on McCormick's model is developed to\u0000numerically predict the spatiotemporal behaviors of the PLC effect in Ti-12Mo\u0000alloy tensile tests at 350 degrees C with strain rates from the order of\u0000$10^{-4}$ s$^{-1}$ to $10^{-2}$ s$^{-1}$. The material parameter identification\u0000procedure is firstly presented in details, and the simulated results are highly\u0000consistent with experimental ones, especially in terms of stress drop\u0000magnitudes and PLC band widths. The distribution of simulated stress drop\u0000magnitudes at a constant tensile velocity (0.01 mm/s) follows a normal\u0000distribution and its peak value is in the range of 26-28 MPa. Furthermore, the\u0000simulated band width slightly fluctuates with the increase of true strain and\u0000its average value is about 1.5 mm. Besides, the staircase behavior of\u0000strain-time curves and the hopping propagation of the PLC band are observed in\u0000Ti-12Mo alloy tensile process, which are related to the strain localization and\u0000stress drop magnitudes.","PeriodicalId":501482,"journal":{"name":"arXiv - PHYS - Classical Physics","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720065","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}
Two critical characteristics for any MEMS resonator are the quality factor ($Q$) and the temperature coefficient of frequency ($TCF$). The connection between $Q$ and $TCF$ is demonstrated here with a phenomenological anharmonic oscillator model. Specifically, it is shown that the same nonlinear terms responsible for the $TCF$ set an upper limit on the resonator's $Q$. A concise formula is found to estimate this loss and is shown to be closely related to Woodruff's formula for Akhiezer damping. The use of this formula is illustrated by extending the model to an important class of MEMS; piezoelectric resonators. Finally, the model is applied to published data for an AlN-on-diamond piezoelectric resonator. The focus here is on MEMS resonators, but the method should apply broadly to any resonance with non-zero $TCF$.
{"title":"Connecting Q to TCF for MEMS and piezoelectric resonators","authors":"S. McHugh","doi":"arxiv-2407.09455","DOIUrl":"https://doi.org/arxiv-2407.09455","url":null,"abstract":"Two critical characteristics for any MEMS resonator are the quality factor\u0000($Q$) and the temperature coefficient of frequency ($TCF$). The connection\u0000between $Q$ and $TCF$ is demonstrated here with a phenomenological anharmonic\u0000oscillator model. Specifically, it is shown that the same nonlinear terms\u0000responsible for the $TCF$ set an upper limit on the resonator's $Q$. A concise\u0000formula is found to estimate this loss and is shown to be closely related to\u0000Woodruff's formula for Akhiezer damping. The use of this formula is illustrated\u0000by extending the model to an important class of MEMS; piezoelectric resonators.\u0000Finally, the model is applied to published data for an AlN-on-diamond\u0000piezoelectric resonator. The focus here is on MEMS resonators, but the method\u0000should apply broadly to any resonance with non-zero $TCF$.","PeriodicalId":501482,"journal":{"name":"arXiv - PHYS - Classical Physics","volume":"65 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720066","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}
Plastic deformation in metallic glasses at room temperature leads to the development of shear bands due to shear localization. In many experiments, shear bands have shown local density variations along their path, with a distinct imbalance in magnitude between local densification and dilation. However, a comprehensive mechanistic understanding or theory to explain this asymmetry has been lacking until now. Here, we introduce a new model that consists of a sequential arrangement of alternating topological 'charges', generating a dipolar field. The resulting microscopic displacement field, when integrated into the deformation gradient tensor, provides an accurate analytical solution for the observed imbalances in the density variations. The implications of this method are discussed, highlighting the potential to elucidate a broader range of observations in shear bands.
{"title":"Unveiling the Asymmetry in Density within the Shear Bands of Metallic Glasses","authors":"Harald Rösner, Arabinda Bera, Alessio Zaccone","doi":"arxiv-2407.07733","DOIUrl":"https://doi.org/arxiv-2407.07733","url":null,"abstract":"Plastic deformation in metallic glasses at room temperature leads to the\u0000development of shear bands due to shear localization. In many experiments,\u0000shear bands have shown local density variations along their path, with a\u0000distinct imbalance in magnitude between local densification and dilation.\u0000However, a comprehensive mechanistic understanding or theory to explain this\u0000asymmetry has been lacking until now. Here, we introduce a new model that\u0000consists of a sequential arrangement of alternating topological 'charges',\u0000generating a dipolar field. The resulting microscopic displacement field, when\u0000integrated into the deformation gradient tensor, provides an accurate\u0000analytical solution for the observed imbalances in the density variations. The\u0000implications of this method are discussed, highlighting the potential to\u0000elucidate a broader range of observations in shear bands.","PeriodicalId":501482,"journal":{"name":"arXiv - PHYS - Classical Physics","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141588044","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}
Linearised relativistic elasticity equations of motion are considered for a rod and a spinning ring encountering a gravitational wave. In the case of the rod, the equations reduce to a wave equation with appropriate boundary conditions. Using Fourier transforms, the resonant frequencies are found and an explicit distributional solution is given, both for a plus- and a cross-polarised gravitational wave. In the case of the spinning ring, the equations are coupled wave equations with periodic boundary conditions. Using a Fourier series expansion, the system of wave equations is recast as a family of ordinary differential equations for the Fourier coefficients, which are then solved via Fourier transforms. The resonant frequencies are found, including simple approximate expressions for slowly rotating rings, and an explicit distributional solution is obtained in the case of the non-spinning ring. Interestingly, it is possible to tune the resonant frequencies by adjusting the angular velocity of the spinning ring.
{"title":"Elastic rods and elastic spinning rings as gravitational wave detectors","authors":"José Natário, Amol Sasane, Rodrigo Vicente","doi":"arxiv-2407.07547","DOIUrl":"https://doi.org/arxiv-2407.07547","url":null,"abstract":"Linearised relativistic elasticity equations of motion are considered for a\u0000rod and a spinning ring encountering a gravitational wave. In the case of the\u0000rod, the equations reduce to a wave equation with appropriate boundary\u0000conditions. Using Fourier transforms, the resonant frequencies are found and an\u0000explicit distributional solution is given, both for a plus- and a\u0000cross-polarised gravitational wave. In the case of the spinning ring, the\u0000equations are coupled wave equations with periodic boundary conditions. Using a\u0000Fourier series expansion, the system of wave equations is recast as a family of\u0000ordinary differential equations for the Fourier coefficients, which are then\u0000solved via Fourier transforms. The resonant frequencies are found, including\u0000simple approximate expressions for slowly rotating rings, and an explicit\u0000distributional solution is obtained in the case of the non-spinning ring.\u0000Interestingly, it is possible to tune the resonant frequencies by adjusting the\u0000angular velocity of the spinning ring.","PeriodicalId":501482,"journal":{"name":"arXiv - PHYS - Classical Physics","volume":"71 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141588045","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}
Schr"odinger-Pauli equation (SP-eq) derived from weakly relativistic approximation (WRA) of Dirac eq, combined with Electromagnetic (EM) field Lagrangian for variational principle, is expected to give a new level of EM response theory. A complete process of this formulation within the second order WRA is given, with explicit forms of charge and current densities, $rho , vec{J}$, and electric and magnetic polarizations, $vec{P}$, $vec{M}$ containing correction terms. They fulfill, not only the continuity equation, but also the relations $nabla cdot vec{P}=-rho, partial vec{P}/partial t + c nabla times vec{M} = vec{J}$, known in the classical EM theory for the corresponding macroscopic variables. This theory should be able to describe all the EM responses within the second order WRA, and the least necessary variables are ${phi, vec{A}, rho, vec{J}}$ (six independent components). From this viewpoint, there emerges a problem about the use of "spin current" popularly discussed in spintronics, because it does not belong to the group of least necessary variables.
从狄拉克方程的弱相对论近似（WRA）导出的薛定谔-保利方程（SP-eq），结合电磁场拉格朗日的变分原理，有望给出电磁响应理论的新水平。本文给出了二阶 WRA 中这一表述的完整过程，其中包括电荷和电流密度（$rho ,vec{J}$）以及电极化和磁极化（$vec{P}$, $vec{M}$）的显式修正项。它们不仅满足连续性方程，还满足$nabla cdot vec{P}=-rho, partial vec{P}/partialt + c nabla times vec{M} = vec{J}$，这些关系在经典电磁理论中对于相应的宏观变量是已知的。这一理论应该能够描述二阶 WRA 内的所有电磁响应，而最小必要变量是 ${phi，vec{A}，rho，vec{J}}$（六个独立分量）。
{"title":"Electromagnetic Response Theory with Relativistic Corrections: Selfconsistency and Validity of Variables","authors":"Kikuo Cho","doi":"arxiv-2407.09570","DOIUrl":"https://doi.org/arxiv-2407.09570","url":null,"abstract":"Schr\"odinger-Pauli equation (SP-eq) derived from weakly relativistic\u0000approximation (WRA) of Dirac eq, combined with Electromagnetic (EM) field\u0000Lagrangian for variational principle, is expected to give a new level of EM\u0000response theory. A complete process of this formulation within the second order\u0000WRA is given, with explicit forms of charge and current densities, $rho ,\u0000vec{J}$, and electric and magnetic polarizations, $vec{P}$, $vec{M}$\u0000containing correction terms. They fulfill, not only the continuity equation,\u0000but also the relations $nabla cdot vec{P}=-rho, partial vec{P}/partial\u0000t + c nabla times vec{M} = vec{J}$, known in the classical EM theory for\u0000the corresponding macroscopic variables. This theory should be able to describe\u0000all the EM responses within the second order WRA, and the least necessary\u0000variables are ${phi, vec{A}, rho, vec{J}}$ (six independent components).\u0000From this viewpoint, there emerges a problem about the use of \"spin current\"\u0000popularly discussed in spintronics, because it does not belong to the group of\u0000least necessary variables.","PeriodicalId":501482,"journal":{"name":"arXiv - PHYS - Classical Physics","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720007","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 problem of relativistic projectiles in a uniform gravitational force field. For the first time, we have found the curve that joins the points of maximum height of all trajectories followed by a projectile in the ultra-relativistic limit. The parametric equations of this curve produce an onion-like curve; in fact, it is one of the loops of a lemniscate-type curve. We also verify that the curve is an ellipse in the nonrelativistic approximation. These two limiting results are obtained by following two slightly distinct approaches. In addition, we calculate the nonrelativistic and ultra-relativistic approximations of the trajectory equation and parametric equations of the trajectory as functions of time. All limiting cases in the article are studied in detail. The content of the article is appropriate for advanced undergraduate students.
{"title":"On the locus formed by the maximum heights of an ultra-relativistic projectile","authors":"Salvatore De Vincenzo","doi":"arxiv-2407.05612","DOIUrl":"https://doi.org/arxiv-2407.05612","url":null,"abstract":"We consider the problem of relativistic projectiles in a uniform\u0000gravitational force field. For the first time, we have found the curve that\u0000joins the points of maximum height of all trajectories followed by a projectile\u0000in the ultra-relativistic limit. The parametric equations of this curve produce\u0000an onion-like curve; in fact, it is one of the loops of a lemniscate-type\u0000curve. We also verify that the curve is an ellipse in the nonrelativistic\u0000approximation. These two limiting results are obtained by following two\u0000slightly distinct approaches. In addition, we calculate the nonrelativistic and\u0000ultra-relativistic approximations of the trajectory equation and parametric\u0000equations of the trajectory as functions of time. All limiting cases in the\u0000article are studied in detail. The content of the article is appropriate for\u0000advanced undergraduate students.","PeriodicalId":501482,"journal":{"name":"arXiv - PHYS - Classical Physics","volume":"41 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141568653","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 a recent Note (Am. J. Phys. 92:397, 2024; arXiv:2309.10826), Vallejo and Bove provide a physical argument based nominally on the second law of thermodynamics as a way of resolving the mathematical question appearing in the title. A remarkable aspect of their argument is that it does not depend on the numerical value of $pi$, because $e^{x} ge x^{e}$ for all positive $x$, with equality occurring only when $x = e$. Moreover, their argument does not depend on the validity of the second law but is rather a limited proof of it for this particular case.
{"title":"Comment on \"Which is greater: $e^π$ or $π^{e}$? An unorthodox physical solution to a classic puzzle\"","authors":"Roderick M. Macrae","doi":"arxiv-2407.09568","DOIUrl":"https://doi.org/arxiv-2407.09568","url":null,"abstract":"In a recent Note (Am. J. Phys. 92:397, 2024; arXiv:2309.10826), Vallejo and\u0000Bove provide a physical argument based nominally on the second law of\u0000thermodynamics as a way of resolving the mathematical question appearing in the\u0000title. A remarkable aspect of their argument is that it does not depend on the\u0000numerical value of $pi$, because $e^{x} ge x^{e}$ for all positive $x$, with\u0000equality occurring only when $x = e$. Moreover, their argument does not depend\u0000on the validity of the second law but is rather a limited proof of it for this\u0000particular case.","PeriodicalId":501482,"journal":{"name":"arXiv - PHYS - Classical Physics","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141720008","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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