This paper presents for the first time two types of metamaterials based on the fragmentation–reconstitution of rotating units in order to produce Poisson's ratio discontinuity at the original state. For both metamaterials, each rotating unit takes the form of a rhombus that comprises eight sub-units. During on-axis stretching, each rhombus fragments into eight rotating sub-units. When the prescribed strain is reversed, these eight sub-units reconstitute back into a single rotating rhombus such that they rotate as a rigid body. Using geometrical construction, the incremental Poisson's ratio was established at the original state. In the case of large deformation, the finite Poisson's ratio was formulated in conjunction with the maximum allowable rotations for full stretching along both axes and for full compression. The family of on-axes Poisson's ratio versus rotational angles for various shape descriptors displays a fork-shaped distribution with discontinuity at the original state. Two major distinguishing factors of these metamaterials—property discontinuity at the original state with constant and variable Poisson's ratio under compression and tension, respectively—allow them to function in ways that cannot be fully performed by conventional materials or even by auxetic materials and metamaterials.
{"title":"Metamaterials with Poisson's ratio discontinuity by means of fragmentation–reconstitution rotating units","authors":"Teik-Cheng Lim","doi":"10.1098/rspa.2023.0442","DOIUrl":"https://doi.org/10.1098/rspa.2023.0442","url":null,"abstract":"This paper presents for the first time two types of metamaterials based on the fragmentation–reconstitution of rotating units in order to produce Poisson's ratio discontinuity at the original state. For both metamaterials, each rotating unit takes the form of a rhombus that comprises eight sub-units. During on-axis stretching, each rhombus fragments into eight rotating sub-units. When the prescribed strain is reversed, these eight sub-units reconstitute back into a single rotating rhombus such that they rotate as a rigid body. Using geometrical construction, the incremental Poisson's ratio was established at the original state. In the case of large deformation, the finite Poisson's ratio was formulated in conjunction with the maximum allowable rotations for full stretching along both axes and for full compression. The family of on-axes Poisson's ratio versus rotational angles for various shape descriptors displays a fork-shaped distribution with discontinuity at the original state. Two major distinguishing factors of these metamaterials—property discontinuity at the original state with constant and variable Poisson's ratio under compression and tension, respectively—allow them to function in ways that cannot be fully performed by conventional materials or even by auxetic materials and metamaterials.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136117645","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove homological stability for two different flavours of asymptotic monopole moduli spaces, namely moduli spaces of framed Dirac monopoles and moduli spaces of ideal monopoles . The former are Gibbons–Manton torus bundles over configuration spaces whereas the latter are obtained from them by replacing each circle factor of the fibre with a monopole moduli space by the Borel construction. They form boundary hypersurfaces in a partial compactification of the classical monopole moduli spaces. Our results follow from a general homological stability result for configuration spaces equipped with non-local data.
{"title":"Homology stability for asymptotic monopole moduli spaces","authors":"Martin Palmer, Ulrike Tillmann","doi":"10.1098/rspa.2023.0300","DOIUrl":"https://doi.org/10.1098/rspa.2023.0300","url":null,"abstract":"We prove homological stability for two different flavours of asymptotic monopole moduli spaces, namely moduli spaces of framed Dirac monopoles and moduli spaces of ideal monopoles . The former are Gibbons–Manton torus bundles over configuration spaces whereas the latter are obtained from them by replacing each circle factor of the fibre with a monopole moduli space by the Borel construction. They form boundary hypersurfaces in a partial compactification of the classical monopole moduli spaces. Our results follow from a general homological stability result for configuration spaces equipped with non-local data.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135810451","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The stream function solution for the inner region Stokes flow, for a locally plane moving fluid interface near the triple point, is derived considering three different boundary conditions: the Navier slip boundary condition (NBC), the super-slip boundary condition and the generalized Navier boundary condition (GNBC). The NBC, incorporating a slip length parameter λ , is a well-known method for regularization in the context of the three-phase dynamic contact line problem. It is demonstrated that the velocity field solution under this boundary condition maintains a C0 continuity at the contact line, resulting in a logarithmic divergence of the pressure at the contact line. By contrast, the super-slip boundary condition establishes a proportional relationship between the wall velocity and the normal derivative of the shear stress, leading to a C1 velocity field. Furthermore, the GNBC, which introduces an uncompensated Young stress to drive the contact line, yields a C2 velocity field. The dominant terms are explicitly derived, and the analytical approach presented here can be extended to other bi-harmonic problems as well.
{"title":"Stream function solutions for some contact line boundary conditions: Navier slip, super slip and the generalized Navier boundary condition","authors":"Yash Kulkarni, Tomas Fullana, Stephane Zaleski","doi":"10.1098/rspa.2023.0141","DOIUrl":"https://doi.org/10.1098/rspa.2023.0141","url":null,"abstract":"The stream function solution for the inner region Stokes flow, for a locally plane moving fluid interface near the triple point, is derived considering three different boundary conditions: the Navier slip boundary condition (NBC), the super-slip boundary condition and the generalized Navier boundary condition (GNBC). The NBC, incorporating a slip length parameter <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>λ</mml:mi> </mml:math> , is a well-known method for regularization in the context of the three-phase dynamic contact line problem. It is demonstrated that the velocity field solution under this boundary condition maintains a <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>0</mml:mn> </mml:msup> </mml:math> continuity at the contact line, resulting in a logarithmic divergence of the pressure at the contact line. By contrast, the super-slip boundary condition establishes a proportional relationship between the wall velocity and the normal derivative of the shear stress, leading to a <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> </mml:math> velocity field. Furthermore, the GNBC, which introduces an uncompensated Young stress to drive the contact line, yields a <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> velocity field. The dominant terms are explicitly derived, and the analytical approach presented here can be extended to other bi-harmonic problems as well.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135849542","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Christopher J. Lustri, Inês Aniceto, Daniel J. VandenHeuvel, Scott W. McCue
Burgers’ equation is an important mathematical model used to study gas dynamics and traffic flow, among many other applications. Previous analysis of solutions to Burgers’ equation shows an infinite stream of simple poles born at t=0+ , emerging rapidly from the singularities of the initial condition, that drive the evolution of the solution for t>0 . We build on this work by applying exponential asymptotics and transseries methodology to an ordinary differential equation that governs the small-time behaviour in order to derive asymptotic descriptions of these poles and associated zeros. Our analysis reveals that subdominant exponentials appear in the solution across Stokes curves; these exponentials become the same size as the leading order terms in the asymptotic expansion along anti-Stokes curves, which is where the poles and zeros are located. In this region of the complex plane, we write a transseries approximation consisting of nested series expansions. By reversing the summation order in a process known as transasymptotic summation, we study the solution as the exponentials grow, and approximate the pole and zero location to any required asymptotic accuracy. We present the asymptotic methods in a systematic fashion that should be applicable to other nonlinear differential equations.
{"title":"Locating complex singularities of Burgers’ equation using exponential asymptotics and transseries","authors":"Christopher J. Lustri, Inês Aniceto, Daniel J. VandenHeuvel, Scott W. McCue","doi":"10.1098/rspa.2023.0516","DOIUrl":"https://doi.org/10.1098/rspa.2023.0516","url":null,"abstract":"Burgers’ equation is an important mathematical model used to study gas dynamics and traffic flow, among many other applications. Previous analysis of solutions to Burgers’ equation shows an infinite stream of simple poles born at <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>t</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mn>0</mml:mn> <mml:mo>+</mml:mo> </mml:msup> </mml:math> , emerging rapidly from the singularities of the initial condition, that drive the evolution of the solution for <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>t</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:math> . We build on this work by applying exponential asymptotics and transseries methodology to an ordinary differential equation that governs the small-time behaviour in order to derive asymptotic descriptions of these poles and associated zeros. Our analysis reveals that subdominant exponentials appear in the solution across Stokes curves; these exponentials become the same size as the leading order terms in the asymptotic expansion along anti-Stokes curves, which is where the poles and zeros are located. In this region of the complex plane, we write a transseries approximation consisting of nested series expansions. By reversing the summation order in a process known as transasymptotic summation, we study the solution as the exponentials grow, and approximate the pole and zero location to any required asymptotic accuracy. We present the asymptotic methods in a systematic fashion that should be applicable to other nonlinear differential equations.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"117 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135810453","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Turing mechanism underpinning pattern formation in reaction–diffusion relies on the interplay between diffusion parameters and reaction kinetics. Diffusion is typically assumed isotropic, however anisotropic diffusion is known to arise in both nature and laboratory conditions. We study how the Turing instability and resulting Turing patterns are modified when the underlying diffusion tensor is both anisotropic and time-dependent, modelling, for instance, chemical species reacting and diffusing through time-varying anisotropic media. We show that the set of unstable wavenumber vectors corresponding to Turing modes evolve in a spatially biased manner under this anisotropy, thereby modifying the spatial scale and structure of any resulting Turing patterns differently along each spatial coordinate. We employ this spatial bias to develop control strategies to modify the shape or even structure of Turing patterns over time. We are able to make minor changes to the aspect ratio of Turing patterns, such as morphing circular Schnakenberg spots into elliptical spots, as well as more major changes to the structure of patterns, for instance converting Gierer–Meinhardt spots into stripes or FitzHugh–Nagumo labyrinthine patterns into target patterns. Our results suggest that time-varying anisotropic media may be used as a tool by which to modify and even control Turing patterns.
{"title":"Modification of Turing patterns through the use of time-varying anisotropic diffusion","authors":"Yaprak Önder, Robert A. Van Gorder","doi":"10.1098/rspa.2023.0487","DOIUrl":"https://doi.org/10.1098/rspa.2023.0487","url":null,"abstract":"The Turing mechanism underpinning pattern formation in reaction–diffusion relies on the interplay between diffusion parameters and reaction kinetics. Diffusion is typically assumed isotropic, however anisotropic diffusion is known to arise in both nature and laboratory conditions. We study how the Turing instability and resulting Turing patterns are modified when the underlying diffusion tensor is both anisotropic and time-dependent, modelling, for instance, chemical species reacting and diffusing through time-varying anisotropic media. We show that the set of unstable wavenumber vectors corresponding to Turing modes evolve in a spatially biased manner under this anisotropy, thereby modifying the spatial scale and structure of any resulting Turing patterns differently along each spatial coordinate. We employ this spatial bias to develop control strategies to modify the shape or even structure of Turing patterns over time. We are able to make minor changes to the aspect ratio of Turing patterns, such as morphing circular Schnakenberg spots into elliptical spots, as well as more major changes to the structure of patterns, for instance converting Gierer–Meinhardt spots into stripes or FitzHugh–Nagumo labyrinthine patterns into target patterns. Our results suggest that time-varying anisotropic media may be used as a tool by which to modify and even control Turing patterns.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136054706","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ergodicity Economics presents an intriguing perspective on the validity of the ergodic hypothesis within economic models and the influence of ergodicity breaking on human decision-making processes. Prior research has illuminated the impact of ergodicity breaking within multiplicative settings. However, the implications of ergodicity breaking within additive dynamics, especially in situations carrying a ‘risk of ruin’ or a complete loss, remain largely unexplored. In our research, we introduce the concept of ‘risk of ruin’ into our decision-making model to examine the effects of non-ergodicity in additive dynamics. Our theoretical framework and experiments show that human decision-makers are sensitive to non-ergodicity within purely additive dynamics. This sensitivity manifests itself in significantly different levels of risk aversion depending on the distance and associated likelihood of ruin. These findings underscore the critical role of time averages in human decision-making, suggesting that humans are less irrational than conventionally assumed in behavioural models rooted in expected values. Drawing on evidence from Ergodicity Economics, incorporating non-ergodicity has the potential to illuminate common trends in decision-making within compounding systems, like multiplicative growth dynamics. Our research underscores a similar potential for understanding decision-making patterns within additive dynamics when the risk of ruin is present.
{"title":"Human decision-making in a non-ergodic additive environment","authors":"A. Vanhoyweghen, V. Ginis","doi":"10.1098/rspa.2023.0544","DOIUrl":"https://doi.org/10.1098/rspa.2023.0544","url":null,"abstract":"Ergodicity Economics presents an intriguing perspective on the validity of the ergodic hypothesis within economic models and the influence of ergodicity breaking on human decision-making processes. Prior research has illuminated the impact of ergodicity breaking within multiplicative settings. However, the implications of ergodicity breaking within additive dynamics, especially in situations carrying a ‘risk of ruin’ or a complete loss, remain largely unexplored. In our research, we introduce the concept of ‘risk of ruin’ into our decision-making model to examine the effects of non-ergodicity in additive dynamics. Our theoretical framework and experiments show that human decision-makers are sensitive to non-ergodicity within purely additive dynamics. This sensitivity manifests itself in significantly different levels of risk aversion depending on the distance and associated likelihood of ruin. These findings underscore the critical role of time averages in human decision-making, suggesting that humans are less irrational than conventionally assumed in behavioural models rooted in expected values. Drawing on evidence from Ergodicity Economics, incorporating non-ergodicity has the potential to illuminate common trends in decision-making within compounding systems, like multiplicative growth dynamics. Our research underscores a similar potential for understanding decision-making patterns within additive dynamics when the risk of ruin is present.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136054812","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
B. D. Goddard, M. Ottobre, K. J. Painter, I. Souttar
Motivated by applications to mathematical biology, we study the averaging problem for slow–fast systems, in the case in which the fast dynamics is a stochastic process with multiple invariant measures . We consider both the case in which the fast process is decoupled from the slow process and the case in which the two components are fully coupled. We work in the setting in which the slow process evolves according to an ordinary differential equation (ODE) and the fast process is a continuous time Markov process with finite state space and show that, in this setting, the limiting (averaged) dynamics can be described as a random ODE (i.e. an ODE with random coefficients).
{"title":"On the study of slow–fast dynamics, when the fast process has multiple invariant measures","authors":"B. D. Goddard, M. Ottobre, K. J. Painter, I. Souttar","doi":"10.1098/rspa.2023.0322","DOIUrl":"https://doi.org/10.1098/rspa.2023.0322","url":null,"abstract":"Motivated by applications to mathematical biology, we study the averaging problem for slow–fast systems, in the case in which the fast dynamics is a stochastic process with multiple invariant measures . We consider both the case in which the fast process is decoupled from the slow process and the case in which the two components are fully coupled. We work in the setting in which the slow process evolves according to an ordinary differential equation (ODE) and the fast process is a continuous time Markov process with finite state space and show that, in this setting, the limiting (averaged) dynamics can be described as a random ODE (i.e. an ODE with random coefficients).","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135655814","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Donello, G. Palkar, M. H. Naderi, D. C. Del Rey Fernández, H. Babaee
Time-dependent basis reduced-order models (TDB ROMs) have successfully been used for approximating the solution to nonlinear stochastic partial differential equations (PDEs). For many practical problems of interest, discretizing these PDEs results in massive matrix differential equations (MDEs) that are too expensive to solve using conventional methods. While TDB ROMs have the potential to significantly reduce this computational burden, they still suffer from the following challenges: (i) inefficient for general nonlinearities, (ii) intrusive implementation, (iii) ill-conditioned in the presence of small singular values and (iv) error accumulation due to fixed rank. To this end, we present a scalable method for solving TDB ROMs that is computationally efficient, minimally intrusive, robust in the presence of small singular values, rank-adaptive and highly parallelizable. These favourable properties are achieved via oblique projections that require evaluating the MDE at a small number of rows and columns. The columns and rows are selected using the discrete empirical interpolation method (DEIM), which yields near-optimal matrix low-rank approximations. We show that the proposed algorithm is equivalent to a CUR matrix decomposition. Numerical results demonstrate the accuracy, efficiency and robustness of the new method for a diverse set of problems.
{"title":"Oblique projection for scalable rank-adaptive reduced-order modelling of nonlinear stochastic partial differential equations with time-dependent bases","authors":"M. Donello, G. Palkar, M. H. Naderi, D. C. Del Rey Fernández, H. Babaee","doi":"10.1098/rspa.2023.0320","DOIUrl":"https://doi.org/10.1098/rspa.2023.0320","url":null,"abstract":"Time-dependent basis reduced-order models (TDB ROMs) have successfully been used for approximating the solution to nonlinear stochastic partial differential equations (PDEs). For many practical problems of interest, discretizing these PDEs results in massive matrix differential equations (MDEs) that are too expensive to solve using conventional methods. While TDB ROMs have the potential to significantly reduce this computational burden, they still suffer from the following challenges: (i) inefficient for general nonlinearities, (ii) intrusive implementation, (iii) ill-conditioned in the presence of small singular values and (iv) error accumulation due to fixed rank. To this end, we present a scalable method for solving TDB ROMs that is computationally efficient, minimally intrusive, robust in the presence of small singular values, rank-adaptive and highly parallelizable. These favourable properties are achieved via oblique projections that require evaluating the MDE at a small number of rows and columns. The columns and rows are selected using the discrete empirical interpolation method (DEIM), which yields near-optimal matrix low-rank approximations. We show that the proposed algorithm is equivalent to a CUR matrix decomposition. Numerical results demonstrate the accuracy, efficiency and robustness of the new method for a diverse set of problems.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136117404","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper reviews macroscopic aspects of the theory of magnetoelastostatics, starting with a brief summary of the experimental and theoretical contributions leading to the development of the current state-of-the-art. It offers some different perspectives than hitherto, with incompressible materials being the main concern. The use of the so-called total energy (density) functions is highlighted along with their associated total stress tensors and succinct forms of the constitutive equations. The symmetry of the total Cauchy stress tensor, which incorporates the non-symmetric Maxwell stress within the material, is emphasized and it is noted that the use of such a Maxwell stress, often appearing in the literature, is thereby avoided. The theory is illustrated for some simple prototype boundary-value problems, specifically the homogeneous deformation of an infinite slab of magnetoelastic material in the presence of a magnetic field and the non-homogeneous extension and inflation of an infinitely long circular cylindrical tube in the presence of either an axial or a circumferential magnetic field.
{"title":"The nonlinear theory of magnetoelasticity and the role of the Maxwell stress: a review","authors":"Luis Dorfmann, Ray W. Ogden","doi":"10.1098/rspa.2023.0592","DOIUrl":"https://doi.org/10.1098/rspa.2023.0592","url":null,"abstract":"This paper reviews macroscopic aspects of the theory of magnetoelastostatics, starting with a brief summary of the experimental and theoretical contributions leading to the development of the current state-of-the-art. It offers some different perspectives than hitherto, with incompressible materials being the main concern. The use of the so-called total energy (density) functions is highlighted along with their associated total stress tensors and succinct forms of the constitutive equations. The symmetry of the total Cauchy stress tensor, which incorporates the non-symmetric Maxwell stress within the material, is emphasized and it is noted that the use of such a Maxwell stress, often appearing in the literature, is thereby avoided. The theory is illustrated for some simple prototype boundary-value problems, specifically the homogeneous deformation of an infinite slab of magnetoelastic material in the presence of a magnetic field and the non-homogeneous extension and inflation of an infinitely long circular cylindrical tube in the presence of either an axial or a circumferential magnetic field.","PeriodicalId":20716,"journal":{"name":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","volume":"81 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135810459","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"综合性期刊","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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