Zero Mach number limit of the compressible Hall-magnetohydrodynamic equations

Mu, Yanmin

doi:10.1007/s00033-015-0604-0

Cited by 62 publications

(5 citation statements)

References 25 publications

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“…A class of planar metamaterials has been recently extensively studied (called pantographic sheets in [17] and studied in detail in [18][19][20][21]) showing a rather exotic mechanical behaviour. Their microstructure can be recognized in figure 1, in which a specimen obtained by using 3D printing technology is shown.…”

Section: Pantographic Lattice With King Post Truss Building Block 21 Mathematical Formulationmentioning

confidence: 99%

“…We can distinguish two families of fibres, which behave as beams, connected to each other by means of cylindrical connections named pivots. Various papers, see for example [17,22,23], show, from a theoretically point of view, through numerical simulation and also by experiments, that this kind of metamaterial behaves elastically under very large deformations; see for example [24] for the case of load applied on single fibres, [25,26] which discuss some non-standard tests such as traction and bending and [27] for a review on pantographic fabrics; see also [28,29] for a general theoretical introduction on discrete system, [30] for non-local structural elements and [31] for fracture phenomena. This peculiar elastic behaviour, which has many potential practical applications, served as a motivation to design the building block unit of this lattice as that illustrated in figure 2 making use of the king post truss scheme; see [32,33].…”

Section: Pantographic Lattice With King Post Truss Building Block 21 Mathematical Formulationmentioning

confidence: 99%

“…with a pair being an η-approximate solution of the equilibrium condition (2.3), the iterative scheme, once the step λ is fixed, is obtained by constructing the η-approximate solution (u j+1 =: u j + u j , λ j+1 := λ j + λ) by using the iterative scheme based on the Riks' arc-length parameter to closely follow the equilibrium path. This effective algorithm is well described in [23] for a nonlinear elastic problem and in [35] for an elasto-plastic problem with some insights into filtering properties of such a procedure.…”

Section: Pantographic Lattice With King Post Truss Building Block 21 Mathematical Formulationmentioning

confidence: 99%

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King post truss as a motif for internal structure of (meta)material with controlled elastic properties

et al. 2017

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One of the most interesting challenges in the modern theory of materials consists in the determination of those microstructures which produce, at the macro-level, a class of metamaterials whose elastic range is many orders of magnitude wider than the one exhibited by ‘standard’ materials. In dell’Isola et al. (2015 Zeitschrift für angewandte Mathematik und Physik 66, 3473–3498. (doi:10.1007/s00033-015-0556-410.1007/s00033-015-0556-4)), it was proved that, with a pantographic microstructure constituted by ‘long’ micro-beams it is possible to obtain metamaterials whose elastic range spans up to an elongation exceeding 30%. In this paper, we demonstrate that the same behaviour can be obtained by means of an internal microstructure based on a king post motif. This solution shows many advantages: it involves only microbeams; all constituting beams are undergoing only extension or compression; all internal constraints are terminal pivots. While the elastic deformation energy can be determined as easily as in the case of long-beam microstructure, the proposed design seems to have obvious remarkable advantages: it seems to be more damage resistant and therefore to be able to have a wider elastic range; it can be realized with the same three-dimensional printing technology; it seems to be less subject to compression buckling. The analysis which we present here includes: (i) the determination of Hencky-type discrete models for king post trusses, (ii) the application of an effective integration scheme to a class of relevant deformation tests for the proposed metamaterial and (iii) the numerical determination of an equivalent second gradient continuum model. The numerical tools which we have developed and which are presented here can be readily used to develop an extensive measurement campaign for the proposed metamaterial.

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Section: Pantographic Lattice With King Post Truss Building Block 21 Mathematical Formulationmentioning

confidence: 99%

Section: Pantographic Lattice With King Post Truss Building Block 21 Mathematical Formulationmentioning

confidence: 99%

Section: Pantographic Lattice With King Post Truss Building Block 21 Mathematical Formulationmentioning

confidence: 99%

See 1 more Smart Citation

King post truss as a motif for internal structure of (meta)material with controlled elastic properties

et al. 2017

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“…See other studies for the study of global existence and temporal decay, and see Fan et al for the study of blow‐up criterion of strong solutions. In Mu, the zero Mach limit of the compressible Hall‐MHD system is studied.…”

Section: Introductionmentioning

confidence: 99%

Liouville‐type theorem for the steady compressible Hall‐MHD system

Zeng

2017

Math Methods in App Sciences

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“…For the incompressible Hall-MHD system, for example, the regularity criteria of the solutions were obtained in [6,11,12,15,26], and the global existence of small solution and global weak solutions were given in [3][4][5]. For the isentropic compressible Hall-MHD system, the local existence of strong solutions, global existence of small solutions were first obtained in [9] and the low Mach number limit problem was discussed in [21,27]. Very recently, the local well-posedness and a blow-up criterion of strong solutions to the 3D compressible full Hall-MHD system (1)-(5) with positive density was obtained in [10].…”

mentioning

confidence: 99%