A Stabilised Micropolar Theory Derived from a Periodic Beam Lattice

Harm Askes; Mariateresa Lombardo; Duc C.D. Nguyen

doi:10.1007/978-3-031-58665-1_11

A Stabilised Micropolar Theory Derived from a Periodic Beam Lattice

Harm Askes^*, Mariateresa Lombardo, Duc C.D. Nguyen

^*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

1 Citation (Scopus)

Abstract

Starting from the finite element equations of a periodic beam lattice, a continuum theory is derived via Taylor series expansions. The continualised equations of motion are of the micropolar type and thus contain stiffness and inertia contributions in terms of translational as well as rotational degrees of freedom. Scrutiny of the underlying energy functionals and of the model’s response to excitation via harmonic waves confirms that the continualised model is unstable. In order to stabilise the model, Padé approximation is applied to the rotational equation of motion. As a result of this process, micro-inertia terms appear in the model that are expressed as the spatial gradients of the standard rotational inertia terms. It is demonstrated by means of an analysis of dispersive waves that the Padé approximation not only stabilises the model but also improves the accuracy with which the continuum model approximates the original discrete model.

Original language	English
Title of host publication	Continuum Models and Discrete Systems - CMDS-14
Editors	François Willot, Dominique Jeulin, François Willot, Justin Dirrenberger, Samuel Forest, Andrej V. Cherkaev
Publisher	Springer
Pages	155-166
Number of pages	12
ISBN (Electronic)	978-3-031-58665-1
ISBN (Print)	978-3-031-58664-4, 978-3-031-58667-5
DOIs	https://doi.org/10.1007/978-3-031-58665-1_11
Publication status	Published - 2024
Event	14th International Symposium on Continuum Models and Discrete Systems, CMDS 2023 - Paris, France Duration: 26 Jun 2023 → 30 Jun 2023 Conference number: 14

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Publisher	Springer
Volume	457
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Conference

Conference	14th International Symposium on Continuum Models and Discrete Systems, CMDS 2023
Abbreviated title	CMDS 2024
Country/Territory	France
City	Paris
Period	26/06/23 → 30/06/23

Keywords

2025 OA procedure
Continualisation
Micropolar theory
Beam lattice

Access to Document

10.1007/978-3-031-58665-1_11

Cite this

Askes, H., Lombardo, M., & Nguyen, D. C. D. (2024). A Stabilised Micropolar Theory Derived from a Periodic Beam Lattice. In F. Willot, D. Jeulin, F. Willot, J. Dirrenberger, S. Forest, & A. V. Cherkaev (Eds.), Continuum Models and Discrete Systems - CMDS-14 (pp. 155-166). (Springer Proceedings in Mathematics and Statistics; Vol. 457). Springer. https://doi.org/10.1007/978-3-031-58665-1_11

Askes, Harm ; Lombardo, Mariateresa ; Nguyen, Duc C.D. / A Stabilised Micropolar Theory Derived from a Periodic Beam Lattice. Continuum Models and Discrete Systems - CMDS-14. editor / François Willot ; Dominique Jeulin ; François Willot ; Justin Dirrenberger ; Samuel Forest ; Andrej V. Cherkaev. Springer, 2024. pp. 155-166 (Springer Proceedings in Mathematics and Statistics).

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abstract = "Starting from the finite element equations of a periodic beam lattice, a continuum theory is derived via Taylor series expansions. The continualised equations of motion are of the micropolar type and thus contain stiffness and inertia contributions in terms of translational as well as rotational degrees of freedom. Scrutiny of the underlying energy functionals and of the model{\textquoteright}s response to excitation via harmonic waves confirms that the continualised model is unstable. In order to stabilise the model, Pad{\'e} approximation is applied to the rotational equation of motion. As a result of this process, micro-inertia terms appear in the model that are expressed as the spatial gradients of the standard rotational inertia terms. It is demonstrated by means of an analysis of dispersive waves that the Pad{\'e} approximation not only stabilises the model but also improves the accuracy with which the continuum model approximates the original discrete model.",

keywords = "2025 OA procedure, Continualisation, Micropolar theory, Beam lattice",

author = "Harm Askes and Mariateresa Lombardo and Nguyen, {Duc C.D.}",

note = "Publisher Copyright: {\textcopyright} The Author(s), under exclusive license to Springer Nature Switzerland AG 2024.; 14th International Symposium on Continuum Models and Discrete Systems, CMDS 2023, CMDS 2024 ; Conference date: 26-06-2023 Through 30-06-2023",

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Askes, H, Lombardo, M & Nguyen, DCD 2024, A Stabilised Micropolar Theory Derived from a Periodic Beam Lattice. in F Willot, D Jeulin, F Willot, J Dirrenberger, S Forest & AV Cherkaev (eds), Continuum Models and Discrete Systems - CMDS-14. Springer Proceedings in Mathematics and Statistics, vol. 457, Springer, pp. 155-166, 14th International Symposium on Continuum Models and Discrete Systems, CMDS 2023, Paris, France, 26/06/23. https://doi.org/10.1007/978-3-031-58665-1_11

A Stabilised Micropolar Theory Derived from a Periodic Beam Lattice. / Askes, Harm; Lombardo, Mariateresa; Nguyen, Duc C.D.
Continuum Models and Discrete Systems - CMDS-14. ed. / François Willot; Dominique Jeulin; François Willot; Justin Dirrenberger; Samuel Forest; Andrej V. Cherkaev. Springer, 2024. p. 155-166 (Springer Proceedings in Mathematics and Statistics; Vol. 457).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

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AB - Starting from the finite element equations of a periodic beam lattice, a continuum theory is derived via Taylor series expansions. The continualised equations of motion are of the micropolar type and thus contain stiffness and inertia contributions in terms of translational as well as rotational degrees of freedom. Scrutiny of the underlying energy functionals and of the model’s response to excitation via harmonic waves confirms that the continualised model is unstable. In order to stabilise the model, Padé approximation is applied to the rotational equation of motion. As a result of this process, micro-inertia terms appear in the model that are expressed as the spatial gradients of the standard rotational inertia terms. It is demonstrated by means of an analysis of dispersive waves that the Padé approximation not only stabilises the model but also improves the accuracy with which the continuum model approximates the original discrete model.

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