PSI - Issue 43

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Procedia Structural Integrity 43 (2023) 65–70 Structural Integrity Procedia 00 (2023) 000–000 Structural Integrity Procedia 00 (2023) 000–000

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© 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under the responsibility of MSMF10 organizers. Abstract We give a simple mechanical motivation for a generalization of the classical Grassmannian considered as a space of m -dimensional linear subspaces of R k to higher-orders cases. Our e ff orts are prolongated to the Weil functor theory, motivated by non-holonomic and semi-holonomic jets, applicable in the description of the Cosserat model. © 2023 The Authors. Published by Elsevier B.V. his is an open access article under the CC BY-NC-ND license (http: // creativec mmons.org / licenses / by-nc-nd / 4.0 / ) er-review under the respons bility of SMF10 organizers. Keywords: jet; bundle functor; Weil functor; Lie group; jet group; principal bundle; homogeneous space PACS: 02.40.-k; 02.20.Qs 2020 MSC: 58A20; 58A32; 58C12 1. Preliminaries 1.1 Mechanical motivation . We give a contribution to the modelling of elementary continuum mechanics concepts by means of modern di ff erential geometry, namely by the concepts like jets, jet groups, principle bundles and frame bundles and some kinds of jet functors and functors related to Weil functors. We generalize the frame bundle to the higher order Grassmannian, which from the geometrical point of view generalizes the projective space (or the line bundle) as well as the classical (i. e.) the first-order Grassmannian. We give the mechanical motivation for such construction. We follow the book Epstein et al (2007) and focus our attention on the properties of local or global uniformity of the first-order and higher-order materials and also on the Cosserat media. As for the last case, there appear also non-holonomic jets as a usefull tool for investigations of such materials and it is useful to apply the Weil theory, which enables to put both of non-holonomic and semi-holonomic jets (appearing when studying connections and homogeneity) under one umbrella. The concept of local or global uniformity corresponds to the locally or globally same material of a body while homogeneity corresponds to the existence of a local or global configuration in which translations act as material isomorphisms discussed next below. Nevertheless investigations of homogeneity or inhomogeneity requires the theory of connections and the concept of inhomogeneity tensor, which is now not our business. For sake of simplicity, consider an elastic material, which means that the response, mostly the stress per unit volume depends only on the present value of the deformation gradient F = κ ◦ κ − 1 0 where κ 0 is the reference configuration and bstract e give a si ple echanical otivation for a generalization of the classical rass annian considered as a space of -di ensional linear subspaces of R k to higher-orders cases. ur e ff orts are prolongated to the eil functor theory, otivated by non-holono ic and se i-holono ic jets, applicable in the description of the osserat odel. 2023 The uthors. Published by Elsevier . . his is an open access article under the - - license (http: // creativec ons.org / licenses / by-nc-nd / 4.0 / ) eer-revie under the responsibility of S F10 organizers. ey ords: jet; bundle functor; eil functor; Lie group; jet group; principal bundle; ho ogeneous space PACS: 02.40.-k; 02.20. s 2020 SC: 58 20; 58 32; 58C12 1. reli inaries 1.1 echanical otivation . e give a contribution to the odelling of ele entary continuu echanics concepts by eans of odern di erential geo etry, na ely by the concepts like jets, jet groups, principle bundles and fra e bundles and so e kinds of jet functors and functors related to eil functors. e generalize the fra e bundle to the higher order rass annian, hich fro the geo etrical point of vie generalizes the projective space (or the line bundle) as ell as the classical (i. e.) the first-order rass annian. e give the echanical otivation for such construction. e follo the book pstein et al (2007) and focus our attention on the properties of local or global unifor ity of the first-order and higher-order aterials and also on the osserat edia. s for the last case, there appear also non-holono ic jets as a usefull tool for investigations of such aterials and it is useful to apply the eil theory, hich enables to put both of non-holono ic and se i-holono ic jets (appearing hen studying connections and ho ogeneity) under one u brella. he concept of local or global unifor ity corresponds to the locally or globally sa e aterial of a body hile ho ogeneity corresponds to the existence of a local or global configuration in hich translations act as aterial iso orphis s discussed next belo . evertheless investigations of ho ogeneity or inho ogeneity requires the theory of connections and the concept of inho ogeneity tensor, hich is no not our business. For sake of si plicity, consider an elastic aterial, hich eans that the response, ostly the stress per unit volu e depends only on the present value of the defor ation gradient κ ◦ κ − 1 0 here κ 0 is the reference configuration and 10th International Conference on Materials Structure and Micromechanics of Fracture Higher-order and Weil generalization of Grassmannian Jiˇr´ı Toma´sˇ a, ∗ a Brno University of Technology, Faculty of Mechanical Engineering, Institute of Mathematics, Technicka´ 2, 612 00 Brno, Czech Republic t I ter ati al fere ce aterials tr ct re a icr ec a ics f ract re i r- r r il r li ti f r i Jiˇr´ı To a´sˇ a, ∗ a Brno niversity of Technology, Faculty of echanical Engineering, Institute of athe atics, Technicka´ 2, 612 00 Brno, Czech Republic

∗ Corresponding author. Tel.: + 420-54114-2627. E-mail address: tomas@fme.vutbr.cz ∗ Corresponding author. Tel.: + 420-54114-2627. E- ail address: to as f e.vutbr.cz

2452-3216 © 2023 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under the responsibility of MSMF10 organizers. 10.1016/j.prostr.2022.12.236 2210-7843 © 2023 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under the responsibility of MSMF10 organizers. 2210-7843 2023 The uthors. Published by Elsevier B. . This is an open access article under the CC BY-NC-ND license (http: // creativecommons.org / licenses / by-nc-nd / 4.0 / ) Peer-review under the responsibility of MSMF10 organizers.