Least Action Principle for Second Gradient Continua and Capillary Fluids: A Lagrangian Approach Following Piola’s Point of View

Dell’Isola, Francesco; Auffray, Nicolas; Eremeyev, Victor A.; Madeo, Angela; Placidi, Luca; Rosi, Giuseppe

doi:10.1007/978-3-319-00263-7_4

Cited by 9 publications

(6 citation statements)

References 127 publications

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“…In a next future, we would like to propose the model in a fully variational setting. It is no trivial but would provide clear methodological advantages (see [50] for an introduction and [51][52][53][54][55] for illustrative cases concerning continua with non-classical properties). To this aim, we introduce two formulations PE1 and PE2 concerning the pseudo-energy concept.…”

Section: Tool Descriptionmentioning

confidence: 99%

Rules governing swarm robot in continuum mechanics

Dell'Erba

2022

Mathematics and Mechanics of Solids

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To determine material body’s deformation is usual to solve Newton’s equation, under assigned boundary conditions; but this is not the only possibility to obtain a plausible description of the phenomenon. In recent years, position-based dynamic (PBD) concept has consolidated its success. The idea has its roots in computer graphic graphics aimed at saving processing time in computing object’s deformation and breakage in video games applications; PBD tries to give a plausible description of the deformation taking into account only the relative point’s positions to describe the action of internal forces. The aim is to obtain an algebraical equation capable of calculating the position of a point using as input the position of some other points. This kinematic approach has the advantage to work with simple algebraical equations; the complexity of the system increases linearly, rather than quadratically, with increasing the point’s number. Working on swarm robotics we have addressed the problem concerning flocking rules, to determine the behaviour of a single element to achieve an assigned configuration of the group. One possibility to perform this task, for each single robot, is to see what some of its neighbours are doing. We then use these rules to reproduce some behaviour of bi-dimensional deformable bodies both consistent with the standard Cauchy model and according to the second-gradient theory. All constitutive properties of the material are hidden within the rules, determining the displacements of the particles each relative to the others. The tool has an advantage in terms of computational cost and is very flexible to be adapted to objects of complex geometry and problems of different nature. The results are encouraging and fracture can be easily managed. However, a connection between the parameters of the tool and the constitutive parameters of the materials is not yet provided and our efforts are addressed in this direction, to validate the tool by a solid classical theory.

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Section: Tool Descriptionmentioning

confidence: 99%

Rules governing swarm robot in continuum mechanics

Dell'Erba

2022

Mathematics and Mechanics of Solids

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“…The personalistic research of the first genius who did invent a theory is not so important: Instead, it is extremely important to understand which was the discovery process that, at first, lead to formulate a novel mathematized mechanical theory: We claim that such a first theory is that of the equilibrium of a lever or that of the functioning of machines 15 . It is clear that learning from history how novel theory were invented may help us to invent newer theories.…”

Section: The Sociological Process Of Knowledge Transmission Needs Car...mentioning

confidence: 99%

“…Indeed, the microstructure of such materials allows for the combination of «simpler forces» to produce, at macro-level, more complex ones. For instance, the microscopically homogeneous materials described by Cauchy continua cannot support distributed couples or double-forces [9][10][11][12] which, instead, can be supported by strain-gradient continua describing, at macro-level, microscopically heterogeneous materials [13][14][15][16][17][18][19]. The description of the mental process, which produced the materialization of forces, is also relevant when one wants to firmly found (especially if he needs to avoid ambiguities and contradictions in their formulation) the mathematical models suitable to describe and predict nonstandard generalized and desired exotic mechanical properties [20][21][22][23][24][25].…”

Section: Introductionmentioning

confidence: 99%

The «materialization» of forces: Why confounding mathematical concept and physical entity makes the design of metamaterials arduous

Dell’Isola

Stilz

2022

Z Angew Math Mech

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“…In case of second gradient we enlarge the set of points with a supplementary shell. Envisaging the possibility to frame the proposed model in a fully variational setting, which is by no means trivial but would provide clear methodological advantages (see [47] for an introduction and [48][49][50][51][52] for illustrative cases concerning continua with non-classical properties), we also like to introduce pseudoenergetic considerations. In the elastic case we can consider the square of the distance between the actual configuration C and the reference configuration C0.…”

Section: The Algorithm: a Quick Resumementioning

confidence: 99%

Swarm robotics and complex behaviour of continuum material

Dell'Erba

2018

Continuum Mech. Thermodyn.

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In swarm robotics, just as for an animal swarm in Nature, one of the aims is to reach and maintain a desired configuration. One of the possibilities for the team, to reach this aim, is to see what its neighbours are doing. This approach generates a rules system governing the movement of the single robot just by reference to neighbour's motion. The same approach is used in position based dynamics to simulate behaviour of complex continuum materials under deformation. Therefore, in some previous works, we have considered a two-dimensional lattice of particles and calculated its time evolution by using a rules system derived from our experience in swarm robotics. The new position of a particle, like the element of a swarm, is determined by the spatial position of the other particles. No dynamic is considered, but it can be thought as being hidden in the behaviour rules. This method has given good results in some simple situations reproducing the behaviour of deformable bodies under imposed strain. In this paper we try to stress our model to highlight its limits and how they can be improved. Some other, more complex, examples are computed and discussed. Shear test, different lattice, different fracture mechanism and ASTM shape sample behaviour have been investigated by the software tool we have developed.

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Least Action Principle for Second Gradient Continua and Capillary Fluids: A Lagrangian Approach Following Piola’s Point of View

Cited by 9 publications

References 127 publications

Rules governing swarm robot in continuum mechanics

Rules governing swarm robot in continuum mechanics

The «materialization» of forces: Why confounding mathematical concept and physical entity makes the design of metamaterials arduous

Swarm robotics and complex behaviour of continuum material

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