An Inverse Method to Get Further Analytical Solutions for a Class of Metamaterials Aimed to Validate Numerical Integrati
We consider an isotropic second gradient elastic two-dimensional solid. Besides, we relax the isotropic hypothesis and consider a D4 orthotropic material. The reason for this last choice is that such anisotropy is the most general for pantographic structu
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Abstract We consider an isotropic second gradient elastic two-dimensional solid. Besides, we relax the isotropic hypothesis and consider a D4 orthotropic material. The reason for this last choice is that such anisotropy is the most general for pantographic structures, which exhibit attracting mechanical properties. In this paper we analyze the role of the external body double force mext on the partial differential equations and we subsequently revisit some analytical solutions that have been considered in the literature for identification purposes. The revisited analytical solutions will be employed as well for identification purposes in a further contribution.
1 Introduction Design of metamaterials (see [25, 35] for recent review papers) is nowadays a very important challenge, and new possibility are available due to an enormously increased capacity of big data analysis. In the present introduction we want to frame this problem in the existing literature and offer some prospects on potentially useful tools.
L. Placidi (B) Faculty of Engineering, International Telematic University Uninettuno, C.so Vittorio Emanuele II, 39, 00186 Rome, Italy e-mail: [email protected] E. Barchiesi Dipartimento di Ingegneria Meccanica e Aerospaziale, Universitá di Roma La Sapienza, Via Eudossiana 18, 00184 Rome, Italy e-mail: [email protected] A. Battista Laboratory of Science for Environmental Engineering, Université de La Rochelle, La Rochelle, France e-mail: [email protected] © Springer Nature Singapore Pte Ltd. 2017 F. dell’Isola et al. (eds.), Mathematical Modelling in Solid Mechanics, Advanced Structured Materials 69, DOI 10.1007/978-981-10-3764-1_13
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Due to the aforementioned computational power, discrete models are becoming increasingly capable of capturing all the important features of continuum mechalics systems (see [2, 14, 41, 56, 60] for some recent numerical and theoretical result). Besides, metamaterials are designed to perform with expected mechanical properties, and the microstructure can be very complicated and difficult to manage from a numerical point of view [17, 20, 40, 59] due to the high computational demand of structures, especially if one wants to include description of impact-like behaviors [15, 19, 63], instabilities [46, 53, 54, 61, 62, 64, 66] and/or surface effects [3, 26] and damage or plastic behaviour [24, 41, 42]. In fact, the size of the microstructure (for nano-sized objects see for instance [4–7]) can be designed in such a way that the number of cells is very high and the effective geometry of the resulting body is complex, see e.g. the geometry of pantographic structures [33, 36, 67] and of truss structures [1, 69]. The numerical simulation of a 3D body with such a geometry goes usually beyond the standard numerical capabilities if one employs standard 3D Cahcy elasticity models. Thus, the necessity to find new models that are able to deal with complex microstructures from a continuum point of view, via e.g. an homogenization c
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