Piezo-electromechanical smart materials with distributed arrays of piezoelectric transducers: Current and upcoming applications

Boutin

et al. 2017

J Elast

124

The well-posedness of the boundary value problems for second gradient elasticity has been studied under the assumption of strong ellipticity of the dependence on the second placement gradients (see, e.g., Chambon and Moullet in Comput. Methods Appl. Mech. Eng. 193:2771-2796 and Mareno and Healey in SIAM J. Math. Anal. 38:103-115, 2006.The study of the equilibrium of planar pantographic lattices has been approached in two different ways: in dell 'Isola et al. (Proc. R. Soc. Lond. Ser. A 472:20150, 2016) a discrete model was introduced involving extensional and rotational springs which is also valid in large deformations regimes while in Boutin et al. (Math. Mech. Complex Syst. 5:127-162, 2017) the lattice has been modelled as a set of beam elements interconnected by internal pivots, but the analysis was restricted to the linear case. In both papers a homogenized second gradient deformation energy, quadratic in the neighbourhood of non deformed configuration, is obtained via perturbative methods and the predictions obtained with the obtained continuum model are successfully compared with experiments. This energy is not strongly elliptic in its dependence on second gradients. We consider in this paper also the important particular case of pantographic lattices whose first gradient energy does not depend on shear deformation: this could be considered either a pathological case or an important exceptional case (see Stillwell et al. in Am. Math. Mon. 105:850-858, 1998 and Turro in Angew. Chem., Int. Ed. Engl. 39:2255Engl. 39: -2259Engl. 39: , 2000. In both cases we believe that such a particular case deserves some attention because of what we can understand by studying it (see Dyson in Science 200:677-678, 1978). This circumstance motivates the present paper, where we address the well-posedness of the planar linearized equilibrium problem for homogenized pantographic lattices. To do so: (i) we introduce a class of subsets of anisotropic Sobolev's space as the most suitable energy space E relative to assigned boundary conditions; (ii) we prove that the considered strain energy density is coercive and positive definite in E; (iii) we prove that the set of placements for which the strain energy is vanishing (the so-called floppy modes) must strictly include rigid motions; (iv) we determine the restrictions on displacement boundary conditions which assure existence and uniqueness of linear static problems. The presented results represent one of the first mechanical applications of the concept of Anisotropic Sobolev space, initially introduced only on the basis of purely abstract mathematical considerations.

Section: Introductionmentioning

confidence: 99%

Linear Pantographic Sheets: Existence and Uniqueness of Weak Solutions

Eremeyev

Boutin

et al. 2017

J Elast

124

“…A review of results on the theoretical foundation of a variational approach for higher gradient theories is [71]. Besides, the use of the methods of metamaterials [72][73][74] like those for pantographic structures [75][76][77] could be considered even to take into account the effects of damage [78][79][80][81][82]. Another possible extension of this model is to include surface effects [83][84][85][86][87][88][89] and the anisotropy induced by the orientation of the NLC directors, as in [90] , or as it is done in granular materials [91].…”

Section: Numerical Simulationsmentioning

confidence: 99%

“Fast” and “slow” pressure waves electrically induced by nonlinear coupling in Biot-type porous medium saturated by a nematic liquid crystal

Rosi

Placidi

2017

Z. Angew. Math. Phys.

“…Also very relevant may be the implications of the results presented here on the understanding of the biomechanics of living tissues: Indeed (see e.g., [26,[31][32][33] and [36][37][38]), the role of fiber reinforcements on the mechanical behavior of growing and reconstructed tissues is attracting some attention. Finally, the consideration of multiphysics effects (as for instance piezo-or flexo-electricity) in systems having the geometry and the kinematics of a pantographic sheet can also disclose new possibilities (see [20,27]). …”

mentioning

confidence: 99%

Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence

Lekszycki

Pawlikowski

et al. 2015

Z. Angew. Math. Phys.

158

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