Metamaterials with relative displacements in their microstructure: technological challenges in 3D printing, experiments and numerical predictions
Tóm tắt
3D printing technology has opened application perspectives which were difficult to imagine only few years ago. In this paper, we show how it is possible to design and print some microstructures in which relative displacements are allowed at micro-level. Some structural elements have been built in the aforementioned structures which can be confidently modeled as perfect pivots or as soft elastic connections. The obtained specimens can be regarded as constituted by very exotic materials, as forecast theoretically. We numerically study the behavior of pantographic structures including soft or nearly perfect pivots in large deformations, and we experimentally observe an enlarged elastic range and peculiar buckling mechanisms. The presented results are extremely promising: We consider now as proven that higher gradient metamaterials can be realized by using microstructures having micro-characteristic lengths of the order on tenth of millimeters.
Tài liệu tham khảo
Abali, B., Müller, W., dell’Isola, F.: Theory and computation of higher gradient elasticity theories based on action principles. Arch. Appl. Mech. 87, 1–16 (2017)
Alibert, J., Della Corte, A.: Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof. Z. Angew. Math. Phys. 66(5), 2855–2870 (2015)
Alibert, J.J., Della Corte, A., Giorgio, I., Battista, A.: Extensional elastica in large deformation as\(\backslash \)gamma-limit of a discrete 1D mechanical system. Z. Angew. Math. Phys. 68(2), 42 (2017)
Alibert, J.J., Seppecher, P., dell’Isola, F.: Truss modular beams with deformation energy depending on higher displacement gradients. Math. Mech. Solids 8(1), 51–73 (2003)
Altenbach, H., Eremeyev, V.: On the constitutive equations of viscoelastic micropolar plates and shells of differential type. Math. Mech. Complex Syst. 3(3), 273–283 (2015)
Auffray, N., dell’Isola, F., Eremeyev, V., Madeo, A., Rosi, G.: Analytical continuum mechanics à la Hamilton-Piola least action principle for second gradient continua and capillary fluids. Math. Mech. Solids 20(4), 375–417 (2015)
Barchiesi, E., Placidi, L.: A review on models for the 3D statics and 2D dynamics of pantographic fabrics. In: Sumbatyan, M.A. (ed.) Wave Dynamics and Composite Mechanics for Microstructured Materials and Metamaterials, pp. 239–258. Springer, Berlin (2017)
Berrehili, Y., Marigo, J.: The homogenized behavior of unidirectional fiber-reinforced composite materials in the case of debonded fibers. Math. Mech. Complex Syst. 2(2), 181–207 (2014)
Bertram, A., Rainer, G.: Gradient materials with internal constraints. Math. Mech. Complex Syst. 4(1), 1–15 (2016)
Bevill, G., Eswaran, S.K., Gupta, A., Papadopoulos, P., Keaveny, T.M.: Influence of bone volume fraction and architecture on computed large-deformation failure mechanisms in human trabecular bone. Bone 39(6), 1218–1225 (2006)
Boutin, C., Giorgio, I., Placidi, L., et al.: Linear pantographic sheets: asymptotic micro-macro models identification. Math. Mech. Complex Syst. 5(2), 127–162 (2017)
Caprino, S., Esposito, R., Marra, R., Pulvirenti, M.: Hydrodynamic limits of the Vlasov equation. Commun. Partial Differ. Equ. 18(5), 805–820 (1993)
Carcaterra, A., dell’Isola, F., Esposito, R., Pulvirenti, M.: Macroscopic description of microscopically strongly inhomogeneous systems: a mathematical basis for the synthesis of higher gradients metamaterials. Arch. Ration. Mech. Anal. 218(3), 1239–1262 (2015)
Carinci, G., De Masi, A., Giardinà, C., Presutti, E.: Hydrodynamic limit in a particle system with topological interactions. Arab. J. Math. 3(4), 381–417 (2014)
Carinci, G., De Masi, A., Giardinà, C., Presutti, E.: Super-hydrodynamic limit in interacting particle systems. J. Stat. Phys. 155(5), 867–887 (2014)
Cuomo, M., dell’Isola, F., Greco, L., Rizzi, N.: First versus second gradient energies for planar sheets with two families of inextensible fibres: investigation on deformation boundary layers, discontinuities and geometrical instabilities. Compos. Part B Eng. 115, 423–448 (2016)
De Masi, A., Merola, I., Presutti, E., Vignaud, Y.: Coexistence of ordered and disordered phases in potts models in the continuum. J. Stat. Phys. 134(2), 243–306 (2009)
De Masi, A., Olla, S.: Quasi-static hydrodynamic limits. J. Stat. Phys. 161(5), 1037–1058 (2015)
Dell’Isola, F., Andreaus, U., Placidi, L.: At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: an underestimated and still topical contribution of gabrio piola. Math. Mech. Solids 20(8), 887–928 (2015)
dell’Isola, F., Della Corte, A., Giorgio, I.: Higher-gradient continua: the legacy of piola, mindlin, sedov and toupin and some future research perspectives. Math. Mech. Solids 22, 852–872 (2016)
dell’Isola, F., Della Corte, A., Giorgio, I., Scerrato, D.: Pantographic 2D sheets: discussion of some numerical investigations and potential applications. Int. J. Non Linear Mech. 80, 200–208 (2016)
dell’Isola, F., Giorgio, I., Andreaus, U.: Elastic pantographic 2D lattices: a numerical analysis on static response and wave propagation. Proc. Estonian Acad. Sci. 64, 219–225 (2015)
dell’Isola, F., Giorgio, I., Pawlikowski, M., Rizzi, N.: Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium. Proc. R. Soc. A 472, 20150790 (2016)
Ferretti, M., Madeo, A., dell’Isola, F., Boisse, P.: Modelling the onset of shear boundary layers in fibrous composite reinforcements by second gradient theory. Z. Angew. Math. Phys. 65(3), 587–612 (2014)
Gabriele, S., Rizzi, N., Varano, V.: A 1D higher gradient model derived from Koiter’s shell theory. Math. Mech. Solids 21, 737–746 (2014)
Ganghoffer, J.: Spatial and material stress tensors in continuum mechanics of growing solid bodies. Math. Mech. Complex Syst. 3(4), 341–363 (2016)
Giorgio, I.: Numerical identification procedure between a micro-cauchy model and a macro-second gradient model for planar pantographic structures. Z. Angew. Math. Phys. 67(4), 95 (2016)
Giorgio, I., Della Corte, A., dell’Isola, F.: Dynamics of 1D nonlinear pantographic continua. Nonlinear Dyn. 88(1), 21–31 (2017)
Giorgio, I., Della Corte, A., dell’Isola, F., Steigmann, D.: Buckling modes in pantographic lattices. C. R. Mec. 344, 487–501 (2016)
Dell Isola, F., Seppecher, P., Della Corte, A.: The postulations á la D’Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results. Proc. R. Soc. A 471, 20150415 (2015)
Javili, A., McBride, A., Mergheim, J., Steinmann, P., Schmidt, U.: Micro-to-macro transitions for continua with surface structure at the microscale. Int. J. Solids Struct. 50(16), 2561–2572 (2013)
Li, J., Slesarenko, V., Galich, P., Rudykh, S.: Oblique shear wave propagation in finitely deformed layered composites. Mech. Res. Commun. 87, 21–28 (2018)
Milton, G., Briane, M., Harutyunyan, D.: On the possible effective elasticity tensors of 2-dimensional and 3-dimensional printed materials. Math. Mech. Complex Syst. 5(1), 41–94 (2017)
Misra, A., Poorsolhjouy, P.: Identification of higher-order elastic constants for grain assemblies based upon granular micromechanics. Math. Mech. Complex Syst. 3(3), 285–308 (2015)
Nadler, B., Papadopoulos, P., Steigmann, D.: Multiscale constitutive modeling and numerical simulation of fabric material. Int. J. Solids Struct. 43(2), 206–221 (2006)
Pideri, C., Seppecher, P.: Asymptotics of a non-planar rod in non-linear elasticity. Asymptot. Anal. 48(1, 2), 33–54 (2006)
Pignataro, M., Rizzi, N., Luongo, A.: Stability, Bifurcation and Postcritical Behaviour of Elastic Structures, vol. 39. Elsevier, New York (2013)
Placidi, L., Andreaus, U., Della Corte, A., Lekszycki, T.: Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients. Z. Angew. Math. Phys. 66(6), 3699–3725 (2015)
Placidi, L., Andreaus, U., Giorgio, I.: Identification of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model. J. Eng. Math. 103, 1–21 (2016)
Placidi, L., Barchiesi, E., Turco, E., Rizzi, N.: A review on 2D models for the description of pantographic fabrics. Z. Angew. Math. Phys. 67(5), 121 (2016)
Placidi, L., El Dhaba, A.: Semi-inverse method à la saint-venant for two-dimensional linear isotropic homogeneous second-gradient elasticity. Math. Mech. Solids 22, 919–937 (2015)
Placidi, L., Greco, L., Bucci, S., Turco, E., Rizzi, N.: A second gradient formulation for a 2D fabric sheet with inextensible fibres. Z. Angew. Math. Phys. 67(5), 114 (2016)
Ruta, G., Varano, V., Pignataro, M., Rizzi, N.: A beam model for the flexural-torsional buckling of thin-walled members with some applications. Thin Walled Struct. 46(7), 816–822 (2008)
Saeb, S., Steinmann, P., Javili, A.: Aspects of computational homogenization at finite deformations: a unifying review from reuss’ to Voigt’s bound. Appl. Mech. Rev. 68(5), 050801 (2016)
Scerrato, D., Giorgio, I., Rizzi, N.: Three-dimensional instabilities of pantographic sheets with parabolic lattices: numerical investigations. Z. Angew. Math. Phys. 67(3), 1–19 (2016)
Scerrato, D., Zhurba Eremeeva, I., Lekszycki, T., Rizzi, N.: On the effect of shear stiffness on the plane deformation of linear second gradient pantographic sheets. ZAMM J. Appl. Math. Mech. Z. Angew. Math. Phys. 96, 1268–1279 (2016)
Sciarra, G., dell’Isola, F., Coussy, O.: Second gradient poromechanics. Int. J. Solids Struct. 44(20), 6607–6629 (2007)
Seppecher, P., Alibert, J.J., dell’Isola, F.: Linear elastic trusses leading to continua with exotic mechanical interactions. In: Journal of Physics: Conference Series, vol. 319, p. 012018. IOP Publishing (2011)
Steigmann, D.: Two-dimensional models for the combined bending and stretching of plates and shells based on three-dimensional linear elasticity. Int. J. Eng. Sci. 46(7), 654–676 (2008)
Steigmann, D., dell’Isola, F.: Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching. Acta. Mech. Sin. 31(3), 372–382 (2015)
Turco, E., dell’Isola, F., Cazzani, A., Rizzi, N.: Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Z. Angew. Math. Phys. 67, 85 (2016). https://doi.org/10.1007/s00033-016-0681-8
Turco, E., Golaszewski, M., Giorgio, I., D’Annibale, F.: Pantographic lattices with non-orthogonal fibres: experiments and their numerical simulations. Compos. B Eng. 118, 1–14 (2017)
Turco, E., Rizzi, N.: Pantographic structures presenting statistically distributed defects: numerical investigations of the effects on deformation fields. Mech. Res. Commun. 77, 65–69 (2016)
Shirani, M., Luo, C., Steigmann, D.J.: Cosserat elasticity of lattice shells with kinematically independent flexure and twist. Continuum Mech. Thermodyn. (2018). https://doi.org/10.1007/s00161-018-0679-x