Abstract
The development of additive manufacturing methods, such as 3D printing, allows the design of more complex architectured materials. Indeed, the main structure can be obtained by means of periodically (or quasi-periodically) arranged substructures which are properly conceived to provide unconventional deformation patterns. These kinds of materials which are ‘substructure depending’ are called metamaterials. Detailed simulations of a metamaterial is challenging but accurately possible by means of the elasticity theory. In this study, we present the steps taken for analyzing and simulating a particular type of metamaterial composed of a pantographic substructure which is periodic in space—it is simply a grid. Nevertheless, it shows an unexpected type of deformation under a uniaxial shear test. This particular behavior is investigated in this work with the aid of direct numerical simulations by using the finite element method. In other words, a detailed mesh is generated to properly describe the substructure. The metamaterial is additively manufactured using a common polymer showing nonlinear elastic deformation. Experiments are undertaken, and several hyperelastic material models are examined by using an inverse analysis. Moreover, a direct numerical simulation is repeated for all studied material models. We show that a good agreement between numerical simulations and experimental data can be attained.
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Abali, B.E.: Thermodynamically Compatible Modeling, Determination of Material Parameters, and Numerical Analysis of Nonlinear Rheological Materials. PhD thesis, Technische Universität Berlin, Institute of Mechanics (2014)
Abali, B.E.: Computational Reality, Solving Nonlinear and Coupled Problems in Continuum Mechanics. Advanced Structured Materials. Springer, Berlin (2017)
Abali, B.E., Müller, W.H., dell’Isola, F.: Theory and computation of higher gradient elasticity theories based on action principles. Arch. Appl. Mech. 87(9), 1495–1510 (2017)
Abali, B.E., Wu, C.-C., Müller, W.H.: An energy-based method to determine material constants in nonlinear rheology with applications. Contin. Mech. Thermodyn. 28(5), 1221–1246 (2016)
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.A.: Analysis of the viscoelastic behavior of plates made of functionally graded materials. ZAMM-Zeitschrift für Angewandte Mathematik und Mechanik 88(5), 332–341 (2008)
Altenbach, H., Eremeyev, V.A. (eds.): Generalized Continua—from the Theory to Engineering Applications, Volume 541 of CISM International Centre for Mechanical Sciences. Springer, Wien (2013)
Altenbach, H., Eremeyev, V.A.: Surface viscoelasticity and effective properties of materials and structures. In: Altenbach, H., Kruch, S. (eds.) Advanced Materials Modelling for Structures, pp. 9–16. Springer, Berlin (2013)
Andreaus, U., Spagnuolo, M., Lekszycki, T., Eugster, S.R.: A Ritz approach for the static analysis of planar pantographic structures modeled with nonlinear Euler–Bernoulli beams. Contin. Mech. Thermodyn. (2018). https://doi.org/10.1007/s00161-018-0665-3
Arruda, E.M., Boyce, M.C.: A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. J. Mech. Phys. Solids 41(2), 389–412 (1993)
Attard, M.M., Hunt, G.W.: Hyperelastic constitutive modeling under finite strain. Int. J. Solids Struct. 41(18–19), 5327–5350 (2004)
Barchiesi, E., Ganzosch, G., Liebold, C., Placidi, L., Grygoruk, R., Müller, W.H.: Out-of-plane buckling of pantographic fabrics in displacement-controlled shear tests: experimental results and model validation. Contin. Mech. Thermodyn. (2018). https://doi.org/10.1007/s00161-018-0626-x
Barchiesi, E., Spagnuolo, M., Placidi, L.: Mechanical metamaterials: a state of the art. Math. Mech. Solids (2018). https://doi.org/10.1177/1081286517735695
Battista, A., Cardillo, C., Del Vescovo, D., Rizzi, N.L., Turco, E.: Frequency shifts induced by large deformations in planar pantographic continua. Nanosci. Technol. Int. J. 6(2), 161–178 (2015)
Battista, A., Del Vescovo, D., Rizzi, N.L., Turco, E.: Frequency shifts in natural vibrations in pantographic metamaterials under biaxial tests. Technische Mechanik 37(1), 1–17 (2017)
Biderman, V.L.: Calculation of rubber parts. Rascheti na prochnost, Moscow (1958)
Boutin, C., dell’Isola, F., Giorgio, I., Placidi, L.: Linear pantographic sheets: asymptotic micro-macro models identification. Math. Mech. Complex Syst. 5(2), 127–162 (2017)
Cuomo, M.: Forms of the dissipation function for a class of viscoplastic models. Math. Mech. Complex Syst. 5(3), 217–237 (2017)
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)
Del Vescovo, D., Giorgio, I.: Dynamic problems for metamaterials: review of existing models and ideas for further research. Int. J. Eng. Sci. 80, 153–172 (2014)
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, 887–928 (2014)
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(4), 852–872 (2017)
dell’Isola, F., Giorgio, I., Pawlikowski, M., Rizzi, N.L.: Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium. Proc. R. Soc. A 472(2185), 1–23 (2016)
dell’Isola, F., Lekszycki, T., Pawlikowski, M., Grygoruk, R., Greco, L.: Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence. Zeitschrift für angewandte Mathematik und Physik 66(6), 3473–3498 (2015)
dell’Isola, F., Seppecher, P., et al.: Pantographic metamaterials: an example of mathematically driven design and of its technological challenges. Contin. Mech. Thermodyn. (2018). https://doi.org/10.1007/s00161-018-0689-8
dell’Isola, F., Steigmann, D., Della Corte, A.: Synthesis of fibrous complex structures: designing microstructure to deliver targeted macroscale response. Appl. Mech. Rev. 67(6), 060804 (2015)
Diyaroglu, C., Oterkus, E., Oterkus, S., Madenci, E.: Peridynamics for bending of beams and plates with transverse shear deformation. Int. J. Solids Struct. 69, 152–168 (2015)
Engelbrecht, J., Berezovski, A.: Reflections on mathematical models of deformation waves in elastic microstructured solids. Math. Mech. Complex Syst. 3(1), 43–82 (2015)
Eremeyev, V.A., Pietraszkiewicz, W.: Material symmetry group and constitutive equations of micropolar anisotropic elastic solids. Math. Mech. Solids 21(2), 210–221 (2016)
Eringen, A.C.: Theory of Micropolar Elasticity. Technical report, DTIC Document (1967)
Eugster, S.R., Hesch, C., Betsch, P., Glocker, C.: Director-based beam finite elements relying on the geometrically exact beam theory formulated in skew coordinates. Int. J. Numer. Methods Eng. 97(2), 111–129 (2014)
Flory, P.J., Rehner Jr., J.: Statistical mechanics of cross-linked polymer networks I. Rubberlike elasticity. J. Chem. Phys. 11(11), 512–520 (1943)
Ganzosch, G., dell’Isola, F., Turco, E., Lekszycki, T., Müller, W.H.: Shearing tests applied to pantographic structures. Acta Polytech. CTU Proc. 7, 1–6 (2016)
Giorgio, I.: Numerical identification procedure between a micro-Cauchy model and a macro-second gradient model for planar pantographic structures. Zeitschrift für angewandte Mathematik und Physik 67(4), 95 (2016)
Giorgio, I., Rizzi, N.L., Turco, E.: Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis. Proc. R. Soc. A 473(2207), 1–21 (2017)
Harrison, P., Clifford, M.J., Long, A.C., Rudd, C.D.: A constituent-based predictive approach to modelling the rheology of viscous textile composites. Compos. A Appl. Sci. Manuf. 35(7–8), 915–931 (2004)
Hoffman, J., Jansson, J., Johnson, C., Knepley, M., Kirby, R.C., Logg, A., Scott, L.R., Wells, G.N.: Fenics (2005). http://www.fenicsproject.org/
Holzapfel, A.G.: Nonlinear Solid Mechanics II. Wiley, New York (2000)
Isihara, A., Hashitsume, N., Tatibana, M.: Statistical theory of rubber-like elasticity. IV (two-dimensional stretching). J. Chem. Phys. 19(12), 1508–1512 (1951)
Itskov, M., Aksel, N.: A class of orthotropic and transversely isotropic hyperelastic constitutive models based on a polyconvex strain energy function. Int. J. Solids Struct. 41(14), 3833–3848 (2004)
James, A.G., Green, A., Simpson, G.M.: Strain energy functions of rubber. I. Characterization of gum vulcanizates. J. Appl. Polym. Sci. 19(7), 2033–2058 (1975)
Julio García Ruíz, M., Yarime Suárez González, L.: Comparison of hyperelastic material models in the analysis of fabrics. Int. J. Cloth. Sci. Technol. 18(5), 314–325 (2006)
Khakalo, S., Balobanov, V., Niiranen, J.: Modelling size-dependent bending, buckling and vibrations of 2D triangular lattices by strain gradient elasticity models: applications to sandwich beams and auxetics. Int. J. Eng. Sci. 127, 33–52 (2018)
Lam, D.C.C., Yang, F., Chong, A.C.M., Wang, J., Tong, P.: Experiments and theory in strain gradient elasticity. J. Mech. Phys. Solids 51(8), 1477–1508 (2003)
Logg, A., Mardal, K.A., Wells, G.N.: Automated Solution of Differential Equations by the Finite Element Method, the FEniCS Book, Volume 84 of Lecture Notes in Computational Science and Engineering. Springer, Berlin (2011)
Marckmann, G., Verron, E.: Comparison of hyperelastic models for rubber-like materials. Rubber Chem. Technol. 79(5), 835–858 (2006)
Martins, P., Natal Jorge, R.M., Ferreira, A.J.M.: A comparative study of several material models for prediction of hyperelastic properties: application to silicone-rubber and soft tissues. Strain 42(3), 135–147 (2006)
Milton, G.W., 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)
Mindlin, R.D., Eshel, N.N.: On first strain–gradient theories in linear elasticity. Int. J. Solids Struct. 4(1), 109–124 (1968)
Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch. Ration. Mech. Anal. 11, 415–448 (1962)
Misra, A., Lekszycki, T., Giorgio, I., Ganzosch, G., Müller, W.H., dell’Isola, F.: Pantographic metamaterials show atypical Poynting effect reversal. Mech. Res. Commun. 89, 6–10 (2018)
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)
Misra, A., Poorsolhjouy, P.: Granular micromechanics based micromorphic model predicts frequency band gaps. Contin. Mech. Thermodyn. 28(1–2), 215–234 (2016)
Mooney, M.: A theory of large elastic deformation. J. Appl. Phys. 11(9), 582–592 (1940)
Nadler, B., Papadopoulos, P., Steigmann, D.J.: Multiscale constitutive modeling and numerical simulation of fabric material. Int. J. Solids Struct. 43(2), 206–221 (2006)
Niiranen, J., Balobanov, V., Kiendl, J., Hosseini, S.B.: Variational formulations, model comparisons and numerical methods for Euler–Bernoulli micro-and nano-beam models. Math. Mech. Solids (2017). https://doi.org/10.1177/1081286517739669
Pideri, C., Seppecher, P.: A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium. Contin. Mech. Thermodyn. 9(5), 241–257 (1997)
Placidi, L., Andreaus, U., Della Corte, A., Lekszycki, T.: Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients. Zeitschrift für angewandte Mathematik und Physik 66(6), 3699–3725 (2015)
Placidi, L., Barchiesi, E.: Energy approach to brittle fracture in strain-gradient modelling. Proc. R. Soc. A 474(2210), 1–19 (2018)
Placidi, L., Barchiesi, E., Battista, A.: An inverse method to get further analytical solutions for a class of metamaterials aimed to validate numerical integrations. In: dell’Isola, F., Sofonea, M., Steigmann, D. (eds.) Mathematical Modelling in Solid Mechanics, pp. 193–210. Springer, Berlin(2017)
Placidi, L., Barchiesi, E., Misra, A.: A strain gradient variational approach to damage: a comparison with damage gradient models and numerical results. Math. Mech. Complex Syst. 6(2), 77–100 (2018)
Placidi, L., Barchiesi, E., Turco, E., Rizzi, N.L.: A review on 2D models for the description of pantographic fabrics. Zeitschrift für angewandte Mathematik und Physik 67(5), 121 (2016)
Placidi, L., Misra, A., Barchiesi, E.: Two-dimensional strain gradient damage modeling: a variational approach. Zeitschrift für angewandte Mathematik und Physik 69(3), 1–19 (2018)
Rivlin, R.S.: Large elastic deformations of isotropic materials iv. further developments of the general theory. Philos. Trans. R. Soc. Lond. A 241(835), 379–397 (1948)
Rosakis, P.: Ellipticity and deformations with discontinuous gradients in finite elastostatics. Arch. Ration. Mech. Anal. 109(1), 1–37 (1990)
Shirani, M., Luo, C., Steigmann, D.J.: Cosserat elasticity of lattice shells with kinematically independent flexure and twist. Contin. Mech. Thermodyn. (2018). https://doi.org/10.1007/s00161-018-0679-x
Soe, S.P., Martindale, N., Constantinou, C., Robinson, M.: Mechanical characterisation of Duraform\(^{\textregistered }\) Flex for FEA hyperelastic material modelling. Polym. Test. 34, 103–112 (2014)
Spagnuolo, M., Barcz, K., Pfaff, A., dell’Isola, F., Franciosi, P.: Qualitative pivot damage analysis in aluminum printed pantographic sheets: numerics and experiments. Mech. Res. Commun. 83, 47–52 (2017)
Steigmann, D.J., dell’Isola, F.: Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching. Acta. Mech. Sin. 31(3), 373–382 (2015)
Thai, H.-T., Vo, T.P., Nguyen, T.-K., Kim, S.-E.: A review of continuum mechanics models for size-dependent analysis of beams and plates. Compos. Struct. 177, 196–219 (2017)
Toupin, R.A.: Theories of elasticity with couple-stress. Arch. Ration. Mech. Anal. 17(2), 85–112 (1964)
Treloar, L.R.G.: The elasticity of a network of long-chain molecules—II. Trans. Faraday Soc. 39, 241–246 (1943)
Turco, E., dell’Isola, F., Cazzani, A., Rizzi, N.L.: Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models. Zeitschrift für angewandte Mathematik und Physik 67(4), 85 (2016)
Turco, E., dell’Isola, F., Rizzi, N.L., Grygoruk, R., Müller, W.H., Liebold, C.: Fiber rupture in sheared planar pantographic sheets: numerical and experimental evidence. Mech. Res. Commun. 76, 86–90 (2016)
Turco, E., Giorgio, I., Misra, A., dell’Isola, F.: King post truss as a motif for internal structure of (meta) material with controlled elastic properties. R. Soc. Open Sci. 4(10), 171153 (2017)
Weber, G., Anand, L.: Finite deformation constitutive equations and a time integration procedure for isotropic, hyperelastic–viscoplastic solids. Comput. Methods Appl. Mech. Eng. 79(2), 173–202 (1990)
Yeoh, O.H.: Some forms of the strain energy function for rubber. Rubber Chem. Technol. 66(5), 754–771 (1993)
Acknowledgements
IG is supported by a grant from the Government of the Russian Federation (No. 14.Y26.31.0031). We thank Prof. Wolfgang H. Müller for fruitful discussions.
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Yang, H., Ganzosch, G., Giorgio, I. et al. Material characterization and computations of a polymeric metamaterial with a pantographic substructure. Z. Angew. Math. Phys. 69, 105 (2018). https://doi.org/10.1007/s00033-018-1000-3
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DOI: https://doi.org/10.1007/s00033-018-1000-3